Motivated by the use of electrostatic assist to improve liquid transfer in gravure printing, we use theory and experiment to understand how electric fields deform thin liquid films near surfaces with cavity-like topographical features. Lubrication theory is used to describe the film dynamics, and both perfect and leaky dielectric materials are considered. For sinusoidal cavities, we apply asymptotic methods to obtain analytical results that relate the film deformation to the other problem parameters. For trapezoidal-like cavities, we numerically solve evolution equations to study the influence of steep topographical features and the spacing between cavities. Results from flow visualization experiments are in qualitative agreement with the theoretical predictions. In addition to being relevant to printing processes, the model problems we consider are also of fundamental interest in and represent novel contributions to the areas of electrohydrodynamics and thin-liquid-film flows.

## I. INTRODUCTION

Electrohydrodynamics is a discipline which describes the interaction of fluids with electric fields. A rigorous understanding of electrohydrodynamic phenomena was developed following the pioneering efforts of Taylor^{1} and Melcher.^{2} Many studies since have examined the influence of electric fields on the shape and stability of liquid interfaces in structures such as liquid bridges,^{3–14} drops,^{1,15–23} jets,^{24–30} and thin films.^{31–34} Electrohydrodynamic flows near interfaces are relevant to applications such as soft lithography,^{31,35–45} manipulation of capillary ridges in coating flows,^{46–50} electrospraying,^{51} electrospinning,^{52} inkjet printing,^{53,54} and improvement of mixing and heat transfer.^{55,56}

Motivated by an application known as electrostatic assist (ESA), we study in this paper the electrohydrodynamic deformation of thin liquid films near surfaces with topography. ESA involves application of an electric field to gravure printing processes in order to remove liquid from cavities tens of microns in size.^{57–59} Gravure is a high-speed roll-to-roll printing process that can be adapted to print electronic devices.^{60–62} A metal cylinder engraved with cavities (gravure roll) picks up ink from a liquid pool. A flexible blade wipes off excess ink from the cavities, leaving behind a thin liquid film on the regions between the cavities, which are known as lands. The ink in the cavities is then transferred to a flexible substrate (Fig. 1(a)). The substrate, known as a web, is often backed by another roll (backing roll). In ESA, a conductive covering is present on the backing roll and an electrostatic potential difference (∼10–100 kV) is applied between it and the gravure roll. The resulting electric field tends to increase the amount of liquid transferred from the cavities to the substrate, leading to significant improvements in print quality (Fig. 1(b)). Despite the importance of ESA, it is not well understood at a fundamental level.

A starting point for developing this understanding is to consider how electric fields deform thin liquid films near surfaces with cavity-like topographical features. In this paper, we develop and study models for such deformations and consider both sinusoidal cavities and trapezoidal-like cavities. Using the former geometry, we apply asymptotic methods to obtain analytical results that relate the film deformation to the other problem parameters. Using the latter geometry, we apply numerical methods to study the influence of steep topographical features and the spacing between cavities. Both perfect dielectric (non-conducting) and leaky dielectric (weakly conducting) liquids are considered. In addition to being relevant to ESA, the model problems we consider are also of fundamental interest in and represent novel contributions to the areas of electrohydrodynamics and thin-liquid-film flows.

In Sec. II of this paper, we present the mathematical model and governing equations. Results for sinusoidal cavities are presented in Sec. III, and results for trapezoidal-like cavities are presented in Sec. IV. Some related experiments have also been performed and these are discussed in Sec. V. Finally, conclusions are given in Sec. VI.

## II. MATHEMATICAL MODEL AND GOVERNING EQUATIONS

We consider the two-dimensional model geometries (sinusoidal and trapezoidal-like) shown in Fig. 2. The cavity is maintained at constant potential *V* and is filled with a Newtonian liquid of constant density *ρ*, viscosity *μ*, and surface tension *γ*. Before application of a potential difference, the liquid is initially at a height *H* above the cavity and has a flat interface with the air above it. The air initially has a thickness *βH* and is bounded from above by a top electrode, which is taken to be grounded. The *x*-*z* coordinate axes lie on the initially flat liquid-air interface, and the cavity is located at *z* = *f*(*x*). The gravitational acceleration is given by ** g** = −

*g*

*e*_{z}, where

*e*_{z}is the unit vector in the

*z*-direction and

*g*is the magnitude of

**. For generality, we will include both gravitational and van der Waals forces in our model.**

*g*Liquids exhibit a wide range of electrical properties, and here we model two common types: perfect dielectrics (polarizable, uncharged, and non-conductive) and leaky dielectrics (polarizable, with surface charge and low conductivity).^{2,15} Air is modeled as a perfect dielectric, with dielectric constant *ϵ*_{1} = 1. We consider cases of both perfect and leaky dielectric liquids with dielectric constant *ϵ*_{2} = *ϵ* and conductivity *σ*. Finally, we represent the height and interfacial charge by *h*(*x*, *t*) and *q*(*x*, *t*), respectively, where *t* denotes time.

