Accepted Manuscript

3D Phase Field Modeling of Electrohydrodynamic Multiphase Flows

Qingzhen Yang, Ben Q. Li, Yucheng Ding

PII: S0301-9322(13)00093-1

DOI: http://dx.doi.org/10.1016/j.ijmultiphaseflow.2013.06.006

Reference: IJMF 1918

To appear in: International Journal of Multiphase Flow

Received Date: 14 January 2013

Revised Date: 12 June 2013

Accepted Date: 15 June 2013

Please cite this article as: Yang, Q., Li, B.Q., Ding, Y., 3D Phase Field Modeling of Electrohydrodynamic Multiphase

Flows, International Journal of Multiphase Flow (2013), doi: http://dx.doi.org/10.1016/j.ijmultiphaseflow.

2013.06.006

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1

(Submitted to Journal of Multiphase flows, January 2013; revised June 2013)

3D Phase Field Modeling of Electrohydrodynamic Multiphase

Flows

Qingzhen Yanga,b

, Ben Q. Lib,*

, Yucheng Dinga

aState Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong

University, Xi’an, Shaanxi 710049, P. R. China

bDepartment of Mechanical Engineering, University of Michigan, Dearborn, Michigan

48128, U.S.A

*Corresponding author: Ben Q. Li (Email: benqli@umich.edu).

ABSTRACT: A 3D phase field model is developed to investigate the

electrohydrodynamic (EHD) two phase flows. The explicit finite difference method,

enhanced by parallel computing, is employed to solve the coupled nonlinear governing

equations for the electric field, the fluid flow field and free surface deformation.

Numerical tests indicate that an appropriate interpolation of densities within the interface

is critical in ensuring numerical stability for highly stratified flows. The 3D phase field

model compares well with the Taylor theory for the deformation of a single dielectric

droplet in an electric field. Computed results show that the deformation of a leaky

dielectric droplet in an electric field undergoes various stages before it reaches the final

oblate shape. This is caused by the free charge relaxation near the fluid-fluid interface.

The coalescence of four droplets in an electric field illustrates a truly 3D deformation

behavior and a complex evolving fluid flow field associated with the participating

droplets. The coalescence is a result of combined actions produced by the global electric

force, the circulatory flows generated by the local electrohydrodynamic stress and the

2

electrically-induced deformation. The 3D phase field model is also applied in modeling

of an electrohydrodynamic patterning process for manufacturing nanoscaled structures, in

which complex 3D flow structures develop as the electrically-induced deformation

evolves.

Keywords: phase field method; leaky dielectric model; electrohydrodynamics; two phase

flow; 3D modeling

1. Introduction

The earliest account of electrohydrodynamic (EHD) research is attributed to Gilbert

(1958) in the 17th century, who observed the conical shape upon placing a charged rod

above a sessile drop. Later, Rayleigh (1882) studied the deformation and the bursting of

charged drops in an electric field. The experiments conducted by Allan and Mason (1962)

eventually motivated Taylor (1966) to propose the concept of “leaky dielectrics”, which

provides an accurate account for the electric charge behavior in a poorly conducting

liquid. Within the framework of the leaky dielectrics, Taylor (1966) developed a

theoretical model for the deformation of a single drop immersed in a host liquid of

different electrical properties in a uniform electric field. Over the last decade, the EHD

flow has received much attention in research community because of its widespread

industrial applications, which, among others, include coating flows (Fermin, 2001),

transport of small liquid samples in microfluidics (Stone et al., 2004), various

microfluidics applications that use the concept of the lab-on-a-chip (Zeng and Korsmeyer,

2004) and micro- or nano-structure formation in soft lithography (Chou et al., 1999;

Chou and Zhuang, 1999; Schaffer et al., 2000; Wu and Russel, 2009; Yang et al., 2013).

As with other complex flow problems, only a limited number of the EHD flow

3

problems within an idealized setting can be solved analytically. Numerical simulations,

more often than not, are required to obtain solutions for the flow systems involving two

fluids with different properties. In numerical solution of two phase flow problems, the

presence of the free surface requires a special treatment. Different numerical schemes are

now available, and the popular ones include the level set method (Adalsteinsson and

Sethian, 1995; Sussman et al., 1994), the volume of fluid method (Hirt and Nichols, 1981;

Nichols et al., 1980; Puckett et al., 1997), and the front tracking method (Tryggvason et

al., 2001). The phase field method, which is based on the coarse-grain averaging concept

from statistical mechanics, has recently emerged as a useful tool for the study of the two

phase flow problems (Anderson et al., 1998; Jacqmin, 1999; Badalassi et al., 2003). A

salient feature of this method lies in its treatment of a moving or free surface as a thin

molecular diffuse layer, thereby enabling the molecular or nanoscaled description of the

collective behavior of molecules within the interface layer. Thus, within the framework

of the phase field modeling, a sharp fluid-fluid interface is represented by a narrow layer

in which the fluids may mix. Tracking of the interface is realized by a conserved order

parameter (or the phase field parameter) that varies continuously over the thin interfacial

layer but otherwise mostly uniform in the bulk fluid phases (Kim, 2012). The concept of

a diffuse interface was first proposed by Van de Waals (1893) long ago, but it has gained

popularity only in recent years as a tool for numerical simulations of two phase flows

(Carlson et al., 2010; Villanueva and Amberg, 2006).

