Thermo-Electrohydrodynamic Patterning in Nanofilms - Langmuir

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Thermo-Electrohydrodynamic Patterning in Nanofilms Hadi Nazaripoor, Charles R. Koch, Mohtada Sadrzadeh, and Subir Bhattacharjee Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b01810 • Publication Date (Web): 25 May 2016 Downloaded from http://pubs.acs.org on May 28, 2016

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Thermo-Electrohydrodynamic Patterning in Nanofilms Hadi Nazaripoor,† Charles R. Koch,∗,† Mohtada Sadrzadeh,† and Subir Bhattacharjee,‡ Department of Mechanical Engineering, 10-232 Donadeo Innovation Center for Engineering, University of Alberta, Edmonton, Alberta, Canada. T6G 1H9, Tel: +1 780 4928821 E-mail: [email protected]

Abstract To improve the electrically assisted patterning process and create smaller sized features with the higher active surface area, the combined thermocapillary-electrohydrodynamic (TC-EHD) instability of liquid nanofilms is considered. First, the 3-D thin film equation is re-derived for non-isothermal films and then the influential factors on the dynamics and stability of thin liquid film are found using linear stability (LS) analysis. Non-linear studies are also conducted to investigate the long-time evolution of the interface using an in-house developed Fortran code employing high order finite difference and adaptive time step solver for the spatial and time derivatives. The number density of pillars (columnar raised structure) formed in 1µm2 area is significantly increased compared to the EHD base-case and nano-sized pillars are created due to the thermocapillary effects. Relative interface area increases of up to 18% due to this pattern ∗

To whom correspondence should be addressed Department of Mechanical Engineering, 10-232 Donadeo Innovation Center for Engineering, University of Alberta, Edmonton, Alberta, Canada. T6G 1H9, Tel: +1 780 4928821 ‡ Water Planet Engineering, 721 Glasgow Ave, Unit D, Inglewood, California, 90301, USA †

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miniaturization are realized. It is also found that increase in the thermal conductivity ratio of layers changes the mechanism of pattern formation resulting in nonuniform and randomly distributed micro pillars being generated.

Introduction Decreasing the feature size on integrated circuits and the fabrication of nano-sized patterns are a significant challenge to the semiconductor industry. 1 The need for powerful microprocessors and cheaper electronic memories has accelerated the research effort on finding novel, fast and inexpensive techniques for creation of nano-sized features. In addition to conventional lithographic techniques (based on the use of electrons, photons or ions) which are commonly used in semiconductor industry, other techniques like nanoimprinting 2,3 and soft lithography, 4 have recently gained attention due to their ability in creation of nano-sized structures. Other alternatives are contactless pattern transfer methods which are based on self-organization 5–8 and surface instabilities. 9,10 In the surface instabilities category, for ultra-thin (thickness less than 100 nm) liquid/molten polymer films, surface instability and subsequent drainage and pattern evolution determine the size and shape of the surface features. Interface instability can be triggered using external forces 11–15 (mechanical, thermal and electrical) as well as intermolecular interactions (van der Waals and polar interactions). 5 Electrically induced instability of thin liquid films is a contactless pattern transfer method which has gained extensive attention due to its ability in creation of novel micro- and nanosized structures ranging from bifocal micro-lens arrays, 16 micro and nano channels, 17 and biomimetic mushroom-shaped microfibers 18 to high-aspect-ratio micropillar arrays. 19 In electrohydrodynamic (EHD) patterning, also called electrically induced patterning (EIP), a thin liquid film is sandwiched between two electrodes and the gap between electrodes is filled with air or a bounding fluid. Electrodes (flat/patterned) are used to apply a transverse (homogeneous/heterogeneous) electric field to the thin film. The mismatch of electrical properties of layers (film and the bounding layer) results in a net electrostatic force application on the thin 2


