Combined thermal and electrohydrodynamic patterning of thin liquid films

Abstract

Both electric fields and temperature gradients can destabilize the surface of a thin liquid film and lead to the self-assembly of patterns composed of pillar-like structures. Such instabilities offer a relatively simple way to tailor the surface topography of coatings, which in turn can be used to influence coating appearance, texture, and functionality. The present work explores how the simultaneous application of an electric field and temperature gradient can be used to further influence thin-liquid-film instabilities. Both perfect and leaky dielectric materials are considered, and lubrication theory is applied to develop a system of nonlinear partial differential equations for the interfacial height and charge. Linear stability analysis of the lubrication equations shows that in perfect dielectric films, thermal effects tend to dominate the process, often rendering the electric field unimportant. However, in leaky dielectric films, both the thermal and electric fields play important roles and together can produce an increase in the growth rate and a reduction in the dominant wavelength of the instability. Nonlinear simulations of the lubrication equations show that the predictions of the linear theory hold even when the interfacial perturbations are no longer small. The effects of viscoelasticity are considered within the linear theory, and it is found that the growth rate of the instability, but not the length scale, depends on the rheological parameters. The findings of this work suggest that the simultaneous use of an electric field and temperature gradient will allow thin films to be patterned at length scales not accessible when only one of these destabilizing forces is used.

Appendix

Shown below are the nonlinear evolution equations for height and charge in a leaky dielectric film:

$$\begin{aligned} \frac{\partial h}{\partial t}&= -\frac{1}{3} (1+h)^3 \frac{\partial ^4 h}{\partial x^4} - (1+h)^2 \frac{\partial ^3 h}{\partial x^3} \frac{\partial h}{\partial x} + \frac{(1+h)^3}{6(h(1-\varepsilon )+1+\beta \varepsilon )^2} \left( \frac{\partial q}{\partial x}\right) ^2 A_1 \nonumber \\&- \frac{(1+h)^2}{6(h(1-\varepsilon )+1+\beta \varepsilon )^3} \frac{\partial ^2 h}{\partial x^2} A_2 + \frac{(1+h)}{(h(1-\varepsilon )+1+\beta \varepsilon )^4} \left( \frac{\partial h}{\partial x} \right) ^2 A_3 \nonumber \\&+ \frac{(1+h)^3}{6(h(1-\varepsilon )+1+\beta \varepsilon )^2} \frac{\partial ^2 q}{\partial x^2} A_4 - \frac{(1+h)^2}{6(h(1-\varepsilon )+1+\beta \varepsilon )^3} \left( \frac{\partial h}{\partial x} \right) \left( \frac{\partial q}{\partial x} \right) A_5 \nonumber \\&+ \frac{1}{2} (1+h)^2 \mathrm {M} \frac{\partial ^2 \theta (z=h)}{\partial x^2} + (1+h)\mathrm {M} \frac{\partial h}{\partial x} \frac{\partial \theta (z=h)}{\partial x}, \end{aligned}$$

