Charge Leakage Mediated Pattern Miniaturization in the Electric Field

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Charge Leakage Mediated Pattern Miniaturization in the Electric Field Induced Instabilities of an Elastic Membrane Mohar Dey,† Dipankar Bandyopadhyay,‡,§ Ashutosh Sharma,*,∥ Shizhi Qian,⊥ and Sang Woo Joo*,† †

School of Mechanical Engineering, Yeungnam University, Gyeongsan 712749, South Korea Department of Chemical Engineering, Indian Institute of Technology Guwahati, 781039, Assam, India § Centre for Nanotechnology, Indian Institute of Technology Guwahati, 781039, Assam, India ∥ Department of Chemical Engineering, Indian Institute of Technology Kanpur, UP 208016, India ⊥ Department of Mechanical and Aerospace Engineering, Old Dominion University, Norfolk, Virginia 23529, United States ‡

ABSTRACT: Electric-field-induced interfacial instabilities of a dielectric elastic membrane confined between a pair of leaky dielectric liquid layers are investigated by linear stability analysis. The role of leaky surroundings is investigated by comparing the instability modes in three different systems: (a) purely dielectric surrounding layers, (b) one leaky and the other pure dielectric layer, and (c) leaky dielectric surrounding layers. The interfaces of the trilayer can become unstable either by a long-wave inphase bending mode or by a finite-wavenumber antiphase squeezing mode. In all the cases, the conditions for the transition from the finite-wavenumber to long-wave mode are compared and contrasted. The study uncovers that the charge leakage in the surrounding layers of the elastic film not only changes the amplitude of deformations at the interfaces but also causes a transition of dominant modes from bending to squeezing. In addition to the presence of the induced dipoles, the augmented destabilizing influence at the interfaces(s) due to free charges from the charge leakage inside the films is found to significantly reduce the length scale. The same effect also reduces the requirement for the minimum field intensity to engender the electro-hydrodynamic (EHD) instabilities in the elastic films. Thus, the study is also relevant in improving the understanding of the fundamental aspects of the electric field induced deformation and rupture of biological membranes in the electrolytes.

1. INTRODUCTION Interfacial instabilities of elastically soft polymer films often result in interesting patterns under the influence of intermolecular interactions1−40 or electric field.41−75 These patterns find important applications in the microelectronic or microfluidic devices, sensors, electrochemical cells, solar panels, and drug delivery modules.76−78 The field-induced deformation of a thin elastic film is also a model in vitro prototype79−82 to investigate the dynamics of biomembranes.83−86 Thus, the deformations of thin membranes have been extensively explored in recent years to scrutinize various fundamental and technological aspects. The stability of ultrathin ( 1 μm) was found to be independent of the material properties.32−34,54 In contrast, the thinner films (hf < 1 μm) develop instabilities with much larger length scales due to the increasing influence of the stabilizing surface tension force.28 A number of previous works related to contact instability suggest that the use of multiple layers of polymers can be useful in controlling the length scales of the patterns fabricated on the surface.30,35−40 Many of the recent experimental41−44 and theoretical45−47 studies reveal the importance of electric field lithography (EFL) as an alternative technique to pattern the thin polymeric films. The external control on the magnitude of the destabilizing field strength, together with the capability to pattern the thicker films makes EFL a unique technique for fabricating micro and nanoscale patterns on polymer films.48−51 For the EFL of the elastic films, the periodicity of the patterns is decided by the imbalance of the destabilizing electrical stress against the Special Issue: Ganapati D. Yadav Festschrift Received: January 26, 2014 Revised: February 27, 2014 Accepted: February 27, 2014

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dx.doi.org/10.1021/ie500378k | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

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transition from the squeezing to the bending mode and vice versa, which is significantly different from a trilayer composed of only dielectric films. The presence of the leaky films can also facilitate pattern miniaturization to the submicrometer scale, as compared to the purely dielectric trilayers. The study also identifies that the field intensity required to initiate instability in elastic membranes is much smaller when surrounded by the leaky dielectric layers, as compared to a single layer of elastic film or a trilayer with dielectric films. Concisely, the study showcases some important fundamental and technological details on the deforming elastic membranes surrounded by electrolytes.

combined stabilizing influence from the capillary and elastic forces.52−55 However, the previous studies indicated that the field intensity required to deform an elastic film is significantly high. In particular, when the gap between the electrodes are of the order or a few micrometers or less, the field intensity can only be increased to a certain limit, which hinders the applicability of this process toward the fabrication of submicrometer patterns. An alternative then would be the use of multiple layers56−58 in which the lesser stabilizing interfacial tension at the embedded interface(s) not only reduces the requirement for the field intensity but also helps in controlling the periodicity of the EFL patterns.59−62 Apart from the use of multiple layers, introduction of leaky dielectric layers63,64 could be another alternative to enhance the intensity of the electric field stress at the interfaces. Recent works predicted a much lower length scales for the EFL instabilities of single layer,55,65,66 confined bilayers,67−73 and free bilayers74,75 composed of leaky dielectric films. The studies highlighted that, in addition to the stresses due to the accumulation of the induced dipoles at the interfaces, the charge leakage inside a film could enforce additional destabilizing stress at the softinterfaces because of the presence of free charges. This effect could significantly reduce the length scale of the patterns to enforce pattern miniaturization. In this study, we consider a trilayer configuration consisted with a thin elastic film confined between a pair of weakly conducting liquid films subjected to an externally applied electric field, as shown in the Figure 1. Linear stability analysis

2. PROBLEM FORMULATION In this work, we consider three different trilayer configurations, (i) a dielectric elastic layer confined between a pair of dielectric liquid layers (PPP), (ii) a dielectric elastic layer sandwiched between a pair of leaky liquid layers (LPL), and (iii) a dielectric elastic layer confined between a dielectric and a leaky liquid layer (LPP). In what follows, we develop a general formulation for LPL configuration, which can be asymptotically reduced to PPP and LPP in the limit where the charge leakage is absent in both the layers and in one of the layers. An elastic film confined between two viscous layers of leaky dielectric fluids is schematically shown in the Figure 1. The liquid films are assumed to be Newtonian fluids with viscosity, μi, dielectric permittivity, εi, and electrical conductivity, σi, where i = 1 and 3 respectively. The elastic film in the middle is assumed to be an incompressible Hookean, perfect dielectric with permittivity ε2, and with shear modulus, G. The x- and z- coordinates are chosen to be parallel and normal to the bottom substrate. The liquid layers are bounded by the anode (z = d) and cathode (z = 0), respectively. The thicknesses of undisturbed lower layer, middle layer, and lower and middle layer combined are denoted by h10, hM (= h20 − h10), and h20, respectively. The position of the lower and upper elastic-viscous interfaces is denoted by h1(x,t), and h2(x, t), respectively. 2.1. Electric Field. The divergence free (∇·Ei = 0) and irrotational (∇ × Ei = 0) electric field in an electro-neutral ith layer can be expressed in terms of the potential function, ψi as, Ei = −∇ψi, which leads to the Laplace equation (∇2ψi = 0) for the ith layer. Here, the subscripts i = 1, 2, and 3 denote the lower, middle, and upper layers, respectively. We enforce a zero-potential (ψ1 = 0) boundary condition at the cathode (z = 0) and ψ3 = ψ at the anode (z = d). The tangential (E1·t1 = E2· t1) and normal (ε0ε1(E1·n1) − ε0ε2 (E2·n1) = q1) components of the electric field are balanced at the lower elastic-viscous interface (z = h1). The normal (ε0ε2(E2·n2) − ε0ε3 (E3·n2) = q2) and tangential (E2·t2 = E3·t2) component balances of electric field are enforced at the upper elastic-viscous interface (z = h2). The unit normal and tangent vectors at the interfaces, z = hj(x, t), are given as nj = [(1 + h2jx)−1/2{− hjx, 1}] and tj = [(1 + h2jx)−1/2{1,hjx}], respectively, where j = 1 and 2, represents the lower and upper interfaces, and qj (x, t), (j = 1 and 2) represents the surface charge density of the free charges at the elastic-viscous interfaces. The subscript x in the expressions denotes differentiation with respect to x. The symbol ε0 denotes the permittivity in free space. 2.2. Electro-hydrodynamic Field. Since we consider films with relatively small thicknesses, inertial and gravitational effects in the liquid layers can be ignored. We also assume electro-neutrality in the bulk (∇·Mi = 0) and the characteristic time for viscous flow is much larger than the electrically

Figure 1. Schematic diagram of the confined leaky trilayer configuration subjected to an applied electrostatic field of potential ψ. The interelectrode distance is denoted by d. The in-phase deformations at the interfaces depict a bending mode of instability whereas an antiphase deformation shows the squeezing mode of instability. The mean and the variable local thicknesses of the lower (viscous) layer are denoted by h10 and h1(x,t), respectively, whereas the same for the combined lower (viscous) and middle (elastic) layers is denoted by h20 and h2(x,t), respectively.

