Electrostatic Contribution to the Stability and the Synergism of Ionic/Nonionic Mixed Micelles in Salt Solutions

Abstract

A phenomenological procedure is presented to analyze the stability and the synergism of ionic/nonionic mixed micelles in salt solutions under three assumptions: (A) The ionic strength is independent of the mole fraction of the ionic species in the micelle, x. (B) The surface charge density varies linearly with x. (C) The nonelectric contribution to the excess free energy per monomer, gex, is given by the regular solution theory (RST) gex,non = βnonx(1 − x), where βnon is a constant. When the electrostatic contribution to gex is written as gex,el = βel(x)x(1 − x), βel(x) is negative at all x but becomes less so as x increases. This feature is valid regardless of the shape of the micelles, whether spherical, rodlike, or platelike. The synergism due to the electrostatic contribution is shown to arise from the positive curvature of the electrostatic free energy per monomer, gel (i.e., d2gel(x)/dx2 > 0). In contrast, the electrostatic free energy per charge, f(x), has a negative curvature: d2f (x)/dx2 ≤ 0. It is found that f(x) can be approximated with a polynomial of the form A(x) = a1x + a2x2 + a3x3, again regardless of the micellar shape. The total excess free energy, gex, defined as (gex,non + gex,el), is found to be gex(x) = −{a1* + a2*(1 + x)}x(1 − x) = β(x)x(1 − x), where a1* = a1 − βnon and a2* = a2 + a3. For 11 mixed micelles, x, a1*, and a2* were determined experimentally using the reported relations between the critical micelle concentration (cmc) and the overall composition. Under the given assumptions, a1* and a2* thus obtained are expected to remain constant even at concentrations higher than the cmc, where the micelle composition is different. For most of the 11 mixed micelles examined, gex was not symmetric with respect to x = 0.5. Instead, the minimum was reached at x < 0.5 and was deeper than that estimated from the symmetric RST. A parameter B1 = ln(cmcI/cmcN) + β(x = 0), where cmcI and cmcN are the cmc values of the pure ionic and nonionic species, is introduced to assess the nonelectric synergism properly.