An improved Monte Carlo method for direct calculation of the density of states

Monte Carlo molecular simulations with FEASST version 0.25.1

FEASST is an open-source Monte Carlo software package for particle-based simulations. This software, which was released in 2017, has been used to study phase equilibrium, self-assembly, aggregation or gelation in biological materials, colloids, polymers, ionic liquids, and adsorption in porous networks. We highlight some of the unique features available in FEASST, such as flat-histogram grand canonical ensemble, Gibbs ensemble, and Mayer-sampling simulations with support for anisotropic models and parallelization with flat-histogram and prefetching. We also discuss how the challenges of supporting a variety of Monte Carlo algorithms were overcome by an object-oriented design. This also allows others to extend classes, which improves software interoperability, as inspired by LAMMPS classes and user packages. This article describes version 0.25.1 with benchmarks, compilation instructions, and introductory tutorials for running, restarting, and testing simulations, user guidelines, software design strategies, alternative interfaces, and the test-driven development strategy.

Flat-Histogram Monte Carlo Simulation of Water Adsorption in Metal-Organic Frameworks

Molecular simulations of water adsorption in porous materials often converge slowly due to sampling bottlenecks that follow from hydrogen bonding and, in many cases, the formation of water clusters. These effects may be exacerbated in metal-organic framework (MOF) adsorbents, due to the presence of pore spaces (cages) that promote the formation of discrete-size clusters and hydrophobic effects (if present), among other reasons. In Grand Canonical Monte Carlo (MC) simulations, these sampling challenges are typically manifested by low MC acceptance ratios, a tendency for the simulation to become stuck in a particular loading state (i.e., macrostates), and the persistence of specific clusters for long periods of the simulation. We present simulation strategies to address these sampling challenges, by applying flat-histogram MC (FHMC) methods and specialized MC move types to simulations of water adsorption. FHMC, in both Transition-matrix and Wang-Landau forms, drives the simulation to sample relevant macrostates by incorporating weights that are self-consistently adjusted throughout the simulation and generate the macrostate probability distribution (MPD). Specialized MC moves, based on aggregation-volume bias and configurational bias methods, separately address low acceptance ratios for basic MC trial moves and specifically target water molecules in clusters; in turn, the specialized MC moves improve the efficiency of generating new configurations which is ultimately reflected in improved statistics collected by FHMC. The combined strategies are applied to study the adsorption of water in CuBTC and ZIF-8 at 300 K, through examination of the MPD and the adsorption isotherm generated by histogram reweighting. A key result is the appearance of nontrivial oscillations in the MPD, which we show to be associated with water clusters in the adsorption system. Additionally, we show that the probabilities of certain clusters become similar in value near the boundaries of the isotherm hysteresis loop, indicating a strong connection between cluster formation/destruction and the thermodynamic limits of stability. For a hydrophobic MOF, the FHMC results show that the phase transition from low density to high density is suppressed to water pressure far above the bulk-fluid saturation pressure; this is consistent with results presented elsewhere. We also compare our FHMC simulation isotherm to one measured by a different technique but with ostensibly the same molecular interactions and comment on observed differences and the need for follow-up work. The simulation strategies presented here can be applied to the simulation of water in other MOFs using heuristic guidelines laid out in our text, which should facilitate the more consistent and efficient simulation of water adsorption in porous materials in future applications.

Phase behavior of the generalized chiral Lebwohl-Lasher model in bulk and confinement

One of the simplest models that is used to study the isotropic-nematic tranition in liquid crystalline systems is the Lebwohl-Lasher model. Several extensions of this model further enhanced its applicability. We combine two of these extensions (a generalization and the inclusion of a chiral term) and study the phase behavior and the nature of phase transitions of the resulting generalized chiral Lebwohl-Lasher model using Monte Carlo simulations. We find that the type (and even the existence) of the transition depends on the combination of the width of the interaction potential, the strength of the chiral part of the interaction, and the geometry of the system. As well, the pitch of the cholesteric bulk phase changes on approaching the phase transition if the interaction width differs from the one of the original Lebwohl-Lasher model. Thus, we identify parameter combinations that allow one to tune the properties of the ordered phase and the nature of the order-disorder transition.

