Monte Carlo simulations of two-dimensional hard core lattice gases

Monte Carlo simulations are used to study lattice gases of particles with extended hard cores on a two-dimensional square lattice. Exclusions of one and up to five nearest neighbors (NN) are considered. These can be mapped onto hard squares of varying side length, lambda (in lattice units), tilted by some angle with respect to the original lattice. In agreement with earlier studies, the 1NN exclusion undergoes a continuous order-disorder transition in the Ising universality class. Surprisingly, we find that the lattice gas with exclusions of up to second nearest neighbors (2NN) also undergoes a continuous phase transition in the Ising universality class, while the Landau-Lifshitz theory predicts that this transition should be in the universality class of the XY model with cubic anisotropy. The lattice gas of 3NN exclusions is found to undergo a discontinuous order-disorder transition, in agreement with the earlier transfer matrix calculations and the Landau-Lifshitz theory. On the other hand, the gas of 4NN exclusions once again exhibits a continuous phase transition in the Ising universality class-contradicting the predictions of the Landau-Lifshitz theory. Finally, the lattice gas of 5NN exclusions is found to undergo a discontinuous phase transition.     

Surface structures of cubo-octahedral Pt-Mo catalyst nanoparticles from Monte Carlo simulations

The surface structures of cubo-octahedral Pt-Mo nanoparticles have been investigated using the Monte Carlo method and modified embedded atom method potentials that we developed for Pt-Mo alloys. The cubo-octahedral Pt-Mo nanoparticles are constructed with disordered fcc configurations, with sizes from 2.5 to 5.0 nm, and with Pt concentrations from 60 to 90 atom %. The equilibrium Pt-Mo nanoparticle configurations were generated through Monte Carlo simulations allowing both atomic displacements and element exchanges at 600 K. We predict that the Pt atoms weakly segregate to the surfaces of such nanoparticles. The Pt concentrations in the surface are calculated to be 5-14 atom % higher than the Pt concentrations of the nanoparticles. Moreover, the Pt atoms preferentially segregate to the facet sites of the surface, while the Pt and Mo atoms tend to alternate along the edges and vertexes of these nanoparticles. We found that decreasing the size or increasing the Pt concentration leads to higher Pt concentrations but fewer Pt-Mo pairs in the Pt-Mo nanoparticle surfaces.     

Microscopic solvation: spectroscopic results vs. Monte Carlo simulations

Van-der-Waals clusters of carbazole (representing the ‘solute’) with up to 40 nitrogen or methane solvent molecules were characterized using two-color resonant two-photon ionization spectroscopy. Features in these spectra (redshift, homogeneous and heterogeneous broadening, etc.) are interpreted as being caused by various static and dynamic effects of the solvent shell surrounding the aromatic substrate. For a better understanding of such effects, Monte Carlo simulations of the clusters were performed: Statics: Using a Monte Carlo simulated annealing minimization procedure, minimum energy structures (local, probably global) were found for the various cluster species. Using a simple empirical additivity rule, spectral shifts are rationalized from these structures.Dynamics: Starting from these minimum configurations, canonical ensemble simulations were carried out in a temperature range from 0 to 35 K. Severalorder-disorder transitions were identified including:

1. orientational isomerization or ‘melting’
2. surface isomerization or decoupling
3. rigid → fluxional transitions
4. full cluster ‘melting’

We present some of our experimental results on the systems carbazole · (N₂) _n and carbazole · (CH₄) _n together with the corresponding simulation data.     

Examining the accuracy of ideal adsorbed solution theory without curve-fitting using transition matrix Monte Carlo simulations

Ideal adsorbed solution theory (IAST) is a well-known approach to predicting multicomponent adsorption isotherms in microporous materials from experimental or simulation data for single-component adsorption. A limitation in practical applications of IAST is that useful calculations often require extrapolation of fitted single-component isotherms beyond the range for which data are available. We introduce a molecular simulation approach in which the intrinsic accuracy of IAST can be examined in a context that avoids any need to perform curve fitting with single-component data. Our approach is based on using transition matrix Monte Carlo to define single-component adsorption isotherms for arbitrary bulk-phase pressures from a single simulation. We apply our approach to several light gas mixtures in silica zeolites and a carbon nanotube to examine the intrinsic accuracy of IAST for these model systems.     

