On the use of Bennett's acceptance ratio method in multi-canonical-type simulations

A common strategy for mapping coexistence curves is to employ multi-canonical (MUCA) sampling to simulate along a macrostate path connecting two phases. Central to this approach is the task of accurately calculating the importance weights used in the MUCA procedure, which are needed for both effective sampling and accurate determination of phase boundaries. The purpose of this study is to develop a strategy for determining the importance weights that is built upon Bennett's optimized acceptance ratio method. This approach is shown to be closely related to transition matrix schemes, and is used to compute the vapor-liquid equilibrium of a Lennard-Jones fluid and the liquid-liquid equilibrium of a n-hexane/n-perfluorohexane mixture. For the Lennard-Jones system, the importance weights as a function of the number of particles "N" (at fixed temperature and volume) are obtained by using Bennett's method to estimate free energy differences between N and N+1 particle systems over the desired range of N values. In this application, the method is found to perform slightly better than a related transition matrix scheme. For the n-hexane/n-perfluorohexane liquid mixture, the method is designed to obtain weights as a function of composition (for fixed temperature, pressure, and total number of particles); in this case, the method is found to outperform the Gibbs ensemble approach.     

Direct calculation of solid-vapor coexistence points by thermodynamic integration: application to single component and binary systems

We present a new thermodynamic integration method that directly connects the vapor and solid phases by a reversible path. The thermodynamic integration in the isothermal-isobaric ensemble yields the Gibbs free energy difference between the two phases, from which the sublimation temperature can be easily calculated. The method extends to the binary mixture without any modification to the integration path simply by employing the isothermal-isobaric semigrand ensemble. The thermodynamic integration, in this case, yields the chemical potential difference between the solid and vapor phases for one of the components, from which the binary sublimation temperature can be calculated. The coexistence temperatures predicted by our method agree well with those in the literature for single component and binary Lennard-Jones systems.     

Direct calculation of solid-liquid coexistence points of a binary mixture by thermodynamic integration

We present a new thermodynamic integration method that directly connects the liquid and the solid phases of a binary mixture by a reversible path. The states along the path are simulated in the isothermal-isobaric semigrand canonical ensemble, in which temperature, pressure, the total number of particles, and the fugacity fractions of the components are held fixed. The thermodynamic integration yields the chemical-potential difference between the two phases for one of the components and this information is then used to locate the solid-liquid coexistence points. The melting temperatures predicted by our method agree well with those predicted by the Gibbs-Duhem integration for a truncated and shifted Lennard-Jones system with a cutoff radius of 2.5sigma.     

Configurational constant pressure molecular dynamics

We propose two new algorithms for generating isothermal-isobaric molecular dynamics. The algorithms are based on an extended phase space dynamics where two extra degrees of freedom, representing the thermostat and the barostat, are included. These new methods adopt a totally different approach towards molecular dynamics simulation in the isothermal-isobaric ensemble. They are fully configurational in the sense that only the particle positions are required in the control of the system temperature and pressure. Following on from the works of Delhommelle and Evans [Mol. Phys., 99, 1825 (2001)] and of Braga and Travis [J. Chem. Phys., 123, 134101 (2005)] concerning configurational canonical dynamics, these new algorithms can be seen as a natural extension to the isothermal-isobaric ensemble. We have validated both of our new configurational isothermal-isobaric schemes by conducting molecular dynamics simulations of a Lennard-Jones fluid and comparing the static and dynamic properties for a single state point. We find that both schemes generate similar results compared with schemes which use kinetic temperature and pressure control. We have also monitored the response of the system to a series of isothermal compressions and isobaric quenches. We find that the configurational schemes performed at least as well as the kinetic based scheme in bringing the system temperature and pressure into line with the set point values of these variables. These new methods will potentially play a significant role in simulations where the calculation of the kinetic temperature and pressure can be problematic. A well known example resides in the field of nonequilibrium simulations where the kinetic temperature and pressure require a knowledge of the streaming velocity of the fluid in order to calculate the true peculiar velocities (or momenta) that enter into their definitions. These are completely avoided by using our configurational thermostats and barostats, since these are independent of momenta. By extending the analysis of Kusnezov et al. [Ann. Phys., 204, 155 (1990)] in order to derive a set of generalized Nose-Hoover equations of motion which can generate isothermal-isobaric dynamics in a number of different ways, we are able to show that both of our new configurational barostats and Hoover's kinetic isothermal-isobaric scheme are special cases of this more general set of equations. This generalization can be very powerful in generating constant pressure dynamics for a variety of systems.     

