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A Parametric Study on Electric field Induced Micro/Nano Patterns in Thin Polymer Film Fenhong Song, Dapeng Ju, Fangwei Gu, Yan Liu, Yuan Ji, Yulin Ren, Xiaocong He, Baoyong Sha, Ben Q Li, and Qingzhen Yang Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b00007 • Publication Date (Web): 15 Mar 2018 Downloaded from http://pubs.acs.org on March 15, 2018
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Langmuir
A Parametric Study on Electric field Induced Micro/Nano Patterns in Thin Polymer Film Fenhong Song1, Dapeng Ju1, Fangwei Gu2,3, Yan Liu2,3, Yuan Ji2,3, Yulin Ren2,3, Xiaocong He2,3, Baoyong Sha3,4, Ben Q. Li5, Qingzhen Yang2,3# 1
School of Energy and Power Engineering, Northeast Electric Power University, Jilin,
Jilin 132012, P. R. China 2
The Key Laboratory of Biomedical Information Engineering of Ministry of Education,
School of Life Science and Technology, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, P. R. China 3
Bioinspired Engineering and Biomechanics Center (BEBC), Xi’an Jiaotong
University, Xi’an, Shaanxi 710049, P. R. China 4
Institute of Basic Medical Science, Xi'an Medical University, Xi'an, Shaanxi 710021,
P. R. China 5
Department of Mechanical Engineering, University of Michigan-Dearborn,
Dearborn, Michigan 48128, United States #
Corresponding authors:
[email protected] ACS Paragon Plus Environment
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Abstract Electric field induced micro/nano patterns in thin polymer film, sometimes referred as electrohydrodynamic patterning, is a promising technique to fabricate micro/nano structures. Extensive attentions have been attracted due to its advantages in micro contact (easy demolding) and low cost. Although considerable work has been done on this technique, including both experimental and theoretical ones, there still appears a requirement for understanding the mechanism of electrohydrodynamic patterning. Thus
we
systematically
studied
the
effect
of
different
parameters
on
electrohydrodynamic patterning with a numerical phase field model. Previous researchers
usually
employed
lubrication
approximation
(i.e.,
long-wave
approximation) to simplify the numerical model. However, this approximation would lose its validity if the structure height is on the same scale or larger than the wavelength, which occurs in most cases. Thus, we abandoned the lubrication approximation and solved the full governing equations for fluid flow and electric field. In this model, the deformation of polymer film is described by the phase field model. As to the electric field, the leaky dielectric model is adopted in which both electrical permittivity and conductivity are considered. The fluid flow together with electric field is coupled together in the framework of phase field. By this model, the effect of physical parameters, such as external voltage, template structure height, and polymer conductivity, are studied in detail. After that, the governing equations are nondimesionalized to analyze the relationship between different parameters. A dimensionless parameter, electrical Reynolds number ER, is defined, for which, a large value would simplify the electric field to perfect dielectric model and a small value leads it to steady leaky model. These findings and results may enhance our understanding of electrohydrodynamic patterning, and may be a meaningful guide for experiments.
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1
Introduction
Electric field can be used to generate micro/nano patterns in thin liquid film. This technique, sometimes referred as electrohydrodynamic patterning, is considered a powerful tool for micro/nano structures fabrication and has attracted rapid growing attentions recently [1]. Compared with conventional techniques, such as photolithography and nano imprint, its merits lie in the following aspects. First, the template and polymer film are non-contact (or micro contact) which eases demolding. Second, this technique is able to overcome the limit of light wavelength [2, 3], which makes this method suitable for micro/nano structure fabrication. In some cases, simple flat templates were employed in this technique to fabricate complex structures [4-6].