### A. Governing equations

Laplace’s equation describes the electrostatic potential *ψ _{i}* in each phase

where *i* = 1 for air and *i* = 2 for liquid. The boundary conditions at the top electrode and cavity are

At the liquid-air interface, we have continuity of the potential

and a jump condition on the electric field

Here, the jump operator ‖....‖ provides the difference between quantities in phases 2 and 1, e.g., ‖*ϵ _{i}ϵ*

_{0}

**∇**

*ψ*‖ =

_{i}*ϵ*

_{2}

*ϵ*

_{0}

**∇**

*ψ*

_{2}−

*ϵ*

_{1}

*ϵ*

_{0}

**∇**

*ψ*

_{1}. In addition,

**is the unit normal vector at the interface pointing into the air, and**

*n**ϵ*

_{0}is the permittivity of free space.

The fluid dynamics in the air are neglected, whereas in the liquid, we have the standard equations governing mass and momentum conservation

where **v** is the velocity vector and *T*_{2} is the total stress tensor for the liquid. We take *T*_{i} = − *p _{i}*

**+**

*δ**μ*[

_{i}**∇v**+ (

**∇v**)

^{T}] +

*M*_{i}, where

*p*is the pressure,

_{i}**is the identity tensor,**

*δ**μ*

_{2}=

*μ*, and

*μ*

_{1}= 0. The Maxwell stress tensor is given by $ M i = \u03f5 i \u03f5 0 [ E i E i \u2212 1 2 ( E i E i ) ] \delta $, where the electric field

*E*_{i}= −

**∇**

*ψ*,

_{i}*ϵ*

_{2}=

*ϵ*, and

*ϵ*

_{1}= 1. The divergence of

*M*_{i}is zero for perfect and leaky dielectric materials,

^{15}so Eq. (7) reduces to the Navier-Stokes equations. Thus, the coupling between hydrodynamics and electrostatics occurs through the boundary conditions at the liquid-air interface.

We apply the no-slip and no-penetration conditions for ** v** at the cavity

At the liquid-air interface, we have the normal stress balance

where *κ* = **∇**_{s}⋅** n** = [(

**−**

*δ***)⋅**

*nn***∇**]⋅

**is the interfacial curvature and**

*n***∇**

_{s}is the surface gradient operator. The tangential stress balance at the liquid-air interface is given by

where ** t** is the tangent vector to the interface.

The temporal evolution of interfacial height and charge is described through the kinematic condition and charge conservation equations

Here, *v _{x}* and

*v*are the two velocity components and all terms are evaluated at the liquid-air interface

_{z}*z*=

*h*(

*x*,

*t*). For perfect dielectrics, Eq. (12) vanishes because

*σ*= 0 and

*q*(

*x*,

*t*) = 0. In this limit, free charge is absent and the liquid can no longer act as a conductor, although it can be polarized by the electric field.

^{15}

### B. Scalings

We first non-dimensionalize the governing equations in Sec. II A by choosing the scales (*L*, *H*, *T*, *V*, *Q*, *W*, *U*, *P*) for the eight system variables (*x*, *z*, or *h*, *t*, *ψ _{i}*,

*q*,

*v*,

_{x}*v*,

_{z}*p*). We then invoke the thin-film (i.e., lubrication) approximation by assuming

*H*/

*L*≪ 1 and neglecting terms of

*O*(

*H*/

*L*) and higher. We must use this simplification with caution because steep slopes in the cavity could invalidate it by creating steep gradients of quantities in the

*x*-direction. However, we note that in past studies on problems involving thin-film flows near cavities, the lubrication approximation has been shown to work quite well even when the cavity slopes are steep.

^{63–66}For brevity, the scaled governing equations are given in the supplementary material.

^{67}The liquid height

*H*and the applied potential

*V*are the only scales that are known at the outset, and the other scales are determined by balancing terms in the governing equations. The jump condition (5) sets the charge scale as

*Q*=

*ϵ*

_{0}

*V*/

*H*. The mass-conservation equation (6) yields

*W*=

*UL*/

*H*as the velocity scale.

For films of thickness ∼100 nm or less, van der Waals forces are known to become significant, creating disjoining pressures that could be positive or negative depending on the materials in the system.^{68–70} To model this case, we incorporate van der Waals forces into the lubrication form of the normal stress balance (9), similar to previous work^{71–74}

where *A*_{1} = ∂*ψ*_{1}/∂*z* and *A*_{2} = ∂*ψ*_{2}/∂*z*. The above equation is given in dimensional form with *A*^{∗} being the dimensional Hamaker constant. Dividing through by $ \u03f5 0 V 0 2 / H 2 $ casts (13) into a convenient dimensionless form

where *p*, *h*, *x*, *f*, *A*_{1}, and *A*_{2} are all dimensionless. The first, second, and third terms contain the ratios *P*/(*ϵ*_{0}*V*^{2}/*H*^{2}), (*γH*^{2}/*L*^{2})/(*ϵ*_{0}*V*^{2}/*H*^{2}), and (*A*^{∗}/*H*^{3})/(*ϵ*_{0}*V*^{2}/*H*^{2}), which determine the importance of pressure, capillary, and van der Waals forces relative to electrostatic forces. Since we are focusing on electrohydrodynamically driven flows, we choose a pressure scale based on the electrostatic stress and set *P* = *ϵ*_{0}*V*^{2}/*H*^{2}. The second ratio (*γH*^{2}/*L*^{2})/(*ϵ*_{0}*V*^{2}/*H*^{2}), equals the inverse of the electric capillary number, defined as *Ca _{E}* = (

*ϵ*

_{0}

*V*

^{2}/

*H*

^{2})/(

*γH*/

*L*

^{2}). The third ratio is a dimensionless Hamaker constant

*A*= (

*A*

^{∗}/

*H*

^{3})/(

*ϵ*

_{0}

*V*

^{2}/

*H*

^{2}).