A new method of semigrand canonical ensemble to calculate first-order phase transitions for binary mixtures

First-order phase transitions of binary mixtures at the given pressure (P) and temperature (T) are studied by taking into account the composition fluctuations. Isothermal-isobaric semigrand canonical ensemble is adopted to find the relations among the total number of molecules, the composition fluctuations and Gibbs free energy density. By combining two identical subsystems of mixtures successively, the free energy density is transformed until being stable and its linear segments represent phase transitions. A new method is developed to calculate the phase equilibriums of binary mixtures. The method handles multiple types and number of phase equilibriums at single time and its solutions are physically justified. One example is shown for calculating the phase diagram of binary Lennard-Jones mixture. It demonstrates that the fluctuations of the total number of molecules in mixtures are fundamental behind phase transitions and the van der Waals loops in Gibbs free energy are reasonable.     

On the computational calculation of surface free energies for the disordered semihexagonal reconstructed Au(100) surface

Previously we developed a general method for calculating the free energy of any surface constrained to a distinct surface excess number/density. In this paper we show how to combine a range of such surfaces, whose free energies have been calculated, to produce an ad hoc semigrand canonical ensemble of surfaces from which ensemble surface properties can be calculated, including the ensemble surface free energy. We construct such an ensemble for the disordered Au(100) semihexagonal reconstructed surface using a Glue model potential at 1000 K and calculate the ensemble surface free energy to be 0.088 18 eVA(2). The ensemble average surface lateral density was found to be 1.375 (with respect to the bulk), which is in agreement with previous grand canonical Monte Carlo studies.     

Phase behavior of a fluid confined in slitlike pores with walls modified by preadsorbed chain molecules

We have studied the microscopic structure, thermodynamics of adsorption, and phase behavior of Lennard-Jones fluid in slitlike pores with walls modified due to preadsorption of chain molecules. The chain species are grafted at the walls by terminating segments. Our theoretical considerations are based on a density functional approach in the semigrand canonical ensemble. The applied constraint refers to the constant number of grafted chain molecules in the pore without restriction of the number of chains at each of the walls. We have observed capillary condensation of Lennard-Jones fluid combined with the change of the distribution of chains from nonsymmetric to symmetric with respect to the pore walls. The phase diagrams of the model are analyzed in detail, dependent on the pore width, length of chains, and grafted density.     

Superficial segregation in nanoparticles: from facets to infinite surfaces

We compare the superficial segregations of the Cu-Ag system for a nanoparticle and for surfaces that are structurally equivalent to each of its facet. Based on a lattice-gas model and within a mean-field formalism, we derive segregation isotherms at various temperatures in the canonical ensemble, i.e., for a given overall solute concentration, and in the semigrand canonical ensemble, i.e., for a given bulk solute concentration. If both processes are very similar for high temperatures, they differ substantially at lower temperatures. Due to the finite-size effect and the indirect coupling between facets and edges, the relative position of the phase transitions of the facets and the corresponding surfaces is inversed when displayed as a function of the solute bulk concentration. Moreover, we show that working in the semigrand canonical ensemble is a much more efficient way to study this phenomenon, although nanoparticles are "canonical" objects in essence.     

Thermodynamics and kinetics of bubble nucleation: Simulation methodology

The simulation of homogeneous liquid to vapor nucleation is investigated using three rare-event algorithms, boxed molecular dynamics, hybrid umbrella sampling Monte Carlo, and forward flux sampling. Using novel implementations of these methods for efficient use in the isothermal-isobaric ensemble, the free energy barrier to nucleation and the kinetic rate are obtained for a Lennard-Jones fluid at stretched and at superheated conditions. From the free energy surface mapped as a function of two order parameters, the global density and largest bubble volume, we find that the free energy barrier height is larger when projected over bubble volume. Using a regression analysis of forward flux sampling results, we show that bubble volume is a more ideal reaction coordinate than global density to quantify the progression of the metastable liquid toward the stable vapor phase and the intervening free energy barrier. Contrary to the assumptions of theoretical approaches, we find that the bubble takes on cohesive non-spherical shapes with irregular and (sometimes highly) undulating surfaces. Overall, the resulting free energy barriers and rates agree well between the methods, providing a set of complementary algorithms useful for studies of different types of nucleation events.     

