Recent Developments in Monte Carlo Simulations of Lattice Models for Polymer Systems

A triangular model of fractal growth with application to adsorptive spin-coating of polymers

Over the last 40 years, applied mathematicians and physicists have proposed a number of mathematical models that produce structures exhibiting a fractal dimension. This work has coincided with the discovery that objects with fractal dimension are relatively common in the natural and human-produced worlds. One particularly successful model of fractal growth is the diffusion limited aggregation (DLA) model, a model as notable for its simplicity as for its complex and varied behavior. It has been modified and used to simulate fractal growth processes in numerous experimental and empirical contexts. In this work, we present an alternative fractal growth model that is based on a growing mass that bonds to particles in a surrounding medium and then exerts a force on them in an iterative process of growth and contraction. The resulting structure is a spreading triangular network rather than an aggregate of spheres, and the model is conceptually straightforward. To the best of our knowledge, this model is unique and differs in its dynamics and behavior from the DLA model and related particle aggregation models. We explore the behavior of the model, demonstrate the range of model output, and show that model output can have a variable fractal dimension between 1.5 and 1.83 that depends on model parameters. We also apply the model to simulating the development of polymer thin films prepared using spin-coating which also exhibit variable fractal dimensions. We demonstrate how the model can be adjusted to different dewetting conditions as well as how it can be used to simulate the modification of the polymer morphology under solvent annealing.

Coil-globule transition in two-dimensional polymer chains in an explicit solvent

The structure of two-dimensional polymer chains in a solvent at different temperatures is still far from being fully understood. Computer simulations of high-density macromolecular systems require the use of appropriate algorithms, and therefore the simulations were carried out using the Cooperative Motion Algorithm. The polymer model studied was exactly two-dimensional, coarse-grained and based on a triangular lattice. The theta temperature and temperature of coil-to-globule transition, and critical exponents were determined. The differences between the structure of such a disk and that of a chain in a dense polymer liquid were shown.

Novel Diffusion Mechanism of Polymers Pinned to an Attractive Impurity

Actual substrates unavoidably possess, to some extent, defects and dirt, which motivate understanding the impact due to their presence. The presence of a substrate naturally breaks symmetries. Additionally, it effectively reduces spatial dimensionality, which favors fluctuation-dominated behavior, but it also provides a multitude of possible interactions. We show evidence of novel behavior in the case of polymer mass transport at a crystalline substrate when a single attractive impurity is present. Specifically, we introduce a model system describing how an attractive impurity pins adsorbed polymers on a substrate. We propose a novel mechanism to explain the size scaling dependence of the diffusion coefficient as D∼N−3/2 for polymers with N monomers. Additionally, the size dependence of the diffusion coefficient scales can be described as D∼N−δ, with δ=1.51 as determined from extensive simulations.

Computational methods in the study of self-entangled proteins: a critical appraisal

The existence of self-entangled proteins, the native structure of which features a complex topology, unveils puzzling, and thus fascinating, aspects of protein biology and evolution. The discovery that a polypeptide chain can encode the capability to self-entangle in an efficient and reproducible way during folding, has raised many questions, regarding the possible function of these knots, their conservation along evolution, and their role in the folding paradigm. Understanding the function and origin of these entanglements would lead to deep implications in protein science, and this has stimulated the scientific community to investigate self-entangled proteins for decades by now. In this endeavour, advanced experimental techniques are more and more supported by computational approaches, that can provide theoretical guidelines for the interpretation of experimental results, and for the effective design of new experiments. In this review we provide an introduction to the computational study of self-entangled proteins, focusing in particular on the methodological developments related to this research field. A comprehensive collection of techniques is gathered, ranging from knot theory algorithms, that allow detection and classification of protein topology, to Monte Carlo or molecular dynamics strategies, that constitute crucial instruments for investigating thermodynamics and kinetics of this class of proteins.

Effect of Composition Asymmetry on the Phase Separation and Crystallization in Double Crystalline Binary Polymer Blends: A Dynamic Monte Carlo Simulation Study

Polymer blends offer an exciting material for various potential applications due to their tunable properties by varying constituting components and their relative composition. Our simulation results unravel an intrinsic relationship between crystallization behavior and composition asymmetry. We report simulation results for nonisothermal and isothermal crystallization with weak and strong segregation strength to elucidate the composition dependent crystallization behavior. With increasing composition of low melting B-polymer, macrophase separation temperature changes nonmonotonically, which is attributed to the nonmonotonic change in diffusivity of both polymers. In weak segregation strength, however, at high enough composition of B-polymer, A-polymer yields relatively thicker crystals, which is attributed to the dilution effect exhibited by B-polymer. When B-polymer composition is high enough, it acts like a "solvent" while A-polymer crystallizes. Under this situation, A-polymer segments become more mobile and less facile to crystallize. As a result, A-polymer crystallizes at a relatively low temperature with the formation of thicker crystals. At strong segregation strength, the dilution effect is accompanied by the strong A-B repulsive interaction, which is reflected in a nonmonotonic trend of the mean square radius of gyration with the increasing composition of the B-polymer. Isothermal crystallization also reveals a strong nonmonotonic relationship between composition and crystallization behavior. Two-step, compared to one-step, isothermal crystallization yields better crystals for both polymers.

