Dielectrophoretic deformation of thin liquid films induced by surface charge patterns on dielectric substrates

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I Introduction
Local build-up of electrostatic charges can potentially cause hazards, especially in high-speed processing of dielectric materials. Examples are dust attraction on solar cells 1 or explosions due to charge build-up in the oil industry. 2 In roll-toroll liquid coating processes, non-uniform static charge densities at the substrate may cause deformations and defects in the nal coating, whereas uniform charges are sometimes utilized to improve the coatability of partially wetting substrates. [3][4][5] It is well known that liquid lms can deform due to an inhomogeneous electric eld. 6 Hateld 7 recognized the application potential of this mechanism for separation processes, which since then has found numerous applications in microbiology and colloid science and technology. 8,9 Chou et al. studied lithographically induced self-assembly of periodic polymer micropillar arrays [10][11][12][13] and achieved feature sizes on the order of 50 nm. 14 They hypothesized that the lm deformations were governed by an electrohydrodynamic mechanism. Schäffer et al. studied the pattern formation and replication in ultrathin polymer lms and bilayers induced by an electric eld applied perpendicular to the polymer lms. [15][16][17] Subsequently, many groups conducted experiments or numerical simulations of instabilities of single or multiple liquid layers sandwiched between two electrodes kept at constant potentials.  Dielectric, leaky dielectric [18][19][20] and conductive materials [45][46][47][48] were considered. Brown et al. investigated the electrostatic deformation of decanol lms on interdigitated electrodes for potential use as voltage-programmable phase gratings. 49,50 In contrast to voltage-controlled dielectrophoresis, much less work has been published on thin lm deformation controlled by static surface charges. Miccio et al. developed a technique for thickness modulation of silicone oil lms utilizing laser light induced charge patterns in photorefractive lithium niobate crystals. 51 Recently, Zhao et al. used a stamping technique to create charge patterns that modulate the dewetting patterns observed aer thermal annealing of ultrathin polymer layers. 52 Earlier, Chudleigh 53 and Engelbrecht 54 created charge patterns on polymer surfaces by applying a voltage across an electrically insulating substrate through a conductive liquid for electret charging. 55 Lyuksyutov et al. 56,57 and Xie et al. 58,59 created nanoscale surface structures in a polymer lm by atomic force microscopy (AFM) with conductive tips. A salient feature of this mechanism is the electric eld induced condensation of a water meniscus between the AFM tip and the polymer surface. 60 In this study, we conducted a comprehensive and quantitative investigation of the deformation of low viscosity perfectly dielectric liquid lms deposited on electrically insulating substrates with static surface charge patterns. We generated the charge patterns by moving an ultrapure water droplet attached to a hollow metallic needle maintained at high voltage along a dielectric substrate 53,54 and characterized the obtained surface charge patterns using an electrostatic voltmeter. 61 Subsequently, we deposited a thin dielectric liquid lm by spincoating and studied its deformation dynamics by interference microscopy. The results were compared to numerical simulations based on the lubrication approximation for a perfect dielectric liquid. Moreover, we performed a scaling analysis of the experimental and numerical results and derived a novel selfsimilar solution describing the dynamics in the case of narrow charge distributions.

A Surface charge deposition
We deposited electrostatic surface charge patterns on dielectric substrates by applying a voltage to a droplet of de-ionized water that was moved in a pre-dened trajectory over the dielectric substrate. Before applying the charge pattern, we de-charged the substrate using an antistatic bar (Simco-Ion MEB), which le residual charge densities of less than 2 mC m À2 on the surface. The substrates were optical quality polycarbonate plates (Bayer, Makrofol DE1-1) of thickness d sub ¼ 750 mm. Fig. 1(a) shows a schematic of the setup. A de-ionized water droplet (Smart2Pure, TKA, resistivity 18.2 MU cm) was supplied from a disposable syringe (BD Plastipak 3 ml) through a polyethylene tube (Braun, Original-Perfusor Line) to a hollow metal needle (brass or stainless steel) that was connected to a high voltage DC power supply (Fug Elektronik, HCN-12500). The substrate was moved with respect to the droplet by a motorized xy-translation stage (Standa, 8MTF) that was equipped with a grounded aluminium plate for supporting the substrates and positioned underneath the stationary needle. The substrate speed U sub ¼ 1-10 mm s À1 was low enough to avoid water loss by viscous entrainment. This process le a surface charge density along the trajectory of the droplet with a magnitude in direct correlation with the voltage V needle supplied to the hollow needle. The width of the charged lines W was close to the droplet size that was mainly dened by the outer diameter of the needle, D needle , to which the droplet was attached. The distance between the needle and the substrate was maintained at approximately 0.5 mm. We measured no change in charge patterns within hours aer deposition. Synchronized control of the translation stages and high voltage power supply as well as acquisition of the signal from the electrostatic voltmeter described in Section II B was achieved by the Labview soware (National Instruments).

