Rheology of polymer brush under oscillatory shear flow studied by nonequilibrium Monte Carlo simulation

The rheological behaviors of polymer brush under oscillatory shear flow were investigated by nonequilibrium Monte Carlo simulation. The grafted chain under oscillatory shear flow exhibited a waggling behavior like a flower, and the segments were found to have different oscillatory phases along the chain contour. Stress tensor was further obtained based on the statistics of sampled configuration distribution functions. The simulation reproduced the abrupt increase of the first normal stress difference N(1) with the flow velocity over a critical value, as observed in the experiment of Klein et al. [Nature (London) 352, 143 (1991)]. However, our simulation did not reproduce the brush thickening with shear velocity increased, which was suggested to be responsible for the abrupt increase of N(1) in the above-mentioned paper. This simulation demonstrates that the increase of normal stress might be an inherent behavior of polymer brush due to chain deformation under flow.     

Monte Carlo simulation of micelle formation in block copolymer solutions

Short block copolymers in selective solvents (bad for A-block, good for B-block) are modeled by flexible bead-spring chains, where beads interact with short range Morse potentials of variable strength. It is shown that already very short chains (N_A = N_B = 2) exhibit a rather well-defined critical micelle concentration (cmc). The mass distribution of the micelles and their gyration tensor components as well as their internal structure are studied. It is shown that the relaxation time increases exponentially with the strength E_AA of the attractive energy between the A-monomers, and thus frozen-in micelles of medium size are obtained when E_AA is chosen too large. Our results are compared to studies of related but somewhat different models.     

Configurational bias Monte Carlo simulation of phase segregation in block copolymer networks

Cross-linked block copolymers are used as adhesives in fiber-reinforced composite material manufactures for automotive applications. Good adhesion between the polymer matrix and fibers in the interphase region is required for the structural integrity of these materials. Experimental evidence indicates that superior adhesion is obtained when phase segregation occurs between the two matrix phase block copolymers. It is therefore desirable to predict the conditions under which phase segregation is expected to occur. Configurational bias Monte Carlo simulations of two-component, trifunctional block copolymer networks were carried out to investigate phase segregation in these materials. The effects of four principal parameters on phase segregation were examined: the weight fractions of the two components, the cross-link length, the connectivity of the network, and the ratio of the square-well interactions. The molecular simulation results confirmed trends observed in laboratory measurements.     

Monte Carlo simulation of block copolymers

Monte Carlo simulations deal with crudely simplified but well-defined models and have the advantage that they treat the statistical thermodynamics of the considered model exactly (apart from statistical errors and problems due to finite size effects). Therefore, these simulations are well suited to test various approximate theories of block copolymer ordering, e.g. the self-consistent field theory. Recent examples of this approach include the study of block copolymer ordering at melt surfaces and confinement effects in thin films, adsorption of block copolymers at interfaces of unmixed homopolymer blends, the phase behavior of ternary mixtures of two homopolymers and their block copolymer, and micelle formation in selective solvents.     

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.     

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.     

Unusual finite size effects in the Monte Carlo simulation of microphase formation of block copolymer melts

Extensive Monte Carlo simulations are presented for the Fried-Binder model of block copolymer melts, where polymer chains are represented as self and mutually avoiding walks on a simple cubic lattice, and monomer units of different kind (A, B) repel each other if they are nearest neighbors (ε_AB > 0). Choosing a chain length N = 20, vacancy concentration Φ_v = 0,2, composition ƒ = 3/4, and a L × L × L geometry with periodic boundary conditions and 8 ≤ L ≤ 32, finite size effects on the collective structure factor S(q) and the gyration radii are investigated. It is shown that already above the microphase separation transition, namely when the correlation length ξ(T) of concentration fluctuations becomes comparable with L, a nonmonotonic variation of both S(q) and the radii with L sets in. This variation is due to the fact that the wavelength λ^*(T) of the ordering (defined from the wavenumber q^* where S(q) is maximal at λ^* = 2 π/q^*) in general is incommensurable with the box. The competition of two nontrivial lengths ξ(T), λ^* (T) with L makes the straigthforward application of finite size scaling techniques impossible, unlike the case of polymer blends. Since also the specific heat is found to have a broad rounded peak near the transition only, locating the transition accurately from Monte Carlo simulations remains an unsolved problem.     