Simulating Enhanced Methane Deliverable Capacity of Guest Responsive Pores in Intrinsically Flexible MOFs

A novel computational procedure, based on the principles of flat-histogram Monte Carlo, is developed for facile prediction of the adsorption thermodynamics of intrinsically flexible adsorbents. We then demonstrate how an accurate prediction of methane deliverable capacity in a metal-organic framework (MOF) with significant intrinsic flexibility requires use of such a method. Dynamic side chains in the framework respond to methane adsorbates and reorganize to exhibit a more conducive pore space at high adsorbate densities while simultaneously providing a less conducive pore space at low adsorbate densities. This "responsive pore" MOF achieves ∼20% higher deliverable capacity than if the framework were rigid and elucidates a strategy for designing high deliverable capacity MOFs in the future.

Flat-Histogram Monte Carlo as an Efficient Tool To Evaluate Adsorption Processes Involving Rigid and Deformable Molecules

Monte Carlo simulations are the foundational technique for predicting thermodynamic properties of open systems where the process of interest involves the exchange of particles. Thus, they have been used extensively to computationally evaluate the adsorption properties of nanoporous materials and are critical for the in silico identification of promising materials for a variety of gas storage and chemical separation applications. In this work we demonstrate that a well-known biasing technique, known as "flat-histogram" sampling, can be combined with temperature extrapolation of the free energy landscape to efficiently provide significantly more useful thermodynamic information than standard open ensemble MC simulations. Namely, we can accurately compute the isosteric heat of adsorption and number of particles adsorbed for various adsorbates over an extremely wide range of temperatures and pressures from a set of simulations at just one temperature. We extend this derivation of the temperature extrapolation to adsorbates with intramolecular degrees of freedom when Rosenbluth sampling is employed. Consequently, the working capacity and isosteric heat can be computed for any given combined temperature/pressure swing adsorption process for a large range of operating conditions with both rigid and deformable adsorbates. Continuous thermodynamic properties can be computed with this technique at very moderate computational cost, thereby providing a strong case for its application to the in silico identification of promising nanoporous adsorbents.

Temperature extrapolation of multicomponent grand canonical free energy landscapes

We derive a method for extrapolating the grand canonical free energy landscape of a multicomponent fluid system from one temperature to another. Previously, we introduced this statistical mechanical framework for the case where kinetic energy contributions to the classical partition function were neglected for simplicity [N. A. Mahynski et al., J. Chem. Phys. 146, 074101 (2017)]. Here, we generalize the derivation to admit these contributions in order to explicitly illustrate the differences that result. Specifically, we show how factoring out kinetic energy effects a priori, in order to consider only the configurational partition function, leads to simpler mathematical expressions that tend to produce more accurate extrapolations than when these effects are included. We demonstrate this by comparing and contrasting these two approaches for the simple cases of an ideal gas and a non-ideal, square-well fluid.

Multivariable extrapolation of grand canonical free energy landscapes

We derive an approach for extrapolating the free energy landscape of multicomponent systems in the grand canonical ensemble, obtained from flat-histogram Monte Carlo simulations, from one set of temperature and chemical potentials to another. This is accomplished by expanding the landscape in a Taylor series at each value of the order parameter which defines its macrostate phase space. The coefficients in each Taylor polynomial are known exactly from fluctuation formulas, which may be computed by measuring the appropriate moments of extensive variables that fluctuate in this ensemble. Here we derive the expressions necessary to define these coefficients up to arbitrary order. In principle, this enables a single flat-histogram simulation to provide complete thermodynamic information over a broad range of temperatures and chemical potentials. Using this, we also show how to combine a small number of simulations, each performed at different conditions, in a thermodynamically consistent fashion to accurately compute properties at arbitrary temperatures and chemical potentials. This method may significantly increase the computational efficiency of biased grand canonical Monte Carlo simulations, especially for multicomponent mixtures. Although approximate, this approach is amenable to high-throughput and data-intensive investigations where it is preferable to have a large quantity of reasonably accurate simulation data, rather than a smaller amount with a higher accuracy.

Predicting structural properties of fluids by thermodynamic extrapolation

We describe a methodology for extrapolating the structural properties of multicomponent fluids from one thermodynamic state to another. These properties generally include features of a system that may be computed from an individual configuration such as radial distribution functions, cluster size distributions, or a polymer's radius of gyration. This approach is based on the principle of using fluctuations in a system's extensive thermodynamic variables, such as energy, to construct an appropriate Taylor series expansion for these structural properties in terms of intensive conjugate variables, such as temperature. Thus, one may extrapolate these properties from one state to another when the series is truncated to some finite order. We demonstrate this extrapolation for simple and coarse-grained fluids in both the canonical and grand canonical ensembles, in terms of both temperatures and the chemical potentials of different components. The results show that this method is able to reasonably approximate structural properties of such fluids over a broad range of conditions. Consequently, this methodology may be employed to increase the computational efficiency of molecular simulations used to measure the structural properties of certain fluid systems, especially those used in high-throughput or data-driven investigations.

