Interaction Force and Isodynamic Curves

Consider two parallel planar dissimilar ion-penetrable membranes 1 and 2 at separation h immersed in a solution containing a symmetrical electrolyte of valence z and bulk concentration n. We take an x-axis as shown in Fig. 13.2 [7-9]. We denote by Ni and Zi, respectively, the density and valence of charged groups in membrane 1 and by N2 and Z2 the corresponding quantities of membrane 2. Without loss of generality we may assume that Zj 0 and Z2 may be either positive or negative and that Eq. (13.1) holds. The Poisson-Boltzmann equations (13.2)-(13.4) for the potential distribution j/(x) are rewritten in terms of the scaled potential y = zeif/IkT as 

Equations (13.67) and (13.68) correspond to the assumption that the potential far inside the membrane is always equal to the Donnan potential. Expressions for yooNi and yDON2 can be derived by setting the right-hand sides of Eqs. (13.59) and (13.61) equal to zero, namely, 

To obtain a relationship between C and h, one must further integrate Eq. (13.73). If y x) passes through a minimum y at some point x = x (0 x h), then further integration of Eq. (13.73) for 0 x Xi yields 

By evaluating integrals (13.75) and (13.76) atx = Xm and x = h, we can obtain relationships between C and h, as follows. If there exists a potential minimum, then 

DONNAN POTENTIAL-REGULATED INTERACTION BETWEEN POROUS PARTICLES 

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