Some numerical studies on the problems of two phase flows in electric fields have been

presented by various researchers. Feng (1996) calculated the equilibrium shape of a leaky

dielectric droplet in an electric field by employing the Galerkin finite element method,

4

coupled with the spine-parameterization of the free surface. The scheme works well for a

single droplet but can be difficult to apply for a system involving multiple droplets (Li,

2006; Huo and Li, 2006). Sherwood (1988) developed a boundary integral approach to

simulate the large deformation of a droplet with finite conductivity in a creeping flow.

The boundary integral model is capable of describing the drop breakup caused by either

the electric stress or the mechanical stress due to the fluid motion. The EHD two phase

flows have also been studied by the level set and volume of fluid methods (Tomar, 2007),

and by a front tracking/finite volume method (Hua, 2008). The latter also includes

various electric models (i.e., leaky dielectric, perfect dielectric and constant charged

models) for the fluids. Recently, Lin et al. (2012) presented a phase field model for the

2D axisymmetric EHD flows and studied the electro-coalescence of two droplets in an

electric field.

This paper presents a 3D phase field model for the EHD two phase flows. The thrust for

the work is derived from the need to develop a comprehensive numerical model for

complex, evolving two phase flows and free surface deformation in the presence of an

external electric field. Such a numerical model is not only of crucial importance to our

fundamental understanding of the physics governing the fluid flow and droplet formation

but also of engineering significance in developing practical guidelines for scalable top-

down fabrication processes for mass production of functional micro- or nano-structures.

For many of these applications and for others such as the electro-coalescence of multiple

droplets, a full 3D representation is required. While considerable amount of work has

been done on the 2D (and/or axisymmetric) EHD two phase flows, there appears to have

been little work reported on the full 3D phase field modeling for the EHD two phase flow

5

problems. An extension of 2D phase field to 3D phase field code can be involved in

practice, though theoretically it may appear straightforward. The data management and

CPU efficiency can be much more demanding for 3D coding and modeling. The

complexity is further compounded by the fact that for a 3D phase field based EHD model,

variables describing multiphysics needs to be solved simultaneously. In what follows, the

mathematical formulation and numerical development of the phase field model for the 3D

EHD flows of leaky dielectric fluids are described. Numerical and parallel computing

performances are discussed and the model validation is carried out by comparing with the

celebrated Taylor theory for droplet deformation in an electric field. The computed

results of 3D EHD two flows are selected from the examples of the electro-coalescence

of multiple droplets and of the EHD patterning processes for micro- or nano-structure

formation.

2. Mathematical Formulations

2.1 Phase field equations

In phase field the free energy density is represented by the phase parameter C, the form

of energy f : [0, 1] → R is taken as

2 21 21 1

12 4

f C C C C , ΩT : =Ω ×(0,T) (1)

where Ω is a bounded domain in R3, with a Lipschitz boundary ∂Ω, and C is the phase

parameter with the property that values of C=1 and C=0 correspond to the two distinctive

phases. Also, in the above equation, γ stands for the surface tension, ξ the interface

thickness and 6 2 a constant (Ding et al., 2007). The first term in Eq. (1) accounts for

the excess free energy due to the inhomogeneous distribution of volume fraction in the

6

interface region, whereas the second term represents the bulk energy density (Cahn and

Hilliard, 1958).

The Cahn-Hilliard equation with convection term, which is a result of the minimization

of the free energy, is employed here to describe the evolution of the phase field parameter

C (Jacqmin, 1999),

0C

u C Mt

, ΩT (2)

where u represents the fluid velocity, M is the phase field mobility and is considered as a

constant in this study, is the chemical potential which is defined f

C

. In this

equation, both convection and diffusion effects are included.

2.2 Electric field equations

For the EHD problems, the Poisson equation may be used to describe the electric field

distribution,

0e

r C V ΩT (3)

where ε0 is the permittivity of vacuum, εr(C) the dielectric constant, V the electric

potential, and ρe the free charge density. In the equation, the magnetic effect is neglected

since for the system under consideration, the electric field changes rather slowly, in

comparison with the diffusion of the electric field.

