Simulation of polymer rheology in an electrically induced micro- or nano-structuring process based on electrohydrodynamics and conservative level set method

Abstract

An electrically induced structuring process, such as an electrohydrodynamic (EHD) approach for fabricating polymeric micro-/nano-structures in various micro-/nano-devices, was performed by applying a voltage to an electrode pair consisting of a planar or structured template and a polymer-coated substrate sandwiching an air gap, resulting in either periodic columns or template-modulated structures. Analytical approaches were explored to characterize this micro-/nano-structuring process, based on a linear thermodynamic instability of the polymer film combining capillary waves and electrostatic forces, leading to a definition of “most unstable wavelength” in relation to various process variables such as external voltage, polymer film thickness, and so on. For mathematical simplicity, the linear stability analysis was only carried out to demonstrate an initiation of the polymer structuring under electrical induction by an infinite planar template, and cannot numerically visualize the evolution of the polymer structure which grows from an initially flat film upwards to the template underside. Therefore, a numerical modelling of such a process, which is capable of demonstrating a full-cycle evolution of the polymer structuring, is desirable to provide an in-depth insight into this electrically induced structuring technique. This paper presents a detailed numerical formulation for simulating the rheological behaviour for this structuring process based on EHD equations and a conservative level set approach. Firstly, a numerical simulation is performed to demonstrate the dynamic evolution of periodic structures for a planar (non-structured) template and compared with the linear instability analysis to validate the effectiveness of the proposed numerical modelling. Then simulations are performed to numerically visualize the evolution of a polymer structure induced by a structured template, with a subsequent discussion about the influences of some critical process variables, such as voltage, air gap, polymer thickness, depth of template patterns, and so on, on the electrohydrodynamic rheology of the polymer.

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1. Introduction

Polymer micro- or nano-structuring is a critical step in the manufacturing process for various micro-/nano-devices, such as microfluidics,¹ organic solar cells,² flexible electronics,³ waveguides for optical communication,⁴ and so on. Various technologies are available for micro- or nano-structuring of a polymer,^5–7 mostly involving a chemical process. Electrically induced structuring techniques, such as a physical process, have been demonstrated to be capable of duplicating periodic pillars, grids, gratings, and so on, at a micro or even a nano scale, onto a dielectric polymer that has been coated on a substrate, with an external voltage exerted between the template and the substrate which act as an electrode pair, sandwiching an air gap.^8–10 Once a voltage is applied on the conductive template and substrate, an electric field is induced which generates an electrostatic force on the air–polymer interface and pulls the polymer upwards non-uniformly to the template underside by overcoming the surface tension, viscous force, and gravity on the polymer, rheologically turning the initially flat polymer into a structure dependent on the template shape. In this process, the template can be either flat,^8,11 generating an array of periodic pillars, or structured,^8,12,13 in an attempt to produce a polymer duplicate of the arbitrarily shaped micro-/nano-structure on the template (Fig. 1).

Fig. 1 Sketch of electrically induced structuring process with a flat template (a) and a structured template (b).

So far, Chou,¹⁴ Schaffer and¹⁵ N. Wu,¹⁶ have studied the electrically induced structuring process (as shown in Fig. 1(a)) for a flat template from an electrodynamic viewpoint based on a linear instability analysis. In this linear formulation, the surface of the rheological polymer was assumed to be planar initially in macro-scale, but bumpy in a nano-scale amplitude and a periodic shape which can be expressed in a Fourier series composed of components of different wavelengths, because of a thermal perturbation.¹⁷ When the polymer film is subjected to an electric field, the component with a specific wavelength, defined as the “most unstable wavelength”, will be dominantly enhanced to grow upwards to the template, being similar to a “resonance”. Finally, an array of polymer columns will be generated with a periodicity equal to the most unstable wavelength.

