Steady State of Electrohydrodynamic Patterning of Micro

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Steady State of Electrohydrodynamic Patterning of Micro/ Nanostructures on Thin Polymer Films Qingzhen Yang,† Ben Q. Li,*,‡ Yucheng Ding,† and Jinyou Shao† †

State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, People’s Republic of China ‡ Department of Mechanical Engineering, University of Michigan, Dearborn, Michigan 48128, United States ABSTRACT: This paper presents a nonlinear numerical study of the electrohydrodynamic (EHD) patterning process as a micro/nanostructuring approach for a thin polymer film. Special interest is focused on the steady state (or equilibrium state) in this study. Numerical results show that the polymer nanostructure height depends strongly on the applied voltage, rising higher with an increasing electric field. Moreover, a critical electric field intensity is discovered, above which the deformed film can grow continuously into a final contact with the template to form a nanostructure conformal the pattern premade on the template. An electric field intensity larger than the critical value is necessary for a faithful replica in EHD patterning; however, breakdown may occur for polymer film under a strong electric field and thus fail the process. In this study, the scale limitation is investigated and some methods are then proposed for scaling down the feature size of the structure. film, a hexagonal holerather than pillararray can be observed in experiments.4 To restrict the region in which the pillar array grows, the flat template is usually marked with a protruding pattern or a surfactant pattern. Specifically for a template with a triangular, rectangular, or circular protrusion, a corresponding shaped pillar array is observed underneath the protrusion, and the lattice structure is determined by the geometry of the protrusion.4 With a large thickness, the polymer film evolves into a series of concentric rings,15 and for an even thicker polymer film, instead of obtaining arrayed pillars, they coarsen into one giant pillar and form a polymer mesa which is a positive replica of the protruding template pattern.16 For patternless template, due to the fact that the nucleation happens randomly during the process, the resulting pattern suffers from large numbers of defects. Therefore, the neatly arranged pillar array on polymer film is frequently distorted and becomes irregular; in other words, it is very difficult to achieve uniform patterns over large areas.17 Based on the reason stated above, the template is more often than not patterned with a predesigned structure on it, and a representative configuration is shown in Figure 1. In this case, the dominant destabilizing factor is the nonuniform electrostatic pressure along the polymer/air interface originating from the structures on the template. Driven by the electrostatic pressure, the polymer will be attracted, against surface tension, to the protrusions. Due to the requirement of mass conservation, the polymer underneath the recess areas is drawn down, and thus a positive replica structure to the template is obtained. Compared with the patternless template case, the site and the morphology of the structure on the polymer film are determined by the patterned

1. INTRODUCTION Fabricating controllable micro/nanostructure is of great importance for the semiconductor industry, surface science, lab on a chip, photovoltaic devices, and so forth.1−3 As a novel technique, electrohydrodynamic (EHD, hereafter) patterning4 can be a promising alternative to the conventional photolithograph, with some distinctive advantages in economy and throughput. Moreover, the size of the structure obtained by this process can overcome the barrier of the wavelength of light. Indeed, its capability of achieving sub-100 nm has been demonstrated in experiments5 by an appropriate scaling of the initial polymer film thickness, the gap between the substrate and the template, and the applied voltage. Recently Li et al. discussed the scale limit of EHD patterning and proved its capability of fabricating sub-30 nm structures by numerical simulation.6 In EHD patterning, a thin polymer film is spuncoated on an electrically conducting substrate and is placed under a template with/without a prepatterned structure with an air gap. Then an external voltage is imposed between the template and the substrate. Under a coaction of the electric field and surface tension, the polymer film deforms into various morphologies and further into a solid micro/nanostructure after photo- or thermocuring.7 For a patternless (i.e., flat) template, the electric field in the capacitor configuration generates an interfacial pressure that, if it is strong enough, destabilizes the polymer film. Thus, by overcoming the stabilizing action of the surface tension, a band of modes is selected and, once the nonlinear process sets in, the fastest growing mode (i.e., most unstable wavelength) dominates.8−11 As a result, the polymer−air interface is perturbed and polymer structures grow until they contact the template. Consequently, the initially flat thin polymer film evolves into a well-organized periodic array that bridges the substrate and the flat template.12−14 Usually pillar array scatters in hexagons, with an adjacent-pillar-space inversely proportional to the applied voltage.8 However, for a thick enough polymer © 2014 American Chemical Society

