Canonical Ensemble Monte Carlo Method Research Articles

A small-system ensemble Monte Carlo simulation of supersaturated vapor: Evaluation of barrier to nucleation

A small-system grand canonical ensemble Monte Carlo method is developed to evaluate cluster size distribution and barrier to the nucleation in a supersaturated Lennard-Jones vapor. The theoretical foundation is a physical cluster theory in which the Stillinger cluster is used as a prototypical physical cluster. Using method of Mayer’s cluster expansion, the cluster–vapor interaction is effectively taken into account. From a separate canonical ensemble Monte Carlo simulation using a test particle method, the averaged volume of the cluster is obtained and is also incorporated in the small-system ensemble simulation. By this implementation our simulation is computationally more efficient compared to that based on the n/v Stillinger cluster theory in that instead of searching the saddle point on a two-dimensional free energy surface (a function of cluster size n and volume v) one needs only to find the peak on a free energy curve (a function of n only). A comparison with the height of barrier obtained from a large-system ensemble Monte Carlo simulation [K. Oh and X. C. Zeng, J. Chem. Phys. 110, 4471 (1999)] shows that omission of the vapor–cluster attraction can cause an overestimation of the height by several kBT.

Phase behavior of a simple fluid confined between chemically corrugated substrates

The phase behavior of a molecularly thin fluid film of Lennard-Jones (LJ)(12,6) fluid confined to a chemically heterogeneous slit-shaped pore was investigated by the grand canonical ensemble Monte Carlo (GCEMC) method. The slit-shaped pore comprises two identical plane-parallel solid substrates, each of which consists of alternating strips of LJ(12,6) solid of two types: strongly (width ${d}_{s})$ and weakly adsorbing (width ${d}_{w})$. With the substrates aligned so that strips of the same type oppose each other, GCEMC was used to compute the local and mean densities of the fluid as well as its isothermal compressibility as functions of substrate separation ${s}_{z}$ and for various degrees of chemical corrugation measured quantitatively in terms of ${c}_{r}\ensuremath{\mathrel{:=}}{d}_{s}{/(d}_{s}{+d}_{w})$. Depending on ${s}_{z}$ and ${c}_{r},$ the confined fluid may consist of inhomogeneous gaslike or liquidlike phases filling the entire volume between the substrates. In addition, liquid ``bridges'' may form as a third phase consisting of stratified liquid stabilized by the ``strong'' strips and separated from two surrounding gaslike regions by an interface. The phase diagram involving all three phases was determined for a mean-field lattice-gas model similar to the one investigated recently by R\ocken and Tarazona [J. Chem. Phys. 105, 2034 (1996)]. The lattice-gas calculations permit a qualitative interpretation of the complex dependence of the GCEMC results on both ${s}_{z}$ and ${c}_{r}.$

Evaluation of Interaction Forces between Macroparticles in Simple Fluids by Molecular Dynamics Simulation

The present article provides the description of the solvation forces between large spheres in a fluid. The molecular dynamics (MD) method was applied to the relatively simple systems in which a pair of structureless macroparticles, either solvophobic or solvophilic, is immersed in a simple fluid of two types, either a soft-sphere or a Lennard-Jones fluid. When a pair of solvophobic macroparticles was in the attractive Lennard-Jones fluid, no dense layer of the solvent particles formed near the surface of the macroparticles and the strong attractive forces were induced between them. In the other combinations of macroparticles and fluids, the dense layers formed and the solvation forces oscillated, exhibiting the attraction and repulsion, whose periodic distance was about the diameter of solvent particles. Our results agreed well with those of the other simulation and theoretical studies with respect to the solvent density profile near a macroparticle and the force–distance profile between macroparticles. The benefit of our approach would be the simplicity in specifying or finding the bulk condition that is in equilibrium with the thin film of molecules between large surfaces. The present method can be applied straightforward to macroparticles immersed in mixtures and complex fluids described by the bead–spring model, to which the conventional grand canonical ensemble Monte Carlo (GCEMC) method is hardly accessible.

