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.     

Brownian dynamics simulation study on the self-assembly of incompatible star-like block copolymers in dilute solution

We study the self-assembly of symmetric star-like block copolymers (A(x))(y)(B(x))(y)C in dilute solution by using Brownian dynamics simulations. In the star-like block copolymer, incompatible A and B components are both solvophobic, and connected to the center bead C of the polymer. Therefore, this star-like block copolymer can be taken as a representative of soft and deformable Janus particles. In our Brownian dynamics simulations, these "soft Janus particles" are found to self-assemble into worm-like lamellar structures, loose aggregates and so on. By systematically varying solvent conditions and temperature, we build up the phase diagram to illustrate the effects of polymer structure and temperature on the aggregate structures. At lower temperatures, we can observe large worm-like lamellar aggregates. Upon increasing the temperature, some block copolymers detach from the aggregate; this phenomenon is especially sensitive for the polymers with less arms. The aggregate structure will be quite disordered when the temperature is high. The incompatibility between the two parts in the star-like block copolymer also affects the self-assembled structures. We find that the worm-like structure is longer and narrower as the incompatibility between the two parts is stronger.     

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.     

Self-assembly behavior of ABA coil-rod-coil triblock copolymers: a Brownian dynamics simulation approach

The self-assembly behavior of ABA coil-rod-coil triblock copolymers in a selective solvent was studied by a Brownian molecular dynamics simulation method. It was found that the rod midblock plays an important role in the self-assembly of the copolymers. With a decrease in the segregation strength, ?(RR), of rod pairs, the aggregate structure first varies from a smecticlike disk shape to a long twisted string micelle and then to small aggregates. The influence of the block length and the asymmetry of the triblock copolymer on the phase behavior were studied and the corresponding phase diagrams were mapped. It was revealed that the variation of these parameters has a profound effect on microstructure. The simulation results are consistent with experimental results. Compared to rod-coil diblock copolymers, the coil-rod-coil triblock copolymers has a larger entropy penalty associated with the interfacial grafting density of the aggregate, leading to a higher ?(RR) value for structural transitions.     

Cell dynamics simulations of block copolymers

Cell dynamics simulations are a powerful tool to simulate kinetic processes in phase separating systems. Here we review the technique and its application to block copolymers. Its advantages and disadvantages compared to other simulation methods for block copolymer structure and dynamics are reviewed. Results on the dynamics of microphase separation and interface propagation, and on the rate of order‐order phase transitions are reviewed. The use of the method to model certain shear‐induced structural and flow effects is also summarised.     

Coarse-grained molecular-dynamics simulations of the self-assembly of pentablock copolymers into micelles

Multiblock polymers in aqueous solution, where one or several blocks are hydrophobic, exhibit a rich variety of phases and states of aggregation. In this paper, we investigate a pentablock system ABCBA, where the B block is always hydrophilic and the A and C blocks have varying degrees of hydrophobicity depending on external conditions. We report coarse-grained molecular-dynamics simulations where the solvent is included explicitly and monomers interact via a 6-9 Lennard Jones potential function. The hydrophobic interaction is modeled by tuning the parameter controlling the strength of the interaction between the hydrophobic monomers and the solvent. We investigate the structure and morphology of the micelles for two concrete situations representing changes in temperature and the pH level. The simulated system is directly relevant to a recently synthesized pentablock system consisting of a triblock Pluronic with an added pH-sensitive end group [B. C. Anderson et al., Macromolecules 36, 1670 (2003)].     

Brownian dynamics simulations on CPU and GPU with BD_BOX

There has been growing interest in simulating biological processes under in vivo conditions due to recent advances in experimental techniques dedicated to study single particle behavior in crowded environments. We have developed a software package, BD_BOX, for multiscale Brownian dynamics simulations. BD_BOX can simulate either single molecules or multicomponent systems of diverse, interacting molecular species using flexible, coarse-grained bead models. BD_BOX is written in C and employs modern computer architectures and technologies; these include MPI for distributed-memory architectures, OpenMP for shared-memory platforms, NVIDIA CUDA framework for GPGPU, and SSE vectorization for CPU.     

