Leslie V. Woodcock

Simulation and approximate formulae for the radial distribution functions of highly asymmetric hard sphere mixtures

The Henderson and Chan (HC) formulae for the contact values of the radial distribution functions (RDFs) of a highly asymmetric hard sphere mixture are reconsidered in light of a recent formula of Roth, Evans and Dietrich for the RDF of a pair of exceedingly large spheres at zero concentration in a solvent of small hard spheres. Two modifications of the HC formulae using this result give a large sphere–large sphere contact value that is considerably smaller than that of the original formulation. These new HC results are compared with the molecular dynamics simulations of Lue and Woodcock, for a diameter ratio of 1:10, supplemented by a few new results that are reported here. The new HC formulae are in much better agreement with the MD results than is the popular Boublik–Mansoori–Carnahan–Starling–Leland formula. Also, some simulation results for the RDFs as functions of separation are reported.

Crystallization of Spheres

Simplified computer models are used to gain insight into more complex real systems. In a reversion of this protocol, a colloidal suspension of submicron spherical particles, approximately hard and uniform, was recently crystallized in space and analyzed for crystal type. The objective was to establish how, and to what structure, hard spheres crystallize without gravity. Computational statistical thermodynamics predicts an equilibrium constant between fcc and hcp of order unity. The microgravity experiments, however, resulted in a random hybrid close-packed structure (rhcp) such that long-range order is two-dimensional. Here we report the mechanism from idealized computer “experiments” for crystallization of spheres from the metastable fluid. Model systems of up to N=64,000 spheres with infinite spatial periodicity have been crystallized in runs of up to 10 billion collisions. When the fluid, initially in a metastable supercooled state at 58% packing, is allowed to nucleate and freeze, a variety of structures emerges. There are three identifiable stages of structural growth: (i) initial nucleation of fcc, rhcp, and also bcc-like (body-centered cubic) local structures; (ii) rapid growth of all incipient nucleites to random stacked two-dimensional hexagonal (rhcp) grains, plus some fcc, to fill the volume; and (iii) relatively slow dissolution of unstable rhcp faces at grain boundaries. Eventually, stable nucleites emerge comprising hexagonal layers, arranged so as to contain predominantly either fcc arrangements of spheres or rhcp, in roughly 50% proportions.

Density functional approach to thermodynamics of self-assembly

A local density functional approximation for predicting the surface crystallization of a thermodynamically small system under gravity is described and tested. Using the model of the classical soft-sphere fluid, the state parameters for such systems are identified. A generalized phase diagram based upon the scaling variables is obtained; systems with the same reduced-state parameters exhibit identical profiles of thermodynamic properties such as density, pressure, and intrinsic chemical potential, measured in the direction of the applied field. The point-thermodynamic approximation of Rowlinson and the local density approximation of the density functional formalism are found to be remarkably accurate. A configurational temperature is defined and shown to agree with the corresponding kinetic temperature for inhomogeneous systems at equilibrium. The structural profiles at the crystal-fluid interface are indicative of a mesolayer of lower density crystal, not seen in the field-free isobaric crystal-liquid interface.

Thermodynamic status of random close packing

An amorphous ground state reminiscent of random close packing (RCP) of spheres terminates a liquid phase that spans all temperatures. On the Gibbs density surface, the liquid phase has bounded by a supercritical percolation line, two-phase liquid–gas coexistence line, and below the triple point, the metastable liquid branch terminating at T = 0 K with the RCP state. There is no “continuity of gas and liquid phases”; they are separated by a supercritical mesophase bounded by percolation transitions. As a consequence, the gas phase cannot be a starting point for liquid-state theory. RCP has a thermodynamic status. For square-well model fluids, evidence from computer simulations shows that the amorphous ground state is the same RCP state of the hard-sphere model. The RCP limiting density state of the hard-sphere fluid, with its reproducible and well-characterized structure, can be obtained by well-defined irreversible and reversible processes which establish a thermodynamic status. Empirical results, within margins of numerical uncertainty, are that the amorphous ground state density corresponds to a packing fraction  = 0.6366 ± 0.0005 (Buffon’s constant 2/π) and a residual entropy per sphere ΔS(0) ∼ kB (Boltzmann constant). We conclude that RCP of spheres is a well-defined state with a thermodynamic status, and a central role in the description of liquid phase equilibria, as originally suggested, 50 years ago, by J.D. Bernal. We postulate that the metastable RCP state is the starting point for a modern theory of the liquid state.

