X-ray Photon Correlation Spectroscopy€¦ · X-ray Photon Correlation Spectroscopy Anders Madsen,...
Transcript of X-ray Photon Correlation Spectroscopy€¦ · X-ray Photon Correlation Spectroscopy Anders Madsen,...
XX--ray Photon Correlation Spectroscopyray Photon Correlation Spectroscopy
Anders Madsen ESRF
Coherence
X-ray Speckle
Photon Correlation Spectroscopy
Applications of XPCS
Coherence
Quantum mechanics probability amplitudes (waves) Optics Youngs double slit experiment interference X-ray (and neutron) scattering
Its all about probability amplitudes and interference
Example Youngs double slit experiment (Thomas Young 1801)[wave-character of quantum mechanical particles (photons)]
Plane mono-chromatic wave
P=|ΣjΦj|2Φ probability amplitudeΦj ~ exp[-i(ωt-klj)]ω=ck k=2πλ lj(Ly)
P(y) ~ cos2(πydλL)∆y=λLd
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Reasons for loss in (visibilityinterferencecoherence)
1) Incoherent superposition of probability amplitudes P=Σj|Φj|2(distinguishable alternatives uncertainty principle)
2) Intensity interference is only observed if event is repeated many times repetition under non-ideal conditions washes outthe visibilityUsing a (partially) coherent X-ray source 2) is a major limitation
Non-ideal conditions Ein Eout kin kout not well defined in the experiment Disorder in the scattering sample Limited detector resolution (temporal and spatial) The source is chaotic
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Light sources
Chaotic sources (spontaneously emitted photons)Lab X-ray generatorsSynchrotron and Neutron sourcesRadioactive nuclei
One-mode sources (stimulated emission Glauber light)Unimodal lasers
Important parameter NC=photons pr coherence volumeNC ~10-3 for typical ESRF undulatorNC ~107 for typical optical laser
Beam direction
lv
lhCoherence volume Vc πlh lv ll 4horizontal vertical and longitudinal (temporal) coherence length
ll
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Chaotic source (ESRF undulator) (spontaneous independent emission in all modes)
Longitudinal (or temporal) coherence
Spatial coherenceAnalogy with Youngs double slit experiment Transverse coherence length (vh) lvh=λLπdvh (~2-150 microm)
0ττ
2
(1) e
|E(t)|τ)(t)E(tE
)(g minus= prop
+τ
At times gtgt τ0 the field amplitudes from a chaotic source are no longer correlated due to the wavelength spread
τ0 = 1(π∆ν) = λ2(cπ∆λ) (~3fs)Longitudinal coherence length ll = cτ0 = λ(π∆λλ) (~1microm)
(Gaussian 1st order time-correlation function)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Important source parameters
Brilliance B = photonssec [source area times solid angle times bandwidth]
Lateral coherence area At=πlhlv4=(λL)2(4πdhdv) (~200 microm2)Νc=photons in Vc (Vc = Αt times ll)
Coherent solid angle ΩC=AtL2=λ2(4πdhdv)= λ216As
Νc=B times τ0 times As times λ216As times ∆νν = Β λ3 (16πc)Coherent intensity Ic = (ΝcVc) times c times At
= B times (λ4)2 times (∆λλ) (~1010 phs)
As=πdhdv4
Coherent intensity decreases with decreasing λ (increasing energy) Coherent scattering is very Brilliance-hungry
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Coherence lengths
λ=21Aring d=11micromVisibility(β) ~ 80
Example Youngs double slit experiment with X-rays
λ=09Aring d=11micromVisibility(β) ~ 30
∆y=λLdLeitenberger et al Physica B 336 36 (2003)
Transverse coherence length (vh) lvh=λLπdvh (~5-250 microm)
Longitudinal coherence length ll = cτ0 = λ2(π∆λ) (~1microm)
Contrast β asymp (coherence volume)(scattering volume)
ESRFvalues
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Setup for coherent X-ray scattering
Coherence parametersTransverse coherence length lhv=λL(πdhv)Longitudinal coherence length ll=λ2(π∆λ)Contrast (degree of coherence) β=β (∆λλ )
IC prop B times λ2
ID10AIc ~2e8 phs (1997)Ic gt1e10 phs (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Fraunhoferdiffraction
from slit
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Coherent scattering What is the information we gain
If coherent light is scattered from a disordered system it gives rise to a random (grainy) diffraction pattern known as speckle A speckle pattern is an inter-ference pattern and related to the exact spatial arrangement of the scatterers in the disordered system (no averaging)
I(Qt) prop Sc(Qt) prop | sum e iQRj(t) |2j in coherence volume Vc=lv lh ll
D Abernathy et al
J Sync Rad 5 37 (1998)
Aerogel SAXSλ=1AringCCD (22 microm pixels)10 micron beamsize
speckle size ~λd
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Longitudinal coherence length and scattering geometry
PLD Path Length Difference lt ll CSAXSPLD = t + BC - AC asymp d tan(2θ)
Bragg geometryPLD asymp 2(micro-1)sin2θ
Grazing Incidence PLD asymp 2Λsinαi
2θAd
Bt
θ
Condition PLD lt ll SAXS d=10microm λ=155Aring ∆λλ=14times10-4 rarr θMAX asymp1o rarr QMAX asymp 015Aring-1
Reflection micro-1Cu= 22microm λ=155Aring ∆λλ=14times10-4
rarr θMAX asymp 5o rarr QMAX asymp 07Aring-1
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the contrast of the setup
Gamma-Poisson distribution of intensity coming from M statistically independent superimposed speckle patterns PM(I)=(MltIgt)MIM-1exp(-MIltIgt)Γ(M)σ2=ltIgt2M 1M=β
Coherent SAXS 8keV
Incoherent fraction
10 micron beam
VycorTM silica glass
Static speckle pattern
contrast β=1M=1281=356M ~ VscatteringVcoherence
Statistical speckle analysisPrinceton Instruments CCD1 Photon gives 879 adu2 Photons give 1758 adu
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the shape of a speckle
200 400 600 800 1000 1200
200
400
600
800
1000
1200
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
vertical direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
horizontal direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
diagonal direction
0 002 0040
002
004
006
008
01
Q
3
1
2
1 2 3
elongatedspeckles
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Pink beam ID10C (∆λλ=15)E=8keV 20microm beamsizeIcoherent gt 1x1011 phs
XPCS What is the information we gain
real space
non-coherent coherent
reciprocal space
ltI(Q)gt I(Qt) I(Qt)
DynamicsKinetics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What do we measure
2)2(
)(
)()()(
Q
QQQ
I
tItIg
ττ
+=
22~)2( |)(|1))((Re1)( tfSg QQQ +=+= ωτ
Temporal intensityauto-correlation function
int int minussdotpropV V
mnmn tibbtf )])()0([exp()()(|)(| rrQQQQ
220 )()]([
1~)(QQ
QSΓ+minus ωω
ω
)(cos)2exp(1)( 02)2( τωττ Γminus+=Qg
)2exp(1)()2( ττ Γminus+=Qg
Dynamic structure factor(Lorentzian shape)
(propagating waves)
(Brownian motion overdamped waves)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Scattering Vector Q [Aring-1]
Length Scale [Aring]Fr
eque
ncy
[Hz]
Energy [eV]
Raman
Brillouin
XPCSPCS
IXS
Spin-Echo
INS
NFS
Other techniques
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Typical example samples with some degree of (quasi) static disorder (gels glasses etc)
To properly calculate g(2) the ensemble averaging ltgtφ must be performed explicitly
Pusey amp van Megen Physica A (1989)
I(t) I(t)
tt
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Solution move the beam around and average g(2) over many exposures
or (much better)
Use a 2D detector and catch many pixels with same |Q|
Medipix-2
Ttp
Ttp
tpp
tItI
tItIQg
leleminuslele
+=
τφτφ
φτ
τ)()(
)()()(
0
)2(
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-equilibrium ie not all ∆t are equivalentg(2) depends on both ∆t and the absolute time (age)
Solution calculate the correlation function without time averaging (two-time correlation function)
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG =
t1
t2
G(t1t2) age
τ
age ta=(t2+t1)2
lag time τ=t2-t1
Only with a 2D detector several Qs at same age can be recorded
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
2d detectors for multi-speckle XPCS
Advantages More efficient data-taking (Mpixels) with 2d detectorBeam induced sample damage minimized
Drawbacks 2d detectors are slow noisy have limited dynamic rangeare not always photon counting have limited resolutionradiation hardness can be a problem
Princeton Instruments Dalsa MedipixMaxipix
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Limitations
sn ratio (more flux)
Speed and area of 2D detectors
Beam damage (minimize exposure)
Medipix-II detector 1kHz ff
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Depletion gel
Inter-particle interaction potential
V(r)Complex phase diagramrgR (01) φ (20) cp (43mgcm3)
2rg
2R
r
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The gel phase is transienthellip
W C K Poon et al Faraday Discuss 112 143 (1999)
delayedsedimentationageltt (latency time)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Coherence
Quantum mechanics probability amplitudes (waves) Optics Youngs double slit experiment interference X-ray (and neutron) scattering
Its all about probability amplitudes and interference
Example Youngs double slit experiment (Thomas Young 1801)[wave-character of quantum mechanical particles (photons)]
Plane mono-chromatic wave
P=|ΣjΦj|2Φ probability amplitudeΦj ~ exp[-i(ωt-klj)]ω=ck k=2πλ lj(Ly)
P(y) ~ cos2(πydλL)∆y=λLd
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Reasons for loss in (visibilityinterferencecoherence)
1) Incoherent superposition of probability amplitudes P=Σj|Φj|2(distinguishable alternatives uncertainty principle)
2) Intensity interference is only observed if event is repeated many times repetition under non-ideal conditions washes outthe visibilityUsing a (partially) coherent X-ray source 2) is a major limitation
Non-ideal conditions Ein Eout kin kout not well defined in the experiment Disorder in the scattering sample Limited detector resolution (temporal and spatial) The source is chaotic
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Light sources
Chaotic sources (spontaneously emitted photons)Lab X-ray generatorsSynchrotron and Neutron sourcesRadioactive nuclei
One-mode sources (stimulated emission Glauber light)Unimodal lasers
Important parameter NC=photons pr coherence volumeNC ~10-3 for typical ESRF undulatorNC ~107 for typical optical laser
Beam direction
lv
lhCoherence volume Vc πlh lv ll 4horizontal vertical and longitudinal (temporal) coherence length
ll
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Chaotic source (ESRF undulator) (spontaneous independent emission in all modes)
Longitudinal (or temporal) coherence
Spatial coherenceAnalogy with Youngs double slit experiment Transverse coherence length (vh) lvh=λLπdvh (~2-150 microm)
0ττ
2
(1) e
|E(t)|τ)(t)E(tE
)(g minus= prop
+τ
At times gtgt τ0 the field amplitudes from a chaotic source are no longer correlated due to the wavelength spread
τ0 = 1(π∆ν) = λ2(cπ∆λ) (~3fs)Longitudinal coherence length ll = cτ0 = λ(π∆λλ) (~1microm)
(Gaussian 1st order time-correlation function)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Important source parameters
Brilliance B = photonssec [source area times solid angle times bandwidth]
Lateral coherence area At=πlhlv4=(λL)2(4πdhdv) (~200 microm2)Νc=photons in Vc (Vc = Αt times ll)
Coherent solid angle ΩC=AtL2=λ2(4πdhdv)= λ216As
Νc=B times τ0 times As times λ216As times ∆νν = Β λ3 (16πc)Coherent intensity Ic = (ΝcVc) times c times At
= B times (λ4)2 times (∆λλ) (~1010 phs)
As=πdhdv4
Coherent intensity decreases with decreasing λ (increasing energy) Coherent scattering is very Brilliance-hungry
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Coherence lengths
λ=21Aring d=11micromVisibility(β) ~ 80
Example Youngs double slit experiment with X-rays
λ=09Aring d=11micromVisibility(β) ~ 30
∆y=λLdLeitenberger et al Physica B 336 36 (2003)
Transverse coherence length (vh) lvh=λLπdvh (~5-250 microm)
Longitudinal coherence length ll = cτ0 = λ2(π∆λ) (~1microm)
Contrast β asymp (coherence volume)(scattering volume)
ESRFvalues
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Setup for coherent X-ray scattering
Coherence parametersTransverse coherence length lhv=λL(πdhv)Longitudinal coherence length ll=λ2(π∆λ)Contrast (degree of coherence) β=β (∆λλ )
IC prop B times λ2
ID10AIc ~2e8 phs (1997)Ic gt1e10 phs (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Fraunhoferdiffraction
from slit
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Coherent scattering What is the information we gain