The free charge conservation can be expressed as (Saville, 1997),

e

eu C Et

ΩT (4)

where ζ(C) denotes the electrical conductivity, and E V is the electric strength. The

second term on the left represents the transport of the free charges by convection, and the

7

term on the right stands for the transport by electromigration. The diffusion effect is

ignored here (Saville, 1997).

There are two limiting cases for Eq. (4). First, if the time scale of charge relaxation

0 rt is much smaller than that of the flow c c ct L U (Lc being the length scale and

Uc the characteristic velocity), then Eq. (4) reduces to

0C V ΩT (5)

Thus, for fluids having a high electrical conductivity, Eq. (5) may be used (Hua et al.,

2008; Lin et al., 2012; Saville, 1997; Tomar et al., 2007). Second, for poorly conductive

materials, tσ>>tc. As a result, the free charge may be neglected and a pure dielectric

model is adequate to describe the electric field (Hua et al., 2008; Lin et al., 2012). For

this limiting case, Eq. (4) becomes redundant, and Eq. (3) recovers to the Laplace

equation. In this study, Eq. (4) is employed since it is a general description of electrical

charge conservation in an electric field.

2.3 Fluid flow equations

The flow field is described by the governing equations of mass and momentum

conservation with the momentum sources originating from the electric field and

interfacial energy. Both phases are considered to be incompressible. Hence the mass

conservation over the whole simulation domain, including both fluid phases and their

interface, can be expressed as (Ding et al.,2007)

0u ΩT (6)

In the EHD system, the electric and the surface tension forces on the interface should be

added. The modified Navier-Stokes equation for variable density and viscosity can be

written as

8

+ e

uC C u u f f

t

ΩT (7)

where the stress tensor П, the electric force ef due to polarization charge and free charge,

and the surface tension force f are given by the following expressions,

,

T

i jp C u u (8)

2 20 0 , 0

1 1

2 2

ee i j rf E E E E E C

(9)

f c (10)

In Eqs. (7) - (10), ρ(C) represents the density of fluid, p the pressure, μ(C) the viscosity,

δi,j identity matrix, and the chemical potential.

2.4 Nondimensionalization

The governing equations above may be nondimensionalized with the characteristic

length Lc, the velocity Uc, and the time tc= Lc/Uc. Here, Uc is determined by balancing the

electrical force and viscous force, Uc=ε0E02Lc/µ0 (Saville, 1997), where µ0 is the

characteristic viscosity which maybe µ1 or µ2 depending on the detail problem. The

nondimensionalization process produces the following dimensionless parameters,

0

0

c cU LRe

0 cU

Ca

c cL U

PeM

20 0r c

E

E LBo

c

CnL

(11)

The well-known Reynolds (Re) number describes the relative importance between the

inertial and the viscous force, the Capillary (Ca) number the relative magnitude of

viscous force and interface tension force, the Peclet (Pe) number the ratio of the

convective over diffusive mass transport, and the electrical Bond number (BoE) the ratio

of the electric over the surface tension force. The Cahn number (Cn) describes the ratio of

9

the interface thickness over the characteristic length. Cn is a reflection of how the

interface is resolved (Kim and Lu, 2006; Lin, 2012). When Cn →0, the phase field model

approaches the sharp interface limit (Jacqmin, 1999). The density, electrical permittivity,

conductivity, and viscosity ratios of the two fluids are: 1 2 , 1 2 , 1 2 ,

1 2 , respectively.

2.5 Numerical Method

The above equations are discretized and solved numerically along with appropriate

boundary conditions for specific problems given below. The finite difference numerical

scheme with explicit time-marching (Zikanov, 2010) is employed to obtain numerical

solutions. The first order terms are discretized by the upwinding scheme and all the

variables are defined at the collocated mesh points (Johnston and Liu, 2002). To speed up

the calculations, the code is enhanced by implementing a parallel computing algorithm.

After discretization, the Poisson equation (Eq. (3)) can be written in the form Ax=b,

where A coefficient matrix, x the unknown, b the force vector, and is solved by the

successive over-relaxation (SOR) method. The iteration follows the scheme:

x(k+1)

=(D+ωL)-1

ωb-[ωU+(ω-1)D]x(k)

, where D, L and U are the diagonal, lower and

upper triangular sub-matrices decomposed from matrix A. To speed up the convergence

rate, the relaxation coefficient is set at ω>1. The governing equation of free charge

conservation (Eq. (4)) is solved by the explicit finite difference scheme. The calculation

starts with the initial volume charge set to zero everywhere. The Navier-Stokes equation

(Eq. (7)) is solved by the standard projection method (Li, 2006; Zikanov, 2010), where

the resulting Poisson equation is solved also by the SOR method. The forth order finite

difference scheme is employed to solve the Cahn-Hilliard equation (Eq. (2)). It worth

10

pointing out that the material parameters εr(C), σ(C), ρ(C), and μ(C) may be taken as a

linear function of phase parameter C within the interface region. Numerical experience

suggests, however, that the harmonic interpolation yields a better, more consistent

approximation, when a large difference in density across the interface exists (Kim, 2012),

viz.