(LSA) is performed after taking into consideration the full descriptions of the Maxwell’s and hydrodynamic stresses for the electric field and flow, respectively. A pair of coupled elasticviscous interfaces was found to deform either in a long-wave bending mode or in a finite-wavenumber squeezing mode with a short wave characteristic.56 Here we investigate the role of the charge leakage in the surrounding liquid layers on the mode selection, the amplitudes of deformation at the interfaces, and the length scale of instability. As shown below, the charge leakage in the surroundings of the membrane can cause a B



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induced or magnetic effects. The Stokes equation of motion along with the Maxwell’s stress tensor, Mi = ε0εi[Ei ⊗ Ei − 0.5(Ei·Ei)I] and the hydrodynamic stress tensor for the Newtonian fluids, σi = μi (∇vi + ∇vTi ) where i = 1 and 3, governs the electro-hydrodynamics (EHD) of the ith liquid layer, ∇·vi = 0

q2t + v3·∇s q2 − q2 n2 ·(n2 ·∇)v3 = −σ3 E3·n2

3. LINEAR STABILITY ANALYSIS 3.1. Base-State Analysis. A quiescent base state, expressed (z) (x) (z) as v(x) i0 = vi0 = pi0 = 0, for the viscous films and ui0 = ui0 = pi0 = 0, for the elastic film, with undisturbed film thicknesses, hi0, and base state interfacial charges qi0, is considered. The base state electric potential ψi0 (i = 1, 2, and 3) is obtained from the governing equation, ψi0zz = 0, which has a general solution,ψi0 = A1iz + A2i. The constants for the base state potentials Aji (j = 1 and 2; i = 1, 2, and 3) can be obtained by applying the boundary conditions: (i) ψ10 = 0, at the cathode (z = 0); (ii) ε0ε1ψ10z − ε0ε2ψ20z = q10 and ψ10 = ψ20 on the lower elasticviscous interface (z = h10); (iii) ε0ε2ψ20z − ε0ε3ψ30z = q20 and ψ20 = ψ30 on the upper elastic-viscous interface (z = h20); and (iv) ψ30 = ψ, at the anode (z = d). The base state charges at the interfaces are obtained by solving the unperturbed charge conservation equations obtained from the eqs 14 − 15, as σ1ψ10z = 0 and σ3ψ30z = 0. On substituting the expressions for the base state potentials in these expressions, we obtain the base state interfacial charges as q10 = −ε0ε2ψ (h20 − h10)−1 and q20 = ε0ε2ψ (h20 − h10)−1. 3.2. Perturbed-State Analysis. The governing equations and the boundary conditions described in section 2 for the EHD field are linearized employing the normal modes vi = vĩ eωt+ikx, ui = ũi eωt+ikx, ψi = ψi0 + ψ̃ i eωt+ikx, pi = pi0 + p̃i eωt+ikx, hi = hi0 + δ̃i eωt+ikx, and qi = qi0 + q̃i eωt+ikx, where ω and k denote the linear growth coefficient and the wavenumber of the disturbance, respectively. The infinitesimal amplitudes of perturbations to the variables vi, ui, ψi, and pi are denoted by the variables vĩ uĩ , ψ̃ i, and p̃i, and are functions of z only. The symbols δĩ and q̃i represent the infinitesimal perturbation of height and charge at the elastic−viscous interfaces. The governing equation for the linear perturbed potential function is, ψ̃ izz − k2ψ̃ i = 0, which has the general solution, ψ̃ i = C1iekz + C2i e−kz. The constants Cji values (i = 1, 2, and 3; j = 1, 2) are obtained by considering the total potential ψi(= ψi0 + ψ̃ i) for the ith layer and using the boundary conditions: (i) at the cathode (z = 0), ψ1 = 0, (ii) at the lower elastic-viscous interface (z = h10), normal (ε0ε1ψ1z − ε0ε2ψ2z = q1) and tangential (ψ1 = ψ2) balances, (iii) at the upper elastic viscous interface (z = h20), normal (ε0ε2ψ2z − ε0ε3ψ3z = q2) and tangential (ψ2 = ψ3) balances, and (iv) at the anode (z = d), ψ3 = ψ. The linearized continuity and momentum equations for the viscous layers (i = 1 and 3) are

(1)

−∇pi + ∇·(σi + Mi) = 0

vi[v(x) i ,

(2)

v(z) i ]

The symbols and pi denote the velocity vector and isotropic static pressure of the ith layer, respectively. The bracketed superscript denotes the x- and z- components of the vector. For an incompressible and Hookean elastic film, τi = G(∇ui + ∇uTi ) where i = 2, the following governing equations describe the EHD, ∇·u i = 0

(3)

−∇pi + ∇·(τi + Mi) = 0.

(4)

(z) The symbol ui [u(x) i , ui ] and pi denote the displacement vector and isotropic pressure of the middle elastic film (i = 2). At the cathode (z = 0) and anode (z = d), the no slip and impermeability boundary conditions are enforced,

v1 = 0

(5)

v3 = 0

(6)

The continuity of velocities, normal and tangential stress balances are enforced as boundary conditions at the lower elastic-viscous interface (z = h1), v1 = u 2t (7) −p1 I + n1·(σ1 + M1) ·n1 + p2 I − n1·(τ2 + M 2) ·n1 = γ21κ1 (8)

t1·(σ1 + M1) ·n1 − t1·(τ2 + M 2) ·n1 = 0

(9)

Here, the subscript t denotes the time derivative of the variable. The symbol γij represents the interfacial tension between the layers i and j, and κj = ∇s·nj represents the curvature of the interface j. At the upper elastic−viscous interface (z = h2), continuity of velocities, normal and tangential stress balances are enforced as boundary conditions, v3 = u 2t (10) −p2 I + n2 ·(τ2 + M 2) ·n2 + p3 I − n2 ·(σ3 + M3) ·n2 = γ32κ2

(11)

t 2 ·(τ2 + M 2) ·n2 − t 2 ·(σ3 + M3) ·n2 = 0

(12)

⎛ d2vi(̃ x) ⎞ ⎟⎟ = 0 −ikpi ̃ + μi ⎜⎜ −k 2vi(̃ x) + dz 2 ⎠ ⎝

(16)

⎛ d2vi(̃ z) ⎞ ⎟⎟ = 0 + μi ⎜⎜ −k 2vi(̃ z) + dz dz 2 ⎠ ⎝

(17)

The kinematic conditions at the interfaces (z = hi(x, t), i = 1 and 2) are expressed as hit +

vi(x)hix

=

vi(z)

−

(13)

Here, the subscripts x and t in the expression denote the derivatives. Since the trilayer is composed of a purely dielectric elastic film (σ2 → 0), the full description of the charge conservation equation at the interfaces (z = hi (x, t), i = 1 and 2) are given by q1t + v1·∇s q1 − q1n1·(n1·∇)v1 = σ1E1·n1

(15)

dpi ̃

ikvi(̃ x) +

dvi(̃ z) =0 dz

(18)