The linear formulation led to an analytical expression for the most unstable wavelength, which has been proved to be in good agreement with the periodicity of arrayed columns experimentally generated by using a flat template.^11,18 However, the analytical formulations were performed for an infinitely expanded flat template and, thus, are not applicable to the edge area of a finite template in real-life engineering, or of a patterned template, for example. In other words, the theoretical analysis is suitable for typical polymer thin films which have a length/width that is much larger than the height, i.e., in the electrically induced structuring process with a flat template. As for the structured template, the width of the template feature is comparable to the polymer height, in which the electric field for the air and polymer cannot be solved by the linear formulation. In consequence, the linear combination of hydrodynamics and electrodynamics can only be used to determine the periodicity of the structures generated and does not allow for revealing the full evolution and final shape of the polymer structure.

A numerical simulation based on non-linear electrohydrodynamics (EHD) can be used to better visualize the dynamic evolution of structures for this polymer forming process, from initiation of the structure to its temporal growth toward the template underside, as demonstrated in our previous publication,¹⁹ where a phase field model was developed to simulate the electrically induced structuring process, and satisfactorily validated against experimental observations widely published by other authors. As an implicit dynamic interface method, a phase field function is defined to a signed distance function to distinguish between two fluids,²⁰ i.e., the interface is represented by the zero value contour, the minus contour on one side of the interface and the positive contour on the other. The phase field model can readily handle topological changes in the air–polymer interface without special treatment, because it is solved in a fixed finite element meshing of the problem domain together with other variables. However, the Cahn–Hilliard equation involved in the phase field model contained a fourth-order spatial derivative, which required an extremely high resolution in the finite element meshing and could therefore lead to a heavy computational effort.²¹ In addition, the main drawback of the phase field method is that it cannot satisfy the law of mass conservation because of the errors from the advection calculation for the phase function, and also because the rate of convergence for the law of mass conservation is proportional to the mesh size, so an extremely small mesh size should be provided to diminish the influence of the poor mass conservation.

Among the dynamic interface tracking methods, the volume of fluid (VOF) method and the level set method are the most commonly used and widely regarded as effective methods in computational fluid dynamics (CFD). Both of them can automatically simulate topological changes without special treatments, just like the phase field method. On the other hand, they are much more economical than the phase field model in terms of computation effort because of their lower-order spatial derivatives and a construction on the mass conservation which is not highly dependent on the mesh size. The VOF method²² has the main advantage of conserving the volumes of the two fluids exactly. The interface is, however, represented by a discontinuity of a color function, defined as the fraction of volume within each cell of one of the fluids; thus it is difficult to reconstruct the interface as well as calculating the surface tension accurately, which depends on the mean curvature of the interface. In contrast, the level set model does not need to reconstruct the interface for every time-step, but there is no built-in mass conservation^23–25—i.e., a small amount of mass is lost or gained, in a similar way to the phase field method. Consequently, in the current work, we adopt the conservative level set model suggested by Olsson and Kreiss,^26,27 which overcomes the drawback of mass conservation but which retains the advantage of being easier to advect the interface and to calculate the curvature with a high order of accuracy. The conservative level set method keeps a diffuse interface with a constant thickness and the fluidic properties are smoothly transformed by the level set function, which is similar to the phase field function but varies from 0 to 1 with the interface denoted by the contour for 0.5.

In this paper, this conservative level set method together with EHD equations is proposed for analysis of the electrically induced structuring process, in which the EHD equations are used to depict an interaction of the electric field and the polymer flow, and the conservative level set formulation is introduced to track the air–polymer interface. Firstly, the numerical simulation for a flat template is tested against the most unstable wavelength to validate effectiveness of the proposed formulation. Then the dynamic evolution of structures for an electrically induced structuring process using a structured template is performed. Finally, the influences of some critical process variables on the forming process with a patterned template, such as voltage, air gap distance, polymer film thickness, and depth of pattern on the template, are discussed based on the electrostatic force by further numerical simulations.

2. Numerical model

In the electrically induced structuring process, the polymer is driven to move upwards under the influence of the electric field, which can be attributed to the interaction of the electric field and the hydrodynamic flow field that dominates the progress evolution, leading to a micro/nano-structure positive to the pattern on the template. With the influence of the electrostatic force on the air–polymer interface, the polymer moves upwards to the template, resulting in a changed morphology of the polymer film, which again generates a different electric field on the interface. Subsequently, the changed electric field can produce a discernable force to drive the further movement of the polymer, leading to different polymer shape. This cycle appears again and again until the polymer structure is deformed positively to the template features. In order to understand the EHD structuring process properly, a numerical model is proposed which couples the hydrodynamic flow field and the electric field.