1 2 1

1 1 1 1=

2i i i

(12)

This is because in general, the harmonic interpolation gives a more accurate account of

flux balance over a cell within which an abrupt change in properties exits (Patankar,

1980).

3. Results and Discussion

3.1 Computational Details

Before the numerical results are presented, computational aspects concerning the

numerical accuracy warrant some discussion. Here, the mesh sensitivity is discussed first,

for which the case of the single leaky droplet deformation under the electric field is

chosen. The electric field configuration and the computational domain are shown in Fig.

1. For the droplet of radius R, a domain of (6R×6R×6R) is chosen for computation. Since

the deformation is symmetric about the planes of x=0 and z=0, a 1/4 domain is needed

(see Fig. 1b). For the problem, the characteristic length is chosen to be the radius of the

droplet, and the parameters for computations are: Re=1, Ca=0.2, Pe=1800, Cn=0.025,

BoE=0.2, λρ=1, λε=2, and λµ=1. For the results presented below, the droplet is labeled as

liquid 1 and the host medium liquid 2. The dimensionless parameters are scaled by the

11

properties of the fluid 2, with Lc=R, the radius of the sphere. This remains true henceforth

unless otherwise indicated.

The computed results are listed in Table 1 for various meshes. Apparently, a mesh of

81×161×81 gives a reasonably good accuracy and is thus used for the computations

discussed below. Studies suggested that a spontaneous shrinkage of the drop occurs if the

mesh is too coarse (Yue et al., 2007; Donaldson et al., 2011). This was also observed

with the simulations in the present study. For the given parameters above, the problem of

artificial droplet shrinkage will not occur with the mesh of 81×161×81.

For the finite difference solution of a two phase flow problem, the variable density

across the fluid-fluid interface can be modeled as either a linear function (Badalassi et al.,

2003; Kim, 2005) or a harmonic function (Khatavkar et al., 2007; Liu and Shen, 2003),

that is,

1 2 1C C C (D1), 1 2

1 1=

C C

C

(D2).

For the convenience of discussion below, D1 and D2 will be used to represent the linear

and harmonic approximations, respectively. Also one of the following interpolation

schemes is used to obtain the value of the density at the center of an element may be

interpolated from the values at the two side points,

1 2 1

1

2i i i (I1),

1 2 1

1 1 1 1=

2i i i

(I2),

where ρi and ρi+1 are the densities at the nodal points. Again for the convenience, I1 and

I2 will be used to denote the two interpolations, respectively.

It is known that for two phase flows with a large density ratio, the local imbalance

between the pressure gradient and the surface tension leads to spurious currents (Lafaurie

12

et al., 1994; Ni, 2009). For some cases, the spurious flow currents gradually disappear

with the time marching forward, as the diffusion effect eventually dominates within the

interface. However, if the surface tension effect is strong, these spurious currents will

exacerbate to cause a disruptive instability in flow simulations. This is further aggravated

if the density ratio of the two fluids is very large and the problem of spurious currents

will not simply disappear just with further refinement of mesh size. To assess the

numerical accuracy of different approximations, four different combinations, D1-I1, D1-

I2, D2-I1 and D2-I2, of the above interpolations are studied. As a testing case, a droplet

of perfect sphericity is considered, with both gravity and the external electric fields set to

zero and with R=0.18mm. Theoretically, the velocity should remain zero at t ≥ 0 for these

conditions. Numerically, however, the spurious currents may occur depending upon the

interpolation functions used for approximating the density variation in the fluid-fluid

interface region. Tests show that for a large density ratio, e.g., ρ1/ρ2=1000, the D1-I1

approximation scheme becomes unstable with time, leading to a disruption in

computation. Clearly, simple arithmetic interpolations do not bode well for a large

density ratio problem. All the other three schemes use the harmonic interpolations,

though differing in details; they give better results. Schemes D2-I1 and D2-I2 create

disturbances in velocity (i.e., spurious currents) initially but these disturbances eventually

disappear as simulation is allowed to continue for sufficiently long time. The D1-I2

approximation, however, performs the best and the spurious current is well suppressed

after a few time steps. These results are shown in Fig. 2.

3.2 Deformation of a Single Droplet (Model Validation)

It is well known that when a droplet and its surrounding fluid are both leaky dielectrics,

13

the droplet can deform into either a prolate ellipsoid or an oblate ellipsoid in a DC

electric field (Taylor, 1966). For the configuration shown in Fig. 1, an analytical

expression can be obtained by Taylor’s theory, which now is widely used as a benchmark

test for numerical computations (Hua et al., 2008; Lin et al., 2012; Tomar et al., 2007).

