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.