Trotter derivation of algorithms for Brownian and dissipative particle dynamics

This paper focuses on the temporal discretization of the Langevin dynamics, and on different resulting numerical integration schemes. Using a method based on the exponentiation of time dependent operators, we carefully derive a numerical scheme for the Langevin dynamics, which we found equivalent to the proposal of Ermak and Buckholtz [J. Comput. Phys. 35, 169 (1980)] and not simply to the stochastic version of the velocity-Verlet algorithm. However, we checked on numerical simulations that both algorithms give similar results, and share the same "weak order two" accuracy. We then apply the same strategy to derive and test two numerical schemes for the dissipative particle dynamics. The first one of them was found to compare well, in terms of speed and accuracy, with the best currently available algorithms.     

Brownian dynamics simulation and experimental study of colloidal particle deposition in a microchannel flow

This paper reports an analysis of the irreversible deposition of colloidal particles from the pressure-driven flow in a microchannel within the framework of DLVO theory. A theoretical model is presented on the basis of the stochastic Langevin equation, incorporating the random Brownian motion of colloidal particles. Brownian dynamics simulation is used to compute the particle deposition in terms of the surface coverage. To validate the theoretical model, experiments are carried out using the parallel-plate flow cell technique, enabling direct videomicroscopic observation of the deposition kinetics of polystyrene latex particles in NaCl electrolytes. The theoretical predictions are compared with the experimental results, and good agreement is found.     

Interfacing Brownian dynamics simulations

Starting from the flux of particles in a Brownian dynamics simulation we derive boundary conditions, which allow us (i) to couple a Brownian dynamics calculation to a reservoir of particles of a given density, i.e., setting up constant density boundary conditions, and (ii) to build an interface between Brownian dynamics and a diffusional treatment of adjacent simulation volumes. With these algorithms it is sometimes possible to dramatically reduce the system size--and therefore the necessary resources--of multiparticle Brownian dynamics calculations. In this paper we give one-dimensional examples which illustrate potential applications and savings.     

Brownian motion of a polymer-bound colloidal particle

A simple model of Brownian motion of a colloidal particle attached to the flat surface by a polymer thread is presented. The model results are discussed in terms of the measurability of the elastic properties of a linear polymer chain in the dispersion medium by observation of the motion of a colloidal particle connected to a fixed point by the polymer. Received: 11 November 1998 Accepted: 9 December 1998     

Fractional Brownian dynamics in proteins

Correlation functions describing relaxation processes in proteins and other complex molecular systems are known to exhibit a nonexponential decay. The simulation study presented here shows that fractional Brownian dynamics is a good model for the internal dynamics of a lysozyme molecule in solution. We show that both the dynamic structure factor and the associated memory function fit well the corresponding analytical functions calculated from the model. The numerical analysis is based on autoregressive modeling of time series.     

Adsorption of charged dendrimers: a Brownian dynamics study

The adsorption of isolated charged dendrimers onto oppositely charged flat surfaces is studied in this work using Brownian dynamics simulations. The dendrimer is modeled as a freely jointed bead-rod chain in which excluded-volume interactions are modeled by a repulsive Lennard-Jones potential and bead-bead and bead-surface electrostatic interactions are described by screened Coulombic potentials. Adsorption behavior is studied as a function of inverse screening length, dendrimer generation, and dendrimer charge distribution. Adsorbed dendrimers adopt a disclike conformation in which they flatten in the direction normal to the surface and expand in the direction parallel to the surface. As the inverse screening length increases, the dendrimer expands in the normal direction and contracts in the parallel direction, adopting a conformation that is more stretched in the normal direction. When the inverse screening length becomes sufficiently large, the dendrimer desorbs and adopts a spherelike conformation. Bead density profiles show that adsorbed dendrimers form a two-layer structure, with one layer corresponding to adsorbed beads and a second, less dense layer corresponding to beads one rod length away from the surface. They also reveal how the distribution of monomers within the dendrimer and near the surface can be tailored by changing various problem parameters. The results presented here are expected to be helpful in providing qualitative guidance for dendrimer design in various applications.     

Brownian dynamics of mixed surfactant micelles

We investigate micelle formation in a system containing two or more different amphiphiles with different geometries using a stochastic molecular-dynamics (MD) simulation method. For a binary system containing two amphiphiles, we calculate the critical micelle concentration (CMC) and cluster distribution for the mixture at several mole fractions and compare the simulation results with those predicted by analytic theories in the dilute limit and with experiments. We find that the CMC obtained from molecular mean-field theory agrees well with our simulation results. Motivated by the industrial use of mixed surfactant systems, we then extend our studies to a system containing six different chain lengths drawn from a Poisson distribution. We find that unlike a binary mixture of amphiphiles, the different species cancel the effects of each other so that the cluster distribution for the mixture has a shape of a system consisted entirely of amphiphiles of length equal to the mean chain length of the Poisson distribution.     