Generalized simulated tempering for exploring strong phase transitions

An extension of the simulation tempering algorithm is proposed. It is shown to be particularly suited to the exploration of first-order phase transition systems characterized by the backbending or S-loop in the statistical temperature or a microcanonical caloric curve. A guided Markov process in an auxiliary parameter space systematically combines a set of parametrized Tsallis-weight ensemble simulations, which are targeted to transform unstable or metastable energy states of canonical ensembles into stable ones and smoothly join ordered and disordered phases across phase transition regions via a succession of unimodal energy distributions. The inverse mapping between the sampling weight and the effective temperature enables an optimal selection of relevant Tsallis-weight parameters. A semianalytic expression for the biasing weight in parameter space is adaptively updated "on the fly" during the simulation to achieve rapid convergence. Accelerated tunneling transitions with a comprehensive sampling for phase-coexistent states are explicitly demonstrated in systems subject to strong hysteresis including Potts and Ising spin models and a 147 atom Lennard-Jones cluster.     

The phase behavior of two-dimensional symmetrical mixtures

Using Monte Carlo simulation methods in the grand canonical and semigrand canonical ensembles, we study the phase behavior of two-dimensional symmetrical binary mixtures of Lennard-Jones particles. We discuss the interplay between the demixing transition in a liquid and the freezing in detail. Phase diagrams for several systems characterized by different parameters describing interactions in the system are presented. It is explicitly demonstrated that different scenarios involving demixing and freezing transitions, described in our earlier paper [A. Patrykiejew and S. Soko?owski, Phys. Rev. E, 81, 012501 (2010)], are possible. In one class of systems, the λ-line representing a continuous demixing transition in a liquid phase starts at the liquid side of either the vapor-liquid or liquid-solid coexistence. The second class involves the systems in which the λ-line begins at the liquid side of the vapor-liquid coexistence, in the lower critical end point, and then terminates at the liquid side of the liquid-solid coexistence, in the upper critical end point. It is also shown that in such systems the solid phase may undergo a demixing transition at the temperature above the upper critical end point.     

Crystal nucleation in binary hard sphere mixtures: a Monte Carlo simulation study

We present calculations of the nucleation barrier during crystallization in binary hard sphere mixtures under moderate degrees of supercooling using Monte Carlo simulations in the isothermal-isobaric semigrand ensemble in conjunction with an umbrella sampling technique. We study both additive and negatively nonadditive binary hard sphere systems. The solid-fluid phase diagrams of such systems show a rich variety of behavior, ranging from simple spindle shapes to the appearance of azeotropes and eutectics to the appearance of substitutionally ordered solid phase compounds. We investigate the effect of these types of phase behavior upon the nucleation barrier and the structure of the critical nucleus. We find that the underlying phase diagram has a significant effect on the mechanism of crystal nucleation. Our calculations indicate that fractionation of the species upon crystallization increases the difficulty of crystallization of fluid mixtures and in the absence of fractionation (azeotropic conditions) the nucleation barrier is comparable to pure fluids. We also calculate the barrier to nucleation of a substitutionally ordered compound solid. In such systems, which also show solid-solid phase separation, we find that the phase that nucleates is the one whose equilibrium composition is closer to the composition of the fluid phase.     

Bridging continuum and statistical thermodynamics via equations of state and the density of states

The connection between molecular force fields and equations of state (EoS) is typically established at the level of predicted quantities, e.g., by comparing simulation data and EoS data. In this paper we show how an EoS can be used to extract the density of states (Omega) of a system thus establishing a deeper connection between EoSs and statistical thermodynamics. We also show how such an EoS Omega can be used to aid molecular simulation methods designed to map out Omega (like the multicanonical approach). Central to the implementation of these ideas is the fact that the configurational Omega is related to thermodynamic properties accessible by an EoS via Boltzmann's equation. Sample calculations are presented for the Omega relevant to isothermal-isobaric and grand canonical ensemble simulations using the hard-sphere system and the Lennard-Jones system as model fluids, and the Carnahan-Starling EoS and a cubic EoS, respectively, as thermodynamic models.     

Enhanced sampling via strong coupling to a heat bath: relationship between Tsallis and multicanonical algorithms

We show that Tsallis and multicanonical statistical mechanics are equivalent under specific conditions and that they describe a system strongly coupled to a heat bath. The concept of the strong coupling to a heat bath, in which energy fluctuation is larger than that in the canonical ensemble [J. Chem. Phys. 119, 7075 (2003)], plays a key role in relating Tsallis formalism to multicanonical formalism. The equivalence between these formalisms allows us to obtain an appropriate q parameter in the Tsallis algorithm to enhance the sampling in the phase space in a manner similar to the multicanonical algorithm. An enhanced sampling in the configurational space by use of the strong coupling formalism is demonstrated in a Lennard-Jones fluid.     