2.1 Hydrodynamic flow field

In order to track the air–polymer interface, a conservative level set method of two-phase flow is adopted to show the movement of the polymer under the electric field with a diffuse interface of thickness of micro-/nano-meters, which separates the two immiscible fluids instead of having a discontinuous interface.^26–28 With the purpose of representing density, viscosity and permittivity discontinuities over the interface and to achieve numerical robustness for the calculation, a smeared-out Heaviside function is used to define the level set function, φ, in which 0 represents one fluid, 1 represents the other, and 0.5 denotes the interface. For simplicity, the level set function can also be regarded as the volume faction of the fluid. Subsequently, the mass density, viscosity coefficient and relative permittivity can be treated as a whole via the level set function φ, i.e., ρ = ρ_air + (ρ_poly − ρ_air)φ, η = η_air + (η_poly − η_air)φ, and ε = ε_air + (ε_poly − ε_air)φ, respectively, with the subscript “air” denoting air and “poly” representing polymer.

The level set function implicitly represents the physical interface advancing with the velocity of the fluid, u, as shown by:

The terms on the left-hand side give the correct motion of the interface, while those on the right hand side are necessary for numerical stability and keeping the thickness of the interface,^26,27 eqn (1) is different from most other level set methods which have the right-hand side as zero. Because it is in the conservative form, the mass is also conservative. The parameter γ determines the amount of reinstallation or stabilization of the level set function, φ. And the parameter ξ determines the thickness of the interface of the region where φ goes from 0 to 1.

As the interface is being tracked with eqn (1), the first step is to initialize the level set function, performed according to the following equation:

in domains initially outside and inside the interface, with d representing the distance of the domain to the initial interface. Here, “inside” refers to the domain where φ < 0.5 and “outside” refers to the domain where φ > 0.5.

At the micro-/nano-scale, the influence of gravity can be neglected because the order of the Bond number (Bo) is about 10⁻¹² to 10⁻⁶,²⁹ defined as Bo = ρgL²/σ where ρ is the mass density, g is the acceleration of gravity, L is the characteristic length, and σ is the surface tension coefficient. Conversely, the surface tension cannot be neglected,³⁰ which can be introduced by a level set function with the formula:²⁸

where n is the unit normal to the interface with the expression n = [(φ/|φ|)]φ = 0.5, and δ is the delta function approximated by a smooth function according to δ = 6|∇φ||φ(1 − φ)|.

Although the electrically induced structuring process is performed on the micro-/nano-scale, the Navier–Stokes equation is also suitable for describing this process for a Knudsen number smaller than 0.1.^30,31 Here, the air and the polymer are considered to be incompressible Newtonian fluids, and the momentum conservation equation and the mass conservation equation are given as follows:³²

in which ρ is the mass density of fluid (air or polymer), u is the velocity, p is the pressure, η is the dynamic viscosity coefficient, F_e is the electrostatic force and F_st is the surface tension. Here, the fluidic priorities are all expressed by the level set function as mentioned previously instead of separate ones for air and polymer.

2.2 Electric field

The electric field in a domain composed of air and polymer can generally be determined from the Maxwell equation as follows:³³

∇(ε₀ε_rE) = q (5)

where ε₀ is the permittivity of a vacuum, ε_r is the relative permittivity of fluid (air or polymer), q is the charge density in the fluid, and E is the electric field.

