Evaluation of the grand-canonical partition function using expanded Wang-Landau simulations. II. Adsorption of atomic and molecular fluids in a porous material

We propose to apply expanded Wang-Landau simulations to study the adsorption of atomic and molecular fluids in porous materials. This approach relies on a uniform sampling of the number of atoms and molecules adsorbed. The method consists in determining a high-accuracy estimate of the grand-canonical partition function for the adsorbed fluids. Then, using the formalism of statistical mechanics, we calculate absolute and excess thermodynamic properties relevant to adsorption processes. In this paper, we examine the adsorption of argon and carbon dioxide in the isoreticular metal-organic framework (IRMOF-1). We assess the reliability of the method by showing that the predicted adsorption isotherms and isosteric heats are in excellent agreement with simulation results obtained from grand-canonical Monte Carlo simulations. We also show that the proposed method is very efficient since a single expanded Wang-Landau simulation run at a given temperature provides the whole adsorption isotherm. Moreover, this approach provides a direct access to a wide range of thermodynamic properties, such as, e.g., the excess Gibbs free energy and the excess entropy of adsorption.     

Phase transitions of two-dimensional dipolar fluids in external fields

In this work, we study condensation phase transitions of two-dimensional Stockmayer fluids under additional external fields using Monte-Carlo (MC) simulations in the grand-canonical ensemble. We employ two recently developed methods to determine phase transitions in fluids, namely Wang-Landau (WL) MC simulations and successive-umbrella (SU) sampling. Considering first systems in zero field (and dipolar coupling strengths μ(2)∕εσ(3) ≤ 6), we demonstrate that the two techniques yield essentially consistent results but display pronounced differences in terms of efficiency. Indeed, comparing the computation times for these systems on a qualitative level, the SU sampling turns out to be significantly faster. In the presence of homogeneous external fields, however, the SU method becomes plagued by pronounced sampling difficulties, yielding the calculation of coexistence lines essentially impossible. Employing the WL scheme, on the other hand, we find phase coexistence even for strongly field-aligned systems. The corresponding critical temperatures are significantly shifted relative to the zero-field case.     

Polynomial path for the calculation of liquid state free energies from computer simulations tested on liquid water

The polynomial path is introduced for the calculation of liquid state free energies. The well-characterized SPC, TIP4P, and MCY water models were used to demonstrate its efficiency, as well as its range of applicability in conjunction with Monte Carlo computer simulations using thermodynamic integration based on Gaussian quadratures. The technique employed is compared with the slow-growth method (another variant of thermodynamic integration), the perturbation method, and the use of the grand-canonical ensemble.     

Determination of surface tension in binary mixtures using transition-matrix Monte Carlo

We present a methodology based on grand-canonical transition-matrix Monte Carlo and finite-size scaling analysis to calculate surface tensions in binary mixtures. In particular, mixture transition-matrix Monte Carlo is first used to calculate apparent, system-size-dependent free-energy barriers separating coexisting fluid phases. Finite-size scaling is then used to extrapolate these values to the infinitely large system limit to determine the true thermodynamic surface tension. A key distinction of the methodology is that it yields the entire isothermal surface-tension curve for a binary mixture in a relatively small number of simulations. We demonstrate the utility of the method by calculating surface-tension curves for three binary Lennard-Jones mixtures. While we have only examined the surface tension of simple fluids in this work, the method is general and can be extended to molecular fluids as well as to determine interfacial tensions of liquid-liquid interfaces.     

Density of configurational states from first-principles calculations: the phase diagram of Al-Na surface alloys.

The structural phases of Al(x)Na(1-x) surface alloys have been investigated theoretically and experimentally. We describe the system using a lattice-gas Hamiltonian, determined from density functional theory, together with Monte Carlo (MC) calculations. The obtained phase diagram reproduces the experiment on a quantitative level. From calculation of the (configurational) density of states by the recently introduced Wang-Landau MC algorithm, we derive thermodynamic quantities, such as the free energy and entropy, which are not directly accessible from conventional MC simulations. We accurately reproduce the stoichiometry, as well as the temperature at which an order-disorder phase transition occurs, and demonstrate the crucial role, and magnitude, of the configurational entropy.     

