Monte Carlo simulation of colloidal membrane filtration: principal issues for modeling

The principal issues involved in developing a Monte Carlo simulation model of colloidal membrane filtration are investigated in this study. An important object for modeling is the physical dynamics responsible for causing particle deposition and accumulation when encountering an open system with continuous outflow. A periodic boundary condition offers a solution to the problem by recirculating continuous flow back through the system. Scaling to full physical dimensions will allow for release of the model from flawed assumptions such as constant cake layer volume fraction and thickness throughout the system. Furthermore, rigorous modeling on a precise scale extends the model to account for random particle collisions with acute accuracy. A major finding of this study proves that forces within the colloidal filtration system are summed and transferred cumulatively through the inter-particle interactions. The force summation and transfer phenomenon only realizes its true value when the model is scaled to full dimensions. The overall strategy for model development, therefore, entails three stages: first, rigorous modeling on a microscopic scale; next, comprehensive inclusion of relevant physical dynamics; and finally, scaling to full physical dimensions.     

Attractive interactions in dispersions of identical charged colloidal particles: a Monte Carlo simulation

Using the Debye-Hückel pair potential, the collective interactions between identical charged particles were examined via Monte Carlo simulations. The results have shown that when the number of charges per particle and the particle volume fraction were sufficiently large, the pair long-range electrostatic repulsion generated an effective attractive interaction between identical charged particles because of many-body effects. Disordered liquid-like structures, ordered crystal-like structures, ordered structures dispersed in disordered ones, and disordered structures dispersed in ordered ones have been found. The structures are dependent on the volume fraction and charge of the particles, as well as on the electrolyte concentration.     

Biomolecule-directed assembly of nanoscale building blocks studied via lattice Monte Carlo simulation

We perform lattice Monte Carlo simulations to study the self-assembly of functionalized inorganic nanoscale building blocks using recognitive biomolecule linkers. We develop a minimal coarse-grained lattice model for the nanoscale building block (NBB) and the recognitive linkers. Using this model, we explore the influence of the size ratio of linker length to NBB diameter on the assembly process and the structural properties of the resulting aggregates, including the spatial distribution of NBBs and aggregate topology. We find the constant-kernel Smoluchowski theory of diffusion-limited cluster-cluster aggregation describes the aggregation kinetics for certain size ratios.     

Mechanism of nanoparticle formation in self-assembled colloidal templates: population balance model and Monte Carlo simulation

Modeling the nanoparticle formation mechanism in water-in-oil microemulsion, a self-assembled colloidal template, has been addressed in this paper by two formalisms: the deterministic population balance equation (PBE) model and stochastic Monte Carlo (MC) simulation. These are based on time-scale analysis of elementary events consisting of reactant mass transport, solid solubilization, reaction, coalescence-exchange of drops, and finally nucleation and growth of nanoparticles. For the first time in such a PBE model, realistic binomial redistribution of molecules in the daughter drops (after coalescence-exchange of two drops) has been explicitly implemented. This has resulted in a very general model, applicable to processes with arbitrary relative rates of coalescence-exchange and nucleation. Both the deterministic and stochastic routes could account for the inherent randomness in the elementary events and successfully explained temporal evolution of mean and variance of nanoparticle size distribution. This has been illustrated by comparison with different yet broadly similar experiments, operating either under coalescence (lime carbonation to make CaCO(3) nanoparticles) or nucleation (hydride hydrolysis to make Ca(OH)(2) nanoparticles) dominant regimes. Our calculations are robust in being able to predict for very diverse process operation times: up to 26 min and 5 h for carbonation and hydrolysis experiments, respectively. Model predictions show that an increase in the external reactant addition rate to microemulsion solution is beneficial under certain general conditions, increasing the nanoparticle production rate significantly without any undesirable and perceptible change in particle size.     

Interaction between small colloidal particles and polymer chains in a semidilute solution: Monte Carlo simulation

 Interaction between flexible-chain polymers and small (nanometric) colloidal particles is studied by Monte Carlo simulation using two-dimensional and three-dimensional lattice models. Spatial distribution of colloidal particles and conformational characteristics of chains in a semidilute solution are considered as a function of the segment adsorption energy, ɛ. When adsorption is sufficiently strong, it induces effective attraction of polymer segments, which results in contraction of macromolecular coils. The strongly adsorbing polymer chains affect the equilibrium spatial distribution of the colloidal particles. The average size of colloidal aggregates <m> exhibits a nontrivial behavior: with ɛ increasing, the value of <m> first decreases and then begins to grow. The adsorption polycomplex formed at strong adsorption exhibits a mesoscopic scale of structural heterogeneity. The results of computer simulations are in a good agreement with predictions of the analytic theory [P.G. Khalatur, L.V. Zherenkova and A.R. Khokhlov (1997) J Phys II (France) 7:543] based on the integral RISM equation technique. Received: 4 August 1997 Accepted: 16 April 1998     

Monte Carlo simulation of DNA electrophoresis

This paper describes an attempt to study the electrophoresis mobility of a DNA molecule in a gel by means of a Monte Carlo simulation. We find that the electrophoresis mobility mu can be well described by the empirical equation mu v kappa 1/N + kappa 2E2 with N being the number of monomers of the model chain and E being the applied field. For small E the data can merge into the linear response result mu = kappa 1/N. The paper also discusses necessary extensions of the present approach.     

