Electrohydrodynamic patterning of ultra-thin ionic liquid films

In the electrohydrodynamic (EHD) patterning process, electrostatic destabilization of the air–polymer interface results in microand nano-size patterns in the form of raised formations called pillars. The polymer film in this process is typically assumed to behave like a perfect dielectric (PD) or leaky dielectric (LD). In this study, an electrostatic model is developed for the patterning of an ionic liquid (IL) polymer film. The IL model has a finite diffuse electric layer which overcomes the shortcoming of assuming infinitesimally large and small electric diffuse layers inherent in the PD and LD models respectively. The process of pattern formation is then numerically simulated by solving the weakly nonlinear thin film equation using finite difference with pseudo-staggered discretization and an adaptive time step. Initially, the pillar formation process in IL films is observed to be the same as that in PD films. Pillars initially form at random locations and their cross-section increases with time as the contact line expands on the top electrode. After the initial growth, for the same applied voltage and initial film thickness, the number of pillars on IL films is found to be significantly higher than that in PD films. The total number of pillars formed in 1 mm area of the domain in an IL film is almost 5 times more than that in a similar PD film for the conditions simulated. In addition, the pillar structure size in IL films is observed to be more sensitive to initial film thickness compared to PD films.


Introduction
Numerous studies have been conducted on the electrohydrodynamic (EHD) process since initial exploration 1 of this technique where a conical shape is formed on a sessile drop aer placing a charged rod close to the top. [2][3][4][5][6] Early studies focused on either perfect conductor (PC) liquids like water and mercury or perfect dielectrics (PD) like benzene. Deformation and the bursting of uid drops in the presence of an electric eld are experimentally examined, 4 resulting in understanding the role of conductivity in behavior of an isolated emulsion drop in an electric eld. 3 This resulted in the theory of leaky dielectric (LD) or poorly conducting materials. EHD ow is being intensively studied due to extensive industrial applications such as de-emulsication, 7 microuidics, 8 lab on a chip 9,10 and so lithography. [11][12][13][14] In electrohydrodynamic lithography (EHL), a molten thin polymer lm which is heated to above its glass transition temperature is subjected to an electric eld to destabilize the lm to create well-controlled micro-and nano-patterns. The lm patterning process relies on the EHD ow in the polymer lm; so this process will be modeled and simulated. The EHD patterning process starts with an initially at (thermal motion is neglected) liquid thin lm that is conned between two electrodes (the thin lm is denoted as h 0 in Fig. 1). The liquid lm is bounded with either air or another polymer lm to ll the gap between electrodes. Applying a transverse electric eld induces the electric pressure (Maxwell stress) at the lm interface that perturbs the pressure balance and enhances the most unstable wavelength of growing instabilities on the lm interface. Various structures form on the polymer surface depending on the initial lling ratio (ratio of initial lm thickness to electrode distance) of the polymer lm, 15 the electric permittivity ratio of layers (lm and bounding media), 16,17 the shape of the electrodes 16,17 and the surface energy of the electrodes and the polymer lm. 18,19 Fig. 1 Schematic view of the thin film sandwiched between electrodes. h ¼ f(x, y, t) is the film height described by the function with the lateral coordinates and time. l is the maximum growing wavelength.
