DONOSTIA INTERNATIONALPHYSICS CENTER

UPV/EHU

CENTRO DE FISICA DE

MATERIALES

DONOSTIA INTERNATIONAL

PHYSICS CENTER

Juan Colmenero

Chain dynamics in dynamically asymmetric

polymer blends

SYSTEMS WITH DYNAMICAL ASYMMETRY Micellar solutions

Water/Proteins

Polymer blends

Colloidal Suspensions

Ions in Molten & Glassy Silicates

Diblock copolymers

Nano-composites Membranes

Polymer solutions

Polymer/plasticizer

SYSTEMS WITH DYNAMICAL ASYMMETRY

POLYMER BLENDS: REPRESENTATIVE EXAMPLE

A. Alegría (Dielectric Spectroscopy) F. Alvarez (Atomistic MD-simulations) A. Arbe (Neutron scattering) M. Brodeck (Atomistic MD-simulations) A. C. Genix (NS and Atomistic MD-simulations) R. Lund (Neutron scattering) A.  J. Moreno (coarse-grained MD-simulations) S. Plaza (Dielectric Spectroscopy)

Chain dynamics in dynamically asymmetric

polymer blends

OUTLINE

1.  Defining ‘asymmetric polymer blends’ 2.  α-relaxation and concentration fluctuations in

asymmetric polymer blends 3.  Chain-dynamics in asymmetric polymer blends 4.  Results from MD-simulations 5.  A Generalized Langevin Equation scenario 6.  Experimental results in this framework 7.  Conclusions

DEFINITION: “Thermodynamically Miscible Polymer Blend”:

two component system molecularly mixed

MISCIBLE POLYMER BLENDS

weak interactions between the two components

Experimental fact: Different segmental dynamics (α-relaxation) for each component in the blend

“DYNAMIC HETEROGENEITY”

100% 100%

50/50

1/T

log τ

τA τB

τA/AB τB/AB

“DYNAMIC HETEROGENEITY”: Consequences

Consequences:

100% 100%

50/50

1/T

log τ

τA τB

τA/AB τB/AB

Tg A Tg B

•  Two dynamic Tg’s in the blend

•  Two equilibrium -non equilibrium (DSC) Tg’s? (“effective Tg’s”)

•  Loss of ‘equilibrium’ behavior for the fast component B

τ ≈ 100s

Tg A/AB Tg B/AB

ASYMMETRIC POLYMER BLENDS AB

1/T

log τ τA τB τA/AB τB/AB

•  Dynamic asymmetry: Very different mobilities of component A (slow) and B (fast) in the blend

Δ(T) = τ (T) A/AB

τ (T) B/AB

Dynamic asymmetry increases as T decreases

•  Composition asymmetry: Concentration rich in the ‘slow’ component <φA> > 0.7 ~

?

Δ 1≈ Δ

α-Relaxation and Concentration Fluctuations in Asymmetric Polymer

Blends: PVME/PSCN

SEGMENTAL DYNAMICS IN ASYMMETRIC POLYMER BLENDS

BLENDS OF FUNCTIONALIZED PS AND PVME

PS Si X

PS

CH3 X≡H (PSHPS) PS(1K) Tg≈ 328K X≡CN (PSCNPS) PS(1K) Tg≈ 328K

1 0 - 2 1 0 - 1 1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 5 1 0 6 1 0 7 0 , 0 0 0 , 0 2 0 , 0 4 0 , 0 6 0 , 0 8 0 , 1 0 0 , 1 2

ε"

f r e q u e n c y [ H z ]

CN: strong dipole moment Asymmetric blends with PVME ΔTg≈78K PS-dynamics dielectrically visible

PVME PSCNPS

PSHPS

0.2

0.3

0.4

2.5 3 3.5 4 4.5

S(Q

=0)-1

(cm

)

1000 / T(K)

-7

-6

-5

-4

-3

-2

-1

0

log[τ max(s)]

0.2

0.4

0.6

0.8

1dCpdT

Thermal concentration fluctuations by SANS variance of the concentration fluctuations

<(δφ)2>~(S(0),V)

α-RELAXATION AND CONCENTRATION FLUCTUATIONS

€

S(q) =1

ro +43Δ2G1 + Cq2PSCN

PVME

PSCN/PVME

1

2

3

4

5

67

0.01 0.1

S(Q

) (cm

-1)

Q(Å-1)

Chain-Dynamics in the non-Entanglement Regime (Rouse) of Asymmetric Polymer

Blends

Short break: ROUSE model

THE ROUSE MODEL

€

r j

€

r j+1€

j

€

j +1

Rouse chain: target chain in a melt of similar chains

•  Constant friction coefficient ξ •  Stochastic random forces

•  Entropic forces (‘springs’)