B Charge pattern characterization
The distribution of charges on the substrate was measured by placing the substrate on the grounded sample holder on the xystage described above and scanning the substrate with an electrostatic voltmeter (Monroe Electronics Isoprobe 244) equipped with a high resolution probe (Isoprobe 1017AEH) at a probesample distance of approximately 0.5 mm. The lateral resolution of this method is estimated 62 to be approximately 0.5-1 mm, depending on the probe-surface distance (see Appendix A.1).
The net surface charge density s was calculated from the bias voltage V bias according to 61 where 3 sub is the relative dielectric permittivity of the substrate (3 sub ¼ 2.9 for polycarbonate) and d sub is the substrate thickness.

C Thin liquid lm preparation and imaging
Aer the creation of a static charge pattern and its quantication, a thin liquid lm of squalane (purity 99%, Aldrich, product number 234311) was spin-coated onto the substrate with a resulting lm thickness of h 0 ¼ 4.1 mm. Squalane was selected due to its low volatility and the very low mobility of charge carriers in alkanes. [63][64][65][66] The viscosity at 23 C is m ¼ 31.9 mPa s, the surface tension 67 g ¼ 31.0 mN m À1 , the density 68 r ¼ 805 kg m À3 and the refractive index is n ¼ 1.452. Within 30 s aer spincoating, the sample was removed from the spin-coater and placed in the microscope (Olympus BX51) equipped with a microscope objective (Olympus, MplanAPO 2.5Â/0.04) and a CCD camera (Pike, Allied Vision Technologies) and interferometric imaging was started. During imaging, the sample was supported by a grounded and anodized aluminium plate, as sketched in Fig. 2. At a distance H tot À d sub ¼ 0.75 mm above the sample a glass window was placed. The glass plate was coated with an electrically conductive and optically transparent indium tin oxide (ITO) layer maintained at grounded potential. This ensured that the electric potential in the system was well dened. We used an illumination setup based on a light emitting diode (Thorlabs, product number LEDC45) with a center wavelength of l ¼ 660 nm and a spectral width of approximately Dl ¼ 30 nm. By analyzing the interference fringe pattern, a cross-section of the height prole h(x) was obtained. Neighboring light and dark fringes represent height differences of l/4n ¼ 114 nm.

III Numerical model
In a long-wave approximation, the height evolution h(x,t) of a nonvolatile liquid lm on a solid substrate is given by the lubrication approximation 69 where m is the viscosity and the augmented pressure consists of capillary pressure, hydrostatic pressure and an electrical contribution P el . We solved eqn (2) and (3) on a onedimensional domain with boundary conditions at x ¼ 0 and the outer domain boundary. We consider one-dimensional surface charge distributions s(x) located at the solid-liquid interface. We assume the material system to be non-conductive and without bulk charge density r c ¼ 0 and the relative dielectric permittivities of the liquid 3 liq and the solid 3 sub are constant and uniform. Consistent with the lubrication approximation for liquid lm dynamics, the two-dimensional Poisson equation for electrostatics Therefore, the electrical potential j is piecewise linear, such We dene the location of z ¼ 0 at the solid-liquid interface. The vertical positions where the potential vanishes (j ¼ 0) are therefore at the bottom plate z ¼ Àd sub and at the conductive glass plate z ¼ H tot À d sub ¼ d air + h(x,t). By using the material parameters and geometrical dimensions of the experimental setup introduced in Fig. 2 and the following electrical boundary conditions and v j air vz The additional pressure P el in the liquid lm induced by the electrostatic eld is determined by the Maxwell stress tensor at the liquid-air interface 26,28,70 We solved eqn (2), (3) and (13)-(15) using the nite element soware COMSOL 3.5a. The width of the computational domain was 6-20 mm and the typical mesh size was 10 mm.

