Effective Debye length in closed nanoscopic systems: A competition between two length scales

The Poisson-Boltzmann equation (PBE) is widely employed in fields where the thermal motion of free ions is relevant, in particular in situations involving electrolytes in the vicinity of charged surfaces. The applications of this non-linear differential equation usually concern open systems (in osmotic equilibrium with an electrolyte reservoir, a semi-grand canonical ensemble), while solutions for closed systems (where the number of ions is fixed, a canonical ensemble) are either not appropriately distinguished from the former or are dismissed as a numerical calculation exercise. We consider herein the PBE for a confined, symmetric, univalent electrolyte and quantify how, in addition to the Debye length, its solution also depends on a second length scale, which embodies the contribution of ions by the surface (which may be significant in high surface-to-volume ratio micro- or nanofluidic capillaries). We thus establish that there are four distinct regimes for such systems, corresponding to the limits of the two parameters. We also show how the PBE in this case can be formulated in a familiar way by simply replacing the traditional Debye length by an effective Debye length, the value of which is obtained numerically from conservation conditions. But we also show that a simple expression for the value of the effective Debye length, obtained within a crude approximation, remains accurate even as the system size is reduced to nanoscopic dimensions, and well beyond the validity range typically associated with the solution of the PBE.