Determination of fluid-phase behavior using transition-matrix Monte Carlo: binary Lennard-Jones mixtures

We present a novel computational methodology for determining fluid-phase equilibria in binary mixtures. The method is based on a combination of highly efficient transition-matrix Monte Carlo and histogram reweighting. In particular, a directed grand-canonical transition-matrix Monte Carlo scheme is used to calculate the particle-number probability distribution, after which histogram reweighting is used as a postprocessing procedure to determine the conditions of phase equilibria. To validate the methodology, we have applied it to a number of model binary Lennard-Jones systems known to exhibit nontrivial fluid-phase behavior. Although we have focused on monatomic fluids in this work, the method presented here is general and can be easily extended to more complex molecular fluids. Finally, an important feature of this method is the capability to predict the entire fluid-phase diagram of a binary mixture at fixed temperature in a single simulation.     

Thermodynamic properties of Ag2O--B2O3 glasses by a modified scale-transformed energy space sampling Monte Carlo method

The density of states of Ag(2)O--B(2)O(3) glasses has been calculated by using a modified scale-transformed energy space sampling algorithm. This algorithm combines the scale-transformed energy space sampling algorithm and the Wang-Landau method. It is shown how the two algorithms can be combined to improve the efficiency of calculation. The thermodynamic properties, in particular the specific heat C(V), of the above-mentioned glass system is studied. At temperatures above 80 K, the value of specific heat C(v) is close to 22 J/mol/K. At low temperatures, the deviations of C(v) from a T(3) behavior are discernible, that is, C(v)/T(3) exhibits a hump at T = 7 K, which is in good agreement with the reported experimental behavior.     

Phase transition of a single star polymer: a Wang-Landau sampling study

Star polymers, as an important class of nonlinear macromolecules, process special thermodynamic properties for the existence of a common connecting point. The thermodynamic transitions of a single star polymer are systematically studied with the bond fluctuation model using Wang-Landau sampling techniques. A new analysis method employing the shape factor is proposed to locate the coil-globule (CG) and liquid-crystal (LC) transitions, which shows a higher efficiency and accuracy than the canonical specific heat function. The LC transition temperature is found to obey the identical scaling law as the linear polymers. However, the CG transition temperature shifts towards the LC transition with the increasing of the arm number. The reason is that for the star polymer a lower temperature is needed for the attractive force to overcome the excluded volume effect of the polymer chain because of its high arm density. This work clearly proves the structural distinction of the linear and star polymers can only affect the CG transition while has no influence on the LC transition.     

Improved Wang-Landau sampling through the use of smoothed potential-energy surfaces

A method is presented to improve the speed of convergence of Wang-Landau simulations as used to calculate the density of states of continuous systems. The density of states is first crudely estimated with calculations employing a smoothed potential-energy surface. This estimate is then used as a seed for subsequent Wang-Landau simulations using the original potential. The performance of the method is demonstrated by employing several simple models, including an analytically solvable harmonic system as well as a Go model of a protein. For all systems considered, the seeded simulations were found to converge significantly faster and with higher accuracy than the standard Wang-Landau simulations.     

Structural properties and thermodynamics of water clusters: a Wang-Landau study

The temperature dependence of structural properties and thermodynamic behavior of water clusters has been studied using Wang-Landau sampling. Four potential models, simple point charge/extended (SPC/E), transferable intermolecular potential 3 point (TIP3P), transferable intermolecular potential 4 point (TIP4P), and Gaussian charge polarizable (GCP), are compared for ground states and properties at finite temperatures. Although the hydrogen bond energy and the distance of the nearest-neighbor oxygen pair are significantly different for TIP4P and GCP models, they approach to similar ground state structures and melting transition temperatures in cluster sizes we considered. Comparing with TIP3P, SPC/E model provides properties closer to that of TIP4P and GCP.     

