Dielectric dispersion of a membrane consisting of a large number of solid bilayers containing mobile ions

Abstract

The complex permittivity and the impedance of a lamellar membrane consisting of a repetition unit of two conducting dielectrics is calculated using the linearized Nernst–Planck–Maxwell electrodynamics previously developed for isolated interfaces, lamellar films with identical cation and anion mobility inside each type of lamella and with identical Nernst distribution coefficients and for homogeneous polymer films with ionic impurities. Any intrinsic dielectric dispersion of the two (ion-free) dielectrics is neglected and periodic boundary conditions are assumed in a infinite array of identical bilayers. The anion and the cation may be of any charge type and their diffusion coefficients may have any values in the two dielectrics. The Nernst distribution coefficient may be different for the cation and the anion within limits determined by the validity of the linearisation procedure. The calculated complex permittivities of the composite material may differ substantially from the permittivities calculated by a simple Maxwell–Wagner–Sillars (MWS) procedure because of the dynamical electrical double layers formed at 1–2 Debye lengths from the interfaces. These polarisation layers give rise to the so-called excess impedance. The MWS results are, however, always exact in the high-frequency limit. Furthermore, the MWS approach is correct for the dielectric loss at low frequencies. A frequency weighted ‘Cole–Cole plot’ of the real and imaginary components of the excess impedance has a characteristic ‘apple shape’ starting in the origin at very low frequencies and moving clockwise around the origin with increasing frequency. In the high-frequency limit, the origin is always approached in the fourth quadrant along a line with slope-1. The low-frequency asymptote of the frequency weighted Cole–Cole plot for the excess impedance is sensitive to the special conditions, however. The Cole–Cole plots of the total membrane impedance (MWS+excess) exhibit sometimes two partially overlapping semi-circular arcs, one for each type of lamella. When the thickness of the lamellae becomes of the same order of magnitude as the Debye length, the distinction between the two relaxations is wiped out and only a single, flattened arc is seen. A model lamellar ‘cellulose acetate’ (CA) membrane, consisting of alternating layers of ‘wet CA phase’ and ‘aqueous microphase’ with NaCl dissolved, is investigated. The parameters of the membrane are chosen quite realistically (except the one-dimensional geometry) as those pertaining to dense CA membranes, in accordance with previous emf and impedance spectroscopical studies of dense and asymmetric cellulose acetate membranes. The usual electrochemical ‘equivalent circuit’ approach, connecting in parallel the membrane resistance and the membrane capacitance, is put to a test. From the time constant related to the maximum negative reactance the so-called ‘geometric capacitance’ is calculated. From this, the ‘geometric’ membrane permittivity of the membrane is found. At low salt concentrations, the geometric permittivity coincides with the low-frequency limit of the membrane permittivity, but with increasing salt concentrations the geometric permittivity increases less rapidly with concentration than does the low-frequency membrane permittivity. The geometric membrane permittivity seems to be a complicated mean value between the low-frequency membrane permittivity and the high-frequency membrane permittivity (independent of concentration), since the characteristic frequency for the impedance relaxation is many orders of magnitude higher than the characteristic frequency of the dielectric relaxation. The former is related to the diffusional relaxation time of the CA layer, the second to the relaxation time in the aqueous microphase.