Ultrathin fluid films confined to a chemically heterogeneous slit-shaped nanopore

The properties of a molecularly thin film of Lennard-Jones (LJ) (12,6) fluid confined to a chemically heterogeneous slit-shaped pore were investigated by the grand canonical ensemble Monte Carlo (GCEMC) method. The slit-shaped pore comprises two identical plane-parallel solid substrates, each of which consists of alternating strips of LJ(12,6) solid of two types: strongly adsorbing and weakly adsorbing. With substrates aligned so that strips of the same type oppose each other, GCEMC was used to compute equilibrium properties of the film as functions of the distance ${s}_{z}$ between the substrates. Results are compared for two well depths ${\ensuremath{\epsilon}}_{\mathrm{fs}}$ of the LJ(12,6) potential between molecules in the film and those in the strongly adsorbing strip. Variations in tensions, mean film density, and isothermal compressibility as functions of ${s}_{z}$ are correlated with structural changes in the film and with its phase behavior. In both cases, when the substrates are sufficiently close together, liquid bridges exist between the opposing strong strips, surrounded by dilute gas over the weak ones. The stronger substrate (i.e., the one with the greater value of ${\ensuremath{\epsilon}}_{\mathrm{fs}}$) is capable of stabilizing a liquidlike phase that fills the whole pore over a certain range of ${s}_{z},$ then abruptly evaporates beyond a critical value of ${s}_{z},$ to leave nanodroplets clinging to the strong strips. For the weaker substrate, however, the liquid bridges collapse to form nanodroplets directly, without the intermediate appearance of a liquid pore-filling phase.

A Grand-Canonical Ensemble Monte Carlo Study of the Effects of Change of Core Size on the Thermodynamic Properties of a Modified Lennard-Jones Fluid

It is shown by consideration of the fundamental theory that when dealing with potentials having hard cores the Grand Canonical Ensemble Monte Carlo (GCEMC) method should not introduce large errors of the type previously found in the use of the Shing-Gubbins (SG) method for estimation of the chemical potential. This is confirmed by making simulation runs using the GCEMC and the Canonical Ensemble Monte Carlo (CEMC) methods on similar systems having the hard-core Lennard-Jones intermolecular potential. Only when the hardcore parameter (α) (where the hard-core diameter is α[sgrave], [sgrave] being the Lennard-Jones length parameter) is increased from 0.9 to 0.95 are appreciable changes in density and reduced potential energy (for a given μ) found. Corresponding, though much larger, changes in the fluctuations quantities fNN (= < N 2> - < N > 2) and fNU (= < NUN > - < N > < UN >), though not in fNU/fNN , also occur.

The impact of discrete wall structure on stratification-induced structural phase transitions in confined films

Molecular mechanisms by which a fluid closely confined between plane-parallel solid walls tends to order itself in layers parallel with the walls (i.e., stratifies) are investigated by a grand canonical ensemble Monte Carlo method. The walls are composed of individual atoms distributed across each wall according to the (100) plane of a face-centered cubic (fcc) crystal. Wall atoms are either rigidly fixed (model A) or thermally coupled (model B) to the film, that is the walls are ‘‘soft’’ on account of intermolecular interactions. As for a film between unstructured (i.e., molecularly smooth) walls [Schoen et al. J. Chem. Phys. 101, 6865 (1994)] stratification is accompanied by a subtle phase transition manifested as a maximum in density fluctuations at the transition point where packing characteristics of film molecules change in transverse directions. Thus, the transition involves phases with different degrees of transverse fcc (100)-like order induced by the walls which act like templates. If the transition involves films comprising only one and two layers, the mechanism of stratification is qualitatively similar to the one previously reported (see above): Well off the transition point the degree of fcc (100)-like order is nearly identical for one- and two-layer films and higher than at the transition point where the film is least ordered. The mechanism of stratification-induced phase transitions is different if it involves thicker films which tend to gain fcc (100)-like order more or less abruptly at the transition point. If wall atoms are not thermally coupled the film may solidify under geometrically favorable conditions. Solidification is not observed in model B under identical thermodynamic conditions.

