M. H. Kalos

A new algorithm for Monte Carlo simulation of Ising spin systems

We describe a new algorithm for Monte Carlo simulation of Ising spin systems and present results of a study comparing the speed of the new technique to that of a standard technique applied to a square lattice of 6400 spins evolving via single spin flips. We find that at temperatures T < Tc, the critical temperature, the new technique is faster than the standard technique, being ten times faster at T = 0.588 Tc. We expect that the new technique will be especially valuable in Monte Carlo simulation of the time evolution of binary alloy systems. The new algorithm is essentially a reorganization of the standard algorithm. It accounts for the a priori probability of changing spins before, rather than after, choosing the spin or spins to change.

Dynamical scaling of structure function in quenched binary alloys

Abstract We study the structure functions S(k, t) obtained from computer simulations of the time evolution of a segregating binary alloy following quenching into the miscibility gap. They are shown to have a simple scaling behavior, S(k, t) ∼ K−3(t) F(k/K(t)). The shape of the function F(x) depends somewhat on the part of the coexistence region into which the quench is made. Comparison with some recent experiments on quenched alloys is quite satisfactory. The time evolution of K−3(t) appears to be linear for late times, consistent with the Lifshitz-Slyozov theory.

“Critical clusters” in a supersaturated vapor: Theory and Monte Carlo simulation

A new thermodynamic analysis is given for the equilibrium between a liquid cluster and the surrounding supersaturated gas phase in afinite constant volume. It is shown that for constant total density and intermediate volume this equilibrium is stable, although it is unstable for very large volume. We show that observation of the critical cluster sizel* then yields information on the surface free energy of the liquid cluster. The accuracy of previous approximate prescriptions for obtaining the free energy of physical clusters is investigated. As an application, the theory is used to analyze Monte Carlo simulations of the two-dimensional lattice gas model at low temperatures. We obtain cluster surface area, diffusivity, and free energy for clusters with 26≥l≥500. It is found that the capillarity approximation is inaccurate forl≥100, but the free energy of small clusters ishigher than the result of classical nucleation theory, in contrast to what one expects from Tolman-like corrections. We interpret these results, deriving low-temperature series expansions for very small clusters, thus showing that the capillarity approximation both underestimates the surface energy and overestimates the surface entropy of very small clusters. Finally, we use our results to give a speculative explanation of recent nucleation experiments. The dependence of the cluster diffusivity on cluster size is tentatively explained in terms of a crossover between two mechanisms yielding different power laws.

Structure of a Liquid-Vapor Interface

The structure of the interface of an argonlike fluid in equilibrium with its own vapor at low temperature is studied using molecular dynamics. The longitudinal pair correlations in the interface are found to be consistent with a simply defined ensemble of local thermodynamic states. However, the transverse correlations exhibit very long-range behavior not predicted by straightforward local thermodynamics. These results strongly suggest that the interface is made up of an ensemble of configurations in each of which the transition from liquid to vapor is locally sharp, but that the transition surface fluctuates strongly in space and time.

THE INTERPRETATION OF STRUCTURE FUNCTIONS IN QUENCHED BINARY ALLOYS

Abstract We study the segregation process in quenched binary alloys by analyzing and comparing the time evolution of the structure function and of the grain distribution obtained from computer simulations on a model system. We find good agreement between cluster sizes and densities determined directly on the computer sample and ones obtained by the Guinier method from the structure function. We then describe a graphical method for determining the scaling behaviour of the structure function S(k, t) which gives good statistics because the whole curve S(k, t) vs k is used. This yields very good agreement between the scaling function (scaled with the Guinier radius) obtained from the computer simulations and from a variety of real experiments. This function shows a universal behaviour independent of the alloy composition, the temperature and even the substance investigated. Our results are also not consistent with the more recent theoretical work (Binder et al., Furukawa et al.) which give alternate derivations and extensions of the Guinier formulas.

Monte Carlo studies of percolation phenomena for a simple cubic lattice

The site-percolation problem on a simple cubic lattice is studied by the Monte Carlo method. By combining results for periodic lattices of different sizes through the use of finite-size scaling theory we obtain good estimates forpc (0.3115±0.0005),β (0.41±0.01),γ (1.6±0.1), andν(0.8±0.1). These results are consistent with other studies. The shape of the clusters is also studied. The average “surface area” for clusters of sizek is found to be close to its maximal value for the low-concentration region as well as for the critical region. The percentage of particles in clusters of different sizesk is found to have an exponential tail for large values ofk forP pc there is too much scatter in the data to draw firm conclusions about the size distribution.

