Instability of a fluctuating biomimetic membrane driven by an applied uniform DC electric field

TL;DR

The study addresses the linear stability of a biomimetic lipid membrane subjected to a perpendicular DC electric field, explicitly incorporating Debye-layer dynamics via an electrokinetic (PNP) framework. By deriving a base state with a nonzero electric field in the Debye layer and performing a linear stability analysis, the authors obtain a dispersion relation where perturbed Debye-layer charges generate destabilizing Maxwell stresses that outweigh stabilizing electrostatic body forces, with asymmetry in electrolyte concentrations further stabilizing by weakening the base field. The analysis introduces key dimensionless groups (e.g., $eta$, $ ho_m$, $Ca$, $oldsymbol{ abla}$, $ ext{and}\, ar{ ext{Γ}}$) and yields a growth-rate formula $4s = s_{ ext{ex}} + s_{ ext{in}} - rac{2}{Ca}(ar{ ext{Γ}} k^2 + k^4)$, highlighting the competing electrostatic mechanisms and the limits of validity of the EK framework relative to the leaky dielectric model. The work suggests nonlinear extensions and finite-$oldsymbol{ abla}$ (nonzero $oldsymbol{ ext{alpha}}$) analyses to bridge EK and LDM regimes and to explore possible pattern formation beyond the linear regime in membrane electrohydrodynamics.

Abstract

The linear stability of a lipid membrane under a DC electric field, applied perpendicularly to the interface, is investigated in the electrokinetic framework, taking into account the dynamics of the Debye layers formed near the membrane. The perturbed charge in the Debye layers redistributes and generates destabilizing Maxwell stress on the membrane, which outweighs the stabilizing contribution from the electrical body force, leading to a net destabilizing effect. The instability is suppressed as the difference in the electrolyte concentration of the solutions separated by the membrane increases, due to a weakened base state electric field near the membrane. This result contrasts with the destabilizing effect predicted using the leaky dielectric model in cases of asymmetric conductivity. We attribute this difference to the varying assumptions about the perturbation amplitude relative to the Debye length, which result in different regimes of validity for the linear stability analysis within these two frameworks.

Figures (8)

• Figure 1: A planar lipid bilayer membrane, with shape denoted by $z=h(x)$, separating electrolyte solutions with different concentrations. The electrodes, with potential $\phi=\pm V$, are held at $z=\pm L$.
• Figure 2: The components of the growth rate in Eq. \ref{['eq:growth rate full']} as a function of the perturbation wavenumber $k$, with the "total" representing the sum of the external, internal, and elastic contributions. We consider symmetric conductivity with $\gamma=1$, $V=2$, $\beta=1$, $Ca=0.2$, $\bar{\Gamma}=0.01$ and $d=0$.
• Figure 3: The effect of $Ca$ on growth rate $s$ as a function of perturbation wavenumber $k$. We consider symmetric conductivity with $\gamma=1$, $V=2$, $\beta=1$, $\bar{\Gamma}=0.01$ for (a) $d=0$ and (b) $d=1$.
• Figure 4: (a)The effect of $\beta$ on the growth rate, with black curve shows the full growth rate as in Eq. \ref{['eq:growth rate full']}, while the blue curve excludes the contribution from the internal traction in Eq. \ref{['eq:internal']}; (b) the effect of $\beta$ on the external and internal contribution to the growth rate in Eq. \ref{['eq:ex in']} at $k=1$; (c) the maximal (full) growth rate $s_{\max}$ as a function of $\beta$. Note that the blue and black curves overlap at $\beta=0,\infty$ in (a). We consider the system with the parameters $V=3$, $\gamma=1$, $d=0$, $Ca=0.5$, and $\bar{\Gamma} =0.01$.
• Figure 5: (a) The effect of $\gamma$ on growth rate $s$ as a function of perturbation wavenumber $k$; (b) maximal growth rate $s_{\mathrm{max}}$ as function of $\gamma$. We consider the system with $V=3$, $\beta=1$, $Ca=0.2$ and $\bar{\Gamma}=0.01$.