Received: Revised: Accepted: Published: 12720

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carried out by using an integrated discontinuous boundary element/finite element formulation. The electric field is calculated by a discontinuous boundary element method, while the free surface deformation at the polymer/air interface is determined by a finite element method. It is found that the steady state does exist in EHD patterning, and for a weak electric field (that is, below a critical intensity) the corresponding equilibrium state results in a structure with a small growing height determined by a combined action of the Laplace pressure caused by surface tension and the electrostatic pressure. For a field strong enough (that is, above a critical value) the polymer film grows into contact with the upper template and, if the template is prepatterned, a conformal nanostructrure can be produced. Therefore, the steady state model can be used to provide an estimate of a threshold value for the applied field intensity. This intensity is needed to fabricate a high structure on the polymer film. A consequent question would arise, can the thin polymer film (usually micro/ nano scale) bear such a strength? The scale limitation of EHD patterning is investigated, and the effect of parameters on the limit is also discussed in detail.

Figure 1. Configuration of EHD pattering. A thin polymer/air film is sandwiched between a substrate (lower electrode) and a template (upper electrode). An external electric field is imposed between the electrode pair.

template, and thus the micro/nanostructures are well regulated by tuning the shape of the template and other parameters. Accordingly, special interests are focused on the patterned template case in this study. Extensive work on EHD patterning has been done by researchers since it was first invented by Chou13 and Schaffer.12 In experiments, various patternings, such as cone-shaped spikes, cylinders, regular pillar arrays, arrays of holes, gratings, rosettes, and concentric rings have been observed depending on the detailed parameters.4,18 Mesa- and microstructures on thick polymer film were experimentally obtained by Liu et al.;19 they also implemented patterning with large aspect ratios (up to 4.5:1) by tuning the parameters. Recently, Chen et al.20 conducted experimental and numerical work on self-encapsulated hollow microstructures via EHD patterning, which are difficult to obtain by other methods. Zhao et al. 21 experimentally implemented the formation of micro/nanostructures induced by electrostatic charges rather than the external electric field. Besides the electrostatic pressure, other external actions can also be employed to generate micro/ nanostructures on a thin polymer film, such as temperature gradient22 and long-range attractive van der Waals force.23 Theoretically, the problem of EHD patterning has been studied largely on the basis of linear stability analysis and lubrication approximation (with a long wave assumption). This linear model is straightforward to implement and can demonstrate the dynamic initiation of the patterning process but fails for a large deformation in its full cycle.24 For more details, interested readers are referred to the review article by Wu and Russel4 in 2009. Recently, this technique has been extended from the polymer/air bilayer case20 to a polymer/ polymer interface,25 and even to a polymer/polymer/air trilayer case.9,26 A similar phenomenon was reported on conductive polymer film27−30 and elastic film under an electric field.31−38 Some hierarchical structures were obtained by Morariu et al.,25 and then the dynamic process of the trilayer case was simulated by Reddy et al. 39 Thaokar’s group developed some sophisticated numerical models which might be more realistic to the experiments, such as linear and weakly nonlinear models with consideration of polymer’s conductivity.27,29,30 They also theoretically studied the performance of polymer film under alternative voltage.40,41 Sharma’s group conducted extensive work on two- and three-dimensional numerical simulation of the dynamic evolution of EHD patterning,42 and they also studied the trilayer case (polymer−polymer−air film39 or polymer−air−polymer film11). Recently they extended the work to the deformation of elastic film under an electric field.31−36 In this paper, a fully nonlinear numerical study on the EHD patterning process is presented, and the steady state (i.e., equilibrium state) is shed light on. Steady state modeling is

