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Numerical Characterization of Electrohydrodynamic Micro- or Nanopatterning Processes Based on a Phase-Field Formulation of Liquid Dielectrophoresis Hongmiao Tian, Jinyou Shao,* Yucheng Ding,* Xiangming Li, and Hongzhong Liu Micro- and Nano-manufacturing Research Center, State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China ABSTRACT: The electrohydrodynamic patterning of polymer is a unique technique for micro- and nanostructuring where an electric voltage is applied to an electrode pair consisting of a patterned template and a polymer-coated substrate either in contact or separated by an air gap to actuate the deformation of the rheological polymer. Depending on the template composition, three processes were proposed for implementing the EHDP technique and have received a great amount of attention (i.e., electrostatic force-assisted nanoimprint, dielectrophoresis−electrocapillary force-driven imprint, and electrically induced structure formation). A numerical approach, which is versatile for visualizing the full evolution of micro- or nanostructures in these patterning processes or their variants, is a desirable critical tool for optimizing the process variables in industrial applications of this structuring technique. Considering the fact that all of these processes use a dielectric and viscous polymer (behaving mechanically as a liquid) and are carried out in ambient air, this Article presents a generalized formulation for the numerical characterization of the EHDP processes by coupling liquid dielectrophoresis (L-DEP) and the phase field of the air−liquid dual phase. More importantly, some major scale effects, such as the surface tension, contact angle, liquid−solid interface slip, and nonNewtonian viscosity law are introduced, which can impact the accuracy of the numerical results, as shown experimentally by our electrical actuation of a dielectric microdroplet as a test problem. The numerical results are in good agreement with or are well explained by experimental observations published for the three EHDP processes.

1. INTRODUCTION

EHDP has received a considerable amount of attention as a micro- or nanostructuring approach because of its unique feature (i.e., the fact that only a minimized external pressure is needed to maintain proper contact or clearance of the template over the liquid polymer) in which the hydraulic flow of polymer is driven dominantly by the pattern-modulated electric field. This is quite contrary to some micro- or nanomolding processes, such as UV imprint lithography 4 and hot embossing,5 where typically a comparatively large external force is required to press a patterned template mechanically against the substrate, possibly leading to poor geometrical integrity in the duplicated structure or even to irreversible damage of the template and substrate,6 making multilayer overlay alignment difficult.7,8 Three processes have been proposed to implement the EHDP technique, including electrostatic force-assisted nanoimprint lithography (EFAN),9 dielectrophoresis−electrocapillary force-driven imprint lithography (DEP-ECF DI),10 and electrically induced structure formation (EISF),11,12 depending on the template composition used, as illustrated in Figure 1.

When an electric voltage is applied between a geometrically structured and conductive template and a conductive planar substrate coated with a dielectric liquid layer, a nonuniform electric field is induced on the air−liquid interface because of the spatial heterogeneity. Correspondingly, a spatially nonuniform electrodynamic force (or Maxwell stress tensor, as it is often called) is generated on the interface and tends to drive the liquid toward the template region of higher electric intensity. This phenomenon, widely referred to as liquid dielectrophoresis (L-DEP), has been explained and comprehensively reviewed by Jones.1−3 Obviously, the liquid will tend to collect preferably toward template protrusions if the template is separated by a small air gap from the liquid surface or to flow into template cavities along cavity sidewalls if the template is kept in contact with the liquid surface. On the basis of this L-DEP behavior, a process called electrohydrodynamic patterning (EHDP) can be devised for polymer micro- or nanostructuring by applying an electric voltage to an electrode pair consisting of a micro- or nanopatterned template and a substrate uniformly coated with a dielectric and rheological polymer, followed by photo- or thermocuring of the deformed polymer. © 2013 American Chemical Society

Received: August 13, 2012 Revised: March 13, 2013 Published: March 18, 2013 4703

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applications can be devised for fabricating microlens arrays,19 for instance. Different explanations have been developed for each of these patterning processes. The application of a dc voltage in EFAN was considered to be able to produce a uniform electrostatic pressure that pulled the template and the substrate toward one another to drive the liquid into template cavities.9 The filling of polymer in template cavities for DEP-ECF DI was attributed to the polymer electrowetting of the dielectric10 (or simply EWOD20), which is due to L-DEP by an electric field and is produced at the solid−liquid−air three-phase contact line along the cavity sidewall. On the basis of the linear instability theory of a thin liquid film that is affected by an initially uniform electric field, a Navier−Stokes formulation considering the electrically induced Maxwell tensor was presented to characterize the EISF process analytically for a nonpatterned (i.e., flat) template,21 leading to the concept of most unstable wavelength, which was then extended to explain the microstructuring dynamics for the EISF process using a patterned template.22 Numerical simulations based on the most unstable wavelength of a freely moving polymer surface were also performed for the EISF process with templates patterned with various periodic microstructures,23,24 demonstrating the dynamic growth of the microstructure from an initially flat polymer surface. However, this numerical approach cannot be extended for DEP-ECF DI or EFAN because the polymer surface cannot move freely as a result of contact constraints by cavity sidewalls of the patterned template, as shown in Figure 1A,B. It should be noted that these three processes or their variants share a common feature (i.e., all using a dielectric and viscous polymer (behaving mechanically as a liquid) and carried out in ambient air (as another fluid)). Obviously, numerical simulation with a formulation that is versatile for the three patterning processes or their variants is highly desirable as a critical tool for understanding the full structural evolution and optimizing the process variables in industrial applications of this EHDP technique. This Article presents a generalized approach to the numerical characterization of the EHDP processes based on a phase-field formulation of L-DEP. The introduction of a phase field into the L-DEP problem domain can deal with the physical interaction at the air−liquid interface well because the air and liquid volumes are treated as one integral domain with continuously varying physical properties (viscosity, mass density, and electric permittivity) across the air−liquid interface, which is considered to have a tiny thickness due to phase diffusion, as demonstrated in our previous publication.25 Because the phase-field function is defined as −1.0 for the air and 1.0 for the polymer, it is straightforward to track the moving air−liquid interface by tracing a zero-value contour of the continuous phase field. More importantly in this Article, we have accounted for some typical scale effects, such as the surface tension, contact angle, liquid−solid interface slip, and non-Newtonian viscosity, which cannot be ignored with respect to the hydrodynamics of polymer manipulated on the micro- or nanoscale.26−29 The numerical simulation is experimentally proven for the electric actuation of a polymer microdroplet on a planar surface, as a test problem, and then performed for the three EHDP processes, which agrees well with and explains some published experimental observations.