The linearized mass and momentum balances for the elastic layer (i = 2) are

(14) C



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⎛ d2u 2̃ (x) ⎞ ⎟⎟ = 0 −ikp2̃ + G⎜⎜ −k 2u 2̃ (x) + dz 2 ⎠ ⎝

(19)

⎛ d2u 2̃ (z) ⎞ ⎟⎟ = 0 + G⎜⎜ −k 2u 2̃ (z) + dz dz 2 ⎠ ⎝

(20)

−

At the upper elastic−viscous interface (z = h20), the linearized forms of the continuity of velocities, the normal and the tangential stress balances, and the kinematic conditions for the height and charge are

dp2̃

iku 2̃ (x)

du ̃ (z) + 2 =0 dz

(21)

dz 4 d4uĩ (z) 4 dz

− 2k

−

2 (z) 2 d vĩ

dz 2

d2u ̃ (z) 2k 2 i2 dz

+ k 4vi(̃ z) = 0 + k 4uĩ (z) = 0

(34)

v3(̃ z) = ωu 2̃ (z)

(35)

p2̃ − p3̃ − 2Gu 2̃ (zz) + 2μ3 v3(̃ zz) + ε0[(ε3A13ψ3̃ z)

The variables p̃1 and p̃3 are eliminated from the linearized governing equations [eqs 16−18] for the two viscous layers and p̃2 from equations [eqs 19−21] to obtain the following biharmonic equations for the viscous layers and the elastic layer, respectively, d4vi(̃ z)

v3(̃ x) = ωu 2̃ (x)

− (ε2A12 ψ2̃ z)] = γ32k 2δ2̃

⎛ ⎞ dψ G(u 2̃ (zx) + iku 2̃ (z)) − μ3 (v3(̃ zx) + ikv3(̃ z)) + ikq20⎜ψ3̃ + 30 δ2̃ ⎟ dz ⎠ ⎝

(22)

δ2̃ = v3(̃ z)|h20 /ω

The general solutions of eqs 22 and 23 are

q2̃ = (24)

uĩ (z) = (B1i + B2i z)ekz + (B3i + B4i z)e−kz

(25)

(26)

v3(̃ x) = v3(̃ z) = 0

(27)

At the lower elastic−viscous interface (z = h10), the linearized forms of the continuity of velocities, the normal and the tangential stress balances, and the kinematic conditions for the height and charge balance are v1̃(x) = ωu 2̃ (x)

(28)

v1̃(z) = ωu 2̃ (z)

(29)

⎤ 1 ⎡ dψ3̃ − ikq20v3(̃ x)⎥ ⎢σ3 ⎦ ω ⎣ dz

(38)

(39)

The 12 homogeneous linear algebraic equations with the unknown constants Bji values (i = 1, 2, and 3; j = 1 to 4) are (x) (z) (x) obtained by substituting the variables ṽ(z) i ,ṽi , ũi , ũi , and p̃i in the boundary conditions [eqs 26−39]. The determinant of the coefficient matrix of the constants leads to a fourth order algebraic dispersion relation, ω = f (k), which is solved using the symbolic package Mathematica to obtain the eigenvalues for ω. The neutral stability conditions are obtained by setting, ω = 0, in the dispersion relation and subsequently obtaining the critical wavenumber, kc, from the resulting algebraic equation. The global maxima of the growth rate ω and the corresponding wavenumber km from the dispersion relation give the dominant growth coefficient (ωm) and the dominant wavelength (λm = 2π/km), respectively. The accuracy of the LSA performed for the simplified LPL model has been verified in the asymptotic limits in Figure 2. The solid and the broken line in this figure show the variation in the growth rate (ω) with the wavenumber (k) plot for two different unstable modes of the existing study on the PPP trilayer.56 The circular symbols indicate that the LSA discussed in the present work can reproduce these unstable modes in the limit with zero conductivities at the

In the above equations, the coefficients Bji values (j = 1 to 4; i = 1, 2, and 3) are constants. The general solutions for ṽ(z) i [eq 24] (x) and ũ(z) i [eq 25] leads to linear expressions for the variables ṽi , (x) ũi and p̃i. The boundary conditions for EHD field are also linearized using the normal modes. The resulting no slip and impermeability conditions at cathode (z = 0) and anode (z = d) are

v1̃(x) = v1̃(z) = 0

(37)

=0

(23)

vi(̃ z) = (B1i + B2i z)ekz + (B3i + B4i z)e−kz

(36)

p1̃ − p2̃ − 2μ1v1̃(zz) + 2Gu 2̃ (zz) + ε0[(ε2A12 ψ2̃ z) − (ε1A11ψ1̃ z)] = γ21k 2δ1̃

(30)

⎛ dψ ⎞ μ1(v1̃(zx) + ikv1̃(z)) − G(u 2̃ (zx) + iku 2̃ (z)) + ikq10⎜ψ1̃ + 10 δ1̃ ⎟ dz ⎠ ⎝ (31)

=0

δ1̃ = v1̃(z)|h10 /ω q1̃ = −

⎤ 1 ⎡ dψ1̃ + ikq10v1̃(x)⎥ ⎢σ1 ⎦ ω ⎣ dz

Figure 2. Plot shows the validated LSA results. The curves (solid and broken lines) correspond to the PPP model and the symbols (circles) correspond to the LPL model in the limiting case when q10 = q20 = 0 and σ1 = σ3 =0. The two curves correspond to the two unstable modes present at ψ = 30 V.

(32)

(33) D



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Figure 3. Bifurcation diagrams for the trilayer configurations. Plots a−c show the variation of kc with ψ,ε2, and hM, respectively. In all the plots, curves 1−3 correspond to LPL, LPP, and PPP models, respectively, with the solid and broken lines representing the bending and squeezing modes of instability denoted by symbols “b” and “s”. In general, the parameters used in all the plots are, G = 0.1 MPa, hM = 0.1 μm, and ε2 = 4 with ψ = 14 and 15 V in panels b and c, respectively.

liquid layers (σ1 = σ3 = 0) together with zero surface charge densities at the interface (q10 = q20 = 0). The LSA could also predict the relative bending and squeezing modes of deformations at the interfaces. Initially, perturbations with amplitudes α2 and δ2 are assumed at the upper elastic-viscous interface (z = h20) in the x- and zdirections as u 2̃ (x) = α2 ,

u 2̃ (z) = δ2

configuration possesses a pair of deformable elastic-viscous interfaces, as a soft elastic layer is bounded between a pair of viscous layers (Figure 1). In such a situation, when the liquid layers are considered to be leaky dielectric materials, application of an external electrostatic field leads to the destabilizing electrical stresses because of free charge accumulation alongside the stresses originating from the induced charge separation. These destabilizing influences compete with the stabilizing forces due to the elastic stiffness of the elastic layer and the capillary forces at the interfaces. A previous study showed that the interfaces of a purely dielectric trilayer can readily deform into a bending mode when exposed to a small destabilizing electric field.56 In comparison, threshold field intensity is essential to initiate the squeezing mode of instability. The twin elastic-viscous interfaces can deform in bending mode with equal amplitudes even when the applied field intensity is vanishingly small because of the absence of any deformation of the elastic film. In such a scenario, only the shorter wavelength modes are stabilized by the capillary forces, while the electric field destabilizes the modes with longer wavelength. In contrast, for an in-phase bending mode with unequal amplitudes of deformations or for an antiphase squeezing mode of deformations at the interfaces, a minimum destabilizing field is necessary to overcome the strain energy required to deform the elastic film. In such cases, the instability shows a finite wavenumber characteristic because the elasto-capillary forces together stabilize a span of unstable modes in the longer and shorter wavelength domains. The electric field can only destabilize the modes of intermediate wavelengths. In the present study, we focus on the destabilizing role of the leaky dielectric surroundings of the membrane. The leaky nature of the surroundings allows accumulation of free charges alongside the induced charge across the interfaces. This could lead to an intensified instability reflected in a reduction in the length scale of deformations. In particular, we focus on the various parameters that regulate the distribution of the free charges at the interfaces to identify the condition under which patterns with smaller periodicity can be achieved. In order to have a comparative outlook on the influence of charge leakage on the length and the time scales alongside the mode selection, three different kinds of trilayers, namely PPP, LPP, and LPL are considered. The typical range of parameters employed for the analysis are, h10 = 1.0 μm, h20 = 1.1 μm, d = 2.1 μm, ε2 = 4, ε1 = ε3 = 80, μ1 = μ3 = 0.001 Pa s, σ1 = σ3 = 10−11 S/m, σ2 = 0, γ21 = γ32 = 0.005 N/m, ψ = 10−50 V, and G = 0.1 MPa. The parameter space resembles a trilayer composed of a cross-linked PDMS (poly dimethylsiloxane) film sandwiched between a pair of water films.