The structure of fluids confined in crystalline slitlike nanoscopic pores

Grand canonical and canonical ensemble Monte Carlo simulation methods are used to study the structure and phase behavior of Lennard-Jones fluids confined between the parallel (100) planes of the face centered cubic crystal. Thin slit pores with a width allowing for the formation of only up to five atomic layers are considered. The phase diagrams of the systems characterized by different pore width as well as by different strength of the fluid-pore walls potential are determined. It is shown that an enormously large number of different phase diagram topologies can occur, depending on the parameters of the problem (pore width, strength of the fluid-pore walls potential, etc).     

Influence of surface chemical heterogeneities on adsorption/desorption hysteresis and coexistence diagram of metastable states within cylindrical pores

Grand canonical Monte Carlo simulations are performed to determine the adsorption/desorption isotherms at different temperatures of a Lennard-Jones fluid confined within a simple model of cylindrical pores presenting chemical heterogeneities. A complex hysteresis loop is observed, showing hysteresis subloops (scanning curves). This is shown to be consistent with the existence of several metastable states (local minima in the system free energy). A recent extension to the Gibbs ensemble technique is then used to calculate the complete coexistence diagram of these local minima.     

Molecular simulation of the phase behavior of fluids and fluid mixtures using the synthetic method

A new molecular simulation procedure is reported for determining the phase behavior of fluids and fluid mixtures, which closely follows the experimental synthetic method. The simulation procedure can be implemented using Monte Calro or molecular dynamics in either the microcanonical or canonical statistical ensembles. Microcanonical molecular dynamics simulations are reported for the phase behavior of both the pure Lennard-Jones fluid and a Lennard-Jones mixture. The vapor pressures for the pure fluid are in good agreement with Monte Carlo Gibbs ensemble and Gibbs-Duhem calculations. The Lennard-Jones mixture is composed of equal size particles, with dissimilar energy parameters (?(2)∕?(1) = 1∕2, ?(12)∕?(1) = 1∕2). The binary Lennard-Jones mixture exhibits liquid-liquid equilibria at high pressures and the simulation procedure allows us to estimate the coordinates of the high-pressure branch of the critical curve.     

Computation of surface tensions using expanded ensemble simulations

A method for the direct simulation of the surface tension is examined. The technique is based on the thermodynamic route to the interfacial tension and makes use of the expanded ensemble simulation method for the calculation of the free energy difference between two inhomogeneous systems with the same number of particles, temperature, and volume, but different interfacial area. The method is completely general and suitable for systems with either continuous or discontinuous interactions. The adequacy of the expanded ensemble method is assessed by computing the interfacial tension of the planar vapor-liquid interface of Lennard-Jones, Lennard-Jones dimers, Gay-Berne, and square-well model fluids; in the latter, the interactions are discontinuous and the present method does not exhibit the asymmetry of other related methods, such as the test area. The expanded ensemble simulation results are compared with simulation data obtained from other techniques (mechanical and test area) with overall good agreement.     

Density functional theory of adsorption in pillared slit-like pores

We propose a density functional theory to describe adsorption of Lennard-Jones fluid in pillared slit like pores. Specifically, the pillars are built of chains that are bonded by their ends to the opposite pore walls. The approach we propose combines theory of quenched-annealed systems and theory of nonuniform fluids involving chain molecules. We compare the results of theoretical predictions with grand canonical ensemble Monte Carlo simulations and compute theoretical capillary condensation phase diagrams for several model systems.     

Recovering the Crooks equation for dynamical systems in the isothermal-isobaric ensemble: a strategy based on the equations of motion

The Crooks equation [Eq. (10) in J. Stat. Phys. 90, 1481 (1998)] relates the work done on a system during a nonequilibrium transformation to the free energy difference between the final and the initial state of the transformation. Recently, the authors have derived the Crooks equation for systems in the canonical ensemble thermostatted by the Nose-Hoover or Nose-Hoover chain method [P. Procacci et al., J. Chem. Phys. 125, 164101 (2006)]. That proof is essentially based on the fluctuation theorem by Evans and Searles [Adv. Phys. 51, 1529 (2002)] and on the equations of motion. Following an analogous approach, the authors derive here the Crooks equation in the context of molecular dynamics simulations of systems in the isothermal-isobaric (NPT) ensemble, whose dynamics is regulated by the Martyna-Tobias-Klein algorithm [J. Chem. Phys. 101, 4177 (1994)]. Their present derivation of the Crooks equation correlates to the demonstration of the Jarzynski identity for NPT systems recently proposed by Cuendet [J. Chem. Phys. 125, 144109 (2006)].