John P. Valleau

Monte Carlo free energy estimates using non-Boltzmann sampling: Application to the sub-critical Lennard-Jones fluid

Abstract The paper reports a Monte Carlo technique for estimation of the free energies of fluids by sampling on distributions designed for this purpose, rather than on the usual Boltzmann distribution. As an illustration of its use, the free energy of a Lennard-Jones fluid in the liquid-vapour coexistence region has been estimated by relating it to that of the inverse-twelve (soft sphere) fluid, which itself shows no condensation.

Electrical double layers. I. Monte Carlo study of a uniformly charged surface

This paper reports grand canonical Monte Carlo computations on the diffuse double layer in an ionic system next to a uniformly charged plane surface. The boundary conditions and the grand canonical techniques are discussed. Calculations were carried out for the 1:1 restricted primitive model at several concentrations and over a range of surface charge densities. The results are compared with the modified Gouy–Chapman theory, and some remarks are also possible with respect to the modified Poisson–Boltzmann and the hypernetted chain approaches. At high concentrations and surface charge densities the counterions are packed closely at the surface and begin to show a layered structure. This results in a large electrostatic potential drop, but only very slight charge oscillations are observed in the solution. None of the theories seems able to describe this behavior.

Monte Carlo Study of the Thermodynamics of Electrolyte Solutions

Monte Carlo techniques have been used to study the primitive model of an ionic solution. The parameters chosen correspond to a simple 1:1 electrolyte dissolved in water at 25°C. Data are derived for the pair distribution functions, excess internal energy and heat capacity, osmotic coefficient, and mean activity coefficient, in the range 0.01–2M. The results are compared with a nonlinear form of the Debye–Huckel theory and with integral equation results in the Percus–Yevick and hypernetted chain approximations. The last appears to be excellent up to 1M.

A Monte Carlo method for obtaining the interionic potential of mean force in ionic solution

A Monte Carlo method for obtaining the solvent‐averaged interionic potential of mean force is described. The potential of mean force is obtained for two charged hard spheres immersed in a dipolar hard sphere solvent. The ions and solvent particles have the same hard sphere diameters and the ions bear single charges of opposite sign. The solvent particles are characterized by a reduced dipole moment μ*=1.0 which corresponds to a dielectric constant e?7.8. The Monte Carlo result is compared with the ’’primitive model’’ and other approximations which have been suggested for the potential of mean force. These approximations are all found to be inadequate for small ionic separations. Some implications with respect to ’’ion pairing’’ are mentioned.

Primitive model electrolytes. I. Grand canonical Monte Carlo computations

Monte Carlo calculations in the grand canonical ensemble are described for coulombic systems, and carried out for 1:1, 2:2, 2:1, and 3:1 aqueous electrolytes in the primitive model with equal ion sizes. Energies and activity coefficients are obtained, and the scope and reliability of the method is discussed.

Monte Carlo Estimation of the Free Energy by Multistage Sampling

A method is described for estimating the free energy and entropy of an assembly of particles. This is done by using Metropolis Monte Carlo techniques to generate energy distributions from which we may calculate the absolute volume of configuration space corresponding to a given energy, and thus the configuration integral. One incidentally obtains the thermodynamic quantities over a wide range of reduced temperature. It is particularly easy to apply the method to particles having hard cores, and calculations are reported for hard spheres with Coulombic forces.

Colloid stability: The forces between charged surfaces in an electrolyte

Monte Carlo methods are presented for the evaluation of the various components of the force between parallel charged surfaces due to the presence between them of an electrolyte, represented by the restricted primitive model. This is a significant problem because some recent theoretical results, and computations on even simpler models, have put in question the adequacy of traditional Derjaguin–Landau–Verwey–Overbeek (DLVO) theory in offering a basic understanding of colloid stability. The methods are applied principally to 2:2 electrolytes at a variety of surface charge densities. We find, in contrast to DLVO theory, that the force between the surfaces resulting from the electrolyte is almost everywhere attractive; in detail these forces turn out to have a remarkably rich behavior resulting from the interaction of their various contributions. The results bolster the perception that we need to take a new look at some basics of colloid science.

Electrical double layers. II. Monte Carlo and HNC studies of image effects

The effect of surface polarization (i.e., image forces) on the properties of electrical double layers is studied by means of Monte Carlo calculations on a primitive model electrolyte next to a planar charged surface bounding a semi‐infinite dielectric. Two cases are considered, that of a conducting material for which an ion is attracted by its own image and that of an insulator for which the self‐image force is repulsive. At low surface charge densities the image forces cause quite dramatic changes in the ionic densities near the wall; the effect on the electrostatic potential is small but increases with surface charge density. The modified Poisson–Boltzmann theory of Outhwaite is quite successful in describing the Monte Carlo results for the range of parameters studied. A screened self‐image model of image effects is also considered for which both Monte Carlo calculations and numerical solution of the HNC equation have been obtained.

Monte Carlo study of a phase‐separating liquid mixture by umbrella sampling

Recently developed nonphysical sampling methods—umbrella sampling—have been used to obtain the free energy and other properties of a binary liquid mixture exhibiting phase separation with an upper critical solution point. The system is a mixture of two identical Lennard‐Jones liquids in which the interactions between the components are characterized by a range parameter σ12 obeying the Lorentz rule, but an energy parameter e12= (1−α) (e11e22)1/2 violating the Berthelot rule. The sampling methods enable one to cover wide ranges of temperature and composition with few Monte Carlo experiments, successfully sampling the metastable regions and obtaining the coexistence curve. The case α=0.25 is studied in detail, and compared with some theoretical predictions. An estimate is made of the minimum value of α required for phase separation at low pressures.

A Monte Carlo study of an electrical double layer

Abstract Results of Monte Carlo experiments on a model of an electrical double layer are reported. The Gouy-Chapman theory as modified by Stern is found to work surprisingly well.