Kinetics of Field-Induced Surface Patterns on PMMA - American

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Kinetics of Field-Induced Surface Patterns on PMMA Jyun-siang Peng,† Fuqian Yang,‡ Donyau Chiang,§ and Sanboh Lee*,† †

Department of Materials Science and Engineering National Tsing Hua University, Hsinchu 300, Taiwan Department of Chemical and Materials Engineering University of Kentucky, Lexington, Kentucky 40506, United States § Instrument Technology Research Center, National Applied Research Laboratories, Hsinchu 30076, Taiwan ‡

ABSTRACT: A simple model was developed to analyze the growth of a liquid pillar under the action of an electric field between two parallel electrodes. A quadratic relationship between time and the diameter of the pillar was obtained. The diameter of the pillar increases with time. Large electric field assists the growth of the liquid pillar, while a liquid with a large viscosity hinders the growth of the liquid pillar. The fieldinduced formation and growth of PMMA pillars on PMMA films were observed using the configuration of a parallel capacitor. Pillars of larger sizes and smaller densities were formed on thicker PMMA films than on thinner PMMA films. The root-mean-square (https://en.wikipedia.org/wiki/Root_ mean_square) diameter of the pillars increases with the increase of the annealing time and annealing temperature. The growth behavior of the pillars can be described by an Arrhenius relation with an activation energy of 24.4 kJ/mol, suggesting that the growth of the pillars is controlled by a thermal activation process.

1. INTRODUCTION The progress in the micro- and nanofabrication techniques has made it possible to fabricate nanoelectronics and microelectromechanical structures. Self-assembly of regular and polymer nano- and microstructures has demonstrated the potential of constructing surface structures on the surface of polymer films, which might not be fabricated through conventional microfabrication techniques. In the heart of the self-assembly of polymer nano- and microstructures is the surface instability of polymer films through the interaction between molecules and surrounding medium, such as van der Waals force,1,2 evaporation, 3,4 electric field,5−7 thermal gradient,8,9 acoustic dispersion force,10 LISA,11,12 and stress and surface tension,13,14 which leads to the motion of contact line and formation of surface structures. Chou et al.11 observed the self-formation of periodic, supramolecular (micrometer scale) pillar arrays in a thin, single-homopolymer film melt between two parallel plates, in which one plate was coated with a smooth, thin, singlehomopolymer film. They suggested that the formation of the pillar arrays is due to the surface instability induced by surface charges. Schäffer et al.15 constructed a capacitor of a wedge shape with one plate coated with a polymer film (PMMA, PS, and PBrS) of liquid state and examined the field-induced surface instability of the polymer film. They measured the characteristic wavelength of the instability. Pattader et al.16 constructed hierarchical, multiscale surface patterns on incompletely cross-linked viscoelastic polydimethylsiloxane (PDMS) films and attributed such behavior to the spatiotemporal interaction between electrodes and electric field. Leach et al.6 studied field-induced surface patterns on © 2016 American Chemical Society

PMMA/PS bilayers and observed the formation of closed-cell structures. Tian et al.17 used a patterned template to form “irregular” surface patterns on mr-NIL 6000E, which was sandwiched between the template and a silicon substrate, under the action of an electric field. They suggested that there exist three regimes for field-induced patterning: (1) spatially undermodulated, (2) spatially fine-modulated, and (3) spatially overmodulated. Wu et al.18 examined the coarsening of PDMS pillars in an electric field and found that average pillar size increases slowly after linear growth, and there is a logarithmic time-dependence of the average pillar size. The field-induced surface patterning on polymer films can be attributed to electrohydro-dynamic (EHD) instability, as reported first by Tonks19 in studying the surface rupture of liquid by an electric field. It is the competition between electric stresses, disjoining pressure and surface tension, that determines the condition for the field-induced surface patterning. Most analyses20,18,21−23 have been based on the theory of viscous flow. Considering the viscoelastic characteristics of polymer, Wu and Chou24 and Tomar et al.25 studied the EHD instability of a confined viscoelastic liquid film sandwiched between parallel plates. They24 suggested that there are two destabilizing mechanisms (electrostatic force and polymer elasticity) determining a resonant phenomenon associated with surface instability. Yang and his co-workers26−28 had analyzed the field-induced surface instability of solid controlled by surface or lattice diffusion. There are few studies Received: April 5, 2016 Published: April 20, 2016 4602