A Writing of surface charges
We deposited surface charge distributions in the form of straight line segments using needles with a diameter D needle of 1.8 and 6 mm. The droplet followed a meandering trajectory over the substrate, during which the needle voltage V needle was switched on and off. A typical two-dimensional measurement of the surface charge distribution consisting of several line segments of approximately 1 cm in length is shown in Fig. 3(a). The white arrows indicate the trajectory followed during surface charge deposition, the needle voltage ranged from 313 to 2500 V. In Fig. 3(b)-(i), several cross-sections of the charge distribution have been plotted, which were obtained using needles of different diameters and different voltages and averaged over a yinterval of 7 mm. The dashed lines in Fig. 3(k) illustrate examples of such cross-sections. The resulting maximum values in this measurement series ranged from approximately 5 to 82 mC m À2 . For comparison, a charge density of 50 mC m À2 corresponds to about 300 elementary charges per mm 2 , which indicates that our continuum lubrication description is appropriate.
The inuence of needle voltage on the center and peak heights of the surface charge density is plotted in Fig. 3( j). The charge density near the edges of the line, s(AED needle /2), is typically higher than at the center, s(0). The measured shape of the surface charge distribution is broadened due to the limited lateral resolution of the electrostatic voltmeter of 0.5-1 mm. Variations in the peak-to-center contrast s(AED needle /2)/s(0) are likely due to slight variation in probe-to-surface distance (see Appendix). The measured peak values monotonically increase with V needle . The dashed lines in Fig. 3(g) represent two independent measurements at U sub ¼ 2.5 mm s À1 and illustrate the repeatability as indicated by the error bar. The dotted lines correspond to experiments performed with different writing speeds in the range of U sub ¼ 1-10 mm s À1 . Within repeatability, the deposited charge density is independent of U sub . Fig. 3(k) shows a contour plot of a line segment written with D needle ¼ 6 mm at a voltage V needle ¼ À2188 V and substrate speed U sub ¼ 2.5 mm s À1 . The white arrows indicate the droplet trajectory. A pronounced asymmetry is observed with respect to the writing direction, which indicates that charge deposition primarily occurs at the receding contact line of the droplet, consistent with the ndings of Engelbrecht. 54 There are many possible mechanisms for charge transfer at a liquid-solid interface. Primarily in so polymeric substrates like poly(dimethylsiloxane) or polyethylene, 71-73 the application of a high voltage leads to the formation of 'water-trees', i.e. the invasion of the liquid into the polymer matrix and a permanent modication of the bulk composition and morphology. Guzenkov and Klimovich 74 studied triboelectric charge generation at liquid-solid interfaces, suggesting that friction plays a role. Nakayama 75 studied triboplasma generation between a diamond tip and an oil-lubricated sapphire disk. In the context of electrowetting, Vallet et al. 76 observed a gas discharge in the vicinity of the contact line of electrolyte droplets. Other groups 77-80 observed contact angle saturation and attributed it to charges trapped in the surface layer of the polymer. Drygiannakis et al. developed a phenomenological model based on the local breakdown of the insulating layer around the contact line. 81 Koopal 82 recently reviewed charging effects at solidliquid interfaces. Using molecular dynamics simulations Liu et al. found that contact angle saturation occurs when the peak electric force near the edge of the drop exceeds the molecular binding force. 83 Knorr et al. 60 performed experiments of charge injection from sharp conducting needles into amorphous polymer layers. They interpreted their results as injection and surface-near accumulation of aqueous ions stemming from eld-induced condensation of a water meniscus between the AFM tip and the polymer surface. Such an accumulation of trapped space charge in polymer dielectrics had been observed earlier by Wintle 84 and Hibma and Zeller. 85 Chudleigh considered the deposition of ions from the electric double layer at the receding contact line. 53 The higher density of surface charges deposited in our experiments at x ¼ AEW/2 is related to the fact that the electric eld strength has a maximum around the contact line of the droplet, 76,86 which leaves higher electrostatic charge in that region, possibly enhanced by eld-induced ion dissociation. 87 Hibma and Zeller 85 observed injection current transients with a time constant far below 1 ms, which may explain why we do not see a signicant dependence on droplet speed for U sub # 10 mm s À1 .

