Current efficiency of an ion-selective membrane: Effect of fixed charge distribution

The effect of fixed charge distribution, C-3, in an ion selective membrane on its current efficiency eta(1) is investigated. An intuitive parameter M-2, which is a measure for the nature of C-3, is proposed, and three families df fixed charge distribution, including Linear, exponential, and trigonometric, are examined. Under the condition that the total amount of fixed charges is constant, M-2 has the positive maximum M-2max if all fixed charges are near the right membrane-liquid interface, and the negative minimum M-2min if all fixed charges are near the left membrane-liquid interface. Any other type of fixed charge distribution has an M-2 value between these two extremes, and if C-3 is uniform, M-2 = 0. It is found that eta is closely related to M-2. In other words, the latter can be used to predict the former. We show that: (i) the value of eta for M-2 < 0 is smaller than that for M-2 = 0; (ii) if M-2 > 0, the variation of eta as a function of M-2 has a maximum; (iii) if M-2 < 0, eta decreases with the increase in current density and reverse is true for M-2 > 0, and (iv) if M-2 = 0, eta is independent of applied current density.