Titration of hydrophobic polyelectrolytes using Monte Carlo simulations

The conformation and titration curves of weak (or annealed) hydrophobic polyelectrolytes have been examined using Monte Carlo simulations with screened Coulomb potentials in the grand canonical ensemble. The influence of the ionic concentration pH and presence of hydrophobic interactions has been systematically investigated. A large number of conformations such as extended, pearl-necklace, cigar-shape, and collapsed structures resulting from the subtle balance of short-range hydrophobic attractive interactions and long-range electrostatic repulsive interactions between the monomers have been observed. Titration curves were calculated by adjusting the pH-pK(0) values (pK(0) represents the intrinsic dissociation constant of an isolated monomer) and then calculating the ionization degree alpha of the polyelectrolyte. Important transitions related to cascades of conformational changes were observed in the titration curves, mainly at low ionic concentration and with the presence of strong hydrophobic interactions. We demonstrated that the presence of hydrophobic interactions plays an important role in the acid-base properties of a polyelectrolyte in promoting the formation of compact conformations and hence decreasing the polyelectrolyte degree of ionization for a given pH-pK(0) value.     

Monte Carlo lattice simulations of the elastic behaviour of nematic liquid crystals

A pairwise additive potential, which approximately reproduces the free energy density for the elastic deformations of a nematic liquid crystal, originally proposed by Gruhn and Hess, has been investigated by simulating the three Freedericksz transitions as well as that of the Schadt-Helfrich cell. The pair potential depends on the three elastic constants K₁, K₂ and K₃ for the splay, twist and bend deformations, respectively. The results of the simulations are compared with the analytical solutions obtained from continuum theory in order to test the accuracy of the model potential at a quantitative level. This comparison is also made for different temperatures to explore the influence of director fluctuations on the elastic behaviour.     

Monte Carlo lattice simulations of the elastic behaviour of nematic liquid crystals

A pairwise additive potential, which approximately reproduces the free energy density for the elastic deformations of a nematic liquid crystal, originally proposed by Gruhn and Hess, has been investigated by simulating the three Freedericksz transitions as well as that of the Schadt-Helfrich cell. The pair potential depends on the three elastic constants K₁, K₂ and K₃ for the splay, twist and bend deformations, respectively. The results of the simulations are compared with the analytical solutions obtained from continuum theory in order to test the accuracy of the model potential at a quantitative level. This comparison is also made for different temperatures to explore the influence of director fluctuations on the elastic behaviour.     

Spatially adaptive lattice coarse-grained Monte Carlo simulations for diffusion of interacting molecules

While lattice kinetic Monte Carlo (KMC) methods provide insight into numerous complex physical systems governed by interatomic interactions, they are limited to relatively short length and time scales. Recently introduced coarse-grained Monte Carlo (CGMC) simulations can reach much larger length and time scales at considerably lower computational cost. In this paper we extend the CGMC methods to spatially adaptive meshes for the case of surface diffusion (canonical ensemble). We introduce a systematic methodology to derive the transition probabilities for the coarse-grained diffusion process that ensure the correct dynamics and noise, give the correct continuum mesoscopic equations, and satisfy detailed balance. Substantial savings in CPU time are demonstrated compared to microscopic KMC while retaining high accuracy.     