If coherent light is scattered from a disordered system it gives rise to a random (grainy) diffraction pattern known as speckle A speckle pattern is an inter-ference pattern and related to the exact spatial arrangement of the scatterers in the disordered system (no averaging)
I(Qt) prop Sc(Qt) prop | sum e iQRj(t) |2j in coherence volume Vc=lv lh ll
D Abernathy et al
J Sync Rad 5 37 (1998)
Aerogel SAXSλ=1AringCCD (22 microm pixels)10 micron beamsize
speckle size ~λd
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Longitudinal coherence length and scattering geometry
PLD Path Length Difference lt ll CSAXSPLD = t + BC - AC asymp d tan(2θ)
Bragg geometryPLD asymp 2(micro-1)sin2θ
Grazing Incidence PLD asymp 2Λsinαi
2θAd
Bt
θ
Condition PLD lt ll SAXS d=10microm λ=155Aring ∆λλ=14times10-4 rarr θMAX asymp1o rarr QMAX asymp 015Aring-1
Reflection micro-1Cu= 22microm λ=155Aring ∆λλ=14times10-4
rarr θMAX asymp 5o rarr QMAX asymp 07Aring-1
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the contrast of the setup
Gamma-Poisson distribution of intensity coming from M statistically independent superimposed speckle patterns PM(I)=(MltIgt)MIM-1exp(-MIltIgt)Γ(M)σ2=ltIgt2M 1M=β
Coherent SAXS 8keV
Incoherent fraction
10 micron beam
VycorTM silica glass
Static speckle pattern
contrast β=1M=1281=356M ~ VscatteringVcoherence
Statistical speckle analysisPrinceton Instruments CCD1 Photon gives 879 adu2 Photons give 1758 adu
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the shape of a speckle
200 400 600 800 1000 1200
200
400
600
800
1000
1200
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
vertical direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
horizontal direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
diagonal direction
0 002 0040
002
004
006
008
01
Q
3
1
2
1 2 3
elongatedspeckles
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Pink beam ID10C (∆λλ=15)E=8keV 20microm beamsizeIcoherent gt 1x1011 phs
XPCS What is the information we gain
real space
non-coherent coherent
reciprocal space
ltI(Q)gt I(Qt) I(Qt)
DynamicsKinetics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What do we measure
2)2(
)(
)()()(
Q
QQQ
I
tItIg
ττ
+=
22~)2( |)(|1))((Re1)( tfSg QQQ +=+= ωτ
Temporal intensityauto-correlation function
int int minussdotpropV V
mnmn tibbtf )])()0([exp()()(|)(| rrQQQQ
220 )()]([
1~)(QQ
QSΓ+minus ωω
ω
)(cos)2exp(1)( 02)2( τωττ Γminus+=Qg
)2exp(1)()2( ττ Γminus+=Qg
Dynamic structure factor(Lorentzian shape)
(propagating waves)
(Brownian motion overdamped waves)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Scattering Vector Q [Aring-1]
Length Scale [Aring]Fr
eque
ncy
[Hz]
Energy [eV]
Raman
Brillouin
XPCSPCS
IXS
Spin-Echo
INS
NFS
Other techniques
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Typical example samples with some degree of (quasi) static disorder (gels glasses etc)
To properly calculate g(2) the ensemble averaging ltgtφ must be performed explicitly
Pusey amp van Megen Physica A (1989)
I(t) I(t)
tt
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Solution move the beam around and average g(2) over many exposures
or (much better)
Use a 2D detector and catch many pixels with same |Q|
Medipix-2
Ttp
Ttp
tpp
tItI
tItIQg
leleminuslele
+=
τφτφ
φτ
τ)()(
)()()(
0
)2(
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-equilibrium ie not all ∆t are equivalentg(2) depends on both ∆t and the absolute time (age)
Solution calculate the correlation function without time averaging (two-time correlation function)
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG =
t1
t2
G(t1t2) age
τ
age ta=(t2+t1)2
lag time τ=t2-t1
Only with a 2D detector several Qs at same age can be recorded
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
2d detectors for multi-speckle XPCS
Advantages More efficient data-taking (Mpixels) with 2d detectorBeam induced sample damage minimized
Drawbacks 2d detectors are slow noisy have limited dynamic rangeare not always photon counting have limited resolutionradiation hardness can be a problem
Princeton Instruments Dalsa MedipixMaxipix
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Limitations
sn ratio (more flux)
Speed and area of 2D detectors
Beam damage (minimize exposure)
Medipix-II detector 1kHz ff
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Depletion gel
Inter-particle interaction potential
V(r)Complex phase diagramrgR (01) φ (20) cp (43mgcm3)
2rg
2R
r
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The gel phase is transienthellip
W C K Poon et al Faraday Discuss 112 143 (1999)
delayedsedimentationageltt (latency time)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Reasons for loss in (visibilityinterferencecoherence)
1) Incoherent superposition of probability amplitudes P=Σj|Φj|2(distinguishable alternatives uncertainty principle)
2) Intensity interference is only observed if event is repeated many times repetition under non-ideal conditions washes outthe visibilityUsing a (partially) coherent X-ray source 2) is a major limitation
Non-ideal conditions Ein Eout kin kout not well defined in the experiment Disorder in the scattering sample Limited detector resolution (temporal and spatial) The source is chaotic
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Light sources
Chaotic sources (spontaneously emitted photons)Lab X-ray generatorsSynchrotron and Neutron sourcesRadioactive nuclei
One-mode sources (stimulated emission Glauber light)Unimodal lasers
Important parameter NC=photons pr coherence volumeNC ~10-3 for typical ESRF undulatorNC ~107 for typical optical laser
Beam direction
lv
lhCoherence volume Vc πlh lv ll 4horizontal vertical and longitudinal (temporal) coherence length
ll
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Chaotic source (ESRF undulator) (spontaneous independent emission in all modes)
Longitudinal (or temporal) coherence
Spatial coherenceAnalogy with Youngs double slit experiment Transverse coherence length (vh) lvh=λLπdvh (~2-150 microm)
0ττ
2
(1) e
|E(t)|τ)(t)E(tE
)(g minus= prop
+τ
At times gtgt τ0 the field amplitudes from a chaotic source are no longer correlated due to the wavelength spread
τ0 = 1(π∆ν) = λ2(cπ∆λ) (~3fs)Longitudinal coherence length ll = cτ0 = λ(π∆λλ) (~1microm)
(Gaussian 1st order time-correlation function)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Important source parameters
Brilliance B = photonssec [source area times solid angle times bandwidth]
Lateral coherence area At=πlhlv4=(λL)2(4πdhdv) (~200 microm2)Νc=photons in Vc (Vc = Αt times ll)
Coherent solid angle ΩC=AtL2=λ2(4πdhdv)= λ216As
Νc=B times τ0 times As times λ216As times ∆νν = Β λ3 (16πc)Coherent intensity Ic = (ΝcVc) times c times At
= B times (λ4)2 times (∆λλ) (~1010 phs)
As=πdhdv4
Coherent intensity decreases with decreasing λ (increasing energy) Coherent scattering is very Brilliance-hungry
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Coherence lengths
λ=21Aring d=11micromVisibility(β) ~ 80
Example Youngs double slit experiment with X-rays
λ=09Aring d=11micromVisibility(β) ~ 30
∆y=λLdLeitenberger et al Physica B 336 36 (2003)
Transverse coherence length (vh) lvh=λLπdvh (~5-250 microm)
Longitudinal coherence length ll = cτ0 = λ2(π∆λ) (~1microm)
Contrast β asymp (coherence volume)(scattering volume)
ESRFvalues
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Setup for coherent X-ray scattering
Coherence parametersTransverse coherence length lhv=λL(πdhv)Longitudinal coherence length ll=λ2(π∆λ)Contrast (degree of coherence) β=β (∆λλ )
IC prop B times λ2
ID10AIc ~2e8 phs (1997)Ic gt1e10 phs (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Fraunhoferdiffraction
from slit
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Coherent scattering What is the information we gain
If coherent light is scattered from a disordered system it gives rise to a random (grainy) diffraction pattern known as speckle A speckle pattern is an inter-ference pattern and related to the exact spatial arrangement of the scatterers in the disordered system (no averaging)
I(Qt) prop Sc(Qt) prop | sum e iQRj(t) |2j in coherence volume Vc=lv lh ll
D Abernathy et al
J Sync Rad 5 37 (1998)
Aerogel SAXSλ=1AringCCD (22 microm pixels)10 micron beamsize
speckle size ~λd
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Longitudinal coherence length and scattering geometry
PLD Path Length Difference lt ll CSAXSPLD = t + BC - AC asymp d tan(2θ)
Bragg geometryPLD asymp 2(micro-1)sin2θ
Grazing Incidence PLD asymp 2Λsinαi
2θAd
Bt
θ
Condition PLD lt ll SAXS d=10microm λ=155Aring ∆λλ=14times10-4 rarr θMAX asymp1o rarr QMAX asymp 015Aring-1
Reflection micro-1Cu= 22microm λ=155Aring ∆λλ=14times10-4
rarr θMAX asymp 5o rarr QMAX asymp 07Aring-1
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the contrast of the setup
Gamma-Poisson distribution of intensity coming from M statistically independent superimposed speckle patterns PM(I)=(MltIgt)MIM-1exp(-MIltIgt)Γ(M)σ2=ltIgt2M 1M=β
Coherent SAXS 8keV
Incoherent fraction
10 micron beam
VycorTM silica glass
Static speckle pattern
contrast β=1M=1281=356M ~ VscatteringVcoherence
Statistical speckle analysisPrinceton Instruments CCD1 Photon gives 879 adu2 Photons give 1758 adu
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the shape of a speckle
200 400 600 800 1000 1200
200
400
600
800
1000
1200
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
vertical direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
horizontal direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
diagonal direction
0 002 0040
002
004
006
008
01
Q
3
1
2
1 2 3
elongatedspeckles
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Pink beam ID10C (∆λλ=15)E=8keV 20microm beamsizeIcoherent gt 1x1011 phs
XPCS What is the information we gain
real space
non-coherent coherent
reciprocal space
ltI(Q)gt I(Qt) I(Qt)
DynamicsKinetics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What do we measure
2)2(
)(
)()()(
Q
QQQ
I
tItIg
ττ
+=
22~)2( |)(|1))((Re1)( tfSg QQQ +=+= ωτ
Temporal intensityauto-correlation function
int int minussdotpropV V
mnmn tibbtf )])()0([exp()()(|)(| rrQQQQ
220 )()]([
1~)(QQ
QSΓ+minus ωω
ω
)(cos)2exp(1)( 02)2( τωττ Γminus+=Qg
)2exp(1)()2( ττ Γminus+=Qg
Dynamic structure factor(Lorentzian shape)
(propagating waves)
(Brownian motion overdamped waves)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Scattering Vector Q [Aring-1]
Length Scale [Aring]Fr
eque
ncy
[Hz]
Energy [eV]
Raman
Brillouin
XPCSPCS
IXS
Spin-Echo
INS
NFS
Other techniques
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Typical example samples with some degree of (quasi) static disorder (gels glasses etc)
To properly calculate g(2) the ensemble averaging ltgtφ must be performed explicitly
Pusey amp van Megen Physica A (1989)
I(t) I(t)
tt
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Solution move the beam around and average g(2) over many exposures
or (much better)
Use a 2D detector and catch many pixels with same |Q|
Medipix-2
Ttp
Ttp
tpp
tItI
tItIQg
leleminuslele
+=
τφτφ
φτ
τ)()(
)()()(
0
)2(
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-equilibrium ie not all ∆t are equivalentg(2) depends on both ∆t and the absolute time (age)
Solution calculate the correlation function without time averaging (two-time correlation function)
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG =
t1
t2
G(t1t2) age
τ
age ta=(t2+t1)2
lag time τ=t2-t1
Only with a 2D detector several Qs at same age can be recorded
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
2d detectors for multi-speckle XPCS
Advantages More efficient data-taking (Mpixels) with 2d detectorBeam induced sample damage minimized
Drawbacks 2d detectors are slow noisy have limited dynamic rangeare not always photon counting have limited resolutionradiation hardness can be a problem
Princeton Instruments Dalsa MedipixMaxipix
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Limitations
sn ratio (more flux)
Speed and area of 2D detectors
Beam damage (minimize exposure)
Medipix-II detector 1kHz ff
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Depletion gel
Inter-particle interaction potential
V(r)Complex phase diagramrgR (01) φ (20) cp (43mgcm3)
2rg
2R
r