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film-bounding fluid interface. The destabilized interface deforms over time and a balance between electrostatic, viscous and interfacial forces leads to the formation of patterns at the interface. 11 Early stages of the interface evolution in the EHD process is analytically predicted by linear stability (LS) analysis providing time and length scales for the growth of instabilities and their sensitivity to the geometrical and EHD properties of the system. It is shown that the maximum wavelength (λmax ) in growth of instabilities matches with the centerto-center distance of pillars 11 and thus this wavelength is used as a characteristic length scale in these systems. In LS analysis, surface tension, viscosity, initial thickness, electric permittivity, conductivity of layers, applied voltage and electrodes spacing are found to affect the patterning process and formation of various surface morphologies. Research in EHD patterning has focused on miniaturization of structures from micron to submicron level, 20–24 reducing the patterning time 25 and fabrication of well ordered structures. 10,26,27 Smaller structures (lower λmax ) are generated via tuning the electrostatic force acting on the interface by changing the electrical properties of either film or the bounding layer. 21–24,28–31 To reduce the structure size, the air gap is replaced with a polymer layer and the interfacial tension is decreased as this is one main barrier in the EHD pattering process. 32–34 In perfect dielectric (PD) films (with no free charge) structure size is reduced by increasing the electrical permittivity of the film which is attributed to an increase in polarizability of the film. In the case of leaky dielectrics (LD), (poorly conductive materials) there is an infinitesmal amount of charges that accumulate at the interface. The charge accumulation adds an extra term to the electrostatic force which results in a higher electrostatic pressure in LD compared to the PD and thus λmax decreases significantly. 21,30 Ionic conductive (IC) materials (finite amount of free ions or charges) are another types of material used in the EIP process. These materials have been first used as a bounding fluid 22,23 to maximize the electric field subjected to the polymer film in a bilayer system by minimizing the potential

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drop at the bounding layer. EHD pattering of ionic liquid films has been recently simulated numerically using an electrokinetic theory instead of PD and LD models. 24,35 Formation of a finite charge diffuse layer at the interface is found to result in the generation of more compact pillars due to a significant increase in the electric force applied at the interface. In thin liquid films resting on a solid substrate, variation of temperature along the interface results in surface tension nonuniformity which breaks the mechanical equilibrium. The resulting interfacial shear stress leads to a convective motion of fluid. Viscosity and mass continuity carry this interfacial motion to the bulk. Along the interface, viscosity tends to stabilize this perturbation. The instability in heated liquid films due to interfacial tension inhemogenities is known as Marangoni or thermocapillary (TC) instability. 36 Dynamics, instability and evolution of heated thin liquid films have be previously investigated using linear and nonlinear stability analysis. 36–41 Acoustic phonos (AP) model 42 and TC flow model 13 are two mechanisms have been proposed to describe nanofilm instability induced by temperature gradient. The AP model is based on the radiation pressure induced by low frequency acoustic phonons and is based on material properties of the nanofilm. Previous experimental studies have shown that the long wave instability presented in TC model is the primary reason for the formation of nanopillar arrays. 13,14 In the EHD patterning, the thin liquid film is typically assumed to be isothermal during its evolution process. LS analysis shows that, lowering the interfacial tension leads to smaller structures since increased temperature reduces the interfacial tension (in commonly used materials in the EHD) Thus combining the TC effect with the EHD process is predicted to result in the formation of smaller sized features. Corbett and Kumar 43 have theoretically analyzed the simultaneous effect of temperature gradient and the EHD in the patterning of both Newtonian and viscoelastic films. For 1-D films, the LS analysis predicts creation of smaller structures when EHD and thermal forces are combined. They have examined and compared their results for both LD and PD electrostatic models. For both PD and

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LD films a decrease in solvent viscosity and Deborah numbers increases the growth rate and consequently results in smaller λmax . They have also shown a similar rate of patterning using nonlinear analysis for a Newtonian film, which deviates from the linear predictions when the interface height reaches 90% of the gap distance. To the best of our knowledge, the dynamics and mechanism of pattern formation in combined TC-EHD assisted patterning of nanofilms using a 3-D nonlinear formulation have not been analyzed. The main goal of this study is to investigate how to create smaller sized pillar arrays (compact features) by integrating the EHD patterning process with the TC model in a long-wave limit formulation. The effects of Marangoni number and thermal conductivity ratio of layers on the time evolution, the shape and the size of structures are investigated using analytical LS analysis and nonlinear numerical simulation.

Figure 1: Schematic view of the thin film sandwiched between electrodes. Electric field and thermal gradient is applied in transverse direction. The z-position of the free interface is described by the function h = f (x, y, t) with the lateral coordinates x, y and time t. The amplitude of the fluctuation with wavelength is not to scale and is magnified. d=100 nm

Problem formulation A schematic of the thin film sandwiched between the two electrodes is shown in Figure 1. The thin liquid film is uniformly heated from below and cooled from top. Thin film evolution using mass and momentum conservation and an energy balance (to capture the 5


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thermocapillary effect) are considered as:

∇.~ui = 0 ρi (

(1)

∂~ui + (~ui .∇~ui )) = −∇Pi + ∇.[µi (∇~ui + (∇~ui )T )] + f~e ∂t ∂Ti ρi c p i ( + ~ui .∇Ti ) = ki ∇2 T ∂t