Stochastic wave function for atomic helium

Abstract A method is given for random sampling which yields a density equal to the groundstate wave function for the helium atom. As an intermediate step, a Greens function for the diffusion operator containing the repulsive interaction is sampled. A technique for generating a density function proportional to ψ 2 is outlined. Attention is paid to the development of estimators with finite variance. Results are shown for the electron density and for 〈 r 2 〉 The latter agrees well with previous numerical results.

Growth of clusters in a first-order phase transition

The results of computer simulations of phase separation kinetics in a binary alloy quenched from a high temperature are analyzed in detail, using the ideas of Lifshitz and Slyozov. The alloy was modeled by a three-dimensional Ising model with Kawasaki dynamics. The temperature after quenching was 0.59Tc, whereTc is the critical temperature, and the concentration of minority atoms wasρ=0.075, which is about five times their largest possible single-phase equilibrium concentration at that temperature. The time interval covered by our analysis goes from about 1000 to 6000 attempted interchanges per site. The size distribution of small clusters of minority atoms is fitted approximately byc1≈(1-ρ)3w(t),c1≈ (1−ρ)4Qlw(t)l(2≤l≤10); wherecl is the concentration of clusters of sizel;Q2,...,Q10 are known constants, the “cluster partition functions”;t is the time; andw(t)=0.015(1+7.17t−1/3). The distribution of large clusters (l≥20) is fitted approximately by the type of distribution proposed by Lifshitz and Slyozov,cl,(t)=−(d/dl)ψ[lnt+pϕ(l/t)], whereϕ is a function given by those authors andψ is defined byψ(x)=Coe−x-C1e−4x/3-C2e−5x/3;C0,C1,C2 are constants determined by considering how the total number of particles in large clusters changes with time.

Clusters, Metastability, and Nucleation: Kinetics of First-Order Phase Transitions

We describe and interpret computer simulations of the time evolution of a binary alloy on a cubic lattice, with nearest neighbor interactions favoring like pairs of atoms. Initially the atoms are arranged at random; the time evolution proceeds by random interchanges of nearest neighbor pairs, using probabilities compatible with the equilibrium Gibbs distribution at temperatureT. For temperatures 0.59Tc, 0.81 Tc, and 0.89Tc, with densityρ of A atoms equal to that in the B-rich phase at coexistence, the density C1 of clusters ofl A atoms approximately satisfies the following empirical formulas: C1 ≈w(1 −ρ)3 andC1, ≈ (1 −ρ)4Q1w1 (2 ⩽l ⩽ 10). Herew is a parameter and we defineQl =∑Ke−βE(K), where the sum goes over all translationally nonequivalentl-particle clusters andE(K) is the energy of formation of the clusterK. Forl > 10,Q1 is not known exactly; so we use an extrapolation formulaQl ≈Aws−ll−α exp(−blσ), wherews is the value ofw at coexistence. The same formula (withw > ws) also fits the observed values of C, (for small values ofl) at densities greater than the coexistence density (forT=0.59Tc): When the supersaturation is small, the simulations show apparently metastable states, a theoretical estimate of whose lifetime is compatible with the observations. For higher supersaturation the system is observed to undergo a slow process of segregation into two coexisting phases (andw therefore changes slowly with time). These results may be interpreted as a more quantitative formulation (and confirmation) of ideas used in standard nucleation theory. No evidence for a “spinodal” transition is found.

Nonequilibrium phase transition in stochastic lattice gases: Simulation of a three-dimensional system

We report results of computer simulations of a three-dimensional lattice gas of interacting particles subject to a uniform external fieldE. The dynamics of the system is given by hoppings of particles to nearby empty sites with rates biased for jumps in the direction ofE. As for the two-dimensional system we find that here too there exists a critical temperature,Tc(E) such that forT < Tc(E) the systems orders in a very anisotropic phase with striplike typical configurations parallel to the field.Tc(E) increases withE but substantially less strongly than in two dimensions. There is a break in the slope of the saturation current atTc(E). Our data are consistent with the critical exponentβ being mean field.