The majority of experimental and theoretical studies are performed considering the liquid thin lm layers to be either perfect dielectrics (PD), 11,13,15,19 with no free charge, or leaky dielectrics (LD), 12,14,20,21 with an innitesimal amount of charges. In ideal PD materials the conductivity is zero; so when a potential difference is applied across the PD material, the electric eld and potential drop depend on the electric permittivity. In LD materials, charges move and accumulate on the interface when an electric eld is applied. When the charge relaxation time (the time required for movement of charges in response to electric eld) is larger than the growth rate of instabilities, LD lms can be modeled by the PD theory. When charges migrate and accumulate at the interface before the pattern formation occurs, then the charge conservation equation must be solved simultaneously with the uid ow equations. The presence of free charges at the interface results in a signicant decrease in the lm structure length scale. 21 In this study the liquid thin lm is assumed to be an ionic liquid. An IL is dened as a salt like material liquid below 100 C or electrolyte (aqueous or nonaqueous). Free ions present in the ILs and in contact with charged surfaces move toward the oppositely charged surfaces and form a diffuse layer, called the double layer (DL). The DL is characterized by its thickness and Debye length. 22 This diffuse layer, associated with the equilibrium charge, is absent in EHD patterning of PCs, PDs and LDs. 11,12,15,20,21,23 How a complete electrokinetic model seamlessly bridges the gap between perfect dielectric and leaky dielectric models was summarized by Zholkovskij et al. 24 How the EK model accounts for double layer connement and overlap in a liquid cell has also been addressed. 25 To study electrically induced interfacial instabilities of ionic (conducting) uid lms using stability analysis, an electrokinetic model was recently presented. 26 These authors used an analytical solution of the linearized Debye Hückel electrokinetic model to assess the ionic layer stability. For low conductivity uids, the Debye length was found to have a considerable effect on the wavelength of the instabilities when the ionic uid layer thickness is comparable to the Debye length. In the previous cases of ionic liquid lms, 26,27 the linearized Poisson equation (the Debye Hückel model) was used for the electrostatic governing equation. In analyses utilizing the Debye Hückel model, care must be taken not to apply too high an electric potential. Furthermore, electric break down of the polymer lm in high electric elds has an intrinsic limitation in nano-scale EHD patterning processes. 27,28 A more detailed numerical study from our group 29 solved the complete Poisson-Nernst-Planck equation for the ionic layer, and came to the conclusion that a full EK model provides a better understanding of the EHD patterning in ionic uids compared to leaky dielectric models.
The dynamics of ions are modeled by ion conservation. 30 Conductivity of an IL depends on the molar concentration of ions and a higher concentration results in higher conductivity of electrolytes. 26 Electrolytes with high molar concentration have a thinner diffuse layer or Debye length. However, the dynamics of charges in a LD material and ions in ILs can be neglected when the process time is much larger than the charge relaxation time and such a system is in quasi-equilibrium. 20 To the best of authors' knowledge, dynamics, instability and the process of pattern formation of IL lms under an applied electric eld for the EHD patterning process has not been studied. In this work, a system consisting of two planar uid layers (thin IL lm below and PD bounding media above) conned between two at electrodes is considered. When a transverse electric eld is applied in the IL lm the free ions move and accumulate close to the charged surfaces (lower electrode and lm interface) and form a diffuse layer.
An electrostatic model is developed for an IL lm in contact with a PD bounding media to obtain the electric potential distribution and consequently the net electrostatic pressure acting on the interface. The electrostatic pressure model is then used in the nonlinear thin lm equation to investigate the dynamics, instability and process of pattern formation in the EHD patterning process. The size and shape of formed structures are compared with previous results for PD lms 11,15,19 of different thicknesses. The contributions of this paper are: (1) developing an electrostatic model for dynamic modeling of the EHD patterning process using ultra-thin IL lms; (2) investigating the dynamics and pattern formation on IL lms. This includes the initial linear stages and further nonlinear stages in growth of instabilities; and (3) evaluating the effects of ionic strength and the IL lm initial thickness on the shape and size of structures that form on the lm.