€

f i (t)

€

ξd r jdt

=3kBT2

r j +1 + r j−1 − 2

r j( ) + f j

€

X p(t) =

1N r j (t)cos pπ

Nj − 12

⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎡

⎣ ⎢ ⎤

⎦ ⎥ j=1

N∑

‘wavelength’ of mode p

€

: Np

Normal coordinates

THE ROUSE MODEL

Correlation function of Rouse modes:

No spatial correlation

No time correlation

Relaxation times Rouse rate

Large wavelength (N/p): slower relaxation times

(the slowest time) = ‘the Rouse time’

€

X p(t)

X p(0) ∝ exp −

tτ p

⎡

⎣ ⎢

⎤

⎦ ⎥

€

X p(t)

X q(0) = 0 p ≠ q

€

X p(t)

X q(0)

€

τ p−1 = 4W sin2 pπ

2N⎡ ⎣ ⎢

⎤ ⎦ ⎥

€

W =3kBTξ2

for

€

p << N

€

τ p ≈1

π 2WNp

⎛

⎝ ⎜

⎞

⎠ ⎟

2

€

N ∝Mw ⇒τ p ∝Mw2

€

τ1

€

τ R

Chain-Dynamics in the non-Entanglement Regime (Rouse) of Asymmetric Polymer

Blends: PMMA/PEO MD-simulations

“Bead-Spring” A/B model MD-simulations

POLY(ETHYLENE OXIDE) / POLY(METHYL METACRYLATE)

PEO PMMA Tg ≈ 210K Tg ≈ 400K ΔTg ≈ 190K

-[CH2-CH2-O-]n -[C-CH2 -] n

C-O-C H 3 O

- =

CH 3 -

20PEO/ 80PMMA

Dynamic asymmetry

Composition asymmetry

PEO and PMMA, narrow molecular weight distributions

PEO and PMMA, deuterated and protonated

PEO - MD-SIMULATIONS: Rouse Mode correlators

Coarse-graining: 1 monomer = 1 blob

Rouse-Mode correlators

€

φpp (t) = Aexp − tτ p

⎛

⎝ ⎜

⎞

⎠ ⎟

β⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

€

β( p)

€

< τ >=τ pβΓ1β

⎛

⎝ ⎜ ⎞

⎠ ⎟

)0()0(/)0()()( pppppp XXXtXt

=φ

Fitting

= -[CH2-CH2-O-]

0

0.2

0.4

0.6

0.8

1

100 101 102 103 104

Φpp

(t)

t (ps)

∑=

⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ −=N

jjp j

Nptr

NtX

1 21cos)(1)( π

Normal coordinates

10-2

100

102

104

106

<τp>

(ps)

300K

400K350K

p-3.75

p-2.1

500K

p-2.45

p-2.7

(x10)

(x0.1)

(x0.01)

0.4

0.6

0.8

1

β

10-2

10-1

100

101

102

1 10 100

<Xp(0

)2 >

N/p

p-2

Number-mode p-dependence

⎟⎟⎠

⎞⎜⎜⎝

⎛−∝

ppp

tXtXτ

exp)0()(

€

β =1€

τ p =1

Wπ 2Np

⎛

⎝ ⎜

⎞

⎠ ⎟

2

€

N→∞

p << N

Rouse prediction

PEO IN PEO/PMMA - MD-SIMS: Rouse Mode correlators

PEO in PEO/PMMA pure PEO

€

Xp2 (0) =

b2

2π 2NNp

⎛

⎝ ⎜

⎞

⎠ ⎟

2

BEAD-SPRING MODEL OF ASYMMETRIC POLYMER BLENDS

Simulations of coarse-grained bead-spring models

BEAD-SPRING POLYMER MODEL 70A/30B (composition asymmetry)

A-chains

“SLOW”

B-chains

“FAST”

(dynamic asymmetry) increases as T decreases

N < Nc ~ 40 Expected Rouse Dynamics

BEAD-SPRING MODEL OF ASYMMETRIC POLYMER BLENDS

Simulations of coarse-grained bead-spring models

BEAD-SPRING POLYMER MODEL 70A/30B (composition asymmetry)

A-chains

“SLOW”

B-chains

“FAST”

(dynamic asymmetry) increases as T decreases

N < Nc ~ 40 Expected Rouse Dynamics

BEAD-SPRING MODEL OF ASYMMETRIC POLYMER BLENDS

Simulations of coarse-grained bead-spring models

BEAD-SPRING POLYMER MODEL 70A/30B (composition asymmetry)