Treadmilling of actin filaments via Brownian dynamics simulations

Actin polymerization is coupled to the hydrolysis of adenosine triphosphate (ATP) into adenosine diphosphate (ADP) and inorganic phosphate (P(i)). Therefore, each protomer within an actin filament can attain three different nucleotide states corresponding to bound ATP, ADP/P(i), and ADP. These protomer states form spatial patterns on the growing (or shrinking) filaments. Using Brownian dynamics simulations, the growth behavior of long filaments is studied, together with the associated protomer patterns, as a function of ATP-actin monomer concentration, C(T), within the surrounding solution. For concentrations close to the critical concentration C(T)=C(T,cr), the filaments undergo treadmilling, i.e., they grow at the barbed and shrink at the pointed end, which leads to directed translational motion of the whole filament. The corresponding nonequilibrium states are characterized by several global fluxes and by spatial density and flux profiles along the filaments. We focus on a certain set of transition rates as deduced from in vitro experiments and find that the associated treadmilling (or turnover) rate is about 0.08 monomers per second.     

Micellization behavior of star-block copolymers in a selective solvent: A Brownian dynamics simulation approach

Micellization behavior of (AB)n type star-block copolymer in a selective solvent for its outer block is investigated by using a Brownian dynamics simulation. Micellar properties are compared in terms of the arm number (n) of star-block copolymer. It is observed that the critical micelle concentration (cmc) shows a minimum when the cmc is plotted against the arm number. The star-block copolymer with longer soluble block shows the cmc minimum at smaller arm number than that with shorter soluble block. Although the star-block copolymer with multiple arms forms more compact core as compared to diblock copolymer, the average aggregation number is inversely proportional to the arm number (approximately 1/n), which implies that the micelle size is invariant with the arm number. Theoretical predictions based on a simple mean field theory agree qualitatively well with the simulation results.     

Dissipative particle dynamics simulations of polymer-protected nanoparticle self-assembly

Dissipative particle dynamics simulations were used to study the effects of mixing time, solute solubility, solute and diblock copolymer concentrations, and copolymer block length on the rapid coprecipitation of polymer-protected nanoparticles. The simulations were aimed at modeling Flash NanoPrecipitation, a process in which hydrophobic solutes and amphiphilic block copolymers are dissolved in a water-miscible organic solvent and then rapidly mixed with water to produce composite nanoparticles. A previously developed model by Spaeth et al. [J. Chem. Phys. 134, 164902 (2011)] was used. The model was parameterized to reproduce equilibrium and transport properties of the solvent, hydrophobic solute, and diblock copolymer. Anti-solvent mixing was modeled using time-dependent solvent-solute and solvent-copolymer interactions. We find that particle size increases with mixing time, due to the difference in solute and polymer solubilities. Increasing the solubility of the solute leads to larger nanoparticles for unfavorable solute-polymer interactions and to smaller nanoparticles for favorable solute-polymer interactions. A decrease in overall solute and polymer concentration produces smaller nanoparticles, because the difference in the diffusion coefficients of a single polymer and of larger clusters becomes more important to their relative rates of collisions under more dilute conditions. An increase in the solute-polymer ratio produces larger nanoparticles, since a collection of large particles has less surface area than a collection of small particles with the same total volume. An increase in the hydrophilic block length of the polymer leads to smaller nanoparticles, due to an enhanced ability of each polymer to shield the nanoparticle core. For unfavorable solute-polymer interactions, the nanoparticle size increases with hydrophobic block length. However, for favorable solute-polymer interactions, nanoparticle size exhibits a local minimum with respect to the hydrophobic block length. Our results provide insights on ways in which experimentally controllable parameters of the Flash NanoPrecipitation process can be used to influence aggregate size and composition during self-assembly.     

Modeling dimerizations of transmembrane proteins using Brownian dynamics simulations

The dimerizations of membrane proteins, Outer Membrane Phospholipase A (OMPLA) and glycophorin A (GPA), have been simulated by an adapted Brownian Dynamics program. To mimic the membrane protein environment, we introduced a hybrid electrostatic potential map of membrane and water for electrostatic interaction calculations. We added a van der Waals potential term to the force field of the current version of the BD program to simulate the short-range interactions of the two monomers. We reduced the BD sampling space from three dimensions to two dimensions to improve the efficiency of BD simulations for membrane proteins. The OMPLA and GPA dimers predicted by our 2D-BD simulation and structural refinement is in good agreement with the experimental structures. The adapted 2D-BD method could be used for prediction of dimerization of other membrane proteins, such as G protein-coupled receptors, to help better understanding of the structures and functions of membrane proteins.     

