A numeric solution to the problem

A numeric solution of a model for diffusive transport of ionic surfactants to an adsorbing interface was recently proposed by MacLeod Radke (1994). The model considers both the diffusion and migration of surfactant, counterions, and background electrolyte in the electric field that develops as the charged surfactant adsorbs at the interface. 

The transport of the three charged species (1 - surfactant, 2 - counterion, 3 - coion) with valence Zj under the effect of an electric potential is given by Eq. (7.75), which reads for the present situation. 

The electrical potential is related to the ion distribution through the Poisson equation 

The initial condition and the boundary conditions far from the interface are, 

As interrelation between potential and surface concentration it is assumed that both the gradient in potential and the surfactant surface concentration are proportional the total surface charge, which gives the boundary condition. 

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From the basic parameters initial concentration of ions, their standard transfer potential, distribution coefficients for neutral components, equilibrium constants of reactions taking place in the system, volume of phases, and temperature, a unique general problem for the Galvani potential difference and distribution concentration of all components was established. A numerical solution to the problem with the help of computer program EXTRA.FIFIl provided a good means for quantitative investigation of the liquid-liquid interface. It is also useful for the study of liquid-liquid extraction, electroextraction, voltammetry at interface of two immiscible electrolyte solutions (ITIES) [15,18], liquid-liquid membrane ion-selective electrodes, biomembrane transport, and other fields of science and engineering. 

Perry, Newman and Cairns (Perry et al., 1998) obtained a numerical solution to the problem of CCL performance and provided the asymptotic analytical solutions for large and small cell current densities jo- Eikerling and Kornyshev (Eikerling and Kornyshev, 1998) derived the explicit analytical solution to the system for the case of small overpotentials in the general case they reported numerical results. 

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