Coarse-grained lattice kinetic Monte Carlo simulation of systems of strongly interacting particles

A general approach is presented for spatially coarse-graining lattice kinetic Monte Carlo (LKMC) simulations of systems containing strongly interacting particles. While previous work has relied on approximations that are valid in the limit of weak interactions, here we show that it is possible to compute coarse-grained transition rates for strongly interacting systems without a large computational burden. A two-dimensional square lattice is employed on which a collection of (supersaturated) strongly interacting particles is allowed to reversibly evolve into clusters. A detailed analysis is presented of the various approximations applied in LKMC coarse graining, and a number of numerical closure rules are contrasted and compared. In each case, the overall cluster size distribution and individual cluster structures are used to assess the accuracy of the coarse-graining approach. The resulting closure approach is shown to provide an excellent coarse-grained representation of the systems considered in this study.     

Prediction of thermodynamic properties of heavy hydrocarbons by Monte Carlo simulation

In this study, Monte Carlo simulation techniques based on the anisotropic united atom (AUA) potential have been used to predict thermodynamic properties, comprising saturation pressures, vaporization enthalpies and liquid densities, at different temperatures for several isoalkanes (2,3-dimethylpentane, 2,4-dimethylpentane), alkylbenzenes (propylbenzene and hexylbenzene), alkyl-substituted cycloalkanes (propylcyclohexane and propylcyclopentane), polycyclic alkanes (trans-decalin), and naphtenoaromatics (tetralin and indan), representing gasoil range fractions of hydrocarbons. This variety of hydrocarbons with different molecular structures served to demonstrate the transferability of the AUA potential parameters. For this purpose, two types of Monte Carlo algorithm were used: the Gibbs ensemble algorithm to predict equilibrium properties at high temperatures, and the NPT algorithm followed by the thermodynamic integration to extend the prediction to lower temperatures. Techniques increasing the efficiency of the algorithms such as configuration bias, reservoir bias, and parallel tempering were also implemented in the algorithms. Based on available experimental information, good predictions of equilibrium properties were obtained for all the hydrocarbon families studied, and small differences between isomers were represented with a good accuracy.     

Modeling flexible amphiphilic bilayers: a solvent-free off-lattice Monte Carlo study

We present a simple, implicit-solvent model for fluid bilayer membranes. The model was designed to reproduce the elastic properties of real bilayer membranes. For this model, we observed the solid-fluid transition and studied the in-plane diffusivity of the fluid phase. As a test, we compute the elastic-bending and area-compressing moduli of fluid bilayer membranes. We find that the computed elastic properties are consistent with the available experimental data.     

Kinetic lattice Monte Carlo simulation of viscoelastic subdiffusion

We propose a kinetic Monte Carlo method for the simulation of subdiffusive random walks on a Cartesian lattice. The random walkers are subject to viscoelastic forces which we compute from their individual trajectories via the fractional Langevin equation. At every step the walkers move by one lattice unit, which makes them differ essentially from continuous time random walks, where the subdiffusive behavior is induced by random waiting. To enable computationally inexpensive simulations with n-step memories, we use an approximation of the memory and the memory kernel functions with a complexity O(log?n). Eventual discretization and approximation artifacts are compensated with numerical adjustments of the memory kernel functions. We verify with a number of analyses that this new method provides binary fractional random walks that are fully consistent with the theory of fractional Brownian motion.     

A Monte Carlo study of amphiphilic dendrimers: spontaneous asymmetry and dendron separation

We study via Monte Carlo simulation the conformation of amphiphilic dendrimers for which terminal monomers (t) and internal monomers (i) interact differently with the solvent (s). Specifically, we have studied g = 3,6 dendrimers as a function of chi(it), chi(is), and chi(ts) (chi is the differential contact energy between the different particles) for parameter values chi(it) = 0, +/-1 and -1 < chi(is), chi(ts) < 1. We have allowed negative chi values in order to model attractive polar interactions (e.g., hydrogen bonding) which are believed to be important in many dendrimer/solvent systems. We find the "phase diagram" of dendrimer conformations to be extremely rich and to be a strong function of g, chi(is), and chi(ts) but only a weak function of chi(it), For chi(is), chi(ts) > 0, we observe dendrimer conformations, such as unimolecular normal micelles and inverted loopy micelles. However, for chi(is) < 0 or chi(ts) < 0, we observe more exotic molecular conformations, for example, the spontaneous development of asymmetry and dendron separation. These properties are analyzed in terms of snapshots as well as more quantitatively in terms of the radii of gyration, radial density profiles, pair-correlation functions, degree of asymmetry, and dendron overlap factor. By exploiting the dramatic conformational changes under different solvent conditions, we suggest the possibility of using amphiphilic dendrimers as stimuli-responsive smart materials.     

