Interfacial instability of thin films in soft microfluidic configurations actuated by electro-osmotic flow

We analyze the interfacial instability of a thin film confined between a rigid surface and a prestretched elastic sheet, triggered by nonuniform electro-osmotic flow. We derive a nonlinear $\text{viscous}-\text{elastic}$ equation governing the deformation of the elastic sheet, describing the balance between viscous resistance, the dielectric and electro-osmotic effects, and the restoring effect of elasticity. Our theoretical analysis, validated by numerical simulations, shows several distinct modes of instability depending on the electro-osmotic pattern, controlled by a nondimensional parameter representing the ratio of electro-osmotic to elastic forces. We consider several limiting cases and present approximate asymptotic expressions predicting the electric field required for triggering of the instability. Through dynamic numerical simulations of the governing equation, we study the hysteresis of the system and show that the instability can result in an asymmetric deformation pattern, even for symmetric actuation. Finally, we validate our theoretical model with finite-element simulations of the two-way coupled Navier equations for the elastic solid, the unsteady Stokes equations for the fluid, and the Laplace equation for the electric potential, showing very good agreement. The mechanism illustrated in this work, together with the provided analysis, may be useful in toward the implementation of instability-based soft actuators for lab-on-a-chip and soft-robotic applications.

Figure 1

Schematic illustration of the examined configuration, showing the coordinate system and relevant physical and geometric parameters. A thin viscous liquid film of initial thickness ${\tilde{h}}_0$ is confined between a rigid surface and a prestretched elastic sheet of length ${\tilde{l}}_m$ and thickness ${\tilde{h}}_m$, supported at its boundaries. nonuniform electro-osmotic flow, induced by an EOF slip velocity ${\tilde{u}}_{\mathrm{EOF}}(\tilde{x},\tilde{t})$, creates negative pressures within the viscous fluid, resulting in $\text{viscous}-\text{elastic}$ interaction, which leads to deformation $\tilde{d}(\tilde{x},\tilde{t})$ of the elastic sheet. Above a certain threshold of the electric field, the system exhibits instability which collapses the elastic sheet onto the floor.

Figure 2

Comparison between asymptotic and numerical results in the case of a gravity-dominant regime. (a) The shape of the steady-state deformation along the $\widehat{\mathit{x}}$ axis for several values of $E_{\mathrm{EOF}}$. Solid lines represent the asymptotic solution Eq. (27) in the case of pure hydrostatic stress, whereas dashed lines represent the numerical solution which takes into account both tension and hydrostatic terms. (b) The growth rate $\sigma$ as a function of $E_{\mathrm{EOF}}/\left|E_{\mathrm{EOF},\mathrm{CR},\mathrm{G}}\right|$, obtained from a linear stability analysis by solving Eq. (24). Gray dots correspond to the pure gravitational contribution, whereas black dots correspond to the case when the tension contribution is included. All calculations were performed using $k=1, \mathcal{B}=0, \mathcal{G}=100$, and $E_{\mathrm{EOF},\mathrm{CR},\mathrm{G}}=-25\pi$.

Figure 3

The effect of gravity on the onset of the instability of a prestretched elastic sheet in a constant current actuation mode. [(a),(b)] The magnitude of the threshold elasto-electro-osmotic number $E_{\mathrm{EOF},\mathrm{CR}}$ and the corresponding maximum deformation $d_{\max ,\mathrm{CR}}$ as a function of $\mathcal{G}$. Dots represent the numerical results including both tension and hydrostatic contributions. Gray line represents the asymptotic solution Eq. (29) for $\mathcal{G}\gg 1$ and black line represents the asymptotic solution Eq. (30) for a wide range of $\mathcal{G}$ values. For $\mathcal{G}\ll 1$, the value of $|E_{\mathrm{EOF},\mathrm{CR}}|$ is independent of $\mathcal{G}$, while for $\mathcal{G}\gg 1$ the value of $|E_{\mathrm{EOF},\mathrm{CR}}|$ scales linearly with $\mathcal{G}$, as given by the asymptotic limit Eq. (30). All calculations were performed using $k=1$ and $\mathcal{B}=0$.

Figure 4

Comparison between the results of a dynamic numerical simulation for constant current and constant voltage actuation modes in a tension-dominant regime. [(a),(b)] The time evolution of the maximum deformation, obtained at the center of the membrane, for several values of $E_{\mathrm{EOF}}$. [(c),(d)] The maximum deformation at steady state as a function of $E_{\mathrm{EOF}}$. Black dots represent the state-state solution of the dynamic simulation Eq. (13), and gray dots represent the solution of the state-state boundary value problem Eq. (23). [(e),(f)] The shape of the steady-state deformation along the $\widehat{\mathit{x}}$ axis, for several values of $E_{\mathrm{EOF}}$. Panels on the left correspond to the case of a constant applied current and panels on the right correspond to the case of a constant applied voltage. All calculations were performed using $\mathcal{B}=\mathcal{G}=0, k=1$, and $h_r={10}^{-2}$.

