Two Parallel Ion-Penetrable Membranes

In the opposite limit of large Kh, it follows from Eq. (18.14) that 

Now consider two parallel identical ion-penetrable membranes 1 and 2 at separation h immersed in a salt-free medium containing only counterions. Each membrane is fixed on a planar uncharged substrates (Fig. 18.2). We obtain the electric potential distribution i/ (x). We assume that fixed charges of valence Z are distributed in the membrane of thickness d with a number density of A (m ) so that the fixed-charge density pgx within the membrane is given by 

We take an x-axis perpendicular to the membranes with its origin on the surface of membrane 1. Let the average number density and the valence of counterions be and —V, respectively. Then we have from electroneutrality condition that 

FIGURE 18.2 Schematic representation of the electrostatic interaction between two parallel identical ion-penetrable membranes (porous plates) of thickness d separated by a distance h between their surfaces. Each membrane is fixed on an uncharged planar substrate. 

Note that becomes proportional to h for h d. We set the equilibrium electric potential j/ x) to zero at points where the volume charge density PeiW resulting from counterions equals its average value —zen). 

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