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$$\begin{aligned} \frac{\partial q}{\partial t}&= \frac{1}{2}q(1+h)^2 \frac{\partial ^4 h}{\partial x^4} - \frac{1}{2} (1+h)^2 \frac{\partial q}{\partial x}\frac{\partial ^3 h}{\partial x^3} - q(1+h)\frac{\partial ^3 h}{\partial x^3}\frac{\partial h}{\partial x} \nonumber \\&+ \frac{q}{2(h(1-\varepsilon ) + 1 +\beta \varepsilon )^4} \left( \frac{\partial h}{\partial x} \right) ^2 B_1 + \frac{(1+h)^2}{2(h(1-\varepsilon )+1+\beta \varepsilon )^2} \left( \frac{\partial q}{\partial x} \right) ^2 B_2 \nonumber \\&+ \frac{q(1+h)^2}{2(h(1-\varepsilon )+1+\beta \varepsilon )^2} \frac{\partial ^2 q}{\partial x^2} B_3 - \frac{q(1+h)}{2(h(1-\varepsilon )+1+\beta \varepsilon )^3} \frac{\partial ^2 h}{\partial x^2} B_4 \nonumber \\&- \frac{(1+h)}{2(h(1-\varepsilon ) + 1+\beta \varepsilon )^3} \frac{\partial h}{\partial x} \frac{\partial q}{\partial x} B_5 + \frac{1+q(h-\beta )}{h(1-\varepsilon )+1+\beta \varepsilon } \sigma \nonumber \\&+ \,\mathrm {M}(1+h) \left( \frac{\partial q}{\partial x} \frac{\partial \theta (z=h)}{\partial x} + q\frac{\partial ^2 \theta (z=h)}{\partial x^2} \right) + \mathrm {M} q\frac{\partial h}{\partial x}\frac{\partial \theta (z=h)}{\partial x}. \end{aligned}$$

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The coefficients of the nonlinear evolution equations are

$$\begin{aligned} A_1&= -2 + 3\beta + h(-7+\beta (10\varepsilon -3)) + 5h^2(\varepsilon -1) + 5\beta ^2 \varepsilon , \end{aligned}$$

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$$\begin{aligned} A_2&= \, 2(1+h)(\varepsilon -1)^2\varepsilon + q (1+\beta )\varepsilon \left[ -1+h(\varepsilon -1) + (4+3\beta )\varepsilon \right] \nonumber \\&+\, q^2 \left[ 3 + 3h^2(\varepsilon -1)^2 + (2+7\beta -\beta ^2)\varepsilon - 3\beta ^3\varepsilon ^2 - 9h^2(\varepsilon -1)(1+\beta \varepsilon ) \right. \nonumber \\&\left. + \,h\left( 9 - (1-16\beta +\beta ^2)\varepsilon + 9\beta ^2\varepsilon ^2 \right) \right] , \end{aligned}$$

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$$\begin{aligned} A_3&= -(1+h)(1+\beta )(\varepsilon -1)^2\varepsilon ^2 - q(1+\beta )^2\varepsilon ^2 \left[ -1 +h(\varepsilon -1) + (2+\beta )\varepsilon \right] \nonumber \\&+\, q^2 \left[ -1 + h^4(\varepsilon -1)^3 - \varepsilon - 4\beta \varepsilon - \varepsilon ^2 - 4\beta \varepsilon ^2 - 6\beta ^2\varepsilon ^2 + \beta ^4\varepsilon ^3\right. \nonumber \\&\left. -\, 4h^3(\varepsilon -1)^2(1+\beta \varepsilon ) - 4h(1+\beta \varepsilon )^3 + 6(\varepsilon -1)h^2(1+\beta \varepsilon )^2 \right] , \end{aligned}$$

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$$\begin{aligned} A_4&= -2(1+\beta )\varepsilon + q\left[ -2 + 3\beta + 5h^2(\varepsilon -1) + 5\beta ^2\varepsilon + h(-7+3\beta -10\beta \varepsilon )\right] , \end{aligned}$$

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$$\begin{aligned} A_5&= (1+\beta )\varepsilon (1+h+8\varepsilon -h\varepsilon +9\beta \varepsilon ) \nonumber \\&+ \,q \left[ 15 - 6\beta + 21h^3(\varepsilon -1)^2 + 8\varepsilon + 31\beta \varepsilon - 19\beta ^2\varepsilon - 21\beta ^3\varepsilon ^2 - 3h^2(\varepsilon -1)(19+\beta (21\varepsilon -2)) \right. \nonumber \\&\left. +\, h\left( 51 - 7\varepsilon + 4\beta (25\varepsilon -3) + \beta ^2\varepsilon (63\varepsilon -19) \right) \right] , \end{aligned}$$