2. MATHEMATICAL MODEL Without loss of generality, the configuration in Figure 1 is considered in this study. The template can be in different morphologies, although the essence of the problem remains the same. A polymer film with an initial thickness of h0 is spun on a substrate, and then an external voltage U is imposed to the electrode pair (template and substrate) with a separating distance d0. The dielectric constant of air (polymer) stands by ε1 (ε2), since ε1 and ε2 are not equivalent, an electrostatic pressure acts across the interface, and it can be laterally spatially modulated by the prestructured template. The interface consequently deforms according to the heterogeneous electric field, which is essentially caused by the patterned structures on the template with a height of Δd. The main focus of the work is the deformation of the polymer/air interface which is represented by h(x). The air (polymer) is considered as an immiscible viscous and incompressible Newtonian fluid and bears a density of ρ1 (ρ2) and viscosity μ1 (μ2). The surface tension of the polymer/air interface is represented by γ, which acts as a resistance against the electrostatic pressure in the process. Subsequently, the governing equations of steady state are given, along with the associated boundary and/or initial conditions. 2.1. Governing Equations and Boundary Conditions. Previous works mainly focused on the dynamic process of EHD patterning; the steady state, however, is also an important issue of EHD patterning and hence is discussed in this paper. In this study, steady state (or equilibrium state) refers to the ultimate deformation of the polymer film and, based on the second law of thermodynamics steady state, corresponds to the minimum value of Gibbs free energy.43 For a flat polymer film under a patternless template, the system is always unstable even under a small voltage and the polymer film evolves into a pillar array bridging the template and substrate.44 Therefore, the steady state of EHD patterning under a flat template is a pillar array and the distance between two adjacent pillars depends on the applied voltage and other parameters.43 This study focuses on the steady state of polymer film under a prepatterned template, in which the height of the polymer structure depends on the applied voltage. In this case, the deformation of polymer film is caused by the nonuniform 12721

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∂Vi =0 ∂n

electrostatic pressure along the polymer/air interface, which origins from the spatially modulated electric field by the structure of the template. Hence the deformation on the polymer film would follow the patterning of the template. For a patterned template, some researchers argued that the periodic length should match the corresponding most unstable wavelength for a faithful replica. The obtained structure periodicity depends on the applied voltage, as it affects the most unstable wavelength. However, it is found in experiments on some conditions that the periodicity of the polymer film follows the template and is independent of the voltage; this can be considered a faithful replica. What conditions can guarantee the faithful replica of a template is still unknown. Some factors leading to the failure of a faithful replica have been found, such as large line/spacing ratio, thick polymer film, and small structure height on the template.19,42,45,46 This study is focused on the case of the faithful replica, in which the structure of the polymer film has the same periodic length as the template. As shown in Figure 1, the configuration is periodic; also, due to symmetry in one period, the computational domain is chosen as only a half-period, which is depicted in Figure 2. By doing this, the computation is much simplified.

∈ ∂Ω1

V1 = V2 ,

ε1

∂V1 ∂V = ε2 2 ∂n ∂n

(1.3)

∈ ∂Ω5|∂Ω5 := Ω1 ∩ Ω 2

(2) σi(1) , j · n ⃗ − σi , j · n ⃗ = γ (∇· s n ⃗ )n ⃗

(2)

where ∇s = ∇ − n⃗(n⃗·∇) is the surface divergence. γ is the surface tension coefficient, and σ(k) i,j is the stress tensor of the fluid k (k = 1 for air, 2 for the polymer), which is a summation of the hydrostatic pressure and electrical stress tensor. Gravity is usually neglected at the micro/nano scale since it is relatively small. In order to simplify the problem, a two-dimensional steady state case is considered first. At steady state, the flow in the normal direction is zero. Actually, a nonzero normal velocity indicates that the interface is still in motion and the state is not steady yet. The polymer and air are assumed to be dielectric materials, which means there are no free charges on the interface, so the tangential velocity is also zero. The flow is quiescent and the viscous contribution is zero; thus the velocity term would vanish in the stress tensor. Substituting the expressions of surface tension stress and electrical stress tensor, eq 2 can be explicitly written as 49

where h(x) is the polymer/air interface, which is boundary 5 in Figure 2. pgag = p(2) − p(1) is the gauge pressure, which represents the pressure difference between the value inside the polymer and that in the ambient air. As stated before, the velocity inside the polymer is zero at steady state, which requires pgag to be a constant everywhere; otherwise flow occurs in the polymer. T stands for the electrostatic pressure, L is the Laplace pressure originating from surface tension, Dn is the normal component of the electric displacement field, and Et is the tangential component of the electric field. The viscosity has no influence on the steady state of polymer film deformation, but it affects the dynamic process significantly. It is worth mentioning that an extra term, M, is added to eq 3, which stands for the polymer−electrode surface interaction via the van der Waals molecular force. In this term, h represents the polymer/air interface measured from the substrate, d is the distance between two electrodes, and A is the Hamaker constant. Physically, this term means a repulsion force exists between the polymer/air interface and the template and between the polymer/air interface and the substrate. The mathematical meaning is that the polymer/air interface is confined between 0 < h(x) < d, which helps to prevent the

