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Study of ER Non-equilibrium Behavior with COMSOL

Cong LI and Luwei ZHOUPhysics Department, Fudan University

Shanghai 200433, P.R.Chinalwzhou@fudan.edu.cn

Presented at the COMSOL Conference 2010 China

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COMSOL: A powerful tool in theoretical study

Lei ZHOU et al., Physics Department, Fudan University.

Metamaterials:Microwave Visible light

1. Theory ‐

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Hard  Soft metamaterialsSingle frequency  Broad band frequenciesOptic Invisibility  Acoustic Invisibility

Negative refraction indices

Y. Gao, et al., PRL 104, 034501 (2010)

Three factors:1. Metal core or shell2. Form chains or columns3. Lamella

3D

Y. Gao, et al., PRL 104, 034501 (2010)

Wavelength 758nm

Jiping HUANG et al., Fudan University.

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Applications

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Background of ER Fluids

• ER (electrorheological) fluids？PM‐ER (polar molecule dominated ER) fluids？

• ER particles + silicon oil

2. Experiment – Equilibrium

LiquidE

Dipole particles

Electrodes

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• Volume fraction fixed,  • Adjust parameters and re‐meshing

1               :            ~100

<<

Z.N. Fang, H.T. Xue, W. Bao, Chem. Phys. Lett. 441 (2007) 314–317.

Aggregated             versus        well dispersedRatio of yield stress

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Local electric field between two spheres

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Local electric field between two ellipsoids

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<<

Wei BAO, et al., J. Phys.- Cond. Mat. 22 (2010) 324105

The yield stress between two short axis chained ellipsoid particles is the largest.

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3.1 Lamellar structures of ER fluids under electric field and shear flow

S. Henley and F. E. Filisco, Inter. J. Mod. Phys. B, 16 (2002) 2286 – 2292.

Bad ER fluid Good ER fluid

= 0E > Ec

= 0> 0 > 0.

.

.

.

Polystyrene ER fluid (A, B) Sulfonated polystyrene

ER fluid (C, D)

E

E

3. Experiment – Nonequilibrium

E > Ec

1111

Experimental Setup

Modified by  Tan P, Liu D.K, Jia Y, Zhou L.W. et al

Haake Mars II rheometer 

Electrorheoscope

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1.6k 1.8k 2kV/mm

400 600 800V/mm

1k 1.2k 1.4kV/mm

Lamellar Structures of a PM-ER Fluidunder Different Electric Fields

13130 2 0 4 0 6 0 8 0 1 0 0 1 2 05 5 0

6 0 0

6 5 0

7 0 0

7 5 0

8 0 0

She

ar S

tress

/ P

a

t / s

Thin ring matters

Simultaneousobservation and comparison of lamellar structure and shear stress of the PM‐ER fluids

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0 20 40 60 80 100 120550

600

650

700

750

800

Sh

ear S

tress

/ Pa

t / s

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0 500 1000 1500 2000 2500

0

200

400

600

800

1000

She

ar s

tress

(Pa)

E (V/mm)

0 200 400 600 800 1000 12000

100

200

300

400

500

600

8001200

1800

E=2200V/mm

She

ar S

tress

(Pa)

Shear Rate(1/s)

0

Simultaneous measurement of ER shear stressand observation of lamellar structures

=300rpm

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• Molecular dynamics  (MD)based on Newton’s second law of motion‐‐ Large amount of calculation, time‐consuming

• Two phase flowbased on Onsager’s  principle with  COMSOL

‐‐ Easy to learn, quick calculation, powerful

Method and Theory

3.2 Simulation:

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Onsager’s Principle

• The Onsager’s principle of minimum energy dissipation rate is about the rules governing the optimal paths of deviation and restoration to equilibrium.

L. Onsager and S. Machlup, Phys. Rev. 91, 1505-1512 (1953). L. D. Landau and E. M. Lifshitz, Statistical Physics, 2nd Ed., London: Addison-Wesley Publishing Co., 1-484 (1969).

( ) ( )F t

tFA

)(2

2

( , )A J V Fs

Minimum

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• The modified Onsager action functional, A

J. W. Zhang, X. Q. Gong, C. Liu, W. J. Wen, P. Sheng, Phys. Rev. Lett. 101, 194503 (2008)

( , )A J V Fs

Onsager’s Principle

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0

1[ ( )] = ( , ) ( ) ( ) ( ) ( )2

( ) ( ) ( ) ( ) ( ) ,2 | |

ij i j

ext

F n x G x y p x n x p y n y dxdy

aE x p x n x dx n x n y dxdyx y

1 12 2 2[ ( ) ( ) ] ( )4 2 2 f sV V J K V V dxs s si j j i n

Free energy

Dissipation

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( ) = ( )sss s s s visc s f s

VV V p K V V

t

( ) = ( )f ff f f f visc s f

VV V p K V V

t

Onsager’s Principle

Continuity equation

Governing 

equations

Navier‐Stokes equation for oil

Navier‐Stokes equation for particles

0 JnVnJn st

J. W. Zhang, X. Q. Gong, C. Liu, W. J. Wen, P. Sheng, Phys. Rev. Lett. 101, 194503 (2008)

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COMSOL Simulationa. Model Establishment

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b. Geometry

2D axial symmetrylength 1mm, width 7.5mm

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c. Parameters

el: Local electric field, ff1&ff2：Conservative force,kk: Stokes drag force density 23

d. Expressions

2= 9 / 2s fK f a

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e. Integration Coupling Variables

Local electric field

Repulsive potential

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dest() operator

• Irrad1=‐((‐2)/(((r‐dest(r))^2+(z‐dest(z))^2)^3)*dest(nn)*dest(pp2))*((sqrt((r‐dest(r))^2+(z‐dest(z))^2)<=10*a)*(sqrt((r‐dest(r))^2+(z‐dest(z))^2)>=2.1*a))

r

dest(r)dest() is a operator to create convolution integral

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e. Boundary Conditions

1 43

2

Oil phase Particle phase Concentration1 Axial symmetry Axial symmetry Symmetry / Insulation2 Logarithmic wall function Wall / No slip Symmetry / Insulation3 Sliding wall / omega*r Sliding wall / omega*r Symmetry / Insulation4 Logarithmic wall function Wall / No slip Symmetry / Insulation

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f. Results

B

COMSOLpatternsimulationsof upper (L) and lower (R) electrodes

Experimentalobservation

MDsimulation

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Shear Stress

0 20 40 60 80 100 120550

600

650

700

750

800

She

ar S

tress

/ P

a

t / s

Integration of the upper boundary

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Conclusion：Static and Dynamic Rings

The angular velocity changesalong the radius. Regions with highvelocity and low velocity exist inthe subdomain.It is the dynamic ring that have

the maximum concentration andvelocity.

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4. Future Work

• Pattern and force with different slip lengths• Quantitative relations between shear stress and lamellar structures

• Relation of patterns and shear stress under AC field

• Different temperature effect ‐‐ All students in soft matter group must study COMSOL Multiphysics

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Expending to biophysics and granules

We should spread COMSOL to China’s western region such as Xinjiang and Gansu

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Thank you very much

State Key Laboratory of Surface Physicsand Physics Department

Fudan University