Quasi-Two-Dimensional Phase Transition of Methane Adsorbed in Cylindrical Silica Mesopores

Using Monte Carlo and molecular dynamics simulations, we examine the adsorption of methane in cylindrical silica mesopores in an effort to understand a possible phase transition of adsorbed methane in MCM-41 and SBA-15 silica that was previously identified by an unexpected increase in the adsorbed fluid density following capillary condensation, as measured by small-angle neutron scattering (SANS) [Chiang, W-S., et al., Langmuir 2016, 32, 8849]. Our initial simulation results identify a roughly 10 % increase in the density of the liquidlike adsorbed phase for either an isotherm with increasing pressure or an isobar with decreasing temperature and that this densification is associated with a local maximum in the isosteric enthalpy of adsorption. Subsequent analysis of the simulated fluid, via computation of bond-orientational order parameters of specific annular layers of the adsorbed fluid, showed that the layers undergo an ordering transition from a disordered, amorphous state to one with two-dimensional hexagonal structure. Furthermore, this two-dimensional restructuring of the fluid occurs at the same thermodynamic state points as the aforementioned densification and local maximum in the isosteric enthalpy of adsorption. We thus conclude that the densification of the fluid is the result of structural reorganization, which is signaled by the maximum in the isosteric enthalpy. Owing to the qualitative similarity of the structural transitions in the simulated and experimental methane fluids, we propose this hexagonal reorganization as a plausible explanation of the densification observed in SANS measurements. Lastly, we speculate how this structural transition may impact the transport properties of the adsorbed fluid.

Predicting low-temperature free energy landscapes with flat-histogram Monte Carlo methods

We present a method for predicting the free energy landscape of fluids at low temperatures from flat-histogram grand canonical Monte Carlo simulations performed at higher ones. We illustrate our approach for both pure and multicomponent systems using two different sampling methods as a demonstration. This allows us to predict the thermodynamic behavior of systems which undergo both first order and continuous phase transitions upon cooling using simulations performed only at higher temperatures. After surveying a variety of different systems, we identify a range of temperature differences over which the extrapolation of high temperature simulations tends to quantitatively predict the thermodynamic properties of fluids at lower ones. Beyond this range, extrapolation still provides a reasonably well-informed estimate of the free energy landscape; this prediction then requires less computational effort to refine with an additional simulation at the desired temperature than reconstruction of the surface without any initial estimate. In either case, this method significantly increases the computational efficiency of these flat-histogram methods when investigating thermodynamic properties of fluids over a wide range of temperatures. For example, we demonstrate how a binary fluid phase diagram may be quantitatively predicted for many temperatures using only information obtained from a single supercritical state.

Renormalized multicanonical sampling in multiple dimensions

This paper extends to two and three dimensions the recently proposed renormalized multicanonical sampling procedure [D. Yevick, Int. J. Mod. Phys. C 27, 1650033 (2016)IJMPEO0129-183110.1142/S0129183116500339], which generates the density of states of a system from that of a smaller system. The accuracy and efficiency of the method is additionally enhanced through a transition matrix Monte Carlo method that incorporates multicanonical sampling. The performance of the resulting procedure is investigated through calculations of the specific heat of the standard Ising model for square lattices with up to 128^{2} spins and for cubic lattices with up to 24^{3} spins. A comparison of the scaling properties to those of other algorithms based on the transition matrix Monte Carlo method demonstrates that the relative advantage of the renormalized procedure increases rapidly with system size.

