Stochastic lattice models for the dynamics of linear polymers

Macromolecular Theory and Simulations, 1997

We investigate the changes in the average chain length of a solution of semi-flexible living polymers between two hard repulsive walls as the width of the slit, D, is varied. Two different Monte Carlo models, that of the 'slithering snake' and of the 'independent monomer states' are employed in order to simulate a polydisperse system of chain molecules confined in a gap which is either closed (with fixed total density), or open and in contact with an external reservoir. It appears that the mean chain length L in a state of equilibrium polymerization depends essentially on the geometry constraints for sufficiently small D. We find that in the case of an open slit the mean length L(D) decreases with D + 0 for flexible chains whereas it grows if the chains are sufficiently stiff. As the width of a closed gap D is decreased, in a three-dimensional gap L(D) gradually decreases for absolutely flexible chains whereas for semi-rigid chains it goes through a minimum at D = 2 and then grows again for D = 1. In two dimensions, in a closed strip the average chain length L(D) for both flexible and rigid macromolecules goes through a sharp minimum and then grows steeply in compliance with a predicted divergence for semi-rigid polymers as D + 0. We attribute the observed discrepancies of our numeric experiments with some recent analytic predictions to the ordering effect of container walls on the polymer solution when chain stiffness and excluded volume interactions are taken into account.

Physical review. E, Statistical, nonlinear, and soft matter physics, 2001

We study the Langevin dynamics of the standard random heteropolymer model by mapping the problem to a supersymmetric field theory using the Martin-Siggia-Rose formalism. The resulting model is solved nonperturbatively employing a Gaussian variational approach. In constructing the solution, we assume that the chain is very long and impose the translational invariance which is expected to be present in the bulk of the globule by averaging over the center of mass coordinate. In this way we derive equations of motion for the correlation and response functions C(t,t') and R(t,t'). The order parameters are extracted from the asymptotic behavior of these functions. We find a dynamical phase diagram with frozen (glassy) and melted (ergodic) phases. In the glassy phase the system fails to reach equilibrium and exhibits aging of the type found in p-spin glasses. Within the approximations used in this study, the random heteropolymer model can be mapped to the problem of a manifold in a...

Macromolecules, 1991

Orientational autocorrelations and cross-correlations are considered for vectors rigidly affixed to bonds subject to configurational transitions in a long polymer chain. Transitions of bonds from one rotational isomeric state to the other are assumed to be dependent on the state of the neighboring bonds. Bond transition rates are obtained from Kramers' expression in the high-friction limit. The friction coefficient affecting a transition is assumed to depend on the location of the bond along the moving sequence in the chain, The joint probability of having a sequence of bonds in one configuration at time zero and in another at time z and the orientational correlation functions are obtained by an efficient matrix multiplication scheme analogous to the matrix generator formalism of the rotational isomeric theory of chain statistics. A sequence of bonds whose length is prescribed by the time window of the experimental technique used is defined as an independent kinetic unit. The stochastic behavior of the latter is assumed to be uncorrelated with the remaining parta of the chain. Calculations are performed for different lengths of independent units, ranging from a few skeletal bonds to segments of the size of a Rouse subchain. Frequency distribution of relaxational modes obtained in this manner agree closely with previous calculations of Fixman by Langevin dynamics. Thus, unlike the Rouse dynamics predictions, the fastest modes of the investigated sub-Rouse regime scale linearly with inverse chain length and the distribution of relaxational frequencies for a given sequence exhibits a pronounced plateau.