Problem formulation
A thin lm system consisting of two layers where the uid layers are assumed to be Newtonian and incompressible with a constant viscosity, m, density, r and electric permittivity 3 is shown in Fig. 1. Unless otherwise indicated, constants and parameters used in the model are listed in Table 1.

Hydrodynamics
The dynamics and evolution of thin polymer lms are described by mass and momentum conservation. For a single layer EHD patterning process, the liquid lm is usually bounded with air. Air can be considered as an inert gas since its viscosity is much less than that of the liquid lm. The liquid layer is thin enough so that gravity effects are negligible. Effects due to inertia are also neglected due to a very small Reynolds number. 20,26 With these assumptions the simplied governing mass and momentum equations of the lm are: Boundary conditions are: (i) No slip condition on the electrodes, z ¼ 0 and z ¼ d No penetration (two media are immiscible): . The subscript i denotes the uid phase (here 1 for the thin polymer lm and 2 for air) and subscripts t, x and y stand for derivatives with respect to time, x and y directions, respectively. k* is the mean surface curvature, g is surface tension, andñ andt i are the normal and tangent vectors of the interface, respectively. 32 The external body force,f e ¼ ÀVf (eqn (2)), accounts for the contribution of external forces to the uid ow. The term f is the conjoining pressure and it is a summation of intermolecular forces (van der Waals and Born repulsive) and electrostatic forces. 15 In eqn (3) the term f vdW is used for the van der Waals forces, dened as f vdW ¼ A l /6ph 3 for the lower electrode. The lower electrode is assumed to have higher energy compared to the polymer lm; therefore the polymer lm is initially stable. Under these conditions the coefficient A l is negative 33 which induces the repulsive van der Waals force. The term f Br is the Born repulsive force, dened as f Br ¼ 8B U /(d À h) 9 for the upper electrode. The Born repulsive force is a short range force and is used to avoid nonphysical penetration of a liquid into the solid phase, mainly for the case in which the interface touches the upper electrode. The coefficient B U is found by setting the conjoining pressure, f, equal to zero at h ¼ d À l 0 to maintain a maximum equilibrium lm thickness (l 0 is the equilibrium distance). The last term in eqn (3), f E , is the electrostatic component of the conjoining pressure which is obtained from Maxwell stress 34 and a more detailed derivation follows.
Applying an electric eld results in the perturbations of the air-polymer interface, h(x, y, t). Initial perturbations grow most quickly with the fastest growing instability wavelength of l, which is much larger than the initial lm thickness, h 0 ; so a "long-wave approximation" 35 can be used to simplify the equations. Using these assumptions results in the "thin lm" equation 19,36 that describes the spatiotemporal evolution of a lm interface as: The term 3mh t is for viscous force present in the system. 13 The term g[h xx + h yy ] is the surface tension force which damps uctuations 33 and minimizes the interface area of the lm and the f term is given in eqn (3).

Electrostatic pressure
To solve the thin lm dynamics it is necessary to have an electrostatic model for f E for PD-PD and PD-IL systems. The derivation of the PD-PD systems is well known and reproduced here for completeness. Throughout this study, it is assumed that electric breakdown does not occur during the EHD patterning process. Using electrolytes, a particular case of ILs, free ions have the ability to migrate or redistribute within the liquid and accumulate on the charged surfaces forming a DL. More details about DL and its regions are presented in the literature. 22,37 The ion conservation equation for the dynamics of free ions in the electrolytes, based on the number concentration of the free monovalent ions (n + and n À ), is: The le hand side of this equation represents the accumulation rate. The rst term on the right hand side is the net ux due to migration and diffusion, respectively. Ion migration is induced by the electric potential gradient, Vj, and the diffusion is the result of concentration gradient Vn AE . The last term, R AE , is the production rate of species due to chemical reactions which is set to zero in this study. m AE is the mobility and e is the elementary charge. The Boltzmann constant k B ¼ 1.3806488 Â 10 À23 (m 2 kg s À2 K À1 ) and T ( K) is temperature. Using denitions for free charge density r f ¼ e(n + À n À ), conductivity s ¼ e(m + n + + m À n À ) and mobility m ¼ D/k B T (D is diffusion coefficient), the ion conservation equation is: This equation is then scaled using nondimensional terms where, s p ¼ mh 0 /3 0 3 1 j 0 2 is the process time and s c ¼ 3 0 3 1 /s is the charge relaxation time. 20 In this study based on the parameters used for simulations, s p and s c vary between 2.35-5.85 (s) and 8.92 Â 10 À11 to 1.05 Â 10 À7 (s), respectively. When the process time is much larger than the charge relaxation time (s p [ s c ), the dynamics of free ions (charges) becomes insignicant 14,26 and le-hand sides of eqn (6) and (7) become zero. In that case the ion conservation equation (eqn (6)) can be simplied to the Poisson equation. 30 where, and n N and j ref are bulk ion number concentration and reference potential, respectively. When the bounding media is air or any PD media, r f ¼ 0, eqn (8) reduces to the Laplace equation. Bulk ion number concentration depends on the molarity of the electrolyte which is dened as n N ¼ 1000N A M where M is the electrolyte molar concentration (mol L À1 ) and N A ¼ 6.022 Â 10 23 mol À1 is the Avogadro number. In eqn (9) the term k B T represents thermal motion while the term e(j À j ref ) is the electrostatic contribution caused by the deviation from the electroneutrality condition (j ref ), respectively. The electrostatics governing equations in the long-wave limit 35 for a bounding layer is and for an IL lm are The boundary conditions for eqn (10), (11a) and (11b) are the applied potential on the lower electrode (j 1 ¼ j l ), grounded upper electrode (j 2 ¼ 0) and the electric potential and displacement continuity (j 1 ¼ j 2 and 3 1 vj is the inverse of Debye length. Solving eqn (10), (12a) and (12b) results in the electric potential distribution within PD and IL layers as follows: where, j s is the interface potential and is determined by the following equation: In ionic liquids, an additional force exists between charged surfaces (lm interface and lower electrode) due to overlapping of DLs, called the osmotic force and dened as F os ¼ ÀdP os / dz. 30,38 The osmotic pressure, P os , has a hydrostatic origin and can be found from the momentum balance in the transverse direction (z). At equilibrium, and assuming that the radius of curvature of the interface is small compared with the IL lm thickness (k*h ( 1), this force is balanced with electrostatic force, F E ¼ Àr (z) (dj/dz), which results in: 39 Thus at a given IL lm thickness there is always a constant difference between osmotic pressure and electrostatic pressure in eqn (16). The term P c can be found from a stress balance at the lm interface. 26 Finally, the term f E in eqn (3), which is the net electrostatic pressure in a PD-IL system is given by 34 Substituting j 1 and j 2 from eqn (13) and (14) into eqn (17) and then using the relationship for j s in eqn (15) result in the following electrostatic pressure for a PD-IL system, Finally, the developed electrostatic pressure (eqn (18)) along with van der Waals, f vdW , and Born repulsion, f Br , pressures are substituted into the thin lm equation (eqn (4)) to nd the dynamics and spatiotemporal evolution of a thin IL lm subjected to a transverse electric eld.