The account of electrohydrodynamic patterning started more than one decade ago and should be contributed to Chou and Schaffer [7-9]. A typical configuration of electrohydrodynamic patterning is illustrated in Figure 1. A structured (or flat) template is laid above a thin liquid polymer film. And an external voltage is imposed between the substrate and template. Usually a voltage of ~100V is applied to a substrate-template distance of ~1μm. If the template is structured, i.e., not flat, a nonuniform electrical would be generated along the polymer/air interface. For dielectric materials, the electrical force originates from the polarized charges on the polymer/air interface due to the difference of electrical permittivity between polymer and air. As to conductive polymers, the electrical force originates from the free charges and also polarized charges [10]. Driven by the nonuniform electrical force, polymer film would deform accordingly. The film keeps evolving and periodic micro/nano patterns are obtained eventually. In most cases, the polymer structures conform to the morphology of templates. For a flat template, the polymer film can also evolve into periodic patterns. This is because the film is unavoidably disturbed by some factors in environment (such as thermal fluctuation), and a small perturbation on the polymer interface would be enlarged by the electric field and evolves into micro/nano scale patterns [9].
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Thus far, different configurations have been adopted for electrohydrodynamic patterning which result into various patterns. For instance, templates with no structure, periodic grid lines and squared protrusions have been used. As to the patterns on polymer film, periodic pillars, grid lines and concentric rings have been successfully obtained [9, 11-13]. Essentially, the above patterns are polymer-air bilayer structure. Some researchers even extend this technique to fabricate trilayer structures, such as polymer-polymer-air or polymer-air-polymer patterns [14-20]. These structures have found widespread applications in various fields, such as micro/nano optics, microfluidics, and biomimetics [21-26].
Besides the experimental attempts and applications exploration, understanding the mechanism is another important aspect of electrohydrodynamic patterning. Studying the dynamic process has attracted considerable attention from researchers ever since electrohydrodynamic patterning was invented [27-31]. However, a direct simulation for the evolution of electrohydrodynamic patterning is of difficulty. The reason lies in the fact that the process involves multiple physics fields, i.e., the electric field and fluid flow field. Besides, the fluid flow contains two fluids (polymer and air) together with the free surface between them. At early stage, researchers used linear model to simulate the dynamic process. In this model, the fluid flow in the polymer film is considered as Poiseuille flow, and thin film lubrication approximation (long-wave approximation) is also employed [32-37]. Besides, all the higher order nonlinear terms are negligible and only linear terms are kept. Such a model is effective in predicting the dynamic process of electrohydrodynamic patterning and can be solved with ease. However, there are still some drawbacks for this model. On one hand, the polymer film is assumed to be slightly deformed, thus it fails for large deformation. On another hand, since all nonlinear terms are dropped, the accuracy of this model is harmed. Thus a fully nonlinear numerical model with high accuracy is required for electrohydrodynamic patterning.
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From the aspect of fluid mechanics, such a problem belongs to the category of multiphase flow. With the advances of computational technology, more numerical methods have been developed for such problems, such as volume of fluid (VOF), level set and phase field [38-43]. Among them, phase field is of favor in two-phase flow problems and has made remarkable achievement in simulating such problems. Compared with the traditional numerical methods for multiphase flow, such as moving mesh method, one major merit of phase field for is that it is remeshing free. Phase field enables moving interface and does not require remeshing when the interface evolves. Besides, it enables some phenomena that may occur in multiphase flow, such as coalescence or breakup of fluids [10]. The moving mesh method may fail to describe such phenomena due to the singularity problem. Moreover, phase field is compatible with other physics fields. Thus far, phase field has been successfully combined with electric field, thermal field, etc. [10]. Thus, phase field appears like a promising candidate to simulate the electrohydrodynamic patterning.
Besides the two-phase flow and the associated polymer film deformation, electric field is also involved in the process of electrohydrodynamic patterning. In the traditional electrodynamics theory, polymers are treated as dielectric materials due to their low conductivity. The electrical permittivity is considered as the influence factor for the electrical behavior, and conductivity is usually ignored. However, extensive experimental results indicate that conductivity although is infinitesimal, it plays an important role in polymer electrical behavior [44]. Thus G. I. Taylor proposed the leaky dielectric model which considers both electrical permittivity and conductivity for poorly conductive materials [45]. In this model, both polarized charges and free charges are allowed to exist in the system [46, 47]. The leaky dielectric model has been widely adopted as it leads to results that are well consistent with experimental observations. Therefore, we would employ the leaky dielectric model in this study to predict the electric field of electrohydrodynamic patterning.
