Electric-field-induced interfacial instabilities of a soft elastic membrane confined between viscous layers

We explore the electric-field-induced interfacial instabilities of a trilayer composed of a thin elastic film confined between two viscous layers. A linear stability analysis (LSA) is performed to uncover the growth rate and length scale of the different unstable modes. Application of a normal external electric field on such a configuration can deform the two coupled elastic-viscous interfaces either by an in-phase bending or an antiphase squeezing mode. The bending mode has a long-wave nature, and is present even at a vanishingly small destabilizing field. In contrast, the squeezing mode has finite wave-number characteristics and originates only beyond a threshold strength of the electric field. This is in contrast to the instabilities of the viscous films with multiple interfaces where both modes are found to possess long-wave characteristics. The elastic film is unstable by bending mode when the stabilizing forces due to the in-plane curvature and the elastic stiffness are strong and the destabilizing electric field is relatively weak. In comparison, as the electric field increases, a subdominant squeezing mode can also appear beyond a threshold destabilizing field. A dominant squeezing mode is observed when the destabilizing field is significantly strong and the elastic films are relatively softer with lower elastic modulus. In the absence of liquid layers, a free elastic film is also found to be unstable by long-wave bending and finite wave-number squeezing modes. The LSA asymptotically recovers the results obtained by the previous formulations where the membrane bending elasticity is approximately incorporated as a correction term in the normal stress boundary condition. Interestingly, the presence of a very weak stabilizing influence due to a smaller interfacial tension at the elastic-viscous interfaces opens up the possibility of fabricating submicron patterns exploiting the instabilities of a trilayer.

Figure 1

Schematic diagram of a confined trilayer under the influence of an electrostatic field of potential $\psi$, applied through an anode and a cathode separated by a distance $d$. The in-phase deformations at the interfaces depict a bending mode whereas an out-of-phase deformation shows the squeezing mode. The mean (local thicknesses) of the lower and upper liquid-liquid interfaces are $h_{10}$ [$h_1(x,t)$] and $h_{20}$ [$h_2(x,t)$], respectively.

Figure 2

Linear stability analysis (LSA) results comparing the general and the simplified [Eq. (54)] analyses of the bending elasticity. The solid line and the symbols (circles) correspond to the finite thickness and small thickness elastic membrane, respectively. The other parameters are $h_{10}$ $=$ 1.0 μm, $h_{20}$ $=$ 1.01 μm, $d$ $=$ 2.01 μm, ${\varepsilon }_1=15$, ${\varepsilon }_2=7$, ${\varepsilon }_3=18$, ${\mu }_1={\mu }_3=0.001$ Pa s, ${\gamma }_{21}={\gamma }_{32}=0$, $\psi =0$, and $G$ $=$ 100 MPa.

Figure 3

LSA results for the symmetric configuration. The solid and broken lines correspond to the bending and squeezing modes. Plots (a)–(c) show the variation of ω with $k$ when $\psi$, $G$, and $\gamma$ are varied. Plots (d)–(f) show the variation of $k_c$ with $\psi$, $G$, and $\gamma$, respectively. Curves 1–3 correspond to $\psi =75$, 104, and 135 V, when $G=0.1$ MPa in plot (a); to $G=0.01$, 0.09, and 1 MPa, when $\psi =100$ V in plot (b); and to ${\gamma }_{21}={\gamma }_{32}=\gamma =0.002$, 0.004, and 0.01 N m${}^{-1}$, when $\psi =100$ V in plot (c). In plot (d), curves 1–3 correspond to $G=0.01$, 0.1, and 1 MPa whereas in plot (e), curves 1–3 correspond to $\psi$ $=$ 75, 100, and 150 V. In plot (f), curves 1–3 correspond to $\psi$ $=$ 100, 125, and 150 V, when $G=0.1$ MPa.

Figure 4

Stability diagrams depicting the stable and unstable regions for the symmetric trilayer. Plots (a) and (b) show the variation of ${\psi }_c$ with $G_c$ when $\gamma$ and $h_{M0}(=h_{20}-h_{10})$ are varied. Plots (c) and (d) show the variation ${\overline{\psi }}_c$ with ${\overline{G}}_c$ when $H_M$${}_0$ and $\Gamma$ are varied keeping $d$ constant. Curves 1–3 correspond to ${\gamma }_{21}={\gamma }_{32}=\gamma =0.005$, 0.05, and 0.5 N/m in plot (a) and to $h_{M0}=0.1$, 0.5, and 2 μm, in plot (b). In plot (c), curves 1–3 correspond to ${\Gamma }_{21}={\Gamma }_{32}=\Gamma =0.0105$, 0.105, and 0.21, and to $H_M$${}_0=0.0471$, 0.095, and 0.23 in plot (d). In plots (a) and (c), $h_{10}=1.0$ μm, $h_{20}=1.1$ μm, and $d=2.1$ μm and in plots (b) and (d), $d=2.1$ μm, and ${\gamma }_{21}={\gamma }_{32}=0.005$ N/m.