Linear stability analysis
Linear stability (LS) analysis is conducted for a thin IL lm (eqn (4)) bounded with a PD bounding layer to predict the characteristic wavelength for growth of instabilities. In LS analysis, the interface height, h, in eqn (4) is disturbed with a small sinusoidal perturbation of the interface, h ¼ h 0 + d exp(ik(x + y) + ut). This perturbation has a wavenumber of k, an amplitude of d and a growth coefficient of u. The resulting nonlinear terms are neglected and the linear dispersion relationship for the fastest growing wave is found as: and the resulting maximum growing wavelength is:

Numerical modeling
Various numerical techniques are used to track free interfaces in general, 40,41 and in the EHD patterning process specically. 12,15,42,43 These techniques are applied as versatile tools to visualize the transient evolution of a thin liquid lm subject to a transverse electric eld. In this study, the thin lm equations, eqn (4) and (3), are solved numerically to obtain the transient behavior and observe the pattern formation process. The nite difference is used to discretize the spatial derivatives in the 4 th order nonlinear partial differential equation (PDE), eqn (4). The resulting differential algebraic equation (DAE) in time is solved by an adaptive time step ordinary differential equation (ODE) solver. 44 A square domain with the length of 4l and periodic boundary conditions are chosen. Initial conditions are set to a small, with an amplitude of 0.005 Â h 0 , random disturbance of the lm interface while maintaining liquid lm volume constant. The spatial grid size of 121 Â 121 is found to be sufficient and used throughout this study.

Results and discussion
The novelty of this work is the addition of the ionic strength of an IL which is determined by IL molarity, M. This is in addition to typical design parameters such as applied voltage j l , electric permittivity of layers 3 and electrode distance d. Hence, a new parameter the ionic strength of IL, which is determined by IL molarity, M, is added. Higher values of M (¼ 0.1 and 0.01 mol L À1 ) account for higher concentration of ions in the layer which results in perfect conducting (PC) behavior whereas lower values of M (¼ 0.01 and 0.000001 mol L À1 ) represent a poorly conductive medium similar to PDs. 26 The effect of molarity on the electrostatic component of conjoining pressure, as the main force, for the pattern formation process is examined.
To show the effects of molarity on the electric potential distribution across the lm thickness, a 50 nm thick IL lm bounded with a 50 nm PD media (e.g. air) is considered. Results for four different values of molarity, M, are presented in Fig. 2. Electric potential distribution from the solution of the Laplace equation in the bounding media (top 50 nm) is linear as expected. For M ¼ 0.01 mol L À1 , electric potential reduction across the IL lm is almost zero and the IL lm behaves like a PC. In this case, the electric eld within the IL lm is zero. For the limiting case of small molarities, M ¼ 0.00001 mol L À1 , the electric potential distribution is similar to that of a PD lm. For IL lms with molarities between these two limiting values, M ¼ 0.001 and 0.0001 mol L À1 , formation of a DL is evident close to the PD-IL interface and lower electrode (see magnied results in Fig. 2). For M ¼ 0.0001 mol L À1 , overlapping DL from the interface and the base affects the electroneutrality condition of the entire IL layer. This is due to the large DL thickness, k À1 , as compared to the lm thickness. Electroneutrality of the bulk is one main assumption in PD and LD models in the literature 11-15,19-21 but does not apply in this case.