The two-phase flow and leaky dielectric model are coupled together in the framework
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of phase field, and the dynamic process of electrohydrodynamic patterning is enabled. In this numerical model, the long-wave approximation is abandoned, as the long-wave approximation would lose its validity for a large deformed polymer film. Instead, the full Navier-Stokes equations for fluid flow and Poisson equation for electric field are solved numerically. In this paper, we focus on the effect of different parameters on the dynamic process. Using this phase field model, the influences of voltage, template structure height and polymer conductivity are discussed in details. We have found that the voltage plays an important role in the dynamic process of electrohydrodynamic patterning. At the end, the governing equations of this numerical model are nondimensionalized and some dimensionless parameters are obtained. With the assistant of dimensionless parameters, the effects of parameters are analyzed. In the fluid flow, it is estimated that gravity is insignificant. Besides, variation and inertial terms can also be ignored due to the small Reynolds number. Thus, the fluid flow can be considered as a steady flow. As to the electric field, the governing equations can also be simplified according to the electrical Reynolds number ER. For a large ER, the electric field will yield to perfect dielectric model. On the other hand, for a small ER the electric field can be described by steady leaky model. These results/analysis could be of help in understanding the mechanism of electrohydrodynamic patterning and in designing the experiments.
2
Mathematical Model
Without losing generality, we consider the electrohydrodynamic patterning process with a template of periodic protrusion as shown in Figure 1. A thin layer of polymer film with a thickness of h0 is coated upon a substrate and a template is then placed above the polymer film. Thus, a dual layer of polymer/air film is confined between the template and substrate. Rather than being flat, the template is structured with a structure height Δd and a periodic length l0. Both the template and substrate are conductive and the distance between them is controlled by the height of spacers which is denoted by d0. An external voltage V0 is imposed between the template and substrate. Driven by the electrical force, the polymer film starts to deform. The
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physical
properties
of
polymer and
air
have significant
effect
on the
electrohydrodynamic process. The density, viscosity, conductivity and electrical permittivity of air (polymer) are represented by ρ1 (ρ2), µ1 (µ2), σ1 (σ2), and ε1 (ε2). The profile of the free surface, i.e., the polymer-air interface, is denoted by h(x).
Various patterns have been observed in experiments and numerical simulations for the electrohydrodynamic patterning. In this study, we focus on the case that the polymer structure conforms well with the template, in other words, one structure forms underneath each protrusion of the template. Thus, we can compute only one period for simplification. The computational domain is illustrated in Figure 2, where numbers 1, 2, 3 and 4 denotes the boundaries.
Essentially, the dynamic process of polymer film evolution is a two-phase flow problem coupled with electric field. When studying the dynamic process, most pioneer researchers would make some simplifications due to its complexity. Usually the long-wave approximation is employed, i.e., the wavelength (periodic length of structures) is assumed much larger than the structure height. Taking this approximation, the governing equations for electric field and fluid flow can be significantly simplified [9, 17, 32, 34, 48]. However this assumption will lose its validity if the polymer film is notably deformed, i.e., the structure height is pronounced. Thus, we abandoned this long-wave approximation in the numerical model and solved the full Navier-Stokes equation and governing equations for electric field. By doing this, the numerical model could enable the large deformation case of EHD patterning. In this study, we use phase field method to simulate such a problem. Phase field is considered as a powerful tool for multiphase flow, moreover it enables multiphysics problem in which other physics field can be incorporated with phase field framework. In the following, the governing equations for phase field, electric field and fluid flow field together with associated boundary conditions are given consequently.
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2.1 The Phase Field Equations In phase field, an extra parameter φ (referred as phase parameter) is introduced to the governing equations. The main function of φ is to distinguish the two phases, i.e., φ=1 represents one phase and φ=0 for another one. As illustrated in figure 2, we use φ=1 for polymer and φ=0 for air. In the classical viewpoint, the polymer and air are separated by an ideal interface with zero thickness. However, in phase field, the polymer/air interface is considered as a thin diffusive layer. In this layer, the phase parameter φ is between 0 and 1. The interface can be revealed by tracing the profile of φ=0.5. By introducing phase parameter φ, other physical parameters, such as density ρ, viscosity µ, can be considered as functions of φ. Thus, the two phases, i.e., polymer and air, together with their interface are governed by unified governing equations [49].