Figure 5

Plot (a) shows the variation of ω with $k$ where curves 1–3 correspond to $\psi$ $=$ 50, 105, and 120 V, respectively, with constant $G$ $=$ 0.1 MPa and ${\varepsilon }_2$ $=$ 2. Plots (b)–(d) show the variations of ${\omega }_m$, ${\lambda }_m$, and ${\delta }_r$ with $\psi$ where curves 1–3 correspond to $G$ $=$ 0.01, 0.1, and 1 MPa when ${\varepsilon }_2$ $=$ 2. The solid and broken lines in plots (a)–(c) correspond to the bending and squeezing modes. The transition potential ${\mathit{\Psi }}_T$ marks the point of transition between the two modes in plots (a), (c), and (d). The downward vertical arrow (2) in plot (d) shows the transition from bending to squeezing mode.

Figure 6

Plot (a) shows the variation of ω with $k$ where curves 1–3 correspond to ${\varepsilon }_2$ $=$ 3.8, 4, and 6 when $G$ $=$ 0.095 MPa and $\psi$ $=$ 150 V. Plots (b)–(d) show the variations of ${\omega }_m$, ${\lambda }_m$, and ${\delta }_r$ with ${\varepsilon }_2$ where curves 1–3 correspond to $\psi$ $=$ 100, 150, and 200 V when $G$ $=$ 0.095 MPa. The solid and broken lines in plots (a)–(c) correspond to the bending and squeezing modes. The transitional dielectric permittivity ${\mathit{\varepsilon }}_{2T}$ marks the point of transition between the two modes in plots (a)–(d). The upward vertical arrows (1–3) in plot (d) show the switching from squeezing to bending mode.

Figure 7

Plot (a) shows the variation of ω with $k$ where curves 1–3 correspond to $G$ $=$ 0.01, 0.09, and 1 MPa, when $\psi$ $=$ 100 V and ${\varepsilon }_2$ $=$ 4. Plots (b)–(d) show the variation of ${\omega }_m$, ${\lambda }_m$, and ${\delta }_r$ with $G$, where curves 1–3 correspond to ${\varepsilon }_2$ $=$ 2, 4, and 7, respectively, at $\psi$ $=$ 100 V. The solid and broken lines in plots (a)–(c) correspond to the bending and squeezing modes. The transitional shear modulus ${\mathbf{G}}_T$ marks the changeover from squeezing to bending mode in plots (a)–(d). The upward vertical arrows (1–3) in plot (d) show the transition from squeezing to bending mode.

Figure 8

Plot (a) shows the variation of ω with $k$ where curves 1–3 correspond to compliance $C$ (=$h_{M0}/G$)=10, 3.3, 0.1 (×10${}^{-12}$) m/Pa, when $\psi$ $=$ 100 V and ${\varepsilon }_2$ $=$ 4. Plots (b)–(d) show the variations of ${\omega }_m$, ${\lambda }_m$, and ${\delta }_r$ with $G$, where curves 1 and 2 correspond to $h_{M0}$ $=$ 10 and 100 nm, respectively. The solid and broken lines in plots (a)–(c) correspond to the bending and squeezing modes. The transitional compliance ${\mathbf{C}}_T$ marks the point of transition between the two modes in plots (a)–(d). The downward vertical arrows (1 and 2) in plot (d) show the switching from bending to squeezing mode.

Figure 9

LSA results for the asymmetric configuration. Plots [(a),(b),(c)], [(d),(e),(f)], [(g),(h),(i)], and [(j),(k),(l)] show the variations of [${\omega }_m$, ${\lambda }_m$, ${\delta }_r$] with $h_r$, ${\varepsilon }_r$, ${\gamma }_r$, and ${\mu }_r$, respectively. In plots (a)–(c) the curves 1–3 correspond to ${\varepsilon }_r=$ 0.5, 1, and 2, respectively, when $h_{M0}=$ 0.1 μm. In plots (d)–(f) the curves 1–3 correspond to ${\mu }_r=$ 0.1, 1, and 10, respectively, when ${\varepsilon }_1=$ 80, ${\varepsilon }_2=$ 4. In plots (g)–(i) the curves 1–3 correspond to ${\varepsilon }_r=$ 0.25, 1, and 4, respectively, when $G$ $=$ 0.095 MPa. Curves 1–3 in plots (j)–(l) represent ${\gamma }_r=$ 0.01, 1, and 10, respectively, when $G$ $=$ 0.095 MPa.

Figure 10

The solid and broken lines correspond to the bending and squeezing modes. Plots (a) and (b) show the variation of $k_c$ with $\Delta \psi$ and $G$, respectively. Curves 1 and 2 in these plots correspond to the cases with fixed applied potential at the elastic-air interfaces and at the electrodes, respectively. In plot (a), $G$ $=$ 0.1 MPa and in plot (b), $\Delta \psi$ $=$ 50 V. For the surface applied potential, ${\gamma }_{21}={\gamma }_{32}=0.05$ N/m and ${\mu }_1={\mu }_3=0$, and for the externally applied potential ${\gamma }_{21}={\gamma }_{32}=0.005$ N/m and ${\mu }_1={\mu }_3=0.001$ Pa s.