Brownian dynamics simulation of protein association

Summary The Brownian Dynamics (BD) method is applied to study the diffusive dynamics and interaction of two proteins, cytochrome c (CYTC) and cytochrome c peroxidase (CYP). We examine the role of protein electrostatic charge distribution in the facilitation of protein-protein docking prior to the electron transfer step, assessing the influence of individual charged amino acid residues. Accurate interaction potentials are computed by iterating the linearized Poisson-Boltzmann (PB) equation around the larger protein CYP. The low dielectric constant inside proteins, electrolyte screening effects and irregular protein surface topography are taken into account. We observe a large ensemble of electrostatically stable encounter complexes seemingly with acceptable geometric requirements for electron transfer rather than a single dominant complex. Stabilities of the large variety of docking complexes are rationalized in terms of generalized charged residue complementarities. However, it is found that the electrostatic interactions giving rise to complex stabilities are somewhat nonspecific in nature. A large series of additional simulations are performed in which individual charged residues on CYTC have been chemically modified. Resulting perturbations of the association rate are significant and qualitatively similar to results observed in comparable kinetics experiments. We therefore demonstrate the potential of the Brownian dynamics method to estimate the effects of site-directed mutagenesis on protein-protein and protein-ligand diffusional association rates.     

Brownian dynamics studies of dilute dispersions

The method of brownian dynamics of used to study the non-equilibrium properties of very dilute colloids electrostatically stabilised in dilute aqueous electrolyte. It is assumed that the colloid is a monodisperse system of structureless spherical particles embedded in a hydrodynamic continuum. Although the particles are interacting electrostatically through a screened Coulomb potential, the dilution is such that effects arising from coupling of hydrodynamic flow can be ignore. Studies of the self-diffusion coefficient and van Howe functions show that after an initial period, during which the particles move essentially independently, the flow properties of the colloids are significantly different from those expected on the basis of free brownian motion.     

Communication: Driven Brownian transport in eccentric septate channels

In eccentric septate channels the pores connecting adjacent compartments are shifted off-axis, either periodically or randomly, so that straight trajectories parallel to the axis are not allowed. Driven transport of a Brownian particle in such a channel is characterized by a strong suppression of the current and its dispersion. For large driving forces, both quantities approach an asymptotic value, which can be analytically approximated in terms of the stationary distribution of the particle exit times out of a single channel compartment.     

Accurate particle size distribution determination by nanoparticle tracking analysis based on 2-D Brownian dynamics simulation

A physical model is presented to simulate the average step length distribution during nanoparticle tracking analysis experiments as a function of the particle size distribution and the distribution of the number of steps within the tracks. Considering only tracks of at least five steps, numerical simulation could be replaced by a normal distribution approximation. Based on this model, simulation of a step length distribution allows obtaining a much more reliable estimation of the particle size distribution, thereby reducing the artificial broadening of the distribution, as is typically observed by direct conversion of step length to particle size data. As this fitting procedure also allowed including data from particles that were followed for a relatively low number of steps, the measurement time could be reduced for particles that are known to be monodisperse. Whereas the inversion is less sensitive towards the particle size distribution width, still similar values were obtained for both the average diameter and standard deviation of a polystyrene latex sample irrespective of the track length, provided that the latter included at least five steps.     

Communication: translational Brownian motion for particles of arbitrary shape

A single Brownian particle of arbitrary shape is considered. The time-dependent translational mean square displacement W(t) of a reference point at this particle is evaluated from the Smoluchowski equation. It is shown that at times larger than the characteristic time scale of the rotational Brownian relaxation, the slope of W(t) becomes independent of the choice of a reference point. Moreover, it is proved that in the long-time limit, the slope of W(t) is determined uniquely by the trace of the translational-translational mobility matrix μ(tt) evaluated with respect to the hydrodynamic center of mobility. The result is applicable to dynamic light scattering measurements, which indeed are performed in the long-time limit.     

Diffusion coefficient of Brownian particle in rough micro-channel

This study examines the feasibility of using of the lattice Boltzmann method to determine how the surface roughness of a quadrate channel affects the diffusion coefficient of Brownian particle(s). The surface was represented by a regular array of spheres. Surface roughness reduced the diffusion coefficient of the Brownian particle(s) because of a change in the velocity autocorrelation function decay and in pressure. Additionally, the neighboring particles increased the diffusion coefficient of Brownian particle.     