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The gel phase is transienthellip
W C K Poon et al Faraday Discuss 112 143 (1999)
delayedsedimentationageltt (latency time)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Light sources
Chaotic sources (spontaneously emitted photons)Lab X-ray generatorsSynchrotron and Neutron sourcesRadioactive nuclei
One-mode sources (stimulated emission Glauber light)Unimodal lasers
Important parameter NC=photons pr coherence volumeNC ~10-3 for typical ESRF undulatorNC ~107 for typical optical laser
Beam direction
lv
lhCoherence volume Vc πlh lv ll 4horizontal vertical and longitudinal (temporal) coherence length
ll
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Chaotic source (ESRF undulator) (spontaneous independent emission in all modes)
Longitudinal (or temporal) coherence
Spatial coherenceAnalogy with Youngs double slit experiment Transverse coherence length (vh) lvh=λLπdvh (~2-150 microm)
0ττ
2
(1) e
|E(t)|τ)(t)E(tE
)(g minus= prop
+τ
At times gtgt τ0 the field amplitudes from a chaotic source are no longer correlated due to the wavelength spread
τ0 = 1(π∆ν) = λ2(cπ∆λ) (~3fs)Longitudinal coherence length ll = cτ0 = λ(π∆λλ) (~1microm)
(Gaussian 1st order time-correlation function)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Important source parameters
Brilliance B = photonssec [source area times solid angle times bandwidth]
Lateral coherence area At=πlhlv4=(λL)2(4πdhdv) (~200 microm2)Νc=photons in Vc (Vc = Αt times ll)
Coherent solid angle ΩC=AtL2=λ2(4πdhdv)= λ216As
Νc=B times τ0 times As times λ216As times ∆νν = Β λ3 (16πc)Coherent intensity Ic = (ΝcVc) times c times At
= B times (λ4)2 times (∆λλ) (~1010 phs)
As=πdhdv4
Coherent intensity decreases with decreasing λ (increasing energy) Coherent scattering is very Brilliance-hungry
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Coherence lengths
λ=21Aring d=11micromVisibility(β) ~ 80
Example Youngs double slit experiment with X-rays
λ=09Aring d=11micromVisibility(β) ~ 30
∆y=λLdLeitenberger et al Physica B 336 36 (2003)
Transverse coherence length (vh) lvh=λLπdvh (~5-250 microm)
Longitudinal coherence length ll = cτ0 = λ2(π∆λ) (~1microm)
Contrast β asymp (coherence volume)(scattering volume)
ESRFvalues
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Setup for coherent X-ray scattering
Coherence parametersTransverse coherence length lhv=λL(πdhv)Longitudinal coherence length ll=λ2(π∆λ)Contrast (degree of coherence) β=β (∆λλ )
IC prop B times λ2
ID10AIc ~2e8 phs (1997)Ic gt1e10 phs (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Fraunhoferdiffraction
from slit
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Coherent scattering What is the information we gain
If coherent light is scattered from a disordered system it gives rise to a random (grainy) diffraction pattern known as speckle A speckle pattern is an inter-ference pattern and related to the exact spatial arrangement of the scatterers in the disordered system (no averaging)
I(Qt) prop Sc(Qt) prop | sum e iQRj(t) |2j in coherence volume Vc=lv lh ll
D Abernathy et al
J Sync Rad 5 37 (1998)
Aerogel SAXSλ=1AringCCD (22 microm pixels)10 micron beamsize
speckle size ~λd
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Longitudinal coherence length and scattering geometry
PLD Path Length Difference lt ll CSAXSPLD = t + BC - AC asymp d tan(2θ)
Bragg geometryPLD asymp 2(micro-1)sin2θ
Grazing Incidence PLD asymp 2Λsinαi
2θAd
Bt
θ
Condition PLD lt ll SAXS d=10microm λ=155Aring ∆λλ=14times10-4 rarr θMAX asymp1o rarr QMAX asymp 015Aring-1
Reflection micro-1Cu= 22microm λ=155Aring ∆λλ=14times10-4
rarr θMAX asymp 5o rarr QMAX asymp 07Aring-1
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the contrast of the setup
Gamma-Poisson distribution of intensity coming from M statistically independent superimposed speckle patterns PM(I)=(MltIgt)MIM-1exp(-MIltIgt)Γ(M)σ2=ltIgt2M 1M=β
Coherent SAXS 8keV
Incoherent fraction
10 micron beam
VycorTM silica glass
Static speckle pattern
contrast β=1M=1281=356M ~ VscatteringVcoherence
Statistical speckle analysisPrinceton Instruments CCD1 Photon gives 879 adu2 Photons give 1758 adu
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the shape of a speckle
200 400 600 800 1000 1200
200
400
600
800
1000
1200
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
vertical direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
horizontal direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
diagonal direction
0 002 0040
002
004
006
008
01
Q
3
1
2
1 2 3
elongatedspeckles
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Pink beam ID10C (∆λλ=15)E=8keV 20microm beamsizeIcoherent gt 1x1011 phs
XPCS What is the information we gain
real space
non-coherent coherent
reciprocal space
ltI(Q)gt I(Qt) I(Qt)
DynamicsKinetics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What do we measure
2)2(
)(
)()()(
Q
QQQ
I
tItIg
ττ
+=
22~)2( |)(|1))((Re1)( tfSg QQQ +=+= ωτ
Temporal intensityauto-correlation function
int int minussdotpropV V
mnmn tibbtf )])()0([exp()()(|)(| rrQQQQ
220 )()]([
1~)(QQ
QSΓ+minus ωω
ω
)(cos)2exp(1)( 02)2( τωττ Γminus+=Qg
)2exp(1)()2( ττ Γminus+=Qg
Dynamic structure factor(Lorentzian shape)
(propagating waves)
(Brownian motion overdamped waves)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Scattering Vector Q [Aring-1]
Length Scale [Aring]Fr
eque
ncy
[Hz]
Energy [eV]
Raman
Brillouin
XPCSPCS
IXS
Spin-Echo
INS
NFS
Other techniques
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Typical example samples with some degree of (quasi) static disorder (gels glasses etc)
To properly calculate g(2) the ensemble averaging ltgtφ must be performed explicitly
Pusey amp van Megen Physica A (1989)
I(t) I(t)
tt
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Solution move the beam around and average g(2) over many exposures
or (much better)
Use a 2D detector and catch many pixels with same |Q|
Medipix-2
Ttp
Ttp
tpp
tItI
tItIQg
leleminuslele
+=
τφτφ
φτ
τ)()(
)()()(
0
)2(
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-equilibrium ie not all ∆t are equivalentg(2) depends on both ∆t and the absolute time (age)
Solution calculate the correlation function without time averaging (two-time correlation function)
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG =
t1
t2
G(t1t2) age
τ
age ta=(t2+t1)2
lag time τ=t2-t1
Only with a 2D detector several Qs at same age can be recorded
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
2d detectors for multi-speckle XPCS
Advantages More efficient data-taking (Mpixels) with 2d detectorBeam induced sample damage minimized
Drawbacks 2d detectors are slow noisy have limited dynamic rangeare not always photon counting have limited resolutionradiation hardness can be a problem
Princeton Instruments Dalsa MedipixMaxipix
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Limitations
sn ratio (more flux)
Speed and area of 2D detectors
Beam damage (minimize exposure)
Medipix-II detector 1kHz ff
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Depletion gel
Inter-particle interaction potential
V(r)Complex phase diagramrgR (01) φ (20) cp (43mgcm3)
2rg
2R
r
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The gel phase is transienthellip
W C K Poon et al Faraday Discuss 112 143 (1999)
delayedsedimentationageltt (latency time)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Chaotic source (ESRF undulator) (spontaneous independent emission in all modes)
Longitudinal (or temporal) coherence
Spatial coherenceAnalogy with Youngs double slit experiment Transverse coherence length (vh) lvh=λLπdvh (~2-150 microm)
0ττ
2
(1) e
|E(t)|τ)(t)E(tE
)(g minus= prop
+τ
At times gtgt τ0 the field amplitudes from a chaotic source are no longer correlated due to the wavelength spread
τ0 = 1(π∆ν) = λ2(cπ∆λ) (~3fs)Longitudinal coherence length ll = cτ0 = λ(π∆λλ) (~1microm)
(Gaussian 1st order time-correlation function)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Important source parameters
Brilliance B = photonssec [source area times solid angle times bandwidth]
Lateral coherence area At=πlhlv4=(λL)2(4πdhdv) (~200 microm2)Νc=photons in Vc (Vc = Αt times ll)
Coherent solid angle ΩC=AtL2=λ2(4πdhdv)= λ216As
Νc=B times τ0 times As times λ216As times ∆νν = Β λ3 (16πc)Coherent intensity Ic = (ΝcVc) times c times At
= B times (λ4)2 times (∆λλ) (~1010 phs)
As=πdhdv4
Coherent intensity decreases with decreasing λ (increasing energy) Coherent scattering is very Brilliance-hungry
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Coherence lengths
λ=21Aring d=11micromVisibility(β) ~ 80
Example Youngs double slit experiment with X-rays
λ=09Aring d=11micromVisibility(β) ~ 30
∆y=λLdLeitenberger et al Physica B 336 36 (2003)
Transverse coherence length (vh) lvh=λLπdvh (~5-250 microm)
Longitudinal coherence length ll = cτ0 = λ2(π∆λ) (~1microm)
Contrast β asymp (coherence volume)(scattering volume)
ESRFvalues
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Setup for coherent X-ray scattering
Coherence parametersTransverse coherence length lhv=λL(πdhv)Longitudinal coherence length ll=λ2(π∆λ)Contrast (degree of coherence) β=β (∆λλ )
IC prop B times λ2
ID10AIc ~2e8 phs (1997)Ic gt1e10 phs (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Fraunhoferdiffraction
from slit
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Coherent scattering What is the information we gain
If coherent light is scattered from a disordered system it gives rise to a random (grainy) diffraction pattern known as speckle A speckle pattern is an inter-ference pattern and related to the exact spatial arrangement of the scatterers in the disordered system (no averaging)
I(Qt) prop Sc(Qt) prop | sum e iQRj(t) |2j in coherence volume Vc=lv lh ll
D Abernathy et al
J Sync Rad 5 37 (1998)
Aerogel SAXSλ=1AringCCD (22 microm pixels)10 micron beamsize
speckle size ~λd
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Longitudinal coherence length and scattering geometry
PLD Path Length Difference lt ll CSAXSPLD = t + BC - AC asymp d tan(2θ)
Bragg geometryPLD asymp 2(micro-1)sin2θ
Grazing Incidence PLD asymp 2Λsinαi
2θAd
Bt
θ
Condition PLD lt ll SAXS d=10microm λ=155Aring ∆λλ=14times10-4 rarr θMAX asymp1o rarr QMAX asymp 015Aring-1
Reflection micro-1Cu= 22microm λ=155Aring ∆λλ=14times10-4
rarr θMAX asymp 5o rarr QMAX asymp 07Aring-1
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the contrast of the setup
Gamma-Poisson distribution of intensity coming from M statistically independent superimposed speckle patterns PM(I)=(MltIgt)MIM-1exp(-MIltIgt)Γ(M)σ2=ltIgt2M 1M=β
Coherent SAXS 8keV
Incoherent fraction
10 micron beam
VycorTM silica glass
Static speckle pattern
contrast β=1M=1281=356M ~ VscatteringVcoherence
Statistical speckle analysisPrinceton Instruments CCD1 Photon gives 879 adu2 Photons give 1758 adu
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the shape of a speckle
200 400 600 800 1000 1200
200
400
600
800
1000
1200
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
vertical direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
horizontal direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
diagonal direction
0 002 0040
002
004
006
008
01
Q
3
1
2
1 2 3
elongatedspeckles
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Pink beam ID10C (∆λλ=15)E=8keV 20microm beamsizeIcoherent gt 1x1011 phs
XPCS What is the information we gain
real space
non-coherent coherent
reciprocal space
ltI(Q)gt I(Qt) I(Qt)
DynamicsKinetics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What do we measure
2)2(
)(
)()()(
Q
QQQ
I
tItIg
ττ
+=
22~)2( |)(|1))((Re1)( tfSg QQQ +=+= ωτ
Temporal intensityauto-correlation function
int int minussdotpropV V
mnmn tibbtf )])()0([exp()()(|)(| rrQQQQ
220 )()]([
1~)(QQ
QSΓ+minus ωω
ω
)(cos)2exp(1)( 02)2( τωττ Γminus+=Qg
)2exp(1)()2( ττ Γminus+=Qg
Dynamic structure factor(Lorentzian shape)
(propagating waves)
(Brownian motion overdamped waves)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Scattering Vector Q [Aring-1]
Length Scale [Aring]Fr
eque
ncy
[Hz]
Energy [eV]
Raman
Brillouin
XPCSPCS
IXS
Spin-Echo
INS
NFS
Other techniques
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Typical example samples with some degree of (quasi) static disorder (gels glasses etc)