DOI: 10.1021/acs.langmuir.6b01304 Langmuir 2016, 32, 4602−4609

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at the sides. ε0 is the dielectric constant of vacuum, and ε (>ε0) is the dielectric constant of the dielectric pillar. Assume that there is a virtual increase of the radius, Δr, of the dielectric pillar, which occurs arbitrarily under the condition of a constant voltage. The work done by the electric stress, 2πrhσrΔr, is equal to the increase of the stored electric energy, that is,

focusing on temporal behavior of the pillars formed between two parallel plates under the action of an electric field. It is known that metallic nanoparticles and submicron structures over a large area can be prepared via the dewetting of a thin solid metal film on a poorly wetting flat substrate.29,30 Ruffino et al.31 reported the formation of Au nanostructures via the dewetting of Au nanofilms on PMMA substrates. Considering the potential applications of surface structures on PMMA substrates in the patterning of metal films for plasmonics and flexible electronics, the temporal evolution of the pillars formed on PMMA [poly(methyl methacrylate)] films between two parallel plates under the action of an electric field is investigated. The effect of temperature is examined. A simple model is developed to analyze the temporal growth of a dielectric pillar sandwiched between two parallel electrodes under the action of an electric field, and a relation between the pillar size and the growth time is established to explain experimental observation.

⎛ V ⎞2 ⎛ V ⎞2 2πrhσr Δr = −πhrε0⎜ ⎟ Δr + πhrε⎜ ⎟ Δr ⎝h⎠ ⎝h⎠ ⎛ V ⎞2 = πhr(ε − ε0)⎜ ⎟ Δr ⎝h⎠

which gives σr =

Figure 1. Schematic of the liquid pillar between two parallel electrodes under the action of an electric voltage: (a) the pillar is in a shape of a cylinder, (b) the growth of the pillar is controlled by the transverse motion of the liquid film, and (c) the growth of the pillar is controlled by the radial flow in the liquid film.

capacitor, consisting of the dielectric pillar and air. An electric voltage of V is applied between the electrodes, which leads to the storage of electric energy in the capacitor. The total stored electric energy, UE, is 2 2 1 ⎛⎜ V ⎞⎟ 1 ⎛V ⎞ ε0 π (b2 − r 2)h + ε⎜ ⎟ πr 2h + UE1 2 ⎝h⎠ 2 ⎝h⎠

⎛ V ⎞2 1 (ε − ε0)⎜ ⎟ ⎝h⎠ 2

(3)

Here, σr is the electric stress applied on the surface of the dielectric, liquid pillar. From eq 3, it is evident that the liquid pillar is experiencing a radial, tensile stress, which tends to increase the radius of the dielectric, liquid pillar. Note that the third term in eq 1 due to the fringing field does not change if the surface of the liquid pillar is not too close to the edges of the plates. For a dielectric, liquid film on the surface of the bottom electrode in a parallel capacitor, applying electric voltage can introduce surface instability and lead to the formation of pillars on the liquid film. The pillars experience radial stress, as analyzed above, which can cause the growth of the pillars in the radial direction. There are two mechanisms for the fieldinduced growth of the pillars sandwiched between two electrodes: (1) the electric stress on the surface of the liquid film causes transverse motion of the liquid film from one electrode to the other electrode, similar to the formation of a pillar (Figure 1b), and (2) the electric stress on the surface of the pillars, as given in eq 3, leading to the radial growth of the pillars in which the liquid flows into the root of the pillars (Figure 1c). When the transverse growth rate is much slower than the radial growth rate, the first mechanism is dominated. On the other hand, when the radial growth rate is much slower than the transverses growth rate, the second mechanism controlled the kinetics. Note that no liquid pillars with height less than the distance between two electrodes were observed in the experimental study. Thus, the analysis is focused on the radial growth of a pillar controlled by the viscous flow of the liquid in the liquid film to the root of the pillar. To simplify the analysis and obtain an analytical solution of the temporal evolution of the pillar, the following assumptions are made: (1) the liquid is a Newtonian fluid, (2) the flow in the liquid film can be approximated as a lubricantion flow, (3) the kinetic energy of the pillar during the radial growth is negligible, (4) the liquid film has the same thickness everywhere, and (5) the radius for the flow of the liquid to the liquid pillar through the pillar root remains unchanged during the radial growth of the liquid pillar. Figure 1c shows the physical model for the growth of a pillar, which consists of a thin disk of a film thickness of δ, outer radius b, and inner radius a which is the radius for the liquid to flow into the pillar. Considering the axisymmetric characteristics and the condition of |b − a| ≫ δ, one can assume that the velocities in the θ and z direction in a cylindrical coordinate system (r, θ, z) are so small in comparison with the velocity in