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 simulation of freely jointed chains with variable bond lengths

Monte Carlo simulation of freely jointed off-lattice chains with variable bond length is usually done with local random displacements of beads and with reptation moves (displacements of a bead along a chain). In dense systems, the acceptance ratio of reptations decreases strongly with density. We discuss versions of reptation moves, which are effective in dense systems. The idea, which comes from lattice systems, is to use a pseudovacancy (walker), which has the same size as a bead of a chain. The walker is attached to a neighbor chain and then another bead of that chain is cleaved. This is equivalent to a reptation move and a nonlocal displacement of the walker and since no free volume is needed, the move can be used with advantage in dense systems. A related technique are cooperative motions, which were introduced by T. Pakula for lattice models, where several chains change their conformation concomitantly. Such cooperative loops are implemented in the Monte Carlo algorithm by creating a temporary walker by cleaving a bead from a chain, moving it with reptations and finally annihilating the walker by attaching it to the same chain it was cleaved from. These moves and the condition of detailed balance are discussed in detail. As an example, we study the integrated autocorrelation time τ_int for the radius of gyration for a two-dimensional system. For reduced densities larger than 0,4, we find that with standard reptations and local bead displacements τ_int increases strongly with density. If reptations with either a permanent or a temporary walker are used in addition to local moves, the integrated autocorrelation time changes only very little with density and very dense systems can still be simulated efficiently.     

Comb copolymer cylindrical brushes containing semiflexible side chains: a Monte Carlo study

The off‐lattice Monte Carlo method is applied to investigate the equilibrium conformations of isolated comb copolymer cylindrical brushes in an athermal solution. The molecules considered consist of a flexible backbone, which is densely grafted with semiflexible side chains. The study focuses on the influence of the degree of intrinsic stiffness, λ_side, of the side chains on the conformational behavior of the molecules. It is demonstrated that with a fixed side chain length, M, the local length scale conformational fluctuations of the backbone increase as a function of λ_side. However, the persistence length, λ, of the cylindrical brush increases considerably with the side chain stiffness, indicating that the backbone becomes more extended at the large length scale. Moreover, as a function of λ_side, there is an increase in the ratio λ/D of the persistence length and the diameter, D, of the brush. This behavior is in good agreement with recent theoretical predictions and provides important new insight in the latest experimental observations.     

Monte Carlo simulation of a lattice model for ternary polymer mixtures

Monte Carlo studies of symmetrical polymer mixturesAB, modelled by selfavoiding walks withN _A =N _B =N steps on a simple cubic lattice, are presented for arbitrary concentrations of vacancies _v in the range from _v =0.2 to _v =0.8 and chain lengthsN64. We obtained the phase diagrams and the equation of state for three choices of the ratio / _AB ( being the energy between monomers of the same kind, _AB being the energy between different monomers). Flory-Huggins theory provides only a qualitative understanding of these results. If the equation of state is fitted with an effective Flory-Huggins parameter _eff , the latter turns out to be strongly dependent on both concentration and temperature.Contributed paper delivered at the Tagung der Deutschen Physikalischen Gesellschaft, Fachausschuß Polymerphysik, Berlin, March 30–April 3, 1987.     

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.     

Monte Carlo studies of tethered chains

Monte Carlo computer simulations of end-tethered chains grafted onto a hard wall have been performed. The chains were modeled as self-avoiding chains on a cubic lattice at athermal solvent conditions. The simulations spanned a wide range of chain lengths, N (100–1000, i.e., up to molecular weights of a few hundred thousands), and anchoring densities, σ (2 × 10⁻⁴ to 0.4), to properly chart the relevant parameter space. It is shown that the reduced surface coverage σ* = σπR is the most appropriate variable that quantitatively determines the mushroom, overlapping mushroom and brush regimes, where R_g is the radius of gyration of a free chain in solution. The simulation data are analyzed to determine the conformational characteristics and shape of the anchored chains and to compare them with the predictions of the analytical self consistent field theory. The strong stretching limit of the theoretical predictions is obtained only for σ* > 8. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47:2449–2461, 2009     