A-chains

“SLOW”

B-chains

“FAST”

(dynamic asymmetry) increases as T decreases

N < Nc ~ 40 Expected Rouse Dynamics

101

102

0.5 1 1.5 2 2.5

Asy

mm

etry

Δ

1/T

10-2

100

102

104

106

< τp>

(ps)

300K

400K350K

p-3.75

p-2.1

500K

p-2.45

p-2.7

(x10)

(x0.1)

(x0.01)

0.4

0.6

0.8

1

β

10-2

10-1

100

101

102

1 10 100

<Xp(0

)2 >

N/p

p-2

100

101

102

103

104

105

106T=0.33T=0.4T=0.5T=0.6T=0.75T=1.0T=1.5

< τp>

(ps)

p-3.5

p-2.2

0.4

0.6

0.8

1

β

0.01

0.1

1

1 10

<Xp(0

)2 >

N/p

p-2.2

PEO in PEO/PMMA pure PEO

Bead-spring N=21 PEO in PEO/PMMA

SUMMARY OF THE RESULTS FROM SIMULATIONS

Strong deviations from simple Rouse behavior for the fast component of the blend, in particular at low T.

Deviations driven by the frozenning of the slow component, i.e., by the increase of the dynamic asymmetry in the system.

0.5

0.6

0.7

0.8

0.9

1

2

2.5

3

3.5

0.3

0.4

0.5

0.6

10 100

€

Δ =τ A

τ B DB

DA

⎛

⎝ ⎜

⎞

⎠ ⎟

Non-exponentiality of Rouse correlators

<β> N=10, low p

Mode-wavelength dependence of

relaxation times (x)

τ ~(N/p)x

time-dependence of mean-squared displacement

(y) <r2(t)>~ty

Asymmetry

0.5

0.6

0.7

0.8

0.9

1

2

2.5

3

3.5

4

0.3

0.4

0.5

101 102 103 104 105 106 107

Asymmetry

€

Δ =τ A

τ B DB

DA

⎛

⎝ ⎜

⎞

⎠ ⎟

PEO in PEO/PMMA

bead-spring

A Generalized Langevin Equation Scenario

GENERALIZED LANGEVIN EQUATION

Values of the parameters and scaling features

Generalized Langevin Equations (GLE)

Memory function due to the slow component

Rouse: Reptation: Prediction (GLE):

€

τ p ∝Np

⎛

⎝ ⎜

⎞

⎠ ⎟

3.5

€

τ p ∝Np

⎛

⎝ ⎜

⎞

⎠ ⎟

2

€

τ p ∝N 3

p2

{ }( ) )()();(0

2

2

tFetvtdrWrdt

rdm jnjn

n

t

ij

j

+−−Γ−−= ∑∫ α

βαβ ττττ∂∂

0.5

0.6

0.7

0.8

0.9

1

2

2.5

3

3.5

0.3

0.4

0.5

0.6

10 100

€

Δ =τ A

τ B DB

DA

⎛

⎝ ⎜

⎞

⎠ ⎟

Non-exponentiality of Rouse correlators

<β> N=10, low p

Mode-wavelength dependence of

relaxation times (x)

τ ~(N/p)x

time-dependence of mean-squared displacement

(y) <r2(t)>~ty

Asymmetry Rouse limit

GLE limit

Rouse limit

GLE limit

0.5

0.6

0.7

0.8

0.9

1

2

2.5

3

3.5

4

0.3

0.4

0.5

101 102 103 104 105 106 107

Asymmetry

€

Δ =τ A

τ B DB

DA

⎛

⎝ ⎜

⎞

⎠ ⎟

PEO in PEO/PMMA

bead-spring

FROM ROUSE TO “CONFINED CHAINS”

Homopolymers

Rouse Entangled chains Crossover

Increasing Mw

Asymmetric blends

Rouse “Confined chains” Crossover

Increasing asymmetry

Slowing down density fluctuations

Frozenning of the slow component

RRM based on GLE

Can the experimental data be described in this theoretical framework?