(2) (3)

Subscript i denotes the fluid phase. The thin film is considered to be a Newtonian, incompressible fluid with a constant electric permittivity. It is assumed that the film thickness is sufficiently thick over the evolution time that the continuum assumption remains valid. Gravity is neglected as the film thickness, h, is less than 100 nm. It is assumed that energy dissipation due to viscous forces is negligible in the energy balance. The hydrodynamic and thermal boundary conditions on the walls:

~u1 = 0 ; T1 = TH

at z = 0

~u2 = 0 ; T2 = TC

at z = d

At the interface (z = h(x, y, t)), they are:

~urelative = 0

~n.k1 ∇T = ~n.k2 ∇T & T1 = T2 with stress balances (normal and tangential) ~n.¯ σ1 .~n − ~n.¯ σ2 .~n = κγ + f~e .~n ~ti .¯ σ1 .~n − ~ti .¯ σ2 .~n = (∇γ + f~e ).~ti The stress tensor is defined as σ ¯ = −P I¯ + µi (∇~ui + (∇~ui )T ) in which µi is the dynamic 6


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viscosity. The external body force is defined as a gradient of a potential, f~e = −∇φ, and term φ is called the conjoining pressure. The normal and tangent vectors of the interface are ~n and ~ti , respectively and κ is the mean interfacial curvature of the film interface. 44 The surface tension, γ, is approximated to have linear 36 dependence to temperature (γ = γ0 − α(T − T0 ) with α > 0) in which α (N/mK) is surface tension gradient and T0 is the reference temperature. For the electrostatic part, it is assumed that the fluid layers behave as PD and the Laplace equation is used to find the electrostatic force acting on the interface.

ε0 εi ∇ 2 ψ i = 0

(4)

The boundary conditions are:

ψ1 = 0 at z = 0

& ψ2 = ψup at z = d

ψ1 = ψ2 & ~n.(ε1 ∇ψ1 ) = ~n.(ε2 ∇ψ2 ) at z = h(x, y, t) where ε1 and ε2 are the relative electric permittivity of fluid layers and ε0 is free space electric permittivity. Finally, to relate the interface height to interfacial velocity components, a kinematic boundary condition is used as follow: 45

w=

∂h ∂h ∂h +u +v at z = h(x, y, t) ∂t ∂x ∂y

(5)

In the EHD (TC) pattern evolution process, the electric field (temperature gradient) destabilizes the interface of the liquid film and bounding fluid, leading to the evolution of interface with the height of h = f (x, y, t). The wavelengths of the growing instabilities, λ, are much larger than the initial film thickness (at initial stages), so a “long-wave approximation” 38,46 is used to simplify the governing equations. Also, inertial effects are negligible in hydrodynamics of nanofilms as Reynolds number is smaller than one. The polymer film (lower layer) is assumed to be more viscous compared to its bounding layer (µ2 /µ1 ≪ 1). 7


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Hence, the dynamic of upper layer is decoupled from thin film which reduces the computational effort in tracking interface height. Applying these assumptions to the governing equations and the aforementioned boundary conditions result in the following “thin film” equation 46 in a long-wave limit form:

3µht + h3 [γ(hxx + hyy ) − φ]x − h2 (φT )x x + h3 [γ(hxx + hyy ) − φ]y − h2 (φT )y y = 0

(6)

In eq.6, the term 3µht , accounts for the viscous damping force and the term γ(hxx + hyy ) stands for the surface tension force which also damps the fluctuations by minimizing the interface area of the film leading to minimum free energy of system. 47 The long wave approximation reduces the complexity of the governing equations; however, it results in a larger λmax compared to the general model. 48 In addition, the long wave approximation is limited to low aspect ratios (height to width) patterns that form during the early stages of film evolution. The sum of excess intermolecular forces is called conjoining/disjoining pressure (force acting on the film interface per unit area) and is defined as φ = φvdW + φBr + φEL . The conjoining/disjoining pressure is the summation of van der Waals, φvdW pressure, φvdW = AL /6πh3 & AL = effective Hamaker conctant 47,49

(7)

Born repulsion, φBr φBr = −8BL /h9 + 8BU /(d − h)9

(8)

and electrostatic pressure (developed by Maxwell stress), φEL 11,50 1 ∂ψ1 ∂ψ2 φEL = ε0 [ε1 ( |z=h )2 − ε2 ( |z=h )2 ] 2 ∂z ∂z

8


(9)