By assuming that the air and the polymer are both purely dielectric (i.e., various free charges in these two media can be ignored), Maxwell's equation can be simplified into the Laplace equation:

∇(ε₀ε_rE) = 0 (6)

When an electric field is applied, the electrostatic force is generated on the air–polymer interface, associated with the Maxwell tension, T^e, expressed as:³⁴

where ε is the permittivity of the fluid (which equals ε₀ε_r here), ρ is the mass density, and I is the identity tensor. Subsequently, the volume formulation of the electrostatic force can be transferred by applying Gauss' law to the Maxwell tension, which is represented as:

The term qE is the Coulomb force, which is the force per unit volume on a medium containing free charge. The second term, the dielectric force, is due to the force exerted on a non-homogenous dielectric medium. The term ∂ε/∂ρ, the electrostrictive force, is regarded as a modification to the fluid pressure. Here, the air and the polymer are purely dielectric and the permittivity is independent of position, where the first term and the last term can be neglected. Consequently, eqn (8) can be simplified to:

which has to be substituted into the momentum equation (i.e., eqn (4)). It can be clearly seen that F_e approaches zero with increasing distance from the air–polymer interface, because at a distance far away from the interface the permittivity becomes a constant, equal to that of the air or the polymer.

Based on eqn (4), (5) and (9), the polymer motion under the influence of an electric field can be shown from the point of view of a fully nonlinear formula. In the implementation of the numerical simulation, the voltage is exerted between the template and the substrate. The boundary of the fluid with the substrate is considered as a no-slip condition, i.e., u⃑ = 0. And the boundary of the fluid with the template is treated as a wetted wall, along which the contact angle θ_w for the fluid is specified, and across which the mass flow is zero. This can be described by n_wall − n cos(θ_w) = 0.²⁶ In addition, a simplification into a two-dimensional problem domain is performed to save computer effort. Here, we use the commercial finite element method software COMSOL Multiphysics® to analyse the motion of the dielectric polymer film subjected to an electric field.

3. Results and discussion

3.1 Flat template

It has been realized that it is possible to fabricate periodic pillars spontaneously when the polymer film is subjected to a uniform electric field as shown in Fig. 1(a). Until now, a definition of the most unstable wavelength is introduced to depict the periodicity of the columns formed based on the thermal perturbation of the film surface, with a formula as:^15,35where E_p is the electric intensity in the polymer domain, V is the applied voltage, and ε_p is the relative permittivity of the polymer. Eqn (10) is obtained with the process variables at the initial stage of the induced process for a flat template. Furthermore, the effectiveness of the most unstable wavelength has been validated by comparing it with the experimental results, but this exercise cannot demonstrate the dynamic evolution of the structuring process.

In this paper, a numerical simulation with a flat template is performed utilizing the model combining EHD theory and the level set method, which is shown in Fig. 2, with the blue region representing the air and the red region denoting the dielectric polymer. In addition, the parameters used in the numerical simulation are listed as follows: the applied voltage is 30 V, the thickness of the polymer film is 150 nm, the air gap is 100 nm, and the relative permittivity, surface tension coefficient, viscosity, and mass density of the polymer are 10, 0.03 N·m⁻¹, 0.34 Pa·s, and 960 kg·m⁻³, respectively. At the initial stage, the surface of the polymer film is planar, as shown in Fig. 2(a). If the voltage remains switched on, the polymer near the edge of the template first moves upwards, and then extends to the region inside the template, i.e., the pillars are formed from the edge to the inside of the template (Fig. 2(b)–(d)), which has been observed in experiments already performed.¹⁸ However, this is not the end of the process as, subsequently, the pillars all contact the template and the polymer then moves along the underside of the template, where the contacting area between the template and the polymer becomes larger and larger, accompanied by a thinner and thinner residual layer thickness, as shown in Fig. 2(e) and (f). If the polymer film thickness and the air gap distance are correct, it can generate a periodic structure without a residual layer, as shown in Fig. 2(f). It was found that the polymeric structuring with a flat template is completed in a short time, which can be attributed to a small viscosity coefficient—much smaller than those quoted in the literature.^8–11 In this numerical simulation, the periodicity is roughly 690 nm (see Fig. 2(f)), which agrees with the most unstable wavelength calculated from eqn (10), which has a value of 740 nm. These results imply that the proposed model combining EHD theory and the level set method is appropriate to describe the electrically induced structuring process.

Fig. 2 Progress evolution of the electrically induced structuring process with a flat template at 0 s (a), 0.005 ms (b), 0.01 ms (c), 0.015 ms (d), 0.025 ms (e) and 0.1 ms (f).