We first define the free energy ffree for the system in study, which takes the Ginzburg-Landau form [10], f free
1 1 2 2 2 1 2 4 2
where ξ represents the two phase interface thickness and λ expresses as 6
(1) 2
,
where γ is the surface tension coefficient. The above equation defines the energy density at each point of the computational domain. As can be seen, this energy is zero at the bulk fluids (polymer and air), where φ=1 or φ=0. It becomes nonzero at the interface where 01, the third term (electromigration effect) in equation (11) is infinitesimal and can be neglected thus there would be no source term for free
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charges. Meanwhile there is no free charges initially and no injected free charges during the process. Therefore, the free charge is always zero. Equation (11) becomes redundant and governing equations of electric field yield to Laplace equation
0 r V * 0 . According to Figure 9, leaky dielectric model approaches to
perfect dielectric model when σ102.
As to the fluid flow, some simplifications can also be made. For a micro/nano flow, the Reynolds number usually is very small. Actually if we substitute the parameters Re
from table 1, we have
0 2 1 V0 2 2
~10-7. Thus for equation (12), the inertial
effect, including the variation term (first term) and convection term (second term), are ignorable. Also the Galilei number Ga
g 0 d 03 -14 2 ~10 , which justifies the 0 2 1 V0
neglect of gravitational force. Thus, the fluid flow is governed by the balance between viscous force, electrical force and surface tension force, i.e., the Navier–Stokes equation can be simplified as,
T 1 0 1 2 1 p u u e E E r 6 2 2 2 Ca Cn 2
(17)
Electrical force is the driving force of the process, and viscous force and surface tension are the resistance which hinder the process. The effect of electrical force and viscous force can be revealed by the expression of characteristic time
tc
2 d 0 2 0 2 1 V02
.
In order to decrease the process time, one needs to increase the voltage V0, using polymer with a high electrical permittivity ε2 or with a low viscosity μ2. Besides, a high electrical conductivity σ2 is also favorable for the process as it may increase the electrical force. As to the effect of surface tension, a polymer with low surface tension γ is of help as it leads to a large Capillary number.
5
Conclusions
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In this study, we proposed a phase field numerical model to simulate the dynamic process of electrohydrodynamic patterning. In this model, the deformation polymer film, which by the way belongs to a two-phase flow problem, is described by phase parameter. As to the electric field, a fully leaky dielectric model is employed which considers both the electrical permittivity and conductivity. Using this phase field model, the dynamic process of electrohydrodynamic patterning is studied with an emphasize on the effect of different parameters. It is found that external voltage can significantly affect the dynamic process. For a low voltage, the polymer film would grow slowly until achieves to its steady state which is a slightly deformed pattern. For a higher voltage, five stages can be observed based on the growth rate of structure height. However for an even higher voltage, only three stages exist during the process. This phenomenon is determined by the competition between electrical force, viscous force, and surface tension. As to the influence of template structure height, it is found that a large structure height can increase the growth rate, i.e., speeding up the process time. However, when the structure height rises up to the template-substrate distance, a further increase of structure height becomes insignificant.
Then the governing equations are nondimensionalized to analyze the relationship between different parameters. Some dimensionless parameters are defined, such as Reynolds number Re, Galilei number Ga, and Electrical Reynolds number ER. For a real case in experiment, usually the Re and Ga are very small, which justifies the ignorance of variation term, inertial term and gravitational force in Navier-Stokes equation. The governing equation thus becomes steady fluid flow (sometimes termed as Stokes flow or low Reynolds number flow). For the electric field, it could also be simplified according to the value of ER. If ER>>1, the governing equation yields to perfect dielectric model and free charges can be ignored. On the contrary, if ER