(1)

where Vi is the electric potential with electric field intensity E⃗ = −V, and subscript i (i = 1, 2) refers to the different domains (air and polymer). The corresponding boundary conditions are listed as follows: V1 = U

∈ ∂Ω3

where ∂Ω is the boundary of the domain Ω, the superscript i indicates the specific boundary as numerated in Figure 2, and n represents the surface normal of the boundary. So far the governing equation and boundary conditions are completed. At steady state, the Gibbs free energy approaches the minimum point, where the total force along the polymer/ air interface is zero. Hence the implementing method for steady state is the force balance.48 About the free surface deformation, the governing equation expresses

Thaokar and Gambhire29 discussed the electrical behavior of polymer film. They derived the performance of polymer film under an electric field for three different cases: κ−1 ≪ h0, κ−1 ≫ h0, and κ−1 ∼ h0, where κ−1 is the electric diffuse layer and h0 is the polymer film thickness. Both conductivity and the dielectric constant can indeed play an important role in determining the electric diffuse layer. In the present study, both air and polymer are assumed to be perfect dielectric materials and conductivities are ignored, which leads to κ−1 ≫ h0. For this specific case, the free charges are zero and electric displacement is continuous across the interface. The flow motion is relatively slow and the applied electric potential is static, indicating that the magnetic effect is negligible. Hence the electric field is described by a Laplace equation:47 ∈ Ωi|i = 1, 2

V2 = 0

(1.2)

(1.4)

Figure 2. Computational domain of steady state. The numbers 1, 2, 3, 4, and 5 represent the boundaries, and ① (②) represents the air (polymer) phase.

∇2 Vi = 0

∈ ∂Ω2 ∪ ∂Ω4|i = 1, 2

(1.1) 12722

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polymer/air interface from penetrating the template or substrate. This pressure generated by the molecular force is referred to as a disjoining pressure, and its specific expression is taken from Oron et al.50 This pressure would become important when the polymer layer approaches infinitely close to (i.e., touches) the template.51,52 With the consideration of this disjoining pressure, the attachment of the polymer with the template, which has been observed in experiments,44 can be successfully modeled. At the two end points of the free surface, the symmetric boundary conditions are imposed (see Figure 2).

dh dx

= x=0

dh dx

Table 1. Parameters or Constants Used in Simulation

=0 x=l

(4)

value 8.85 × 10−12 F/m 1 2 25 mN/m 480 nm 1600 nm 1600 nm 640 nm 1 10−20 J

A set of curves are depicted in Figure 3 which represent the steady state of the polymer/air interface under different electric

where l is the base length, which is the length of boundary 3 in Figure 2. In eq 3, pgag is a constant everywhere inside the polymer. To determine the specific value of this constant, an auxiliary equation is required which is expressed as follows:

∫0

parameter permittivity of vacuum (ε0) dielectric constant of air (ε1) dielectric constant of polymer (ε2) surface tension coefficient (γ) thickness of polymer (h0) periodic length of template (l0) distance between two electrodes (d0) structure height on template (Δd) line/spacing ratio of template Hamaker constant (A)

l

h(x) dx = V0

(5)

where V0 is the initial volume of the polymer fluid. Equation 5 describes the volume conservation of polymer film. 2.2. Numerical Method. The detailed numerical procedure is described elsewhere,53−55 and only a brief outline is given here. For the steady state, although various numerical methods are available to solve the electric field and its associated properties, the boundary element method (BEM) is chosen in this study, owing to its merit of being geometrically flexible in a multidomain problem.56 This method discretizes only the boundaries of the computational domain and is hence computationally more efficient with a better accuracy. Moreover, the discontinuous boundary elements are employed to treat the discontinuity of the normal component of the electric field at the geometric corners.53,55 The governing equation of free surface deformation (eq 3) is solved by the standard Galerkin finite element method.57 Quadratic elements are employed to accurately describe the free polymer/air interface. Equations 1 and 3 are coupled together. Specifically, the electric field depends on the geometry of the polymer/air interface and on the other hand the interface is determined by the distribution of electrostatic pressure along it. In the scheme, eqs 1 and 3 are solved sequentially and the iterative procedure repeats itself until the convergence is achieved within a preset small tolerance.