Solvation pressure of simple fluids in spherical pores

Computer simulations via the grand canonical ensemble Monte Carlo method have been carried out to investigate the equilibrium properties of simple fluids distributed between the external bulk phase and the pore phase. Solvation pressures and pore density profiles are computed for the system of hard-sphere fluids inside a spherical hard-wall model pore. For dilute concentrations, the resulting solvation pressures are approximately equal to those in the bulk phase. However, at a moderately dense concentration of nB* = 0.4, a weakly oscillatory behavior of the solvation pressure starts to emerge. The apparent maximum and minimum peaks are exhibited in the case of nB* = 0.6 due to the local density inhomogeneities in the pore phase.

Open-system Monte Carlo simulations of Xe in NaA

The grand canonical ensemble Monte Carlo method is used to determine the adsorption isotherm, energy, entropy, occupancy distribution, and density distribution of Xe atoms adsorbed in zeolite NaA, which is modeled as a single alpha cage and as a network of connected cages. The isotherms exhibit multiple plateaus, indicating the existence of certain favored loadings. These plateaus can be predicted by considering constant loading energy and density distributions found previously. A linear regression method of determining the constant loading entropy of Xe atoms is presented. The entropy is a smooth function of loading, which suggests that the structure of the adsorbed Xe is also smoothly varying. The pressures needed to obtain complete loading are quite large and beyond the range of conventional experiment. However, the simulated isotherm and occupancy distribution compare well with the available experimental data. Structural modifications of NaA in the form of cation removal are investigated. A cation-poor model NaA adsorbs fewer Xe at low chemical potential due to its smaller electric field, but has a larger overall adsorption capacity due to its greater accessible volume.

Monte Carlo calculation of chemical potential for the Stockmayer fluid

The chemical potential of the Stockmayer fluid is computed by the canonical ensemble Monte Carlo methods of Widom, Bennett, and Han. Both spherical cutoff and metallic Ewald sum techniques are employed. For the states studied it is found that the Ewald sum provides little improvement over the simpler spherical cutoff, that Bennett’s and Han’s methods produce essentially the same results, and that for certain states Widom’s method is less accurate than Bennett’s or Han’s. The accuracy of the chemical potential calculated via the three methods was verified with Adams’ version of the grand canonical ensemble Monte Carlo method.

Monte Carlo simulation of hard chain–hard sphere mixtures in slitlike pores

The structure of mixtures of hard chains (modeled as a pearl necklace of freely jointed hard spheres) and hard spheres in slitlike pores is studied using a canonical ensemble Monte Carlo method. Simulation results for the density profiles in pores with wall separations varying from two to ten hard sphere diameters are presented at overall volume fractions of 0.12 and 0.34, and the effect of pore size, chain concentration, and chain length on the structure of the chains is investigated. It is found that the chains are depleted at the wall at the lower density, but enhanced at the wall (relative to the center of the pore) at the higher density; this depletion increases as the chain length is increased. It is found that the enhancement of monomers and depletion/enhancement of chains at the wall becomes more marked as the pore size is increased. As all the pores we study are integer multiples of the bead diameter, we do not observe the oscillatory variation of the wall density with wall separation which is expected at high densities. The fluid is uniform in the middle of the larger pores, but is not uniform anywhere in the smaller pores. At the same total packing fraction, increasing the mole fraction of the monomers tends to remove the chains from the wall and also make the oscillations in the chain profile more pronounced. The density of end sites of the chain at the wall is higher than it is for middle sites. In the smaller pores, the chains are found to be flattened in the direction perpendicular to the walls; in large pores the fluid in the middle of the pore is essentially uniform.

Monte Carlo study of physical adsorption: Comparison of the critical properties for two- and three-dimensional models of adsorption

Abstract The grand canonical ensemble Monte Carlo method is used to investigate the critical properties of two- and three-dimensional models of adsorption on a structureless solid surface. It is shown that the critical temperature of the gas-liquid-like phase transition in the adsorbed film is strongly dependent on the strength of the gas-solid interaction.