Evaluation of the grand-canonical partition function using expanded Wang-Landau simulations. IV. Performance of many-body force fields and tight-binding schemes for the fluid phases of silicon

We extend Expanded Wang-Landau (EWL) simulations beyond classical systems and develop the EWL method for systems modeled with a tight-binding Hamiltonian. We then apply the method to determine the partition function and thus all thermodynamic properties, including the Gibbs free energy and entropy, of the fluid phases of Si. We compare the results from quantum many-body (QMB) tight binding models, which explicitly calculate the overlap between the atomic orbitals of neighboring atoms, to those obtained with classical many-body (CMB) force fields, which allow to recover the tetrahedral organization in condensed phases of Si through, e.g., a repulsive 3-body term that favors the ideal tetrahedral angle. Along the vapor-liquid coexistence, between 3000 K and 6000 K, the densities for the two coexisting phases are found to vary significantly (by 5 orders of magnitude for the vapor and by up to 25% for the liquid) and to provide a stringent test of the models. Transitions from vapor to liquid are predicted to occur for chemical potentials that are 10%-15% higher for CMB models than for QMB models, and a ranking of the force fields is provided by comparing the predictions for the vapor pressure to the experimental data. QMB models also reveal the formation of a gap in the electronic density of states of the coexisting liquid at high temperatures. Subjecting Si to a nanoscopic confinement has a dramatic effect on the phase diagram with, e.g. at 6000 K, a decrease in liquid densities by about 50% for both CMB and QMB models and an increase in vapor densities between 90% (CMB) and 170% (QMB). The results presented here provide a full picture of the impact of the strategy (CMB or QMB) chosen to model many-body effects on the thermodynamic properties of the fluid phases of Si.

Extending the parQ transition matrix method to grand canonical ensembles

Phase coexistence properties as well as other thermodynamic features of fluids can be effectively determined from the grand canonical density of states (DOS). We present an extension of the parQ transition matrix method in combination with the efasTM method as a very fast approach for determining the grand canonical DOS from the transition matrix. The efasTM method minimizes the deviation from detailed balance in the transition matrix using a fast Krylov-based equation solver. The method allows a very effective use of state space transition data obtained by different exploration schemes. An application to a Lennard-Jones system produces phase coexistence properties of the same quality as reference data.

Self-assembly of trimer colloids: effect of shape and interaction range

Trimers with one attractive bead and two repulsive beads, similar to recently synthesized trimer patchy colloids, were simulated with flat-histogram Monte Carlo methods to obtain the stable self-assembled structures for different shapes and interaction potentials. Extended corresponding states principle was successfully applied to self-assembling systems in order to approximately collapse the results for models with the same shape, but different interaction range. This helps us directly compare simulation results with previous experiment, and good agreement was found between the two. In addition, a variety of self-assembled structures were observed by varying the trimer geometry, including spherical clusters, elongated clusters, monolayers, and spherical shells. In conclusion, our results help to compare simulations and experiments, via extended corresponding states, and we predict the formation of self-assembled structures for trimer shapes that have not been experimentally synthesized.

Relation between pore size and the compressibility of a confined fluid

When a fluid is confined to a nanopore, its thermodynamic properties differ from the properties of a bulk fluid, so measuring such properties of the confined fluid can provide information about the pore sizes. Here, we report a simple relation between the pore size and isothermal compressibility of argon confined in such pores. Compressibility is calculated from the fluctuations of the number of particles in the grand canonical ensemble using two different simulation techniques: conventional grand-canonical Monte Carlo and grand-canonical ensemble transition-matrix Monte Carlo. Our results provide a theoretical framework for extracting the information on the pore sizes of fluid-saturated samples by measuring the compressibility from ultrasonic experiments.

Computational study of trimer self-assembly and fluid phase behavior

The fluid phase diagram of trimer particles composed of one central attractive bead and two repulsive beads was determined as a function of simple geometric parameters using flat-histogram Monte Carlo methods. A variety of self-assembled structures were obtained including spherical micelle-like clusters, elongated clusters, and densely packed cylinders, depending on both the state conditions and shape of the trimer. Advanced simulation techniques were employed to determine transitions between self-assembled structures and macroscopic phases using thermodynamic and structural definitions. Simple changes in particle geometry yield dramatic changes in phase behavior, ranging from macroscopic fluid phase separation to molecular-scale self-assembly. In special cases, both self-assembled, elongated clusters and bulk fluid phase separation occur simultaneously. Our work suggests that tuning particle shape and interactions can yield superstructures with controlled architecture.

