Electrical Response of Materials Containing Space Charge with Discharge at the Electrodes

The system considered consists of a solid or liquid material containing mobile positive and negative charges between two identical plane electrodes separated by a distance $l$⁠. The results obtained apply also for a single working electrode, without specific adsorption, and an indifferent electrode. Uni‐univalent, nonrecombining positive and negative charges, usually of equal mobility, are assumed. Frequency and transient responses are compared in the linear regime for two different boundary conditions. One condition requires complete blocking of positive and negative charges at the electrodes; the other requires complete blocking for charges of one sign but allows free discharge of those of opposite sign. The working electrode is thus reversible, or Ohmic, for charges of one sign. The method of solution includes electromigration effects, does not require electroneutrality anywhere, and leads to results satisfying Poisson's equation exactly everywhere for all times and frequencies. These results apply to solids and liquids with either electronic or ionic conduction. They are relevant to low‐conductivity liquids, as in Kerr and liquid‐crystal‐display cells, to ordinary solid and liquid electrolytes, and at least to some extent to fused salts. Although the frequency and transient responses involve continuous distributions of relaxation times, three main relaxation times, $τD, MτD$⁠, and $M2τD$⁠, occur. Here $τD$ is the dielectric relaxation time of the charge‐containing material and $M ≡ l / 2LD$⁠, where $LD$ is the appropriate Debye length. Results are obtained for any value of $M$ but are summarized here for the usual experimental condition $M ≫ 1$⁠. At high normalized frequencies, $Ω ≡ ωτD ≫ 1$⁠, the system parallel capacitance associated with mobile charge is proportional to $ω−3/2$ for both blocking and discharge conditions. The series capacitance is proportional to $ω−1/2$ for these conditions. It is shown, for the first time in the completely blocking situation, that this high‐frequency response is associated with a Warburg, or diffusion, impedance. An equivalent circuit is obtained for the discharge case with a maximum of frequency‐independent elements and a minimum frequency dependence of remaining elements. It involves a dc path and two distinct Warburg impedances, occurring in different frequency ranges. Cole–Cole plots of the effective complex dielectric constant associated with the “interface” elements of the equivalent circuit yield, for both boundary conditions, curves very close to that arising from a Davidson–Cole continuous distribution of relaxation times with distribution parameter $β = 0.5$⁠. In the limit of large $M$⁠, blocking frequency response becomes identical with such Davidson–Cole response. Interface transient response is found in both cases to involve regions where the current decay is proportional to $t−1/2$⁠, but that in the discharge case can occur over a very long time range extending to $t ∼ 5 × 10−4 M2τD$⁠. For large $M$⁠, this limiting time may be measured in days or months. The long‐time limit of the transient charge is finite and consistent with the low‐frequency limiting capacitance of the system. Although the $Ω → 0$ space‐charge capacitance in the blocking case is independent of $l$⁠, the corresponding dominant capacitance in the discharge case is extrinsic and directly proportional to $l$⁠, making potentiostatic measurements inapplicable at very long times. It is about $M / 12$ times larger than the blocking‐case capacitance and arises from charge diffusion (not space charge) in a finite length, the entire bulk region. Tremendously large capacitances may thus appear at ultra‐low frequencies since $M$ may often be as large as 10⁶. Some applications and limitations of the present results to electrolyte situations are discussed, and the important effects of a finite $l$ are emphasized. Faradaic and non‐Faradaic processes do not separate clearly in the present discharge case, but it is suggested that when an excess of indifferent electrolyte is present in addition to the discharging ion such separation may be a good approximation. An approximate equivalent circuit is proposed for this situation which differs in important ways from those heretofore employed by electrochemists.