Study on the electric double layer of a cylindrical reverse micelle with functional theoretical approach

Abstract

The iterative method in functional analysis is applied to looking for a solution of the Poisson-Boltzmann equation in order to describe the problems of the distribution of the potentials in the electric double layer (EDL) inside the water pool for a cylindrical inverse micelle. Potentials as a function of the position of a particular point in EDL are computed, which display a quantitative agreement with those from earlier calculation of Debye and Hückel in the case of low potentials. But it is also shown that in the higher-potential range the iterative calculations can give more accurate results. These results indicate the utility of this functional analysis technique in the description of the properties of EDL for a cylindrical inverse micelle.

Closed-form solution of mid-potential between two parallel charged plates with more extensive application

“…It has been shown in previous studies [31,32] that the iterative method in functional theory can be used to obtain the analytical solution of Eq. ( 1) since the PB equation satisfies norm axioms of the functional theory and Lipschitz condition.…”

“…( 6) and the first-order iterative analytical solution (see Eq. ( 7)) presented by Wang et al, [32] the second-order iterative analytical solution can be obtained as Eq. ( 8).…”

“…( 7) and ( 8) for varying φ 0 and κh from 1 to 10 and to 10 respectively. Wang et al [31,32] stated that the iterative method only needs a first or second-order iteration to find an analytical solution without any restriction. However, it can be seen from Fig.…”

“…In previous decades, many studies on analytical solution of PB equation were carried out. [23][24][25][26][27][28][29][30][31][32][33][34][35] For two infinite parallel charged plates, Luo et al [25,26] obtained the analytical solutions of the PB equation on the assumption that φ 0 is much higher than 1 or κh is much less than 1. Assuming that the dimensionless surface potential is very close to the mid-plane potential, Wang et al [27] presented an analytical solution.…”

“…Assuming that the dimensionless surface potential is very close to the mid-plane potential, Wang et al [27] presented an analytical solution. Wang et al [31,32] employed the iterative method in functional theory to obtain analytical solutions of the PB equation, and stated that this approach can be used to construct an accurate and simple analytical expression at any potential. Xing [33] solved the nonlinear PB equation by introducing Weierstrass elliptic functions, and obtained the midplane potential analytical expressions which are applicable to small and large interparticle distance at low surface potential, and so on.…”