The evolution of the induced process with a flat template can be attributed to the distributions of the electric field and the electrostatic force on the air–polymer interface at the initial stage, which are demonstrated in Fig. 3(a) and (b), respectively, and correspond to that of Fig. 2(a). Once a voltage is applied, the electric field is built up simultaneously, resulting in a corresponding electrostatic force. Fig. 3(a) and (b) illustrate that the distributions of electric field and the electrostatic force are almost the same, except for the magnitude, indicating that the initial electric field on the air–polymer interface is the critical factor, which determines the electrostatic force and subsequently affects the structure formed. At the initial stage of the induction of the process, there is a gradient of the electrostatic force on the edge of the template, meaning that the resultant force cannot achieve a balanced state. However, the distribution of the electrostatic force in the inner region of the template is almost planar—i.e., the resultant force can acquire balance. Consequently, the polymer film will move upwards at the region near the edge of the template first. Once the region of the polymer near the edge of the template almost touches the template, the electrostatic force distribution also changes, as shown in Fig. 3(c), corresponding to the curve shown in Fig. 2(b). The two peaks on the sides corresponding to the two pillars shown in Fig. 2(b) have a larger order of magnitude than that in Fig. 3(b), since the distance between the pillars and the template becomes smaller. Except for the two peaks, the positions of the gradients of the electrostatic force are transferred to the region near the pillars formed first, shown in the circles marked in Fig. 3(c). Consequently, the region near the pillars formed first is the next to generate pillars. This process is repeated again and again until all of the pillars are formed, leading to a phenomenon where the pillars are generated from the edge to the inside of the template.

Fig. 3 Distributions of the electric field (a) and the electrostatic force (b) electrostatic force corresponding to Fig. 2(a) and (c) electrostatic force corresponding to Fig. 2(b).

3.2 Structured template

With a flat template, it is difficult to fabricate arbitrary structures, such as grids or gratings. An alternative approach is to duplicate the structure with a structured template, resulting in micro-/nano-features corresponding to the pattern on the template. Here, an electrically induced structuring process was performed with a patterned template using the following process parameters: voltage: 50 V, air gap: 200 nm, polymer thickness: 200 nm, line-to-spacing ratio: 1 : 1 with a line width of 200 nm; the other parameters were identical to those used in the process illustrated by Fig. 2. When an electric field is applied, the electrical distribution on the air–polymer interface is sinusoidal instead of planar (Fig. 4(b)), in accordance with the structured template (Fig. 4(a)), which demonstrates that the pattern on the template determines the electrical distribution on the polymer film surface at an initial stage, resulting in the large electric intensity corresponding to the protrusions on the template and the small intensity in the cavity. According to eqn (9), the electrostatic force has an almost square relationship with the electric field , leading to the identical distribution of the electrostatic force. Subsequently, the region with a larger electrostatic force will first move upwards and a positive feedback relationship between the polymer height and the electric intensity can promote the motion of the polymer, i.e., a larger electric field drives the polymer to move higher, and the higher polymer generates a larger electric field. This effect drives the polymer flow as the distribution of electric field and generates the structure coinciding with features on the template.

Fig. 4 (a) Sketch of the electrically induced structuring process with a patterned template and (b) the electrical distribution with the structured template on the initial stage of the induced process.

Fig. 5 illustrates the dynamic evolution of an electrically induced structuring process with a patterned template. In the beginning, the polymer surface is planar (Fig. 5(a)). Under the influence of a modulated electric field, the region with a large electric intensity will flow first, as shown in Fig. 5(b), and during this process the feedback relationship of the polymer height and the electric intensity can promote the movement of the polymer film. Subsequently, the mass under the cavity of the template will move to that under the protrusions arising from the law of mass conservation, as shown in Fig. 5(c). Similarly to that of the flat template, it is not the end of the process when the polymer touches the template, as shown in Fig. 5(d) and (e): the contact area becomes larger and larger, and the thickness of the residual layer becomes thinner and thinner simultaneously. When the process is at the stage shown in Fig. 5(e), the anticipated micro-/nano-structure can be obtained by curing the polymer. Here, the structuring with a patterned template is still completed in a short time, which can also be attributed to the small viscosity coefficient of the polymer in similar fashion to that obtained with a flat template.