By this theory, the deformation of the droplet by the electric forces obeys the following

relation (Taylor, 1966; Torza et al., 1971),

1 2 1 2 1 2

2

1 2

/ , / , /9

16 2 /

dEfL B Bo

DL B

(13)

where L and B are the length of the parallel and perpendicular axes of the ellipsoid with

respect to the applied electric field, and the subscripts 1 and 2 denote the droplet and the

surrounding host fluid, respectively. The function, 1 2 1 2 1 2/ , / , /df , represents the

discriminating function,

2

1 21 1 1 1 1 1 1

2 2 2 2 2 2 2 1 2

2 33, , 1 2

5 1df

(14)

where the droplet takes a spherical shape for fd=0, a prolate shape for fd <0 and an oblate

shape for fd >0.

Computations were performed on a cube Ω = [0,l ]x[0,l ]x[0,l ] where l = 6R, and the

boundary conditions are taken as the same as in Lin et al. (2012), the difference being

that here the 3D calculations are considered. The boundary conditions are listed as

follows,

0 0 0,V V u y l T

0 0 0 0,V u y T

0 0 0 0 0,2 2

V l lu n x x z z T

n

14

0 0 0 0,p C

Tn n n

where ∂Ω is the boundary of computational domain. Initially the velocity is set to zero

and the phase field is C=1 for the droplet and C=0 elsewhere. Because of symmetric

conditions on the z=l/2 and x=l/2 planes, only a quarter of the domain needs to be

discretized. The calculations below used a mesh of 81x161x81. The original radius of the

droplet R=l/6, and it is also used as the characteristic length. Other parameters are: Re=1,

Ca=0.2, Pe=1800, Cn=0.025, BoE=0.2, λρ=1, λε=2 and λµ=1.

Fig. 3 compares the results obtained from the present 3D phase field calculations and

from the analytical solution by Taylor’s theory. As it is seen, for small droplet

deformations, the agreement between the 3D numerical and the analytical solutions is

excellent. For large deformations, however, the numerical model deviates from the

analytical solution. This is expected in that the Taylor theory is based on the small

perturbation analysis, which is valid only for small droplet deformations.

3.3 Transient Behavior of Electrically-induced Droplet Deformation

While the analytical solution is for a steady state deformation, the phase field model

presented is fully 3D and transient, which allows us to capture the transient behavior of a

droplet as it undergoes deformation. Knowledge on the transient development of droplet

deformation can be of great value in analyzing the underlying physics governing the

electrohydrodynamics of a deforming droplet. One interesting case is presented in Fig. 4,

where the final (or equilibrium) droplet shape is oblate. The droplet deformation,

however, for this case undergoes several different phases as the droplet deformation

transits in time to reach its equilibrium shape from its initial condition. The transient

15

development of the deformation experiences the following stages: it starts with a

spherical shape, evolves to a prolate ellipsoid, returns to spherical, and finally becomes

oblate. The detailed flow velocity field, flow streamlines and droplet deformation are

plotted in Fig. 4, displaying different stages of droplet deformations. The parameters used

for the calculations are: Re=4, Ca=0.8, Pe=7200, Cn=0.025, BoE=0.8, λρ=1, λε=5, λµ=1

and λζ=0.2. The conductivity of droplet and surrounding liquid are 4.0×10-11

S/m and

2.0×10-10

S/m, respectively. Analysis of the simulation data reveals that this transition in

deformed shapes is a result of electric charge interacting with the fluid flow field. At the

onset of the deformation process, there is no free charge and, the whole system behaves

like pure dielectric materials. The droplet deformation is dominated by the dielectric

forces, which stretches the droplet along the electric field and thus leads to a prolate

ellipsoidal shape. This has been shown by both experimental measurements and

theoretical analyses (Hua et al., 2008; Tomar et al., 2007). As the time goes by, free

charges starts to accumulate at the droplet-fluid interface as demanded by Eq. (4), and the

additional Coulomb force gradually comes into play, which acts to compress the droplet

along the two poles. As a result, the droplet returns to the spherical shape, and then

evolves continuously and eventually into the final oblate shape. It is important to stress

that the transition will not occur if Eq. (5) were used to model the free charge distribution.

This is because by Eq. (5) the free charge distribution is assumed to be established

instantaneously and no charge relaxation will occur.

3.4 Electro-Coalescence of Multi Droplets

The phenomenon of a dielectric droplet or particle moving in a non-uniform electric

field is referred to as dielectrophoresis (Pohl, 1978). By the theory of dielectrophoresis,

16

two dielectric droplets placed in the direction parallel to the electric field would attract

each other and coalesce. Two leaky dielectric droplets, however, can respond to an

applied electric field in different ways. They may be pushed apart or pulled together

depending upon the detailed flow and electric field conditions (Baygents et al., 1998).

