Xerrada a Aachen l'any 2007 sobre ferrofluids
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STRUCTURE FORMATION STRUCTURE FORMATION IN FERROFLUID MONOLAYERS: IN FERROFLUID MONOLAYERS:
ThEORY AND COMpUTER SIMULATIONS.ThEORY AND COMpUTER SIMULATIONS.
S. KantorovichS. Kantorovich,, C. Holm, C. Holm, J.J. CerdàJ.J. Cerdà
Ural State University, Ekaterinburg
Max-Plank Institute for Polymer Research
FERROFLUIDS
Ferrofluid:Ferrofluid: stable colloidal suspension of sub-domain magnetic particles in a liquid carrier. The particles, which have an average size of about 10 nm, are coated with a stabilizing dispersing agent (surfactant) which prevents particle agglomeration even when a strong magnetic field gradient is applied to the ferrofluid.
The surfactantThe surfactant must be matched to the carrier type and must overcome the attractive van der Waals and magnetic forces between the particles.
A typical ferrofluid may contain by volumecontain by volume 5% magnetic solid, 10% surfactant and 85% carrier.
FERROFLUID MONOLAYERS
Absence of real space experi-mental information in 3D.
Recent direct-observation of chain and ring formation in monolayers. Direct comparison to experiments is feasible. Quasi-two dimensional systems can exhibit a behavior different from 2D or 3D systems.
OVERVIEW
DF DF ThEORYThEORY
∑
∑
∞
=
∞
=
−−+
+−−
=
5
1
)(ln8
1)(ln)(
)(ln8
1
)(ln)(),(
n
n
nWesnfnfkT
nQe
sngngkTfgF
ρπ
ρπ
Check the degree of correctness of the
theoretical formalism.
Analyze the process of microstructure formation, and phase behavior: gain
physical insight.
MD SIMULATIONSMD SIMULATIONS
MODELIzATION OF ThE FERROFLUID MONOLAYER
Quasi-two dimensional systemQuasi-two dimensional system
-Position particles: 2D.Position particles: 2D.
-Dipoles: free 3D rotation.Dipoles: free 3D rotation.
-Single magnetic domain.Single magnetic domain.
Short range interactionShort range interactionSurfactant layer, oleic acid (2 nm) Magnetic core, Fe3O4
Us (r
12 )
r12σm/2 σ/2
WCA Potential
kkBBTT
≈ ≈ σσ
Tk
U
Tk B
dd
B
)2,1(
2
1
4 3210 −==
σπµµµλ
<1 12
monolayertheofareaTotal
areastionseccrossparticleofSum=Φ
λλ= 1.54 ... 4.99φφ= 0.01 ... 0.25
Range parameters
MODELIzATION OF ThE FERROFLUID MONOLAYER
−= 2
12
122121213
12
0,,
3,1
4)2,1(
r
rμrμμμ
rπµ
ddU
� 1
12r12
Long-range interaction Long-range interaction Control ParametersControl Parameters
-Area fraction, Area fraction, ΦΦ..-Dipolar Coupling, Dipolar Coupling, λλ
UUdddd(1,2)(1,2)
- Point dipoles at the CM of particles.- Point dipoles at the CM of particles.
COMpUTER SIMULATIONS
Simulation results
N=1000N=1000λλ=4.99=4.99
COMpUTER SIMULATIONS
COMpUTER SIMULATIONS
1 unit = 10 1 unit = 10 nmnm
COMpUTER SIMULATIONS
1 unit = 10 1 unit = 10 nmnm
∑
∑
∞
=
∞
=−−+−
−=
51)(ln
81)(ln)()(ln
81
)(ln)(),(nn
nWesnfnfkTnQ
e
sngngkTfgF φπφ
π
snnfnng
nn
φ=+ ∑∑∞
=
∞
= 51
)()(
ThEORETICAL MODEL: DENSITY FUNCTIONAL AppROACh.
Present LimitationsPresent Limitations
- Monodisperse system.
- Intra-cluster: only nearest neighbors interactions.
- Chains and rings.
- No inter-cluster interactions.
Excluded Area Interactions
22σσ
22σσ
σσ
- Excluded area: partially.
ThEORETICAL MODEL: EqUILIbRIUM SURFACE FRACTIONS.