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The electric potential distribution in each layer (ψ1 (z) and ψ2 (z)) is governed by Laplace equation for the PD films (eq.4). The resulting electrostatic pressure acting on the interface is given by, ε1 ψup 1 φEL = − ε1 ε0 ( − 1)[ 2 ε2 h(1 − εε21 ) +

ε1 ] d ε2

2

(10)

A Born repulsive force is used to avoid nonphysical fluid penetration to solid substrates (upper and lower electrodes). The coefficients: BL and BU in eq.8 are found by setting φ equal to zero at a cutoff distance of l0 from solid surfaces. 51,52 This helps to carry out the simulation in both short and long time thin film evolution. Heating the thin liquid film results in an interfacial tension gradient along the interface. The thermocapillary effect is included in “thin film” equation with the inclusion of an extra term φT . For the case of thermal conduction in the thin film and the bounding layer as a dominant mode of heat transfer. It is found as: 3α∇T φT = 2

kr d (1 − kr )[h(1 − kr ) + kr d]

(11)

Where, kr = k1 /k2 is the relative thermal conductivity of layers, α (N/mK) is the surface tension gradient in the linear approximation 36 and ∇T = TH − TC . The convective mode of heat transfer between film interface and the bounding layer is negligible (Biot number ≪ 0.1) as the film thickness in the EHD pattering process is in micron and submicron ranges. Linear stability analysis Initially, the investigation of nanofilm dynamics uses LS analysis. This analytical technique is used to predict the characteristic wavelength for growth of instabilities. In the EHD patterning process it is shown that the center to center distance of pillars (raised columnar structure) are characterized with λmax resulting from LS analysis. 11 The interface height, h, in eq.6, is replaced with a small sinusoidal perturbation of the interface about its initial 9


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value, h = h0 +ǫ exp (i(κx + κy) + st). In this relation, κ is the wave number, s is the growth coefficient and ǫ is the small amplitude of sinusoidal perturbation. After substituting this relation into eq.6 and eliminating the nonlinear terms, the following dispersion relation is found:

s+

h0 2 κ2 h0 γ 2 h0 ∂φEL ∂φT [ κ + | h0 + |h ] = 0 µ 3 3 ∂h ∂h 0

(12)

The dominant wave number, κmax , corresponding to the fastest growing wave is:

κmax =

"

h0 ∂φEL T | + ∂φ | 3 ∂h h0 ∂h h0 −2h0 γ 3

# 21

(13)

The maximum wavelength for growth of instabilities is defined as λmax = 2π/κmax . In the case of isothermal liquid film the term,

∂φT | , ∂h h0

is zero in the κmax relation. 11,26 The

lateral size of patterning (center to center distance of pillars) is analytically predicted by the maximum wavelength of λmax . Numerical method To numerically simulate the dynamics and morphology of interface, the thin film equation is nondimensionalized using following scaling factors: 1

Ls = (γǫ22 h30 /0.5ǫ0 ǫ1 (ǫ1 − ǫ2 )ψup 2 ) 2

(14a)

Ts = 3µγǫ22 h30 /[0.5ǫ0 ǫ1 (ǫ1 − ǫ2 )ψup 2 ]

(14b)

Φs = 0.5ǫ0 ǫ1 (ǫ1 − ǫ2 )ψup 2 /(ǫ22 h20 )

(14c)

where, Ls is used for spatial coordinates (X = x/Ls , Y = y/Ls ), Ts for time (T = t/Ts ) and ΦS for conjoining pressure (Φ = φ/Φs ). The length and time scaling factors are chosen based on isothermal EHD case and kept constant unless otherwise indicated to allow for 10


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comparison with the the same scaling. The interface height is scaled with the initial mean film thickness (H = h/h0 ). The resulting nondimensionalized thin film equation is:

HT + [H 3 ([HXX + HY Y ] − Φ)X − H 2 (ΦT )X ]X + [H 3 ([HXX + HY Y ] − Φ)Y − H 2 (ΦT )Y ]Y = 0

(15)

with ΦT given by:

ΦT =

¯ kr d M (1 − kr )[h(1 − kr ) + kr d]

¯ = In scaled form of thermal pressure, term M

3α∇T 2h0 Φs

(16)

is called modified Marangoni number

which is the ratio of the interfacial force gradient to electrostatic force. For the heated nanofilms with no applied electric field (ψup = 0), the thermocapillary and van der Waals forces are dominant and following scalings based on the LS analysis (details found in 37,38 ) for the base-case are used: 1

Ls = h0 2 /(2πγ/AL ) 2

(17a)

Ts = AL 2 /(12π 2 µγh0 5 )

(17b)

Φs = 2πh0 3 /AL

(17c)