Figure 1. Three EHDP processes (with arrows indicating the vertically projected Maxwell tensor at the liquid surface). (A) Electrostatic forceassisted nanoimprint lithography (EFAN), where the template is moveable toward the substrate after the template is in uniform contact with the polymer; (B) dielectrophoresis−electrocapillary force-driven imprint lithography (DEP-ECF DI), where a fixed position of the template is maintained relative to the substrate after the template is in uniform contact with the polymer; and (C) electrically induced structure formation (EISF), where a fixed clearance is maintained between the template and the substrate.

For EFAN (Figure 1A), the template pattern is dielectric and backed with a conductive planar plate and is kept in contact with the polymer while a dc voltage capable of generating micro- or nanostructure with good consistency over a large area is applied. This ability makes it potentially applicable as a manufacturing technique for various sectors of nanotechnology.13,14 Being similar to EFAN, DEP-ECF DI (Figure 1B), as explored by us to produce micro- or nanopatterns with high aspect ratios uniformly over a large area, uses a template pattern that is electrically conductive and coated with a thin dielectric layer and is maintained in contact with the polymer while a voltage is applied. The ability to produce structures of high aspect ratio on the micro- or nanoscale (and therefore over a significantly increased actual surface area) is especially useful for in vitro cell or tissue manipulation, microfluidics components, and various chemical sensors.15−17 Obviously, the cavity sidewalls on the structured template for EFAN and DEP-ECF DI serve as the constraining borders that mold the polymer into exact duplicates of the template structure; therefore, the mass production of the micro- or nanostructure can be precisely and economically performed. EISF (Figure 1C) uses a conductive and patterned template that is kept at a fixed clearance over the polymer surface with an air gap and generates a spatially nonuniform electrodynamic force on the polymer surface when a voltage is applied. Although EISF may not be applicable for generating structures with high aspect ratios,18 it allows for easy removal of the patterned template from the cured structure because no contact between the cured structure and the cavity sidewall of the template is involved. Also because the structure generated by EISF tends to have a curvature that is well determined by the electric modulation and the surface tension, some interesting 4704

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of 4.1641γ whereas a 99% variation in φ corresponds to a thickness of 7.4859γ.33 Besides, λ and γ are related to the surface tension coefficient σ by the formula σ = (2(2)1/2)/3λ/γ. The total free energy Ffree is an integration of the free-energy density on the problem domain Ω; therefore,

2. NUMERICAL MODELING The numerical simulation proposed in this Article is to trace the air−polymer dual-phase interface, which moves dynamically and varies geometrically under the influence of an electric field. Because the template and substrate for the EHDP processes are considered to be solid boundaries for the problem domain, the dynamically varying shape of the air−polymer interface illustrates the temporal evolution of the polymer structures. The numerical model proposed here couples L-DEP and the phase field, where L-DEP depicts the effect of the electric field on the air and polymer and the phase field of the dual-phase flow describes the movement of the air−polymer interface. 2.1. Phase-Field Formulation of Dual-Phase Flow. In a phase-field formulation for the dual-phase flow of air and polymer,30−34 the air−polymer interface is considered to be diffusive physically and to have a tiny thickness (usually on the micrometer or nanometer scale, which should be much smaller than size of the problem domain). Therefore, the problem domain, composed of the air and the polymer with distinct physical properties, can be treated as an integral domain with continuous physical properties by introducing a continuous phase-field function φ, which varies from −1 for air to 1 for the polymer, with the highest gradient appearing in a region of the air−polymer interface. Moreover, volumetric fractions can be defined as Vf1 = (1 − φ)/2 and Vf2 = (1 + φ)/2 for the air and the polymer, respectively. Correspondingly, the physical properties in the problem domain as a whole can be represented in terms of φ

Ffree(φ) = +

G=

(1)

where subscripts air and poly denote the air and polymer, respectively, with ρ, ε, and η representing the mass density, permittivity, and viscosity, respectively, for the problem domain as a whole. The mass conservative law for the problem domain can then be represented in terms of the Cahn−Hilliard equation for dual-phase diffusion as follows30

1 λ 2 λ |∇φ|2 + (φ − 1)2 2 4γ

⎡ ∂Ffree φ(φ 2 − 1) ⎤ = λ⎢ −∇2 φ + ⎥ ∂φ ⎦ ⎣ γ2

⎧ ∂u ⃗ + ρ u ⃗ · ∇u ⃗ ⎪ρ ⎪ ∂t ⎪ = −∇p + η∇2 u ⃗ + 1 η∇(∇·u ⃗) + f e + ρg ⎨ 3 ⎪ ⎪ ∂(ρu ⃗) + ∇·(ρu ⃗) = 0 ⎪ ⎩ ∂t

(2)

where ∇ stands for a gradient operator, u⃗ stands for the fluid velocity, μ stands for the fluid mobility determining the diffusivity, and G stands for the chemical potential. The second term on the left-hand side of the equation represents the convection flux, and the right-hand term is the diffusion flux. The chemical potential G is a partial differential of the total free energy Ffree with respect to φ (i.e., G = (∂Ffree)/(∂φ)), and the free-energy density f free, for the problem domain consisting of air and liquid can be expressed in a Ginzburg−Landau form as follows32,34−38 ffree (φ) =

(4)

(5)

