Zeta potential and discrete vs. uniform surface charges.

Dear Sir: There has been a wide spread use of surface charges, a, since Abramson and Muller (1933) first calculated them from the zeta potential, V, obtained from electrokinetic measurements. It has been convenient to express these results in terms of an average separation, b, between ionic charges, e, on the surface. The recent use of charges absorbed on cell membranes as significant factors in ion permeability behavior (Gilbert and Ehrenstein, 1969) has raised again the pragmatic question: Are the charge separations given by the assumption of uniformity useful approximations in cases for which the charges are assumed to be discrete? CALCULATIONS It is assumed that the net free charges are at the interface between aqueous and lipid phases with dielectric constants (a, Xm and Debye reciprocal lengths Ka, Km with K2 = 87rne2/EkT. Further sinh Ve/kT is approximated by Ve/kT to give the Poisson-Boltzman equations d2V/dx2 = KV (1) for the plane problem and Ild I2dV\ K 2v r2 drTr d-r)=K (2 in the spherical case. For uniform distribution with mKm << EaKa and in the aqueous phase V= Vexp(-KX) (3) gives V = 47ra/EK at the interface. Elementary charges with the spacing bo in a hexagonal array given an average charge density a-= 2e/V/3 bo2 and the zeta potential eK 87r 210 4 e K0 K2bo2 (4) The coefficient of eKIE is shown as a function of Kbo in Fig. 1, (0).