Effect of a non-equilibrium adsorption layers

It is known that the surface energy depends not only on the composition of the surface layer, but also on that of the bulk phases [130]. To formulate the Gibbs law for the non-equilibrium chemical potential, additional so-called cross-chemical potentials (the partial derivatives of the surface free energy with respect to the component concentrations in the bulk phases) have been introduced. Rusanov and Prokhorov [131] derived the Gibbs equation and the expression for the free energy of the surface layer in terms of the ordinary chemical potentials by dividing the transition layer adjacent to the surface into n thin layers. For each layer an equilibrium state was assumed. The expression for surface energy was derived by the summation of the equilibrium equations over all these layers. Further, the expression for the additional contribution to the surface tension due to the non-equilibrium diffusion layer was derived in [48, 132] 

Here Ay is the non-equilibrium surface tension jump, x is the co-ordinate normal to the surface, s is the thickness of the layer assumed to be in a non-equilibrium adsorption state. 

An estimation of Ay obtained from Eq. (4.86) shows that it is essential to this non-equilibrium effect into account when Cq S-IO mol/cm, which coincides very good with the criterion for the non-diffusional adsorption kinetics in the Langmuir model discussed above. If we consider the von Szyszkowski-Langmuir equation, then the adsorption layer in equilibrium obeys 

Calculations show that the model of a non-equilibrium surface layer is an alternative to kinetic-controlled adsorption models. On the basis of the purely diffusion-controlled adsorption mechanism the proper consideration of a non-equilibrium diffusion layer leads to a satisfactory agreement between theory and experimental data for various studied systems, systematically demonstrated for the short-chain alcohols [132], The non-equilibrium model is applicable in the concentration range from 10 to 10 mol/cm at different values of the Langmuir constant at- For l 10 mol/cm a consideration of non-equilibrium layer effects is not necessary. For ai 10 mol/cm and large surfactant concentration the Ay values calculated from the proposed theory do not compensate the discrepancy to the experimental data so that other mechanisms have to be taken into account. An empirical formula also proposed in [132] for the estimation of the non-equilibrium surface layer thickness leads to a better agreement with experimental data, however this expression restricts the validity of the non-equilibrium surface layer model as alternative to non-diffusional adsorption kinetics. 

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