The expression above for G has to be substituted into eq 2 to solve the Cahn−Hilliard equation. When the phase-field method is utilized to describe the motion of the air and polymer under an electric field, an initial phase field has to be defined with the formulas φair = tanh((d)/ (21/2γ)) for the air and φpoly = −tanh((d)/(21/2γ)) for the polymer, respectively, on the basis of an analytical steady-state solution of eq 2 for a straight, nonmoving interface,33 where d is the distance away from the air−polymer interface. The radius of gyration for most polymers in a liquid form, τ, which is the mean square root of the distances of all repeating units in a molecular chain from the center of mass of the chain, is generally only several nanometers, as estimated from the point of view of molecular physics.39 Because the characteristic length of the involved phase-field domain, Lc, is much larger than this radius for the micro- or nanostructuring experimentally implemented so far for the three processes, the Knudsen number, Kn, which is defined as Kn = τ/Lc, can usually become less than 0.1.26 Therefore, the movement of the fluid can be depicted by the Navier−Stokes equation of momentum and mass conservation as follows25

ε = εairVf1 + εpoly Vf2

∂φ + ∇·(u ⃗φ) = ∇·(μ∇G) ∂t

λ 2 (φ − 1)2 dΩ 4γ

Hence, the chemical potential can be expressed as follows:

ρ = ρair Vf1 + ρpoly Vf2

η = ηair Vf1 + ηpoly Vf2

∫Ω ffree (φ) dΩ = ∫Ω 12 λ |∇φ|2

(6a, 6b)

where ρ and η are the mass density and viscosity represented in terms of the phase-field function as in eq 1, respectively. p is the hydraulic pressure in the fluid, and fe denotes the volumetric eletrodynamic force generated by an electric field, as will be introduced in section 2.2. It is worthwhile to note that the air should be treated as a compressible fluid, especially for the DEP-ECF DI and EFAN processes where the air can be trapped in spatially enclosed micro- or nanocavities on the template. Therefore, the air’s mass density is considered to be dependent on the hydraulic pressure p, as represented by the ideal gas law ρair = pM/RT, where M is the air’s molecular weight, R is the gas constant, and T is the temperature. 2.2. Electrodynamics for L-DEP. Because the air and polymer involved in EHDP are both dielectric, free electric charges and the magnetic effect can be assumed to be neglected. The electric field, therefore, is governed by Gauss’s law as represented in the following Maxwell equation40

(3)

where λ is the mixing energy density and γ is a nanometer-scale capillary width representative of the thickness of the air− polymer interface (i.e., a measure of the thickness of the diffusive interface). Empirically, γ should be so determined by trial and error that a 90% variation in φ occurs over a thickness 4705

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∇·(εE ⃗) = 0 ε = ε0εr

(7)

cos(θ) = cos(θ0) +

where ε0 is the vacuum permittivity, εr is the relative permittivity of the problem domain (air and polymer), and E⃗ is the electric field. With the introduction of the phase function in section 2.1, ε is treated as being continuous across the air− polymer interface by interpolating between the air and polymer as expressed in eq 1. Because the electric field is a gradient with electric potential V (i.e., E⃗ = −∇V), eq 7 can be transformed to the Laplace equation: −∇·(ε∇V ) = 0

n ⃗ ·∇φ = cos(θ)|∇φ|

(9)

where I ⃗ is a unit stress tensor and E is the electric intensity. Correspondingly, a volumetric electrodynamic force fe, equivalent to the Maxwell stress tensor Te, can then be obtained by Gauss’s law as follows f e = ∇·T e = ρf E ⃗ −

1 2 E ∇ε 2

(10)

where ρf is the density of free charges. Because the polymer and air are assumed to be dielectric media with no volumetric free charges, eq 10 becomes 1 f = − E 2 ∇ε 2

(13)

where n⃗ is the normal to the solid wall. Equation 13 has to be satisfied by φ on the whole fluid−solid interface as a boundary condition for the phase field. Furthermore, a hydrodynamic assumption of the dynamic contact angle is adopted to deal with the contact angle near the moving contact lines44−46 in which the microscope angle is considered to retain a static value that is determined by the electric field and can be different from the dynamic contact angle that is experimentally observable in the macroscopic regime as a result of the viscous bending of liquid−gas interface within the mesoscopic regime. As is widely known, in the conventional macroscale regime the liquid velocity can be assumed to have a zero component tangential to the solid surface. However, in the micro- or nanoscale regime a slip velocity for the liquid, uw, has to be considered at the solid surface (i.e., the template and substrate area in contact with the polymer). uw can be calculated in the following equation by introducing an experimentally measurable and nanometer-scale slip length Ls (Navier slip model)47

(8)

⎛ ∂ε ⎞⎤ 1 2⎡ E ⎢ε − ρ ⎜ ⎟⎥I ⃗ 2 ⎣ ⎝ ∂ρ ⎠⎦

(12)

where θ0 is the initial contact angle. The contact angle θ is related to the phase-field function φ in the form38

Once the electric field is determined, the electrodynamic force generated at the air−polymer interface can be deduced from the Maxwell stress tensor Te as represented by41 ⃗ ⃗− T e = εEE

Fl σ

u w = Ls

∂u ∂n

(14)

where n is the normal to the solid interface and inward to the liquid. This means that a nonzero tangential speed has to be specified at the solid surface as a boundary condition for the fluid velocity. The slip-length boundary condition has been suggested, resulting in an effective slip only near the contact line, which weakens the local singularity and allows the contact line to slip along the solid boundary.48 Furthermore, the advantage of the Navier slip model is the fact that it specifies a slip length instead of relying on the mesh-dependent effective length of the numerical discrimination, decreasing the maximum shear rate at the contact line, but then a further reduction in mesh size is needed to realize the convergence. On the basis of numerous studies on multiscale fluid mechanics,49,50 for Kn ≤ 0.001, an assumption of continuum for the fluid (i.e., the air and liquid polymer) is still appropriate and the flow can be analyzed using the Navier−Stokes equations (eqs 6a, 6b) with the conventional nonslip boundary condition. However, for 0.001 ≤ Kn ≤ 0.1 (commonly referred to as the slip-flow regime), an effect called rarefaction starts to influence the flow and the Navier−Stokes equations can still be employed, yet with a boundary condition of nonzero tangential slip velocity along the solid walls. As mentioned previously, the radius of gyration of the polymer used is usually several nanometers, a domain with a characteristic length ranging from tens of nanometers to several micrometers in the slip-flow regime (0.001 ≤ Kn ≤ 0.1), for which the slip-velocity boundary condition (eq 14) should be used for the Navier−Stokes equations (eqs 6a, 6b). In our numerical formulation, the characteristic length is taken as the smallest of the line or trench widths or the diameter of the holes on the template. As the feature size decreases, the shear rate in the fluid increases significantly,51 leading to non-Newtonian viscosity.