Fig. 5 Dynamic evolution of electrically induced structuring process with a patterned template at 0 s (a), 0.01 ms (b), 0.02 ms (c), 0.03 ms (d) and 0.06 ms (e).

The total process of electrically induced structuring, consisting of a flat or patterned template, can be described in three stages: initial stage, generating stage, and quasi-static stage. When the voltage is applied to the electrode pair of the template and substrate, the polymer film is initially flat, and this can be defined as the initial stage. Subsequently, the flat polymer will move upwards under the influence of the modulated electric field, and this is defined as the generating stage. Until the polymer makes contact with the template, the polymer can continue to move along the protrusions, and this is defined as the quasi-static stage. Once the process has reached the quasi-static stage, the polymer can be cured to obtain the expected structure. By actively controlling the relationship of these three stages, a structure with small or large curvature, different height, residual layer thickness, contact area between the polymer and the template, and so on, can be obtained.

The growing rate of the polymer construction shown in Fig. 5 is demonstrated in Fig. 6, which shows how the different polymer morphology can be obtained by curing the polymer at different stages, as in a sinusoidally shaped curve or in a structure with no residual layer. According to the slope of polymer height versus process time, the generating stage can be divided into two stages, consisting of a growing stage and an extending stage, corresponding to stage (I) and (II), as shown in Fig. 6. The first stage is the growing stage, which demonstrates the greatest slope, in which the polymer film is growing towards the template protrusions until contacting the template, as shown in Fig. 5(b)–(d). The extending stage is the second stage, with a smaller slope, during which the polymer is creeping along the bottom surface of the template due to the wettability of the polymer on the template surface, simultaneously, the residual layer becomes thinner and thinner. When the polymer height stays constant, the process is under the quasi-static stage, implying that there is deformation with no residual layer. Consequently, the polymer with specific morphology can be obtained by curing at different stages. Here, the polymer height can be detected by the leaky current between the template and the substrate,³⁶ in which the different heights have a specific current.

Fig. 6 Variation of the polymer height in the structuring using a patterned template.

3.3 Comparison of the level set method with the phase field method

In order to demonstrate the superiority of the level set method, a comparison of the level set method with the phase field approach is set out in Table 1, in which the VOF method is not discussed because of the difficulty of reconstructing the interface as well as calculating the surface tension accurately. It can be clearly seen that, for the same problem with the identical finite element mesh, the numbers of degrees of freedom for the phase field method is much higher than that of the level set method, and the computer memory and time for the phase field method results are also much larger than those of level set method, consisting of the structuring with a flat template and that with a structured template. Consequently, the proposed model based on the EHD and conservative level set method is proposed as more suitable for depicting the process of electrically induced structuring.

Table 1 Comparison of the level set method and the phase field method

Type of structuring	Method	Degrees of freedom	Memory	Time
Flat template	Phase field	48 401	1.45 G	2408 s
	Level set	39 199	1.12 G	1079 s
Structured template	Phase field	53 717	1.24 G	1060 s
	Level set	43 502	1.16 G	479 s

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3.4 Influence of process variables

As mentioned previously, the electrically induced structuring process with a patterned template can duplicate an arbitrary structure positive to the template features, which has attracted much more attention compared to that with a flat template. Consequently, the influence of the process parameters on the forming process with a patterned template is explored in this section. It has been proved that the electric field and electrostatic force on the air–polymer interface at the initial stage can determine the final form of the structure, and through that the influence of the process parameters can be studied. In the forming process with a patterned template, the driving factor is the difference of the electrostatic force on the polymer surface corresponding to the protrusions and the cavities on the template, which can be attributed to the difference of the square of the electric intensity, i.e., ΔE², according to eqn (9). Fig. 7 illustrates the parameter ΔE² versus voltage, air gap distance, polymer film thickness, and the depth, width and separating of features on the template, respectively represented in Fig. 7(a)–(f), where the process variables are identical to those shown in Fig. 5 except for the one discussed.