(40)

In consequence to these deformations, the linearized x- and zcomponents of the displacements at the lower elastic-viscous interface (z = h10) α1 and δ1 are written as u 2̃ (x) = α1 ,

u 2̃ (z) = δ1

(41)

Using the boundary conditions eqs 28, 29, 34, and 35, we obtain the following expressions, v1̃(x) = ωα1 ,

v1̃(z) = ωδ1

(42)

v3(̃ x) = ωα2 ,

v3(̃ z) = ωδ2

(43)

The coefficients Bji values (j = 1 to 4; i = 1, 2, and 3) in the (z) (z) solution for ṽ(z) 1 , ũ2 , and ṽ3 are obtained from the boundary conditions, (i) at z = 0 and z = d, no slip and impermeability conditions [eqs 26 and 27] and (ii) at z = h10 and h20, the continuity of x- and z- component of velocities [eqs 28−29 and eqs 34−35]. For a set of system parameters, replacing the maximum linear growth coefficient (ωm) and the corresponding wavenumber (km) into the expressions for Bji values (j = 1 to 4; i = 1, 2, and 3) converts them to a function of only four parameters, α1, α2, δ1, and δ2. Following this, any three out of the remaining four unused boundary conditions, (i) tangential stress balances at z = h10 and h20 [eqs 31 and 37], and (ii) normal stress balances at z = h10 and h20 [eq 30 and 36], can be used to obtain the ratio δr (= δ2/δ1). The magnitude, |δr| = 1, signifies that both interfaces deform with equal amplitude whereas |δr| > 1 (|δr| < 1) indicates a larger deformation at the upper (lower) elastic-viscous interface. The condition, δr > 0 (δr < 0), signifies an in-phase (antiphase) bending (squeezing) mode.

4. RESULTS AND DISCUSSION In the presence of an external electric field, the interfaces of leaky dielectric fluids are acted upon by coupled stresses originating from the accumulation of the free charges together with the induced charge separation. This is in contrast with the behavior of similar dielectric liquids, in which the electrical stresses at the interfaces develop only due to the accumulation of induced dipoles. In the present work, the trilayer E



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Figure 4. LSA results for the three trilayer configurations. Panels a−c, d−f, and g−i show the growth rate (ω) vs wavenumber (k) plots to show the effect of variation of applied potential ψ, dielectric permittivity, ε2, and thickness, hM, of the elastic film, on the growth rates of instability, respectively, for the LPL [a, d, g], LPP [b, e, h] and PPP [c, f, i] configurations. In panels a−c, curves 1−3 correspond to ψ = 10, 16, 20 V, respectively, when ε2 = 4 and hM = 0.1 μm. In panels d−f, curves 1−3 correspond to ε2 = 4, 5, 6, respectively, when ψ = 20 V and hM = 0.1 μm. In panels g−i, curves 1−3 correspond to hM = 0.08, 0.1, 0.15 μm, respectively, when ε2 = 4 and ψ = 20 V. In all the plots, the solid and broken lines correspond to the bending and squeezing modes of instability denoted by symbols “b” and “s”.

trilayer can even be unstable when the applied potential is ∼10 V, which is rather remarkable as compared to the potential requirement to destabilize a nonslipping elastic film on a rigid surface under the influence of an external electrostatic field.54 In Figure 4, the influence of free charge concentration on the growth rate (ω), wavenumber (k), and on the mode selection has been explored. The plots [(a) − (c)], [(d) − (f)] and [(g) − (i)] in Figure 4 show the variations in ω with k when ψ, ε2, and hM are varied, respectively. In order to highlight the effects of charge leakage, we show the plots for LPL [column 1, plots a, d, and g], LPP [column 2, plots b, e, and h], and PPP [column 3, plots c, f, and i] trilayers under similar condition. Curves 1−3 in the plots b and c show that as compared to a PPP trilayer [plot c], increase in the q10 at one of the elastic− viscous interfaces of a LPP trilayer [plot b] by increasing ψ can significantly increase the growth rate and shift the length scale toward the smaller wavelength regime. In comparison, Figure 4a suggests that increase in q10 and q20 at both the elastic viscous interfaces of a LPL trilayer can further increase ω and reduce the length scale as compared to similar PPP and LPP trilayers when ψ is increased. The solid and the broken lines in the Figure 4c show that under this condition the elastic− viscous interfaces can only undergo a bending mode of deformation while the squeezing mode is stable. The solid and the broken lines in the Figure 4b suggest that introduction of the electrical stress because of the free charge at any of the interfaces could force a subdominant squeezing mode of

The expression for the base-state charges at the two interfaces (q10 = −ε0ε2ψ (h20 − h10)−1 and q20 = ε0ε2ψ (h20 − h10)−1) can be useful in the interpretation of the results in which the free charge densities (q10 and q20) are found to increase with the increase in the applied potential (ψ) and the dielectric permittivity (ε2) of the elastic layer whereas the same variables are found to reduce with the increase in the elastic film thickness (hM). Plots a−c in Figure 3 show the neutral stability plots when ψ, ε2, and hM are varied, respectively. Curves 1−3 in this figure correspond to the LPL, LPP, and PPP trilayers, respectively. It may be noted here that the symbols ‘b’ and ‘s’ in the plots denote the curves corresponding to bending and squeezing modes, respectively. Curve 1 in plot a highlights that the critical voltage (ψc) required to initiate any of the squeezing (curves 1s−3s) or bending (curves 1b−3b) modes of instabilities at the elastic-viscous interfaces is minimum for a LPL trilayer as compared to a LPP (curve 2) or a PPP (curve 3) trilayer. Further, curve 2 shows that even the LPP trilayers require smaller critical field intensity as compared to the PPP trilayers (curve 3). The curves confirm that the free charge accumulation at any of the interfaces or at both interfaces can increase the destabilizing electrical stress when exposed to an external electrostatic field, which can enforce the elastic film to deform at a much lower field intensity. Plots b and c suggest that for a fixed ψ, the free charge accumulation can also destabilize a thinner elastic film having lower dielectric permittivity. The figure conveys that an elastic film in a LPL F



