MATHEMATICAL THEORY OF ELECTRO-CAPILLARY SURFACES

In the 1960s fluid mechanician G. I. Taylor undertook a series of beautiful experiments that looked at electrostatically actuated capillary surfaces. While his interest was in the interaction of water droplets in electrified clouds, his work has found surprising applications in many fields such as chemistry [FMMWM89] and space propulsion [MGA98]. In particular, his investigation in [Tay68] of the coalescence of soap bubbles maintained at different electric potentials has become of great practical relevance in the field of microelectromechanical systems [PeBe03, §7]. A generalized version of his system consists of a soap film $\Sigma$ with fixed boundary $\partial \Omega$ held at voltage $V$ and suspended at a distance $h$ above a rigid ground plate. In this way the potential difference creates an attractive force between the two components and deflects the soap bubble down towards the bottom membrane (see Figure 1). The most impactful and surprising observation of this work was that when the voltage difference is increased beyond some critical value $V^*$, the top film snaps down on to the bottom plate or gets “pulled-in”.

In my research, I study the equilibrium shape of the deflected soap film as a function of the applied voltage $V$ and the gap distance $h$. Similar to conventional methods for finding the shape of a soap bubble, this system can be modeled variationally. That is, if the equilibrium shape of the soap film is $\Sigma = (x,y,u(x,y))$, where $u:{\Omega}\subseteq \mathbf{R}^2 \to (-h,\infty)$ is $C^1_0(\Omega)$, then $\Sigma$ minimizes the total energy \begin{equation}\label{eq:total_energy1} \mathcal{E}[u,\psi] = \gamma \mathcal{A}[u] + \frac{\eps_0}{2} \mathcal{U}[u,\psi], \end{equation} where \[ \mathcal{A}[u] = \iint_{\Omega} \sqrt{1+ |\grad u|^2} \, dx \, dy, \quad \mathcal{U}[u,\psi] = \iint_{\Omega}\int_{-h}^{u}{|{\nabla}\psi|^2}\, dz\, dx \, dy. \] Here $\gamma$ is the surface tension of the soap bubble, $\eps_0$ is the permittivity of free space and $\psi$ is the unknown electrostatic potential. The first component of energy \eqref{eq:total_energy1} is the free surface energy of the film and the second is electrostatic potential energy of the electric field. By nondimensionalizing \eqref{eq:total_energy1} via the transformations \[ u(x,y) \mapsto h \, u(x/L,y/L), \ \ \psi(x,y,z) \mapsto V \, \psi(x/L,y/L,z/h), \] where $L= \max\{ |x|: x\in \partial \Omega\}$, and then minimizing the result, we find that dimensionless optimizers $(u,\psi)$ of \eqref{eq:total_energy1} must necessarily satisfy the following coupled system of nonlinear partial differential equations: \begin{equation}\label{eq:general_sys_pde_ab} \eps^2 \Delta_{\mathbf{R}^2} \psi + {\psi}_{zz} = 0,\ \ \Div \frac{\nabla u}{ \sqrt{ 1 +\eps^2| \nabla u |^2}} = \lambda \big( \eps^2 \left| \nabla_{\!\mathbf{R}^2} \psi \right|^2 + {{\psi}_{z}}^2\big) \qquad \end{equation} where the first PDE holds in $\Omega \times \{-1 < z < u(x,y)\}$ and the second holds in $\Omega \times \{z = u(x,y)\}$. The nonnegative dimensionless parameters $\eps$ and $\lambda$ are defined as $\eps = h/L$ and $\lambda = {\eps_0 V^2 L^2}/{2 \gamma h^3}$, respectively. Specifically, $\eps$ measures the shape of the device and $\lambda$ measures the relative strengths of the surface and electrostatic forces. Also from