B Dielectrophoretic deformation of thin liquid lms
Aer deposition and characterisation of the surface charge density, we applied a thin liquid lm of squalane on the sample by spin-coating. We monitored the evolution of the height prole of the liquid lm by interference microscopy. An example of an interference picture is shown in Fig. 4(a). The time evolution of the cross-section through the fringe pattern indicated by the dashed black line in Fig. 4(a) is visualized in Fig. 4(b). Local maxima develop at x ¼ AEW/2, i.e. the edges of the charged line and lm thickness depressions (minima) form at x z AEW, which deepen and widen in time. Fig. 4(c) shows height proles determined from the microscope images through fringe analysis. In this example, the center line lm thickness h center ¼ h(x ¼ 0,t) becomes smaller during the rst 200 s and thereaer increases again.

C Inuence of surface charge prole s(x)
The actual shape of the charge density s(x) determines the evolution of the liquid lm prole h(x,t). Because of the limited lateral resolution of the measurement probe, we investigate the inuence of the charge distribution on the lm deformation numerically. Fig. 5(c) and (d) show a quantitative comparison of height proles extracted from the experimental results of Fig. 4(b) at t ¼ 300 s with numerical simulations corresponding to two different surface charge proles s(x), plotted in Fig. 5(a) and (b). The symbols in Fig. 5(a) and (b) correspond to the raw data of the surface charge measurements. The lines represent analytical functions s 1 (x) and s 2 (x), used as input for the numerical simulations. Detailed information on s 1 and s 2 is given in Appendix A.2. The charge distribution s 1 (x) is chosen to faithfully represent the charge measurement data, whereas s 2 (x) exhibits a pronounced maximum at the edge of the line x ¼ AEW/ 2, corresponding to the observations described in Section IV A. The height and width of this maximum can be variedthough not independentlywithout changing the resulting deformation of the liquid lm signicantly. Fig. 5(b), (d) and (f) contain multiple solid lines, according to the sets of values specied in Appendix A.2. In principle one could identify the realistic combination by monitoring for instance Dh max at very early times (10-100 ms), which however is outside the range of accessible times for our experiments. Consequently, given our experimental restrictions and reproducibility we cannot distinguish between these parameter settings. The symbols in Fig. 5(c) and (d) correspond to the experimental data, whereas the solid lines represent the numerically determined lm height proles at t ¼ 300 s. Fig. 5(e) and (f) present the time evolution of the maximum, minimum and center line lm thickness, dened in Fig. 4(c). The symbols represent experimental data and the lines represent numerical simulations. Fig. 5(g) shows the numerical simulations of Fig. 5(b), (d) and (f), visualized as an interference pattern using the procedure described in Appendix A.3 for direct comparison with Fig. 4(b). The surface charge distribution s 1 results in a simulated height prole that is in poor agreement with the measured prole in Fig. 5(c). The amplitude of deformation is much lower than experimentally observed; the center height is too high, the peak amplitude is too low and the minimum is not deep enough. The agreement remains comparably poor for different times, as shown in Fig. 5(e). The experimental center lm thickness h center shows an initial decrease, whereas the simulated value increases monotonically. In contrast, the surface charge distribution s 2 (x) results in excellent agreement between the experimental and numerical lm height proles, not only at t ¼ 300 s [ Fig. 5(d)] but also over the entire time interval considered in Fig. 5(f). The simulated center lm thickness now reproduces the experimentally observed local minimum at t z 200 s. The occurrence of this minimum, i.e. the initial decrease of the center line thickness, is primarily due to the positive charge density gradient towards the peaks at x ¼ AEW/2. This induces a ow away from the center until a decrease in capillary pressure reverses the trend.
The simulations reproduce the trend in the experimental data very well, but the exact numerical values of the deformation are not captured accurately. According to eqn (13)- (15), P el scales as s 2 , which amplies the experimental uncertainty associated with measurements of s(x) as discussed in the context of Fig. 3(g). Moreover, the charge distribution may not only change in magnitude proportional to needle voltage, but also somewhat in shape. The depths of the center h center and outer minima h min generally agree better with the simulations than the maximum values. This indicates that the uncertainty in the charge distributions s(x) is predominant in the vicinity of the peak values in s(x). Within reproducibility, the experimental data in Fig. 6(d) do not show any dependence on charge density s. The numerical curve for s ¼ 30 mC m À2 apparently saturates at t z 1000 s, roughly coinciding with the time at which ÀDh min approaches its maximum possible value, h 0 , i.e. h min / 0 in Fig. 6(c).