Monte Carlo calculation of the thermodynamic properties of water

The Monte Carlo method and parallel computing are used to calculate the thermodynamic properties of water (density, heat capacity, compressibility, thermal expansion coefficient, and static dielectric constant) in a wide range of temperatures (from 70 K to 530 K) at constant (atmospheric) pressure. Four groups of computational experiments are carried out, each for its own model of the water molecule: TIP3P (Jorgensen et al., 1983), SPC/E (Berendsen et al., 1987), TIP4P/2005 (Abascal&Vega, 2005), and TIP5P-E (Rick, 2004). An additional calculation based on the replica exchange method is conducted for the TIP4P/2005 model. A comparison of the calculated properties of water with experimental data suggests that the TIP4P/2005 model can provide highly realistic computer simulation results for water and aqueous solutions.     

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.     

Monte Carlo simulation for the micellar behavior of amphiphilic comb-like copolymers

Micellar behaviors in 2D and 3D lattice models for amphiphilic comb-like copolymers in water phase and in water/oil mixtures were simulated. A dynamical algorithm together with chain reptation movements was used in the simulation. Three-dimension displaying program was pro-grammed and free energy was estimated by Monte Carlo technigue. The results demonstrate that reduced interaction energy influences morphological structures of micelle and emulsion stems greatly; 3D simulation showing can display more direct images of morphological structures; the amphiphilic comb-like polymers with a hydrophobic main chain and hydrophilic side chains have lower energy in water than in oil.     

Critical phenomena in polymer mixtures: Monte Carlo simulation of a lattice model

A lattice model of a symmetrical binary (AB) polymer mixture is studied, modelling the polymer chains by self-avoiding walks withN _A =N _B =N steps on a simple cubic lattice. If a pair of nearest neighbour sites is taken by different monomersAB orBA, an energy _ab is won; if the pair of sites is taken by anAA or aBB pair, an energy is won, while the energy is reduced to zero if at least one of the sites of the pair is vacant. To allow enough chain mobility, 20% of the lattice sites are vacancies. In addition to local motions of the chain segments we use a novel grand-canonical simulation technique:A chains are transformed intoB chains and vice versa, keeping the chemical potential difference fixed. The phase diagram is obtained forN=4, 8,16 and 32; the critical behaviour is analysed by finite-size scaling methods. It is shown that the critical exponents are those of the Ising model (=0.32,=0.63) rather than those of the Flory-Huggins meanfield theory (==1/2). Implications of these results for real polymers are briefly discussed.Paper presented at the Frühjahrstagung 1986, Polymerphysik, der Deutschen Physikalischen Gesellschaft, Kaiserslautern, March 12–14, 1986.     

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.     

Self-assembled structures of block copolymers in selective solvents reproduced by lattice Monte Carlo simulation

A lattice Monte Carlo (MC) simulation was applied to the study of block copolymers in selective solvent or amphiphilic surfactant solution on the segment level, hydrodynamic interactions being neglected. The code was found to be very efficient, employing a partial reptation mode as the elementary movement of the self-avoiding lattice chains. Typical self-assembled structures of block copolymers such as micelle, lamellae, hexagonal cylinder and bicontinuous networks have been successfully reproduced without any priori specification of structure. Order–disorder and order–order transitions of diblock copolymers are systematically studied by adjusting the temperature, the concentration or the block length ratio in a series computer simulations. The structural differences between micelles composed of ABA and BAB triblock copolymers are also explicitly revealed by direct visualisation of the underlying chain configurations. The simulation results are consistent with the experimental observations in the literature. This simulation approach is thus a very useful tool in the extensive investigation of self-assembled structures. It has the advantage that both micro-domains and chain configurations can be studied with only a comparatively modest call on computational resources.     

Simulation of aggregating particles in complex flows by the lattice kinetic Monte Carlo method

We develop and validate an efficient lattice kinetic Monte Carlo (LKMC) method for simulating particle aggregation in laminar flows with spatially varying shear rate, such as parabolic flow or flows with standing vortices. A contact time model was developed to describe the particle-particle collision efficiency as a function of the local shear rate, G, and approach angle, θ. This model effectively accounts for the hydrodynamic interactions between approaching particles, which is not explicitly considered in the LKMC framework. For imperfect collisions, the derived collision efficiency [?=1 - ∫(0)(π/2) sinθ exp(-2cotθΓ(agg)/G)dθ] was found to depend only on Γ(agg)∕G, where Γ(agg) is the specified aggregation rate. For aggregating platelets in tube flow, Γ(agg)=0.683 s(-1) predicts the experimentally measured ε across a physiological range (G = 40-1000 s(-1)) and is consistent with α(2b)β(3)-fibrinogen bond dynamics. Aggregation in parabolic flow resulted in the largest aggregates forming near the wall where shear rate and residence time were maximal, however intermediate regions between the wall and the center exhibited the highest aggregation rate due to depletion of reactants nearest the wall. Then, motivated by stenotic or valvular flows, we employed the LKMC simulation developed here for baffled geometries that exhibit regions of squeezing flow and standing recirculation zones. In these calculations, the largest aggregates were formed within the vortices (maximal residence time), while squeezing flow regions corresponded to zones of highest aggregation rate.     

