Experimental studies have shown that modification of a smooth substrate by chemical means can enforce spontaneous droplet migration toward regions of stronger liquid adhesion. Similar behavior may be expected by thermal modulation of the liquid-solid interfacial energy. Detailed studies of this method of propulsion are lacking since thermocapillary stresses at the gas-liquid interface typically obfuscate the smaller contribution arising from the variation in interfacial tension at the liquid-solid interface. Using molecular dynamics simulations, we examine the behavior of a partially wetting polymer droplet traversing a smooth substrate subject to a uniform lateral temperature gradient. As desired, the motion of the droplet is dominated by the variable liquid-surface interaction energy and not the thermocapillary stress at the free surface. The droplet center of mass is shown to follow a biased diffusion process as the average trajectory translates toward the cooler side of the substrate. We examine the molecular motion internal to the droplet as well as the fluid structure near the variable energy wall as a function of the parameters governing the fluid-wall potential and the applied thermal gradient.

1. Droplet Propulsion by Thermal Modulation of
the Liquid-Solid Interfacial Energy
Nikolai V. Priezjev
Prof. Sandra M. Troian
Microfluidic Research & Engineering Laboratory
Dept. of Chemical Engineering
Princeton University
http://www.princeton.edu/~priezjev
Priezjev and Troian, preprint (2004)

2. Outline
Princeton University
Fluid droplet
• Isothermal wall conditions:
free diffusion of droplet center mass
• Details of molecular dynamics model
• Droplet motion in presence of temperature gradient
Estimation of thermocapillary
force at liquid vapor interface
Variation in the effective
surface adhesive energy
Temperature variation T(x)
Why is the droplet moving?
• Conclusions

3. Interaction potentials:
Molecular Dynamics Simulation Model
ij
i i i
i j i
V
my m y f
y≠
∂
+ Γ = − +
∂
∑
1
friction coefficient
Gaussian random forceif
τ −
Γ =
=
Modified Langevin thermostat for atoms:wall
6 12
LJ
2
2
FENE o 2
o
r r
V (r) 4
1 r
V (r) kr ln 1
2 r
ε
σ σ
− −
⎡ ⎤⎛ ⎞ ⎛ ⎞
= −⎢ ⎥⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
⎛ ⎞
= −⎜ ⎟
⎝ ⎠
0.9 wall-fluid interactionwfε ε=
-3
0.8 wall densitywρ σ=
cut-offr 2.5σ=
Princeton University
0T(x)=T (1 x)α−
Temperature
gradient
Bead-spring model: 10-mers
0 Bwhere T =1.0 kε
Nfluid=3600
max
min
T ~1.4
T ~ 0.6
B
B
k
k
ε
ε

4. Isothermal wall conditions
CM CM
2/
/
1
=
2t
for
D x (t)-x (t )
t-t τ
0 ?α ≠
CM i
1
1
x x
fluidN
ifluidN =
= ∑
Applied T0: 0.6 – 1.3 ε ⁄ kB Diffusion coefficient:
• Diffusion follows
Arrhenius dependence
Princeton University
• For typical duration
of simulation 104τ
droplet center mass
diffuses about 10σ
for T=1.0 ε⁄kB
Arrhenius law:
( )BD(T) exp -E k T∼
E = 4.4 ε
0α =

5. Droplet Motion in Presence of Temperature Gradient
-
-
-1
1
1
1
-1
-
= 0.0
= 0.0
= 0
= 0.0148
.0085
127
= 0.0
017
σ
σ
σ
σ
σ
α
α
α
α
α
0T(x)=T (1 x)α−
0 BT =1.0 kε
time
CM i
1
1
x x
fluidN
ifluidN =
= ∑
• Faster motion for
higher gradient α
• Small gradient α:
trajectory is liner
with superimposed
fluctuations
• Large gradient α:
trajectory becomes
curved (slow motion
at right cold end)
Princeton University
High THigh T
Low T

6. Why is the Droplet Moving?
Estimation of thermocapillary force:
F = LVd dT
Sin dS
dT dx
γ
θ∫
θ
-2
-2
=
(T=0.6)
( 1.3)
surface tension decreases monotonically
from =
to
for the thin polymer vapor interface
1.28
0.54
LV
LV
LV T
εσ
εσ
γ
γ
γ = =
( )T x
dF
pulling stress interface
Fluid droplet
n
Estimation of surface tension
Polymer fluid
( ) constantT x =
~ 20σ
( )LV Tγ
LVγ
Broska et al., Langmuir, `93
Ford and Nadim, Phys. Fluids, `94
TEST:
free diffusionfluidF N =
3
010 recoverfluidNF α⋅ = ≠
(1)=Typ O εical force is lsma lF σ

7. Droplet Diffusion Influenced by Adhesion Energy Variation
wf wfε ( ) ε (1 )x xδ= +
0 BT =1.0 k is constantε
= surface energy gradientδ
• Faster motion for
higher gradient δ
• Trajectory for
δ=0.1 σ-1
is similar to motion at
temperature gradient
α=0.0127σ−1CM i
1
1
x x
fluidN
ifluidN =
= ∑
Isothermal wall conditions:
Brochard, Langmuir 5, 432 (1989)
Chaudhury and Whitesides, Science 256, 1539 (1992)
Strong
attraction
Weak
attraction

8. Conclusions:
Princeton University
Using molecular dynamics simulations, we have investigated the behavior
of polymeric droplet under imposed temperature gradient.
http://www.princeton.edu/~priezjev
• Diffusion of the droplet’s center of mass follows Arrhenius law.
• Faster CM droplet motion for higher temperature gradient.
• Thermocapillary effects at the liquid/vapor interface play a minor role.
• The motion of the droplet is dominated by the variable liquid-surface
interaction energy. Drift velocity of the center of mass grows linearly
with surface energy gradient.