Frenkel-Ladd Method Research Articles

On the stability of a quasicrystal and its crystalline approximant in a system of hard disks with a soft corona.

Using computer simulations, we study the phase behavior of a model system of colloidal hard disks with a diameter σ and a soft corona of width 1.4σ. The particles interact with a hard core and a repulsive square-shoulder potential. We calculate the free energy of the random-tiling quasicrystal and its crystalline approximants using the Frenkel-Ladd method. We explicitly account for the configurational entropy associated with the number of distinct configurations of the random-tiling quasicrystal. We map out the phase diagram and find that the random tiling dodecagonal quasicrystal is stabilised by entropy at finite temperatures with respect to the crystalline approximants that we considered, and its stability region seems to extend to zero temperature as the energies of the defect-free quasicrystal and the crystalline approximants are equal within our statistical accuracy.

Efficient approach to the free energy of crystals via Monte Carlo simulations.

We present a general approach to compute the absolute free energy of a system of particles with constrained center of mass based on the Monte Carlo thermodynamic coupling integral method. The version of the Frenkel-Ladd approach [J. Chem. Phys. 81, 3188 (1984)]JCPSA60021-960610.1063/1.448024, which uses a harmonic coupling potential, is recovered. Also, we propose a different choice, based on one-particle square-well coupling potentials, which is much simpler, more accurate, and free from some of the difficulties of the Frenkel-Ladd method. We apply our approach to hard spheres and compare with the standard harmonic method.

A comparative study of methods to compute the free energy of an ordered assembly by molecular simulation

We present a comparative study of methods to compute the absolute free energy of a crystalline assembly of hard particles by molecular simulation. We consider all combinations of three choices defining the methodology: (1) the reference system: Einstein crystal (EC), interacting harmonic (IH), or r(-12) soft spheres (SS); (2) the integration path: Frenkel-Ladd (FL) or penetrable ramp (PR); and (3) the free-energy method: overlap-sampling free-energy perturbation (OS) or thermodynamic integration (TI). We apply the methods to FCC hard spheres at the melting state. The study shows that, in the best cases, OS and TI are roughly equivalent in efficiency, with a slight advantage to TI. We also examine the multistate Bennett acceptance ratio method, and find that it offers no advantage for this particular application. The PR path shows advantage in general over FL, providing results of the same precision with 2-9 times less computation, depending on the choice of a common reference. The best combination for the FL path is TI+EC, which is how the FL method is usually implemented. For the PR path, the SS system (with either TI or OS) proves to be most effective; it gives equivalent precision to TI+FL+EC with about 6 times less computation (or 12 times less, if discounting the computational effort required to establish the SS reference free energy). Both the SS and IH references show great advantage in capturing finite-size effects, providing a variation in free-energy difference with system size that is about 10 times less than EC. This result further confirms previous work for soft-particle crystals, and suggests that free-energy calculations for a structured assembly be performed using a hybrid method, in which the finite-system free-energy difference is added to the extrapolated (1/N→0) absolute free energy of the reference system, to obtain a result that is nearly independent of system size.

Confirmation of the Random Tiling Hypothesis for a Decagonal Quasicrystal

Mechanisms that stabilize quasicrystals are much discussed but not finally resolved. We confirm the random tiling hypothesis and its predictions in a fully atomistic decagonal quasicrystal model by calculating the free energy and the phason elastic constants over a wide range of temperatures. The Frenkel-Ladd method is applied for the phonon part, and an approach of uncorrelated phason flips is applied for the configurational part. When lowering the temperature, a phase transition to an approximant occurs. Close to the transition temperature, one of the phason elastic constants becomes soft.

Phase diagram of hard tetrahedra

Advancements in the synthesis of faceted nanoparticles and colloids have spurred interest in the phase behavior of polyhedral shapes. Regular tetrahedra have attracted particular attention because they prefer local symmetries that are incompatible with periodicity. Two dense phases of regular tetrahedra have been reported recently. The densest known tetrahedron packing is achieved in a crystal of triangular bipyramids (dimers) with a packing density of 4000/4671 ≈ 85.63%. In simulation a dodecagonal quasicrystal is observed; its approximant, with periodic tiling (3.4.3(2).4), can be compressed to a packing fraction of 85.03%. Here, we show that the quasicrystal approximant is more stable than the dimer crystal for packing densities below 84% using Monte Carlo computer simulations and free energy calculations. To carry out the free energy calculations, we use a variation of the Frenkel-Ladd method for anisotropic shapes and thermodynamic integration. The enhanced stability of the approximant can be attributed to a network substructure, which maximizes the free volume (and hence the wiggle room) available to the particles and facilitates correlated motion of particles, which further contributes to entropy and leads to diffusion for packing densities below 65%. The existence of a solid-solid transition between structurally distinct phases not related by symmetry breaking--the approximant and the dimer crystal--is unusual for hard particle systems.

Entropic Stabilization of Tunable Planar Modulated Superstructures

The self-assembly of novel [corrected] ordered structures with nanoparticles has recently received much attention. Here we use computer simulations to study a two-dimensional model system characterized by a simple isotropic interaction that could be realized with building blocks on the nanoscale. We find that the particles arrange themselves into hexagonal superstructures of twin boundaries whose superlattice vector can be tuned reversibly by changing the temperature. Thermodynamic stability is confirmed by calculating the free energy with a combination of thermodynamic integration and the Frenkel-Ladd method. Different contributions to the free energy difference are discussed.