D Inuence of initial lm thickness h 0 and line width W
The comparison between experiments and simulations, presented in the previous subsection, constitutes a general validation of the numerical model. In the following sections, we study the inuence of different geometric parameters on the thin lm evolution by numerical simulations. Fig. 7(a)-(c) show the inuence of the initial lm thickness h 0 on the normalized change in the lm thickness of the maximum at x ¼ AEW/2, the minimum at x z W and the center at x ¼ 0. At early times t < 1 s, the rate of deformation increases with lm thickness. From t T 50 s the curves for different values of h 0 in Fig. 7(a) and (c) essentially collapse. For early times, the development of the maximum and minimum height follows a linear time dependence, as indicated by the dashed black lines. The center line height Dh center (t)/h 0 exhibits a local minimum in time, which occurs earlier but becomes shallower for higher lm thickness h 0 .
The inuence of the line width W is illustrated in Fig. 8. Fig. 8(a) and (b) show the time evolution of the height prole for two values of W ¼ 0.4 and 4 mm. In Fig. 8(c)-(e), the inuence of W on the normalized change in the lm thickness of the maximum at x ¼ AEW/2, the minimum outside the charged line and the center at x ¼ 0 is shown. For small values of W, the two maxima at x ¼ AEW/2 observed at early times merge into a single peak due to the negative capillary pressure caused by the close proximity of the peaks. For wider lines, the two peaks remain for the entire simulated time interval. The center line height Dh center (t)/h 0 exhibits a local minimum in time, which occurs later and becomes deeper for larger W. The times at which the local minima occur in Fig. 7(b) and 8(d) correspond to power laws t min $ h 0 À3 and $W 3.8 , respectively, as indicated by the arrows in Fig. 7(b) and 8(d).

