Gerassimos Orkoulas

Critical point estimation of the Lennard-Jones pure fluid and binary mixtures

The apparent critical point of the pure fluid and binary mixtures interacting with the Lennard-Jones potential has been calculated using Monte Carlo histogram reweighting techniques combined with either a fourth order cumulant calculation (Binder parameter) or a mixed-field study. By extrapolating these finite system size results through a finite size scaling analysis we estimate the infinite system size critical point. Excellent agreement is found between all methodologies as well as previous works, both for the pure fluid and the binary mixture studied. The combination of the proposed cumulant method with the use of finite size scaling is found to present advantages with respect to the mixed-field analysis since no matching to the Ising universal distribution is required while maintaining the same statistical efficiency. In addition, the accurate estimation of the finite critical point becomes straightforward while the scaling of density and composition is also possible and allows for the estimation of the line of critical points for a Lennard-Jones mixture.

Parallel Markov chain Monte Carlo simulations

With strict detailed balance, parallel Monte Carlo simulation through domain decomposition cannot be validated with conventional Markov chain theory, which describes an intrinsically serial stochastic process. In this work, the parallel version of Markov chain theory and its role in accelerating Monte Carlo simulations via cluster computing is explored. It is shown that sequential updating is the key to improving efficiency in parallel simulations through domain decomposition. A parallel scheme is proposed to reduce interprocessor communication or synchronization, which slows down parallel simulation with increasing number of processors. Parallel simulation results for the two-dimensional lattice gas model show substantial reduction of simulation time for systems of moderate and large size.

Acceleration of Markov chain Monte Carlo simulations through sequential updating

Strict detailed balance is not necessary for Markov chain Monte Carlo simulations to converge to the correct equilibrium distribution. In this work, we propose a new algorithm which only satisfies the weaker balance condition, and it is shown analytically to have better mobility over the phase space than the Metropolis algorithm satisfying strict detailed balance. The new algorithm employs sequential updating and yields better sampling statistics than the Metropolis algorithm with random updating. We illustrate the efficiency of the new algorithm on the two-dimensional Ising model. The algorithm is shown to identify the correct equilibrium distribution and to converge faster than the Metropolis algorithm with strict detailed balance. The main advantages of the new algorithm are its simplicity and the feasibility of parallel implementation through domain decomposition.

A Monte Carlo study of the freezing transition of hard spheres

A simulation method for fluid-solid transitions, which is based on a modification of the constrained cell model of Hoover and Ree, is developed and tested on a system of hard spheres. In the fully occupied constrained cell model, each particle is confined in its own Wigner-Seitz cell. Constant-pressure simulations of the constrained cell model for a system of hard spheres indicate a point of mechanical instability at a density which is about 64% of the density at the close packed limit. Below that point, the solid is mechanically unstable since without the confinement imposed by the cell walls it will disintegrate to a disordered, fluid-like phase. Hoover and Ree proposed a modified cell model by introducing an external field of variable strength. High values of the external field variable favor configurations with one particle per cell and thus stabilize the solid phase. In this work, the modified cell model of a hard-sphere system is simulated under constant-pressure conditions using tempering and histogram reweighting techniques. The simulations indicate that as the strength of the field is reduced, the transition from the solid to the fluid phase is continuous below the mechanical instability point and discontinuous above. The fluid-solid transition of the hard-sphere system is determined by analyzing the field-induced fluid-solid transition of the modified cell model in the limit in which the external field vanishes. The coexistence pressure and densities are obtained through finite-size scaling techniques and are in good accord with previous estimates.

Communication: Tracing phase boundaries via molecular simulation: An alternative to the Gibbs–Duhem integration method

Precise simulation of phase transitions is crucial for colloid/protein crystallization for which fluid-fluid demixing may be metastable against solidification. In the Gibbs-Duhem integration method, the two coexisting phases are simulated separately, usually at constant-pressure, and the phase boundary is established iteratively via numerical integration of the Clapeyron equation. In this work, it is shown that the phase boundary can also be reproduced in a way that avoids integration of Clapeyron equations. The two phases are simulated independently via tempering techniques and the simulation data are analyzed according to histogram reweighting. The main output of this analysis is the density of states which is used to calculate the free energies of both phases and to determine phase coexistence. This procedure is used to obtain the phase diagram of a square-well model with interaction range 1.15σ, where σ is the particle diameter. The phase boundaries can be estimated with the minimum number of simulations. In particular, very few simulations are required for the solid phase since its properties vary little with temperature.