Balancing terms in the components of the momentum conservation equation (7) leads to the generation of two more dimensionless groups. These are the electro-viscous number *EV* = *ϵ*_{0}*V*^{2}/*μWL*, which is the ratio of electrostatic to viscous forces, and the gravity number *Gr* = *ρgH*^{3}/*ϵ*_{0}*V*^{2}, which is the ratio of gravitational to electrostatic forces. We note that *Ca _{E}* sets the length scale

*L*, and

*EV*sets the velocity scale

*W*. Unless otherwise noted, we set

*Ca*and

_{E}*EV*to 1. We note that

*Ca*= (

_{E}*L*/

*H*)

^{3}

*Ca*, where

*Ca*=

*μW*/

*γ*is the standard capillary number. The standard lubrication approximation assumes (

*L*/

*H*)

^{3}

*Ca*∼ 1,

^{71}consistent with our choice of

*Ca*. The characteristic time scale can be inferred from the kinematic condition (11) as

_{E}*T*=

*H*/

*U*. The other dimensionless parameters of importance are

*β*, which denotes the initial air gap relative to the initial liquid thickness, and the dimensionless conductivity

*S*=

*Lσ*/(

*Uϵ*

_{0}), which shows up in the charge conservation equation (12).

### C. Evolution equations

In the lubrication limit, the governing equations can be reduced to a pair of coupled nonlinear partial differential equations

The functions *f*_{1}(*h*(*x*, *t*), *q*(*x*, *t*), *f*(*x*)) and *f*_{2}(*h*(*x*, *t*), *q*(*x*, *t*), *f*(*x*)) are extremely lengthy and given in the supplementary material.^{67}

Examination of evolution equations (15) and (16) shows that when *A* = 0, they are exactly satisfied by *h*(*x*, *t*) = 0 and *q*(*x*, *t*) = 1/*β* for leaky dielectric liquids for any cavity shape *f*(*x*). This means that the presence of topography on the substrate has no influence on the shape of a leaky dielectric interface. It can be shown that at steady state, the entire leaky dielectric liquid is at a constant potential *V*(*x*, *t*) = 1 because the interfacial charge suppresses the electric field within the liquid.^{15,75} Consequently, all the potential drop occurs in the air between the liquid-air interface and top electrode, creating a uniform normal electric field of strength 1/*β* in the air gap. There is thus a uniform electrostatic stress at the liquid-air interface, which can lead to interfacial instability in systems where topography is absent, as studied in earlier related literature.^{33,34,75} Therefore, for leaky dielectrics, we expect that a flat liquid-air interface may be unstable even though it is a steady solution when topography is present. In contrast, a flat interface is not a steady solution for perfect dielectrics when topography is present (*f*(*x*)≠0), implying that interface deformation is always expected on application of an external potential difference.

## III. DEFORMATION OF LIQUIDS NEAR SINUSOIDAL CAVITIES

### A. Problem statement and methods

We consider a cavity with a sinusoidal geometry (Fig. 2(a)) described by

where *ϕ*_{1} is a parameter that controls the depth of the cavity and *k* controls the periodicity. When, *ϕ*_{1} = 0, the problem reduces to the well-studied case of a thin liquid film on a flat substrate, for which the steady-state solution for leaky dielectrics is *h*(*x*) = *h*_{0} = 0 and *q*(*x*) = *q*_{0} = 1/*β*. For perfect dielectrics, we simply have *h*(*x*) = *h*_{0} = 0. These solutions are stable only when *Gr* > *Gr _{c}*, where

*Gr*is a critical gravity number. Standard linear stability analysis can be employed to derive an analytical expression for

_{c}*Gr*when

_{c}*ϕ*

_{1}= 0 (see Ref. 76 and supplementary material

^{67}). These expressions also appear in the results from our asymptotic analysis, as will be discussed below.

For |*ϕ*_{1}| ≪ 1, asymptotic methods can be applied to obtain analytical expressions for steady interface shapes and thus gain physical insight into the interface deformation. Taylor series are applied to expand the interfacial height and charge around *h*_{0} and *q*_{0}

Here, $ h 1 ( x ) = \u2202 h ( x , \varphi 1 ) / \u2202 \varphi 1 \varphi 1 = 0 $, $ h 2 ( x ) = \u2202 2 h ( x , \varphi 1 ) / \u2202 \varphi 1 2 \varphi 1 = 0 $, $ q 1 ( x ) = \u2202 q ( x , \varphi 1 ) / \u2202 \varphi 1 \varphi 1 = 0 $, and $ q 2 ( x ) = \u2202 2 q ( x , \varphi 1 ) / \u2202 \varphi 1 2 \varphi 1 = 0 $. The expressions in (18) and (19) are substituted into the evolution equations (15) and (16) at steady state to yield

