Nonuniform Electro-osmotic Flow Drives Fluid-Structure Instability

We demonstrate the existence of a fluid-structure instability arising from the interaction of electro-osmotic flow with an elastic substrate. Considering the case of flow within a soft fluidic chamber, we show that above a certain electric field threshold, negative gauge pressure induced by electro-osmotic flow causes the collapse of its elastic walls. We combine experiments and theoretical analysis to elucidate the underlying mechanism for instability and identify several distinct dynamic regimes. The understanding of this instability is important for the design of electrokinetic systems containing soft elements.

Figure 1

Illustration of the configuration used for experiments and modeling. (a) Image of the experimental device, (b) exploded isometric view showing its different layers, and (d) two-dimensional model and key parameters used for the theoretical analysis. The device consists of a microfluidic chamber chemically functionalized to produce nonuniform EOF. The chamber’s floor is fixed while its ceiling can move vertically, supported by an elastic sheet. Upon application of an electric field, flow is driven from the center to the edges of the chamber, resulting in negative pressure and downward motion of the ceiling. (c) We monitor the height of the ceiling $\tilde{h}(\tilde{t})$ in time by capturing the change in the point spread function (PSF) of beads deposited on top of it, and observe its collapse onto the bottom surface when instability is triggered.

Figure 2

Time evolution of the fluidic gap resulting from nonuniform electro-osmotic flow. (a) The experimentally observed temporal evolution of the film thickness $\tilde{h}(\tilde{t})$ for various values of $\tilde{E}$ and (b) the corresponding theoretically predicted evolution of $h(t)$ for various values of $\beta$, indicating three distinct dynamic behaviors. Below the threshold values of ${\tilde{E}}_{\mathrm{CR}}=-2.23\times {10}^4\mathrm{V}{\mathrm{m}}^{-1}$ or ${\beta }_{\mathrm{CR}}=-1.129$, the ceiling approaches a stable steady-state height. Setting $\tilde{E}$ or $\beta$ to slightly above their threshold values ($\tilde{E}=-2.26\times {10}^4\mathrm{V}{\mathrm{m}}^{-1}$ and $\beta =-1.145$) triggers instability characterized by a bottleneck period. For higher values of $\tilde{E}$ or $\beta$ (e.g., for $\tilde{E}=-5\times {10}^4\mathrm{V}{\mathrm{m}}^{-1}$ or $\beta =-2.53$) the bottleneck phase disappears and the ceiling immediately collapses onto the rigid floor. Error bars in (a) indicate a 95% confidence of the mean position of nine beads measured at each time point. The grayed-out region highlights two experiments in which the driving field differs by only 1.3%, capturing the threshold of instability and leading to a longer bottleneck due to the proximity to this value.

Figure 3

Theoretical and experimental observation of a threshold electric field for instability. (a) The experimentally observed height $\tilde{h}$ at the end of each experiment, as a function of applied electric field $\tilde{E}$. Error bars indicate a 95% confidence on the mean based on four experiments [10]. (b) The theoretically predicted steady-state height $h$ as a function of $\beta$. The red dashed line represents the results of a linear stability analysis, whereas black dots represent the results of the dynamic simulation. At ${\beta }_{\mathrm{CR}}=-1.129$ the stable equilibrium intersects an unstable solution and disappears at a saddle-node bifurcation [10]. Above the threshold values of ${\tilde{E}}_{\mathrm{CR}}=-2.23\times {10}^4\mathrm{V}{\mathrm{m}}^{-1}$ and ${\beta }_{\mathrm{CR}}=-1.129$, no equilibrium solutions exist and the system exhibits instability that collapses the ceiling onto the floor. Gray lines in both subfigures were added to guide the eye.

Figure 4

Stability map showing three distinct regions as a function of $\beta$ and $\phi$. The light gray region corresponds to a stable downward motion due to negative EOF-induced pressure, the dark gray region corresponds to a stable upward motion where the dielectric force overcomes the EOF-induced negative pressure, and the white region corresponds to an unstable collapse of the plate toward the floor. The inset presents the steady-state height of the plate as a function of $\beta$ for $\phi =0.4$, showing the transition between regimes as $\beta$ is made more negative.