Continuum Mechanics and Thermodynamics, 2017

Representing polymers by random walks on a lattice is a fruitful approach largely exploited to study configurational statistics of polymer chains and to develop efficient Monte Carlo algorithms. Nevertheless, the stretching and the folding/unfolding of polymer chains within the Gibbs (isotensional) and the Helmholtz (isometric) ensembles of the statistical mechanics have not been yet thoroughly analysed by means of the lattice methodology. This topic, motivated by the recent introduction of several single-molecule force spectroscopy techniques, is investigated in the present paper. In particular, we analyse the force-extension curves under the Gibbs and Helmholtz conditions and we give a proof of the ensembles equivalence in the thermodynamic limit for polymers represented by a standard random walk on a lattice. Then, we generalize these concepts for lattice polymers that can undergo conformational transitions or, equivalently, for chains composed of bistable or two-state elements (that can be either folded or unfolded). In this case, the isotensional condition leads to a plateau-like force-extension response, whereas the isometric condition causes a sawtooth-like force-extension curve, as predicted by numerous experiments. The equivalence of the ensembles is finally proved also for lattice polymer systems exhibiting conformational transitions.

Europhysics Letters (EPL), 2004

We use an off-lattice bead-spring model of a self-avoiding polymer chain immersed in a 3-dimensional quenched random medium to study chain dynamics by means of a Monte-Carlo (MC) simulation. The chain center of mass mean-squared displacement as a function of time reveals two crossovers which depend both on chain length N and on the degree of Gaussian disorder ∆. The first one from normal to anomalous diffusion regime is found at short time τ1 and observed to vanish rapidly as τ1 ∝ ∆ −11 with growing disorder. The second crossover back to normal diffusion, τ2, scales as τ2 ∝ N 2ν+1 f (N 2−3ν ∆) with f being some scaling function. The diffusion coefficient DN depends strongly on disorder and drops dramatically at a critical dispersion ∆c ∝ N −2+3ν of the disorder potential so that for ∆ > ∆c the chain center of mass is practically frozen. The time-dependent Rouse modes correlation function Cp(t) reveals a characteristic plateau at ∆ > ∆c which is the hallmark of a non-ergodic regime. These findings agree well with our recent theoretical predictions.

Theoretical and Mathematical Physics

Chemical Physics Letters, 2005

While lattice models are used extensively for macromolecules (synthetic polymers proteins, etc.), calculation of the absolute entropy, S, and the free energy, F, from a given Monte Carlo (MC) trajectory is not straightforward. Recently, we have developed the hypothetical scanning MC (HSMC) method for calculating S and F of fluids. Here we extend HSMC to self-avoiding walks on a square lattice and discuss its wide applicability to complex polymer lattice models. HSMC is independent of existing techniques and thus constitutes an independent research tool; it provides rigorous upper and lower bounds for F, which can be obtained from a very small sample and even from a single chain conformation.

2012

Abstract: This report deals with phase transition in Bond Fluctuation Model (BFM) of a linear homo polymer on a two dimensional square lattice. Each monomer occupies a unit cell of four lattice sites. The condition that a lattice site can at best be a part of only one monomer ensures self avoidance and models excluded volume effect. We have simulated polymers with number of monomers ranging from 10 to 50 employing Boltzmann and non-Boltzmann Monte Carlo simulation techniques.

e-Polymers, 2004

It is well known that the properties of polymeric materials depend strongly upon their chemical structure. Other more specific factors that may be related to the chemical structure also determine the macroscopic behaviour of such materials, namely the relative position of the different segments of the polymeric chain, the molecular architecture (molecular weight distribution, branching, copolymer organisation, cross-linking extent, etc.), the crystalline environment and the pressure/temperature conditions. All these factors have a common impact in the material: they are strongly correlated to the mobility on the molecular level. That is why a huge amount of work has been devoted to the study of translational/rotational mobility that occurs within the polymeric chains. This review is intended to provide a brief survey on such kinds of mobilities, how they can be studied and what are their main characteristics. Examples on systems studied in our groups will be provided, obtained by di...

Physical Review Letters, 2007

We present a neutron scattering investigation on a miscible blend of two polymers with greatly different glass-transition temperatures T g . Under such conditions, the nearly frozen high-T g component imposes a random environment on the mobile chain. The results demand the consideration of a distribution of heterogeneous mobilities in the material and demonstrate that the larger scale dynamics of the fast component is not determined by the average local environment alone. This distribution of mobilities can be mapped quantitatively on the spectrum of local relaxation rates measured at high momentum transfers.