e

(11)

which has to be substituted into the momentum conservation equation (i.e., eqs 6a, 6b). Obviously, fe approaches zero sharply with respect to the distance from the air−polymer interface because farther away from the air−polymer interface ε becomes constant (equal to either εair or εpoly). 2.3. Scale Effects. Various scale effects, which have often been omitted in the conventional macroscale regime, can significantly influence electrohydrodynamic behaviors for the EHDP processes performed on the micro- or nanoscale. Surface tension, contact angle, liquid−solid interface slip, and non-Newtonian viscosity are considered to be the mechanically dominating scale effects,26−29 which are introduced in our numerical simulation. In the phase-field formulation, the surface tension (acting on the air−polymer interface) is considered to be acting volumetrically throughout the diffusive thickness of the air− polymer interface, calculated in terms of the phase-field function φ and chemical potential G by fst = G∇φ. This volumetric surface tension (approaching zero with the distance from the air−polymer interface) also has to be added to the right-hand side of the momentum conservative equations (eqs 6a, 6b). The L-DEP-generated hydraulic pressure on the three-phase contact line (TCL)42 (i.e., Pe = −1/2 ε0εp(εp − 1)Ep2) tends to change the contact angle. The L-DEP pressure can be transformed to a linear force by Fl = ∫ S Pe dl, where dl is the unit length of the TCL. Then a variation of the electrically influenced contact angle θ can be calculated by the following Lippmann−Young equation43 4706

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Figure 2. Comparison between experimental and numerical results for the electric actuation of a dielectric microdroplet. (A) Photographs of the droplet taken experimentally for voltages of 0 and 1000 V, respectively. (B) i and ii show the numerically obtained droplet shapes by phase-field color mapping, and iii shows the numerical result at a voltage of 1000 V, obtained without considering scale effects such as the liquid−solid interface slip, varying wetting angle, and non-Newtonian viscosity law.

initially. The voltage was applied between the substrate and the needle during the electric actuation, while the variation of contact angle and the air−polymer interface was measured by a video device (OCA20, by Dataphysics, Germany). The liquid polymer had a relative permittivity of 6.0, a viscosity of 0.2 Pa·s, an initial surface tension of 0.032 N/m, and a slip length of 100 nm estimated based on the published literature.47,53 The initial contact angles of the polymer were measured to be 78° for zero voltage and 45° for a voltage of 1000 V, respectively, as shown in Figure 2Ai,ii. For this test problem, the problem domain for our numerical simulation was considered to be axis-symmetric. Figure 2Bi,ii shows the corresponding droplet shape numerically obtained for voltages of 0 and 1000 V, respectively. Obviously, satisfactory agreement has been reached between the experimental test and the numerical simulation proposed in this Article. If scale effects such as the liquid−solid interface slip, varying wetting angle, and non-Newtonian viscosity are neglected (i.e., by assuming a constant viscosity, a constant contact angle, and a zero slip length on the liquid−solid interface), then the numerical approach leads to only a slight decrease in the contact angle for a voltage of 1000 V, as shown in Figure 2Biii, but to an obvious increase in the amount of polymer on the conductive needle, deviating tremendously from the experimental observation in Figure 2Aii. This indicates that the omission of the scale effects in the electrohydrodynamics for the micro- or nanoscale regime can negatively impact the accuracy of the numerical simulation to a significant extent. In a practical simulation, the diameter of the droplet is 850 μm and the velocity is small, corresponding to a Knudsen number smaller than 0.001 and a low shear−strain rate, leading to the negligibility of the slip velocity and the non-Newtonian viscosity. Therefore, the scale effect can be attributed mainly to the electrically varied contact angle.

This is especially true for a macromolecular composed polymer, for which the viscosity is usually dependent on the shear rate and can be represented by the following equation (for Carreau fluid, a type of generalized Newtonian fluid)52 η = η∞ + (η0 − η∞)[1 + (ξγ )̇ 2 ](m − 1)/2

(15)

where η0 is the zero-shear-rate viscosity, η∞ is the infinite-shearrate viscosity, ξ and m are material-specific constants, and γ̇ is the shear strain rate, respectively. In detail, γ̇0 is the critical shear strain rate, below which the Carreau fluid behaves as a Newtonian fluid and above which the power-law viscosity becomes dominant.26,52 The numerical simulation in this Article involves the simultaneous solution of the Cahn−Hilliard equation (eq 2), the Navier−Stokes equation (eq 6), and the Laplace equation (eq 8), with an initialized phase field. The iterative solution of this coupled problem can be carried out by the finite element method in COMSOL Multiphysics v4.2 commercial software in which the governing equations, physical properties, and boundary conditions described in the preceding are defined explicitly and symbolically and then automatically solved by the Galerkin approach. Once the phase-field function φ is obtained, the moving air−liquid interface can be easily identified by tracing a zero-value contour of the phase-field function or by simply color mapping the phase-field function, providing a visualization of the polymer’s deformation step by step.