Electrostatic interaction
The value of interface potential, j s , a component of the electrostatic conjoining pressure, depends on the molarity of the IL lm as well as the electric permittivity ratio of layers, 3 1 /3 2 , and the lm thickness, h. Interface potential versus interface height for four molarity values of M ¼ 0.01, 0.001, 0.0001, and 0.00001 mol L À1 are compared in Fig. 3 for a xed applied potential, j l , electric permittivity of layers, 3 1 and 3 2 , and electrode distance,    (18)). For the case in which the PD bounding layer thickness is comparable with the IL lm thickness (h ¼ 50 nm in Fig. 3), a higher slope for interface potential occurs for ILs with low molarity indicating that a small change in interface height changes the surface potential considerably. Fig. 3 conrms that at a constant interface height, the rate of potential drop becomes more signicant for lower molarities. On the other hand, as the molarity increases, interface potential becomes more insensitive to the interface height which is similar to the behavior of PC lms. In the present work ILs are considered as materials whose heightpotential relationship lies between the limits of M ¼ 0.01 and 0.00001 mol L À1 .
The external force in eqn (2) (dened as a gradient of potential, ÀVf) also includes the force due to variation of potential with respect to local lm thickness, À vf vh Vh. This gradient is called the spinodal parameter and lm instability due to this force is called spinodal instability. 15 At a given location, when vf vh \0, a uid ows from regions with lower thickness to higher thickness (negative diffusion) leading to growth of instabilities. To investigate the instability of an IL lm subjected to an electric eld, the term f in eqn (3) is simplied to include only f E as the electrostatic component since this is dominant compared to other interactions for the EHD patterning. 15 The variations of electrostatic conjoining pressure and the spinodal parameter with interface height for IL lms of different molarities are shown in Fig. 4. A negative value of f E shows that the electrostatic force pushes the interface toward the upper electrode, oen called disjoining force. By increasing the lm thickness the electrostatic force is also increased for all three molarity values. Films with higher ion concentration, M ¼ 0.01 and 0.001 mol L À1 , experience higher electrostatic forces compared to the low concentration case of M ¼ 0.00001 mol L À1 . This difference is more apparent at higher lm thicknesses (h 0 > 50 nm), i.e. thicker lms experience a higher force (more negative) than the thinner ones. This force accelerates growth of instabilities at initial stages to enhance the growth rate of structures at later stages.

Scaling results
For thin lm time evolution simulation, LS analysis is used to scale length X ¼ x/L s and Y ¼ y/L s , time T ¼ t/T s and conjoining The interface height is scaled with mean initial lm thick-

Perfect dielectric lmsbaseline
To provide insight into the growth of instabilities in the EHD pattern formation process, PD lms with no free ions (r f ¼ 0) are simulated as a baseline for comparison to IL simulations. Since PD simulation has been done 15,19 the accuracy and capability of the numerical method can be checked. First the dynamics and spatiotemporal evolution of the liquid lm in response to the transverse electric eld are investigated. Nondimensional structure height variations over time and 3-D snapshots of a PD lm interface, with an initial lm thickness of  h 0 ¼ 30 nm, are presented in Fig. 5. Tracking the structure height over time shows that the initial random perturbations rst grow and when the interface touches the upper electrode for the rst time, the minimum height decreases until the lm thickness becomes zero. The 3-D snapshots of the PD lm interface for four stages of the pattern formation process are shown in Fig. 5(a-d). Fig. 5(a) indicates initial random perturbations and formation of bicontinuous structures. At this stage, ridge fragmentation occurs and isolated islands are formed. The uid ows from regions with lower thickness to higher thickness and eventually pillars are formed. Pillars initially have a cone shape (Fig. 5(a)) but become more columnar over time (Fig. 5(b)). The pillar height and width increase over time until they reach the upper electrode ( Fig. 5(b)). Then as the contact line expands, the pillars grow in cross-section (Fig. 5(c and d)). The results in Fig. 5 match known results 15,19 and provide a validation for the code.