Monte Carlo simulation of osmotic equilibria

We present a Metropolis Monte Carlo simulation algorithm for the Tpπ-ensemble, where T is the temperature, p is the overall external pressure, and π is the osmotic pressure across the membrane. The algorithm, which can be applied to small molecules or sorption of small molecules in polymer networks, is tested for the case of Lennard-Jones interactions.     

Monte Carlo simulation of polymer adsorption

We developed and employed the incremental gauge cell method to calculate the chemical potential (and thus free energies) of long, flexible homopolymer chains of Lennard-Jones beads with harmonic bonds. The free energy of these chains was calculated with respect to three external conditions: in the zero-density bulk limit, confined in a spherical pore with hard walls, and confined in a spherical pore with attractive pores, the latter case being an analog of adsorption. Using the incremental gauge cell method, we calculated the incremental chemical potential of free polymer chains before and after the globual-random coil transitions. We also found that chains confined in attractive pores exhibit behaviors typical of low temperature physisorption isotherms, such as layering followed by capillary condensation.     

Order-parameter-based Monte Carlo simulation of crystallization

A Monte Carlo simulation method is presented for simulation of phase transitions, with emphasis on the study of crystallization. The method relies on a random walk in order parameter Phi(q(N)) space to calculate a free energy profile between the two coexisting phases. The energy and volume data generated over the course of the simulation are subsequently reweighed to identify the precise conditions for phase coexistence. The usefulness of the method is demonstrated in the context of crystallization of a purely repulsive Lennard-Jones system. A systematic analysis of precritical and critical nuclei as a function of supercooling reveals a gradual change from a bcc to a fcc structure inside the crystalline nucleus as it grows at large degrees of supercooling. The method is generally applicable and is expected to find applications in systems for which two or more coexisting phases can be distinguished through one or more order parameters.     

Monte Carlo simulation of hard spheroids

We present Monte Carlo simulations of the equation of state and radial distribution function for a model fluid composed of hard spheroids.     

Collective-variable Monte Carlo simulation of DNA

Monte Carlo simulations have been carried out on DNA oligomers using an internal coordinate model associated with a pseudorotational representation of sugar repuckering. It is shown that when this model is combined with the scaled collective variable approach of Noguti and Go, much more efficient simulations are obtained than with simple single variable steps. Application of this method to a DNA oligomer containing a recognition site for the TATA-box binding protein leads to striking similarities with results recently obtained from a 1-ns molecular dynamics simulation using explicit solvent and counterions. In particular, large amplitude bending fluctuations are observed directed toward the major groove. Conformational analysis of the Monte Carlo simulation shows clear base sequence effects on conformational fluctuations and also that the DNA energy hypersurface, like that of proteins, is complex with many local, conformational substates. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 2001–2011, 1997     

Phase behavior of colloidal hard tetragonal parallelepipeds (cuboids): a Monte Carlo simulation study

The impact of particle geometry on the phase behavior of hard colloidal tetragonal parallelepipeds (TPs) was studied by using Monte Carlo simulations in continuum space. TPs or "cuboids" of aspect ratios varying from 0.25 to 8 were simulated by approximating their shapes with multisite objects, i.e., via rigid clusters of hard spheres. Using equation of state curves, order parameters, radial distribution functions, particle distribution functions along three directions, and visual analysis of configurations, an approximate phase diagram for the TPs was mapped out as a function of aspect ratio (r) and volume fraction. For r > 3 and intermediate concentrations, the behavior of the TPs was similar to that of spherocylinders, exhibiting similar liquid crystalline mesophases (e.g., nematic and smectic phases). For r = 1, a cubatic phase occurs with orientational order along the three axes but with little translational order. For 1 < r < 4, the TPs exhibit a cubatic-like mesophase with a high degree of order along three axes where the major axes of the particles are not all aligned in the same direction. For r < 1, the TPs exhibit a smectic-like phase where the particles have rotational freedom in each layer but form stacks with tetratic order. The equation of state for perfect hard cubes (r = 1) was also simulated and found to be consistent with that of the rounded-edge r = 1 TPs, except for its lack of discontinuity at the cubatic-solid transition.     

Monte Carlo simulation of catalytic CO oxidation

A simple Monte Carlo model of the CO oxidation on a single-crystal catalyst surface is presented. The simulation model considers the following elementary reaction steps:

1. (1) chemisorption of a CO molecule, its surface migration and possible desorption

2. (2) physisorption of an O₂ molecule to a precursor state and its subsequent dissociative chemisorption

3. (3) activated reaction of adsorbed O and CO (the Langmuir - Hinshelwood reaction mechanism), formation of CO₂ and its rapid desorption.

