The problem of the moving contact line between two immiscible fluids on a smooth surface is revisited using molecular dynamics (MD) and continuum simulations. In MD simulations a finite slip is allowed by choosing incommensurate wall-fluid densities and weak wall-fluid interaction energies. The shear stresses and velocity fields are extracted carefully in the bulk fluid region as well as near the moving contact line. In agreement with previous studies, we found slowly decaying partial slip region away from the contact line. In steady-state shear flows we extract the friction coefficient along the liquid-solid interface, the local slip length, and the dynamic contact angle. The MD results show that both dynamic contact angle and slip velocity near the contact line increase with increasing the capillary number (Ca). Also, at high Ca the break up of fluid-fluid interface is observed. The slip boundary conditions near the moving contact line extracted from MD simulations were then used in the continuum solution of the Navier-Stokes equation in the same geometry to reproduce velocity profiles and the shape of the fluid-fluid interface.

1. Slip boundary conditions for the moving contact line in
molecular dynamics and continuum simulations
Anoosheh Niavarani and Nikolai V. Priezjev
Department of Mechanical Engineering
Michigan State University
Movies and preprints @ http://www.egr.msu.edu/~niavaran
• A. Niavarani and N. V. Priezjev, “Modeling the combined effect of surface roughness and shear
rate on slip flow of simple fluids”, Physical Review E 81, 011606 (2010).
• A. Niavarani and N. V. Priezjev, “The effective slip length and vortex formation in laminar flow
over a rough surface”, Physics of Fluids 21, 052105 (2009).

2. Introduction
Question: How to model the moving contact line problem in a shear flow using molecular
dynamics and continuum methods?
Equilibrium:
sθ : Contact angle
γ : Surface tension
γFluid 1 Fluid 2
sθ1γ 2γ
2 1cos sγ θ γ γ= −
sd θθ ≠
Fluid 1
Fluid 2
γ
2γ1γ
U
• No-slip boundary condition leads to divergence of
energy dissipation (unphysical)
• Contact line singularity is regularized by introduction
of the slip region near the contact line.
Moving contact line:
2
( cos sin )r c d
r
θ
µ
τ θ θ= −
Huh and Scriven, J. Colloid Interface Sci. 35, 85 (1971)
Department of Mechanical Engineering Michigan State University
Young’s equation

3. • The Navier model describes the slip boundary condition at the solid/liquid interface:
Linear relation outside contact line:
U
Contact line
• The friction coefficients can be estimated from the
molecular dynamics simulation away from the contact line.
(cos cos )CL contact suβ γ θ θ= −• At the contact line
• Our goal is to use molecular dynamics
simulations to estimate the stress tensors,
friction coefficient, and flow profiles and
determine the correct boundary condition
for continuum modeling.
What is the boundary condition near the moving contact line?
,slipuτ β= 0/ Lβ µ=
L0
h
Solid wall
U
x
z
uslip
u(z)
Ren and E, Physics of Fluids 19, 022101 (2007)
Qian, Wang, and Sheng, Phys. Rev. E 68, 016306 (2003)
β
,slipu β
,contact CLu β
Department of Mechanical Engineering Michigan State University

4. 1−=δ
12 ww εε =
w2γ
90=sθ
Fluid 1 Fluid 2 Fluid 1
1 2γ γ=
12 2 ww εε =
Fluid 1 Fluid 2 Fluid 1
1 2γ γ> 130≈sθ
γFluid 1 Fluid 2
sθ1γ 2γ
2 1cos sγ θ γ γ= −
Immiscible fluids














−





=
612
4)(
σ
δ
σ
ε
rr
rVLJ
ij
i i i
i j i
V
my m y f
y≠
∂
+ Γ =− +
∂
∑ 
1
τ −
Γ =
BT=1.1 kε
( ) ( ) 2 ( )i i Bf t f t mk T t tδ′ ′= Γ −
if
2 1/2
( )mτ σ ε=
σ LJ molecular length scale
ε LJ energy scale
LJ time scale
Lennard-Jones potential:
Langevin Thermostat
Friction coefficient
Equation of motion:
Gaussian random force
3
81.0 −
= σρFluid density
Details of the molecular dynamics simulations
x
z
sθ
sθ

5. 2
|| )2.07.3()]()([ −
⊥ ±=−= ∫ εσττγ drrrSurface tension
xxτ
yyτ
zzτ
Extracting normal stresses from molecular dynamics
σ/x
σ/z
U = 0Equilibrium 12 ww εε =1 2γ γ=
x
z
,xxτ yyτ
zzτ
• To calculate the surface tension, the normal stresses are estimated accurately using a modified
Irving-Kirkwood relation.
• The surface tension from molecular dynamics simulations is then used in continuum simulations.