Crossover from a Kosterlitz-Thouless phase transition to a discontinuous phase transition in two-dimensional liquid crystals

Liquid crystals in two dimensions do not support long-range nematic order, but a quasinematic phase where the orientational correlations decay algebraically is possible. The transition from the isotropic to the quasinematic phase can be continuous and of the Kosterlitz-Thouless type, or it can be first order. We report here on a liquid-crystal model where the nature of the isotropic to quasinematic transition can be tuned via a single parameter p in the pair potential. For p<p(t), the transition is of the Kosterlitz-Thouless type, while for p>p(t), it is first order. Precisely at p=p(t), there is a tricritical point where, in addition to the orientational correlations, also the positional correlations decay algebraically. The tricritical behavior is analyzed in detail, including an accurate estimate of p(t). The results follow from extensive Monte Carlo simulations combined with a finite-size scaling analysis. Paramount in the analysis is a scheme to facilitate the extrapolation of simulation data in parameters that are not necessarily field variables (in this case, the parameter p), the details of which are also provided. This scheme provides a simple and powerful alternative for situations where standard histogram reweighting cannot be applied.

Length-scale crossover of the hydrophobic interaction in a coarse-grained water model

It has been difficult to establish a clear connection between the hydrophobic interaction among small molecules typically studied in molecular simulations (a weak, oscillatory force) and that found between large, macroscopic surfaces in experiments (a strong, monotonic force). Here, we show that both types of interaction can emerge with a simple, core-softened water model that captures water's unique pairwise structure. As in hydrophobic hydration, we find that the hydrophobic interaction manifests a length-scale dependence, exhibiting distinct driving forces in the molecular and macroscopic regimes. Moreover, the ability of this simple model to capture both regimes suggests that several features of the hydrophobic force can be understood merely through water's pair correlations.

Density of states-based molecular simulations

One of the central problems in statistical mechanics is that of finding the density of states of a system. Knowledge of the density of states of a system is equivalent to knowledge of its fundamental equation, from which all thermodynamic quantities can be obtained. Over the past several years molecular simulations have made considerable strides in their ability to determine the density of states of complex fluids and materials. In this review we discuss some of the more promising approaches proposed in the recent literature along with their advantages and limitations.

Understanding free-energy perturbation calculations through a model of harmonic oscillators: theory and implications to improve the sampling efficiency by molecular simulation

Free-energy perturbation calculation is frequently used to calculate free-energy differences because it is easy to implement and the computation is fast. However, the calculation is subject to large inaccuracies in some circumstances due to the insufficient sampling of the relevant tails of the energy-difference distributions. Here we expand this knowledge of insufficient sampling into a two-dimensional (2D) energy space using a model of harmonic oscillators. We show analytically the relation between the energies of the sampling system and those of the desired target energy spaces, which provide the basis to understand the difficulties in free-energy perturbation calculations. We clarify the reasons of the inaccurate calculation in the different harmonic cases that stem from the spatial separations of the reference and the target energy pairs located in the two-dimensional energy space. The potential-energy space introduced into this 2D energy-space model provides additional clues to improve the sampling efficiency. Based on this understanding, we propose two ways to calculate the free-energy differences using the two schemes of the distribution method. We show that the distribution method implemented in the appropriate energy space--the energy-difference space and the potential-energy space, respectively--can improve the calculation of free energies in different circumstances. This analysis implies that the sampling can be improved if it is directed toward the appropriate region in the potential-energy space, which is easily implemented in various types of free-energy calculations. To test this, we calculate the free-energy surface of alanine dipeptide in gas phase and in aqueous phase, respectively. We demonstrate that the free-energy surface calculation is improved when the biased sampling of the potential energy is integrated into the sampling scheme.

A method for computing the solubility limit of solids: application to sodium chloride in water and alcohols

We present an adaptable method to compute the solubility limit of solids by molecular simulation, which avoids the difficulty of reference state calculations. In this way, the method is highly adaptable to molecules of complex topology. Results are shown for solubility calculations of sodium chloride in water and light alcohols at atmospheric conditions. The pseudosupercritical path integration method is used to calculate the free energy of the solid and gives results that are in good agreement with previous studies that reference the Einstein crystal. For the solution phase calculations, the self-adaptive Wang-Landau transition-matrix Monte Carlo method is used within the context of an expanded isothermal-isobaric ensemble. The method shows rapid convergence properties and the uncertainty in the calculated chemical potential was 1% or less for all cases. The present study underpredicts the solubility limit of sodium chloride in water, suggesting a shortcoming of the molecular models. Importantly, the proper trend for the chemical potential in various solvents was captured, suggesting that relative solubilities can be computed by the method.