A phase field model for binary
fluid-surfactant system
Kuan-Yu Chen (陳冠羽)
Advisor: Ming-Chih Lai (賴明治)
Department of Applied Mathematics,
National Chiao Tung University, Taiwan
Outline of this talk
 Introduction
 Mathematical formulations
- model for binary fluid system
- model for surfactant
 Numerical schemes
 Validations and Results
 Conclusion and future works
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Introduction
 1D / 2D Problem
 surfactant (surface active agent)
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 Example
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Model formulation
 Cahn-Hilliard surface free energy *
G ( )   [
2
|  |2  f ( )]dx
2
 φ : phase function (order parameter), 0<=φ<=1
 ε : interface width scale
 f(φ) is the bulk energy density
1
f ( )  (  1) 2  2
4
J. W. Cahn and J. E. Hilliard, “Free energy of a nonuniform system. I. Interfacial energy,” J. Chem.
Phys 28, 258 (1958).
J. Kim, “Numerical simulations of phase separation dynamics in a water-oil-surfactant system,” J.
Colloid & Int. Sci. 303, 272 (2006)
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 Cahn-Hilliard equation

G
   (M 
)    ( M  ( f '( )   2 2 ))
t

   G

0
on 
n n 
 Mφ is the mobility
 Mass conservation
d
G
 G
 dx   t dx     (M   )dx   M 
ds  0

dt

n 

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 Property of Cahn-Hilliard energy
dG ( )


2
   
 f '( )
dx
dt
t
t

2 2
  [ f '( )     ]
dx
t
   G
 G 
(

 0 on )

dx
n n 
 t
G
G

  (M  
)dx


G 2
  M  | 
| dx  0

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 Coupling binary-fluid & surfactant energy
G ( , )   [
2
2
|  |2  f ( ) 
 ( )
2
( |  |) 2
  ( )( ln   (1  ) ln(1  ))]dx
 α ~ O(ε2), potential term coefficient
 θ ~ O(ε2), entropy term coefficient
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 Coupling binary-fluid & surfactant system

G
   (M  
)
t

1

2
   ( M  [ f '( )       [( |  |)
]])

|  |

G
   ( M 
)
t

   ( M [ ( |  |)   (ln   ln(1  ))])
   G   G


0
n n  n 
on 
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 Surfactant equation

H
   [ M 
]    ( M [ ( |  |)   (ln   ln(1   ))])
t

 H
 0 on 
n 
 MΓ is the mobility
(M   (1  ))
 Mass conservation
d
H
 H
dx   t dx    (M 
)dx   M 
ds  0

dt

n 

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 Simplified surfactant equation

G
G
   [ M 
]    [(1  )
]
t


   [(1  )( ( |  |)  (1  ) (ln   ln(1  )))]

   [(1  )( |  |)]    [(1  )(
)]
(1  )
   [(1  )( |  |)]   2
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 Property of Coupled energy
dG ( , )
 G   G 


dx
dt
 t  t
G
G G
G

  (M 
)
  ( M 
)dx




G 2
G 2
  M  | 
| M  | 
| dx  0


   G   G
(


 0 on )
n n  n 
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Numerical scheme
 Phase field Equation
 n 1   n
 M   2  n 1
t
 G n1
 n
n 1
n
2 2 n 1
  [ ]  f '( )        [
]   2 n1

|  |
 Let L be Standard Laplacian discretization :


n
 1
  n 1  


I

M
L
t





t
n 1 
 2
    
 n 
n
(



)
L
I
f
'(

)




[

] 



|  | 

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 Neumann Boundary Condition => cosine transform
 1
 t I
 2
  
 h 2 Lh

Lh    n 1  
2
R.H .S .1
h



n 1


R.H .S .2 
I     

M
2
 h2
 n 1


 2 
h



I  Lh ( 2 Lh )  
   Lh R.H .S .2 
R.H .S .1


h
M

 tM 


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 Surfactant Equation
 n1   n
   [(1  )( |  |)]n   2 n1
t
 n

1
 n1
  I   L         [(1  )( |  |)]n  BC 
 t

 t

 Using similar manner in phase field solver
   n1  
1

I

L


R
.
H
.
S
.3
h
2
 t
 

 
h
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Validations & Results
 Convergence Test (1D)
domain: 0<= x <= 2π
initial: φ(x)=0.3 + 0.01*cos(6x),
Γ(x)=0.1 + 0.03*exp(-(x-π)2/0.52)
parameters: ε2=α=θ=0.0001
test on T=1, dt~O(dx2)
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mesh size phase function (φ)
L2-error
order
surfactant (Γ)
L2-error
order
128
256
2.4225x10-3
1.5623x10-3
5.5523x10-4 2.1253 3.4942x10-4 2.1606
512
1.0695x10-4 2.3760 7.3528x10-5 2.2485
comparison based on finest mesh (1024)
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 Time evolution

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 Mass & Energy
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 Sample Test (2D)
domain: 0<= x <= 2π, 0<= y <= 2π
initial:
φ(x,y)=0.3 + 0.01*cos(6x)*cos(6y),
Γ(x,y)=0.1 + 0.03*exp(-((x-π)2 +(y-π)2 )/0.52)
parameters: ε=α=θ=0.04
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 Time evolution
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 Mass & Energy
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Conclusion and future works
 We develop a phase field model for binary fluid-
surfactant system.
 We propose a simple numerical scheme for our
model, which keeping the mass conservation and
energy decreasing properties.
 Challenge : Coupled with fluid dynamics
(i.e. Navier-Stokes systems)
 Other possible formulations for binary fluid-surfactant
system ?
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Undergoing Work
 Incompressible Navier-Stokes Equation
with binary fluid-surfactant system.
u
 ( )(  u u)  P   ( ( )(u  uT ))
t
  c ( )(  (1  )   s  s |  |2 )
 u0
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 Binary-fluid & surfactant system under flow field

G
 U ε     ( M  
)
t

1

2
   ( M  [ f '( )       [( |  |)
]])

|  |

G
 U ε     ( M 
)
t

   ( M [ ( |  |)   (ln   ln(1  ))])
   G   G


0
n n  n 
on 
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Thanks for your attention
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