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$$\begin{aligned} B_1&= -2q(1+\beta )^2\varepsilon ^2 \left[ -2 - 2h(\varepsilon -1) + (3+\beta )\varepsilon \right] - (1+h)(\varepsilon -1)^2 \varepsilon \left( -1 + h(\varepsilon -1) + (3+2\beta )\varepsilon \right) \nonumber \\&+\, q^2 \left[ -2 + 2h^4(\varepsilon -1)^3 - (3+10\beta +\beta ^2)\varepsilon - (3 + 12\beta + 17\beta ^2 + 2\beta ^3)\varepsilon ^2 + 2\beta ^4\varepsilon ^3 \right. \nonumber \\&-\,8h^3(\varepsilon -1)^2(1+\beta \varepsilon ) + h^2(\varepsilon -1)\left( 12 + (1+26\beta +\beta ^2)\varepsilon + 12\beta ^2\varepsilon ^2 \right) \nonumber \\&\left. -\,2h \left( 4 + (1+14\beta +\beta ^2)\varepsilon + \beta (1+14\beta +\beta ^2)\varepsilon ^2 + 4\beta ^3\varepsilon ^3 \right) \right] , \end{aligned}$$

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$$\begin{aligned} B_2&= -(1+\beta )\varepsilon + 2q\left[ -1 + 2\beta + 3h^2(\varepsilon -1) + 3\beta ^2\varepsilon + h(-4+2\beta -6\beta \varepsilon )\right] , \end{aligned}$$

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$$\begin{aligned} B_3&= -(1+\beta )\varepsilon + q \left[ -1 + 2\beta + 3h^2(\varepsilon -1) + 3\beta ^2\varepsilon + h(-4+2\beta -6\beta \varepsilon )\right] , \end{aligned}$$

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$$\begin{aligned} B_4&= (1+h)(\varepsilon -1)^2\varepsilon + 2q(1+\beta )^2\varepsilon ^2 \nonumber \\&+\, q^2 \left[ 2 + 2h^3(\varepsilon -1)^2 + \varepsilon + 4\beta \varepsilon - \beta ^2\varepsilon - 2\beta ^3\varepsilon ^2 - 6h^2(\varepsilon -1)(1+\beta \varepsilon ) \right. \nonumber \\&\left. \qquad \qquad +\, h\left( 6- (1-10\beta +\beta ^2)\varepsilon + 6\beta ^2\varepsilon ^2 \right) \right] , \end{aligned}$$

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$$\begin{aligned} B_5&= (1+h)(\varepsilon -1)^2\varepsilon + 6q(1+\beta )^2\varepsilon ^2 \nonumber \\&+ \, q^2 \left[ 10 - 2\beta + 12h^3(\varepsilon -1)^2 + 5\varepsilon + 20\beta \varepsilon - 9\beta ^2\varepsilon - 12\beta ^3\varepsilon ^2 - 2h^2(\varepsilon -1)(17+\beta (18\varepsilon -1)) \right. \nonumber \\&\left. \qquad \qquad + \,h\left( 32 - 5\varepsilon + 9\beta ^2\varepsilon (4\varepsilon -1) + \beta (58\varepsilon -4)\right) \right] , \end{aligned}$$

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$$\begin{aligned} \frac{\partial \theta (z=h)}{\partial x}&= \frac{-\kappa (\beta +1)}{(h(\kappa -1)+\beta +\kappa )^2}\frac{\partial h}{\partial x}, \end{aligned}$$

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$$\begin{aligned} \frac{\partial ^2 \theta (z=h)}{\partial x^2}&= -\frac{\kappa (\beta +1)}{(h(\kappa -1)+\beta +\kappa )^2} \frac{\partial ^2 h}{\partial x^2} + \frac{2\kappa (\kappa -1)(1+\beta )}{(h(\kappa -1)+\beta +\kappa )^3} \left( \frac{\partial h}{\partial x} \right) ^2. \end{aligned}$$

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