To properly calculate g(2) the ensemble averaging ltgtφ must be performed explicitly
Pusey amp van Megen Physica A (1989)
I(t) I(t)
tt
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Solution move the beam around and average g(2) over many exposures
or (much better)
Use a 2D detector and catch many pixels with same |Q|
Medipix-2
Ttp
Ttp
tpp
tItI
tItIQg
leleminuslele
+=
τφτφ
φτ
τ)()(
)()()(
0
)2(
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-equilibrium ie not all ∆t are equivalentg(2) depends on both ∆t and the absolute time (age)
Solution calculate the correlation function without time averaging (two-time correlation function)
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG =
t1
t2
G(t1t2) age
τ
age ta=(t2+t1)2
lag time τ=t2-t1
Only with a 2D detector several Qs at same age can be recorded
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
2d detectors for multi-speckle XPCS
Advantages More efficient data-taking (Mpixels) with 2d detectorBeam induced sample damage minimized
Drawbacks 2d detectors are slow noisy have limited dynamic rangeare not always photon counting have limited resolutionradiation hardness can be a problem
Princeton Instruments Dalsa MedipixMaxipix
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Limitations
sn ratio (more flux)
Speed and area of 2D detectors
Beam damage (minimize exposure)
Medipix-II detector 1kHz ff
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Depletion gel
Inter-particle interaction potential
V(r)Complex phase diagramrgR (01) φ (20) cp (43mgcm3)
2rg
2R
r
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The gel phase is transienthellip
W C K Poon et al Faraday Discuss 112 143 (1999)
delayedsedimentationageltt (latency time)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Important source parameters
Brilliance B = photonssec [source area times solid angle times bandwidth]
Lateral coherence area At=πlhlv4=(λL)2(4πdhdv) (~200 microm2)Νc=photons in Vc (Vc = Αt times ll)
Coherent solid angle ΩC=AtL2=λ2(4πdhdv)= λ216As
Νc=B times τ0 times As times λ216As times ∆νν = Β λ3 (16πc)Coherent intensity Ic = (ΝcVc) times c times At
= B times (λ4)2 times (∆λλ) (~1010 phs)
As=πdhdv4
Coherent intensity decreases with decreasing λ (increasing energy) Coherent scattering is very Brilliance-hungry
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Coherence lengths
λ=21Aring d=11micromVisibility(β) ~ 80
Example Youngs double slit experiment with X-rays
λ=09Aring d=11micromVisibility(β) ~ 30
∆y=λLdLeitenberger et al Physica B 336 36 (2003)
Transverse coherence length (vh) lvh=λLπdvh (~5-250 microm)
Longitudinal coherence length ll = cτ0 = λ2(π∆λ) (~1microm)
Contrast β asymp (coherence volume)(scattering volume)
ESRFvalues
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Setup for coherent X-ray scattering
Coherence parametersTransverse coherence length lhv=λL(πdhv)Longitudinal coherence length ll=λ2(π∆λ)Contrast (degree of coherence) β=β (∆λλ )
IC prop B times λ2
ID10AIc ~2e8 phs (1997)Ic gt1e10 phs (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Fraunhoferdiffraction
from slit
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Coherent scattering What is the information we gain
If coherent light is scattered from a disordered system it gives rise to a random (grainy) diffraction pattern known as speckle A speckle pattern is an inter-ference pattern and related to the exact spatial arrangement of the scatterers in the disordered system (no averaging)
I(Qt) prop Sc(Qt) prop | sum e iQRj(t) |2j in coherence volume Vc=lv lh ll
D Abernathy et al
J Sync Rad 5 37 (1998)
Aerogel SAXSλ=1AringCCD (22 microm pixels)10 micron beamsize
speckle size ~λd
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Longitudinal coherence length and scattering geometry
PLD Path Length Difference lt ll CSAXSPLD = t + BC - AC asymp d tan(2θ)
Bragg geometryPLD asymp 2(micro-1)sin2θ
Grazing Incidence PLD asymp 2Λsinαi
2θAd
Bt
θ
Condition PLD lt ll SAXS d=10microm λ=155Aring ∆λλ=14times10-4 rarr θMAX asymp1o rarr QMAX asymp 015Aring-1
Reflection micro-1Cu= 22microm λ=155Aring ∆λλ=14times10-4
rarr θMAX asymp 5o rarr QMAX asymp 07Aring-1
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the contrast of the setup
Gamma-Poisson distribution of intensity coming from M statistically independent superimposed speckle patterns PM(I)=(MltIgt)MIM-1exp(-MIltIgt)Γ(M)σ2=ltIgt2M 1M=β
Coherent SAXS 8keV
Incoherent fraction
10 micron beam
VycorTM silica glass
Static speckle pattern
contrast β=1M=1281=356M ~ VscatteringVcoherence
Statistical speckle analysisPrinceton Instruments CCD1 Photon gives 879 adu2 Photons give 1758 adu
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the shape of a speckle
200 400 600 800 1000 1200
200
400
600
800
1000
1200
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
vertical direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
horizontal direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
diagonal direction
0 002 0040
002
004
006
008
01
Q
3
1
2
1 2 3
elongatedspeckles
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Pink beam ID10C (∆λλ=15)E=8keV 20microm beamsizeIcoherent gt 1x1011 phs
XPCS What is the information we gain
real space
non-coherent coherent
reciprocal space
ltI(Q)gt I(Qt) I(Qt)
DynamicsKinetics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What do we measure
2)2(
)(
)()()(
Q
QQQ
I
tItIg
ττ
+=
22~)2( |)(|1))((Re1)( tfSg QQQ +=+= ωτ
Temporal intensityauto-correlation function
int int minussdotpropV V
mnmn tibbtf )])()0([exp()()(|)(| rrQQQQ
220 )()]([
1~)(QQ
QSΓ+minus ωω
ω
)(cos)2exp(1)( 02)2( τωττ Γminus+=Qg
)2exp(1)()2( ττ Γminus+=Qg
Dynamic structure factor(Lorentzian shape)
(propagating waves)
(Brownian motion overdamped waves)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Scattering Vector Q [Aring-1]
Length Scale [Aring]Fr
eque
ncy
[Hz]
Energy [eV]
Raman
Brillouin
XPCSPCS
IXS
Spin-Echo
INS
NFS
Other techniques
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Typical example samples with some degree of (quasi) static disorder (gels glasses etc)
To properly calculate g(2) the ensemble averaging ltgtφ must be performed explicitly
Pusey amp van Megen Physica A (1989)
I(t) I(t)
tt
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Solution move the beam around and average g(2) over many exposures
or (much better)
Use a 2D detector and catch many pixels with same |Q|
Medipix-2
Ttp
Ttp
tpp
tItI
tItIQg
leleminuslele
+=
τφτφ
φτ
τ)()(
)()()(
0
)2(
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-equilibrium ie not all ∆t are equivalentg(2) depends on both ∆t and the absolute time (age)
Solution calculate the correlation function without time averaging (two-time correlation function)
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG =
t1
t2
G(t1t2) age
τ
age ta=(t2+t1)2
lag time τ=t2-t1
Only with a 2D detector several Qs at same age can be recorded
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
2d detectors for multi-speckle XPCS
Advantages More efficient data-taking (Mpixels) with 2d detectorBeam induced sample damage minimized
Drawbacks 2d detectors are slow noisy have limited dynamic rangeare not always photon counting have limited resolutionradiation hardness can be a problem
Princeton Instruments Dalsa MedipixMaxipix
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Limitations
sn ratio (more flux)
Speed and area of 2D detectors
Beam damage (minimize exposure)
Medipix-II detector 1kHz ff
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Depletion gel
Inter-particle interaction potential
V(r)Complex phase diagramrgR (01) φ (20) cp (43mgcm3)
2rg
2R
r
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The gel phase is transienthellip
W C K Poon et al Faraday Discuss 112 143 (1999)
delayedsedimentationageltt (latency time)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Coherence lengths
λ=21Aring d=11micromVisibility(β) ~ 80
Example Youngs double slit experiment with X-rays
λ=09Aring d=11micromVisibility(β) ~ 30
∆y=λLdLeitenberger et al Physica B 336 36 (2003)
Transverse coherence length (vh) lvh=λLπdvh (~5-250 microm)
Longitudinal coherence length ll = cτ0 = λ2(π∆λ) (~1microm)
Contrast β asymp (coherence volume)(scattering volume)
ESRFvalues
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Setup for coherent X-ray scattering
Coherence parametersTransverse coherence length lhv=λL(πdhv)Longitudinal coherence length ll=λ2(π∆λ)Contrast (degree of coherence) β=β (∆λλ )
IC prop B times λ2
ID10AIc ~2e8 phs (1997)Ic gt1e10 phs (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Fraunhoferdiffraction
from slit
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Coherent scattering What is the information we gain
If coherent light is scattered from a disordered system it gives rise to a random (grainy) diffraction pattern known as speckle A speckle pattern is an inter-ference pattern and related to the exact spatial arrangement of the scatterers in the disordered system (no averaging)
I(Qt) prop Sc(Qt) prop | sum e iQRj(t) |2j in coherence volume Vc=lv lh ll
D Abernathy et al
J Sync Rad 5 37 (1998)
Aerogel SAXSλ=1AringCCD (22 microm pixels)10 micron beamsize
speckle size ~λd
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Longitudinal coherence length and scattering geometry
PLD Path Length Difference lt ll CSAXSPLD = t + BC - AC asymp d tan(2θ)
Bragg geometryPLD asymp 2(micro-1)sin2θ
Grazing Incidence PLD asymp 2Λsinαi
2θAd
Bt
θ
Condition PLD lt ll SAXS d=10microm λ=155Aring ∆λλ=14times10-4 rarr θMAX asymp1o rarr QMAX asymp 015Aring-1
Reflection micro-1Cu= 22microm λ=155Aring ∆λλ=14times10-4
rarr θMAX asymp 5o rarr QMAX asymp 07Aring-1
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the contrast of the setup
Gamma-Poisson distribution of intensity coming from M statistically independent superimposed speckle patterns PM(I)=(MltIgt)MIM-1exp(-MIltIgt)Γ(M)σ2=ltIgt2M 1M=β
Coherent SAXS 8keV
Incoherent fraction
10 micron beam
VycorTM silica glass
Static speckle pattern
contrast β=1M=1281=356M ~ VscatteringVcoherence
Statistical speckle analysisPrinceton Instruments CCD1 Photon gives 879 adu2 Photons give 1758 adu
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the shape of a speckle
200 400 600 800 1000 1200
200
400
600
800
1000
1200
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
vertical direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
horizontal direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
diagonal direction
0 002 0040
002
004
006
008
01
Q
3
1
2
1 2 3
elongatedspeckles
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Pink beam ID10C (∆λλ=15)E=8keV 20microm beamsizeIcoherent gt 1x1011 phs
XPCS What is the information we gain
real space
non-coherent coherent
reciprocal space
ltI(Q)gt I(Qt) I(Qt)
DynamicsKinetics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What do we measure
2)2(
)(
)()()(
Q
QQQ
I
tItIg
ττ
+=
22~)2( |)(|1))((Re1)( tfSg QQQ +=+= ωτ
Temporal intensityauto-correlation function
int int minussdotpropV V
mnmn tibbtf )])()0([exp()()(|)(| rrQQQQ
220 )()]([
1~)(QQ
QSΓ+minus ωω
ω
)(cos)2exp(1)( 02)2( τωττ Γminus+=Qg
)2exp(1)()2( ττ Γminus+=Qg
Dynamic structure factor(Lorentzian shape)
(propagating waves)
(Brownian motion overdamped waves)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Scattering Vector Q [Aring-1]
Length Scale [Aring]Fr
eque
ncy
[Hz]
Energy [eV]
Raman
Brillouin
XPCSPCS
IXS
Spin-Echo
INS
NFS
Other techniques
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Typical example samples with some degree of (quasi) static disorder (gels glasses etc)
To properly calculate g(2) the ensemble averaging ltgtφ must be performed explicitly
Pusey amp van Megen Physica A (1989)
I(t) I(t)
tt
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Solution move the beam around and average g(2) over many exposures
or (much better)
Use a 2D detector and catch many pixels with same |Q|
Medipix-2
Ttp
Ttp
tpp
tItI
tItIQg
leleminuslele
+=
τφτφ
φτ
τ)()(
)()()(
0
)2(
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-equilibrium ie not all ∆t are equivalentg(2) depends on both ∆t and the absolute time (age)