For perfect dielectrics, *C*_{1} = 0 and *C*_{2} = 0 for all *f*(*x*). We have presented detailed expressions for *K*_{1}, *K*_{2}, *C*_{1}, and *C*_{2} in the supplementary material^{67} for the specific case of a sinusoidal cavity (Eq. (17)) and also for arbitrary *f*(*x*). If a solution exists for Eqs. (20) and (21) for arbitrary non-zero *ϕ*_{1}, we must have *K*_{1} = 0, *K*_{2} = 0, *C*_{1} = 0, and *C*_{2} = 0. These ordinary differential equations are solved analytically using Mathematica to yield *h*_{1}(*x*), *h*_{2}(*x*), *q*_{1}(*x*), and *q*_{2}(*x*).

We also solve the coupled evolution equations (15) and (16) numerically to calculate steady interface shapes in order to compare them with results from the asymptotic study. The initial condition is always assumed to be a flat uncharged interface and the interface is tracked with time until it reaches steady state. Periodic boundary conditions are imposed at the ends of the domain. The equations are discretized over the domain with a fourth-order centered finite-difference scheme, and the time integration is done with a commercial package DDASPK 3.0.^{77} A domain length of 2*π* was chosen to simulate a region corresponding to a single sinusoidal cavity, and 500 nodes uniformly distributed across the length were sufficient for mesh-independence of the results. Mass conservation was also checked to ensure that all simulations have a net mass change of less than 0.01%.

### B. Results

The first-order solution for perfect dielectrics is

where we have used the subscript *p* to denote perfect dielectric. The second-order solution is

where *D*_{1}, *D*_{2}, *D*_{3}, *D*_{4}, and *D*_{5} are given in the Appendix. When *A* = 0, Eq. (23) simplifies to

Inspection of these expressions provides physical insight into the interface deformation. The first-order expression (22) is directly proportional to cos(*kx*), which means that the interface deformation will mimic the cavity topography for shallow cavities. As the cavity becomes deeper, the second-order correction (23) becomes more important and brings with it a dependence on cos(2*kx*), which has double the wavenumber of the cavity. The cos(2*kx*) function is out of phase with the cos(*kx*) function at the center of the cavity. Therefore, as *ϕ*_{1} increases, we expect the interface shape to deviate from an almost perfect cosine shape to one which is flatter at the center.

We also note that the expressions (22) and (23) become singular when *Gr* = *Gr*_{c,p} = ((−1+*ϵ*)^{2}*ϵ* + 3*A*(1+*βϵ*)^{3} − *k*^{2}(1+*βϵ*)^{3})/(1+*βϵ*)^{3}, which correspond exactly to the perfect dielectric linear stability limit when topography is absent (i.e., *ϕ*_{1} = 0). This is expected because the asymptotic expansions hold only when *Gr* > *Gr*_{c,p}, and begin to break down close to this limit. The expression for *Gr*_{c,p} reflects a balance between destabilizing electrostatic forces, stabilizing surface-tension and gravitational forces, and van der Waals forces (stabilizing if *A* < 0 and destabilizing if *A* > 0). In the linear stability analysis, *k* corresponds to the wavenumber of the disturbance to the liquid-air interface, whereas in the asymptotic analysis presented here, *k* corresponds to the wavelength of the substrate topography. The value of *Gr* = *Gr*_{c,p} is largest for *k* = 0; non-zero values of *k* lead to smaller values of *Gr* = *Gr*_{c,p}. The smaller values of *Gr*_{c,p} correspond to larger potential differences needed to destabilize the interface, which arise because stabilizing surface-tension forces become stronger with smaller wavelengths (larger *k*).

In Fig. 3, we compare the asymptotic predictions with numerical solutions of the full evolution equation (15). (For perfect dielectrics the charge evolution equation (16) is absent.) In principle, the asymptotic solutions are expected to work accurately only for small *ϕ*_{1}. We have tested their performance for *ϕ*_{1} ≪ 1 and *ϕ*_{1} ∼ 1. Figure 3(a) shows that the first-order expressions are able to capture the interface shape when *ϕ*_{1} ≪ 1, and the second-order correction further improves the accuracy of the match. As *ϕ*_{1} is increased so that *ϕ*_{1} = 1, the second-order terms contribute significantly to the interface shape (Fig. 3(b)), but substantial deviations from the simulations appear. When *ϕ*_{1} is further increased to a value of 2, the second-order asymptotics cannot even qualitatively capture the shape. The second-order correction dominates over the first-order term and we see an inversion of the predicted interface profile at the center (Fig. 3(c)). The accuracy of the asymptotic approximations can be improved by including higher-order terms in the Taylor expansion in (18) and (19) but these have not been worked out due to their length.

For leaky dielectrics, the first-order asymptotic expressions are

where we have used the subscript *l* to denote leaky dielectric. Second-order expressions can also be formulated and have been included in the supplementary material^{67} due to their length.