Figure 3. Profiles of polymer/air interface. The applied electric field intensity E is varied to investigate its effect on the deformation. The intensities are 3.8 × 107, 5.8 × 107, 6.7 × 107, 7.3 × 107, 7.7 × 107, and 7.8 × 107 V/m correspondingly.

field intensities. To underline the effect of the electric field intensity, all other parameters are kept the same as in Table 1. As can be seen, a curve with a small deformation is obtained under a weak intensity (see the curve of 3.8 × 107 V/m). With the increase of intensity, the deformation of the polymer/air interface gets severe consequently. Referring to Figure 3, the deformation gets pronounced when the electric field intensity is 6.7 × 107 V/m or above. It is plain to understand since a stronger intensity generates a larger electrostatic pressure along the interface and needs a larger Laplace pressure to balance it, thus resulting in a greater deformation. Another phenomenon observed is that the deformation is sensitive to intensity when it gets large. Specifically, a small increase of intensity causes a large rise of interface deformation; see the curves of 7.7 × 107 and 7.8 × 107 V/m for instance. This nonlinear effect is caused by electrical interaction between the template and the polymer/ air interface when they are getting close. For simplification, only a half-periodic interface is calculated as presented in Figure 3. In order to gain a global view of the entire system, the interface is extended by periodicity over several periods after obtaining the results. Some representative examples are depicted in Figure 4. Three different values of intensity are chosen here: 5.8 × 107, 6.9 × 107, and 7.8 × 107 V/m. For the case with intensity of 5.8 × 107 V/m, the resulting equilibrium shape is an interface with small ripples on it. When the intensity increases to 6.9 × 107 V/m, the deformation becomes larger as illustrated in Figure 4b. For an

3. RESULTS AND DISCUSSION 3.1. Steady State of EHD Patterning. With the numerical algorithm stated above, the governing equations of steady state can be discretized and solved accordingly. For convenience, a set of default parameter values are specified as listed in Table 1. These values will remain the same in the following sections unless otherwise indicated. The electric field is characterized by the electric field intensity inside the polymer: E=

ε1U ε1h0 + ε2(d0 − h0 − Δd /2)

(6)

where U is the applied voltage; ε1 and ε2 stand for dielectric constants of air and the polymer. A strong electric field is represented by a larger intensity value E. 12723

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Figure 5. Contours of polymer/air interface under strong electric field intensity. The intensities are 7.9 × 107, 8.6 × 107, 9.6 × 107, 1.06 × 108, and 1.15 × 108 V/m correspondingly.

geometric constraint. If the intensity is further increased, the polymer film starts to spread along the bottom of template and the residual layer of polymer film gets thinner due to the requirement of mass conservation. As what was done before, the contours of the polymer/air interface are extended over multiple periods for a better perspective. The obtained results are presented in Figure 6, and

Figure 4. Polymer/air interface profiles extended to multiperiods. Electric field intensity is also varied: (a) 5.8 × 107, (b) 6.9 × 107, and (c) 7.8 × 107 V/m.

even stronger intensity of 7.8 × 107 V/m, the interface evolves into a notable deformation. As can be seen in Figure 4, the template is also displayed to indicate the geometric relationship between the interface morphology and the template. For each period of the template, one feature forms on the polymer film with the bulge part corresponding to the protrusion on the template, leading to a positive replication of the structure on the template. The reason lies in the fact that the electrostatic pressure is stronger at this point. Throughout the numerical results in this section, the existence of steady state in the EHD patterning process has been demonstrated. To be specific, a small deformed polymer film is obtained when the imposed intensity is weak and a larger one is obtained under a stronger intensity. This indicates that the polymer film cannot grow into a high structure under a weak intensity even after a long time. 3.2. Critical Electric Field Intensity of EHD Patterning Process. The existence of steady state has been demonstrated in section 3.1, and this part subsequently focuses on the steady state under a strong intensity. Figure 5 exhibits some representative contour shapes of the polymer surface as a function of electric field intensity, and other parameters are taken from Table 1. Different from the previous cases, the deformation under a strong intensity is considerably large. As shown in Figure 5, the polymer has already touched the template. Physically, this means the polymer film grows up out of the stability region and eventually is stopped by the geometric constraint. Actually, the main purpose of introducing disjoining pressure between the template and the polymer interface is to resolve this template−polymer contact case. This disjoining pressure (the term M in eq 3) is negligible when the deformation is small since the interface is close to neither the template nor the substrate. When the polymer approaches the template, disjoining pressure comes into play. Generally speaking, for a strong intensity, the polymer film keeps evolving until it touches the template and gets ceased due to