2. PROBLEM FORMULATION Consider a dielectric, liquid pillar of radius r and length h, which is sandwiched between two circular, parallel electrodes of radius b, as shown in Figure 1a. The structure forms an electric

UE =

(2)

(1)

where the first term on the right side represents the electric energy stored in air, the second term on the right represents the electric energy stored in the liquid pillar, and the last term on the right represents the electric energy due to the fringing field 4603


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Figure 2. Flowchart for the surface treatment of the Si plates for top electrode of a parallel capacitor.

the r direction that they are negligible to a first order approximation. Thus, the equations describing the flow of the fluid become

∂ (r vr) = 0 ∂r

The boundary conditions are

(4)

for mass conservation, and −

⎛1 ∂ ∂σ ⎞ ∂p σ +⎜ (rσ ) − θθ + rz ⎟ = 0 ⎝ r ∂r rr ∂z ⎠ ∂r r

1 ∂p − =0 r ∂θ −

(5)

for mechanical equilibrium. Here, vr is the velocity in the r direction, p is pressure, and σrr, σθθ, and σrz are the stress components. For |b − a| ≫ δ, there are

∂vr ∂z

(14)

p = 0 at r = b

(15)

p = pa at r = a

(16)

(17)

Substituting eq 9 into eq 10 and using the boundary conditions, one obtains the velocity as vr =

(8)

2 1 ∂p 1 (ε − ε0) ⎛⎜ V ⎞⎟ z(z − 2δ) z(z − 2δ) = 2η ∂r 4η ln(b/a) ⎝ h ⎠ r

(18)

With the first-order approximation, there are σrr = σθθ = 0 and σrz = η

∂vr = 0 at z = δ ∂z

⎛ V ⎞2 1 pa ≈ − (ε − ε0)⎜ ⎟ ⎝h⎠ 2

(7)

∂vr ∂v v ∂v < < r and r < < r ∂r ∂z r ∂z

(13)

The condition of (15) assumes that the stress at the outer radius of the film is completely relaxed. If the contribution of the surface tension of the liquid pillar is negligible to the stress balance at the surface of the pillar, one has

(6)

∂p 1 ∂ + (rσrz) = 0 ∂z r ∂r

vr = 0 at z = 0

From eq 18, the rate of the liquid flowing through the root of the liquid pillar, V̇ , is found as (9)

V̇ = −

with η being the viscosity of the liquid. The equations for mechanical equilibrium become ∂σ ∂p − + rz = 0 ∂r ∂z

(10)

1 ∂p =0 r ∂θ

(11)

∂p =0 ∂z

(12)

∫0

∫0

δ

2πr vr dz =

2 π (ε − ε0) ⎛⎜ V ⎞⎟ 2η ln(b/a) ⎝ h ⎠

δ

z(z − 2δ) dz =

3 2 π (ε − ε0)δ ⎜⎛ V ⎟⎞ 3η ln(b/a) ⎝ h ⎠

(19)