E Scaling analysis and self-similar solutions
In the early phase, the height prole h ¼ h 0 + Dh, where Dh ( h 0 , deviates very little from the prescribed initial condition h ¼ h 0 , which allows linearization of the lubrication equation. Moreover, the electrical pressure P el $ s 2 (x) is almost independent of Dh and the capillary and hydrostatic contributions are negligible. Consequently, an approximate solution to the lubrication equation is given by Dh $ h 0 3 f 00 (x)t, which is represented by the dashed lines in Fig. 7(a). Here, f (x) h s 2 (x). Motivated by the presence of sharp peaks of the charge density at the line edges, we now consider an innitely narrow electrical pressure distribution where the function F(h) is determined by eqn (13)-(15) and we introduce the following non-dimensional quantities For a uniform initial condition h(t ¼ 0) ¼ h 0 or equivalently h ( t ¼ 0) ¼ 1, the hydrostatic pressure contribution is negligible initially, until the resulting lm thickness perturbation spreads much further than the capillary length ' c h ffiffiffiffiffiffiffiffiffiffiffiffiffiffi g=ðrgÞ p . Consequently, in this regime the non-dimensionalization of eqn (2) and (3) where we dened P el h W 2 P el /(gh 0 ). We consider small perturbations We seek a self-similar solution for the function 4 h h 1 / t b in the reduced coordinate hh x/ t a , which results in where we made use of the scaling property of the delta function Consequently, a self-similar solution exists for the exponents a ¼ b ¼ 1/4 with the corresponding ordinary differential equation Fig. 9 shows solutions of eqn (2) and (3) represented in terms of the non-dimensional lm deformation 4(h) for nine points in time and a delta-like charge distribution s 3 (see Appendix). The nine curves, corresponding to eight decades in time, essentially collapse as expected from the self-similar solution. The shape of the curves qualitatively resembles the experimental and numerical solution depicted in Fig. 5(d).
In view of this self-similar solution, one would expect power law relationships t min $ h 0 À3 W 4 in Fig. 7(b) and 8(d), which is in excellent agreement with the observed scaling behavior. These scalings coincide with those of capillary redistribution dynamics, 88,89 because there is only a relatively small variation in s between the peak positions |x| < W/2 and away from the charged line |x| > W/2.
The dashed line in Fig. 6(d) corresponds to a power law with exponent 0.22, which is close to the value of 1/4, that is also expected for capillary redistribution in thin liquid lms 90 in the regime Dh/h 0 ( 1. Since the non-dimensional deformation Dh/h 0 for the data corresponding to s ¼ 30 mC m À2 approaches one, the agreement is better for smaller values of s. The dashed lines in Fig. 6(a) and (c) correspond to power laws Dh max $ t 0.3 . The reasons for the slight deviation from the exponent 0.25 expected from the self-similar solution are that the condition Dh ( h 0 is not perfectly fullled in all cases, that the inuence of the transition from the linear regime still lingers on and that the two interior minima in the region |x| < W/2 begin to merge at t z 10 s. For our case of a peak in the charge distribution that has a smallbut nitewidth w peak , the transition between the linear regime Dh $ t and the self-similar regime Dh $ t 1 = 4 can be estimated to occur when the width of the self-similar solution exceeds w peak . According to the denition of the reduced coordinate h ¼ x/ t a , we expect the transition time to scale as t trans $ 3mw peak 4 /(gh 0 3 ), i.e. to be independent of W, which is consistent with Fig. 7(c) and 8(e).

V Summary
We investigated the impact of static surface charge distributions on the deformation of thin lms of a dielectric liquid on a solid substrate by means of experiments and numerical simulations. We deposited patterns of surface charges by dragging a droplet over the substrate in a pre-dened trajectory, while attached to a needle, maintained at high voltage. We measured the surface charge distribution using an electrostatic voltmeter and characterized the charge density prole as a function of the needle voltage and diameter. Subsequently, we spin-coated a thin lm of a dielectric liquid onto the charged substrate and measured the time evolution of the lm thickness distribution by interference microscopy. We obtained good agreement between experiments and numerical simulations based on the lubrication approximation and a dielectrophoretic pressure. We systematically investigated the inuence of the charged line width, initial lm thickness and surface charge magnitude and elucidated the observed scaling behavior by means of a selfsimilar solution.
Appendix A 1 Inuence of probe-to-surface distance Fig. 10 shows several measurements of the same surface charge density prole s(x), scanned at different distances H between the charged substrate and electrostatic voltmeter probe. The charge was deposited using a needle of diameter D needle ¼ 6 mm at a voltage D needle ¼ À1250 V. The peaks at the edges of the charge distribution, x ¼ AE W/2, are more pronounced in measurements with smaller probe-to-surface distance.

Expressions for charge distributions s(x)
The analytical expression for the charge distribution s 1 (x) presented in Fig. 5 with s 0 ¼ 17 mC m À2 , w slope ¼ 0.2 mm and s slope ¼ 8 mC m À2 and W is a variable. For the simulations presented in the paper, the values w peak ¼ 50 mm and s peak ¼ 37 mC m À2 were used. Table 1 shows examples of alternative combinations yielding indistinguishable results.