Physical adsorption of gases at high pressure

The adsorption of a (modified) Lennard-Jones gas on a plane, rigid solid surface is studied using the grand canonical ensemble Monte Carlo method for a wide range of physical conditions. The shapes of the adsorption isotherms are in good qualitative agreement with those observed experimentally. Multilayers adsorption is observed at temperatures from just below to well above the critical temperature, Tc . As the pressure is increased for T⪷Tc the adsorbed film tends to become thick in the sense that the outer adsorbed layer becomes highly deloealized and asymmetric.

Physical adsorption of gases at high pressure

The grand canonical ensemble Monte Carlo method is used to simulate the behaviour of a marginally supercritical Lennard-Jones gas near a graphite surface. The results reproduce the cusp-like shape of the experimentally observed adsorption isotherms. Even at these high temperatures, the computations clearly show the occurrence of multilayer adsorption; up to four layers are discernible at sufficiently high pressures.

Finite ion size effects in the electrical double layer—A Monte Carlo study

The grand canonical ensemble Monte Carlo method is applied to monovalent and divalent symmetric primitive model electrolytes between plane charged surfaces. Results for continuous and discrete surface charge distributions are compared with the Gouy–Chapman theory.

Computer simulation of reactive liquids in chemical equilibrium

A grand canonical ensemble Monte Carlo method applicable to reactive liquid mixtures in chemical equilibrium is described. The results of preliminary calculations in which the Br 2, Cl 2, BrCl equilibrium system is modeled using the new technique are reported.

Solvation forces in simple dense fluids. II. Effect of chemical potential

The grand canonical ensemble Monte Carlo method is used to calculate the distribution of liquid molecules and the solvation force between two parallel solid surfaces separated by a simple liquid. Neither this distribution nor the solvation force are very sensitive to changes in the chemical potential (i.e., bulk density) of the interstitial liquid.

A Monte Carlo study of fluid water in contact with structureless walls

A canonical ensemble Monte Carlo method has been used to examine the structure of fluid water in the space between two walls, each of which exerted a 9–3 potential. The intermolecular pair potential between the water molecules was that proposed by Rowlinson from the properties of ice and steam. The simulation involved 216 water molecules at a temperature of 298 K and employed periodic boundary conditions in the two directions parallel to the surface. In addition to the usual thermodynamic quantities we have collected singlet distributions for the oxygen and hydrogen atoms and the dipole vector, and pair distributions in three laminar regions between the walls for interatomic and dipole pairs. The thermodynamic and structural properties of our non-uniform sample are compared with those of uniform bulk water from simulation data.

Calculation of solvation forces between solid particles immersed in a simple liquid

Two simple approximations are used to calculate the solvation force between two plane parallel Lennard-Jones solids immersed in a Lennard-Jones liquid. The approximations consist of constructing the liquid singlet distribution function between the solid surfaces at finite separation from the distribution function adjacent to an isolated surface. The results are compared with the forces calculated by the Grand Canonical Ensemble Monte Carlo method.

The grand canonical ensemble Monte Carlo method applied to the electrical double layer

The grand canonical ensemble Monte Carlo method is applied to the electrical double layer, using a discrete surface charge distribution and the restricted primitive model for the electrolyte. A systematic examination of the effects of the dimensions of the basic Monte Carlo cell shows that the usual minimum image technique, for determining the potential energy of the system, must be supplemented by long range corrections. Alternatively, very large dimensions or numbers of ions must be employed. The calculated ion distribution functions are compared with the Gouy–Chapman theory for bulk electrolyte concentrations up to 0.2 C/m2. The Gouy–Chapman theory compares very well with the Monte Carlo results for electrolyte concentrations up to 0.1 mole/dm3, but quantitatvive differences appear at 1 mole/dm3. These differences are more pronounced at 2 mole/dm3 for which the Monte Carlo results indicate an inversion of the positive and negative charge distribution functions.

The grand canonical ensemble Monte Carlo method applied to electrolyte solutions

A grand canonical ensemble Monte Carlo method is developed for application to electrolyte solutions. The method proves to be of comparable accuracy and speed to the conventional NVT Monte Carlo method for electrolytes but has the added advantage of being able to fix the chemical potential. This latter point is vital for the study of surface phenomena. Application of the method to 1:1 and 2:2 primitive model electrolytes is made as is a comparison with the results of approximate statistical mechanical treatments.