A Monte Carlo simulation of grafted poly(ethylene oxide) chains

Conformational changes of a simplified model of grafted poly(ethylene oxide) (PEO) chains were simulated using an off-lattice Monte Carlo model. A random-walk scheme was used in our simulations. The initial polymer structure was modeled with molecular mechanics and models of grafted polymer chains were built using programs developed in our laboratory. During the simulation, all bond angles and bond lengths were kept fixed while the dihedral angles of backbones were changed to search for energy-favorite conformations. Torsional energy, van der Waals interaction, and Coulombic interaction were considered. Periodic boundary conditions were implemented. In addition, the solvent quality was simulated implicitly by modifying the Lennard-Jones 12–6 van der Waals expression. Each PEO chain, 50-monomer long, was represented with a united-atom model. Eight series of simulations with varying solvent quality, simulation temperature, and Coulombic interaction were carried out. For each series, nine different initial grafting densities of grafted PEO chains were considered. Five different conformations were simulated at each grafting density. The calculated system energies, scaling properties, and atom density profiles were studied. Changes in solvent quality produced different structural behaviors. As the grafting density increased, there was a mushroom-to-brush transition, and the scaling property of average layer thickness was dependent on the grafting density.     

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.     

Study on structure formation of short polyethylene chains via dynamic Monte Carlo simulation

Monte Carlo (MC) simulations of structure formation for short polyethylene chains at low temperature are performed based on a recent developed method that uses coarse-grained chains on a high coordination lattice. Local short-range interactions based on rotational isomeric state (RIS) model and long-range interactions obtained from Lennard–Jones (LJ) potential are introduced during the simulation. Properties evaluated from the simulations are the mean square dimensions, anisotropy of the radius of gyration tensor, local conformation determined by the occupancy of trans state and orientation correlation functions, energy of the system, and chain packing reflected by the pair correlation functions and structure factors. All of these parameters reveal an ordering process that produces an approximation to a hexagonal crystal phase. The hexagonal structure is imposed by the presence of a diamond lattice underlying the high coordination lattice on which the simulation is performed. Folding of the chains in the crystal is mandatory, because they have fully extended lengths in excess of the dimension of the simulated periodic box. Nevertheless, the simulations demonstrate that a high degree of crystallinity can be achieved in reasonable computer time. The simulation technique should be applicable to other choices of periodic boundary conditions that do not affect the results as strongly as in the present case.     

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.     

Wavelet-accelerated Monte Carlo sampling of polymer chains

We introduce a new method for coarse-graining polymer chains, based on the wavelet transform, a multiresolution data analysis technique. This method, which assigns a cluster of particles to a coarse-grained bead located at the center of mass of the cluster, reduces the complexity of the problem significantly by dividing the simulation into several stages, each with a small fraction of the number of beads in the overall chain. At each stage, we compute the distributions of coarse-grained internal coordinates as well as potential functions required for subsequent simulation stages. We show that, with this wavelet-accelerated Monte Carlo method, coarse-grained Gaussian and self-avoiding random walks can reproduce results obtained from atomistic simulations to a high degree of accuracy in orders of magnitude less time. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 897–910, 2005     

Dynamic Monte Carlo simulation of aggregation of nanoparticles in the presence of diblock copolymer

The aggregation of hydrophobic nanoparticles in the presence of diblock copolymers is investigated using dynamic Monte Carlo simulation on a simple cubic lattice. One nanoparticle occupies one lattice site, one block copolymer (A(m)B(m)) occupies 2m sequentially linked sites with m segments of A and m segments of B, and solvents are represented by any unoccupied sites. All of them are self-avoiding and nearest-neighbor interactions are considered. A compact big aggregate, dispersed aggregates wrapped by polymer chains, and an ordered lamellar structure are obtained by varying the concentration of copolymer. The structures are seen to be controlled by competing forces between the interaction of copolymer with nanoparticles and the self-assembly of copolymer in solution. The critical concentration of copolymer needed to form the lamellar structure, C(p,L), decreases with the chain length. It is also found that C(p,L) decreases roughly linearly with the concentration of nanoparticles C(n), which can be approximately expressed as C(p,L)=0.764-0.857C(n) when m=2. The simulation demonstrates that addition of diblock copolymer can effectively control the aggregation of nanoparticles and lead to the formation of a variety of nanostructures.