Fig. 7 Influence of process variables on the electrically induced structuring process with a patterned template consisting of voltage (a), air gap distance (b), polymer film thickness (c), depth of pattern on the template (d), width of pattern on the template (e), and separating of pattern on the template (f).

Fig. 7(a) shows that the parameter ΔE² (i.e., the driving force) is increased with the incremental increase of the voltage, which implies that increasing the applied voltage is advantageous to promote the movement of the polymer film. Fig. 7(b) shows that the driving force is decreased with an increase in the air gap distance between the template and the polymer film surface, implying that to decrease the air gap it might be useful to increase the driving force. However, the decrease of air gap distance also means that the growing space for the polymer film becomes smaller—i.e., a smaller air gap restricts the structured height. The parameter ΔE² is decreased with the increment of the polymer film thickness (Fig. 7(c)), indicating a smaller thickness is useful to increase the driving force. However, a smaller thickness also means less polymer mass, which may lead to insufficient polymer to generate the anticipated structure. The variation of ΔE² with the variation in depth of the pattern on the template is shown in Fig. 7(d), where it can be clearly seen that the driving force increases with depth, but almost becomes a constant once the depth exceeds a specific value. It can be seen that the driving force is almost constant when the aspect ratio of the template pattern is larger than 1, indicating that there is no need to fabricate a pattern with a higher depth for the purpose of supplying a larger ΔE². Fig. 7(e) demonstrates the influence on ΔE² of the width of the template features, where ΔE² increases with the increment of the pattern width, implying that it is difficult to duplicate a structure with a small size. The influence of separation between the features of the template is shown in Fig. 7(f), in which it can be seen that ΔE² is increased as the separation increases, i.e., the structure with a sparse distribution is easily fabricated.

Comparing the influence of process variables in Fig. 7, the order of variables can be listed as follows, based on the slope of the curves plotted from the largest to the smallest: air gap, voltage, polymer film thickness, and the properties of features on the template. In summary, the driving force for the electrically induced structuring process with a patterned template can be enhanced by increasing voltage, depth, width and separation of pattern on the template or decreasing the air gap distance and polymer film thickness, in which two preconditions must be guaranteed: (a) the exerted electric intensity must be smaller than the breakdown value of the polymer; and (b) the ratio of the air gap to the polymer film thickness must supply sufficient growing space and polymer mass to deform the polymer structure positively to the template features.

4. Conclusion

This paper attempts to provide another insight into the electrically induced structuring process for the micro-/nano- structuring of a photo- or thermally-curable fluidic polymer, from the viewpoint of mechanical analysis of a numerical simulation. A model combining EHD theory and conservative level set method is proposed to visualize the evolution of a dielectric polymer film subjected to an electric field. Based on that model, more numerical analysis of the electrically induced process has been performed using a flat template and a patterned one, in which the reasonableness of the model is validated by comparing the numerical simulated result of a flat template with the analytical formula. In the evolutionary process, the electric field is the essential factor, which determines the electrostatic force distribution and then affects the structure formed. In summary, the structuring process can be ascribed to three stages: the initial stage, the generating stage, and the quasi-static stage, where different structures can be obtained by adopting a process where the polymer film is cured at a specific stage when the leaky current between the template and the substrate is detected. The comparison between the level set method and the phase field approach, in relation to numbers of degrees of freedom, computer memory usage and time taken, demonstrates the superiority of the proposed model based on the EHD and the level set method. Finally, the influence of process variables are discussed based on the electrostatic force for the structuring process with a patterned template, which implies that increasing the voltage, depth, width and separation of pattern on the template or decreasing the air gap distance and polymer film thickness can all enhance the driving force, together with a precondition that guarantees that the electric intensity will be smaller than the breakdown value for polymer used and a proper ratio of air gap to polymer film thickness, in order to supply sufficient growing space and polymer mass to duplicate the anticipated structure.

Acknowledgements

This work was supported by the Major Research Plan of NSFC on Nanomanufacturing (grant No. 91323303), the Funds of NSFC (Grant Nos. 51175417 and 51275401), and Program for New Century Excellent Talents in University (NCET-13-0454).

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