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And the surface tension can be deduced from chemical potential and the gradient of phase
field function, expressed as follows
[23]
135
(7)
The movement of polymer in micro-/nano- scale can be depicted by Navier-Stokes
equations.
[24, 25]
Besides, it is assumed that the fluid is Newton fluid and incompressible. Further
combining the electrostatic force and phase field, the specific Navier-Stokes equations are
represented as equation (8). 140
( ) ( )
0
T
ef st
u
u u p u u F g F
t
u
(8)
where ρ, η are the density and viscosity of fluid deduced from phase field, respectively, u is the
velocity, and p is the hydraulic pressure that enforces the incompressibility of fluid, while F
ef
, F
st
,
ρg denote the electrostatic force, surface tension, and gravity, respectively.
In this paper, the commercial FEM software (COMSOL Multiphysics) has been utilized to 145
introduce the model formula above to simulate the evolution of the electrically induced pattering
process by coupling the electric field, phase field and flow field with FEM method, accompanied
with a simplification into two dimensional problem domain proceeded to save the computer efforts.
Besides, the influence of the external factors on the strong/weak modulation and the method to
guarantee the modulation under strong modulation via adjusting the external factors is studied. 150
2 Results and discussion
2.1 Dimensionless variables revealing the strong/weak modulation
The sketch of electrically induced patterning technique to fabricate arbitrary structures is
shown in Fig. 1, in which the template and the substrate is acting as an electrode pair, Li(i=1,2,3)
is the width of the protrusions of the template, the distance from the undersurface of the protrusion 155
to the polymer surface is the air gap distance, V is the applied voltage. It is assumed that V is 40V,
air gap distance is 2μm, thickness of polymer is 1μm, separating distance between protrusions is
5μm, L1, L2 and L3 is 20μm, 1μm, and 5μm, respectively. And the properties about the polymer
is mass density 960kg/m3, viscosity coefficient 0.34Pa·s, surface tension on the polymer/air
0.03N·m. Here Li (i=1,2,3) is different to study the evolution of the arbitrary formation. 160
V
L1
L2
L3
Template
Polymer
Substrate
图 1 任意结构电诱导成形示意图
Fig.1 Sketch of electrically induced patterning technique to fabricate arbitrary structures
The consequent electrical distribution on the polymer/air interface at the initial stage and the
detailed evolution of induced process is shown in Fig. 2, with red region denoting polymer and 165
blue region representing air. When a voltage is applied, the polymer surface is planar at the initial
stage (shown in Fig. 2b(i)). After an interval, the polymer begins to overcome the surface tension,
viscosity and gravity to move upwards in the influence of the electrostatic force (shown in Fig.
2b(ii)). The final state is shown in Fig. b(iii), with periodic structures corresponding to the
protrusion, L1, single structure to L3, and no structure to L2. In addition, the electrical distribution 170
on the polymer/air interface at the initial stage is shown in Fig. 2a. Here it illustrates that: (1) the
larger width of protrusion presents a larger electric intensity, in which the electric intensity of L1