6. Distribution of the shear stress along the lower wall in equilibrium (U=0)
Snapshot of the atoms near the contact line Tangential stress along the lower wall
x
z
10σ
• The tangential stresses along the lower wall is calculated from LJ forces per unit area
between wall atoms and fluids molecules.
• The negative and positive stresses, within 5σ from the contact line, are due to a reduced
density in the fluid/fluid interfacial region.
/x σ
xzτ
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7. sd θθ ≠
dθ
U
Dynamic contact angle and shape of interface in steady-state shear flow
Fluid 1
Fluid 2
γ
2γ1γ
τσ /05.0=U
0.1 /U σ τ=
U = 0
σ/x
• As the wall speed (Capillary number) increases,
the contact angle becomes larger.
• The dynamic contact angle is .90dθ >
Dynamic contact angle vs wall speed
U
x
z
dθ
dθ
dθ
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8. 240σ/x0
σ/z
25
σ/x
σ/z
0 240
25
U
U
τσ /1.0=U
• The flow velocities are computed from the time averaging of instantaneous molecule speeds
in small spatial bins over a long period of time.
Extracting macroscopic velocities from molecular dynamics
Velocity profiles in the first fluid layer
First fluid layer
σ/x
slipu
x
z
• In each fluid phase the slip velocity of the first fluid layer increases near the contact line
(symbols), but the overall slip velocity is smaller than wall speed at the contact line (dashed line).
Fluid 1 Fluid 2

9. U
0 240
25
σ/x
σ/x
Velocity profile in the first fluid layer
slipu
Slip velocity and the friction coefficient from molecular dynamics
/z σ
U
First fluid layer
Velocity profile along the z direction
( )u z
σ/x
τσ /01.0=U
0.05 /U σ τ=
0.1 /U σ τ=
τσ /01.0=U
0.05 /U σ τ=
0.1 /U σ τ=
• The slip velocity near the contact line becomes
larger with increasing the wall speed.
• The velocity profiles are linear and the slip length
is calculated from a linear fit to the profiles.
0 2.2L σ=
0/ 0.9Lβ µ= =
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10. τ3000Movie length ~
Motion of the contact line at high shear rates
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• At higher capillary numbers (higher U) the contact line undergoes an unsteady motion.
• High stresses at the contact line lead to a pronounced curvature of the fluid-fluid
interface and a breakup of a continuous fluid phase.
0.2 /U σ τ= 2 1w wε ε=http://www.egr.msu.edu/~niavaran

11. 0
u
x
∂
=
∂
0v =
2
( ( ) )
u
u u p u f
t
ρ µ
∂
+ ⋅∇ = −∇ + ∇ +
∂
0u∇⋅ =
( )k k k
k
f n x x Sγκ δ=− − ∆∑
  
1
( )t t t
k k kx x t u+
= + ∆
  
( )k ij k
ij
u u x xδ= −∑
   
Interface location
predicted by marker points
Details of the continuum modeling of the moving contact line
Navier-Stokes equation
applied on the fixed grids
Boundary conditions U
U
• Away from the contact point (single phase
fluid): Navier Slip
B.C.:
xz slipuτ β=
, , CLγ β β extracted from molecular dynamics simulations
• At the contact point (the marker point):
(cos cos )CL contact suβ γ θ θ= −
dim
1
( ( ) )1
( ) (1 cos )
2
m m k
k k
m
x x
x x if x x d
d d
π
δ
=
−
−= + − ≤∏
 
   
• Near the contact point: a distribution function
interpolates the velocities between the contact
point and single phase fluid
slipu slipu
contactu
The system size and the flow properties are the same as in the molecular dynamics method
x
z

12. 0.01U =
0.05U =
0.1U =
0.01U =
0.05U =
0.1U =
Molecular dynamics
results=dashed lines
Dynamic contact angle and flow profiles near the moving contact line
0.05U =
0.1U =
0.01U =
x
z
dθ
dθ
dθ
σ/x
slipu
Velocity profiles in the first fluid layer
• The slip velocity and the contact angle increase
at higher wall speeds.
• The continuum results agree well with molecular
dynamics simulations.
Molecular dynamics
results=dashed line
Dynamic contact angle vs wall speed
U
dθ

13. • The slip boundary conditions near the moving contact line extracted from MD simulations were
used in the continuum solution of the Navier-Stokes equation in the same geometry to reproduce
velocity profiles and the shape of the fluid-fluid interface.
• The MD results show that both dynamic contact angle and slip velocity near the contact line
increase with increasing the capillary number (Ca).
• At higher capillary numbers (higher U) the contact line undergoes an unsteady motion. High
stresses at the contact line lead to a pronounced curvature of the fluid-fluid interface and a
breakup of a continuous fluid phase.
Important conclusions
Department of Mechanical Engineering Michigan State University