3. TEST PROBLEM To prove the effectiveness of the numerical approach proposed, the electric actuation of a polymer microdroplet is determined experimentally and compared with the corresponding numerical simulation, as shown in Figure 2. This test, which uses a dielectric liquid polymer, is quite similar to the classical approach employed for measuring the contact angle for electrowetting on a dielectric (EWOD),20 which usually uses a conductive or electrolytic liquid instead. Consequently, the deformation of a dielectric liquid polymer under an electric field can also be regarded as a specific case of the polymer for electroweting. In our experiment, a dielectric polymer droplet with a diameter of roughly 850 μm was dripped on a conductive substrate (doped silicon) coated with a dielectric SiO2 layer of 200 nm in thickness, and a conductive needle (Pt) with a diameter of 100 μm was then immersed into the droplet

4. RESULTS AND DISCUSSION 4.1. EFAN Process. The polymer’s physical properties and process variables used for the numerical simulation are the same as those available in Chou’s publication,9 as listed in Table 1. In their particular experiment, the substrates were semiconductive Si wafers and the template consisted of a conductive and transparent ITO backup and a nanopatterned SiO2 layer. Voltage was applied between the substrate coated with a photocurable liquid polymer film and the ITO backup after the 4707

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pressure or force per unit area of the surface) by the formula Pdep = Te·n⃗. Under the combination of these two driving pressures, the polymer surface rises to the highest centrally in the cavity, as shown in Figure 3Aii. This tends to cause the polymer to reach the center of the bottom of the cavity first, compressing the trapped air toward the corner of the cavity, as shown in Figure 3Aiii. The final shape of polymer is determined by a static equilibrium between the hydraulic pressure of trapped air and the electrostatic pressure. Obviously, the liquid polymer in the cavity, after photo- or thermocuring, will become a pillar trimmed along its top edge with a residual layer of 22 nm, as can be seen in Figure 3Aiv. This observation agrees closely with what can be identified from the SEM images presented by Chou.9 Interestingly, because the polymer surface is deformed to a convex curvature before reaching the bottom of the cavity, this process can be used to generate a microlens array of convex curvature if the cavity diameter is scaled up to the micrometer regime.54 In the numerical simulation, the air was treated as a compressible fluid in the cavity of the template, and the incompressible polymer was considered to flow upward under the influence of the electric field, with the template being moveable toward to the substrate and treated as a vertically free-moving rigid boundary. The EFAN process was proposed on the basis of a belief that the dc voltage generated an electrostatic pressure P esa equivalent to an externally applied mechanical pressure as in a conventional nanoimprint process, which attracted the template and substrate toward one another to drive the polymer into the template cavities, whereas the L-DEP pressure Pdep on the polymer surface was not considered. To determine which of the two pressures can be dominating to actuate the polymer flow, the hydrodynamic behaviors have been simulated for Pdep and Pesa being effective alone separately and in combination, as shown in Figure 3B,C.

Table 1. Physical and Geometrical Data Used for EFAN and DEP-ECF DI Processes simulation parameters

EFAN

DEP-ECF DI

voltage relative permittivity of polymer surface tension coefficient constant viscosity of polymer mass density of polymer thickness of polymer relative permittivity of dielectric layer cavity width on the template cavity depth on the template initial contact angle

40 V 4 0.03 N/m 0.3 Pa·s 960 kg/m3 70 nm 4 130 nm 72 nm 80°

200 V 41 0.03 N/m 0.34 Pa·s 960 kg/m3 20 μm 4 30 μm 50 μm 80°

template was in contact with the substrate without a small pressure. Figure 3A shows the progressive evolution of polymer at the cross-section of nanogratings with the geometric data listed in Table 1. In the numerical simulation, it has been assumed that the air−polymer interface is planar at the initial contact of the template with the polymer (Figure 3Ai). Once the dc voltage is applied, the top polymer surface moves progressively upward with a spatially nonuniform velocity. Because the template is moveable relative to the substrate, two forces will be effective for driving the polymer flow: the L-DEP force Fdep acting on the polymer surface, which is due to the Maxwell stress expressed in eq 11, and the electrostatic attraction Pesa between the template and the substrate, as given by Chou in eq 169 εr 2ε0 2 E (16) 2 To compare with the electrostatic attraction Pesa, the L-DEP force is also transformed into the surface force density (i.e., Pesa =

Figure 3. (A) Progressive evolution of polymer at a voltage of 40 V, with a final polymer shape that looks like a pillar trimmed on its top edge (iv). (B) Rising height of the central polymer surface with respect to the timescale under different driving pressures (L-DEP effective alone, electrostatic template−substrate attraction effective alone, and their combination effective). (C) Filling of the polymer at the same template position affected by different forces (i.e., two pressures combined (i), Pdep alone (ii), and Pesa alone (iii), respectively). The color-mapped phase-field function is equal to −1.0 for the blue region (air) and +1.0 for the red region (polymer). 4708

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Figure 3B illustrates the rising height of the central polymer surface with respect to the scaled time (i.e., the electrically actuating time scaled to the time taken for the central polymer surface to reach the bottom of the cavity under the combined effect of Pdep and Pesa). Obviously, the L-DEP pressure (as depicted by the dotted line) that tends to pull the polymer surface upward in the cavity can play a more dominant role in driving the polymer’s flow (with a faster filling speed) compared to the electrostatic attraction between the template and substrate (centerlined), which tends to push the polymer surface upward. Figure 3C shows the filling of the polymer at the same template position affected by different forces (i.e., two forces combined (i), Pdep alone (ii), and Pesa alone (iii), respectively). Obviously, Pdep being effective alone can generate a polymer surface curvature that is almost equal to that generated by the two pressures combined, whereas Pesa being effective alone deforms the polymer surface into a significantly smaller curvature. This again indicates that the L-DEP pressure plays a more dominant role in determining the polymer filling in the template cavity. If the polymer used in EFAN has a finite electric conductivity, then an improved variant to EFAN can be devised by using an ac voltage with a proper frequency specific to the dielectric polymer instead of the dc voltage because the L-DEP pressure can also be significantly influenced by the frequency of the voltage.1−3 The curved shape of the air−polymer interface can be attributed to the variation of the electric field on the air− polymer interface because the driving force is deduced from the electric intensity, as expressed in eq 11. Figure 4A illustrates the variation of the electric distributions on the air−polymer interface with the rising polymer height. Initially, the rising polymer height is 0 nm (as shown by the solid line in Figure 4A), and the corresponding electric intensity near the template sidewall is larger than that at the center of the cavity. However, the polymer velocity near the sidewall is not the largest because of the friction of the sidewall (introduced by the slip-velocity boundary condition). Subsequently, the polymer in the center of the cavity tends to move faster, leading to a larger electric intensity in the central area (as shown by the dashed line for a rising height of 40 nm). With the increasing polymer rise, the difference between the electric intensity in the center of the cavity and that at the sidewall becomes more and more significant (as shown by the dotted line for a rising height of 50 nm), leading to a larger curvature of the polymer surface and the earliest contact of the central polymer with the bottom of the template. Because the characteristic length of the gratings is scaled down to a cavity width of 130 nm, the influence of the slip velocity boundary condition cannot be ignored, as shown in Figure 4B,C, where the polymer surfaces obtained for the slip and the nonslip boundary conditions become essentially different as the polymer reaches the bottom of the cavity. Under the slip boundary condition, the polymer rising height at the template sidewall is 37 nm when the polymer reaches the bottom of the cavity in the template (Figure 4B), whereas the polymer rise computed near the template sidewall is only 15 nm under the nonslip boundary condition (Figure 4C). This implies that the slip boundary condition does significantly affect the simulated movement of the polymer. 4.2. DEP-ECF DI Process. Although DEP-ECF DI is different from EFAN only in that the micro- or nanostructured template for the former is conductive and coated with a dielectric layer, the electrohydrodynamic deformation of