The electro-coalescence of two droplets was simulated numerically by the phase field

model (Lin et al., 2012), assuming a 2D axisymmetry. In this study, the 3D phase field

simulation was conducted of the coalescence of four identical liquid droplets of radius R

immersed in a host fluid of different material properties under the influence of electric

field. The computational domain was (l×l×l/2) with l = 6R and initially the four droplets

were placed in the plane of z=l/4, as shown in Fig. 5. Two adjacent droplets were

separated by 0.5R in the x direction and R in the y direction. Because of symmetry, only

1/4 of the domain or (l/2×l×l/4) was used and discretized into a mesh of 101×201×51.

For the case shown in Fig. 5, λζ =0.25 and λε=2 were chosen, with the electrical

conductivities of the two phases being 5*10-9

S/m and 2*10-8

S/m, respectively. Other

parameters, and the boundary and initial conditions are the same as given in the previous

section.

The electro-coalescence involves the following three different interactions (Baygents et

al., 1998). First, the global electrical force on droplet exists due to the dielectrophoresis

effect, even though the droplets are free of native charge. This force causes the droplet

mass center to move in translation. Second, the flow circulation outsides the droplet

accompanying the internal circulation, which is produced by the distributed tangential

electric stress, will produce a hydrodynamic force. It either enhances or impedes the

coalescence process depends on the flow circulation pattern. Third, the local electric field

17

induces the droplet deformation and thus changes the distance of droplets.

Fig. 5 shows Computed results for the coalescence of four liquid droplets in an electric

field applied in the y (or vertical) direction. The droplets first deform into an oblate shape

(t*=54) by the leaky dielectric effect (Eq. (14)). The deformation elongates the droplets in

the x (or horizontal direction) and causes the collision of the two droplets in the same row

(t*=84). As a result, the four droplets are combined into two bigger ones (t

*=90), each of

which continues to deform (t*=120). These resulting two large droplets then gradually

move together (t*=348) and eventually coalesce into a single large droplet (t

*=354, 360,

420).

The above dynamically evolving events in essence are a manifestation of different

mechanisms driving the electrically-induced coalescence in the parallel and normal

directions of the applied electric field. Let us first consider the coalescence in the normal

direction. By symmetry, the like electric charges accumulate at the two droplets in the

same row. As the like electric charges repel each other, the two droplets are pushed away

from each other by the global electric force. The circulatory flows outside a droplet move

away from the droplet, which produces an action to push apart the two droplets. The only

action that favors the coalescence is the droplet deformation or elongation in the

horizontal direction, which reduces the separation between the droplets in the same row

and eventually causes them to collide with each other.

For the electro-coalescence in the direction parallel to the electric field, however, the

situation is different. The global electric force favors translation of the drops towards

each other, because the opposite charges induced on the nearest surfaces of two droplets

produce an attraction between the two. Since λζ < λε, the circulatory flow outside the

18

droplet, which is caused by local tangential electric stress, moves towards the droplet and

thus enhances the coalescence. One the other hand, by Eq. (14), the droplets deform into

an oblate ellipsoid and this oblate deformation takes place along the horizontal direction,

which increases the clearance between the two droplets aligned in the vertical direction

(or parallel to the electric field) and thus further separates the droplets apart, an action

unfavorable for coalescence. The droplets eventually coalesce under a combined action

of the global electric force and the circulatory flow, which overpowers the unfavorable

effect caused by deformation.

Fig. 6 shows the 3D flows at the dimensionless times t*= 84 and t

*=348 at which the

coalescence is taking place in the normal and parallel directions, respectively. For a

single droplet at equilibrium, the circulatory flows are present inside and outside the

droplet. Similar flow structures are also observed in Fig. 4c, and also in 2D calculations

(Hua et al., 2008; Lin et al., 2012; Taylor, 1966; Tomar et al., 2007). This is caused by

the tangent electric stress (Saville, 1997) and occurs with the droplets placed far from

each other. The corresponding flow structures also evolve, as the droplets are pulled

together. As shown in Fig. 6a and its accompanying 3D views, when two droplets are in

contact (t*= 84), the circulatory flows inside the droplets are diminished. The flow

streamline illustrates that the two droplets move towards each other and the coalescence

is manifested by the exchange of the fluid between the two droplets. Fig. 6b shows the

evolution of the 3D flow structures both inside and outside the droplets just before these

two larger droplets coalesce together. It is important to note that in order to observe the

coalescence in the direction normal to the applied field, the droplets need to be placed

close enough to each other so that deformation-caused collision occurs to affect the

19

coalescence. On the other hand, as it was demonstrated by simulation, if the droplets are

placed horizontally (or normal to the applied field) at a distance of R, they will drift apart

from each other from their initial positions.