The nondimensionalized thin film equation (Eq.15) is solved numerically to find the spatiotemporal behavior of heated thin liquid film interface under the application of a transverse electric field. The spatial derivatives are discretized using high order finite difference (FD) and the resulting sets of time differential algebraic equations (DAE) are solved with adaptive time step ODE solver of DDASSL (in SLATEC library). In all simulations, a square spatial domain with periodic boundary conditions is considered. Simulations are initialized with random perturbation of interface conserving volume. The numerical scheme is examined for grid independency by using four different uniform grid sizes (Nx × Ny ) of 61×61, 91×91, 11


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121×121 and 151×151 grid points. Grid size with 91 grid points in each direction was found to be sufficient compromising between CPU time and accurate tracking of the interface. A list of constants and simulation parameters are presented in Table 1. Table 1: Constants and simulation parameters. Parameter viscosity of liquid film (µ) interfacial tension (γ) interfacial tension gradient (α) free space electric permittivity (ε0 ) electric permittivity of the liquid film (ε1 ) electric permittivity of the bounding layer (ε1 ) applied voltage (ψup ) thermal conductivity of film (k1 ) temperature difference (TH − TC ) thermal conductivity ratio (k1 /K2 ) initial film thickness (h0 ) electrodes distance (d) effective Hamaker constant (AL ) cut off distance (l0 )

Value 1 Pa s 0.048 N/m 48 × 10−5 N/mK 8.85×10−12 C/V m 2.5 (−) 1 (−) 0−20 V 0.18 W/mK 0 − 100 K 2 − 10 (−) 26 − 55 nm 100 nm −1.5 ×10−21 J 1−3 nm

Results and discussion A thin liquid film interface destabilizes and deforms when it is subjected to an electric field and/or a temperature gradient. In electrically assisted patterning, a variety of structures form depending on the initial filling ratio (ratio of initial film thickness to electrodes distance), electric permittivity ratio of the layers, the number of layers, etc. 10,11,22,26,30,33,53–64 From LS analysis, increasing the applied voltage results in smaller size patterns, however, experimental studies 22 show that electric breakdown occurs in either the film or the bounding layer when the applied electric field passes the electric breakdown limit of the material. Experimentally, the electric breakdown results in formation of larger size structures and unevenly distributed patterns in experiments which are not captured in the theoretical pre12


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Figure 2: (a)Electrostatic and (b)thermal pressure, left axis, and spinodal parameter, right axis, distributions versus film thickness, effects of (a) applied voltage,ψup (b) modified ¯ and thermal conductivity ratio, kr . Marangoni number, M dictions (λexp ≫ λLS ). Therefore, creating ordered smaller structure (at a submicron level) is a limitation of the EHD patterning technique. Although different solutions have been suggested using multi-layer films with PD and/or LD 10,61 and IC 22–24 properties, minimizing the size of structures in the EHD process is still an open research area. In the EHD process, viscosity and interfacial tension tend to flatten the interface (i.e. damping forces). Using low viscous polymers is found to make the EHD patterning process faster. Lowering the process time from 24 hours to 2 hours in some cases. 25 Lowering the interfacial tension by using thermal gradient results in thermocapillry instability of the film. The dynamics and pattern formation in heated thick (with gravity effects) liquid films 36,38–41 and nanofilms (with negligible gravity effects) 12–14,42 have been extensively studied. Here the focus is on combining thermocapillary effects with EHD to pattern nanofilms. The goal is to find techniques to effectively reduce the structure size. The variation of the electrostatic, φEL (N.m−2 ), and thermal , φT (N.m−1 ), pressures with film thickness is compared in Figure 2 and provides insight into destabilizing factors. Negative values for φ indicate that the interface is pulled toward the upper electrode. In these cases, higher pressure acts on the interface when film is thicker implying that both electrostatic and thermal forces tend to amplify the perturbations. Thicker regions experi-

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Figure 3: Growth rate distribution versus wave number (a) effect of modified Marangoni ¯ = 4.5. number, (b) thermal conductivity ratio. ψup =5 V, (a) kr = 2, (b) M ence higher (upward) forces compared to the thinner regions and so instabilities grow. The impact of applied voltage on φEL , and the effects of modified Marangoni number and thermal conductivity ratio on φT are presented in Figures 2(a) and 2(b), respectively. In Figure 2(a), higher applied voltage (5→20 V) results in higher electrostatic pressure as expected. The increase in Marangoni number increases the thermal pressure over the range of film thickness as shown in Figure 2(b). However, increasing the thermal conductivity ratio of layers, kr , (from 2 to 5) results in lower absolute value of φT (for example, in h= 50 nm, the | φT | decreases from 0.096 to 0.03 N.m−1 ). The maximum wave number κmax from LS analysis is found to be a function of the gradient of electrostatic and thermal pressure with respect to the film thickness (eq. 13) and these gradients are called spinodal parameters. 5 In Figure 2, the spinodal parameters are also plotted (dotted lines and blue) versus the film thickness. The negative values of electrostatic and thermal gradients demonstrate their destabilizing effect and the negative flow diffusion. 5 Higher values of