)exp()(8
11
)(
)exp(8
11
)( 1
nnwsn
nf
nqs
ng
n
n
µφπ
µφπ
−=
−= −
µ - Lagrange multiplier to be found from the mass balance equation
MICROSTRUCTURE ANALYSIS: 2ND VIRIAL COEFFICIENTS
3.4811.86108.62
4.15×103
B233
3.7328.95580.50
3.72×104
1.103.8035.501.40×103
2.022.593.284.07
B222B223λ
Q2D: 3D dipoles, but 2D sample
3D dipoles&sample
2D dipoles& sample
Quasi 2D geometry changes the ferrofluid microstructure effective interactions are weaker
MICROSTRUCTURE ANALYSIS: TRACkINg CLUSTERS…
0
0
≥
≥
≤
122121
21
c12
r,μr,μ
μ,μ
rr
The eye (distance criterion) can be misleading
<1
r12
12
Entropy criterionEntropy criterion
MICROSTRUCTURE ANALYSIS: MICROSTRUCTURE ANALYSIS: bRANChED STRUCTURESbRANChED STRUCTURES
VS ChAINS&RINgSVS ChAINS&RINgS
MICROSTRUCTURE ANALYSIS: NEIghbOURS.
0 0.05 0.1 0.15 0.20
0.2
0.4
0.6
0.8
1
area fraction
X0
0 0.05 0.1 0.15 0.20
0.1
0.2
0.3
0.4
0.5
area fraction
X1
0 0.05 0.1 0.15 0.20
0.2
0.4
0.6
area fraction
X2
λλ=2.59=2.59λλ=3.28=3.28
TheoryTheorySimulationsSimulations
λλ=3.28=3.28 λλ=3.28=3.28λλ=3.28=3.28
φφ=0.05=0.05 φφ=0.15=0.15φφ=0.01=0.01
λλ=2.59=2.59λλ=3.28=3.28
λλ=2.59=2.59λλ=3.28=3.28
0 0.05 0.1 0.15 0.20
0.2
0.4
0.6
0.8
area fraction
X0
0 0.05 0.1 0.15 0.20
0.1
0.2
0.3
0.4
0.5
area fraction
X1
0 0.05 0.1 0.15 0.20
0.2
0.4
0.6
0.8
1
area fraction
X2
λλ=4.07=4.07
λλ=4.99=4.99
TheoryTheorySimulationsSimulations
λλ=4.99=4.99 λλ=4.99=4.99λλ=4.99=4.99
φφ=0.05=0.05 φφ=0.15=0.15φφ=0.01=0.01
λλ=4.07=4.07
λλ=4.99=4.99
λλ=4.07=4.07
λλ=4.99=4.99
MICROSTRUCTURE ANALYSIS: NEIghbOURS.
MICROSTRUCTURE ANALYSIS: CLUSTER SIzE.
0 0.1 0.22
2.5
3
area fraction
aver
age
clu
ster
siz
e
Theory (Excluded Area)Theory (Excluded Area)Theory (No Excluded Area)Theory (No Excluded Area)
SimulationsSimulations
λλ=1.54=1.54
λλ=2.02=2.02
λλ=1.54=1.54
λλ=2.02=2.02
Theory (Excluded Area)Theory (Excluded Area)Theory (No Excluded Area)Theory (No Excluded Area)
SimulationsSimulations
SS=2.59=2.59
===3.28=3.28
λλ=2.59=2.59
λλ=3.28=3.28
0 0.1 0.22
3
4
5
area fraction
aver
age
clus
ter
size
MICROSTRUCTURE ANALYSIS: CLUSTER SIzE.
0 0.1 0.2
5
10
area fraction
aver
age
clus
ter
size
Theory (Excluded Area)Theory (Excluded Area)Theory (No Excluded Area)Theory (No Excluded Area)
SimulationsSimulations
λλ=4.02=4.02
MICROSTRUCTURE ANALYSIS: CLUSTER SIzE.
The cut-off of the dipolar interaction (intra, and inter-cluster) could be the in a The cut-off of the dipolar interaction (intra, and inter-cluster) could be the in a large extend the cause of the mismatch between theory and simulations at large extend the cause of the mismatch between theory and simulations at large values of the dipolar coupling constant large values of the dipolar coupling constant λλ.. NEXT REFINEMENTNEXT REFINEMENT
MICROSTRUCTURE ANALYSIS: CLUSTER SIzE.