Ionic liquid lms
Addition of free ions to the liquid lm is found to increase the electrostatic force signicantly (Fig. 4). EHD patterning of IL lms is investigated in this section. Thin lm height and resulting structures as a function of time using 3-D snapshots of an IL lm interface, with M ¼ 0.001 mol L À1 and initial lm thickness of h 0 ¼ 30 nm, are shown in Fig. 6. Tracking of structure height indicates that the rst pillar forms aer a relatively long time (stage (a)) compared to the total time required for termination of the pillar formation process (stages (a-d)) and compared to PD lms (Fig. 5). The same sequence of pillar formation in IL lms compared to PD lms is observed; however, the pillars in IL lms have a smaller cross-section than those in PD lms (Fig. 5 and 6). For the IL lm, pillars are initially formed on the random locations and enlarged in crosssection with time as the contact line expands on the top electrode. The growth of pillars aer touching the upper electrode depletes the liquid around pillars and forms ring-like depressions around them which are apparent in stage (b).
Increasing the initial lm thickness of an IL lm from h 0 ¼ 30 to 50 nm results in faster growth of instabilities and ultimately a faster pattern formation process (Fig. 7). However, these patterns are less stable since neighboring pillars tend to coalesce at later stages ( Fig. 7(d)). Faster pattern formation and coalescence of pillars at higher initial lm thicknesses are also observed in PD lms.
To further understand the potential of IL lms for making smaller sized patterns in the EHD patterning process, molarity is increased. The effects of molarity and initial lm thickness on  the number of pillars in the EHD patterning process are presented in Fig. 8. For a xed area, having a larger number of pillars means formation of smaller sized pillars on the IL lm interface. The effect of ionic conductivity of IL lms on the number of pillars and how fast they form is investigated by varying the molarities from M ¼ 0.00001 to 0.01 mol L À1 . The results are compared to those of a PD lm of the same initial thickness, h 0 ¼ 30 nm, in Fig. 8(a). In Fig. 8(a) the number of pillars as a function of nondimensional time is plotted. The maximum value of each curve is the nal (or maximum) number of pillars formed under quasi-stable conditions. Under the same applied voltage, the lowest nal number of pillars is formed on the PD lm (16 pillars). 20, 38, 41 and 43 nal pillars are formed for the IL lms with molarities of M ¼ 0.00001, 0.0001, 0.001 and 0.01 mol L À1 . When the molarity of IL lms is increased ten times (from M ¼ 0.00001 to 0.0001 mol L À1 ), the nal number of pillars almost doubled, but further increase in molarity did not have a signicant effect on the nal number of pillars ( Fig. 8(a)). In addition, it is apparent in Fig. 8(a) that the time gap between the formation of rst and the nal pillars in the IL lm decreases with increasing molarity.
Initial lm thickness also affects the nal number of pillars as shown in Fig. 8(b). Nondimensional time, T, is normalized by initial lm thickness cubed; so a modied time is dened as T* ¼ Th 0 3 , to cancel the effect of lm thickness. Thus three initial lm thicknesses of 20, 30 and 50 nm are selected at a constant molarity, M ¼ 0.0001 mol L À1 and pillar formation as a function of T* is plotted in Fig. 8(b). Based on the time axis T*, lms with higher initial thickness and the same interfacial tension and viscosity 13 have faster time evolution which is similar to that observed in PD lms. 19 It is important to note that the above results show that the number of pillars formed on the interface from beginning to the nal number of pillars is under quasi-steady conditions. It has been observed that a coarsening of structure in PD lms with high initial thickness occurs in further later stages beyond these simulations. 13,15 The coarsening of structure in IL lms and related coarsening mechanism will be discussed later. The simulation results show that the number of pillars on the interface increases as the initial lm thickness increases as shown in Fig. 8. The number and density of pillars for three initial thicknesses for both IL and PD lms are given in Table 2. For the IL lm the nal number of pillars formed on the interface is increased signicantly from 22 to 52 when the initial lm thicknesses increased from 20 to 50 nm. The number of pillars formed in 1 mm 2 area of the domain is dened as the density of pillars and is given in Table 2. The pillar density increases for both PD and IL lms as a function of the initial lm thickness with IL lms having almost 5 times more pillar density than PD lms.
To gain further insight into the 2-D spatiotemporal evolution of an IL lm, a lm with 30 nm thickness and a molarity of M ¼   0.00001 mol L À1 is presented in Fig. 9(a-e) using interface height contours. Fig. 9(a) shows an early stage of the pattern formation process and contains an initial bicontinuous structure, isolated islands and a rst columnar structure formation in one snapshot. Isolated islands from the previous stage are developed and converted to a columnar structure as shown in Fig. 9(b). The contact lines expanded on the top electrode and pillars are expanded in cross-section. Depletion of the liquid around pillars and formation of a ring-like depression around them are more obvious at this stage. The later stages of pillar formation are identied in Fig. 9(c) and (d) by dotted circles.
Here coalescence of previously formed pillars is seen. Two neighboring pillars, dotted 1 and 2 in Fig. 9(c), merge and form one larger size pillar, 3 in image (d). The coalescence of neighboring pillars continues over time and pillars in Fig. 9(d) (dotted circle) merge to an even larger size pillar with an oval cross-section as shown in the dotted circle in Fig. 9(e). The dynamics and structure formation on the interface over time for PD and IL lms can also be compared using 1-D spatiotemporal evolution of 50 nm IL lms at Y ¼ 0 (i.e. midplane). For an initial lm thickness h 0 ¼ 50 nm, IL lms with a with molarity of M ¼ 0.001 and 0.0001 mol L À1 are compared to a PD lm and the results are presented in Fig. 10. Increasing time is shown going upward in the subplots. Increasing the ionic conductivity of lms (from zero in the case of PD lms) results in pillars with smaller diameter and consequently denser structures as seen when comparing Fig. 10(a)-(c). The expansion of the contact line on the top electrode is also visible with this 1-D spatiotemporal evolution of the interface, Fig. 10a(ii-iii), b(iii-iv) and c(iv-v). Two mechanisms of pillar coalescence in the EHD patterning process are observed. The rst one is Ostwald ripening of neighboring pillars in a PD lm (Fig. 10a(iv)-a(vi)) in which an incomplete smaller pillar merges with a larger size pillar. 13 The second mechanism is collision of neighboring pillars in an IL lm with a molarity of M ¼ 0.001 mol L À1 (Fig. 10c(v)-c(vii)). In this case, completely formed pillars with almost the same size merge and form larger size pillars. This type of coarsening is also observed in PD lms, h 0 /d < 0.5, which have uniformly distributed pillars. 15,27 An intermediate stage between the PD lm and a highly ionic conductive lm is shown in Fig. 10b(i)-b(vii) where the number of pillars formed is increased compared with the PD lm.