∂φEL ∂h

and

∂φT ∂h

result in higher values of

¯ at a constant kr = 2 κmax which means smaller structure sizes will be formed. Increasing M increases the absolute value of

∂φT ∂h

¯ = 4.5, a and φT . However, for the case of constant M

decrease in kr does not necessarily lead to an increase in thermal force but depends on the 14


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¯ = 4.5 and film thickness. In other words, thicker regions of film (h> 75nm) in case of M T ¯ = 4.5 and kr = 5 experience much higher thermal forces ( ∂φ ) compared to the case of M ∂h

kr = 2, whereas the behaviour is opposite for their thinner regions (h< 75nm). In Figure 3, the LS analysis results for the growth rate of instabilities s, is plotted as a function of wave number, κ. The isothermal film (pure EHD) is compared to the heated film (combined EHD and thermocapillary effects) in Figure 3(a). The peaks in growth rate variation indicate the maximum growth rates (smax ) and their corresponding maximum wave number (κmax ) in growth of instabilities. The LS analysis confirms that the addition of thermal gradient to the system results in application of higher force to the film interface and ¯ = 4.5 the increase in κmax is more than a smaller structure (higher κmax ). For the case of M 30%. The effect of thermal conductivity ratio of layers, kr , on κmax is also investigated using LS analysis and the results are shown in Figure 3(b). Maximum wave number occurs for the lower ratio kr = 2 which agrees with the thermal pressure results presented earlier. Similar trends, decrease in characteristic wave length of instabilities with increase in Marangoni number and decrease in the thermal conductivity ratio, for the combined EHD and TC instabilities in the 1D film are observed. 43 Electrohydrodynamic (EHD) and Thermocapillary (TC) instabilities (base-cases) The dynamics and 3-D snapshots of thin liquid film morphology in both isothermal and non-isothermal films have been investigated and the results are presented in Figure 4 as a base-case. The interface pattern is presented in the quasi-steady stage in which the formed pillars noticeably do not merge for a long period of time. The interface maximum and minimum heights are tracked over time to demonstrate: (i) the thin liquid film drainage time, (ii) the time it takes the film interface to touch the upper electrode (growth rate of instabilities) and (iii) the rate of formation of pillars on the interface. The significant difference in non-dimensional times in Figures 4a(i) and 4b(i) is due to the different scalings (see eq.14a-14c and 17a-17c) used to nondimensionalize thin film equation for these two

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Figure 4: Base-case EHD and thermocapillary instabilities, Interface height change versus time and 3-D snapshot of the film interface for a(ii) EHD and b(ii) thermocapillary basecases at their quasi-steady stage. In EHD base-case: ψup =5 V and λ = 5.34 µm. In ¯ = 4.05 , AL = −1.51 × thermocapillary base-case (using scalings 17a- 17c): ψup =0 V, M −17 10 J and λ = 0.6 µm. cases. The negative diffusion (negative spinodal parameter) of fluid flow from thinner to thicker region of film results in raised columnar structures (pillars) on the interface (Figure 4 a,b (ii)). When pillars touch the upper electrode their height is limited to the electrodes distance. The results show that growth rate of the interface is higher in the EHD case compared to the thermocapillary case. For both cases, over time more fluid flows to the core region and this entrainment enlarges the pillar and deepen the surrounding ring. Formed pillars in the EHD base-case have a more cylindrical shape whereas in the TC (Marangoni) case pillars have a more conical shape. In both cases, four pillars with circular cross section form in the 2λ × 2λ computation spatial domain.