RENORMALIZED ROUSE MODELS & NEUTRON SCATTERING

Mean-squared segment correlation function

Valid for non-exponential Cp(t) if the modes are orthogonal

NEUTRON SCATTERING

Rouse mode correlators

€

Cp (t) =

X p (0)

X p (t)

X p (0)

X p (0)

GAUSSIAN APPROXIMATION

€

= R m (t) −

R n (0)[ ]2

€

= 6DRt + n −mb2 +4Nb2

π1p2cos pπm

N⎛

⎝ ⎜

⎞

⎠ ⎟ cos

pπnN

⎛

⎝ ⎜

⎞

⎠ ⎟ 1−Cp (t)[ ]

p=1

N

∑

€

Schain (Q,t) =1N

exp i Q R n (t) −

R m (0)[ ]{ }

n,m∑

€

Schain (Q,t) =1N

expn,m∑ −

16Q2

⎡

⎣ ⎢ ⎤

⎦ ⎥

φnm(t)

φnm(t)

φnm(t)

€

Schain (Q,t) =1N

expn,m∑ −

16Q2

⎡

⎣ ⎢ ⎤

⎦ ⎥ φnm(t)

CHECKING THE GAUSSIAN APPROXIMATION: PEO MD-SIMULATIONS PEO: Neutrons and Simulation results Rouse behavior

φnm(t) calculated from the non-exponential Cp(t) €

Schain (Q,t) directly calculated from the MD-simulations

Good agreement!!

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40

S chai

n(Q,t)

t (ns)

Q=0.2Å-1

Q=0.3Å-1

0

0.2

0.4

0.6

0.8

1

10-3 10-2 10-1 100 101 102

S chai

n(Q,t)

t (ns)

Q=0.2Å-1

Q=0.3Å-1

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40

S chai

n(Q,t)

t (ns)

Q=0.2Å-1

Q=0.4Å-1

0

0.2

0.4

0.6

0.8

1

10-3 10-2 10-1 100 101 102

S chai

n(Q,t)

t (ns)

Q=0.2Å-1

Q=0.4Å-1

€

Schain (Q,t) =1N

expn,m∑ −

16Q2

⎡

⎣ ⎢ ⎤

⎦ ⎥ φnm(t)

CHECKING THE GAUSSIAN APPROXIMATION: PEO/PMMA MD-SIMULATIONS PEO/PMMA (20/80) blend

φnm(t) calculated from the non-exponential Cp(t) €

Schain (Q,t) directly calculated from the MD-simulations

Strong disagreement!!

0

0.2

0.4

0.6

0.8

1

10-2 10-1 100 101 102 103 104 105

S chain(Q,t)

t

Q=2.3Å-1

-0.2

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

S chain(Q,t)

t

Q=2.3Å-1€

Schain (Q,t) =1N

expn,m∑ −

16Q2

⎡

⎣ ⎢ ⎤

⎦ ⎥ φnm(t)

CHECKING THE GAUSSIAN APPROXIMATION: BEAD-SPRING MD-SIMULATIONS A/B (70/30) blend

φnm(t) calculated from the non-exponential Cp(t) €

Schain (Q,t) directly calculated from the MD-simulations

Strong disagreement!!

-12

-11

-10

-9

-8

-7

-6

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4

log[< τ>(s)]

log[Q(Å-1)]

CHAIN DYNAMICS OF PEO IN PEO/PMMA: NEUTRON SCATTERING Incoherent measurements

Sself(Q,t) BS

pure PEO

PEO/PMMA

resolution 10-3

10-2

10-1

100

-15 -10 -5 0 5 10 15

scat

tere

d in

tens

ity (a

rb. u

nits) Q=0.56Å-1

ω(µeV)

Single chain dynamics Schain(Q,t) NSE

€

β < 0.5 (Rouse) Pure Rouse bad description!

Strong deviations from normal Rouse behavior

€

Sself (Q,t) = exp − tτ self

⎛

⎝ ⎜

⎞

⎠ ⎟

β⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

Q-4

pure PEO 350K

T=350K

PEO in PEO/PMMA

300K

350K

RENORMALIZED ROUSE MODELS & DIELECTRIC SPECTROSCOPY

End-to-End correlation function

€

Φee (t) =

R N (t)

R N (0)

R N2 (0)

=8π 2

1p2p:odd

∑ Cp (t)

€

R N

€

µ

€

µ ||

€

µ ⊥

Valid also for non-exponential Cp(t)

€

Φee (t) directly calculated from the MD-simulations

€

Φee (t) calculated from Non-exponential Cp(t)

Bead-spring A/B (70/30) blend

0

0.2

0.4

0.6

0.8

1

10-3 10-1 101 103 105

Φ(t)

t

T:0.330.400.450.500.600.751.0

RENORMALIZED ROUSE MODELS & DIELECTRIC SPECTROSCOPY

End-to-End correlation function

measured by dielectric spectroscopy

Type-A polymers

“Normal mode frequency domain”

€

ε"(ω )εo −ε∞

=8π 2

1p2p:odd

∑ ϕp (ω)