Solution calculate the correlation function without time averaging (two-time correlation function)
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG =
t1
t2
G(t1t2) age
τ
age ta=(t2+t1)2
lag time τ=t2-t1
Only with a 2D detector several Qs at same age can be recorded
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
2d detectors for multi-speckle XPCS
Advantages More efficient data-taking (Mpixels) with 2d detectorBeam induced sample damage minimized
Drawbacks 2d detectors are slow noisy have limited dynamic rangeare not always photon counting have limited resolutionradiation hardness can be a problem
Princeton Instruments Dalsa MedipixMaxipix
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Limitations
sn ratio (more flux)
Speed and area of 2D detectors
Beam damage (minimize exposure)
Medipix-II detector 1kHz ff
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Depletion gel
Inter-particle interaction potential
V(r)Complex phase diagramrgR (01) φ (20) cp (43mgcm3)
2rg
2R
r
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The gel phase is transienthellip
W C K Poon et al Faraday Discuss 112 143 (1999)
delayedsedimentationageltt (latency time)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Setup for coherent X-ray scattering
Coherence parametersTransverse coherence length lhv=λL(πdhv)Longitudinal coherence length ll=λ2(π∆λ)Contrast (degree of coherence) β=β (∆λλ )
IC prop B times λ2
ID10AIc ~2e8 phs (1997)Ic gt1e10 phs (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Fraunhoferdiffraction
from slit
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Coherent scattering What is the information we gain
If coherent light is scattered from a disordered system it gives rise to a random (grainy) diffraction pattern known as speckle A speckle pattern is an inter-ference pattern and related to the exact spatial arrangement of the scatterers in the disordered system (no averaging)
I(Qt) prop Sc(Qt) prop | sum e iQRj(t) |2j in coherence volume Vc=lv lh ll
D Abernathy et al
J Sync Rad 5 37 (1998)
Aerogel SAXSλ=1AringCCD (22 microm pixels)10 micron beamsize
speckle size ~λd
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Longitudinal coherence length and scattering geometry
PLD Path Length Difference lt ll CSAXSPLD = t + BC - AC asymp d tan(2θ)
Bragg geometryPLD asymp 2(micro-1)sin2θ
Grazing Incidence PLD asymp 2Λsinαi
2θAd
Bt
θ
Condition PLD lt ll SAXS d=10microm λ=155Aring ∆λλ=14times10-4 rarr θMAX asymp1o rarr QMAX asymp 015Aring-1
Reflection micro-1Cu= 22microm λ=155Aring ∆λλ=14times10-4
rarr θMAX asymp 5o rarr QMAX asymp 07Aring-1
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the contrast of the setup
Gamma-Poisson distribution of intensity coming from M statistically independent superimposed speckle patterns PM(I)=(MltIgt)MIM-1exp(-MIltIgt)Γ(M)σ2=ltIgt2M 1M=β
Coherent SAXS 8keV
Incoherent fraction
10 micron beam
VycorTM silica glass
Static speckle pattern
contrast β=1M=1281=356M ~ VscatteringVcoherence
Statistical speckle analysisPrinceton Instruments CCD1 Photon gives 879 adu2 Photons give 1758 adu
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the shape of a speckle
200 400 600 800 1000 1200
200
400
600
800
1000
1200
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
vertical direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
horizontal direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
diagonal direction
0 002 0040
002
004
006
008
01
Q
3
1
2
1 2 3
elongatedspeckles
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Pink beam ID10C (∆λλ=15)E=8keV 20microm beamsizeIcoherent gt 1x1011 phs
XPCS What is the information we gain
real space
non-coherent coherent
reciprocal space
ltI(Q)gt I(Qt) I(Qt)
DynamicsKinetics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What do we measure
2)2(
)(
)()()(
Q
QQQ
I
tItIg
ττ
+=
22~)2( |)(|1))((Re1)( tfSg QQQ +=+= ωτ
Temporal intensityauto-correlation function
int int minussdotpropV V
mnmn tibbtf )])()0([exp()()(|)(| rrQQQQ
220 )()]([
1~)(QQ
QSΓ+minus ωω
ω
)(cos)2exp(1)( 02)2( τωττ Γminus+=Qg
)2exp(1)()2( ττ Γminus+=Qg
Dynamic structure factor(Lorentzian shape)
(propagating waves)
(Brownian motion overdamped waves)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Scattering Vector Q [Aring-1]
Length Scale [Aring]Fr
eque
ncy
[Hz]
Energy [eV]
Raman
Brillouin
XPCSPCS
IXS
Spin-Echo
INS
NFS
Other techniques
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Typical example samples with some degree of (quasi) static disorder (gels glasses etc)
To properly calculate g(2) the ensemble averaging ltgtφ must be performed explicitly
Pusey amp van Megen Physica A (1989)
I(t) I(t)
tt
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Solution move the beam around and average g(2) over many exposures
or (much better)
Use a 2D detector and catch many pixels with same |Q|
Medipix-2
Ttp
Ttp
tpp
tItI
tItIQg
leleminuslele
+=
τφτφ
φτ
τ)()(
)()()(
0
)2(
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-equilibrium ie not all ∆t are equivalentg(2) depends on both ∆t and the absolute time (age)
Solution calculate the correlation function without time averaging (two-time correlation function)
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG =
t1
t2
G(t1t2) age
τ
age ta=(t2+t1)2
lag time τ=t2-t1
Only with a 2D detector several Qs at same age can be recorded
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
2d detectors for multi-speckle XPCS
Advantages More efficient data-taking (Mpixels) with 2d detectorBeam induced sample damage minimized
Drawbacks 2d detectors are slow noisy have limited dynamic rangeare not always photon counting have limited resolutionradiation hardness can be a problem
Princeton Instruments Dalsa MedipixMaxipix
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Limitations
sn ratio (more flux)
Speed and area of 2D detectors
Beam damage (minimize exposure)
Medipix-II detector 1kHz ff
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Depletion gel
Inter-particle interaction potential
V(r)Complex phase diagramrgR (01) φ (20) cp (43mgcm3)
2rg
2R
r
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The gel phase is transienthellip
W C K Poon et al Faraday Discuss 112 143 (1999)
delayedsedimentationageltt (latency time)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Fraunhoferdiffraction
from slit
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Coherent scattering What is the information we gain
If coherent light is scattered from a disordered system it gives rise to a random (grainy) diffraction pattern known as speckle A speckle pattern is an inter-ference pattern and related to the exact spatial arrangement of the scatterers in the disordered system (no averaging)
I(Qt) prop Sc(Qt) prop | sum e iQRj(t) |2j in coherence volume Vc=lv lh ll
D Abernathy et al
J Sync Rad 5 37 (1998)
Aerogel SAXSλ=1AringCCD (22 microm pixels)10 micron beamsize
speckle size ~λd
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Longitudinal coherence length and scattering geometry
PLD Path Length Difference lt ll CSAXSPLD = t + BC - AC asymp d tan(2θ)
Bragg geometryPLD asymp 2(micro-1)sin2θ
Grazing Incidence PLD asymp 2Λsinαi
2θAd
Bt
θ
Condition PLD lt ll SAXS d=10microm λ=155Aring ∆λλ=14times10-4 rarr θMAX asymp1o rarr QMAX asymp 015Aring-1
Reflection micro-1Cu= 22microm λ=155Aring ∆λλ=14times10-4
rarr θMAX asymp 5o rarr QMAX asymp 07Aring-1
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the contrast of the setup
Gamma-Poisson distribution of intensity coming from M statistically independent superimposed speckle patterns PM(I)=(MltIgt)MIM-1exp(-MIltIgt)Γ(M)σ2=ltIgt2M 1M=β
Coherent SAXS 8keV
Incoherent fraction
10 micron beam
VycorTM silica glass
Static speckle pattern
contrast β=1M=1281=356M ~ VscatteringVcoherence
Statistical speckle analysisPrinceton Instruments CCD1 Photon gives 879 adu2 Photons give 1758 adu
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the shape of a speckle
200 400 600 800 1000 1200
200
400
600
800
1000
1200
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
vertical direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
horizontal direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
diagonal direction
0 002 0040
002
004
006
008
01
Q
3
1
2
1 2 3
elongatedspeckles
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Pink beam ID10C (∆λλ=15)E=8keV 20microm beamsizeIcoherent gt 1x1011 phs
XPCS What is the information we gain
real space
non-coherent coherent
reciprocal space
ltI(Q)gt I(Qt) I(Qt)
DynamicsKinetics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What do we measure
2)2(
)(
)()()(
Q
QQQ
I
tItIg
ττ
+=
22~)2( |)(|1))((Re1)( tfSg QQQ +=+= ωτ
Temporal intensityauto-correlation function
int int minussdotpropV V
mnmn tibbtf )])()0([exp()()(|)(| rrQQQQ
220 )()]([
1~)(QQ
QSΓ+minus ωω
ω
)(cos)2exp(1)( 02)2( τωττ Γminus+=Qg
)2exp(1)()2( ττ Γminus+=Qg
Dynamic structure factor(Lorentzian shape)
(propagating waves)
(Brownian motion overdamped waves)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Scattering Vector Q [Aring-1]
Length Scale [Aring]Fr
eque
ncy
[Hz]
Energy [eV]
Raman
Brillouin
XPCSPCS
IXS
Spin-Echo
INS
NFS
Other techniques
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Typical example samples with some degree of (quasi) static disorder (gels glasses etc)
To properly calculate g(2) the ensemble averaging ltgtφ must be performed explicitly
Pusey amp van Megen Physica A (1989)
I(t) I(t)
tt
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Solution move the beam around and average g(2) over many exposures
or (much better)
Use a 2D detector and catch many pixels with same |Q|
Medipix-2
Ttp
Ttp
tpp
tItI
tItIQg
leleminuslele
+=
τφτφ
φτ
τ)()(
)()()(
0
)2(
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-equilibrium ie not all ∆t are equivalentg(2) depends on both ∆t and the absolute time (age)
Solution calculate the correlation function without time averaging (two-time correlation function)
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG =
t1
t2
G(t1t2) age
τ
age ta=(t2+t1)2
lag time τ=t2-t1
Only with a 2D detector several Qs at same age can be recorded
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
2d detectors for multi-speckle XPCS
Advantages More efficient data-taking (Mpixels) with 2d detectorBeam induced sample damage minimized
Drawbacks 2d detectors are slow noisy have limited dynamic rangeare not always photon counting have limited resolutionradiation hardness can be a problem
Princeton Instruments Dalsa MedipixMaxipix
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Limitations
sn ratio (more flux)
Speed and area of 2D detectors
Beam damage (minimize exposure)
Medipix-II detector 1kHz ff
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Depletion gel
Inter-particle interaction potential
V(r)Complex phase diagramrgR (01) φ (20) cp (43mgcm3)
2rg
2R
r
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The gel phase is transienthellip
W C K Poon et al Faraday Discuss 112 143 (1999)
delayedsedimentationageltt (latency time)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Coherent scattering What is the information we gain
If coherent light is scattered from a disordered system it gives rise to a random (grainy) diffraction pattern known as speckle A speckle pattern is an inter-ference pattern and related to the exact spatial arrangement of the scatterers in the disordered system (no averaging)
I(Qt) prop Sc(Qt) prop | sum e iQRj(t) |2j in coherence volume Vc=lv lh ll
D Abernathy et al
J Sync Rad 5 37 (1998)
Aerogel SAXSλ=1AringCCD (22 microm pixels)10 micron beamsize
speckle size ~λd
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Longitudinal coherence length and scattering geometry
PLD Path Length Difference lt ll CSAXSPLD = t + BC - AC asymp d tan(2θ)
Bragg geometryPLD asymp 2(micro-1)sin2θ
Grazing Incidence PLD asymp 2Λsinαi
2θAd
Bt
θ
Condition PLD lt ll SAXS d=10microm λ=155Aring ∆λλ=14times10-4 rarr θMAX asymp1o rarr QMAX asymp 015Aring-1
Reflection micro-1Cu= 22microm λ=155Aring ∆λλ=14times10-4
rarr θMAX asymp 5o rarr QMAX asymp 07Aring-1
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the contrast of the setup