Figure 5

The effect of the elasto-electro-osmotic number on the transient behavior and the magnitude of the contact width $\mathrm{\Delta }x$. (a) The time evolution of the deformation field for $E_{\mathrm{EOF}}=-50$ in the case of a constant applied voltage. In total, 10 profiles at equally spaced time steps between $t=0$ and a steady state at $t=0.058$ are shown. (b) The contact width $\mathrm{\Delta }x$, representing the part of the elastic sheet where $h=h_r$, as a function of $|E_{\mathrm{EOF}}|$. All calculations were performed using $\mathcal{G}=\mathcal{B}=0, k=1,$ and $h_r={10}^{-2}$.

Figure 6

Hysteresis for the onset of instability. [(a),(b)] The maximum deformation $d_{\max }$ at steady state as a function of $E_{\mathrm{EOF}}$ for a constant current (a) and a constant voltage (b) actuation modes. [(c),(d)] The magnitude of the threshold elasto-electro-osmotic number $E_{\mathrm{EOF},\mathrm{CR}}$ as a function of $\mathcal{G}$ for a constant current (c) and a constant voltage (d) actuation modes, resulting from two different initial states of the system. Black dots represent the results obtained from an initially flat elastic sheet and gray dots represent the results obtained from an initially deformed elastic sheet which adhered to the floor, corresponding to $d_{\max }=-0.99$. Black dashed lines in panels (a) and (b) represent unstable equilibrium solutions obtained from Eq. (23). Black solid lines in panels (c) and (d) represent the asymptotic solutions for $\mathcal{G}\gg 1$, showing that in this case $|E_{\mathrm{EOF},\mathrm{CR}}|$ scales linearly with $\mathcal{G}$. All calculations were performed using $k=1, \mathcal{B}=\mathcal{G}=0,$ and $h_r={10}^{-2}$.

Figure 7

The effect of bending on the onset of the instability of a prestretched elastic sheet. [(a),(b)] The magnitude of the threshold elasto-electro-osmotic number $E_{\mathrm{EOF},\mathrm{CR}}$ and the corresponding maximum deformation $d_{\max ,\mathrm{CR}}$ as a function of $\mathcal{B}$. Dots represent the numerical results, while gray and black lines represent the asymptotic solutions Eqs. (31) and (32). Gray and black symbols correspond to constant current and constant voltage actuation modes, respectively. All calculations were performed using $k=1$ and $\mathcal{G}=0$.

Figure 8

Investigation of symmetric and asymmetric deformation patterns resulting from a symmetric actuation with a constant applied voltage, in the case of a strongly prestretched elastic sheet. [(a),(c),(e)] The time evolution of the deformation field for $E_{\mathrm{EOF}}=32$ (a), $E_{\mathrm{EOF}}$=$-200$ (c), and $E_{\mathrm{EOF}}=-275$ (e). [(b),(d),(f)] The time evolution of the corresponding pressure field. The insets in [(b),(d),(f)] present the pressure distribution in the fluid, as the film thickness reduces to $h=h_r$. All calculations were performed using $k=3$ and $\mathcal{B}=\mathcal{G}=0$.

Figure 9

[(a),(b)] The resulting flow streamlines at steady state underneath the deformed elastic sheet (represented by black thick lines) for $E_{\mathrm{EOF}}=-200$ (a) and $E_{\mathrm{EOF}}=-275$ (b). Insets show magnification of the narrow regions where $h=h_r={10}^{-2}$. Gray lines represent the streamlines and the arrows indicate the flow direction. All calculations were performed using $k=3$ and $\mathcal{B}=\mathcal{G}=0$.

Figure 10

Comparison of finite-element simulation results and theoretical model predictions for the case of a constant applied voltage. (a) The maximum deformation at steady state as a function of $E_{\mathrm{EOF}}$. Black dots represent the theoretical model predictions, whereas red crosses represent the results of the finite-element simulation. (b) The time evolution of the maximum deformation, obtained at the center of the membrane, for several values of $E_{\mathrm{EOF}}$. [(c),(d)] The time evolution of the deformation field for $E_{\mathrm{EOF}}=-11$ (c) and $E_{\mathrm{EOF}}=-12$ (d). Gray solid lines represent the theoretical model predictions and black dashed lines represent the finite-element simulation results. All calculations were performed using the values from Tables 2 and 3, with $k=1$.

Figure 11

(a) The variation of the growth rate $\sigma$ with $E_{\mathrm{EOF}}$ for wave numbers $k=1$, 2, and 3, in a tension-dominant regime. (b) The variation of the growth rate $\sigma$ with $k^2$ for $E_{\mathrm{EOF}}=-5,-25,$ and $-75$, in a tension-dominant regime. All calculations were performed using $\mathcal{B}=\mathcal{G}=0$. Dotted lines are added to guide the eye.

Figure 12

Comparison of finite-element simulation results and theoretical model predictions for the case of a constant applied current. (a) The maximum deformation at steady state as a function of $E_{\mathrm{EOF}}$. Black dots represent the theoretical model predictions, whereas red crosses represent the results of the finite-element simulation. (b) The time evolution of the maximum deformation for several values of $E_{\mathrm{EOF}}$. [(c),(d)] The time evolution of the deformation field for $E_{\mathrm{EOF}}=-7$ (c) and $E_{\mathrm{EOF}}=-8$ (d). Gray solid lines represent the theoretical model predictions and black dashed lines represent the finite-element simulation results. All calculations were performed using the values from Tables 2 and 3, with $k=1$.