Figure 4. (A) Electric field at the air−polymer interface for different rising heights of polymer as the central polymer reaches the bottom of the cavity. (B, C) Structures numerically obtained by using slip and nonslip boundary conditions, respectively.

polymer in template cavities can be distinct. In our implementation of the DEP-ECF DI process, the template was initially brought into proper contact with the polymer surface by a small pressure and then kept at a fixed position relative to the substrate while a voltage was applied. Obviously only the L-DEP force will become effective in the template cavities, acting on the polymer surface and often accompanied by the trapping of compressible air. The polymer’s physical parameters and process variables used for our numerical simulation are the same as those used in the experiment performed by Shao and Ding,10 listed in Table 1. Figure 5A illustrates the progressive evolution of the polymer in a cylindrical microhole. For the initial contact of the template with the polymer, the polymer surface has been assumed to be planar, as shown in Figure 5Ai. Once the voltage is exerted, the polymer is pulled into the microhole by the L-DEP force (Figure 5Aii,iii) gradually to a filling depth at which equilibrium is finally reached between the L-DEP force and the aerial 4709

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Figure 5. (A) Progressive evolution of polymer in a cylindrical microhole for DEP-ECF DI at an rms voltage of 200 V. (B) Final shape of polymer in microcavities of varying size (i) and of spatially varying line/spacing ratio (ii), both at an rms voltage of 150 V, which indicates that the DEP-ECF DI process tends to generate a practically consistent height for microstructures with nonuniform size and nonperiodic spacing. The color-mapped phasefield function is equal to −1.0 for the blue region (air) and +1.0 for the red region (polymer).

microstructures was also proven in our published experiment in a comparison to conventional imprint lithography.10 The electric field at the air−polymer interface is shown in Figure 6A. When the polymer filling depth is 0 μm (solid line), the electric intensity near the sidewall is largest, similar to that of EFAN. But the influence of the friction of the sidewall on the microscale is different from that in EFAN on the nanoscale, leading to a higher velocity near the sidewall. As the filling

pressure by the trapped air, as shown in Figure 5Aiv. Interestingly the polymer surface is concave, with a final filling height of 39 μm. It can also be seen that the polymer curvature tends to decrease with the increasing filling height. This agrees well with our published experimental observations.10 The generated concave polymer surface is a unique feature that can be applied for the generation of a microlens or micromirror array of concave curvature by using a microhole-arrayed template.55 In our numerical simulation, it has been assumed that compressible air is trapped in the cavity of the template and that incompressible polymer flows into the template cavity, accompanied by lateral polymer flow in the fixed gap between the template and the substrate to simulate the practical scenario as explained in Figure 1B. Figure 5Bi illustrates the final shape of the polymer in two microcavities with different characteristic lengths (at a voltage of 150 V). The filling height of polymer in the large cavity is almost equal to that in the small cavity. This observation is quite contrary to those obtained for the conventional imprint lithography where the filling height for a larger microcavity is larger than that for a smaller one.56,57 Considering the sizedependent viscosity represented in eq 15, we can explain this by the fact that the hydrodynamic resistance increases with the decreasing feature size as a result of the increasing shearing rate of the polymer in conventional imprint lithography whereas the size-dependent L-DEP force in a small cavity tends to cancel the correspondingly increased hydrodynamic resistance in the DEP-ECF DI process. Figure 5Bii shows the final shape of the polymer in a cluster of gratings with the same characteristic but a different line/ spacing ratio. Although such irregularly distributed microstructures can cause a spatially inconsistent filling height of polymer in conventional molding, as found in Jay’s experiment,58 the DEP-ECF DI process has generated a practically uniform filling height of polymer. This means that DEP-ECF DI can be better used to mold microstructures with nonuniform size and nonperiodic spacing. The ability of DEP-ECF DI to generate a uniform height for molded

Figure 6. (A) Electric field at the air−polymer interface for different filling depths. (B) Final shape of polymer with the voltage of 0 V, which implies that the natural capillary effect cannot effectively drive the filling of the template cavity with polymer. 4710