Gamma-Poisson distribution of intensity coming from M statistically independent superimposed speckle patterns PM(I)=(MltIgt)MIM-1exp(-MIltIgt)Γ(M)σ2=ltIgt2M 1M=β
Coherent SAXS 8keV
Incoherent fraction
10 micron beam
VycorTM silica glass
Static speckle pattern
contrast β=1M=1281=356M ~ VscatteringVcoherence
Statistical speckle analysisPrinceton Instruments CCD1 Photon gives 879 adu2 Photons give 1758 adu
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the shape of a speckle
200 400 600 800 1000 1200
200
400
600
800
1000
1200
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
vertical direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
horizontal direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
diagonal direction
0 002 0040
002
004
006
008
01
Q
3
1
2
1 2 3
elongatedspeckles
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Pink beam ID10C (∆λλ=15)E=8keV 20microm beamsizeIcoherent gt 1x1011 phs
XPCS What is the information we gain
real space
non-coherent coherent
reciprocal space
ltI(Q)gt I(Qt) I(Qt)
DynamicsKinetics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What do we measure
2)2(
)(
)()()(
Q
QQQ
I
tItIg
ττ
+=
22~)2( |)(|1))((Re1)( tfSg QQQ +=+= ωτ
Temporal intensityauto-correlation function
int int minussdotpropV V
mnmn tibbtf )])()0([exp()()(|)(| rrQQQQ
220 )()]([
1~)(QQ
QSΓ+minus ωω
ω
)(cos)2exp(1)( 02)2( τωττ Γminus+=Qg
)2exp(1)()2( ττ Γminus+=Qg
Dynamic structure factor(Lorentzian shape)
(propagating waves)
(Brownian motion overdamped waves)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Scattering Vector Q [Aring-1]
Length Scale [Aring]Fr
eque
ncy
[Hz]
Energy [eV]
Raman
Brillouin
XPCSPCS
IXS
Spin-Echo
INS
NFS
Other techniques
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Typical example samples with some degree of (quasi) static disorder (gels glasses etc)
To properly calculate g(2) the ensemble averaging ltgtφ must be performed explicitly
Pusey amp van Megen Physica A (1989)
I(t) I(t)
tt
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Solution move the beam around and average g(2) over many exposures
or (much better)
Use a 2D detector and catch many pixels with same |Q|
Medipix-2
Ttp
Ttp
tpp
tItI
tItIQg
leleminuslele
+=
τφτφ
φτ
τ)()(
)()()(
0
)2(
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-equilibrium ie not all ∆t are equivalentg(2) depends on both ∆t and the absolute time (age)
Solution calculate the correlation function without time averaging (two-time correlation function)
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG =
t1
t2
G(t1t2) age
τ
age ta=(t2+t1)2
lag time τ=t2-t1
Only with a 2D detector several Qs at same age can be recorded
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
2d detectors for multi-speckle XPCS
Advantages More efficient data-taking (Mpixels) with 2d detectorBeam induced sample damage minimized
Drawbacks 2d detectors are slow noisy have limited dynamic rangeare not always photon counting have limited resolutionradiation hardness can be a problem
Princeton Instruments Dalsa MedipixMaxipix
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Limitations
sn ratio (more flux)
Speed and area of 2D detectors
Beam damage (minimize exposure)
Medipix-II detector 1kHz ff
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Depletion gel
Inter-particle interaction potential
V(r)Complex phase diagramrgR (01) φ (20) cp (43mgcm3)
2rg
2R
r
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The gel phase is transienthellip
W C K Poon et al Faraday Discuss 112 143 (1999)
delayedsedimentationageltt (latency time)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Longitudinal coherence length and scattering geometry
PLD Path Length Difference lt ll CSAXSPLD = t + BC - AC asymp d tan(2θ)
Bragg geometryPLD asymp 2(micro-1)sin2θ
Grazing Incidence PLD asymp 2Λsinαi
2θAd
Bt
θ
Condition PLD lt ll SAXS d=10microm λ=155Aring ∆λλ=14times10-4 rarr θMAX asymp1o rarr QMAX asymp 015Aring-1
Reflection micro-1Cu= 22microm λ=155Aring ∆λλ=14times10-4
rarr θMAX asymp 5o rarr QMAX asymp 07Aring-1
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the contrast of the setup
Gamma-Poisson distribution of intensity coming from M statistically independent superimposed speckle patterns PM(I)=(MltIgt)MIM-1exp(-MIltIgt)Γ(M)σ2=ltIgt2M 1M=β
Coherent SAXS 8keV
Incoherent fraction
10 micron beam
VycorTM silica glass
Static speckle pattern
contrast β=1M=1281=356M ~ VscatteringVcoherence
Statistical speckle analysisPrinceton Instruments CCD1 Photon gives 879 adu2 Photons give 1758 adu
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the shape of a speckle
200 400 600 800 1000 1200
200
400
600
800
1000
1200
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
vertical direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
horizontal direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
diagonal direction
0 002 0040
002
004
006
008
01
Q
3
1
2
1 2 3
elongatedspeckles
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Pink beam ID10C (∆λλ=15)E=8keV 20microm beamsizeIcoherent gt 1x1011 phs
XPCS What is the information we gain
real space
non-coherent coherent
reciprocal space
ltI(Q)gt I(Qt) I(Qt)
DynamicsKinetics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What do we measure
2)2(
)(
)()()(
Q
QQQ
I
tItIg
ττ
+=
22~)2( |)(|1))((Re1)( tfSg QQQ +=+= ωτ
Temporal intensityauto-correlation function
int int minussdotpropV V
mnmn tibbtf )])()0([exp()()(|)(| rrQQQQ
220 )()]([
1~)(QQ
QSΓ+minus ωω
ω
)(cos)2exp(1)( 02)2( τωττ Γminus+=Qg
)2exp(1)()2( ττ Γminus+=Qg
Dynamic structure factor(Lorentzian shape)
(propagating waves)
(Brownian motion overdamped waves)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Scattering Vector Q [Aring-1]
Length Scale [Aring]Fr
eque
ncy
[Hz]
Energy [eV]
Raman
Brillouin
XPCSPCS
IXS
Spin-Echo
INS
NFS
Other techniques
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Typical example samples with some degree of (quasi) static disorder (gels glasses etc)
To properly calculate g(2) the ensemble averaging ltgtφ must be performed explicitly
Pusey amp van Megen Physica A (1989)
I(t) I(t)
tt
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Solution move the beam around and average g(2) over many exposures
or (much better)
Use a 2D detector and catch many pixels with same |Q|
Medipix-2
Ttp
Ttp
tpp
tItI
tItIQg
leleminuslele
+=
τφτφ
φτ
τ)()(
)()()(
0
)2(
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-equilibrium ie not all ∆t are equivalentg(2) depends on both ∆t and the absolute time (age)
Solution calculate the correlation function without time averaging (two-time correlation function)
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG =
t1
t2
G(t1t2) age
τ
age ta=(t2+t1)2
lag time τ=t2-t1
Only with a 2D detector several Qs at same age can be recorded
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
2d detectors for multi-speckle XPCS
Advantages More efficient data-taking (Mpixels) with 2d detectorBeam induced sample damage minimized
Drawbacks 2d detectors are slow noisy have limited dynamic rangeare not always photon counting have limited resolutionradiation hardness can be a problem
Princeton Instruments Dalsa MedipixMaxipix
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Limitations
sn ratio (more flux)
Speed and area of 2D detectors
Beam damage (minimize exposure)
Medipix-II detector 1kHz ff
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Depletion gel
Inter-particle interaction potential
V(r)Complex phase diagramrgR (01) φ (20) cp (43mgcm3)
2rg
2R
r
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The gel phase is transienthellip
W C K Poon et al Faraday Discuss 112 143 (1999)
delayedsedimentationageltt (latency time)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the contrast of the setup
Gamma-Poisson distribution of intensity coming from M statistically independent superimposed speckle patterns PM(I)=(MltIgt)MIM-1exp(-MIltIgt)Γ(M)σ2=ltIgt2M 1M=β
Coherent SAXS 8keV
Incoherent fraction
10 micron beam
VycorTM silica glass
Static speckle pattern
contrast β=1M=1281=356M ~ VscatteringVcoherence
Statistical speckle analysisPrinceton Instruments CCD1 Photon gives 879 adu2 Photons give 1758 adu
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the shape of a speckle
200 400 600 800 1000 1200
200
400
600
800
1000
1200
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
vertical direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
horizontal direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
diagonal direction
0 002 0040
002
004
006
008
01
Q
3
1
2
1 2 3
elongatedspeckles
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Pink beam ID10C (∆λλ=15)E=8keV 20microm beamsizeIcoherent gt 1x1011 phs
XPCS What is the information we gain
real space
non-coherent coherent
reciprocal space
ltI(Q)gt I(Qt) I(Qt)
DynamicsKinetics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What do we measure
2)2(
)(
)()()(
Q
QQQ
I
tItIg
ττ
+=
22~)2( |)(|1))((Re1)( tfSg QQQ +=+= ωτ
Temporal intensityauto-correlation function
int int minussdotpropV V
mnmn tibbtf )])()0([exp()()(|)(| rrQQQQ
220 )()]([
1~)(QQ
QSΓ+minus ωω
ω
)(cos)2exp(1)( 02)2( τωττ Γminus+=Qg
)2exp(1)()2( ττ Γminus+=Qg
Dynamic structure factor(Lorentzian shape)
(propagating waves)
(Brownian motion overdamped waves)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Scattering Vector Q [Aring-1]
Length Scale [Aring]Fr
eque
ncy
[Hz]
Energy [eV]
Raman
Brillouin
XPCSPCS
IXS
Spin-Echo
INS
NFS
Other techniques
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Typical example samples with some degree of (quasi) static disorder (gels glasses etc)
To properly calculate g(2) the ensemble averaging ltgtφ must be performed explicitly
Pusey amp van Megen Physica A (1989)
I(t) I(t)
tt
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Solution move the beam around and average g(2) over many exposures
or (much better)
Use a 2D detector and catch many pixels with same |Q|
Medipix-2
Ttp
Ttp
tpp
tItI
tItIQg
leleminuslele
+=
τφτφ
φτ
τ)()(
)()()(
0
)2(
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-equilibrium ie not all ∆t are equivalentg(2) depends on both ∆t and the absolute time (age)
Solution calculate the correlation function without time averaging (two-time correlation function)
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG =
t1
t2
G(t1t2) age
τ
age ta=(t2+t1)2
lag time τ=t2-t1
Only with a 2D detector several Qs at same age can be recorded
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
2d detectors for multi-speckle XPCS
Advantages More efficient data-taking (Mpixels) with 2d detectorBeam induced sample damage minimized
Drawbacks 2d detectors are slow noisy have limited dynamic rangeare not always photon counting have limited resolutionradiation hardness can be a problem
Princeton Instruments Dalsa MedipixMaxipix
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Limitations
sn ratio (more flux)
Speed and area of 2D detectors
Beam damage (minimize exposure)
Medipix-II detector 1kHz ff
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Depletion gel
Inter-particle interaction potential
V(r)Complex phase diagramrgR (01) φ (20) cp (43mgcm3)
2rg
2R
r
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The gel phase is transienthellip
W C K Poon et al Faraday Discuss 112 143 (1999)
delayedsedimentationageltt (latency time)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What is the shape of a speckle
200 400 600 800 1000 1200
200
400
600
800
1000
1200
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
vertical direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
horizontal direction
0 002 0040
002
004
006
008
01
Q
20 40 60
10
20
30
40
50
60
70
20 40 60
10
20
30
40
50
60
70
minus5 0 5
x 10minus4
0
001
002
003
004
005
∆Q
diagonal direction
0 002 0040
002
004
006
008
01
Q
3
1
2
1 2 3
elongatedspeckles
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Pink beam ID10C (∆λλ=15)E=8keV 20microm beamsizeIcoherent gt 1x1011 phs
XPCS What is the information we gain
real space
non-coherent coherent
reciprocal space
ltI(Q)gt I(Qt) I(Qt)
DynamicsKinetics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What do we measure
2)2(
)(
)()()(
Q
QQQ
I
tItIg
ττ
+=
22~)2( |)(|1))((Re1)( tfSg QQQ +=+= ωτ
Temporal intensityauto-correlation function