3.5 Electrohydrodynamic Patterning

The EHD Patterning is a novel technique to produce the controllable polymer

micro/nano-patterns (Chou et al., 1999; Chou and Zhuang, 1999; Schaffer et al., 2000;

Wu and Russel, 2009; Yang et al., 2013). In this process, the polymer/air or

polymer/polymer film is sandwiched between a pair of electrodes as shown in Fig. 7a,

with the upper-electrode (i.e. template) being patterned (e.g. Fig. 7b). The polymer/air

interface deforms according to a modulated electrical force resulting from the patterned

structure, thereby producing a micro- or nano-structure that conforms well with the

template. Usually the scale of EHD patterning is on the order of hundreds of nanometers

or larger. At this scale, the continuum assumption is still valid (Koplik and Banavar,

1995). The 3D phase field model, presented above, can be used to simulate the EHD

patterning process (Kim and Lu, 2006a; Kim and Lu, 2006b; Tian et al., 2011). The

characteristic length is chosen as the distance d0 of the two electrodes (see Fig. 7a). The

properties of the air and polymer differ dramatically, and for modeling the air/polymer

system, λρ=1000, λε=3, λµ=1000 and λζ=104 are chosen. The conductivity for polymer is

taken as 10-5

S/m. The template is patterned with a periodic array of cubic protrusions, as

shown in Fig. 7b. The patterned feature has a periodic length l0 of 200nm, with

d0=100nm, Δd=50nm, and the cross section of the protrusion being 100nm*100nm.

Taking the advantage of the periodicity, the 3D simulations were conducted within a

computational domain of l0×(d0+Δd)×l0, which is the size of one extruded cubic structure.

20

The periodic conditions are applied in the horizontal directions. The equi-potentials and

the no slip flow conditions are set at the electrodes, with the bottom electrode acting as

the ground electrode. The simulation used a mesh of 101*76*101.

Fig. 8 illustrates the transient development of structures as they occurs in the EHD

patterning process. The initial polymer/air interface is flat and the interface starts to

deform under the action of the electric force between the polymer surface and the

template, which is generated by the electric voltage applied between the two electrodes.

This is shown in Fig. 8b. The onset of the perturbation in the interface continues to grow

under the action of the electric force, as illustrated in Figs. 8c and 8d, forming the

structure in conformation with the pattern on the template. The structure is eventually

pulled by the electric force until the top of the polymer structure touches the template and

then flattened out by the template (see Figs. 8e and 8f). In Figs 8d and e, four ridges on

each patterned structure can be observed, each of which corresponds to one corner of the

cubic extrusion on the template. Fig. 8f is the top-down view of Fig. 8e, which shows the

EHD patterned structure conforms well with the template. Simulations also show that the

growing rate of the structure height accelerates with time, which is attributed to the fact

that the electric force becomes stronger when the polymer/air interface is pulled closer to

the template.

Detailed flow pattern during the process is depicted in Fig. 9, where the electric force

pulls up the polymer layer towards the upper electrode. As the polymer fluid rises up in

the middle, the fluid far away from the extruded electrode is depressed because of

requirement for mass conservation. As is evident by the flow steam lines, a large rotating

eddy entrains the liquid to the center region. Fig. 9b shows the corresponding 3D flow

21

structure, which indicates that a full 3D model is necessary in faithfully representing the

EHD patterning process for the micro- or nano-structure formation.

3.6 Parallel Computing

The 3D phase field model for the EHD flow problem described above can be

computationally very intensive. Parallel computing is applied to improve the computing

efficiency. Parallel computing performance of the code was evaluated with the EHD

patterning problem as the testing case. The in-house Fortran code for the 3D phase field

modeling was parallelized with the openMP paradigm. The test used the same mesh of

101*76*101 and other parameters are also kept the same as in the previous section.

Computations were done on a AMD Opteron Linux platform, which has 48

microprocessors with 64GB Ram capable of high performance parallel computing. The

same platform was also used to obtain the computed results presented above.

The parallel computing performance for the 3D phase field model calculations is shown

in Fig. 10. Clearly, the more numbers of CPUs are used, the less time is required for

computation. However, it appears that the most drastic time reduction occurs with the

increase up to 10 CPUs, which give an improvement of 6 times in computational

efficiency. A further increase in number of the CPU processors becomes less effective,

largely because the CPU time is increasingly consumed by the inter-communication and

data-exchange between different processors. By using 10 CPUs, one EHD patterning case

would take several days to finish.

