Numerical finding of border separating limited and overlimiting currents in a semiconducting electric membrane

Abstract

One of the possible physical mechanisms of transition from limited to overlimiting currents in the electrolyte located between two electric membranes considered from theoretical point of view. This mechanism can be named electrokinetic instability. The boundary separating the limited and overlimiting modes is found numerically, according to the stability or instability of one-dimensional state of equilibrium. We can find the conclusion to develop new numerical method of solving the problem. Finding of the nonequilibrium one-dimensional equilibrium state as well as its research on stability is conducted numerically using $\tau$-version of Galerkin's method. The system of Chebyshev's polynomials is used as a complete system of basis functions. Boundary eigenvalue problem is described by linearized system of ordinary differential Nernst-Planck-Poisson-Stokes equations with the appropriate boundary conditions, which is reduced to generalized algebraic eigenvalue problem after projection on the basis functions: $\text{det}(A+\lambda B)=0$, where $A$, $B$ - the complex matrixes, $\lambda$ - eigenvalues. If the real part of at least one eigenvalue of discrete spectrum is positive, then one-dimensional equilibrium position corresponding to the limited currents is unstable, the regime changes and the transition to overlimiting currents takes place. Physically, by this transition two ion transport mechanism, diffusion and electromigration, are complemented by the third mechanism - advection. Numerical data of the work is well corresponded to the results obtained analytically for small Debye numbers.