Figure 6. Polymer/air interface profiles under strong electric field intensity. The contours are extended to multiperiods. From (a) to (c), the applied external voltages are 7.9 × 107, 9.6 × 107, and 1.15 × 108 V/m correspondingly.

it is clearly shown that the features on the polymer film are well consistent with a preconfigured template. For each value of electric field intensity, 7.9 × 107, 9.6 × 107, or 1.15 × 108 V/m, the equilibrium state of polymer film is a contacting profile with template in every case. The template−polymer contact part gains its area with the increase of intensity and the contour of 12724

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the interface changes from a smooth cosine wave (7.9 × 107 V/ m) to a periodic step curve with steep edges (1.15 × 108 V/m). Comparing the steady state of 7.8 × 107 V/m and that of 7.9 × 107 V/m, it can be seen that a small increase of electric field intensity causes a dramatic change of polymer film morphology. This intriguing phenomenon would be the main focus and is investigated in this section. First, the structure height is employed to characterize the deformation; specifically, it is defined as the difference of the highest point and the lowest one along the interface. After obtaining the steady state of the polymer film, the relationship of the structure height and the electric field intensity can be further obtained with ease and is depicted in Figure 7. As can be seen, the height increases with

down of EHD patterning. Discussion on the scale limit of EHD patterning is the main focus of this section, and some methods are then proposed to scale down the structure. To investigate the scale limit of EHD patterning, first the periodic length l0 (as shown in Figure 1) is chosen to characterize the scale of the structure. As stated before, an intensity stronger than the critical value is required for a faithful structure formation. In this case, the corresponding electric field intensity is required to be smaller than the dielectric strength of polymer: Ec ≈

ε1Uc ≤ E break ε1h0 + ε2(d0 − h0 − Δd /2)

(7)

where Ec represents the critical electric field intensity and Ebreak is the strongest electric field that the polymer can bear (i.e., dielectric strength). For the given parameters in Table 1, the critical intensity is 7.8 × 107 V/m, which means that a structure with periodicity of 1600 nm can be obtained if the dielectric strength of the polymer film is larger than this value; otherwise breakdown occurs and the patterning process would fail. In the same procedure, the scale limit is obtained for different parameters. The critical intensity is a function of the configuration scale, dielectric constant, and surface tension, Ec(l0, d0, h0, Δd, ε2, γ), and its specific value can be obtained by a numerical method for given parameters. To reduce the number of independent variables, the values of d0/l0, Δd/l0, and h0/l0 are fixed as 1.0, 0.4, and 0.3 respectively for different scale l0. Thus the critical intensity becomes a function of the structure scale, dielectric constant, and surface tension, Ec(l0, ε2, γ). For a given dielectric strength, the scale limit can be further determined by eq 7 and the results are shown in Figure 8. As

Figure 7. Relationship of structure height and electric field intensity. The curve is divided into three different phases represented by A, B, and C correspondingly.

the intensity for the reason stated before. The whole curve can be divided into three parts: A, B, and C. In phase A, a small deformed structure is obtained under a weak intensity, and the deformation is getting severe with increasing intensity. If the intensity is strong enough, polymer grows up until contacting with the ceiling (i.e., template); this phase corresponds to part C in the curve. In phase C, the structure height is less than the electrode distance (960 nm for this case) which is due to the repulsion effect of disjoining pressure. For an even stronger intensity, the height can be further increased as the residual layer is thinned. Phase B is the intermediate stage, and no corresponding steady state can be obtained. This particular value of the intensity is defined as the critical intensity, and its specific value is 7.8 × 107 V/m with the parameters in Table 1. The definition of critical intensity has significant meaning for experiments: it is the minimum intensity to generate a high enough structure that bridges the template and substrate. Heier et al. conducted some experiments on the steady state of EHD patterning.44 They found that a small deformed structure is obtained under a weak intensity and, for a strong intensity, the polymer film grows until it is in contact with the template. All those results are consistent with our numerical prediction. 3.3. Scale Limit of EHD Patterning. Usually a high structure bridging the template and substrate is favorable in experiments, and based on the above analysis, an intensity greater than the critical value Ec is necessary to guarantee this structure. Theoretically, the scale of the EHD patterning structure can be infinitesimal, as long as the resolution of the template is high enough and the intensity is greater than the critical value. In a real system, however, electrical breakdown may occur under a strong intensity, which hinders the scaling

