Glenn M. Torrie

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.

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.

Stochastic trajectory simulation of iodine recombination in liquids

A stochastic trajectory simulation of iodine recombination in dense liquid solvents is presented. The calculations utilize a mean force potential which contains direct I–I interactions as well as solvent structure effects. Dynamical solvent effects are accounted for by a random force and friction coefficient. The time dependent probability of reaction for two initially separated radicals is determined. The choice of initial separations and atomic velocity distributions is appropriate for secondary recombination. The results of this study show the importance of including the strong direct chemical forces between the I atoms; the validity of simple diffusion equation approaches can thus be assessed. Effects due to solvent structure are quantitatively examined and are interpreted in terms of ’’caging’’ in dense fluids. The computer simulation results are also compared with the solution of the Smoluchowski equation for this problem and effects due to friction coefficient variation are discussed.

Self‐avoiding walks interacting with an interface

This paper is concerned with self‐avoiding walks with at least one vertex in an interfacial plane and interacting with the interface via a short range potential. We present some rigorous results on the form of the partition function and examine exact enumeration data for several lattices in an attempt to locate the point at which adsorption sets in.

Excluded volume effects in the stabilization of colloids by polymers

A self‐avoiding walk on a lattice, confined between two planes, is examined as a model of the stabilization of a colloidal system by adsorbed polymer. If the walk interacts with neither plane, there is a repulsive force on the planes due to the decrease in the volume of configuration space available to the walk. This effect is larger for a self‐avoiding walk than for a random walk. If the walk interacts with only one of the planes, with an attractive potential, there is a decrease in the magnitude of the stabilizing effect as the temperature decreases, but the force on the planes remains repulsive. If the walk interacts with both planes, with an attractive potential, there is a stabilizing effect at high temperatures but a destabilization (sensitized flocculation) at low temperatures.

Asymptotic behavior of the mean‐square lengths of self‐avoiding walks terminally attached to a surface

We present some exact enumeration data on the mean‐square lengths of self‐avoiding walks on a lattice confined to a half‐space (or half‐plane) with the walk originating on the dividing plane (or line). We argue that the exponent characterising the asymptotic behavior is unchanged by this spatial constraint.

Branched polymers with a prescribed number of cycles

The authors consider the number of lattice animals with exactly c cycles. For weakly embeddable (bond) clusters they show rigorously that the growth constant is independent of c and they derive upper and lower bounds on the critical exponents for each value of c. They use series analysis methods to estimate the critical exponents for c=1 and 2 in two and three dimensions and find that the critical exponent does depend on the number of cycles. Evidence from series analysis results and from expansions in inverse powers of the dimension suggests that, in the case of strong embeddings (i.e. site clusters) the growth constant is independent of c and the corresponding values of the critical exponents are identical to the values for weak embeddings (i.e. bond clusters). They discuss the relationship of these results to the field theory prediction that the critical exponent is independent of cycle fugacity.

Exact enumeration of neighbour-avoiding walks on the tetrahedral and body-centred cubic lattices

The numbers and mean square lengths of short, neighbour-avoiding walks on the tetrahedral and body-centred cubic lattices have been determined exactly. Using standard extrapolation techniques, estimates have been made of the connective constants and mean square length exponents for these walks. The estimate of the mean square length exponent is 1.22, but a value of 1.20 also appears plausible.

The surface susceptibility exponent for the polymer problem

Series analysis methods are used to estimate the surface susceptibility exponent for the polymer analogue of a magnet with a free surface, on the square and simple cubic lattices. Our exponent estimates are slightly larger than the surface scaling predictions for both lattices.