Brownian dynamics simulations of polyelectrolyte adsorption in shear flow

Brownian dynamics simulations are used to study the adsorption of an isolated polyelectrolyte molecule onto an oppositely charged flat surface in the absence and the presence of an imposed shear flow. The polyelectrolyte is modeled as a freely jointed bead-rod chain where excluded volume interactions are incorporated by using a hard-sphere potential. The total charge along the backbone is distributed uniformly among all the beads, and the beads are allowed to interact with one another and the charged surface through screened Coulombic interactions. The simulations are performed by placing the molecule a fixed distance above the surface, and the adsorption behavior is then studied as a function of screening length. In the absence of an imposed flow, the chain is found to lie flat and extended on the adsorbing surface in the limit of weak screening, whereas in the limit of strong screening it desorbs from the surface and attains free-solution behavior. For intermediate screening, only a small portion of the chain adsorbs and it becomes highly extended in the direction normal to the surface. An imposed shear flow tends to orient the chain in the direction of flow and also leads to increased contact of the chain with the surface.     

Brownian dynamics simulations of polyelectrolyte adsorption onto topographically patterned surfaces

The effect of patterned surface topography on the adsorption of single polyelectrolyte molecules is explored using Brownian dynamics simulations. The polyelectrolyte is modeled as a free-draining, freely jointed bead-rod chain, and electrostatic interactions are incorporated using a screened Coulombic potential with excluded volume interactions accounted for by the repulsive part of a Lennard-Jones potential. Topography consisting of periodically spaced valleys of square cross section separated by flat hills is considered. Chain conformations are characterized for a wide range of valley widths, depths, and spacings as well as for several different types of surface charge distributions. Depending on the parameter values describing the topography, the chains are found to adopt conformations ranging from flat and extended to those associated with bridge-, brush-, or semi-bridge-like structures. The formation of these structures is rationalized on the basis of a free-energy model that takes into account the increase in free energy due to entropic confinement, excluded volume interactions, and chain stretching as well as the decrease in free energy due to bead-surface electrostatic attraction. The results of this work are expected to be useful in designing patterned surface topography to control the conformations of adsorbed polyelectrolyte molecules.     

Brownian dynamics simulations of polyelectrolyte adsorption onto charged patterned surfaces

The adsorption of single polyelectrolyte molecules onto surfaces decorated with periodic arrays of charged patches was studied using Brownian dynamics simulations. A free-draining, freely jointed bead-rod chain was used to model the polyelectrolyte, and electrostatic interactions were incorporated using a screened Coulombic potential with the excluded volume accounted for by a hard-sphere potential. The simulations predicted that the polyelectrolyte lies close to the adsorbing surface if the patch length, surface charge density, and screening length are sufficiently large. Chain conformations were found to be very sensitive to patch length, patch spacing, and the nature of the charge on adjacent patches. This is due both to the size of the polymer relative to patch length and spacing and to the structure of the electric field near the surface. In some cases, the component of the radius of gyration parallel to the surface can be made smaller than its free-solution value, which is contrary to what is observed for a uniformly charged surface. Isolated charged patches were also considered, and significant adsorption was observed above a critical surface charge density. The results demonstrate how polyelectrolyte conformations can be controlled by the design of the charged patches and may be useful for applications in which adsorbed polyelectrolyte films play a key role.     

Domain growth dynamics and Freedericksz transitions of liquid crystals: Brownian dynamics simulations on lattices

A Brownian dynamics simulation using the Lebwohl-Lasher (LL) nematogen model is developed to investigate liquid crystals (LCs) on two-dimensional lattices. According to the examination of defect annihilation and domain growth, this methodology appears to be a successful approach in studying the LC dynamics, its main advantage being that the time evolution in the simulation mimics the physical time, in contrast to related modellings published recently. In addition, the Freedericksz transition of an LC under static magnetic fields is reproduced. Further analysis of the simulation outputs and the associated continuum theory of LC elasticity reveals that the LL potential can, by analogy, be obtained from the Frank energy. As a result, the methodology of the combination of Brownian dynamics simulation with the LL model is proven valid and can be extended to study the director dynamics of LCs subjected to external fields.     