Figure 8. Scale limit of EHD patterning. (a) Surface tension is kept as γ = 25 mN/m, and the dielectric constant ε2 is taken as 2.0, 3.0, and 5.0. (b) Dielectric constant of polymer is ε2 = 2.0, and the surface tension γ is taken as 25, 10, and 5 mN/m.

clearly depicted, the scale limit decreases with the dielectric strength of polymer, since this polymer material can stand a stronger intensity, and thus leads to a smaller size structure. When the dielectric constant ε2 = 2.0 and surface tension γ = 25 mN/m, for instance, the scale limit is 1 μm when the dielectric 12725

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strength Ebreak = 1 × 108 V/m, while it is 100 nm with Ebreak = 3.1 × 108 V/m. According to Paschen’s law,58 the dielectric strength is not a constant but is a variable depending on the scale. To be specific, it increases with decreasing scale, which is favorable for scaling down the micro/nanostructures. Another issue is that field electron emission may occur under an ultrahigh electric field59 (usually 109V/m), which should be avoided in experiments even if the polymer can stand such a strong field. Besides the dielectric strength, the surface tension and dielectric constant are also important factors affecting the scale limit. As can be seen from Figure 8, increasing the dielectric constant and decreasing the surface tension can obviously scale down the limit of the structure size. In experiments, the air gap is sometimes replaced by another polymer and this polymer/ polymer case can generate a smaller structure as the surface tension is reduced compared with the polymer/air interface.22 3.4. Analytical Formula of the Critical Intensity. The nonlinear numerical method proposed above can accurately describe the steady state of EHD patterning and predict the critical intensity. However, for a fast estimation of the approximated value of critical intensity, an analytical formula is more convenient. As stated before, the governing equation is expressed in eq 3. In order to simplify the governing equation and obtain the analytical solution, the lubrication approximation is employed, ∇h ≪ 1. The lubrication approximation is valid only for a small deformation, where the structure height is much smaller than the periodic length. Although it would fail for a large deformed polymer film, it still leads to a feasible analytical solution to estimate the polymer film deformation and is widely employed by researchers.4,8,12 The tangential electrical component Et is relatively small compared with the normal component En due to the lubrication approximation. If the deformation is small, disjoining pressure can be ignored compared with electrostatic pressure and Laplace pressure, as the interface is far from the template or the substrate. Thus, the governing eq 3 yields a simple form: γ ∇2 h + pgag +

⎛1 1 1⎞ ε0Dn 2⎜ − ⎟ = 0 2 ε2 ⎠ ⎝ ε1

Under the lubrication approximation, the normal component of electrical displacement Dn simplifies to Dn =

+∞

(11)

⎫−1 ⎬ − ε0ε1ε2(ε2 − ε1) E ⎪ ⎭ 2 2⎪

(12)

The negative sign on the right-hand side of eq 12 represents the phase difference between the polymer structure and the template. As can be seen, the structure height is a function of the polymer film thickness, periodic length, structure height on the template, dielectric constant, surface tension, and applied voltage. Equation 12 can be employed to estimate the steady state of EHD patterning. The structure height would go to infinity when the denominator approaches zero. Although the critical voltage represents the value with a large structure height (e.g., the polymer contacts the template), the voltage corresponding to a zero denominator can be considered as the critical value approximately, which is expressed as Ec =

⎤ ⎞ ε ⎡ ⎛ Δd 2π γ 1 ⎢ε2⎜d0 − − h0⎟ + ε1h0 ⎥ ⎝ ⎠ ⎦ l0(ε2 − ε1) ε0ε2 ⎣ 2 (13)

For the given parameters in Table 1, the calculated critical intensity by eq 13 is 2.1 × 108 V/m, which is much larger than the nonlinear numerical result (7.8 × 107 V/m). The discrepancy between them is caused by the ignoring the nonlinear effects in the analytical solution. To account for the nonlinear effects and make the analytical formula more realistic, the expression is corrected by adding a coefficient Cr:

(8)

Ec = Cr

⎤ ⎞ ε ⎡ ⎛ Δd 2π γ 1 ⎢ε2⎜d0 − − h0⎟ + ε1h0 ⎥ ⎠ ⎦ l0(ε2 − ε1) ε0ε2 ⎣ ⎝ 2 (14)

Nonlinear numerical simulations are conducted for different parameters, and the results are compared with the analytical solution. Then the coefficient Cr is determined by minimizing the deviation between numerical and analytical results:

(9)