The determination of the thermodynamic characteristics of Lennard-Jones fluids under isobaric conditions by the method of molecular functions

Integral equation theory for partial distribution functions was used to suggest an effective method for calculating the derivative of radial distribution function with respect to temperature under isobaric conditions. The thermodynamic expansion, compression, and pressure coefficients and isobaric and isochoric heat capacities were calculated for Lennard-Jones fluids. The calculation results were in close agreement with the known thermodynamic relations.     

An off-lattice Wang-Landau study of the coil-globule and melting transitions of a flexible homopolymer

The Wang-Landau Monte Carlo approach is applied to the coil-globule and melting transitions of off-lattice flexible homopolymers. The solid-liquid melting point and coil-globule transition temperatures are identified by their respective peaks in the heat capacity as a function of temperature. The melting and theta points are well separated, indicating that the coil-globule transition occurs separately from melting even in the thermodynamic limit. We also observe a feature in the heat capacity between the coil-globule and melting transitions which we attribute to a transformation from a low-density liquid globule to a high-density liquid globule.     

Wang-Landau algorithm: a theoretical analysis of the saturation of the error

In this work we present a theoretical analysis of the convergence of the Wang-Landau algorithm [Phys. Rev. Lett. 86, 2050 (2001)] which was introduced years ago to calculate the density of states in statistical models. We study the dynamical behavior of the error in the calculation of the density of states. We conclude that the source of the saturation of the error is due to the decreasing variations of the refinement parameter. To overcome this limitation, we present an analytical treatment in which the refinement parameter is scaled down as a power law instead of exponentially. An extension of the analysis to the N-fold way variation of the method is also discussed.     

A consistent calculation of the chemical potential for dense simple fluids

A general method to calculate the excess chemical potential betamuex, that is based on the Kirkwood coupling parameter's dependence of the correlation functions, is presented. The expression for the one particle bridge function B(1)r is derived for simple fluids with spherical interactions. Only the knowledge of the bridge function B(2)r is required. The accuracy of our approach is illustrated for a dense hard sphere fluid. As far as B(2)r is considered as exact, B(1)r is found to be, at high densities, the normalized bridge function -B(2)rB(2)(r=0). This expression ensures a consistent calculation of the excess chemical potential by satisfying implicitly the Gibbs-Duhem constraint. Only the pressure-consistency condition is necessary to calculate the structural and thermodynamic properties of the fluid.     

Resummed thermodynamic perturbation theory for central force associating potential. Multi-patch models

A resummed thermodynamic perturbation theory for associating fluids with multiply bondable central force associating potential is extended for the fluid with multiple number of multiply bondable associating sites. We consider a multi-patch hard-sphere model for associating fluids. The model is represented by the hard-sphere fluid system with several spherical attractive patches on the surface of each hard sphere. Resummation is carried out to account for blocking effects, i.e., when the bonding of a particle restricts (blocks) its ability to bond with other particles. Closed form analytical expressions for thermodynamical properties (Helmholtz free energy, pressure, internal energy, and chemical potential) of the models with arbitrary number of doubly bondable patches at all degrees of the blockage are presented. In the limiting case of total blockage, when the patches become only singly bondable, our theory reduces to Wertheim's thermodynamic perturbation theory (TPT) for polymerizing fluids. To validate the accuracy of the theory we compare to exact values, for the thermodynamical properties of the system, as determined by Monte Carlo computer simulations. In addition we compare the fraction of multiply bonded particles at different values of the density and temperature. In general, predictions of the present theory are in good agreement with values for the model calculated using Monte Carlo simulations, i.e., the accuracy of our theory in the case of the models with multiply bondable sites is similar to that of Wertheim's TPT in the case of the models with singly bondable sites.     