Brownian dynamics of grafted polymer chains: time dependent properties

Results of computer simulations of polymer layers consisting of chains grafted by one end on an unpenetrable plane are presented. Characteristics of translational and rotational motion of different chain segments and correlation functions of chain radii were calculated both for single layers at different grafting densities s and for two interacting layers at different distances D between parallel grafting planes. Two values of grafting density were used in the latter case. The behavior of different correlation times as function of s and D and the interplay between the interpenetration of the brushes and rotational and translational motion are discussed. Both relaxation functions and mean square displacements are discussed in terms of stretched exponentials, and the behavior of the resulting “Kohlrausch exponents” γ is presented in detail.     

Event-driven Brownian dynamics for hard spheres

Brownian dynamics algorithms integrate Langevin equations numerically and allow to probe long time scales in simulations. A common requirement for such algorithms is that interactions in the system should vary little during an integration time step; therefore, computational efficiency worsens as the interactions become steeper. In the extreme case of hard-body interactions, standard numerical integrators become ill defined. Several approximate schemes have been invented to handle such cases, but little emphasis has been placed on testing the correctness of the integration scheme. Starting from the two-body Smoluchowski equation, the authors discuss a general method for the overdamped Brownian dynamics of hard spheres, recently developed by one of the authors. They test the accuracy of the algorithm and demonstrate its convergence for a number of analytically tractable test cases.     

Brownian dynamics simulations of associating diblock copolymers

A novel coarse-grained computational model for associating polymers is proposed that is based on a Gaussian "blob" representation of the polymer chains. The model allows a large number of model polymers to be simulated at moderate computational cost over a wide packing fraction range using the Brownian dynamics, BD, technique. The attraction of the hydrophobic part of the polymer to those on other molecules can lead to strong aggregation of the polymer molecules in real systems, and this is included in the model by an attractive potential felt by the Gaussian blobs to a common "nodal" point that represents the center of the micelle. Attention here is confined to model AB diblock copolymers in which the hydrophilic block, A, has a much higher mass than the hydrophobic moiety, B, which leads to relatively small aggregation numbers, Nagg, of approximately 8. The aggregation number at low packing fractions is found to increase with packing fraction, as observed in experiments, with a functional form that closely follows a simple theory derived here that is based on entropy-derived mean-field terms for the free-energy change associated with the incorporation of the polymer molecule into the micelle. The computational model exhibits an extremely low critical micelle concentration (cmc), and micelles with Nagg approximately 5 are observed at the lowest packing fractions, phi, simulated ( approximately 10-4), which is consistent with experiment. The long-time self-diffusion coefficient of the polymers (and hence micelles) decreases logarithmically with packing fraction, and the viscosity increased with concentration according to the Huggins equation. The spherical blob coarse graining results in the simulable time scales being longer than the Rouse time of the chain, and hence for the nonassociating polymers the intrinsic viscosity is an input parameter in the model. The introduction of association leads to the partial inclusion of the intrinsic viscosity in the simulation and has an effect on the computed Huggins coefficient, kH, which is found to be approximately 6 in those cases.     

Atomistic Brownian dynamics simulation of peptide phosphorylation

We report the implementation of an all-atom Brownian dynamics simulation model of peptides using the constraint algorithm LINCS. The algorithm has been added as a part of UHBD. It uses adaptive time steps to achieve a balance between computational speed and stability. The algorithm was applied to study the effect of phosphorylation on the conformational preference of the peptide Gly-Ser-Ser-Ser. We find that the middle serine residue experiences considerable conformational change from the C(7eq) to the alpha(R) structure upon phosphorylation. NMR (3)J coupling constants were also computed from the Brownian trajectories using the Karplus equation. The calculated (3)J results agree reasonably well with experimental data for phosphorylated peptide but less so for doubly charged phosphorylated one.     

Brownian dynamics simulation of grafted polymer brushes

We present results of computer simulations by the method of Brownian dynamics of polymeric brushes attached to impenetrable planes. For testing both model and method we have used one polymer brush attached to a repulsive plane and compare some results with Monte Carlo results of Lai and Binder on the bond fluctuation model. We have also studied two polymeric brushes attached to two parallel planes at different distances between planes, and investigate the interplay between the interpenetration of the brushes and the configurational properties of the grafted chains.     

Melting in two-dimensional Yukawa systems: a Brownian dynamics simulation

We studied the melting behavior of two-dimensional colloidal crystals with a Yukawa pair potential by Brownian dynamics simulations. The melting follows the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) scenario with two continuous phase transitions and a middle hexatic phase. The two phase-transition points were accurately identified from the divergence of the translational and orientational susceptibilities. Configurational temperatures were employed to monitor the equilibrium of the overdamped system and the strongest temperature fluctuation was observed in the hexatic phase. The inherent structure obtained by rapid quenching exhibits three different behaviors in the solid, hexatic, and liquid phases. The measured core energy of the free dislocations, E(c) = 7.81 ± 0.91 k(B)T, is larger than the critical value of 2.84 k(B)T, which consistently supports the KTHNY melting scenario.