The mass balance gives V̇ =

πDh dD 2 dt

(20)

in which D is the diameter of the pillar. Subsituting eq 20 in eq 19 yields the temporal evolution of the pillar as 4604


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3 2 4 (ε − ε0)δ ⎛⎜ V ⎞⎟ (t − t0) 3η ln(b/a)h ⎝ h ⎠

Fisher Scientific). The surface topology of the surface structures formed on the PMMA films was characterized by an optical microscope (Olympus Optical Co.) and an atomic force microscope (AFM) (Dimension ICON, Bruker) using a tapping mode. The spring constant of the AFM tips was ∼42 N/m, and the resonance frequency was 330 kHz. The characteristics of the surface patterns were analyzed, using the AFM software, Gwyddion (Czeh Metrology Institute), and ImageJ (National Institute of Health).

(21)

Here, D0 is the diameter of the pillar at the incubation time of t0. Before the incubation time, the pillar experiences the nucleation process and is presented as a nucleus. There is a quadratic relationship between the growth time and the diameter of the pillar, which is different from the results given by Wu et al.,18 who analyzed the growth of a liquid pillar through the transverse motion of the liquid film.

4. RESULTS AND DISCUSSION The FTIR spectra of the polymer are depicted in Figure 3. The polymer displayed the stretching mode of single C−H bond

3. EXPERIMENTAL DETAIL The material used in this work was PMMA (Acros Organics) with average molecular weight of 35000 g/mol and the polydispersity index of 1.2. Polymer solutions were prepared by dissolving PMMA in chloroform. The polymer solutions were stirred for 2 h at 25 °C and were then filtered using PTFE (polytetrafluoroethylene) membranes (Pall, New York) with the pore size of ∼200 nm to remove undissolved particles. The silicon substrates used in this work were p-type Si (100) wafers with a thickness of 525 μm (Summit-Tech, Taiwan). Using the microelectronic fabrication process, spacers of dielectric SiO2 with 670 nm in thickness and 1 mm in width were prepared. The Si wafers with the spacers of 0.5 cm apart were sliced into plates of 1 × 1 cm2. To increase electric conduction, a sputter coater (Pelco SC-6, Ted Pella) was used to coat an Au film of ∼40 nm in thickness on the rear side of the Si plates at a pressure of 0.5 mbar and an electric current of 0.5 mA. PMMA thin films with molecular weight of 35000 g/mol were spincoated on the Si plates, using a spin-coating process at 25 °C. PMMA thin films of different thicknesses in the range of 50−80 nm were obtained, using different solution concentrations. The spin-coating was performed on a spin coater SWIENCO PM-490 at 4000 ± 50 rpm for 20 s. The Si plates with the PMMA films were used as the bottomelectrode in parallel capacitors. The Si plates for the top-electrodes of the parallel capacitors were surface-treated using the approach of forming a self-assembled monolayer. The sequences of the surface treatment are shown in Figure 2, which include (1) immersion of the Si plates in a piranha solution for 30 min, (2) ultrasonification in acetone and DI water for 10 min, respectively, (3) hydroxylation in a piranha solution at 90 °C for 30 min, (4) spin-clean, using DI water, at a spin speed of 5000 rpm, (5) dehydration at 70 °C for 8 h, (6) ultrasonification in a solution consisting of toluene (JT Baker Chemical Co.) and 0.5 wt % OTS [noctadecyltrichlorosilane (90%+, Aldrich Chemical Co., Inc.)] for 1 min and then stationary immersion in the solution for 9 min, (7) spinclean, using toluene, at a spin speed of 5000 rpm. The surface treatment led to the formation of single molecular layer of noctadecyltrichlorosilane on the surface of the Si plates which helps the separation of the top electrode from the PMMA pillars induced by the electric field. Similar to the Si plates for the bottom electrode, an Au film of ∼40 nm in thickness was coated on the rear side of the Si plates after the surface treatment at a pressure of 0.5 mbar and an electric current of 0.5 mA. Parallel capacitors were constructed using the Si plates with the PMMA films as the bottom electrode (cathode) and the Si plates with a single molecular layer of n-octadecyltrichlorosilane as the top electrode (anode). A DC power (PPS-2018A, Taiwan) was used to apply an electric field between two electrodes in a furnace (DV-303 Channel, Taiwan) in air. The temperature was in the range from 150 to 180 °C. To determine the sizes of the pillars formed, the capacitors were removed immediately from the furnace after finishing the field-assisted process and quenched by spreading air. This process limited any possible changes of shapes of the pillars due to viscoelastic flow and allowed the observation of the geometrical configuration of the pillars. The FTIR (Fourier transform infrared spectroscopy) of the received polymer was performed using Nicolet Nexus 320 FTIR (Thermo