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instability, which is found to be stable in case of the PPP trilayer. The solid and the broken lines in Figure 4a show that the free charge accumulation can lead to a dominant squeezing mode (curve 3s) even at much lower field intensity. It may be noted here that when a LPL trilayer is exposed to electric field the elastic−viscous interfaces tend to attract each other because of the presence of the opposite free charges at the two interfaces. This particular influence is expected to engender a dominant squeezing mode for a LPL trilayer when the applied field intensity is small. In case of the LPP trilayer, accumulation of one type of free charge at one of the interfaces leads to an induced attractive influence with the noncharged interface, which leads to the subdominant squeezing mode. In comparison, for the PPP trilayer absence of these effects ensures that the squeezing mode is stable at lower field intensities. The solid lines in the Figure 4a−c suggest that even the bending mode of instability is empowered significantly as the attractive electrostatic interaction increases at the elasticviscous interfaces with free charge accumulation from PPP to LPP to LPL. However, the broken line confirms that this influence is more profound on the squeezing mode, which even becomes the dominant mode in a LPL trilayer at 20 V. Panels d−f in Figure 4 suggest that the free charge accumulation at the interfaces can also be increased by increasing ε2 for a LPP and LPL trilayer as compared to a PPP trilayer. The solid and the broken curves in Figure 4f suggest that when ψ is fixed to 20 V the PPP trilayer shows an exclusively unstable bending and a stable squeezing mode of instability at different ε2. In contrast, under similar conditions, the LPP trilayer show the appearance of a subdominant squeezing mode at higher values of ε2 with a dominant bending mode of instability (Figure 4e) However, the curves 2s and 3s in Figure 4d show that for a LPL trilayer a dominant squeezing mode can be observed due to larger attractive force at the elastic−viscous interfaces originating from the larger free charge accumulation at higher ε2. Again, from PPP to LPP to LPL although the bending mode gets stronger with the free charge accumulation at the elastic-viscous interfaces, simultaneously the squeezing mode also gets stronger with this transition and hence remains the dominant mode in the LPL trilayer as seen from Figure 4d. Panels g−i in Figure 4 suggest that the reduction in hM has similar influence as increase in ψ and ε2. The curves in these figures suggest that as the elastic film becomes thinner the attractive force at the deformable elastic−viscous interfaces owing to the charge accumulation becomes stronger to engender a dominant squeezing mode for a LPL trilayer (curves 2s and 3s in Figure 4g). However, this influence gets pacified as one of the liquid layers become dielectric (LPP) or for a PPP trilayer when both the liquid layers are purely dielectric materials. Concisely, the results shown in Figure 4 conveys that the presence of weakly conductive bounding liquid layer(s) can enforce an additional attractive electrostatic interaction between the soft elastic-viscous interfaces of a trilayer due to free charge accumulation when exposed to an external electric field. This effect is strong enough not only to increase the strength of the bending mode of instability but also to fuel up a dominant squeezing mode of instability even at smaller applied field intensities. This effect is found to influence the length and the time scales of instabilities significantly, as discussed in greater detail in the following. Figure 5 shows the variations of the dominant growth coefficient (ωm) and the corresponding wavelength (λm) with ψ

Figure 5. LSA results for the three trilayer configurations. Plots a−b, c−d, and e−f show the variations of [ωm,λm] with ψ, ε2, and hM, respectively. In all plots a−f, curves 1−3 represent the LPL, LPP, and PPP models, respectively, with the solid and the broken lines corresponding to the bending and squeezing modes of instability denoted by symbols “b” and “s”. In general, the parameters used in all the plots are ψ = 20 V, hM = 0.1 μm, and ε2 = 4.

[plots a and b], ε2 [plots c and d], and hM [plots e and f], respectively. In all these plots, curves 1−3 correspond to LPL, LPP, and PPP trilayers, respectively .Plots a, c, and e clearly show that ωm is the lowest for the PPP trilayer, intermediate for the LPP trilayer, and the highest for the LPL trilayer. The curves indicate that indeed the free charge accumulation at the interfaces expedites the electric field induced instabilities with the introduction of the leaky layers. Plots b, d, and f suggest that when the interfaces are unstable by a bending mode the LPL trilayer (solid line) shows the smallest λm whereas the PPP trilayer (broken line) shows smallest λm for a squeezing mode of instability. Curves 1−3 in plots b, d, and f show that λm is lowest at the point of transition from dominant bending [curves 1b−3b] to dominant squeezing mode of instability. For example, the curve 1b in plot b shows that, for a LPL trilayer, the bending mode dominates the instability at lower ψ. With increase in the free charge accumulation at higher ψ, λm progressively reaches a minimum before a transition from dominant bending to dominant squeezing mode takes place. The broken curve 1s in the plot b depicts that λm progressively increases with increase in the free charge accumulation for the squeezing mode. Interestingly, the curves 1s − 3s in plot b suggest that λm for the LPL and LPP trilayers are even larger as compared to the PPP trilayer. It may be noted here that a squeezing mode of instability always require a larger energy penalty owing to the antiphase deformation of the interfaces of the elastic film, as compared to similar in-phase bending deformation of the elastic film. Thus, for a PPP trilayer the length scale progressively increase as the deformation at the interfaces become larger with increase in the applied field intensity. In case of LPL and LPP trilayers, the electrodes G



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Figure 6. LSA results for the three trilayer configurations.Plots a−c show the variations in the normalized length sclae (λM/hM) with variation in ψ, ε2, and hM, respectively. For the given range of parameters ψ, ε2, and hM, the base state surface charge density q0 = −q10 = q20 varies as shown in the plots. In plots a−c, curves 1−3 represent the LPL, LPP, and PPP configurations, respectively, with the solid and the broken lines corresponding to the bending and squeezing modes of instability denoted by symbols “b” and “s”. In general, the parameters used in all the plots are ψ = 20 V, hM = 0.1 μm, and ε2 = 4.

Figure 7. LSA results for the three trilayer configurations. Plots a−c show the variation of δr with ψ, ε2, and hM, respectively. In all the plots, curves 1−3 represent the LPL, LPP, and PPP models, respectively with the solid and the broken lines corresponding to the bending and squeezing modes of instability, and the arrows show the switchover from one such mode to another. In general, the parameters used in all the plots are, ψ = 20 V, hM = 0.1 μm, and ε2 = 4.

especially when the films were thinner even at a larger external field strength.54,56 Figure 6 shows the transition from the longer to shorter wavelength regime with increase in the free charge densities at the interfaces. Plots a−c show the variation in the normalized dominant wavelength (λm/hM) with the variations in ψ, ε2, and hM, respectively. Again, curves 1−3 in the plots correspond to LPL, LPP, and PPP trilayer, respectively, and the symbols ‘b’ and ‘s’ denote the curves corresponding to bending and squeezing modes, respectively. The free charge densities (q0 = −q10 = q20) with the variation in ψ, ε2, and hM is also shown in the plots. The plots suggest that at the intermediate values of ψ, ε2, and hM where the transition from squeezing to bending or vice versa takes place the length scale is of the order of ∼200 nm for a LPL, which is a signature of a short-wave instability. Interestingly, this short-wave instability can be engendered even when the applied potential is of the order of 20 V. Further, the plots depict that at higher or lower values ψ, ε2, and hM the instability progressively shifts toward the longer wavelength regime for both the bending (solid line) and the squeezing (broken line) modes of instability. Concisely, the figure shows that the presence of enhanced electrical stresses at the coupled elastic-viscous interfaces facilitates in bringing down the length scales of patterns to the submicrometer range even at very low external field intensity. Plots a−c in Figure 7 show the variation in the relative amplitudes of deformations (δr) with the variations in ψ, ε2, and hM, respectively. Again, curves 1−3 in the plots correspond to LPL, LPP, and PPP trilayer, respectively, and the symbols ‘b’ and ‘s’ denote the curves corresponding to bending and squeezing modes, respectively. The plots depict that for a LPL or a PPP trilayer the deformations are symmetric across the interfaces (δr = 1 or −1) whereas under similar condition the

possess similar charge as the interfaces when free charges accumulate, which in turn enforces a repulsive influence between the electrodes and the elastic−viscous interfaces. The length scale is the net outcome of the attractive influence between the interfaces and the repulsive influence between the interfaces and the electrodes. In this case the increase in λm is a signature of the enhancement of the repulsive influence between the interface and the electrodes. For the LPL and LPP trilayers, plots d and f show a similar trend for λm as depicted in plot b when the free charge accumulation at the interfaces is enhanced with increasing in ε2 and reduction in hM. However, for the PPP trilayer, the dominant growth coefficient ωm decreases and the corresponding wavelength λm increases with increase in ε2 (curve 3b, Figure 5c, d). This change can be attributed to the two different phenomena dictating the time and length scales of instabilities in the trilayer configuration depending on the presence or absence of leaky viscous layers. In the presence of leaky dielectric liquid layers, increase in the dielectric permittivity increases the surface charge density at the interface(s) and thus, leads to increased electrical stresses. But in the case of pure dielectrics (PPP trilayer), the change in the dielectric permittivity influences the dielectric contrast at the interfaces which is reduced with increase in ε2, thus decreasing the destabilizing electrical stresses at the interfaces, leading to reduction in growth rates of instability. Importantly, plots b, d, and f together show that even at a moderate field intensity the dominant length scale can be reduced to as low as 200 nm employing an LPL trilayer with a 100 nm elastic film. This is remarkable as compared to a similar PPP trilayer or a single nonslipping elastic layer deforming under an external electric field. This is because, in such situations, the minimum length scale observed was of the order of 3hM to 4hM or even larger H



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the contours of the critical destabilizing electrostatic field ψc corresponding to variation in ε2 and hM. Images a and b convey that at a moderately high electric field potential ψ, and small values of the elastic film thickness hM, patterns of the order of a few hundred nanometers can be obtained from this kind of trilayer. Importantly, in this regime the instability is indeed of short-wave in nature. Image c indicates that the critical electric field (ψc) to initiate instability in a LPL trilayer is rather moderate. Briefly, Figure 9 provides an estimate for the future experiments on the requirement of the applied field strength and the typical length scales for the electric field induced instabilities in a LPL trilayer.