New approach to molecular dynamics for a non-pairwise Hamiltonian

In order to be able to simulate complex systems of non-pairwise additive interactions, a new computational approach, nth nearest neighbour network (n-NNN), has been proposed. In this new method, the continuous force acting on the central atom from its neighbours is a discretized multidimensional function based on the positions of the neighbours and stored in the computer memory. The memorized force is reused if an identical cluster neighbourhood is encountered once again. Here, the performance of the new method is evaluated for the 12–6 Lennard-Jones fluid and found to give reasonably accurate values for the thermodynamic properties. The algorithm is just as fast with many-body forces, as it is with pairwise additivity. The efficiency of the algorithm is demonstrated by applying it to MD simulations that explicitly incorporate three-body forces, and then comparing the computer time with the same simulation performed with conventional MD methods. The new n-NNN approach, although fast and accurate, is dependent on large amounts of computer memory. Suggestions are made to further improve the method.

Neural Network Approach in Computer Simulations

In order to be able to simulate complex systems of non‐pairwise additive interactions, a new computational approach, n‐th Nearest Neighbor Network (n‐NNN) method, has been developed. In this new method, the force acting on the central particle due to its neighbors is discretized based on the positions of the neighbors and memorized so that it can be reused if an identical cluster neighborhood is encountered once again by any particle during the simulation. The performance of the new method is evaluated for the 12‐6 Lennard‐Jones fluid and it is found to give reasonably accurate values for the thermodynamic properties. The efficiency of the new method is tested by applying it to computer simulations which explicitly included three‐body forces and comparing the computer time with the same simulations performed with conventional methods.

Artificial Intelligence In Materials Science: Application to Molecular and Particulate Simulations

We illustrate how emerging methods in artificial intelligence (AI) may be useful in materials science. Historically, these methods were developed in the area of materials process control and, more recently, in the nascent field of materials discovery. However, machine intelligence is of much broader import and our primary objective here is to illustrate how such methods may be used to circumvent some serious roadblocks in the computer simulation of a significant class of computationally hard problems in materials science. This is illustrated by a new approach to solving the dynamics of the N-body problem for large numbers of objects of essentially arbitrarily complex geometry or interaction potential. The approach, based on a particulate artificial neural net dynamics algorithm (PANNDA) is more than two orders of magnitude faster than existing methods when applied to large systems and is only marginally slower (∼10%) than the theoretical lower limiting case of hard spheres. In this method an artificial neural net is trained to predict accurately the time to next collision for binary encounters spanning the Hilbert space of relative positions, orientations and momenta (linear and angular). This approach, which can be extended to soft complex systems, enables construction of exact, albeit numerical, models for the thermodynamic, transport and non-equilibrium properties of very large ensembles of hard or soft objects of arbitrarily complex shape or interaction potential. Our results open up the possibility of immediate application to an usually wide spectrum of contemporary computationally intractable “hard” problems ranging from granular materials with asperities through inclusion of complex many-body terms in the intermolecular interaction in molecular dynamics calculations of complex fluids and polymers.

Computer simulation of glass formation from simple model liquids

Computer simulation techniques have been applied to the process of glass formation from simple model atomic and molecular liquids. Owing to the economic restrictions of ultrafast quench rates the „computer glass transition“ as characterised by the change in apparent heat capacity is seen to be so diffuse that applications to quasi-thermodynamic transitions as observed in the laboratory are severely limited. Preliminary results for a simulation of a small periodic diatomic molecular liquid (modelling chlorine) are compared with data on model atomic liquids. The influence on glass formation of the form of the interaction potential, the periodic sample size and the quench rate are discussed and the structural properties of the quenched states are compared.