An examination of the relatively simple first-order expressions (25) and (26) provides valuable insight into the effect of conductivity. In the absence of van der Waals forces (*A* = 0), the first-order corrections become zero. The second-order corrections can also be shown to reduce to zero. Hence, a steady leaky dielectric interface is found to remain flat with a uniform base-state charge *q*_{0}(*x*) = 1/*β*. This makes the steady state in the presence of a cavity exactly the same as that of a thin film on a flat surface, which is in accordance with our earlier observations in Sec. II C. However, the stability of the flat interface is a function of *Gr*. The first-order asymptotic expressions become singular at *Gr* = *Gr*_{c,l} = (1 + 3*Aβ*^{3} − *k*^{2}*β*^{3})/*β*^{3}, which correspond to the critical gravity number calculated by linear stability analysis when *ϕ*_{1} = 0. Thus, the asymptotic expansions are valid only for *Gr* > *Gr*_{c,l}, similar to the situation with perfect dielectrics discussed earlier. We note that although the first-order expressions are independent of the conductivity and dielectric constant, these quantities appear in high-order corrections. This is consistent with the fact that for leaky dielectrics, the potential difference in the air is much larger than that in the liquid and it is this difference that drives the interface deformation.

The first-order expressions also provide insight into the role of van der Waals forces in interface deformation. In Eqs. (25) and (26), the numerators are proportional to *A* and thus reflect the influence of van der Waals forces. The denominators consists of four terms corresponding to contributions from electrostatic, van der Waals, gravitational, and capillary forces, respectively. Thus, the expressions in Eqs. (25) and (26) provide a measure of the relative importance of van der Waals forces with respect to the sum of all the forces acting on the liquid film. In the limit of small *A*, these expressions tend to zero. In the limit of large *A*, we recover *h*_{1l}(*x*) = cos(*kx*) and *q*_{1l} = cos(*kx*)/*β*^{2}, which indicates that the dependence of deformation on the van der Waals forces diminishes as those forces increase in magnitude.

We note that for typical values of the dimensional Hamaker constant (∼10^{−20}–10^{−19} J), thin films of thickness ∼100 nm, and applied potential of ∼1 to 10 kV, *A* is very large, ∼10^{10} and higher, while *Gr* is extremely small, ∼10^{−13} to 10^{−11}. Therefore, in this limit, electrostatic effects are not significant unless the applied potential is extremely high, ∼10^{9} V or more. Also, gravitational forces are negligible relative to the other forces and for all practical purposes *Gr* = 0. Thus, the asymptotic study reveals that for shallow cavities (when the asymptotic results are most accurate), one would expect to see an interface profile largely unaffected by electrostatic effects or the magnitude of *A*. Since we are primarily interested in probing electrostatic effects in this system, we set *A* = 0 in our analysis for the rest of this paper and consider moderately thick films of ∼1*μ*m and higher. For a film thicknesses of 1, 10, and 100 μm, potential differences of ∼0.03, 1, and 30 V, respectively, correspond to *O*(1) values of *Gr*.

## IV. DEFORMATION OF LIQUIDS NEAR TRAPEZOIDAL-LIKE CAVITIES

We now turn to the case of trapezoidal-like cavities (Fig. 2(b)) to study the influence of steep topographical features and the spacing between cavities. In addition, trapezoidal-like cavities are often used in gravure printing. To create trapezoidal-like shapes, we employ arctangent functions, *f*(*x*) = (*Y*(arctan((−*G*/2 + *x*)/*D*) − arctan((*G*/2 + *x*)/*D*)))/*π* − 1.0. The advantage of using arctangent functions is that the cavity depth (*Y*), steepness (*D*), and width (*G*) can be varied independently. Arctangent functions have been used in prior studies to systematically study the influence of surface topography on thin-film flows.^{46,48,63,64,72,78–80}

The nonlinear evolution equations (15) and (16) are solved numerically as described in Sec. III A. Unless otherwise noted, a domain length of 54 is chosen and 3000 uniformly distributed nodes are used. This allows simulation of a cavity that is effectively isolated; the case of interacting cavities will considered at the end of this section.

### A. Interface shapes for perfect dielectrics

We begin by considering the case of perfect dielectrics, starting from a liquid-air interface that is initially flat. When the applied potential difference is below a critical value (i.e., *Gr* > *Gr _{c}*), surface-tension and gravity forces balance the destabilizing electrostatic forces and produce a steady interface shape.

Fig. 4(a) shows typical interface shapes at several different times. There is a high initial rate of growth characterized by a rapid change in interface height, accompanied by a slow outward spreading. The liquid-air interface is lifted up near the corners of the cavity and dips at the center. This shape can be understood by examining the electric field, which sets the magnitude of the electrostatic forces, which in turn act only normal to the interface in perfect dielectrics. The electric field is stronger near the cavity corners than near the cavity center, where the same potential drop occurs over a larger gap. This gradient in the electric field sets up a pressure gradient that drives a liquid flow from the center of the cavity to the cavity corners, causing the interface to rise there and drop in the canter. Evidence of this is shown in Fig. 4(b), where we have plotted the different components that contribute to ∂*h*/∂*t* in Eq. (15). (The forms of these different components are shown in the supplementary material.^{67}) The electrostatic terms that contribute to ∂*h*/∂*t* cause it to be positive near the cavity corners and negative near the cavity center. This is balanced by both the capillary and gravitational forces, which act to try to restore the interface to a flat shape.