Brownian dynamics simulation study of self-assembly of amphiphiles with large hydrophilic heads

We have studied the effect of shape of an amphiphilic molecule on micellization properties by carrying out stochastic molecular dynamics simulation on a bead-spring model of amphiphiles for several sizes of hydrophilic head group with a fixed hydrophobic tail length. Our studies show that the effect of geometry of an amphiphile on shape and cluster distribution of micelles is significant. We find the critical micelle concentration increases with the increasing size of the hydrophilic head. We demonstrate that the onset of micellization is accompanied by (i) a peak in the specific heat as found earlier in the simulation studies of lattice models, and (ii) a peak in the characteristic relaxation time of the cluster autocorrelation function. Amphiphiles with larger hydrophilic head form smaller micelles with sharper cluster distribution. Our studies are relevant to the controlled synthesis of nanostructures of desired shapes and sizes using self-assembling properties of amphiphiles.     

Synthesis and dissipative particle dynamics simulation of cross-linkable fluorinated diblock copolymers: self-assembly aggregation behavior in different solvents

Developing microstructures, such as low molecular aggregates, spherical micelles and multi-compartment micelles, is an expanding area of research in Materials Science. By applying an atom transfer radical polymerization (ATRP) process to cross-linkable fluorinated diblock copolymers and analyzing the data we are able to demonstrate the potential for developing films with different micro-structures for additional biological research. Applying the Dissipative Particle Dynamic (DPD) Method, Transmission Electron Microscopy (TEM) and Scanning Electron Microscopy (SEM) techniques to cross-linkable fluorinated diblock copolymers of (methyl methacrylate-co-hydroxyethyl methacrylate-co-butyl methacrylate)-b-2-(perfluoroalkyl)ethyl methacrylate (MMA-co-HEMA-co-BMA-b-FMA) we were able to analyze the structures and their relationships to the aggregation of various microstructure formations through the use of various solvents in the process. For the self-assembly of the cross-linkable diblock copolymer in solutions, the DPD simulation results are only in qualitative agreement with experimental data of aggregate morphologies and sizes. This suggests an improved approach to creating materials and methods necessary for studying microstructures in films used in other research areas. Our work examines whether using selective solvents can be easily extended to prepare aggregates with different morphologies, which is an effective shortcut to obtain films with different microstructures. DPD simulation can be considered as an adjunct to experiments and provides other valuable information for the experiment.     

Synthesis of AB2 star-shaped miktoarm copolymers and their crystallization behavior

A small molecule (GMS-SA₂) with one alkyl chain and two terminal carboxyl groups was synthesized successfully by the reaction of glyceryl monostearate (GMS) with excess succinic anhydride (SA). Then, GMS-SA₂ was used as a coupling agent to condensate with polyethylene glycols (PEG) of different molecular weight or polyethylene glycol monomethyl ether (PEGm) in the presence of stannous octoate as catalyst and diphenyl ether as azeotropic agent. The AB₂ star-shaped miktoarm copolymers were obtained successfully and were characterized by ¹H-NMR, DSC, GPC, XRD, FTIR and polarizing microscopy. The results of DSC and XRD measurements indicate that the crystallization temperature and the melting temperature of the AB₂ star-shaped miktoarm copolymers are different from those of the corresponding linear PEGs, because the existing of GMS-SA₂ alters their crystalline growth velocity and the perfect degree of crystals. It is very important to control the crystal morphology of star-shaped copolymers by introducing the miktoarm into the starshaped polymers and adjusting its content in star-shaped polymers. __________ Translated from Chemical Journal of Chinese Universities, 2007, 28(7): 1365–1370 [译自: 高等学校化学学报]     

Reactions on cell membranes: comparison of continuum theory and Brownian dynamics simulations

Biochemical transduction of signals received by living cells typically involves molecular interactions and enzyme-mediated reactions at the cell membrane, a problem that is analogous to reacting species on a catalyst surface or interface. We have developed an efficient Brownian dynamics algorithm that is especially suited for such systems and have compared the simulation results with various continuum theories through prediction of effective enzymatic rate constant values. We specifically consider reaction versus diffusion limitation, the effect of increasing enzyme density, and the spontaneous membrane association/dissociation of enzyme molecules. In all cases, we find the theory and simulations to be in quantitative agreement. This algorithm may be readily adapted for the stochastic simulation of more complex cell signaling systems.