LPP trilayer can show asymmetric deformations at the interfaces (1 > δr > − 1). The plots also show that for the LPL and the PPP trilayer δr are similar when the liquid properties are similar for both the bending (solid line) and squeezing (broken line) modes of instabilities under the influence of the external electrostatic field. In contrast, for the LPP trilayer the deformation is more at the interface where the free-charge accumulation takes place as compared to the other interface. In such a situation, bending and squeezing modes of instabilities with dissimilar δr are observed at the interfaces. Figure 8 shows the parametric plots for a symmetric trilayer where the domains of stability or instability have been identified

5. CONCLUSIONS Linear stability analysis has been employed to investigate the electric-field induced instabilities at the coupled elastic−viscous interface in a trilayer configuration containing an elastic film sandwiched between two weakly conducting liquid layers. The present work can be considered as a step ahead in the direction of modeling systems similar to the biological membranes where the biomembrane having elastic properties is confined between conducting aqueous solutions. Major findings are as follows: (i) The present analysis confirms the presence of two modes of instability for the trilayer configuration under the influence of an external electric field, viz., the in-phase bending, and antiphase squeezing mode of instability. The bending mode has long-wave nature and is present even at very small destabilizing fields, whereas the finite wavenumber squeezing mode appears only when either the stabilizing field becomes very weak or the destabilizing field strength becomes too high. A transition from squeezing to bending and vice versa is possible with the variation in the free charge accumulation at the interfaces. This effect also leads to the change in the amplitudes of deformation. (ii) The length scale for the bending mode is found to decay whereas that for the squeezing mode is found to increase with increase in the free charge accumulation. The reduction in the length scale for the bending mode takes place because of the attractive electrostatic interaction between the elastic−viscous interfaces during the in phase deformation. In contrast, the increase in the length scale for the squeezing mode is attributed to the requirement of larger energy penalty for the antiphase deformation of the elastic membrane together with the

Figure 8. Stability diagram depicting the stable and unstable regions for the symmetric trilayer configurations. Plot shows the variation of 1/2 ψ*c [= ψcε1/2 0 /(γ hM) ] with G* c (= Gc/γ hM) where the curves 1, 2, and 3 correspond to LPL, LPP, and PPP models respectively. In general, the parameters used in this plot are hM = 0.1 μm and γ = γ21 = γ32 = 0.005 N m1−.

by varying the nondimensional parameters ψc* [ = ψcε1/2 0 /(γ hM) 1/2] and Gc*(= Gc/γhM) keeping the other system parameters constant. The regions marked “U” and “S” represents the unstable and stable zones. Curves 1−3 correspond to the LPL, LPP, and PPP configurations, respectively. It is evident from this figure that the LPL and LPP configuration can be unstable for parameter ranges where the PPP system remains stable. Again, the figure corroborate that the deformations at the interfaces of an elastic layer can be initiated at a lower applied field intensity for soft elastic membranes by introducing weakly conducting bounding liquid layers. Figure 9 shows the contour plots for the LPL trilayer varying different parameters. Images a and b denote the contours of the dominant wavelength λm and normalized dominant wavelength λm/hM in the plane where ψ and hM are varied. Image c shows

Figure 9. Contour plots for the symmetric LPL configuration. Images a and b depict the variation of dominant wavelength λm and normalized dominant wavelength λm/hM with variation in elastic film thickness, hM and the applied potential,ψ, respectively. Image c shows the contour plot for the critical potential ψc with variation in thickness, hM and dielectric permittivity, ε2 of the middle elastic film. In general, the parameters used in this plot are G = 0.1 MPa and ε2 = 4. I



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(12) Fisher, L. S.; Golovin, A. A. Nonlinear stability analysis of a twolayer thin liquid film: Dewetting and autophobic behavior. J. Colloid Interface Sci. 2005, 291 (2), 515−528. (13) Lenz, R. D.; Kumar, S. Instability of confined thin liquid film trilayers. J. Colloid Interface Sci. 2007, 316 (2), 660−670. (14) Bandyopadhyay, D.; Sharma, A.; Rastogi, C. Dewetting of the thin liquid bilayers on topographically patterned substrates: formation of microchannel and microdot arrays. Langmuir 2008, 24 (24), 14048−14058. (15) Danov, K. D.; Paunov, V. N.; Alleborn, N.; Raszillier, H.; Durst, F. Stability of evaporating two-layered liquid film in the presence of surfactant. I. The equations of lubrication approximation. Chem. Eng. Sci. 1998, 53 (15), 2809−2822. (16) Danov, K. D.; Paunov, V. N.; Stoyanov, S. D.; Alleborn, N.; Raszillier, H.; Durst, F. Stability of evaporating two-layered liquid film in the presence of surfactant. II. Linear analysis. Chem. Eng. Sci. 1998, 53 (15), 2823−2837. (17) Fisher, L. S.; Golovin, A. A. Instability of a two-layer thin liquid film with surfactants: Dewetting waves. J. Colloid Interface Sci. 2007, 307 (1), 203−214. (18) Paunov, V. N.; Danov, K. D.; Alleborn, N.; Raszillier, H.; Durst, F. Stability of evaporating two-layered liquid film in the presence of surfactant. III. Non-linear stability analysis. Chem. Eng. Sci. 1998, 53 (15), 2839−2857. (19) Nepomnyashchy, A. A.; Simanovskii, I. B. Decomposition of a two-layer thin liquid film flowing under the action of Marangoni stresses. Phys. Fluids 2006, 18 (11), 112101. (20) Nepomnyashchy, A. A.; Simanovskii, I. B. Convective flows in a two-layer system with a temperature gradient along the interface. Phys. Fluids 2006, 18 (3), 032105. (21) Nepomnyashchy, A. A.; Simanovskii, I. B. Dynamics of ultrathin two-layer films under the action of inclined temperature gradients. J. Fluid Mech. 2009, 631, 165−197. (22) Pototsky, A.; Bestehorn, M.; Merkt, D.; Thiele, U. Morphology changes in the evolution of liquid two-layer films. J. Chem. Phys. 2005, 122 (22), 224711. (23) Sarkar, J.; Shenoy, V.; Sharma, A. Patterns, forces, and metastable pathways in debonding of elastic films. Phys. Rev. Lett. 2004, 93 (1), 018302. (24) Kumar, S.; Matar, O. K. Dewetting of thin liquid films near soft elastomeric layers. J. Colloid Interface Sci. 2004, 273 (2), 581−588. (25) Matar, O. K.; Gkanis, V.; Kumar, S. Nonlinear evolution of thin liquid films dewetting near soft elastomeric layers. J. Colloid Interface Sci. 2005, 286 (1), 319−332. (26) Chung, J. Y.; Kim, K. H.; Chaudhury, M. K.; Sarkar, J.; Sharma, A. Confinement-induced instability and adhesive failure between dissimilar thin elastic films. Eur. Phys. J. E: Soft Matter 2006, 20 (1), 47−53. (27) Ghatak, A.; Chaudhury, M. K.; Shenoy, V.; Sharma, A. Meniscus instability in a thin elastic film. Phys. Rev. Lett. 2000, 85 (20), 4329− 4332. (28) Gonuguntla, M.; Sharma, A.; Mukherjee, R.; Subramanian, S. A. Control of self-organized contact instability and patterning in soft elastic films. Langmuir 2006, 22 (16), 7066−7071. (29) Monch, W.; Herminghaus, S. Elastic instability of rubber films between solid bodies. Europhys. Lett. 2001, 53 (4), 525−531. (30) Mukherjee, R.; Pangule, R. C.; Sharma, A.; Banerjee, I. Contact instability of thin elastic films on patterned substrates. J. Chem. Phys. 2007, 127 (6), 064703. (31) Shenoy, V.; Sharma, A. Pattern formation in a thin solid film with interactions. Phys. Rev. Lett. 2001, 86 (1), 119−122. (32) Shenoy, V.; Sharma, A. Stability of a thin elastic film interacting with a contactor. J. Mech. Phys. Solids 2002, 50 (5), 1155−1173. (33) Gonuguntla, M.; Sharma, A.; Sarkar, J.; Subramanian, S. A.; Ghosh, M.; Shenoy, V. Contact instability in adhesion and debonding of thin elastic films. Phys. Rev. Lett. 2006, 97 (1), 018303. (34) Gonuguntla, M.; Sharma, A.; Subramanian, S. A. Elastic contact induced self-organized patterning of hydrogel films. Macromolecules 2006, 39 (9), 3365−3368.