Polymer-centered theory in comparison with surfactant-centered theory: a lattice Monte Carlo study

Lattice Monte Carlo simulation is used to study micellization of both pure and surfactant-polymer mixture, with an emphasis on cluster size distribution. The amphiphile molecule is of the type H(4)T(4) where the H (head) monomers like the solvent molecules and the T (tail) monomers are solvophobic. To compare polymer- and surfactant-centered theories, copolymers with the structure of (H(4)T(4))(5) are used instead of homopolymers. Since above copolymer molecules have the structural unit like the structure of surfactant molecules, it is possible to study the competition between the binding of surfactant molecules to the copolymer and the micellization in a copolymer-free solution. Results show that, first, surfactant molecules bind to the copolymer molecules, and not until copolymers are saturated do micelles form. Furthermore, it is shown that for the model used in this paper, the polymer-centered theory is more appropriate than the surfactant-centered theory, and finally the cooperative nature of cluster formation on copolymers is also discussed.     

Monte Carlo simulation and thermodynamic integration applied to protein charge transfer

We introduce a combination of Monte Carlo simulation and thermodynamic integration methods to address a model problem in free energy computations, electron transfer in proteins. The feasibility of this approach is tested using the ferredoxin protein from Clostridium acidurici. The results are compared to numerical solutions of the Poisson-Boltzmann equation and data from recent molecular dynamics simulations on charge transfer in a protein complex, the NrfHA nitrite reductase of Desulfovibrio vulgaris. Despite the conceptual and computational simplicity of the Monte Carlo approach, the data agree well with those obtained by other methods. A link to experiments is established via the cytochrome subunit of the bacterial photosynthetic reaction center of Rhodopseudomonas viridis.     

A Monte Carlo simulation of the inhomogeneous Lebwohl-Lasher lattice model

A Monte Carlo computer simulation of a modified Lebwohl-Lasher model is presented. The model consists of a set of interaction centres placed at the sites of a cubic lattice. The angular part of the pair potential is a second Legendre polynomial of the relative orientation between the two particles, like that of the Lebwohl-Lasher model. Each particle interacts with its six nearest neighbours with an attractive anisotropic potential differing in strength for the four horizontal and the two vertical neighbours. Various values of the in-plane to out-of-plane coupling ratio δ have been considered, i.e. δ = 0.75, 0.5, 0.1, 0.0. The latter case corresponds to the limiting situation of a two-dimensional lattice. Systems with a 1000 particles have been simulated for δ = 0.75, 0.5 and 0.1 while a sample of 3600 particles has been investigated for the two-dimensional lattice. Comparisons are made with available simulations and with mean field theory. We find that the molecular field theory predictions worsen as the effective coordination number is decreased. Energy, specific heat, second and fourth rank order parameters have been evaluated for the various models. We also present, for the first time, a way of approximately reconstructing the pair distribution, G(r₁₂, ω₁₂), using maximum entropy and second and fourth rank two particle order parameters.     

Monte Carlo estimation of kinetic parameters in polymerization reactions

A general method for estimating kinetic parameters in polymerization reactions using Monte Carlo simulation to represent the models of the reactions is developed. From a statistical point of view, the procedure is a Bayesian one in which a posterior probability density surface (PPDS) is calculated for points on a grid in the parameter space. A smoothing function is fitted to the PPDS, then a posterior probability region, which is similar to a confidence region, is calculated for the parameters. An application to a relatively trivial example, the Mayo–Lewis copolymerization model is shown in detail. Many other potential applications are suggested.     

Monte Carlo simulation studies of polymer systems

Polymers molecules in solution or melt are more or less flexible and continuously change their shape and size. Thus, characteristic properties of the system fluctuate around statistical mean values which are dependent on the concentration of the solution, on the quality of the solvent used, and on the specific structure of the molecules, e.g. linear or star-branched. The most direct approach to these quantities on a molecular level are computer simulations. Due to restrictions of computer power fully atomistic simulations of macromolecules are presently still at the beginning but several arguments justify the use of simplified models. The most efficient way dealing with polymer systems are Monte Carlo simulations based on lattice chains, at least as long as static properties are of interest only. In the present paper a short introduction to the field is given and selected examples are presented in order to demonstrate the usefulness of these methods.     

Confinement effect in diffusion-controlled stepwise polymerization by Monte Carlo simulation

Diffusion-controlled stepwise polymerization of a linear polymer confined in nanoscopic slits is simulated through a Monte Carlo approach. A noticeable influence of the confinement on the kinetics is found. The confinement modifies both the spatial pair distribution function and the diffusive properties of the polymers. As a consequence, the confined system can show either faster or slower reaction kinetics with respect to the bulk system, depending on the strength of intermolecular interactions. The predicted polydispersity of the polymer is in agreement with recent theories of diffusion-controlled stepwise polymerization, and can be slightly affected by the confinement.