The changes in the activation energy of reaction and in the adsorption energy of CO resulting from the interactions between adsorbed species are also considered. The model makes possible to monitor temperature programmed reaction spectra or reaction spectra obtained during changes of the ratio of the partial pressures of CO and O₂. The results of simulations for a Pd(111) single-crystal plane are compared with experiment.     

Path-integral Monte Carlo simulation of the recombination of two Al atoms embedded in parahydrogen

We report the use of path-integral Monte Carlo (PIMC) simulations in the study of the stability against recombination of two Al atoms trapped in solid parahydrogen (pH2) at 4 K. The many-body interactions involving open-shell Al atoms are described with a pairwise additive Hamiltonian model. To estimate the lifetime against recombination, we use PIMC simulations to define an effective potential averaged over the position of the pH2 molecules, followed by a transition-state treatment. Different initial embedding sites are explored. If the initial substitution sites are within a distance of approximately 13 bohrs, the Al atoms will significantly distort the lattice structure to allow recombination, with an accompanying release of energy during the process. For substitution distances longer than approximately 14 bohrs, the dispersion of Al atoms is shown to be metastable, with lifetimes varying from approximately 30 min to several days. The electronic anisotropy is a factor that helps to stabilize the dispersion.     

Reverse Monte Carlo modeling of the structure of colloidal aggregates

In this work we present results for the structure of aerogels coming from the diffusion-limited cluster aggregation simulation method. Pair distribution functions and structure factors, resulting from simulation, were considered as experimental input for reverse Monte Carlo modeling. The modeling yielded structural models with pair distribution functions and structure factors nearly identical to the results of the simulations. Particle configurations from both the simulations and reverse Monte Carlo modeling have been analyzed in terms of the distribution of the number of neighbors. It is suggested that the reverse Monte Carlo method, when applied to the structure factor, may be a suitable technique for the interpretation of experimental scattering data on colloidal aerogels.     

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.     

Monte Carlo simulation study of droplet nucleation

A new rigorous Monte Carlo simulation approach is employed to study nucleation barriers for droplets in Lennard-Jones fluid. Using the gauge cell method we generate the excess isotherm of critical clusters in the size range from two to six molecular diameters. The ghost field method is employed to compute the cluster free energy and the nucleation barrier with desired precision of (1-2)kT. Based on quantitative results obtained by Monte Carlo simulations, we access the limits of applicability of the capillarity approximation of the classical nucleation theory and the Tolman equation. We show that the capillarity approximation corrected for vapor nonideality and liquid compressibility provides a reasonable assessment for the size of critical clusters in Lennard-Jones fluid; however, its accuracy is not sufficient to predict the nucleation barriers for making practical estimates of the rate of nucleation. The established dependence of the droplet surface tension on the droplet size cannot be approximated by the Tolman equation for small droplets of radius less than four molecular diameters. We confirm the conclusion of ten Wolde and Frenkel [J. Chem. Phys. 109, 9901 (1998)] that integration of the normal component of the Irving-Kirkwood pressure tensor severely underestimates the nucleation barriers for small clusters.     

Monte Carlo simulation of pulsed laser polymerization

Pulsed laser polymerization (PLP) has been simulated using a Monte Carlo procedure. From the results of numerous simulations it has been shown that the molecular weight distribution (MWD) consists primarily of two superimposed distributions. One distribution, a relatively broad background, represents the termination reactions during the dark period; the other, a rather sharply peaked distribution, represents the termination reactions occurring as a consequence of the large number of small radicals produced during the laser pulse. The postulate that the inflection point on the sharp peak can be used to calculate that the propagation rate constant was tested and found to be accurate to within 3%. The relative position of the broad and sharp distributions on the chain length scale determines the qualitative appearance of the overall MWD and is in turn governed by the rate of photoinitiation and the relative values of termination and propagation rate constants. This explains the qualitatively different shapes of MWD which have been experimentally observed. Finally, it is shown that the occurrence of chain length dependent termination reactions precludes the use of an analytical expression to deduce quantitative or qualitative information about the termination reaction from PLP data.     

Donnan potential of dilute colloidal dispersions: Monte Carlo simulations

Donnan equilibrium between a salt-free colloidal dispersion and an electrolyte solution has been investigated by Monte Carlo simulations. The Donnan potential is directly calculated by considering two compartments separated by a semipermeable membrane. In order to understand the role played by colloid–ion interactions, the influences of colloidal characteristics, including particle size R, intrinsic particle charge Z, counterion valency z_c, and concentration c_p, on Donnan potential Ψ_D and effective charge Z_eff are examined. Our simulations show that the electroneutrality condition is not followed in both compartments and the Donnan potential arises due to the net charge density. The Donnan potential grows by increasing c_p and Z_eff and by decreasing dielectric constant ε_r, i.e., Ψ_DZ_effc_p/ε_r approximately. Note that the effective charge varies with R,Z,c_p,ε_r and z_c as well. When the salt concentration is increased, the net charge density is lowered and thus the Donnan potential decays accordingly. The validity of the classical theory based on the Nernst equation and the electroneutrality assumption is also examined. In general, the simulation results at high-salt condition can be well represented by such mean-field theory.