22

4. Conclusions

This paper has presented a phase field computational methodology for the numerical

solution of 3D electrohydrodynamic two phase flow problems. The mathematical

formulation consists of a set of coupled nonlinear equations, including the Navier-Stokes

equations for fluid motion, the convective Cahn-Hilliard equation for the free surface and

phase field, and the Poisson equation for the electric field distribution in pure and/or

leaky dielectric fluids. These equations are discretized by the finite difference method

along with the explicit time marching scheme, enhanced by a parallel computing

algorithm. For the problems of two fluids with a large disparity in fluid densities, an

appropriate interpolation of densities across the free surface is essential to ensure the

numerical stability. Numerical efficiency can be drastically improved via parallel

computing, but the most gains are obtained with the use of up to 10 CPUs, when the code

is parallelized with openMP.

The 3D model is validated by comparing with the Taylor model for a droplet

deformation in an electric field. It is found that while a leaky dielectric droplet deforms

into an oblate ellipsoid as its final equilibrium shape in an electric field, the transient

development in deformation entails the transition from the initial spherical to the prolate

and then to the intermediate spherical and eventually the final oblate shape. This is

different from the deformation behavior of a pure dielectric droplet, the difference being

attributed to the relaxation of free charges in the interface region. The simulation for the

coalescence of four droplets in an electric field reveals a truly 3D deformation and fluid

flow field. Different mechanisms of electro-coalescence in the directions parallel and

perpendicular to the electric field are discussed. The usefulness of the 3D phase field

23

model is further illustrated by its application in modeling the electrically-induced two

phase flow phenomena associated with an electrohydrodynamic patterning process for

micro- or nano-structure formation.

Acknowledgments

This work is supported by NSFC (Grant no. 90923040) and the National Basic Research

Program of China (Grant no. 2009CB724202). The first author also acknowledges the

partial support of the University of Michigan at Dearborn as a visiting Ph.D. student.

Helpful discussion with Professor H. A. Stone of Princeton University on the subject of

electro-coalescence is also gratefully acknowledged.

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Table 1. The mesh sensitivity of leaky droplet deformation*

Mesh Deformation

(D) Error

51×101×51 0.062658 12.7%

61×121×61 0.070084 2.36%

81×161×81 0.070191 2.21%

91×181×91 0.071029 1.04%

101×201×101 0.071779

* The case 101×201×101 is used as a base for comparison

Fig. 1. Schematic illustration of an EHD two phase flow problem: (a) a droplet in an

electric field generated by two electrodes and (b) the computational domain.

28

Fig. 2. Spurious currents at time t*=10

-3 as calculated by different approximation

schemes. The parameters used for calculations (1–droplet and 2–host fluid): λρ=1000,

λµ=1000, Ca=0.18. All these figures are shown at the cross section z=0. It should be

mentioned that the velocity is characterized by Uc=µ1/ρ1R for this case.

Fig. 3. Comparison of the deformation D between the analytical solutions and the 3D

phase field numerical results for BoE=0.2, λε=ε1/ ε2=2 and λµ=µ1/ µ2=1.

29

Fig. 4. Transient development of droplet deformation, where the cross section at z=0 is

shown rather the 3D shape along with the fluid flow field and phase field. The initially

spherical shape is spherical (not shown). The deformation undergoes through the prolate

first (a), then the spherical (b), and then eventually the oblate shape (c). The

dimensionless time for (a-c) are t*=1.8, 8.46 and 360, respectively. The corresponding

deformation parameters are: D=0.14, D≈0 and D=−0.26.

Fig. 5. Snap shots of the coalescence process of four droplets in an electric field (applied

in the vertical direction) at times: t*= 0, 54, 84, 90, 120, 348, 354, 360, and 420. The

parameters used for computations: Re=4, Ca=0.8, Pe=7200, Cn=0.025, BoE=0.8, λρ=1,

λε=2, λµ=1 and λζ=0.25.

30

Fig. 6. 3D view of the flow field distribution as the four droplets undergo electro-

coalescence and at two different times: t*= 84 (a) and t

*=348 (b).

31

Fig. 7. (a) Configuration for the EHD patterning and (b) the shape of the template.

Fig. 8. Numerical simulation of the EHD nanopatterning process – evolution of the

polymer/air interface as it deforms by an applied electric field. The corresponding

dimensionless times for (a) to (e) are: t*=0, 8, 12, 16, and 20.

32

Fig. 9. The flow field and the steamline for the EHD nanopatterning at t*=16, as

computed from the 3D phase field model: (a) the cut-pane view at the middle plane, and

(b) the 3D flow structure.

Fig. 10. Parallel computing performance of the 3D phase field model for EHD two

phase flow problems: computing time per time step vs. number of CPUs. Inset shows

the log-log plot.

Highlights

We present a 3D phase field model for two phase electrohydrodynamic flows.

The transient behavior of a leaky droplet deformation under electric field is investigated.

We study the four droplets electro-coalescence

We compute electrically-induced two phase flow associated with nanostructure patterning.

*Highlights (for review)