Equilibrium concentration fluctuations in open reactive systems

We discuss the calculation of local concentration fluctuations, at constant volume and temperature, for an open, non-interacting, chemically-reactive fluid. One conclusion is that even for ideal mixtures, fluctuations cannot be calculated using a grand-canonical theory if the open volume element constitutes a non-negligible part of the total system. It is also shown that the calculation of equilibrium fluctuation for interacting molecules is, in general, not given by the ordinary grand-canonical result. There does exist a coupling between reaction progress fluctuations and concentration fluctuations due to migration.     

Performance evaluation of third-order thermodynamic perturbation theory and comparison with existing liquid state theories

To evaluate the performance of a recently proposed third-order thermodynamic perturbation theory (TPT), we employ the third TPT for calculation of thermodynamic properties such as compressibility factor, internal energy, excess chemical potential, gas-liquid coexistence curve, and critical properties of several fluids. By comparing the third-order TPT results with corresponding simulation data available in literature and supplied in the present report and theoretical results from several other theoretical approaches, one concludes that the third-order TPT is, in general, more accurate than other approaches such as Barker-Henderson second-order TPT using a macroscopic compressibility approximation (MCA-TPT), self-consistent Ornstein-Zernike approach, Monte Carlo perturbation theory, and a specially devised equation of state. Specifically, the third-order TPT can predict quantitatively a double critical phenomena of gas-liquid transition and a low-density liquid (LDL)-high-density liquid (HDL) transition associated with a soft core (SC) potential fluid very satisfactorily, but the predictions for the LDL-HDL transition based on the second-order MCA-TPT are quantitatively very bad or qualitatively incorrect. The failure of the second-order MCA-TPT for the SC fluid can be ascribed to the facts that for the SC potential the second-order and third-order terms of the perturbation expansion are not small quantities and that the second-order term is underestimated by the MCA. It is concluded that the present third-order version of the TPT is reliable for varying model fluids.     

Perturbed chain-statistical associating fluid theory extended to dipolar and quadrupolar molecular fluids

The perturbed chain statistical associating fluid theory (PC-SAFT) is extended to polar molecular fluids, namely dipolar and quadrupolar fluids. The extension is based on the perturbation theory for polar fluids by Stell and co-workers. Appropriate expressions are proposed for dipole-dipole, quadrupole-quadrupole, and dipole-quadrupole interactions. Furthermore, induced dipole interactions are calculated explicitly in the model. The new polar PC-SAFT model is relatively complex; for this purpose, a truncated polar PC-SAFT model is proposed using only the leading term in the polynomial expansion for polar interactions. The new model is used for the calculation of thermodynamic properties of various quadrupolar pure fluids. In all cases, the agreement between experimental data and model predictions is very good.     

A new approach for efficient simulation of Coulomb interactions in ionic fluids

We propose a simplified version of local molecular field (LMF) theory to treat Coulomb interactions in simulations of ionic fluids. LMF theory relies on splitting the Coulomb potential into a short-ranged part that combines with other short-ranged core interactions and is simulated explicitly. The averaged effects of the remaining long-ranged part are taken into account through a self-consistently determined effective external field. The theory contains an adjustable length parameter sigma that specifies the cutoff distance for the short-ranged interaction. This can be chosen to minimize the errors resulting from the mean-field treatment of the complementary long-ranged part. Here we suggest that in many cases an accurate approximation to the effective field can be obtained directly from the equilibrium charge density given by the Debye theory of screening, thus eliminating the need for a self-consistent treatment. In the limit sigma-->0, this assumption reduces to the classical Debye approximation. We examine the numerical performance of this approximation for a simple model of a symmetric ionic mixture. Our results for thermodynamic and structural properties of uniform ionic mixtures agree well with similar results of Ewald simulations of the full ionic system. In addition, we have used the simplified theory in a grand-canonical simulation of a nonuniform ionic mixture where an ion has been fixed at the origin. Simulations using short-ranged truncations of the Coulomb interactions alone do not satisfy the exact condition of complete screening of the fixed ion, but this condition is recovered when the effective field is taken into account. We argue that this simplified approach can also be used in the simulations of more complex nonuniform systems.