€

P ∝ R N

€

ϕp (ω)

Non-exponential Cp(t)

€

Φee (t) =

R N (t)

R N (0)

R N2 (0)

=8π 2

1p2p:odd

∑ Cp (t)

0

0.01

0.02

0.03

10-1 100 101 102 103 104 105 106

ε ''

Freq. [Hz]

Normal mode

α-relaxation

Non-Debye

POLYISOPRENE

RENORMALIZED ROUSE MODELS & DIELECTRIC SPECTROSCOPY

POLYISOPRENE / POLY-TERT-BUTYLSTYRENE PI PtBS

Tg ≈ 200K Tg ≈ 370K ΔTg ≈ 170K

35 PI / 65 PtBS

Dynamic asymmetry

Composition asymmetry

-[CH2-CH=C-CH2]n CH3

-

Mn = 2700g/mol, Mw/Mn =1.07

Mn = 2300g/mol Mw/Mn =1.06

-[CH2-CH]n

CH3

- -

C -

CH3 H3C

RENORMALIZED ROUSE MODELS & DIELECTRIC SPECTROSCOPY

35 PI / 65 PtBS

Normal mode of PI

0.001

0.01

-1 0 1 2 3 4 5 6

ε"( ω

)

log(f[Hz])

T(K): 280 300 320 360 290 310 340 0.5

1

1.5

2

HWHM

0.001

0.002

0.003

0.004

2.8 3.2 3.6 4 4.4 4.8

dCp/d

T

1000 / T(K)

PI in PI/PtBS

Bulk PI

Tg(PtBS/blend)

RENORMALIZED ROUSE MODELS & DIELECTRIC SPECTROSCOPY

Phenomenological approach guided by RRM and simulation results

Non-exponential Rouse correlators Cp(t)

€

= exp −tτ p

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

β⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

Mode-number scaling of correlation times τp ∝ p-x x ≥ 2

€

Φee (t) =8π

1p2p:odd

∑ exp −tpx

τ1

⎛

⎝ ⎜

⎞

⎠ ⎟

β⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

Are x and β correlated?

€

Cp (t) ≈ exp −3KBTpπNb⎛

⎝ ⎜

⎞

⎠ ⎟ 2 dt '

ξo + Δξo(t ')0

t

∫⎡

⎣ ⎢

⎤

⎦ ⎥ ≈ exp −3KBTA

pπNb⎛

⎝ ⎜

⎞

⎠ ⎟ 2

t β⎡

⎣ ⎢

⎤

⎦ ⎥

RENORMALIZED ROUSE MODEL (K.S. Schweizer, J. Chem. Phys. 91 (1989) 5802)

Low-p limit

Stretched KWW-form

N/p scaling parameter x and β determined independently!

Normal mode correlators

€

Cp (t) ≈ exp −tτ p

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

β⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

€

τ p ∝pπNb⎛ ⎝ ⎜

⎞ ⎠ ⎟ −2β

with

€

x = 2 β

(simulation results)

€

Cp (t) =

X p (0)

X p (t)

X p (0)

X p (0)

€

~ tβ

2

2.5

3

3.5

4

4.5

2 2.5 3 3.5 4 4.5

2/<β>

x (p-scaling)

RENORMALIZED ROUSE MODEL

N/p scaling parameter x and β determined independently!

€

x = 2 β

bead-spring p=1

PEO in PEO/PMMA fully atomistic

0

0.002

0.004

0.006

0.008

10-1 100 101 102 103 104 105 106

normal mode α-relaxation

f(Hz)

T = 270Kε"

(ω)

RENORMALIZED ROUSE MODELS & DIELECTRIC SPECTROSCOPY

Working in the frequency domain results 35 PI / 65 PtBS

€

ε"(ω)ε o −ε∞

=8π

1p2p:odd

∑ ϕ(ω,Δ pω,x)

€

Δ pω :FWHM of p-correlator

2

2.5

3

3.5

4

1.2 1.4 1.6 1.8 2 2.2

τ1/τ3τ1/τ5τ3/τ5

FWHM

x

0

0.002

0.004

0.006

0.008 normal mode

ε"(ω) T = 330K

CHAIN-DYNAMICS IN ASYMMETRIC POLYMER BLENDS

CONCLUSIONS

•  Strong deviations from Rouse behavior for the unentangled fast component driven by the dynamic asymmetry in the system (frozenning of the slow component) instead of by the length of the chains.

•  Scaling features similar to those predicted by GLE for entangled chains.

•  Normal mode dielectric data in asymmetric polymer blends can be well described in this framework.