int int minussdotpropV V
mnmn tibbtf )])()0([exp()()(|)(| rrQQQQ
220 )()]([
1~)(QQ
QSΓ+minus ωω
ω
)(cos)2exp(1)( 02)2( τωττ Γminus+=Qg
)2exp(1)()2( ττ Γminus+=Qg
Dynamic structure factor(Lorentzian shape)
(propagating waves)
(Brownian motion overdamped waves)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Scattering Vector Q [Aring-1]
Length Scale [Aring]Fr
eque
ncy
[Hz]
Energy [eV]
Raman
Brillouin
XPCSPCS
IXS
Spin-Echo
INS
NFS
Other techniques
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Typical example samples with some degree of (quasi) static disorder (gels glasses etc)
To properly calculate g(2) the ensemble averaging ltgtφ must be performed explicitly
Pusey amp van Megen Physica A (1989)
I(t) I(t)
tt
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Solution move the beam around and average g(2) over many exposures
or (much better)
Use a 2D detector and catch many pixels with same |Q|
Medipix-2
Ttp
Ttp
tpp
tItI
tItIQg
leleminuslele
+=
τφτφ
φτ
τ)()(
)()()(
0
)2(
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-equilibrium ie not all ∆t are equivalentg(2) depends on both ∆t and the absolute time (age)
Solution calculate the correlation function without time averaging (two-time correlation function)
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG =
t1
t2
G(t1t2) age
τ
age ta=(t2+t1)2
lag time τ=t2-t1
Only with a 2D detector several Qs at same age can be recorded
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
2d detectors for multi-speckle XPCS
Advantages More efficient data-taking (Mpixels) with 2d detectorBeam induced sample damage minimized
Drawbacks 2d detectors are slow noisy have limited dynamic rangeare not always photon counting have limited resolutionradiation hardness can be a problem
Princeton Instruments Dalsa MedipixMaxipix
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Limitations
sn ratio (more flux)
Speed and area of 2D detectors
Beam damage (minimize exposure)
Medipix-II detector 1kHz ff
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Depletion gel
Inter-particle interaction potential
V(r)Complex phase diagramrgR (01) φ (20) cp (43mgcm3)
2rg
2R
r
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The gel phase is transienthellip
W C K Poon et al Faraday Discuss 112 143 (1999)
delayedsedimentationageltt (latency time)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS What is the information we gain
real space
non-coherent coherent
reciprocal space
ltI(Q)gt I(Qt) I(Qt)
DynamicsKinetics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What do we measure
2)2(
)(
)()()(
Q
QQQ
I
tItIg
ττ
+=
22~)2( |)(|1))((Re1)( tfSg QQQ +=+= ωτ
Temporal intensityauto-correlation function
int int minussdotpropV V
mnmn tibbtf )])()0([exp()()(|)(| rrQQQQ
220 )()]([
1~)(QQ
QSΓ+minus ωω
ω
)(cos)2exp(1)( 02)2( τωττ Γminus+=Qg
)2exp(1)()2( ττ Γminus+=Qg
Dynamic structure factor(Lorentzian shape)
(propagating waves)
(Brownian motion overdamped waves)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Scattering Vector Q [Aring-1]
Length Scale [Aring]Fr
eque
ncy
[Hz]
Energy [eV]
Raman
Brillouin
XPCSPCS
IXS
Spin-Echo
INS
NFS
Other techniques
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Typical example samples with some degree of (quasi) static disorder (gels glasses etc)
To properly calculate g(2) the ensemble averaging ltgtφ must be performed explicitly
Pusey amp van Megen Physica A (1989)
I(t) I(t)
tt
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Solution move the beam around and average g(2) over many exposures
or (much better)
Use a 2D detector and catch many pixels with same |Q|
Medipix-2
Ttp
Ttp
tpp
tItI
tItIQg
leleminuslele
+=
τφτφ
φτ
τ)()(
)()()(
0
)2(
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-equilibrium ie not all ∆t are equivalentg(2) depends on both ∆t and the absolute time (age)
Solution calculate the correlation function without time averaging (two-time correlation function)
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG =
t1
t2
G(t1t2) age
τ
age ta=(t2+t1)2
lag time τ=t2-t1
Only with a 2D detector several Qs at same age can be recorded
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
2d detectors for multi-speckle XPCS
Advantages More efficient data-taking (Mpixels) with 2d detectorBeam induced sample damage minimized
Drawbacks 2d detectors are slow noisy have limited dynamic rangeare not always photon counting have limited resolutionradiation hardness can be a problem
Princeton Instruments Dalsa MedipixMaxipix
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Limitations
sn ratio (more flux)
Speed and area of 2D detectors
Beam damage (minimize exposure)
Medipix-II detector 1kHz ff
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Depletion gel
Inter-particle interaction potential
V(r)Complex phase diagramrgR (01) φ (20) cp (43mgcm3)
2rg
2R
r
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The gel phase is transienthellip
W C K Poon et al Faraday Discuss 112 143 (1999)
delayedsedimentationageltt (latency time)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
What do we measure
2)2(
)(
)()()(
Q
QQQ
I
tItIg
ττ
+=
22~)2( |)(|1))((Re1)( tfSg QQQ +=+= ωτ
Temporal intensityauto-correlation function
int int minussdotpropV V
mnmn tibbtf )])()0([exp()()(|)(| rrQQQQ
220 )()]([
1~)(QQ
QSΓ+minus ωω
ω
)(cos)2exp(1)( 02)2( τωττ Γminus+=Qg
)2exp(1)()2( ττ Γminus+=Qg
Dynamic structure factor(Lorentzian shape)
(propagating waves)
(Brownian motion overdamped waves)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Scattering Vector Q [Aring-1]
Length Scale [Aring]Fr
eque
ncy
[Hz]
Energy [eV]
Raman
Brillouin
XPCSPCS
IXS
Spin-Echo
INS
NFS
Other techniques
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Typical example samples with some degree of (quasi) static disorder (gels glasses etc)
To properly calculate g(2) the ensemble averaging ltgtφ must be performed explicitly
Pusey amp van Megen Physica A (1989)
I(t) I(t)
tt
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Solution move the beam around and average g(2) over many exposures
or (much better)
Use a 2D detector and catch many pixels with same |Q|
Medipix-2
Ttp
Ttp
tpp
tItI
tItIQg
leleminuslele
+=
τφτφ
φτ
τ)()(
)()()(
0
)2(
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-equilibrium ie not all ∆t are equivalentg(2) depends on both ∆t and the absolute time (age)
Solution calculate the correlation function without time averaging (two-time correlation function)
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG =
t1
t2
G(t1t2) age
τ
age ta=(t2+t1)2
lag time τ=t2-t1
Only with a 2D detector several Qs at same age can be recorded
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
2d detectors for multi-speckle XPCS
Advantages More efficient data-taking (Mpixels) with 2d detectorBeam induced sample damage minimized
Drawbacks 2d detectors are slow noisy have limited dynamic rangeare not always photon counting have limited resolutionradiation hardness can be a problem
Princeton Instruments Dalsa MedipixMaxipix
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Limitations
sn ratio (more flux)
Speed and area of 2D detectors
Beam damage (minimize exposure)
Medipix-II detector 1kHz ff
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Depletion gel
Inter-particle interaction potential
V(r)Complex phase diagramrgR (01) φ (20) cp (43mgcm3)
2rg
2R
r
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The gel phase is transienthellip
W C K Poon et al Faraday Discuss 112 143 (1999)
delayedsedimentationageltt (latency time)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Scattering Vector Q [Aring-1]
Length Scale [Aring]Fr
eque
ncy
[Hz]
Energy [eV]
Raman
Brillouin
XPCSPCS
IXS
Spin-Echo
INS
NFS
Other techniques
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Typical example samples with some degree of (quasi) static disorder (gels glasses etc)
To properly calculate g(2) the ensemble averaging ltgtφ must be performed explicitly
Pusey amp van Megen Physica A (1989)
I(t) I(t)
tt
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Solution move the beam around and average g(2) over many exposures
or (much better)
Use a 2D detector and catch many pixels with same |Q|
Medipix-2
Ttp
Ttp
tpp
tItI
tItIQg
leleminuslele
+=
τφτφ
φτ
τ)()(
)()()(
0
)2(
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-equilibrium ie not all ∆t are equivalentg(2) depends on both ∆t and the absolute time (age)
Solution calculate the correlation function without time averaging (two-time correlation function)
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG =
t1
t2
G(t1t2) age
τ
age ta=(t2+t1)2
lag time τ=t2-t1
Only with a 2D detector several Qs at same age can be recorded
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
2d detectors for multi-speckle XPCS
Advantages More efficient data-taking (Mpixels) with 2d detectorBeam induced sample damage minimized
Drawbacks 2d detectors are slow noisy have limited dynamic rangeare not always photon counting have limited resolutionradiation hardness can be a problem
Princeton Instruments Dalsa MedipixMaxipix
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Limitations
sn ratio (more flux)
Speed and area of 2D detectors
Beam damage (minimize exposure)
Medipix-II detector 1kHz ff
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Depletion gel
Inter-particle interaction potential
V(r)Complex phase diagramrgR (01) φ (20) cp (43mgcm3)
2rg
2R
r
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The gel phase is transienthellip
W C K Poon et al Faraday Discuss 112 143 (1999)
delayedsedimentationageltt (latency time)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Typical example samples with some degree of (quasi) static disorder (gels glasses etc)
To properly calculate g(2) the ensemble averaging ltgtφ must be performed explicitly
Pusey amp van Megen Physica A (1989)
I(t) I(t)
tt
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Solution move the beam around and average g(2) over many exposures
or (much better)
Use a 2D detector and catch many pixels with same |Q|
Medipix-2
Ttp
Ttp
tpp
tItI
tItIQg
leleminuslele
+=
τφτφ
φτ
τ)()(
)()()(
0
)2(
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-equilibrium ie not all ∆t are equivalentg(2) depends on both ∆t and the absolute time (age)
Solution calculate the correlation function without time averaging (two-time correlation function)
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG =
t1
t2
G(t1t2) age
τ
age ta=(t2+t1)2
lag time τ=t2-t1
Only with a 2D detector several Qs at same age can be recorded
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
2d detectors for multi-speckle XPCS
Advantages More efficient data-taking (Mpixels) with 2d detectorBeam induced sample damage minimized
Drawbacks 2d detectors are slow noisy have limited dynamic rangeare not always photon counting have limited resolutionradiation hardness can be a problem
Princeton Instruments Dalsa MedipixMaxipix
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Limitations
sn ratio (more flux)
Speed and area of 2D detectors
Beam damage (minimize exposure)
Medipix-II detector 1kHz ff
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Depletion gel
Inter-particle interaction potential
V(r)Complex phase diagramrgR (01) φ (20) cp (43mgcm3)
2rg
2R
r
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The gel phase is transienthellip
W C K Poon et al Faraday Discuss 112 143 (1999)
delayedsedimentationageltt (latency time)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-ergodicity ie ltgtt ne ltgtφ (time average different from ensemble average)
systems that do not sample the entire allowed set of spatial configurations
Solution move the beam around and average g(2) over many exposures
or (much better)
Use a 2D detector and catch many pixels with same |Q|
Medipix-2
Ttp
Ttp
tpp
tItI
tItIQg
leleminuslele
+=
τφτφ
φτ
τ)()(
)()()(
0
)2(
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-equilibrium ie not all ∆t are equivalentg(2) depends on both ∆t and the absolute time (age)
Solution calculate the correlation function without time averaging (two-time correlation function)
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG =
