The interaction between charged surfaces in 1:1, 1:2, and 2:2 electrolyte solutions at various concentrations have been calculated using the anisotropic hypernetted chain (HNC) theory and Monte Carlo (MC) simulations. For divalent counterions, the surface interaction has an attractive minimum at short separations. This minimum turns more attractive at increasing electrolyte concentration, while the interaction at somewhat larger separations becomes oscillatory. The agreement between the HNC and the MC results is excellent for the 1:2 and 2:2 electrolyte systems. For 1:1 electrolytes the surface interaction is repulsive except at high concentrations at which it shows a weak attractive minimum. The HNC and the MC results agree quantitatively except in a few angstroms wide region at short separations, where the agreement is only qualitative due to a slight difference in the contributions from the hard core interactions. This is a consequence of the neglect of the short-range bridge function in the HNC approximation.

An electric double layer is studied by means of Monte Carlo simulations and mean-field theory. The counterions of the uniformly charged surfaces are modeled as flexible polyelectrolytes. For this particular model system it turns out that the traditional double layer repulsion becomes attractive for a wide range of systems. The main reason for this attraction is an entropically driven bridging mechanism, and its magnitude is significant compared to ordinary double layer or van der Waals forces. The polyelectrolyte Poisson–Boltzmann theory developed here behaves in a qualitatively correct manner, also predicting an attractive interaction extending over several nanometers. These results may have some relevance to technical and biological systems, where sometimes puzzling force behavior is seen in the presence of polyelectrolytes.

Ca/Na montmorillonite and natural Wyoming bentonite (MX-80) have been studied experimentally and theoretically. For a clay system in equilibrium with pure water, Monte Carlo simulations predict a large swelling when the clay counterions are monovalent, while in presence of divalent counterions a limited swelling is obtained with an aqueous layer between the clay platelets of about 10 A. This latter result is in excellent agreement with X-ray scattering data, while dialysis experiments give a significantly larger swelling for Ca montmorillonite in pure water. Obviously, there is one "intra-lamellar" and a second "extra-lamellar" swelling. Montmorillonite in contact with a salt reservoir containing both Na(+) and Ca(2+) counterions will only show a modest swelling unless the Na(+) concentration in the bulk is several orders of magnitude larger than the Ca(2+) concentration. The limited swelling of clay in presence of divalent counterions is a consequence of ion-ion correlations, which reduce the entropic repulsion as well as give rise to an attractive component in the total osmotic pressure. Ion-ion correlations also favor divalent counterions in a situation with a competition with monovalent ones. A more fundamental result of ion-ion correlations is that the osmotic pressure as a function of clay sheet separation becomes nonmonotonic, which indicates the possibility of a phase separation into a concentrated and a dilute clay phase, which would correspond to the "extra-lamellar" swelling found in dialysis experiments. This idea also finds support in the X-ray scattering spectra, where sometimes two peaks corresponding to different lamellar spacings appear.

Both natural and synthetic polyelectrolytes form strong complexes with a variety of proteins. One peculiar phenomenon is that association can take place even when the protein and the polyelectrolyte carry the same charge. This has been interpreted as if the ion-dipole interaction can overcome the repulsive ion-ion interaction. On the basis of Monte Carlo simulations and perturbation theory, we propose a different explanation for the association, namely, charge regulation. We have investigated three different protein-polymer complexes and found that the induced ionization of amino acid residues due to the polyelectrolyte leads to a surprisingly strong attractive interaction between the protein and the polymer. The extra attraction from this charge-induced charge interaction can be several kT and is for the three cases studied here, lysozyme, alpha-lactalbumin, and beta-lactoglobulin, of the same magnitude or stronger than the ion-dipole interaction. The magnitude of the induced charge is governed by a response function, the protein charge capacitance Z2-Z2. This fluctuation term can easily be calculated in a simulation or measured in a titration experiment.