Optimization of 89Zr production using Monte Carlo simulations

Zirconium-89 (⁸⁹Zr) can be produced in a cyclotron by focusing the proton beam on an yttrium-89 (⁸⁹Y) foil target. Optimal combination of beam energy and target assembly configuration enables maximum production of ⁸⁹Zr while minimizing the formation of contaminant nuclides such as ⁸⁸Zr and ⁸⁸Y to allow efficient and effective radiopharmaceutical labeling. Accurate modeling of the proton beam and the target is therefore an essential step to assure the best beam and target specification. We used the radiation transport code MCNPX to simulate the transport of protons through the irradiation assembly and the nuclear reaction code TALYS to obtain the production cross sections of various nuclides from proton-⁸⁹Y reactions. Results from simulating the irradiation of 14 mm diameter targets with aluminum (Al) degrader at 19.8 mA for 1 h suggested that the 0.15 mm thick one would produce 227 MBq while the 0.3 mm thick one would produce 413 MBq of ⁸⁹Zr with less than 1 % uncertainty. These results show excellent agreement with experimental work in literature. This work provides the basis for further experimental and theoretical assessments of the use of ⁸⁹Zr as radiopharmaceutical labels.     

Monte Carlo simulations of single crystals from polymer solutions

A novel "anisotropic aggregation" model is proposed to simulate nucleation and growth of polymer single crystals as functions of temperature and polymer concentration in dilute solutions. Prefolded chains in a dilute solution are assumed to aggregate at a seed nucleus with an anisotropic interaction by a reversible adsorption/desorption mechanism, with temperature, concentration, and seed size being the control variables. The Monte Carlo results of this model resolve the long-standing dilemma regarding the kinetic and thermal roughenings, by producing a rough-flat-rough transition in the crystal morphology with increasing temperature. It is found that the crystal growth rate varies nonlinearly with temperature and concentration without any marked transitions among any regimes of polymer crystallization kinetics. The induction time increases with decreasing the seed nucleus size, increasing temperature, or decreasing concentration. The apparent critical nucleus size is found to increase exponentially with increasing temperature or decreasing concentration, leading to a critical nucleus diagram composed in the temperature-concentration plane with three regions of different nucleation barriers: no growth, nucleation and growth, and spontaneous growth. Melting temperatures as functions of the crystal size, heating rate, and concentration are also reported. The present model, falling in the same category of small molecular crystallization with anisotropic interactions, captures most of the phenomenology of polymer crystallization in dilute solutions.     

Monte Carlo simulations on mesophase formation using dipolar Gay-Berne model

We report the results of a Monte Carlo simulation of polar particles interacting via the Gay-Berne potential combining dipole-dipole interactions. Simulations were carried out on a system of 256 particles with either a zero dipole moment or longitudinal dipole moment located at the centre of the molecule. The system was found to spontaneously form nematic, smectic and crystal phases from an isotropic phase with a random configuration as temperature was decreased, irrespective of values of the dipole moment. The results do not give any indication of a net polarization even in the system with a strong dipole moment (μ* = 2.00). The transition temperature from the isotropic to nematic phase is not sensitive to the value of the dipole moment within the limits of statistical error, while the transition from the nematic to smectic phase depends on the strength of dipole moment. At lower temperatures forming the smectic or the crystal phase, the translational order along the director increases with increasing dipole moment. The dipolar interactions contribute to the long range ordering.     

Thermodynamic properties of van der Waals fluids from Monte Carlo simulations and perturbative Monte Carlo theory

Computer simulations have been performed for fluids with van der Waals potential, that is, hard spheres with attractive inverse power tails, to determine the equation of state and the excess energy. On the other hand, the first- and second-order perturbative contributions to the energy and the zero- and first-order perturbative contributions to the compressibility factor have been determined too from Monte Carlo simulations performed on the reference hard-sphere system. The aim was to test the reliability of this "exact" perturbation theory. It has been found that the results obtained from the Monte Carlo perturbation theory for these two thermodynamic properties agree well with the direct Monte Carlo simulations. Moreover, it has been found that results from the Barker-Henderson [J. Chem. Phys. 47, 2856 (1967)] perturbation theory are in good agreement with those from the exact perturbation theory.     