t1
t2
G(t1t2) age
τ
age ta=(t2+t1)2
lag time τ=t2-t1
Only with a 2D detector several Qs at same age can be recorded
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
2d detectors for multi-speckle XPCS
Advantages More efficient data-taking (Mpixels) with 2d detectorBeam induced sample damage minimized
Drawbacks 2d detectors are slow noisy have limited dynamic rangeare not always photon counting have limited resolutionradiation hardness can be a problem
Princeton Instruments Dalsa MedipixMaxipix
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Limitations
sn ratio (more flux)
Speed and area of 2D detectors
Beam damage (minimize exposure)
Medipix-II detector 1kHz ff
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Depletion gel
Inter-particle interaction potential
V(r)Complex phase diagramrgR (01) φ (20) cp (43mgcm3)
2rg
2R
r
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The gel phase is transienthellip
W C K Poon et al Faraday Discuss 112 143 (1999)
delayedsedimentationageltt (latency time)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Challenges
Non-equilibrium ie not all ∆t are equivalentg(2) depends on both ∆t and the absolute time (age)
Solution calculate the correlation function without time averaging (two-time correlation function)
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG =
t1
t2
G(t1t2) age
τ
age ta=(t2+t1)2
lag time τ=t2-t1
Only with a 2D detector several Qs at same age can be recorded
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
2d detectors for multi-speckle XPCS
Advantages More efficient data-taking (Mpixels) with 2d detectorBeam induced sample damage minimized
Drawbacks 2d detectors are slow noisy have limited dynamic rangeare not always photon counting have limited resolutionradiation hardness can be a problem
Princeton Instruments Dalsa MedipixMaxipix
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Limitations
sn ratio (more flux)
Speed and area of 2D detectors
Beam damage (minimize exposure)
Medipix-II detector 1kHz ff
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Depletion gel
Inter-particle interaction potential
V(r)Complex phase diagramrgR (01) φ (20) cp (43mgcm3)
2rg
2R
r
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The gel phase is transienthellip
W C K Poon et al Faraday Discuss 112 143 (1999)
delayedsedimentationageltt (latency time)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
2d detectors for multi-speckle XPCS
Advantages More efficient data-taking (Mpixels) with 2d detectorBeam induced sample damage minimized
Drawbacks 2d detectors are slow noisy have limited dynamic rangeare not always photon counting have limited resolutionradiation hardness can be a problem
Princeton Instruments Dalsa MedipixMaxipix
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Limitations
sn ratio (more flux)
Speed and area of 2D detectors
Beam damage (minimize exposure)
Medipix-II detector 1kHz ff
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Depletion gel
Inter-particle interaction potential
V(r)Complex phase diagramrgR (01) φ (20) cp (43mgcm3)
2rg
2R
r
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The gel phase is transienthellip
W C K Poon et al Faraday Discuss 112 143 (1999)
delayedsedimentationageltt (latency time)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Limitations
sn ratio (more flux)
Speed and area of 2D detectors
Beam damage (minimize exposure)
Medipix-II detector 1kHz ff
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Depletion gel
Inter-particle interaction potential
V(r)Complex phase diagramrgR (01) φ (20) cp (43mgcm3)
2rg
2R
r
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The gel phase is transienthellip
W C K Poon et al Faraday Discuss 112 143 (1999)
delayedsedimentationageltt (latency time)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Depletion gel
Inter-particle interaction potential
V(r)Complex phase diagramrgR (01) φ (20) cp (43mgcm3)
2rg
2R
r
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The gel phase is transienthellip
W C K Poon et al Faraday Discuss 112 143 (1999)
delayedsedimentationageltt (latency time)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The gel phase is transienthellip
W C K Poon et al Faraday Discuss 112 143 (1999)
delayedsedimentationageltt (latency time)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
hellipbut the structure is constant in the accessible Q-range
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Two time analysis
φφ
φ)()(
)()()(
21
2121 tQItQI
tQItQIttQG = Kohlrausch-Williams-Watts (KWW)
g(2)(τ) =1+exp(-2(Γτ)γ)
γlt1
γ~1
γgt1
age
τ=t1-t2
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
A jamming transition
(a) young gel(b) fully aged gel
relaxation timevs age
stretching exponentvs age
Phys Rev E 76 010401(R) (2007)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Jamming
Characteristic features
crowded media intermittent dynamics lt∆x2gt prop t2 (or Q prop Γ) cooperative behavior aging
Micro-collapses
Kohlrausch-Williams-Watts (KWW) g(2)(t) =1+exp(-2(Γt)γ) with γ~15
Bouchaud amp Pitard EPJE 6 231 (2001)
V Trappe et al Nature 411 772 (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The Glass Transition(solid like behavior of liquids)
Characterized by several distinct relaxations in the super-cooled state TltTm
Non-Arrhenius behavior of the viscosity
Form glasses below Tg depending on the cooling rate
From Pablo G Debenedetti and Frank H Stillinger Nature (2001)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Recent results on surface dynamics
Capillary waves on highly viscous liquids Γ = 2γη Q
What happens as ηinfin ie at the transition from asupercooled liquid to a glass
What happens with the shear response as the liquid solidifies
RHEOLOGY (modulus=stressstrain)
Liquid deforms cont under stressSolid equilibrium deformation under stress
Liquid stress relaxes under a constant strainSolid constant stress level under constant strain
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Grazing incidence XPCS
beamstop
αi lt αc
deflectingmirror
sample(liquid or solid)
monochromaticsynchrotron beam
Specklepattern
slits
2D detector
pinholeaperture
coherent intensity Ic prop Bλ2
Intensity fluctuations of the speckle pattern reflect the dynamics of the sample
2)(
)()0()(
tqI
tqIqItqg =XPCS Intensity auto-correlation function
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
All functions are simple exponential decays
Characteristic times change 5 ordersof magnitude from 280 to 214K (Tg ~ 205K)
)2exp(1)()2( ττ Γminus+=Qg
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on supercooled poly-propylene glycol (PPG)
qηγ2
=Γ
Newtonian liquid
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Kelvin-Voigt model(viscoelastic solid)
Maxwell-Debye model(viscoelastic liquid)
Over-damped capillary wavesNewtonian liquid
η
EE η
ηγ2q
=Γ
)()( ωωηω iG = EiG += 0)( ωηω
ωτηωηi+
=1
)( 0
ωηωη
iE
+= 0)(
002 ηηγ Eq
+=Γ
0
)0(ηE
=Γ
This is what we observeeven far above Tg
1
0 21
2
minus
+=Γ
Gqq γ
ηγ
0)0( =Γ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Models for viscoelasticity
Liquids near the glass transition Viscosity and elasticity become important viscoelasticity
Combined Maxwell-Debye and Kelvin-Voigt model
EiG += )()( ωωηω
ωτηωηi+
=1
)( 0Maxwell Debye E Elasticity fromVoigt-Kelvin
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Γ Eη for q 0
Γ G(infin)η for q infin
q0=2Eγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
CW dynamics on poly-propylene glycol (PPG)
10
0
0)(2)(1
2)( minus
infin
++
+=Γ
Gqqqq γ
ηγ
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Viscoelasticity of poly-propylene glycol (PPG)
XPCS of CWs is a genuine way to measurelow frequency response of glass forming liquids
Low frequency elasticity of (supercooled) liquids is an extremely controversialtopic in the literature and non-invasive experimental methods are missing
η~exp(A(T-T0)) VTF law
E~exp(A(T-T0)) VTF law
Chushkin Caronna and Madsen EPL (in press)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Next topic Nano-particles in supercooled liquids
Or should I stop here
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dynamics of nano-particles in a glass forming solvent
Samples Nano-spheres in a glass forming solvent
Silica R=250nm
Solvent Propanediol (TG ~ 170K)Liquid at elevated temp (η ~ 01 Pas 290K)
httpwwwesrffrnewsspotlightspotlight39spotlight39xpcsAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Simple Diffusion and Hyper-Diffusion
Ballistic motion x = vt lt[x(t)-x(0)]2gt = ∆x2 = v2t2 Q prop 1x prop 1t = Γ
Brownian (random walker) motion lt[r(t)-r(0)]2gt = 6D0t Q2 prop Γ
D0 self-diffusion constant
D0=kBT6πηR (Einstein 1905)
g(2) ~ 1 + exp( -2Γt ) Γ = D0Q2 for Brownian motion
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Dispersion relations
Γ[s-1
]
QR
Silica nano-particles (1 vol) in supercooled propanediol
n=2(Brownian motion)
n1(Hyper diffusion)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Correlation functions
Results QR=675
240K simple exponential decay
205K decay faster than exponential
γ=1
γasymp17
Fits with empirical model (KWW) g(2) =1+exp(-2(Γt)γ)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
XPCS results n = 2
n = 1
Lines are guides to the eyeAnders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Model for particle dynamics
2)2( |)(|1)( tQftQg +=Continuous time random walk model
suminfin
==
0)()()(
Nt NQhNPtQf
)exp()( RQ sdotminus= αiNNQh
R
α=1 h(N+1)=h(N)h(1) (ballistic motion)
α=12 Brownian motion (simple diffusion)
αlt12 sub-diffusionαgt12 hyper diffusion
P(R) distribution of R (Gaussian)lt|R|gt=δ
))(exp()( 00
NttNP
Nt
ΓΓminus=
MSD ~ t2α
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
The fits with the KWW form f(t) =exp(-(Γt)γ) are perfect The model explains the KWW exponent γ
α=1(ballistic)
Model for particle dynamics
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Caronna Chushkin Madsen and Cupane PRL 100 055702 (2008)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Results
Tg=170K (by DSC)
The transition from simple diffusion into event dominated hyper-diffusion is caused by elastic relaxations of the solvent kicking the particles
Those excitations become more important than diffusion at around 12Tg
This again shows that the solid like response of supercooled liquids dominates as the viscosity gets high close to Tg (to be continued)
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
Further reading
Magnetic colloids J Appl Cryst 40 250 (2007) J Phys Cond Mat 18 S2697 (2006)
X-ray fluorescence correlation spectroscopy J Appl Cryst 40 283 (2007)
Charge stab colloids J Chem Phys 126 104905 (2007) Phys Rev Lett 96 138303 (2006)
SrTiO3 displacive transition Phys Rev Lett 98 105501 (2007)
Nano-particles on the surface of thin polymer films Phys Rev Lett 98 047801 (2007)
Dynamical heterogeneity in colloidal gels Phys Rev E 76 051404 (2007)
Transient gelation Phys Rev E 76 010401(R) (2007)
Partially wetting thin liquid films Phys Rev Lett 99 096104 (2007)
Bulk fluctuations in a lamellar phase Phys Rev E 74 031706 (2006)
Coarsening dyn in alloys Phys Rev B 73 180101 (2006) Phys Rev E 74 041107 (2006)
Glass-glass transition in attractive colloids Phys Rev E 74 051504 (2006)
Ordering Kinetics in a Cu-Pd Alloy Phys Rev B 72 144201 (2005)
Grazing incidence XPCS J Sync Rad 12 786 (2005)
XPCS Microrheology J Phys Condens Matter 17 L279 (2005)
Dynamics of fluctuations in smectic membranes Phys Rev E 72 011704 (2005)
XPCS Journal of Alloys and Compounds 362 3 (2004)
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008
General conclusion on XPCS
XPCS is the X-ray counterpart to PCS (DLS) with the obvious advantages of X-rays over visible light
Larger Q-range Surface sensitivity Penetration power
No (or only little) multiple scattering
Difficulties arise from the fact that the synchrotron is a chaotic source and intensity is lost when a coherent beam is defined
Brilliance hungry technique but samples are often radiation sensitive
Interesting perspectives with X-ray lasers coming on-line butXPCS at storage ring based sources (quasi DC) will remain competitive
Anders Madsen X-ray coherence lecture series ESRF June 30 2008