Monte Carlo simulations of lattice models for single polymer systems

Single linear polymer chains in dilute solutions under good solvent conditions are studied by Monte Carlo simulations with the pruned-enriched Rosenbluth method up to the chain length N~O(10(4)). Based on the standard simple cubic lattice model (SCLM) with fixed bond length and the bond fluctuation model (BFM) with bond lengths in a range between 2 and √10, we investigate the conformations of polymer chains described by self-avoiding walks on the simple cubic lattice, and by random walks and non-reversible random walks in the absence of excluded volume interactions. In addition to flexible chains, we also extend our study to semiflexible chains for different stiffness controlled by a bending potential. The persistence lengths of chains extracted from the orientational correlations are estimated for all cases. We show that chains based on the BFM are more flexible than those based on the SCLM for a fixed bending energy. The microscopic differences between these two lattice models are discussed and the theoretical predictions of scaling laws given in the literature are checked and verified. Our simulations clarify that a different mapping ratio between the coarse-grained models and the atomistically realistic description of polymers is required in a coarse-graining approach due to the different crossovers to the asymptotic behavior.

Nanoscale crosslinking in thermoset polymers: a molecular dynamics study

Nanoscale crosslinking analysis of thermoset polymer.

Effect of block asymmetry on the crystallization of double crystalline diblock copolymers

Monte Carlo simulation on the crystallization of double crystalline diblock copolymer unravels an intrinsic relationship between block asymmetry and crystallization behaviour. We model crystalline A-B diblock copolymer, wherein the melting temperature of A-block is higher than that of the B-block. We explore the composition dependent crystallization behaviour by varying the relative block length with weak and strong segregation strength between the blocks. In weak segregation limit, we observe that with increasing the composition of B-block, its crystallization temperature increases accompanying with higher crystallinity. In contrast, A-block crystallizes at a relatively low temperature along with the formation of thicker and larger crystallites with the increase in B-block composition. We attribute this non-intuitive crystallization trend to the dilution effect imposed by B-block. When the composition of the B-block is high enough, it acts like a "solvent" during the crystallization of A-block. A-block segments are more mobile and hence less facile to crystallize, resulting depression in crystallization temperature with the formation of thicker crystals. At strong segregation limit, crystallization and morphological development are governed by the confinement effect, rather than block asymmetry. Isothermal crystallization reveals that the crystallization follows a homogeneous nucleation mechanism with the formation of two-dimensional crystals. Two-step, compared to one-step isothermal crystallization leads to the formation of thicker crystals of A-block due to the dilution effect of the B-block.

Monte Carlo simulations of a polymer chain conformation. The effectiveness of local moves algorithms and estimation of entropy

A linear chain on a simple cubic lattice was simulated by the Metropolis Monte Carlo method using a combination of local and non-local chain modifications. Kink-jump, crankshaft, reptation and end-segment moves were used for local changes of the chain conformation, while for non-local chain rearrangements the "cut-and-paste" algorithm was employed. The statistics of local micromodifications was examined. An approximate method for estimating the conformational entropy of a polymer chain, based on the efficiency of the kink-jump motion respecting chain continuity and excluded volume constraints, was proposed. The method was tested by calculating the conformational entropy of the undisturbed chain, the chain under tension and in different solvent conditions (athermal, theta and poor) and also of the chain confined in a slit. The results of these test calculations are qualitatively consistent with expectations. Moreover, the obtained values of the conformational entropy of self avoiding chain with ends fixed over different separations, agree very well with the available literature data.