Conclusions
The system under study consists of two planar uid layers, thin ionic liquid (IL) lm and bounding media which are conned with at electrodes. When a transverse electric eld is applied to an IL lm, the free ions move and accumulate close to the charged surfaces. This results in the formation of diffuse layers, called double layers, which are absent in perfect dielectric (PD) and leaky dielectric (LD) models. An electrostatic model is developed for conned IL lms in contact with PD bounding media via coupling of linearized Poisson-Boltzmann and Laplace equations to obtain the electric potential distribution and consequently the net electrostatic pressure acting on the interface.
The developed IL model is not limited to the assumption of having an innitesimally large or small electric diffuse layer that is inherited in the PD and LD models, respectively. This model is coupled to the nonlinear thin lm equation and solved numerically to investigate the dynamics, instability and process of pattern formation in the EHD patterning process. Based on this model, ionic liquids are dened to be materials having properties between two limiting cases of PC and PD materials based on their ionic strength. IL lms experience a much higher electrostatic pressure compared to similar PD lms (with the same electric permittivity) and an increase in ionic strength results in higher electrostatic pressure. PD lms are modeled rst to investigate the accuracy of the numerical scheme and to provide a well known baseline case to compare the results with IL thin lms. Finally IL lms of various molarities are numerically simulated and the structure and number of pillars are compared to those of PD lms. Early stages of the pillar formation process in IL lms are found to be similar to those of PD lms and electrically induced instability growth is caused by negative diffusion. Pillars are initially formed on the random locations and enlarge in crosssection over time as the contact line expands on the top electrode. The pillar formation process occurs faster in IL lms and is further accelerated with increasing ionic strength. Generally, smaller structures are found in IL lms when compared to similar PD lms and are increased with initial lm height. In the case of increasing h 0 from 20 to 50 nm, the IL lm with M ¼ 0.0001 mol L À1 gets 5 times more structures per 1 mm 2 compared to a PD lm.