Figure 3. FTIR spectra of a PMMA film.

around 2856−2970 cm−1, the stretching mode of double CO bond around 1700−1744 cm−1, the bending mode of single C− H bond around 1438 cm−1, and the stretching mode of C−O− C around 11151 cm−1. These results are in accord with those for pure PMMA,32,33 which suggests that the received polymer is chemically the same as pure PMMA. The surface roughness of the prepared PMMA films was characterized using AFM. The root-mean-square roughness was in the range of 1−3 nm. The prepared PMMA films were relatively smooth. There were no significant surface features which can influence the surface organization on the surface of the polymer films under the action of an electric field. Figure 4 shows optical images of the field-induced pillars on the surface of PMMA films of 50, 60, 70, and 80 nm, respectively, for different heating times at a temperature of 170 °C under an electric voltage of 30 V. It is evident that the radii of the pillars increase with increasing the annealing time, similar to the observation by Wu et al.,18 suggesting the continuous flow of the PMMA into the pillars after the formation of the pillars. There exists a driving force which counterbalances the increase in surface energy. Comparing the sizes of the pillars formed on the PMMA films of different thicknesses under the same conditions (electric voltage and annealing temperature), one can note that pillars of larger sizes and smaller densities were formed on thicker PMMA films than on thinner PMMA films. This is likely due to the fact that there is less disjoining pressure acting on the surface of thicker films, which requires less electric stress to initiate the surface instability, and there is enough PMMA being allowed to continuously flow into the pillars. The geometric characteristics of the pillars on the PMMA films were characterized by an atomic force microscope (Dimension ICON, Bruker) using a tapping mode. Figure 5 shows AFM images of the field-induced pillars on the surface of PMMA films of 50, 60, 70, and 80 nm, respectively, for different annealing times at a temperature of 170 °C under an electric voltage of 30 V. The AFM images reveal the same trends as observed from the optical images. The pillars are presented in 4605


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The temporal evolution of the root-mean-square diameters of the pillars at a temperature of 170 °C under an electric voltage of 30 V is depicted in Figure 6. The root-mean-square

Figure 6. Temporal evolution of the pillars formed on PMMA films of 50, 60, 70, and 80 nm (temperature: 170 °C; electric voltage: 30 V).

diameter of the pillars formed on all four PMMA films of different thicknesses increase with the increase of the annealing time, suggesting that there exists a large driving force which continuously forces the PMMA to flow into the space between the two parallel plates for the experimental conditions used. According to eq 21, one can approximately express the relation between the root-mean-square (https://en.wikipedia. org/wiki/Root_mean_square) diameter of the pillars and the annealing time as

Figure 4. Optical images of the field-induced pillars on the surface of PMMA films of 50, 60, 70, and 80 nm, respectively, for different heating times (temperature: 170 °C; electric voltage: 30 V).