We have also examined influence of cavity geometry on the interface shape and briefly summarize the results here. Increasing the cavity depth and steepness leads to increases in the magnitude of the interface deformation because of the stronger electric field gradients that are produced. Increasing the cavity width does not lead to significant changes in interface deformation. This can be understood by recognizing that making the cavity wider does not significantly modify the electric field gradients near the cavity corners, which are what drive the interface deformation.

Finally, we note that if *Gr* is smaller than a critical value, *Gr _{c}*, the interface grows until it contacts the top electrode and no steady interface shapes are found. For the calculations shown in Fig. 4,

*Gr*≈ 1. In contrast, in the absence of the cavity and all other parameters being the same,

_{c}*Gr*

_{c,p}≈ 0.3 for

*k*= 0. This implies that when the cavity is present, a weaker electric field is needed to destabilize the interface. This can be rationalized by recognizing that when the cavity is present, the gap between the liquid-air interface and the top electrode becomes smaller as

*Gr*decreases (due to interface deformation), thereby increasing the magnitude of the destabilizing electrostatic forces. In contrast, when the cavity is absent and

*Gr*is decreased, the steady liquid-air interface remains flat (and becomes unstable below

*Gr*

_{c,p}).

### B. Interface shapes for leaky dielectrics

We now turn to the case of leaky dielectrics, for which a flat interface with constant charge (*h*(*x*, *t*) = 0, *q*(*x*, *t*) = 1/*β*) is a steady solution (see Sec. II C). If *Gr* is sufficiently large, then small-amplitude perturbations to the interface decay and this steady solution are recovered. If the interface perturbation amplitude is sufficiently large, however, the perturbation will grow until it hits the top electrode. As *Gr* becomes smaller, a smaller perturbation amplitude is required to initiate growth of the interface. In contrast to perfect dielectrics, no steady states with non-flat interfaces are observed.

Figure 5 shows results from a simulation where a small-amplitude random perturbation was given to the interface. The perturbation is generated by a linear combination of about 700 Fourier modes with random amplitudes (all <0.1), wavelengths (varying between 10^{−3} and 10), and phases (between 0 and 2*π*). A notable difference from perfect dielectrics is that the interface can grow at a single point. This occurs because if there is a localized accumulation of charge of sufficient magnitude at the interface, large normal electrostatic forces will be generated and cause the interface to rise at that point. Tangential electrostatic forces at the interface will be present as well, but these become overwhelmed by the normal forces as the interface gets closer to the top electrode.

### C. Two-cavity interactions

So far, we have considered electrostatically induced meniscus deformation over an isolated cavity. However, in actual gravure printing processes, many cavities are present and the flows near them may interact with each other. As a first step toward understanding these multi-cavity interactions, we consider what happens near two adjacent cavities. A sum of two arctangent functions is used to describe the cavity shape: *f*(*x*) = (*Y*(arctan((−*G*/2 + *x*)/*D*) − arctan((*G*/2 + *x*)/*D*)))/*π* + (*Y*(arctan((−*G*/2 + *x* − *X*)/*D*) − arctan((*G*/2 + *x* − *X*)/*D*)))/*π* − 1.0. Here, the separation between the cavities is denoted by *X*. Beginning from an initially flat interface, we run transient simulations to track the liquid height until it no longer changes at steady state. For these simulations, we have doubled both the domain length and the number of nodes (compared to single cavity cases). All other simulation parameters are unchanged.

Fig. 6 shows results for the case of perfect dielectrics where *Gr* is large enough so that a steady state is attained. At short times, the interfaces near each cavity appear to deform independently. At later times, however, they interact and the interface grows in the region between the cavities. The smaller the value of the cavity separation *X*, the sooner this interaction occurs.

For leaky dielectrics, the behavior of perturbations to the interface is similar to that in the case of a single cavity. Small-amplitude perturbations to the interface decay above a critical value of *Gr*. However, if the perturbation amplitude is sufficiently large or if *Gr* is below a critical value, then perturbations will grow. Figure 7 shows an example of a interface in which perturbations have grown. Here, the accumulation of charge enhances the force on the interface, causing it to grow rapidly. Indeed, the rate of growth is so rapid that the interfaces do not have time to merge and form a large peak above the region between the cavities. As with single cavities, the interface eventually hits the top electrode and no steady states are observed like those in perfect dielectrics.

## V. EXPERIMENTS

To complement the theoretical results of Secs. III and IV, flow visualization experiments have been performed. Experiments were conducted with a flat substrate and with a substrate having a long groove (Fig. 8). The use of a long groove rather than a short (in the *y*-direction) cavity allows for a more consistent comparison with our calculations which are for two-dimensional geometries.

### A. Experimental setup

The groove was made on one side of a 25 mm × 13 mm × 6 mm aluminum block by electrical discharge machining. The cross-section of the groove is a trapezoid with height 1.0 mm, bottom length 0.9 mm, and top length 1.5 mm (Fig. 9). The groove was sealed with a metal slab and transparent glass plate at either end. A smooth aluminum block was used for the experiments on the flat substrate.