repulsive interaction between the elastic−viscous interfaces and the electrodes. The length scale is found to be lowest exactly at the point where the transition from the bending to squeezing mode of instability takes place. (iii) An elastic membrane can be made unstable at smaller field intensity when surrounded by a pair of leaky dielectric liquid layers. The additional destabilizing electrical stresses arising from the free charge accumulation at the interfaces is the reason behind this phenomenon. The confinement of an elastic film by two thin leaky dielectric liquids leads to enhanced electrical stresses at the two viscous-elastic interfaces, which is a result of accumulation of free charges coupled with induced charge separation at the interface(s). This results in a very high destabilizing field even at a very low applied potential, which can be exploited for the pattern miniaturization, cell electro-poration, cell fusion, and cellinjection where there is an upper limit of the applied voltage bias.

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AUTHOR INFORMATION

Corresponding Authors

*Tel.: 91-512-2597026. E-mail: [email protected]. *Tel.: 82-53-810-3239. Fax: 82-53-810-2062. E-mail: swjoo@ ynu.ac.kr. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by grant 2011-0014246 of the National Research Foundation of Korea.

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REFERENCES

(1) Higgins, A. M.; Jones, R. A. L. Anisotropic spinodal dewetting as a route to self-assembly of patterned surfaces. Nature 2000, 404, 476− 478. (2) Segalman, R. A.; Green, P. F. Dynamics of rims and the onset of spinodal dewetting at liquid/liquid interfaces. Macromolecules 1999, 32 (3), 801−807. (3) Sharma, A. Relationship of thin film stability and morphology to macroscopic parameters of wetting in the apolar and polar systems. Langmuir 1993, 9 (3), 861−869. (4) Sharma, A.; Khanna, R. Pattern formation in unstable thin liquid films. Phys. Rev. Lett. 1998, 81 (16), 3463−3466. (5) Thiele, U.; Velarde, M. G.; Neuffer, K. Dewetting: Film rupture by nucleation in the spinodal regime. Phys. Rev. Lett. 2001, 87 (1), 016104. (6) Wyart, F. B.; Martin, P.; Redon, C. Liquid/liquid dewetting. Langmuir 1993, 9 (12), 3682−3690. (7) Bandyopadhyay, D.; Gulabani, R.; Sharma, A. Instability and dynamics of thin liquid bilayers. Ind. Eng. Chem. Res. 2005, 44 (5), 1259−1272. (8) Bandyopadhyay, D.; Sharma, A. Nonlinear instabilities and pathways of rupture in thin liquid bilayers. J. Chem. Phys. 2006, 125 (5), 054711. (9) de Silva, J. P.; Geoghegan, M.; Higgins, A. M.; Krausch, G.; David, M. O.; Reiter, G. Switching layer stability in a polymer bilayer by thickness variation. Phys. Rev. Lett. 2007, 98 (26), 267802. (10) Leopoldes, J.; Damman, P. From a two-dimensional chemical pattern to a three-dimensional topology through selective inversion of a liquid−liquid bilayer. Nat. Mater. 2006, 5 (12), 957−961. (11) Bandyopadhyay, D.; Sharma, A. Dewetting pathways and morphology of unstable thin liquid bilayers. J. Phys. Chem. B 2008, 112 (37), 11564−11572. J



Article

(35) Pangule, R. C.; Banerjee, I.; Sharma, A. Adhesion induced mesoscale instability patterns in thin PDMS-metal bilayers. J. Chem. Phys. 2008, 128 (23), 234708. (36) Ru, C. Q. Surface wrinkling of two mutually attracting elastic thin films due to van der Waals forces. J. Appl. Phys. 2001, 90 (12), 6098−6104. (37) Sharma, A.; Gonuguntla, M.; Mukherjee, R.; Subramanian, S. A.; Pangule, R. C. Self-organized meso-patterning of soft solids by controlled adhesion: Elastic contact lithography. J. Nanosci. Nanotechnol. 2007, 7 (6), 1744−1752. (38) Tomar, G.; Sharma, A.; Shenoy, V.; Biswas, G. Surface instability of confined elastic bilayers: Theory and simulations. Phys. Rev. E 2007, 76 (1), 011607. (39) Yoon, J.; Ru, C. Q.; Mioduchowski, A. Surface instability of a bilayer elastic film due to surface van der Waals forces. J. Appl. Phys. 2005, 98 (11), 113503. (40) Bandyopadhyay, D.; Dinesh Sankar Reddy, P.; Sharma, A. Electric field and van der Waals force induced instabilities in thin viscoelastic bilayers. Phys. Fluids 2012, 24 (7), 074106. (41) Deshpande, P.; Sun, X. Y.; Chou, S. Y. Observation of dynamic behavior of lithographically induced self-assembly of supramolecular periodic pillar arrays in a homopolymer film. Appl. Phys. Lett. 2001, 79 (11), 1688−1690. (42) Harkema, S.; Steiner, U. Hierarchical pattern formation in thin polymer films using an electric field and vapor sorption. Adv. Funct. Mater. 2005, 15 (12), 2016−2020. (43) Lin, Z. Q.; Kerle, T.; Baker, S. M.; Hoagland, D. A.; Schaffer, E.; Steiner, U.; Russell, T. P. Electric field induced instabilities at liquid/ liquid interfaces. J. Chem. Phys. 2001, 114 (5), 2377−2381. (44) Schaffer, E.; Thurn-Albrecht, T.; Russell, T. P.; Steiner, U. Electrically induced structure formation and pattern transfer. Nature 2000, 403, 874−877. (45) Reddy, P. D. S.; Bandyopadhyay, D.; Sharma, A. Self-organized ordered arrays of core-shell columns in viscous bilayers formed by spatially varying electric fields. J. Phys. Chem. C 2010, 114 (49), 21020−21028. (46) Srivastava, S.; Bandyopadhyay, D.; Sharma, A. Embedded microstructures by electric-field-induced pattern formation in interacting thin layers. Langmuir 2010, 26 (13), 10943−10952. (47) Verma, R.; Sharma, A.; Kargupta, K.; Bhaumik, J. Electric field induced instability and pattern formation in thin liquid films. Langmuir 2005, 21 (8), 3710−3721. (48) Voicu, N. E.; Harkema, S.; Steiner, U. Electric-field-induced pattern morphologies in thin liquid films. Adv. Funct. Mater. 2006, 16 (7), 926−934. (49) Voicu, N. E.; Ludwigs, S.; Steiner, U. Alignment of lamellar block copolymers via electrohydrodynamic-driven micropatterning. Adv. Mater. 2008, 20 (16), 3022−3027. (50) Wu, N.; Pease, L. F.; Russel, W. B. Toward large-scale alignment of electrohydrodynamic patterning of thin polymer films. Adv. Funct. Mater. 2006, 16 (15), 1992−1999. (51) Wu, N.; Russel, W. B. Micro- and nano-patterns created via electrohydrodynamic instabilities. Nano Today 2009, 4 (2), 180−192. (52) Bandyopadhyay, D.; Sharma, A.; Shankar, V. Electric-field- and contact-force-induced tunable patterns in slipping soft elastic films. Epl 2010, 89 (3), 36002. (53) Bandyopadhyay, D.; Sharma, A.; Thiele, U.; Reddy, P. D. Electric-field-induced interfacial instabilities and morphologies of thin viscous and elastic bilayers. Langmuir 2009, 25 (16), 9108−9118. (54) Sarkar, J.; Sharma, A.; Shenoy, V. B. Electric-field induced instabilities and morphological phase transitions in soft elastic films. Phys. Rev. E: Stat. Nonlin. Soft Matter Phys. 2008, 77 (3), 031604. (55) Tomar, G.; Shankar, V.; Sharma, A.; Biswas, G. Electrohydrodynamic instability of a confined viscoelastic liquid film. J. NonNewtonian Fluid Mech. 2007, 143 (2−3), 120−130. (56) Dey, M.; Bandyopadhyay, D.; Sharma, A.; Qian, S.; Joo, S. W. Electric-field-induced interfacial instabilities of a soft elastic membrane confined between viscous layers. Phys. Rev. E 2012, 86 (4), 041602.