D = D0 +α(t − t0)1/2

(22)

where ⎛ 4 (ε − ε )δ 3 ⎞1/2 ⎛ V ⎞ 0 ⎟ ⎜ ⎟ α∝⎜ ⎝ 3η ln(b/a)h ⎠ ⎝ h ⎠

(23)

⟨D⟩ is the root-mean-square diameter of the pillars, ⟨D0⟩ is the root-mean-square diameter of the pillars at the incubation time of t0, t is the annealing time, and α is the parameter inversely proportional to the viscosity of the polymer. Eq 22 is used to curve-fit the experimental data, and the fitting curves are also included in Figure 6. The fitting parameters of (⟨D0⟩, α, and t0) in the units of (μm, μm/min1/2, and min) are (0.53, 0.039, and 4.5), (0.61, 0.038, and 4.5), (0.67, 0.052, and 4.5), and (0.83, 0.039, and 4.5) for the film thicknesses of 50, 60, 70, and 80 nm, respectively. Note that the confidence of the value of the parameter of α for the curve-fitting is larger than 0.95. The incubation time is independent of film thickness. The rootmean-square diameter of the pillar nuclei at the incubation time of 4.5 min increases with the increase of the film thickness, as expected. This is due to the fact that more PMMA is available to form the pillars. The parameter of α is in a range from 0.038 to 0.052 μm/min1/2 and is generally independent of the film thickness. This result suggests that the mobility of the polymer chains during the growth of the pillars is independent of the film thickness. It is known that the mobility of the polymer chains is temperature-dependent. One expects that the growth of the pillars is also temperature-dependent. To examine the temperature-dependent growth of the pillars, the field-induced growth of the pillars on PMMA films of 70 nm were performed in a temperature from 150 to 180 °C under the action of an electric

Figure 5. AFM images of the field-induced pillars on the surface of PMMA films of 50, 60, 70, and 80 nm, respectively, for different heating times (temperature: 170 °C; electric voltage: 30 V).

the form of cylinder, and the difference between the crosssectional area of the top surface and that of the bottom surface of the pillars is not detectable. The height of the pillars is in a range from 180 to 200 nm, independent of the initial thickness of the PMMA films. This result suggests that the pillars start to form on the surface of the PMMA, as expected, grow in the length direction to be in contact with the top electrode, and then grow in the radial direction. 4606


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Langmuir field of 30 V. Figure 7 shows the temporal evolution of the pillars formed on PMMA films of 70 nm at four annealing

Figure 7. Temporal evolution of the pillars formed on PMMA films of 70 nm at different temperatures (electric voltage: 30 V).

temperatures. For the same annealing time, the average diameter of the pillars increases with the increase of the annealing temperature due to a larger mobility (smaller viscosity) of polymer chains at a higher temperature. Eq 22 is used to curve-fit the experimental data, and the fitting curves are also included in Figure 7. It is evident that eq 22 can be used to describe the growth of the pillars for the experimental temperatures used. Using the fitting results, the temperature dependence of the parameter of α is depicted in Figure 8. The curve-fitting of the

Figure 9. FFT diffraction patterns of the pillars formed on the PMMA films of four different thicknesses at four different temperatures (annealing time: 70 min; electric voltage: 30 V).

diffraction patterns of the pillars formed on the PMMA films of four different thicknesses at four different temperatures under the action of an electric field of 30 V. The FFT diffraction patterns exhibit ring-like patterns. The ring-like patterns indicates that the spatial distribution of the pillars is narrow, and the temperature and film thickness have no significant effect on the self-assembly of the pillars for the experimental conditions used. From the FFT diffraction patterns, one can determine the variation of the intensity with wavenumber of the surface patterns. Figure 10 shows the variation of the characteristic

Figure 8. Temperature dependence of the parameter α for the growth of the pillars on the PMMA films of 70 nm at different temperatures (electric voltage: 30 V).

experimental data reveals that the parameter of α is an exponential function of temperature, following the Arrhenius relation as

α = 1.43e−1470/ T

Figure 10. Variation of the characteristic wavenumber with film thickness for the surface patterns formed at four different temperatures (annealing time: 70 min; electric voltage: 30 V).

(24)

wavenumber with the film thickness for the surface patterns formed at four different temperatures (with the annealing time of 70 min and an applied electric voltage of 30 V). The characteristic wavenumber decreases with the increase of the film thickness and the increase of the annealing time under the same annealing temperature and electric voltage, which is qualitatively in accord with the variation of the average density of the pillars on the film thickness. The pillars formed on a PMMA film of a larger film thickness have a larger characteristic wavelength than those formed on a PMMA film of a smaller film thickness.