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depth increases (see the dashed line for a filling depth of 19 μm and the dotted line for a filling depth of 33 μm), the difference between the electric intensity at the cavity sidewall and that in the center of the cavity becomes less significant. Thus, the curvature of the polymer surface decreases with the polymer rising height. The final shape of the polymer with a voltage of 0 V is shown in Figure 6B, the polymer height near the sidewall is about 5 μm, and the height in the center of the cavity is roughly 0 μm, implying that the polymer is truly driven by the L-DEP force and the effect of the natural capillary (for a voltage of 0 V) can be neglected. The difference between DEP-ECF DI and EFAN can also be seen from the variation of the average electric intensity on the polymer surface with respect to the polymer’s rising height, as shown in Figure 7. In the DEP-ECF DI process, the average

height in EFAN is opposite, as shown in Figure 7B. Therefore, the upward movement of polymer can be initiated more easily at the beginning but becomes slow in the later stage for DEPECF DI. In contrast, the upward movement of polymer in EFAN tends to initiate slowly and becomes fast as the polymer rises into the template cavities. Obviously, because only a comparatively small voltage is desirable in avoiding an electric breakdown of the structured and dielectric template, this may possibly lead to an L-DEP force that is insufficient to drive the polymer to move upward in EFAN at the beginning. To fabricate structures with a high aspect ratio by EFAN, a small external force may be necessary for driving the polymer to an initial height in template cavities, combined with the electric field to drive the polymer fully into template cavities. 4.3. EISF Process. In the EISF process, a thin polymer is initially spin coated onto a conductive substrate as one of the electrodes. Then a conductive and flat or patterned template, as another electrode, was mounted with a fixed clearance over the substrate, leaving an air gap. The assembly was heated to above the glass-transition temperature of the polymer, and a proper voltage was then applied and maintained for a proper time. Micro/nano structuring by a flat template or a patterned one was investigated, leading to periodic pillars or structures of polymer conformal to the template pattern, respectively. The polymer properties and processing parameters for numerical simulation in the case of a flat template are listed in Table 2. As Table 2. Physical and Geometrical Data Used in the EISF process simulation parameters

flat

patterned

voltage relative permittivity of polymer surface tension coefficient constant viscosity of polymer mass density of polymer thickness of polymer air gap distance protrusion width on the template protrusion depth on the template

30 V 41 0.03 N/m 0.3 Pa·s 960 kg/m3 150 nm 100 nm

42 V 2.5 0.03 N/m 0.3 Pa·s 960 kg/m3 35 nm 90 nm 200 nm 170 nm

for structuring by a patterned template, the polymer’s properties and process variables are the same as those available in Schaffer’s experiment for the nanostructuring of periodic gratings,11 as also listed in Table 2. Figure 8A illustrates the dynamic structural evolution of a polymer’s periodic pillars induced by a flat template. Initially, the polymer film is planar under the influence of surface tension. Once a voltage is exerted, the polymer film is electrically driven to move upward, with the polymer rise starting at the two edges of the template and stretching toward the template center (i), gradually deformed into a pillar array with an initial periodicity of 550 nm (ii), which is the same as the most unstable wavelength analytically calculated.21 The movement of the polymer can be attributed to the spatially irregular variation of the electric field on the air−polymer interface (as shown in Figure 9A). In the initial stage, a tiny bump (if looking at half of the symmetric template) is produced on the planar polymer surface near the edge of the template, corresponding to an electric field distribution illustrated by the solid line. Consequently, the bump continues to grow near the edge, generating the first pillar near the edge. When the first pillar reaches a height of 28 nm, the

Figure 7. Variation of average electric intensity on the air−polymer interface with respect to the polymer’s rising height for (A) DEP-ECF DI and (B) EFAN.

electric intensity initially increases to the highest and then decreases as the polymer fills the template cavity (Figure 7A), which can be attributed to an enhancement by the thin dielectric layer coated on the conductive structure. In detail, the region with the highest electric intensity is that closest to the conductive tip, and thus the electric intensity on the polymer surface is not the greatest in the initial stage for the dielectric layer coated on the template. In the process of the polymer moving upward, the distance from the polymer surface to the conductive tip is ordered from large to small and then to large, leading to the variation in electric intensity shown in Figure 7A. The trend in the average electric intensity versus the polymer 4711

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Figure 9. Electric field on the air−polymer interface for (A) a flat template and (B) a patterned template.

influence of the nonuniform electric field modulated by the template, the polymer begins to move upward toward the template protrusions, leading to a wavy shape of the polymer surface (i). As the voltage is sustained, the polymer continues to flow upward until contacting the template protrusions (ii) and then stretching horizontally on the template protrusions, leading to a structure composed of nearly vertical gratings (iii) with a grating width of 120 nm, which is roughly equal to that extracted from the experimental results.11 The polymer flow can also be ascribed to the variation of the electric field that is spatially modulated by the template structure as shown in Figure 9B, where a nonuniform electric field is generated by a patterned template on the air−polymer interface in the initial stage (solid line), leading to polymer accumulation at the protrusions of the template where the highest electric intensity occurs. Subsequently, the polymer rise under the protrusions can in turn induce an increased electric intensity because the air gap in this region is reduced, as shown by the dashed line for a polymer rising height of 17 nm. This trend continues until the polymer grows up to the ceiling of the template protrusions (as shown by the dotted line). Therefore, this process can duplicate micro- or nanostructures geometrically positive to the patterns on the template. It is also interesting to find that increasing the voltage from 42 to 60 V can lead to the growth of secondary gratings, as shown in Figure 8C. The primary gratings (those grown under the template protrusions) become thinner to allow for the growth of the secondary gratings beside the former to satisfy the polymer’s mass conservation. Sharma’s experiment also demonstrated the same phenomenon.60 The secondary gratings

Figure 8. (A) Progressive structural evolution of polymer in the case of a flat template. (B, C) Progressive structural evolution of polymer in the case of a patterned template at voltages of 42 and 60 V, respectively. The color-mapped phase-field function is equal to −1.0 for the blue region (air) and +1.0 for the red region (polymer).

corresponding electric intensity on it becomes significant and influences the nearby polymer region to generate the second pillar with a height of 10 nm (dashed line). Similarly, as the height of the pillar increases to 132 nm so that the polymer reaches the template, the electric intensity corresponding to the first and second pillars are all increased (dotted line), in which the second and third pillars have heights of 49 and 10 nm, respectively. This trend will spread from the edge of the flat template to its center until all of the pillars are formed. However, as the electric field sustains, the polymer pillar in contact with the template ceiling will continue to move and the neighboring pillars will merge (iii), leading to a final structure as shown in iv. This agrees quite well with the experimental observations by Harkema59 and can be explained from the viewpoint of free energy (i.e., the polymer tends to coalesce because the merged polymer has a lower free energy). Figure 8B demonstrates the evolution of electrically induced polymer under a grating-patterned template. Under the 4712