Prediction of membrane protein structures by replica-exchange Monte Carlo simulations: case of two helices

We test our prediction method of membrane protein structures with glycophorin A transmembrane dimer and analyze the predicted structures in detail. Our method consists of two parts. In the first part, we obtain the amino-acid sequences of the transmembrane helix regions from one of existing WWW servers and use them as an input for the second part of our method. In the second part, we perform a replica-exchange Monte Carlo simulation of these transmembrane helices with some constraints that indirectly represent surrounding lipid and water effects and identify the predicted structure as the global-minimum-energy state. The structure obtained in the case for the dielectric constant epsilon=1.0 is very close to that from the nuclear magnetic resonance experiments, while that for epsilon=4.0 is more packed than the native one. Our results imply that the helix-helix interaction is the main driving force for the native structure formation and that the stability of the native structure is determined by the balance of the electrostatic term, van der Waals term, and torsion term, and the contribution of electrostatic energy is indeed important for correct predictions. The inclusion of atomistic details of side chains is essential for estimating this balance accurately because helices are tightly packed.     

Elastic properties of soft sphere crystal from Monte Carlo simulations

Elastic properties of faced centered cubic (fcc) crystals composed of soft spheres, interacting through potentials of the form u(r) ~ r(-n), have been investigated by Monte Carlo (MC) simulations. It is shown that both the softness parameter (n(-1)) and temperature strongly influence the elastic properties of the studied system. The simulations show explicitly that when T > 0 the elastic constants of the hard sphere crystal can be obtained by taking the limit n --> infinity of soft spheres. When T --> 0 for any finite n, the elastic constants of the soft spheres tend to those of the static model. At all temperatures and softness parameters studied here, n, the Poisson's ratio in [110] (perpendicular direction) is negative.     

Direct calculation of solid-liquid equilibria from density-of-states Monte Carlo simulations

A density-of-states Monte Carlo method is proposed for simulations of solid-liquid phase equilibria. A modified Wang-Landau density-of-states sampling approach is used to perform a random walk in regions of potential energy and volume relevant to solid-liquid equilibrium. The method provides a direct estimate of the relative density of states [Omega(U,V)] and thus the relative free energy within these regions, which is subsequently used to determine portions of the melting curve over wide ranges of pressure and temperature. The validity and usefulness of the method are demonstrated by performing crystallization simulations for the Lennard-Jones fluid and for NaCl.     

Equilibrium states of self-assembly systems: Monte Carlo simulations

We investigated the equilibrium states of the self-assembly of amphiphilic molecules in water. The amphiphiles are represented by chains of the type H1T4, where H is the hydrophilic part of the molecule and T is its hydrophobic portion formed by four monomers. We have performed Monte Carlo simulations on a two-dimensional lattice, in which each water molecule occupies a single site, and the amphiphiles occupy five sites of the lattice. We have determined the aggregate distribution curves for the system at low concentration and fixed temperature. We have shown that the criterion to determine the equilibrium states of the system, based on the stabilization of energy curves as a function of the simulation time, is not reliable. The best way to ensure that the equilibrium state was reached was to follow the route to equilibrium of all aggregate sizes of the system.     

Monte Carlo simulations of polymer dynamics: Recent advances

A brief review is given of applications of Monte Carlo simulations to study the dynamical properties of coarse-grained models of polymer melts, emphasizing the crossover from the Rouse model toward reptation, and the glass transition. The extent to which Monte Carlo algorithms can mimic the actual chain dynamics is critically examined, and the need for the use of coarse-grained rather than fully atomistic models for such simulations is explained. It is shown that various lattice and continuum models yield qualitatively similar results, and the behavior agrees with the findings of corresponding molecular dynamics simulations and experiments, where available. It is argued that these simulations significantly enhance our understanding of the theoretical concepts on the dynamics of dense macromolecular systems. © 1997 John Wiley & Sons, Inc.     

Fast protein structure prediction using Monte Carlo simulations with modal moves

Using normal modes to generate torsion space moves in Monte Carlo simulations of peptides and proteins is not a new idea; nevertheless, despite its power it has not received widespread application. We show that such a "Modal Monte Carlo" approach is an efficient tool for ab initio predictions of small-protein structures. We apply this method to the Trp cage, a 20-residue polypeptide designed to fold rapidly into a structure that includes tertiary contacts, despite its short length. We achieve a high-quality ab initio structure prediction in about 2 orders of magnitude less computation time than state of the art molecular dynamics techniques.