Revisiting the Frenkel-Ladd method to compute the free energy of solids: The Einstein molecule approach

In this paper a new method to evaluate the free energy of solids is proposed. The method can be regarded as a variant of the method proposed by Frenkel and Ladd [J. Chem. Phys. 81, 3188 (1984)]. The main equations of the method can be derived in a simple way. The method can be easily implemented within a Monte Carlo program. We have applied the method to determine the free energy of hard spheres in the solid phase for several system sizes. The obtained free energies agree within the numerical uncertainty with those obtained by Polson et al. [J. Chem. Phys. 112, 5339 (2000)]. The fluid-solid equilibria has been determined for several system sizes and compared to the values published previously by Wilding and Bruce [Phys. Rev. Lett. 85, 5138 (2000)] using the phase switch methodology. It is shown that both the free energies and the coexistence pressures present a strong size dependence and that the results obtained from free energy calculations agree with those obtained using the phase switch method, which constitutes a cross-check of both methodologies. From the results of this work we estimate the coexistence pressure of the fluid-solid transition of hard spheres in the thermodynamic limit to be p*=11.54(4), which is slightly lower than the classical value of Hoover and Ree (p*=11.70) [J. Chem. Phys. 49, 3609 (1968)]. Taking into account the strong size dependence of the free energy of the solid phase, we propose to introduce finite size corrections, which allow us to estimate approximately the free energy of the solid phase in the thermodynamic limit from the known value of the free energy of the solid phase with N molecules. We have also determined the free energy of a Lennard-Jones solid by using both the methodology of this work and the finite size correction. It is shown how a relatively good estimate of the free energy of the system in the thermodynamic limit is obtained even from the free energy of a relatively small system.

High-pressure phase diagram of the exp-6 model: The case of Xe

We investigated numerically the high-temperature--high-pressure phase diagram of xenon as modeled through the exp-6 interaction potential, which is thought to provide a reliable description of the thermal behavior of rare gases under extreme conditions. We performed a series of extensive NVT Monte Carlo simulations which, in conjunction with exact computation of the solid free energy by the Frenkel-Ladd method, allowed us to precisely locate the freezing and melting thresholds at each temperature. We find that, under isothermal compression, the exp-6 fluid freezes directly into a fcc solid; however, above 4500 K, an intermediate bcc phase becomes stable in a narrow range of pressures. The chemical potential of the hcp phase never significantly differs from that of the fcc solid of equal $T$ and $P$, though the former is found to be slightly greater than the latter. We discuss our results in the light of previous numerical studies of the same model system and of the experimental data available for xenon.

Three-Dimensional Hard Dumbbell Solid Free Energy Calculation Via the Fluctuating Cell Model

Determination of the solid–liquid phase transition point of a molecular substance requires calculation of the free energy in both phases. Progress has been made on this problem by modeling molecules as fused hard spheres and adding attraction and electric multipole moments perturbatively. The solid free energy of hard heteronuclear dumbbells of bond length L*, used to model diatomic molecules, can in principle be calculated exactly via the Frenkel–Ladd method, but this is computationally intensive. Use of Lennard Jones–Devonshire fixed cells to calculate free energy is much simpler computationally but is an approximation. The fluctuating cell model is investigated as an alternative intermediate method which is still computationally simpler than the Frenkel–Ladd method. As was found earlier in two dimensions, for small L* the simple cell model is in better agreement with Frenkel–Ladd than the fluctuating cell model, but for larger L* the fluctuating cell model is in better agreement. The probability distributions of free volumes are also analyzed and show different functional behavior for near-zero bond length and appreciable bond length.

Efficient Monte Carlo simulations using a shuffled nested Weyl sequence random number generator.

The pseudorandom number generator proposed recently by Holian et al. [B. L. Holian, O. E. Percus, T. T. Warnock, and P. A. Whitlock, Phys. Rev. E 50, 1607 (1994)] is tested via Monte Carlo computation of the free energy difference between the defectless hcp and fcc hard sphere crystals by the Frenkel-Ladd method [D. Frenkel and A. J. C. Ladd, J. Chem. Phys. 81, 3188 (1984)]. It is shown that this fast and convenient for parallel computing generator gives results in good agreement with results obtained by other generators. An estimate of high accuracy is obtained for the hcp-fcc free energy difference near melting.

Evaluation of harmonic methods for calculating the free energy of defects in solids.

The free energy of defects in solids is computed with Monte Carlo simulation techniques and by various approximate techniques. The results are compared in order to determine the accuracy and range of applicability of the various approximate methods. The systems studied are a bulk crystal, a vacancy, a (100) free surface, and a \ensuremath{\Sigma}5(310)/[001] symmetric tilt boundary in Cu described by embedded-atom method potentials. The Monte Carlo simulations employ both the Frenkel-Ladd method and thermodynamic integration. The approximate techniques include quasiharmonic calculations and the recently proposed local harmonic method. The results indicate that the harmonic methods significantly underestimate the temperature variation of the defect-free energies for this potential. The discrepancies become large for temperatures above about half of the melting point.