Surfactants, interfaces and pores : a theoretical study

The aim of this study was to investigate the behavior of surfactants in porous media by theoretical means. The influence of curvature of a surface on the adsorption has been studied with a mean field lattice (MFL) model, as developed by Scheutjens and Fleer. An analytical theory has been developed to interpret the MFL results. The chapters three and four, which form the core of this thesis, have been devoted to the background and the outcomes of both theories. These theories contain various approximations and therefore limitations. In the flanking chapters two and five attempts to overcome two of these approximations have been described. First, the MFL theory considers water as built up from isotropic monomers. As a consequence, the theory cannot predict the characteristic behavior of water. An alternative model could be the Besseling theory, which is based on the quasi- chemical approach. Some elaborations of this water model have been reported in chapter two. Second, the MFL theory, as used in the chapters three and four, always assumes homogeneous surfactant layers, which is inherent to its mean field approximation. However, it is well known that adsorption layers of surfactants often consist of discrete aggregates. The Herzfeld model has been chosen to study the discrete nature of the adsorbed layer. The theory and its outcomes, dealing with aggregation and ordering behavior of surfactant aggregates at interfaces, have been described in the last chapter.In chapter two a lattice model for water, developed by Besseling, has been extended by incorporation of the electrostatic interactions of the water molecules with each other and with an external electrostatic field. This could have been the first step towards a better description of water near charged interfaces or in charged pores and electrosorption phenomena. The waterwater and water-field interactions have been treated with the reaction field approach of Onsager. Expressions have been obtained for the dielectric constant of the water in an external field.At low field strengths, the predicted permittivity is close to the experimental one. The temperature dependence has also been reproduced. The dipolar correlation factor has been obtained by using the Clausius-Mossoti equation for the refractive index and the Kirkwood-Fröhlich expression for the dielectric constant. The predicted correlation factor and its temperaturedependence agree well with experimental data. However, the field strength behavior of the model is unexpected. Before saturation the predicted permittivity passes through a maximum. By modifying the hydrogen-bond interaction the saturation could be manipulated. However, this latter procedure does not have a sound physical origin. Somehow, if water molecules orient in large amounts, their interactions change. Therefore, investigations with this theory were not continued and much more simple models for water had to be chosen to study surfactant adsorption.An analytical theory for nonionic surfactants in hydrophilic cylindrical pores has been developed in chapter three. The adsorption has been approximated with a phase transition model. Above a certain surfactant concentration a monolayer of isolated molecules converts into a bilayer. With the help the thermodynamics of phase transitions, the surfactant chemical potential at phase transition could be related to the curvature of the pore. The shift in this chemical potential due to the curvature is in first approximation proportional to the curvature constant of the bilayer. A molecular model, mean field type has been used to interpret this curvature energy. Both the curvature energy and the surface tension can be calculated from the excess grand potential density profile. The curvature constant has been calculated from the profile of a flat layer, which is allowed as long as this profile is not very sensitive to the curvature. An equation, which relates the chemical potential at phase transition, the curvature, the structure of the layer and the affinity, has been derived.Our model predicts that the chemical potential of phase transition decreases with decreasing pore radius. The adsorbed bilayer becomes more stable when the pore - radius decreases. Experiments confirmed these trends. If the affinity of the adsorbed layer for the surface increases, the curvature influence on the chemical potential of phase transition increases.To test the analytical theory and to obtain generic knowledge about the influence of curvature on surfactant adsorption, MFL calculations have been performed, which have been described in chapter four. Contrary to the analytical model, the MFL theory allows changes in the structure of the adsorbed layer with curvature.The position of the phase transition in a curved system has been calculated as a function of the adsorption energy and the size of the tails and the headgroups. No matter what parameters were used, the MFL calculations always predicted that the surfactant chemical potential of phase transition decreases with decreasing pore radius, which confirmed the outcome of the analytical model. Especially the adsorption energy turned out to have a strong influence on the sensitivity of phase transition for the curvature. The shift in the chemical potential of the phase transition becomes stronger with increasing adsorption energy, as has also been predicted with the analytical theory.The most important approximation of the analytical theory, which has been tested with MFL calculations, is the curvature independency of the structure of the adsorbed layer. The surface tension of an adsorbed bilayer has been calculated as a function of the curvature for different chemical potentials. The curvature constant has been obtained as a function of the chemical potential by fitting these curves and calculating it from the excess grand potential density profile of a flat layer. It turned out that the last procedure, which is also used in the analytical theory, underestimates the value of the curvature constant. As long as this constant has a considerable value, the error made by this procedure does not effect the essential physics. Therefore it may be concluded that the analytical theory still captures the important physics despite its severe assumptions.Both the analytical theory and the MFL model neglect the existence of discrete surfactant aggregates at the surface. To remedy this shortcoming, in chapter 5 the Herzfeld model has been applied to the adsorption of rod-like aggregates at a solid- water interface. The adsorbed layer was represented as a collection of rod-like polydisperse particles embedded in a monolayer of surfactants. An equilibrium condition has been derived, stating that the intrinsic excess grand potential of a rod of a given length plus its hard rod chemical potential has to be zero. The intrinsic excess grand potential has been divided into cap and body contributions, which are in principle the only two input parameters of the model. To calculate the hard rod chemical potential, the Herzfeld lattice model has been used. Rods are represented as rectangles. These rectangles have been placed on a square lattice. As a consequence the number of possible orientations of a rod is two.By combining the Herzfeld model and the equilibrium condition, expressions for the length distributions in both directions have been obtained. With these distributions expressions were derived for the total number of rods, the average aspect ratio and the standard deviation of this aspect ratio, all in both directions. The distributions have exponential forms, with decay parameters equal to the average aspect ratio minus one and the standard deviation of the aspect ratio. The close relationship between the average aspect ratio and its standard deviation has made clear that size fluctuations are very important in systems with large rods.Calculations have been performed on isotropic systems. Adsorption isotherms have been calculated for different cap Helmholtz energies. These isotherms show that the surfactants already adsorb in large amounts when the aggregates as such are not stable. However the collection of aggregates is stabilized by the translational entropy. If the caps become more unfavorable the co-operativity of the adsorption increases, because the length of the adsorbed aggregates increases. Adsorption isotherms have also been obtained for systems, which are allowed to order. It has been shown that at a certain surfactant chemical potential a second order isotropic- nematic phase transition occurs. After this transition the growth of the aggregates is promoted in the direction of alignment and inhibited in the direction perpendicular to that.The isotropic-nematic transition line has been calculated. It turned out that ordering can take place at much lower surface densities of rods when the average aspect ratio increases. The cap fraction is inversely proportional to the average aspect ratio minus one to the power two at the transition line. Although this expression could not be derived, the close resemblance with an equation, derived for monodisperse rods, confirmed that it is an exact outcome of the model. With the formula found, a relation between surface density at the transition line and the Helmholtz energy of the end caps has been derived, which showed that the nernatic ordering takes place at lower rod densities when the caps become more unfavorable.In the end of my study it has become more and more evident that the phase behavior of surfactants at interfaces should be at least as rich as in solution. As experimental techniques to investigate the structure of micelles at a surface are becoming available (e.g. AFM), we expect that the possible morphologies will become known to us in increasing detail in the near future. This thesis may assist the explorations, directed to fill the gaps in our knowledge of the behavior of surfactants at the solid-water interface, and thus allows us to use surfactants more effective in its applications.