Pump oil (Welch Duoseal Vacuum Oil) was used in all experiments. Its dielectric constant was measured to be 9.94 ± 0.83 using a Scientifica 870 Dielectric Constant Meter. The oil conductivity was measured to be 0 ± 0.4 *μ*S/cm using a Vernier Conductivity Probe. These small conductivities motivate us to use the prefect dielectric model when analyzing the results. The reported oil viscosity is 150 cP, the density was determined to be 728 ± 1 kg/m^{3}, and the surface tension was measured to be 0.03 N/m ±0.01 mN/m using a Wilhelmy plate apparatus (Kruss Digital Tensiometer K10ST).

The oil was first deposited onto the substrate with a dropper to form a film >1 mm thick. Then, excess oil was removed using a home-built apparatus, which moves a metal doctor blade across the substrate at a fixed angle, height, and speed. A second aluminum block was used as the top electrode and its separation from the substrate was controlled by plastic shims (Artus Corp.). Due to variability in the initial deposition of the oil, oil film thicknesses after doctoring were not reproducible from experiment to experiment. In addition, lateral variations in film thickness exist.

For experiments conducted at low voltages (<350 V), a low-voltage DC supply (Welch Scientific) with an increment of 10 V was used. For higher voltage experiments, a Spellman SL-130 Series High Voltage Power Supply capable of providing up to 20 kV DC voltage, with an increment of 0.1 kV, was connected between the two electrodes. As a safety measure, high resistances were included in the circuit to limit the maximum possible current to <1 mA.

For the flow visualizations, a camera (NEC Navitar zoom 6000 system) was set up to observe the liquid-air interface through the end sealed with the glass slide. Lighting was provided at the other end of the aluminum base. A similar setup is used for the flat substrate. The potential difference was systematically increased from zero until the liquid interface deformed and hit the top electrode. The minimum potential at which the interface contacted the top electrode was recorded as the critical voltage.

### B. Results

We begin with the results for the flat substrate. Table I lists the results from experiments with three different mean dimensionless air gaps *β*. The dimensional mean air gaps and film thicknesses are also shown, and these are based on ten measurements across the field of view given by the camera. The experimental values of the critical voltage (*V*_{c,e}) are shown in Table I along with the corresponding theoretical predictions from linear stability analysis (*V*_{c,th}) for perfect dielectrics.^{76}

Film thickness (
. μm) | Air gap (
. μm) |
. β |
. V_{c,e} (kV) |
. V_{c,th} (kV) | Aspect ratio
. H/L |
---|---|---|---|---|---|

379.39 ± 16.93 | 136.97 ± 10.40 | 0.36 ± 0.04 | 0.1 | 0.07 | 0.09 |

148.11 ± 17.44 | 372.35 ± 13.75 | 2.54 ± 0.22 | 0.4 | 0.24 | 0.57 |

123.90 ± 12.70 | 502.58 ± 4.40 | 4.10 ± 0.44 | 0.6 | 0.37 | 0.93 |

Film thickness (
. μm) | Air gap (
. μm) |
. β |
. V_{c,e} (kV) |
. V_{c,th} (kV) | Aspect ratio
. H/L |
---|---|---|---|---|---|

379.39 ± 16.93 | 136.97 ± 10.40 | 0.36 ± 0.04 | 0.1 | 0.07 | 0.09 |

148.11 ± 17.44 | 372.35 ± 13.75 | 2.54 ± 0.22 | 0.4 | 0.24 | 0.57 |

123.90 ± 12.70 | 502.58 ± 4.40 | 4.10 ± 0.44 | 0.6 | 0.37 | 0.93 |

As can be seen in Table I, the experimental and theoretical results agree best when the aspect ratio is smallest, as might be expected. (The aspect ratio is based on the mean value of the liquid film thickness, *H*, and a value of $L= \gamma H 3 / \u03f5 0 V 2 $ determined using the experimental value of the critical voltage.) The theoretical predictions for the critical voltage are consistently smaller than the experimental values. This difference may be due to variations in the film thickness (caused by a non-level surface or imperfections in the confining surfaces, for example), a breakdown of the lubrication approximation, or three-dimensional effects that are not considered in the model. Flow visualizations (Fig. 10) reveal that pillar-like structures form above the critical voltage. Unfortunately, the limited field of view of the camera (and spreading of the liquid once it contacts the top electrode) prevented us from obtaining reliable information about pillar spacing.

We now turn to the results for the substrate with the groove. Table II lists the results from experiments with three different mean air gaps *β*. Also shown are the experimental values of the critical voltage (*V*_{c,e}) and the values determined from numerical simulations (*V*_{c,sim}). The simulations are carried out using the method for perfect dielectrics described in Sec. IV using a cavity shape similar to that in the experiments. As with the flat surfaces, the absolute difference between the model predictions and the experimental values for the critical voltage is smaller for smaller aspect ratios, consistent with what would be expected from lubrication theory. The predicted voltage is smaller than the experimental value at the smallest air gap, then becomes larger at a larger air gap. For the largest air gap, the simulations do not predict any contact with the top electrode. In this case, the electric field is weakened to such a large extent that the interface grows very slowly, while also spreading outward due to capillarity. The two peaks at the cavity edges move toward the ends of the domain as time progresses and begin to interact with each other. Unfortunately, increasing the length of simulation domain did not eliminate this issue, and further increases require prohibitively long simulation times. In addition to the large aspect ratio, the steep walls of the cavity also contribute to the breakdown of the lubrication model.

