Complex permittivity of a conducting, dielectric layer containing arbitrary binary Nernst–Planck electrolytes with applications to polymer films and cellulose acetate membranes

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Abstract
The theory of Trukhan [Sov. Phys. Solid State (Engl. Transl.), 1963, 4, 2560] for the calculation of the complex permittivity of a conducting dielectric film containing a binary 1 : 1 electrolyte is critically reviewed. It is shown that the final result of Trukhan is correct, in spite of some errors and odd procedures in the original derivation. The method of Trukhan is changed to a more concise procedure which is better suited to generalisations, and the theory is generalised to binary electrolytes of any charge type. The dimensionless excess impedance (over and above the Maxwell–Wagner–Sillars impedance) and the complex relative permittivity are functions of the type of electrolyte, the dimensionless frequency, the ratio of the ionic diffusion coefficients and the film thickness scaled by the Debye length. The complex relative permittivity has a Debye-like relaxation, but for very different values of the two diffusion coefficients and for films which are thin compared with the Debye length, one can clearly distinguish two dielectric relaxations, one for each ion. The peak in the dielectric loss corresponding to the ion of higher valency, is dominating in thin films. When the film is thicker, (or when the electrolyte concentration is increased) the low-frequency peak diminishes relative to the high-frequency peak. The maximum in the loss tangent is positioned at higher frequencies than the maximum in the dielectric loss. Therefore it is often an advantage to fit the loss tangents to actual measurements, since measurements at low frequencies are generally more difficult. Two applications of the theory are treated: the low-frequency dielectric relaxation of a dry copolymer of vinylidene cyanide and vinyl acetate with conducting ions (T. Furukawa, M. Date, K. Nakajima, T. Kosaka and I. Seo, Jpn. J. Appl. Phys., 1986, 25, 1178) and a wet membrane of dense cellulose acetate (I. W. Plesner, B. Malmgren-Hansen and T. S. Sørensen, J. Chem. Soc., Faraday Trans., 1994, 90, 2381). The use of the so-called ‘constant-phase element’ is discussed and criticised. The constant-phase element leads to infinite dissipation at zero frequency in contrast with the generalised Trukhan theory (and common sense). The constant-phase element is just superficially fitting the data in a limited range of frequency and no physical information (e.g. diffusion coefficients) is obtained from the fitted parameters of such an element.