Communication: A simple method for simulation of freezing transitions

Despite recent advances, precise simulation of freezing transitions continues to be a challenging task. In this work, a simulation method for fluid-solid transitions is developed. The method is based on a modification of the constrained cell model which was proposed by Hoover and Ree [J. Chem. Phys. 47, 4873 (1967)]. In the constrained cell model, each particle is confined in a single Wigner-Seitz cell. Hoover and Ree pointed out that the fluid and solid phases can be linked together by adding an external field of variable strength. High values of the external field favor single occupancy configurations and thus stabilize the solid phase. In the present work, the modified cell model is simulated in the constant-pressure ensemble using tempering and histogram reweighting techniques. Simulation results on a system of hard spheres indicate that as the strength of the external field is reduced, the transition from solid to fluid is continuous at low and intermediate pressures and discontinuous at high pressures. Fluid-solid coexistence for the hard-sphere model is established by analyzing the phase transition of the modified model in the limit in which the external field vanishes. The coexistence pressure and densities are in excellent agreement with current state-of-the-art techniques.

Parallel canonical Monte Carlo simulations through sequential updating of particles

In canonical Monte Carlo simulations, sequential updating of particles is equivalent to random updating due to particle indistinguishability. In contrast, in grand canonical Monte Carlo simulations, sequential implementation of the particle transfer steps in a dense grid of distinct points in space improves both the serial and the parallel efficiency of the simulation. The main advantage of sequential updating in parallel canonical Monte Carlo simulations is the reduction in interprocessor communication, which is usually a slow process. In this work, we propose a parallelization method for canonical Monte Carlo simulations via domain decomposition techniques and sequential updating of particles. Each domain is further divided into a middle and two outer sections. Information exchange is required after the completion of the updating of the outer regions. During the updating of the middle section, communication does not occur unless a particle moves out of this section. Results on two- and three-dimensional Lennard-Jones fluids indicate a nearly perfect improvement in parallel efficiency for large systems.

Spatial updating grand canonical Monte Carlo algorithms for fluid simulation: generalization to continuous potentials and parallel implementation.

Spatial updating grand canonical Monte Carlo algorithms are generalizations of random and sequential updating algorithms for lattice systems to continuum fluid models. The elementary steps, insertions or removals, are constructed by generating points in space either at random (random updating) or in a prescribed order (sequential updating). These algorithms have previously been developed only for systems of impenetrable spheres for which no particle overlap occurs. In this work, spatial updating grand canonical algorithms are generalized to continuous, soft-core potentials to account for overlapping configurations. Results on two- and three-dimensional Lennard-Jones fluids indicate that spatial updating grand canonical algorithms, both random and sequential, converge faster than standard grand canonical algorithms. Spatial algorithms based on sequential updating not only exhibit the fastest convergence but also are ideal for parallel implementation due to the absence of strict detailed balance and the nature of the updating that minimizes interprocessor communication. Parallel simulation results for three-dimensional Lennard-Jones fluids show a substantial reduction of simulation time for systems of moderate and large size. The efficiency improvement by parallel processing through domain decomposition is always in addition to the efficiency improvement by sequential updating.

Acceleration of Monte Carlo simulations through spatial updating in the grand canonical ensemble

A new grand canonical Monte Carlo algorithm for continuum fluid models is proposed. The method is based on a generalization of sequential Monte Carlo algorithms for lattice gas systems. The elementary moves, particle insertions and removals, are constructed by analogy with those of a lattice gas. The updating is implemented by selecting points in space (spatial updating) either at random or in a definitive order (sequential). The type of move, insertion or removal, is deduced based on the local environment of the selected points. Results on two-dimensional square-well fluids indicate that the sequential version of the proposed algorithm converges faster than standard grand canonical algorithms for continuum fluids. Due to the nature of the updating, additional reduction of simulation time may be achieved by parallel implementation through domain decomposition.

Precise simulation of the freezing transition of supercritical Lennard-Jones

The fluid-solid transition of the Lennard-Jones model is analyzed along a supercritical isotherm. The analysis is implemented via a simulation method which is based on a modification of the constrained cell model of Hoover and Ree. In the context of hard-sphere freezing, Hoover and Ree simulated the solid phase using a constrained cell model in which each particle is confined within its own Wigner-Seitz cell. Hoover and Ree also proposed a modified cell model by considering the effect of an external field of variable strength. High-field values favor configurations with a single particle per Wigner-Seitz cell and thus stabilize the solid phase. In previous work, a simulation method for freezing transitions, based on constant-pressure simulations of the modified cell model, was developed and tested on a system of hard spheres. In the present work, this method is used to determine the freezing transition of a Lennard-Jones model system on a supercritical isotherm at a reduced temperature of 2. As in the case of hard spheres, constant-pressure simulations of the fully occupied constrained cell model of a system of Lennard-Jones particles indicate a point of mechanical instability at a density which is approximately 70% of the density at close packing. Furthermore, constant-pressure simulations of the modified cell model indicate that as the strength of the field is reduced, the transition from the solid to the fluid is continuous below the mechanical instability point and discontinuous above. The fluid-solid transition of the Lennard-Jones system is obtained by analyzing the field-induced fluid-solid transition of the modified cell model in the high-pressure, zero-field limit. The simulations are implemented under constant pressure using tempering and histogram reweighting techniques. The coexistence pressure and densities are determined through finite-size scaling techniques for first-order phase transitions which are based on analyzing the size-dependent behavior of susceptibilities and dimensionless moment ratios of the order parameter.