Film thickness (
. μm) | Air gap (
. μm) |
. β |
. V_{c,e} (kV) |
. V_{c,sim} kV | Aspect ratio
. H/L |
---|---|---|---|---|---|

203.81 ± 10.08 | 299.12 ± 12.86 | 1.47 ± 0.12 | 0.4 | 0.21 | 0.48 |

214.49 ± 13.03 | 454.94 ± 7.61 | 2.13 ± 0.12 | 0.5 | 0.89 | 0.59 |

93.53 ± 5.74 | 573.93 ± 9.84 | 6.18 ± 0.42 | 1.3 | No contact | 2.31 |

Film thickness (
. μm) | Air gap (
. μm) |
. β |
. V_{c,e} (kV) |
. V_{c,sim} kV | Aspect ratio
. H/L |
---|---|---|---|---|---|

203.81 ± 10.08 | 299.12 ± 12.86 | 1.47 ± 0.12 | 0.4 | 0.21 | 0.48 |

214.49 ± 13.03 | 454.94 ± 7.61 | 2.13 ± 0.12 | 0.5 | 0.89 | 0.59 |

93.53 ± 5.74 | 573.93 ± 9.84 | 6.18 ± 0.42 | 1.3 | No contact | 2.31 |

Figure 11 shows flow visualizations with the groove geometry. The camera is at a slight angle to the groove and the back-lighting causes light to tunnel through the groove by reflecting off its walls. This causes the oil in the groove to be lighter and the groove walls to be visible. The oil film above the filled groove, however, does not allow light to pass through and appears black. Despite the discrepancy of the measured critical voltages with simulation predictions, the visualizations show that interface growth occurs toward the edges of the groove, as predicted by the simulations in Sec. IV A. This indicates that the lubrication model at least qualitatively succeeds in capturing the experimental observations. The lack of quantitative agreement may be due to the reasons discussed above for the flat substrate. Full two- and three-dimensional numerical simulations in which the lubrication approximation is relaxed may be helpful in resolving this issue.

## VI. CONCLUSIONS

In this work, we have used theory and experiment to understand the fundamentals of how thin liquid films are deformed by electric fields near surfaces having topographical features. When topography is present, a flat interface is not a steady state for prefect dielectrics but it is for leaky dielectrics (unless the film is so thin that van der Waals forces become important). Asymptotic analysis reveals how the interface deformation depends on various problem parameters and indicates that for small-amplitude sinusoidal cavities, the interface deformation mimics the cavity shape.

For trapezoidal cavities and perfect dielectrics, the interface deforms near the cavity edges, rising toward the top electrode. Interface behavior can be dramatically different with leaky dielectrics, where charge accumulation can cause the interface to rise rapidly at only one point if an initially flat interface is perturbed. When multiple cavities are present, the interfaces near different cavities can interact with each other, causing a large peak to form in the region between the cavities in the case of perfect dielectrics. For leaky dielectrics, the interface evolution is more rapid and merging may not occur before the interface contacts the top electrode. Flow visualization experiments yield results in qualitative agreement with the theoretical predictions for perfect dielectrics near single cavities, with the discrepancy perhaps arising from a breakdown of the lubrication approximation.

This work was motivated by the use of ESA in gravure printing, where the applied voltages are much larger and the cavities are much smaller than those used in our experiments. Results from our model suggest that under these conditions, the interface will rapidly contact the web (Fig. 1(a)). If contact occurs at the cavity edges first before the liquid in the middle has time to make contact, an air bubble could be trapped, leading to degradation of print quality. In addition, if the cavities are too closely spaced, the interface could contact the web in the region between the cavities, leading to deposition of ink in places it is not desired. These issues would be expected to be less pronounced if the ink is conductive, suggesting that such inks may be particularly advantageous when ESA is applied.

To develop predictive models of ESA, it would be instructive to perform two- and three-dimensional numerical simulations using the model geometries considered here as this would allow one to overcome the limitations of the lubrication approximation. The model considered here is also limited by the assumption that a lubrication film exists on the lands (Fig. 2). This assumption facilitates application of the lubrication approximation. However, there could be cases where a cavity is partially filled, leading to the presence of contact lines, and these situations could be treated with the numerical simulations described above. Finally, there is the question of liquid dynamics after contact with the top surface.^{81–85} After contact, a liquid bridge will form and its stretching and eventual break-up in the presence of the electric field will determine the effectiveness of ESA. Nevertheless, the present work provides a foundation for thinking about these issues as well as contributing novel results of fundamental interest in the areas of electrohydrodynamics and thin-liquid-film flows.

## Acknowledgments

This work was supported through the Industrial Partnership for Research in Interfacial and Materials Engineering at the University of Minnesota. S.K. acknowledges support from the Department of Energy under Award No. DE-FG02-07ER46415. We are also grateful for resources from the University of Minnesota Supercomputing Institute. Finally, we thank Wieslaw Suszynski and Melissa Cabak for assistance with the flow visualization experiments.