equilibrium with the ion concentrations at the outer boundary A quantitative theory for dynamic surface tension has been deof the double layer. This model is discussed in Appendix A. veloped for the diffusion-controlled and diffusion-convective-con-Borwanker and Wasan (31) use the isotherm of Davies (53trolled models without any ''quasi-equilibrium'' hypotheses. A rig-55) to consider the diffusion-controlled transport of ionic orous model considers both the diffusion and the migration of surfactant to a double layer in local quasi-equilibrium with surfactant and electrolytes in the electrical field that develops as the surfactant and the kinetic-controlled transport of an ionic the charged surfactant adsorbs at a planar interface. The surface surfactant to the interface. In fact, the authors (31) extended concentration of surfactant is calculated by using the coupled Poisthe model of Ward and Tordai to describe transport of an son and Nernst-Planck equations. An unknown electrical potenionic surfactant. In recent papers (46, 51) a model for transtial in the Nernst-Planck equation is found from the integral form of the charge-balance equation. It is shown that the proposed port of an ionic surfactant, counterions, and background elecsystem of equations describes over a wide range of time the nontrolytes takes into account the effects of the surface charge steady-state process in the bulk for both ionic surfactants and accumulation at an interface. The model (46, 51) considers electrolytes. The relaxation equation of F( g(t)) ร
log[(g 0 0 the diffusion and migration of surfactant and ions in the g(t))/(g(t) 0 g e )] ร
n log(t/t rel ) is suitable to describe the dybuilding electric field that develops as surface charge at the namic surface tension process over a wide range of times for diffuinterface. On the basis of a numerical scheme, the authors sion-controlled and diffusion-convective-controlled adsorption. develop solutions for the coupled Nernst-Planck and Pois-The simple formula is derived to calculate the parameters of n, t rel son equations. The electrical migration in the Nernst-Planck (the relaxation time), and D eff (the effective diffusion coefficients) equation is described by using the electrical potential, f r from the experimental data represented in the form of the relax-(see Eq. [B1]) due to the electrical charge in the double ation function F(t) versus log( t). The analytical expressions are layer. However, as shown in the present paper, an unknown obtained to describe the dynamic surface tension for short and long times by using the diffusion-controlled and diffusion-convective-electrical potential, f(x, t), in the Nernst-Planck equation controlled adsorption obeying linear, rectangular, and arbitrary (see Eq. [10b]) does not equal the electrical potential, f r , adsorption isotherms.