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Statistical model for self-assembly of trimesic acid molecules into homologous series of flower phases

The statistical three-state model is proposed to describe the ordering of triangular TMA molecules into flower phases. The model is solved on a rescaled triangular lattice, assuming following intermolecular interactions: exclusion of any molecules on nearest neighbor sites, triangular trio H-bonding interactions for molecules of the same orientation on next-nearest neighbor sites, and dimeric H-bonding interactions for molecules of different ("tip-to-tip") orientations on third-nearest neighbor sites. The model allows us to obtain the analytical solution for the ground state phase diagram with all homologous series of flower phases included, starting with the honeycomb phase (n=1) and ending with the superflower structure (n=∞). Monte Carlo simulations are used to obtain the thermodynamical properties of this model. It is found that phase transitions from disordered to any of the flower phases (except n=1) undergo via intermediate correlated triangular domains structure. The transition from the disordered phase to the intermediate phase is, most likely, of the first order, while the transition from the intermediate to the flower phase is definitely first order phase transition. The phase diagrams including low-temperature flower phases are obtained. The origin of the intermediate phase, phase separation, and metastable structures are discussed.

Elastic modulus of a polymer nanodroplet: theory and experiment

We redevelop a theoretical model that, in conjunction with atomic force microscopy (AFM), can be used as a noninvasive method for determination of the elastic modulus of a polymer nanodroplet residing on a flat, rigid substrate. The model is a continuum theory that combines surface and elasticity theories for prediction of the droplet's elastic modulus, given experimental measurement of its adsorbed height. Utilization of AFM-measured heights for relevant droplets reported in the literature and from our own experiments illustrated the following: the significance of both surface and elasticity effects in determining a polymer droplet's spreading behavior; the extent of a continuum theory's validity as one approaches the nanoscale; and a droplet size effect on the elastic modulus.

Effects of compositional asymmetry in phase behavior of ABA triblock copolymer melts from Monte Carlo simulation

We simulate ABA triblock copolymer melts using a lattice Monte Carlo method, known as cooperative motion algorithm, probing various degrees of compositional asymmetry. Selected order-disorder transition lines are determined in terms of the segment incompatibility, quantified by product χN , and the triblock asymmetry parameters, α and β. We correlate the results of the simulation with the self-consistent field theory and an experimental study of polyisoprene-polystyrene-polyisoprene triblock melt by Hamersky and coworkers. In particular, we confirm the mean-field prediction that for highly asymmetric triblocks the short A -block is localized in the middle of the B -domain due to an entropic advantage. This results in the middle block relaxation and is consistent with the experimental data indicating that as the relatively short A -blocks are grown into AB diblock, from the B -block side, the order-disorder transition temperature is considerably depressed.

Phase transitions of a single polymer chain: A Wang-Landau simulation study

A single flexible homopolymer chain can assume a variety of conformations which can be broadly classified as expanded coil, collapsed globule, and compact crystallite. Here we study transitions between these conformational states for an interaction-site polymer chain comprised of N=128 square-well-sphere monomers with hard-sphere diameter sigma and square-well diameter lambdasigma. Wang-Landau sampling with bond-rebridging Monte Carlo moves is used to compute the density of states for this chain and both canonical and microcanonical analyses are used to identify and characterize phase transitions in this finite size system. The temperature-interaction range (i.e., T-lambda) phase diagram is constructed for lambda<or=1.30. Chains assume an expanded coil conformation at high temperatures and a crystallite structure at low temperatures. For lambda>1.06 these two states are separated by an intervening collapsed globule phase and thus, with decreasing temperature a chain undergoes a continuous coil-globule (collapse) transition followed by a discontinuous globule-crystal (freezing) transition. For well diameters lambda<1.06 the collapse transition is pre-empted by the freezing transition and thus there is a direct first-order coil-crystal phase transition. These results confirm the recent prediction, based on a lattice polymer model, that a collapsed globule state is unstable with respect to a solid phase for flexible polymers with sufficiently short-range monomer-monomer interactions.

How to define variation of physical properties normal to an undulating one-dimensional object

One-dimensional flexible objects are abundant in physics, from polymers to vortex lines to defect lines and many more. These objects structure their environment and it is natural to assume that the influence these objects exert on their environment depends on the distance from the line object. But how should this be defined? We argue here that there is an intrinsic length scale along the undulating line that is a measure of its stiffness (i.e., orientational persistence), which yields a natural way of defining the variation of physical properties normal to the undulating line. We exemplify how this normal variation can be determined from a computer simulation for the case of a so-called bottle-brush polymer, where side chains are grafted onto a flexible backbone.

Versatile approach to access the low temperature thermodynamics of lattice polymers and proteins

We show that Wang-Landau sampling, combined with suitable Monte Carlo trial moves, provides a powerful method for both the ground state search and the determination of the density of states for the hydrophobic-polar (HP) protein model and the interacting self-avoiding walk (ISAW) model for homopolymers. We obtain accurate estimates of thermodynamic quantities for HP sequences with >100 monomers and for ISAWs up to >500 monomers. Our procedure possesses an intrinsic simplicity and overcomes the limitations inherent in more tailored approaches making it interesting for a broad range of protein and polymer models.