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Figure 5: Interface height change versus time (a) and spatiotemporal evolution of interface b(i-iv) in combined EHD-Marangoni instabilities. Applied voltage, ψup =5 V, initial film thickness, h0 = 26 nm, relative thermal conductivity, kr = 2 and modified Marangoni num¯ = 0.22. Nondimensional times, T = (i)1.2×104 , (ii)1.5×104 , (iii)1.7×104 (iv)2.0×104 ber, M and (v)7.0×104 . λ = 5.34 µm. Combined EHD-Thermocapillary (EHD-TC) instabilities: In order to investigate the addition of the thermocapillary effects on the morphological evolution of EHD nanofilms, the modified thin film equation in a long-wave limit (eq.15) is solved numerically. Modified Marangoni and thermal conductivity ratio are found as effective parameters from pressure distribution and LS analysis results (Figures 2 and 3). The effects of Marangoni number and thermal conductivity ratio on the dynamics and film evolution process are investigated next. Applied voltage is chosen as low as 5 V, to both avoid the electric breakdown in nanofilm and the bounding layer 22 and to also enable capturing the thermocapillary contribution in the patterning formation. The main spatiotemporal evolution milestones of thin liquid film under combined effect of electrostatic and Marangoni forces are summarized in Figure 5. The thin film undergoes

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Figure 6: Interface height change versus time (a) and spatiotemporal evolution of interface b(i-iv) in combined EHD-Marangoni instabilities. Applied voltage, ψup =5 V, initial film thickness, h0 = 26 nm, relative thermal conductivity, kr = 2 and modified Marangoni ¯ = 2.26. Nondimensional times, T = (i)171, (ii)200 (iii)240 and (iv)494×104 . number, M Domain size, (2λ × 2λ) = 10.7 µm2 . the following five major stages. Initial disturbances are reorganized and small amplitude bicontinuous structures are formed (stage i). Fragmentation occurs at long ridges and coneshape pillars are formed (stages i and ii). The size of pillars is increased and isolated pillars with a hexagonal pattern are created (stages iii and iv). At quasi-steady state of patterning process, pillars have circular cross-section. Lastly, the coarsening begins which results in uneven distribution and larger pillars over the domain. The coarsening process and coalescence of neighbor pillars will be discussed in detail later. ¯ = 0.22 (Figure 5) to M ¯ = 2.26 When the modified Marangoni number is increased from M (Figure 6), while holding all electrostatic parameters constant, it is found that the interface evolution time decreases considerably due to increased Marangoni effect. The lower evolution time and consequently faster fabrication process has also been experimentally reported when low viscous polymer films is used in the EHD pattering process. 25 Although the mechanism 18


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of pattern formation remains the same (Figure 6 b(i-iv)), pillars with a smaller size are formed leading to more compact structure fabrication when comparing Figure 5 to Figure 6. This decreasing trend in structure size qualitatively matches with those previously found for the 1D films. 43 To investigate the effect of the thermal conductivity ratio of the film and the bounding layer on the interface dynamics, the spatiotemporal evolution of interface varying kr = 2 to 10 is presented in Figure 7. As mentioned earlier, decreasing the relative thermal conductivity of layers (lower kr ) increases the the thermal pressure at the interface. In addition, based on LS analysis, as kr decreases the maximum wave number of the induced instabilities increases. This behavior is confirmed by the spatiotemporal evolution of interface as shown in Figure 7. According to this Figure, initial instabilities first grow and form a bicontinuous structure (stage i). Then fragmentation begins and spike shape pillars randomly grow on the interface and touch the upper electrode (stage ii). At this stage, pillars formed early get bigger and merge to neighbor pillars before complete pillar formation over the domain occurs (see Figure 7 b(ii) and image M-ii). This early coarsening process avoids the formation of hexagonal pattern structure (see Figure 7 b(ii, iii)). The final structure in Figure 7 b(iv) consists of unstable non-uniform sized pillars which are randomly distributed over the domain. This behavior might be attributed to a significant increase in the thermal force for thicker regions of film when kr is high which leads to higher and faster flow diffusion to the thicker regions (early formed pillars) whereas the flow diffusion is slower in other thinner regions (not completely formed pillars). The exponential increase in the thermal pressure and force is shown and compared for kr = 2 and 5 in Figure 2 (b). When Figure 7 is examined closely, two coarsening mechanisms are found: (a) collision of neighbor pillars with the same size which mainly occurs at the early stage of coarsening (compare region A in images M-iii and M-iv), and (b) Ostwald ripening which is merging of smaller pillars into larger size ones occurring predominantly in later stages of coarsening (compare region B in images M-iii and M-iv).

