Weakly nonlinear analysis of the electrohydrodynamic instability of a charged membrane

The effect of nonlinear interactions on the linear instability of shape fluctuations of a flat charged membrane immersed in a fluid is analyzed using a weakly nonlinear stability analysis. There is a linear instability when the surface tension reduces below a critical value for a given charge density, because a displacement of the membrane surface causes a fluctuation in the counterion density at the surface, resulting in an additional Maxwell normal stress at the surface which is opposite in direction to the stress caused by surface tension. The nonlinear analysis shows that at low surface charge densities, the nonlinear interactions saturate the growth of perturbations resulting in a new steady state with a fluctuation amplitude determined by the balance between the destabilizing electrodynamic force and surface tension. As the surface charge density is increased, the nonlinear terms destabilize the perturbations, and the bifurcation is subcritical. There is also a significant difference in the predictions of the approximate Debye-Huckel and more exact Poisson-Boltzmann equations at high charge densities, with the former erroneously predicting that the bifurcation is supercritical at all charge densities.