As shown in eq 23, the parameters, ε, ε0, δ, b, a, V, and h are temperature-independent with the exception of η. The parameter of α is inversely proportional to the square root of the viscosity of the polymer. The motion of the polymer chains under the action of an electric field is controlled by a thermal activation process. From eq 24, the thermal activation energy is found to 24.4 kJ/mol. To analyze the spatial characteristics of the pillars formed under the action of the electric field, two-dimensional (2D) fast Fourier transform of the surface patterns was performed using ImageJ (National Institute of Health). Figure 9 shows the FFT 4607


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quadratic relationship between the growth time and the diameter of the pillar was obtained. The square of the pillar diameter is a linear function of the ratio of the square of the electric voltage to the viscosity of the liquid film. Large electric field will accelerate the growth of the liquid pillar, and the liquid with low viscosity will have less resistance to the viscous flow for the growth of the liquid pillar. The field-induced formation and growth of PMMA pillars on PMMA films were performed using the configuration of a parallel capacitor. Pillars of larger sizes and smaller densities formed on thicker PMMA films than on thinner PMMA films. The root-mean-square diameter of the pillars formed on all PMMA films of different thicknesses increases with the increase of the annealing time, and the temporal evolution of the pillars is qualitatively in accord with the relation derived. The temperature-dependence of the temporal evolution of the pillars follows the Arrhenius relation with an activation energy of 24.4 kJ/mol. The FFT diffraction patterns of the surface patterns formed on the PMMA films exhibit ring-like patterns, suggesting that the size distribution of the pillars is narrow. The characteristic wavenumber of the surface patterns decreases with the increase of film thickness and the increase of annealing time under the same annealing temperature and electric voltage. The variation of the spatial wavenumber with the film thickness cannot be simply attributed to the surface instability induced by electric field and van der Waals interaction for the initiation of surface patterns. It might involve the coalescence of the pillars during growth.

There are various analyses focusing on the dependence of spatial wavelength on electric field intensity, using the stability analysis,18−23 for field-induced surface patterns on the surface of polymer films. In general, the spatial wavelength, λ, is a function of applied voltage and initial film thickness, hp,which can be expressed as18−23 λ = 2π

3/2 γh3 [εp − (εp − 1)hp/h] (εp − 1)V ε0εp

(25)

where ε0 is the dielectric constant of vacuum, εp is the relative dielectric constant of polymer film, hp is the thickness of polymer film, and γ is the surface tension of polymer film. It is known that spatial wavenumber, k, is inversely proportional to spatial wavelength length as k = 2π/λ. There is k=

ε0εp 3

(εp − 1)V

γh [εp − (εp − 1)hp/h]3/2

(26)

For PMMA, the relative dielectric constant is 3.6. For h = 670 nm and a constant voltage of 30 V used in this work, the variation of the spatial wavenumber with the film thickness is depicted in Figure 11. The spatial wavenumber increases with

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel/Fax: 886-3-5719677. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was financially supported by the Ministry of Science and Technology, Taiwan.

■

Figure 11. Thickness dependence of wavenumber (temperature: 200 °C).

REFERENCES

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the decrease of the film thickness in contrast to the result of eq 26. It needs to be pointed out that Verma et al.34 considered the contribution of van der Waals interaction and electrostatic interaction and noted that the spatial wavelength decreases with the film thickness, which is also in contrast to the results shown in Figure 11. Such a result indicates that the surface patterns formed cannot be simply attributed to the surface instability induced by the electric field and van der Waals interaction for the initiation of surface patterns. It might involve the coalescence of the pillars during growth.

5. CONCLUSION Surface structures of small scales have great potential in various scientific and technological applications. Using electric field to form surface patterns on polymer films has become a viable method to the fabrication of surface nanostructures. A simple model for the growth of a liquid pillar under the action of an electric field between two parallel electrodes was developed. A 4608


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Kinetics of Field-Induced Surface Patterns on PMMA - American

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