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Figure 7: Interface height change versus time (a) and spatiotemporal evolution of interface b(i-iv) in combined EHD-Marangoni instabilities. Applied voltage, ψup =5 V, initial film thickness, h0 = 26 nm, relative thermal conductivity, kr = 10, and modified Marangoni ¯ = 4.5. Nondimensional times, T = (i)560, (ii)649 (iii)693 and (iv)793. Domain number, M size, (2λ × 2λ) = 10.7 µm2 . Improving the EHD patterning process in nanofilms by creating smaller sized pillars or more compact features by adding thermocapillary instability is the main goal of this study. So far it is shown that the modified Marangoni number and the thermal conductivity ratio affect both pattern formation time period and shape and size of the final structures. To quantify the compactness of the formed patterns, the number density of pillars formed in 1 µm2 area of thin film is needed. Additionally the improvement in active surface area of the film is needed as this is of interest in data storage applications and surface analysis. The effect of modified Marangoni number on density of pillars and active surface area is now

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examined.

Figure 8: 3-D snapshots of the interface at quasi-steady state (i-iv). Variation of (a) relative surface area and number density of pillars and (b) characteristic wavelength with modified Marangoni number. (i) EHD base-case, applied voltage, ψup =5 V. Modified Marangoni ¯ = (ii)0.22, (iii)2.26 , (iv)4.5. Initial film thickness, h0 = 26 nm, relative thermal number, M conductivity, kr = 2 and λ = 5.34 µm. The thin film equation (eq.15) is numerically solved for different modified Marangoni numbers and 3-D snapshot of the pattern structure at the quasi-steady states are presented in Figure8 (i-iv). The base-case EHD patterning (Figure8 (i)) results show that in (4λ × 4λ) domain, 16 pillars formed which are hexagonally ordered over the domain. Adding ¯ increased from 0.22 to 4.5, in Figure 8 (ii)-(iv)) results in an thermocapillary effects (M ¯ = 4.5 a dense structure of increase in total number of pillars forming on the interface. At M cone shape pillars packed in smaller domain of (2λ × 2λ) in Figure 8 (iv) is observed. The number density of pillars is defined as total number of completely formed pillars in 1 µm2 area. Increasing the Marangoni number from 0 to 4.5 increases the number density of pillars significantly from 0.03 to 3.2 − a factor of 100. In addition, the center to center distance 21


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length decreases from ∼5 µm to ∼0.3 µm. The thermocapillary effect also decreased the cross section area of pillars, as can be observed in Figure 8 (i-iv). The surface area of the interface is also calculated and its relative value with respect to the flat film interface at initial stage are plotted. Results show that the surface area is increased from 0.31% to 18.1% by adding the thermocapillary effects compared with traditional (isothermal film) EHD patterning.

Conclusions Corbett and Kumar 43 showed, for the first time, the characteristic length scale in growth of instabilities decreases in the thermal-EHD patterning using both the LS and nonlinear analysis in 1D films. To extend that, the dynamics, morphological evolution and drainage time of liquid nanofilm (thickness< 100 nm) subjected to a combined thermal-electrical field are studied considering a 2D lubrication-theory-based model. The thin film equation that includes thermal effects is re-derived in a long-wave limit and solved for both short time, linear stages, and long time, nonlinear stages. The LS analysis predicts a significant effect of the modified Marangoni number (relative strength of interfacial tension gradient force to the electrostatic force) and the thermal conductivity ratio of layers on the maximum wavelength on the growth of instabilities which, qualitatively matches the 1D film results. 43 The thin film equation is then numerically solved to numerically explore the nonlinear interface dynamics stages. Thermally and electrically induced base-case patterns are used to validate the numerical scheme. Then effects of Marangoni number and thermal conductivity ratio of layers on the shape and size of structures are investigated. Increasing the relative strength of thermocapillary force or, induced by the Marangoni effect, results in much smaller sized pillars while the mechanism of pattern formation remains unchanged. However, the thermal conductivity ratio is found to have an inverse effect on the pattern formation process. A coarsening mechanism is found to initiate at the early stages of pattern formation for high thermal conductivity ratios that prevents formation of uniform and hexagonal ordered pillars.

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This is attributed to a drastic increase in the thermocapillary force applied to thicker region of films at higher thermal conductivity ratios. In general, the smaller structures are formed when the thermocapillary effects are combined with the EHD patterning mechanism. The number density of pillars formed in 1 µm2 of domain area is increased from 0.03 to 3.2 (over one hundredfold) resulting in nano-sized pillar formation. The relative surface area of the patterned film (compared to initial flat stage) is increased up to 18% for combined TC-EHD, whereas this increase is about 1% in the EHD base-case which is of interest in the data storage applications in this nano-sized structures. The relative surface area (compared to initial flat surface area) of patterned film is increased by about 1% in the EHD base-case whereas this increase is up to 18% for combined TC-EHD which is of interest in the data storage applications in this nano-sized structures.

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