(57) Reddy, P. D. S.; Bandyopadhyay, D.; Sharma, A. Electric-fieldinduced instabilities in thin liquid trilayers confined between patterned electrodes. J. Phys. Chem. C 2012, 116 (43), 22847−22858. (58) Roberts, S. A.; Kumar, S. Electrohydrodynamic instabilities in thin liquid trilayer films. Phys. Fluids 2010, 22 (12), 122102. (59) Amarandei, G.; Beltrame, P.; Clancy, I.; O’Dwyer, C.; Arshak, A.; Steiner, U.; Corcoran, D.; Thiele, U. Pattern formation induced by an electric field in a polymer-air-polymer thin film system. Soft Matter 2012, 8 (23), 6333−6349. (60) Atta, A.; Crawford, D. G.; Koch, C. R.; Bhattacharjee, S. Influence of electrostatic and chemical heterogeneity on the electricfield-induced destabilization of thin liquid films. Langmuir 2011, 27 (20), 12472−12485. (61) Leach, K. A.; Gupta, S.; Dickey, M. D.; Willson, C. G.; Russell, T. P. Electric field and dewetting induced hierarchical structure formation in polymer/polymer/air trilayers. Chaos 2005, 15 (4), 047506. (62) Morariu, M. D.; Voicu, N. E.; Schaffer, E.; Lin, Z. Q.; Russell, T. P.; Steiner, U. Hierarchical structure formation and pattern replication induced by an electric field. Nat. Mater. 2003, 2 (1), 48−52. (63) Melcher, J. R.; Taylor, G. I. ElectrohydrodynamicsA review of role of interfacial shear stresses. Annu. Rev. Fluid Mech. 1969, 1, 111− 146. (64) Saville, D. A. Electrohydrodynamics: The Taylor−Melcher leaky dielectric model. Annu. Rev. Fluid Mech. 1997, 29, 27−64. (65) Pease, L. F.; Russel, W. B. Electrostatically induced submicron patterning of thin perfect and leaky dielectric films: A generalized linear stability analysis. J. Chem. Phys. 2003, 118 (8), 3790−3803. (66) Wu, N.; Pease, L. F., 3rd; Russel, W. B. Electric-field-induced patterns in thin polymer films: Weakly nonlinear and fully nonlinear evolution. Langmuir 2005, 21 (26), 12290−12302. (67) Craster, R. V.; Matar, O. K. Electrically induced pattern formation in thin leaky dielectric films. Phys. Fluids 2005, 17 (3), 032104. (68) El-Sayed, M. F.; Moussa, M. H. M.; Hassan, A. A. A.; Hafez, N. M. Electrohydrodynamic instability of two thin viscous leaky dielectric fluid films in a porous medium. ISRN Appl. Math. 2011, 2011, 498718. (69) Gambhire, P.; Thaokar, R. M. Electrohydrodynamic instabilities at interfaces subjected to alternating electric field. Phys. Fluids 2010, 22 (6), 064103. (70) Gambhire, P.; Thaokar, R. M. Linear stability analysis of electrohydrodynamic instabilities at fluid interfaces in the ″small feature″ limit. Eur. Phys. J. E: Soft Matter 2011, 34 (8), 84. (71) Gambhire, P.; Thaokar, R. M. Role of conductivity in the electrohydrodynamic patterning of air−liquid interfaces. Phys. Rev. E 2012, 86 (3), 036301. (72) Roberts, S. A.; Kumar, S. AC electrohydrodynamic instabilities in thin liquid films. J. Fluid Mech. 2009, 631, 255−279. (73) Shankar, V.; Sharma, A. Instability of the interface between thin fluid films subjected to electric fields. J. Colloid Interface Sci. 2004, 274 (1), 294−308. (74) Goldberg-Oppenheimer, P.; Mahajan, S.; Steiner, U. Hierarchical electrohydrodynamic structures for surface-enhanced raman scattering. Adv. Mater. 2012, 24 (23), Op175−Op180. (75) Goldberg-Oppenheimer, P.; Mahajan, S.; Steiner, U. Hierarchical electrohydrodynamic structures for surface-enhanced raman scattering. Adv. Mater. 2012, 24 (23), Op175−Op180. (76) Langer, R.; Tirrell, D. A. Designing materials for biology and medicine. Nature 2004, 428, 487−492. (77) Nie, Z. H.; Kumacheva, E. Patterning surfaces with functional polymers. Nat. Mater. 2008, 7, 277−290. (78) Squires, T. M.; Quake, S. R. Microfluidics: Fluid physics at the nanoliter scale. Rev. Mod. Phys. 2005, 77 (3), 977−1026. (79) Ambjornsson, T.; Lomholt, M. A.; Hansen, P. L. Applying a potential across a biomembrane: Electrostatic contribution to the bending rigidity and membrane instability. Phys. Rev. E 2007, 75 (5), 051916. K



Article

(80) Lacoste, D.; Cosentino Lagomarsino, M.; Joanny, J. F. Fluctuations of a driven membrane in an electrolyte. Epl 2007, 77 (1), 18006. (81) Lacoste, D.; Menon, G. I.; Bazant, M. Z.; Joanny, J. F. Electrostatic and electrokinetic contributions to the elastic moduli of a driven membrane. Eur. Phys. J. E: Soft Matter 2009, 28 (3), 243−264. (82) Lekkerkerker, H. N. W. Contribution of the electric double layer to the curvature elasticity of charged amphiphilic monolayers. Phys. A 1989, 159 (3), 319−328. (83) Chou, T.; Jaric, M. V.; Siggia, E. D. Electrostatics of lipid bilayer bending. Biophys. J. 1997, 72 (5), 2042−2055. (84) Schwalbe, J. T.; Vlahovska, P. M.; Miksis, M. J. Lipid membrane instability driven by capacitive charging. Phys. Fluids 2011, 23 (4), 041701. (85) Winterhalter, M.; Helfrich, W. Effect of surface charge on the curvature elasticity of membranes. J. Phys. Chem. 1988, 92 (24), 6865− 6867. (86) Ziebert, F.; Bazant, M. Z.; Lacoste, D. Effective zero-thickness model for a conductive membrane driven by an electric field. Phys. Rev. E: Stat. Nonlin. Soft Matter Phys. 2010, 81 (3), 031912.

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