Disjoining pressure of thin films stabilized by nonionic surfactants.
In this article an attempt is made to derive a comprehensive theory of the disjoining pressure of thin liquid films, stabilized by low molecular nonionic surfactants. We accounted for effects playing a role in the case of surfactants with spherical hydrophilic heads: (i) The thermal fluctuations of the adsorbed surfactant molecules, due to the fact that the energy of adsorption of a -CH(2)- group is approximately equal to the average thermal energy k(B)T; (ii) The contribution of the collisions between molecules adsorbed on different surfaces; (iii) The restriction imposed on the fluctuation of the molecules by the presence of a second surface situated at a small distance h from the interface where the molecules are adsorbed; (iv) The volume of the hydrophilic heads, which expels part of the water molecules from the film region; (v) The equilibrium between the molecules adsorbed at the film surfaces and at the menisci surrounding the film. The adsorption on the film surfaces has two main effects. First, the concentration of solute inside the film region becomes larger than in the bulk solution and this will push the solvent toward the film thus creating an osmotic pressure (the disjoining pressure), which tends to increase the film thickness. Second, the higher concentration inside the film and the collisions between the polar heads lead to higher chemical potential, which pushes the surfactant toward the meniscus. We treated these effects by modifying adequately the Hildebrand-Scatchard theory for the osmotic pressure of concentrated solutions. The partition function of the surfactant, needed for this calculation, was found by deriving an expression for the configurational integral, based on virial expansion. The surface equations of state of Helfand, Frisch and Lebowitz and Volmer were critically analyzed and then generalized, by using the partition function obtained by virial expansion, to permit the derivation of partition functions of the surfactant molecules in the film. A simple thermodynamic approach was developed and applied to derive expressions for the disjoining pressure, Pi, and the chemical potential of the surfactant molecules in the film, mu. They were used to calculate numerically Pi and mu and analyze their dependence on the film thickness h and the surface coverage theta. It turned out that Pi has completely different behavior above and below h=2d, where d is the diameter of the hydrophilic head. For thick films, with h>2d, the decay of Pi is initially exponential (due mainly to the thermal fluctuations of the adsorbed molecules), followed by a long tail, proportional to h(-2), due to the contribution of the osmotic pressure of the displaced solvent molecules. At h<2d the collisions between the molecules adsorbed at different surfaces are hindered, which leads to a steady decrease of the contribution due the interaction between the molecules. The overall result of these effects is the appearance of a maximum of Pi at h=2d. It is very large (it may reach 1000 atm and even more) and depends strongly on the surfactant adsorption. To facilitate the application and the analysis of the theory, we derived several simpler asymptotic expressions. One of them is virial expansion, which is valid for small surface coverage and has the advantage of being independent of the adsorption model. The other asymptotic expression is applicable at h>2d, which is the region where the stabilization of the film occurs. We compared our theory with the simpler theory of Israelachvili and Wennerström. It turned out that while both theories lead to decay of Pi vs. h, the numerical results and the shape of the curves are usually very different. The experimental data, which could be used to verify our theory, are scarce, but we found reasonable agreement with the data of Lyle and Tiddy for bilayers of C(12)EO(4). The data of Parsegian et al. for lipid bilayers also confirmed qualitatively some of our theoretical conclusions.