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alignment in imprint lithography. Opt. Laser Technol. 2012, 44, 446−451. (9) Liang, X.; Zhang, W.; Li, M.; Xia, Q.; Wu, W.; Ge, H.; Huang, X.; Chou, S. Y. Electrostatic force-assisted nanoimprint lithography (EFAN). Nano Lett. 2005, 5, 527−530. (10) Li, X. M.; Shao, J. Y.; Tian, H. M.; Ding, Y. C.; Li, X. M. Fabrication of high-aspect-ratio microstructures using dielectrophoresis-electrocapillary force-driven UV-imprinting. J. Micromech. Microeng. 2011, 21. (11) Schaffer, E.; Thurn-Albrecht, T.; Russell, T. P.; Steiner, U. Electrically induced structure formation and pattern transfer. Nature 2000, 403, 874−877. (12) 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, 1688−1690. (13) Boltau, M.; Walheim, S.; Mlynek, J.; Krausch, G.; Steiner, U. Surface-induced structure formation of polymer blends on patterned substrates. Nature 1998, 391, 877−879. (14) Lin, T. C.; Huang, L. C.; Huang, C. C.; Chao, C. Y. Formation of self-assembled periodic grooves via thermal drawing lithography for alignment layers in liquid crystal devices. Soft Matter 2011, 7, 270− 274. (15) Discher, D. E.; Janmey, P.; Wang, Y. L. Tissue cells feel and respond to the stiffness of their substrate. Science 2005, 310, 1139− 1143. (16) Sidorenko, A.; Krupenkin, T.; Aizenberg, J. Controlled switching of the wetting behavior of biomimetic surfaces with hydrogelsupported nanostructures. J. Mater. Chem. 2008, 18, 3841−3846. (17) McAlpine, M. C.; Ahmad, H.; Wang, D. W.; Heath, J. R. Highly ordered nanowire arrays on plastic substrates for ultrasensitive flexible chemical sensors. Nat. Mater. 2007, 6, 379−384. (18) Dickey, M. D.; Raines, A.; Collister, E.; Bonnecaze, R. T.; Sreenivasan, S. V.; Grant, W. C. High-aspect ratio polymeric pillar arrays formed via electrohydrodynamic patterning. J. Mater. Sci. 2008, 43, 117−122. (19) Lee, Y. J.; Kim, Y. W.; Kim, Y. K.; Yu, C. J.; Gwag, J. S.; Kim, J. H. Microlens array fabricated using electrohydrodynamic instability and surface properties. Opt. Express 2011, 19, 10673−10678. (20) Mugele, F.; Baret, J. C. Electrowetting: from basics to applications. J. Phys.: Condens. Mat. 2005, 17, R705−R774. (21) Schaffer, E.; Thurn-Albrecht, T.; Russell, T. P.; Steiner, U. Electrohydrodynamic instabilities in polymer films. Europhys. Lett. 2001, 53, 518−524. (22) Verma, R.; Sharma, A.; Kargupta, K.; Bhaumik, J. Electric field induced instability and pattern formation in thin liquid films. Langmuir 2005, 21, 3710−3721. (23) Wu, N.; Pease, L. F.; Russel, W. B. Electric-field-induced patterns in thin polymer films: weakly nonlinear and fully nonlinear evolution. Langmuir 2005, 21, 12290−12302. (24) Deshpande, P.; Pease, L. F.; Chen, L.; Chou, S. Y.; Russel, W. B. Cylindrically symmetric electrohydrodynamic patterning. Phys. Rev. E 2004, 70. (25) Tian, H.; Shao, J.; Ding, Y.; Li, X.; Li, X. Numerical studies of electrically induced pattern formation by coupling liquid dielectrophoresis and two-phase flow. Electrophoresis 2011, 32, 2245−2252. (26) Lee, Y. H.; Sin, H. C.; Kim, N. W. Impact of slip and contact angle on imprinting pressure in nanoimprint lithography. J. Vac. Sci. Technol., B 2009, 27, 590−596. (27) Voronov, R. S.; Papavassiliou, D. V.; Lee, L. L. Boundary slip and wetting properties of interfaces: correlation of the contact angle with the slip length. J. Chem. Phys. 2006, 124. (28) Morihara, D.; Nagaoka, Y.; Hiroshima, H.; Hirai, Y. Numerical study on bubble trapping in UV nanoimprint lithography. J. Vac. Sci. Technol., B 2009, 27, 2866−2868. (29) Yao, D. G.; Kim, B. Simulation of the filling process in micro channels for polymeric materials. J. Micromech. Microeng. 2002, 12, 604−610.

are obviously undesirable from the viewpoint of structure duplication conformal to the template.

5. CONCLUSIONS The proposed formulations can be defined explicitly and symbolically with COMSOL Multiphysics commercial software and then solved automatically by the finite element method based on the Galerkin approach. The introduction of the phase field makes it straightforward to track the air−polymer interface by tracing a zero-value contour of the phase-field function or just by color mapping the phase field for visualization. It has also been found that the scale effects such as the liquid−solid interface slip, L-DEP wetting angle, and non-Newtonian viscosity of the polymer can tremendously impact the accuracy of the numerical simulation performed on the micro- or nanoscale involved. The numerical simulations performed for the three typical EHDP processes (i.e., EFAN, DEP-ECF DI, and EISF) agree well with the corresponding published experimental observations, proving the effectiveness and versatility of the proposed approach as a tool for characterizing the full evolution of the polymer micro- or nanostructures and for optimizing the process variables for industrial applications of the EHDP technique.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by the NSFC Major Research Plan on Nanomanufacturing (grant no. 90923040) and by the National Basic Research Program of China (grant no. 2009CB724202). Thanks are due to Professor Ben Q. Li (University of Michigan at Dearborn) for his valuable suggestions with respect to our mathematical formulation of the phase-field problem.

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REFERENCES

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