⎛ 2πnx ⎞ ⎛ 2πx ⎞ ⎟ ≈ h0 + h1 cos⎜ ⎟ ⎝ l0 ⎠ ⎝ l0 ⎠

∑ hn cos⎜ n=1

= ε2E

h(x) ε2

⎡ ⎤ 2 h1 = −⎢ε0ε12ε2 2(ε2 − ε1) ΔdE2 ⎥ ⎣ ⎦ π 2 ⎧ ⎡ ⎛ ⎤ ⎪ ⎛ 2π ⎞ ⎞ Δd ⎨γ ⎜ ⎟ ε12⎢ε2⎜d0 − − h0⎟ + ε1h0 ⎥ ⎪ ⎝ ⎠ ⎦ 2 ⎩ ⎝ l0 ⎠ ⎣

+∞ ⎛ Δd ⎞ 2 1 ⎛ nπ ⎞ ⎛ 2nπx ⎞ ⎟ + Δd ∑ sin⎜ ⎟ cos⎜ d (x ) = ⎜d 0 − ⎟ ⎝ 2 ⎠ ⎝ l0 ⎠ ⎝ 2 ⎠ π n=1 n

h(x) = h0 +

+

Substituting eqs 9−11 into the governing eq 8, the amplitude of the structure height can be obtained.

The geometry of the template boundary d(x) and the interface h(x) can be decomposed into a set of Fourier series:

⎛ 2πx ⎞ ⎛ Δd ⎞ 2 ⎟ + ≈ ⎜d 0 − Δd cos⎜ ⎟ ⎝ 2 ⎠ π ⎝ l0 ⎠

V0 d(x) − h(x) ε1

Cr = (10)

where n is the simple harmonic number of the Fourier series and hn is the varying component of the interface. The amplitude of the n-order (n > 1) waves is smaller than the amplitude of the first-order wave. As the simple harmonic number (n) increases, the value of the n-order waves approaches the real value. For simplification, the Fourier series are truncated to first order. h0 represents the initial polymer film thickness, and h1 is the amplitude of the structure and is the variable that needs to be solved.

0.86 − h0 /d0 1.42 + 0.04ε2 /ε1

(15)

After correction, the critical voltage predicted by the analytical formula is displayed in Figure 9 together with the numerical results. As evidenced in Figure 9, the analytical solutions are well consistent with numerical ones in a wide range. Accordingly, eq 14 provides a faithful prediction of the critical voltage of EHD patterning which is very straightforward to implement. Figure 9 also reveals that some treatments are effective to lower the critical intensity, such as increasing the dielectric constant of the polymer, decreasing the surface 12726

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Article

formula is proposed to predict the critical voltage, which is a straightforward-implementing method without much loss of accuracy, and it is proved to agree well with the numerical results in a wide range. All these results could provide a meaningful guide for experiments.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: (313) 593-5241. Fax: (313) 593-3851. Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work is financially supported by the Major Research Program of NSFC on Nanomanufacturing (Grant 91323303), NSFC Fund (Grants 51175417 and 51275401), and Program for New Century Excellent Talents in University (NCET-130454).

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REFERENCES

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Figure 9. Critical electric field intensity predicted by both the analytical method (eq 14) and nonlinear numerical method. (a) The dielectric constant of polymer ε2 is varied. (b) The initial thickness of polymer film h0 is varied. (c) The surface tension coefficient of polymer/air interface γ is varied. Other parameters are the same as in Table 1.

tension coefficient, or increasing the initial thickness of polymer film.

4. CONCLUSIONS In this study, fully nonlinear modeling for the steady state of EHD patterning for micro/nanostructure formation has been conducted. A coupled discontinuous boundary element/finite element model was developed to represent the steady state of an EHD patterning process. The structure height of polymer film is found to be closely dependent on the applied voltage, increasing with a stronger field. A critical electric field intensity is discovered during simulation, the minimum value above which the EHD deformation causes the polymer film to bridge the template and the substrate, forming a nanostructure conformal the template if prepatterned. The electric field intensity needs to be greater than this critical value for a faithful replica of the structure on the polymer. However, for a thin polymer film, breakdown occurs under a strong electric field and thus prevents the miniaturization of the structure. A scale limit exists for given parameters, and it is explored in detail and the feasibility to fabricate a sub-100 nm structure is proved by optimizing the parameters, such as the dielectric constant, surface tension, and dielectric strength. At last an analytical 12727

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