Flexible polymers in a nematic medium: a Monte Carlo simulation

Polymers in random media: An introduction

Polyelectrolytes, polymers in poor solvents, polymers mixed with particles, and other systems with attractions and repulsions show formation of globules/structures in equilibrium or in flow. To study the flow behavior of such systems, we developed a simple coarse-grained model with short ranged attractions and repulsions. Polymers are represented as charged bead-spring chains and they interact with oppositely charged colloids. Neglecting hydrodynamic interactions, we study the formation of compact polymer structures called globules. Under certain conditions, increase in shear rate decreases the mean first passage time to form a globule. At other conditions, shear flow causes the globules to breakup, similar to the globule-stretch transition of polymers in poor solvents.

An exhaustive analysis of the shape properties of discrete self-avoiding random walks is presented. The dependence of the main parameters of the probability distributions from the length of the walk and the differences between these distributions for conventional and self-avoiding walks are studied. By means of high-precision Monte Carlo simulations it is shown that the characteristics of the shape of self-avoiding random walks, when regarded as a function of the walk length, present the expected asymptotic behaviour. The differences with conventional random walks depend upon the observable considered: the probability distributions of the principal inertia eigenvalues of self-avoiding walks spread around the most probable value more widely than the corresponding distributions for unrestricted walks, while in the cases of the asphericity or ratios of inertia eigenvalues, the distributions for self-avoiding walks are somewhat more peaked than their counterparts for conventional random walks. Analytical expressions for the probability distributions of inertia moment ratios and the two-dimensional asphericity are given. For common random walks there is an excellent agreement between these analytical distributions and the Monte Carlo data. In this work it is established that this concordance is maintained if the same analytical distributions are applied to the self-avoiding case. This means that within the precision of the simulations the functional form of the mentioned distributions does not vary when passing from conventional to self-avoiding walks.

We show that the ‘erasing-larger-loops-first’ (ELLF) method, which was first introduced for erasing loops from the simple random walk on the Sierpiński gasket, does work also for non-Markov random walks, in particular, self-repelling walks to construct a new family of self-avoiding walks on the Sierpiński gasket. The one-parameter family constructed in this method continuously connects the loop-erased random walk and a self-avoiding walk which has the same asymptotic behavior as the ‘standard’ self-avoiding walk. We prove the existence of the scaling limit and study some path properties: The exponent $ν$ governing the short-time behavior of the scaling limit varies continuously in the parameter. The limit process is almost surely self-avoiding, while it has path Hausdorff dimension $1/ν $, which is strictly greater than $1$.

We present the main types of phase diagrams obtained with flexible polymers in nematic solvents. Data deduced from experimental phase diagrams are compared with two theoretical descriptions.

Random walks on self-avoiding walks (SAWs) are studied using Monte Carlo techniques on a square lattice (with nearest-neighbour hopping along the chain and between SAW points which are nearest neighbours on the embedding lattice). The average of the square of the end-to-end distance for random walks of t steps on SAWs of length N is fitted to the scaling forms (Rt2) varies as Ndelta tk (for t or approximately=Ntheta ), where theta approximately=2 nu s/k; nu s being the average end-to-end distance exponent for SAWs. The observed value of the exponent delta is supported by the authors' real space renormalisation group result for the conductivity of SAW chains. The exponent k has been related to the 'effective' fractal dimension of the SAW chain.

ABSTRACTWe present a theoretical study of liquid crystalline polymers in nematic solvents. We focus on the following topics: (i) Rod-like chains confined between plates, (ii) Flexible chains in a slit (iii) Nematic brushes and double-brushes (iv) Anchoring transitions. Confined and grafted chains in nematic solvents can exhibit a variety of interesting effects, including quasi-piezoelectricity and a lowering of the critical voltage for the Fréedericksz transition.

We consider the phenomena of entanglement of the two interacting self‐avoiding walks (SAW) situated in a member of the three‐dimensional Sierpinski Gasket (SG) fractal family. We focus our attention to determine number of point contacts between the two SAW paths M, which turns out to be a set of power laws whose characteristics depend predominantly on the interactions between SAW steps. The phase diagrams have been establised and corresponding values of the contact critical exponents φ, associated with the two‐path mutual contacts, have been found.

The self-organized criticality on the random fractal networks has many motivations, like the movement pattern of fluid in the porous media. In addition to the randomness, introducing correlation between the neighboring portions of the porous media has some nontrivial effects. In this paper, we consider the Ising-like interactions between the active sites as the simplest method to bring correlations in the porous media, and we investigate the statistics of the BTW model in it. These correlations are controlled by the artificial "temperature" T and the sign of the Ising coupling. Based on our numerical results, we propose that at the Ising critical temperature T_{c} the model is compatible with the universality class of two-dimensional (2D) self-avoiding walk (SAW). Especially the fractal dimension of the loops, which are defined as the external frontier of the avalanches, is very close to D_{f}^{SAW}=4/3. Also, the corresponding open curves has conformal invariance with the root-mean-square distance R_{rms}∼t^{3/4} (t being the parametrization of the curve) in accordance with the 2D SAW. In the finite-size study, we observe that at T=T_{c} the model has some aspects compatible with the 2D BTW model (e.g., the 1/log(L)-dependence of the exponents of the distribution functions) and some in accordance with the Ising model (e.g., the 1/L-dependence of the fractal dimensions). The finite-size scaling theory is tested and shown to be fulfilled for all statistical observables in T=T_{c}. In the off-critical temperatures in the close vicinity of T_{c} the exponents show some additional power-law behaviors in terms of T-T_{c} with some exponents that are reported in the text. The spanning cluster probability at the critical temperature also scales with L^{1/2}, which is different from the regular 2D BTW model.

The scanning method is a computer simulation technique for macromolecules suggested recently. The method is described here in detail and its applicability (in contrast to other simulation techniques) to a wide range of chain models is discussed. It is argued that for most of these models the scanning method constitutes the most efficient tool for estimating the entropy. The method is applied to self-avoiding walks (SAWs) (of N≤399 steps) on both a three-choice square lattice and a five-choice simple cubic lattice and the results for the entropy, the end-to-end distance, the radius of gyration, and other quantities of interest are found to be in very good agreement with the results obtained by other numerical techniques. In particular, our calculations support the law of divergence for the persistence length of SAWs on two-dimensional lattices suggested recently by Grassberger. However, for the simple cubic lattice, the persistence length is found by us to be constant.

We describe a new algorithm, the reflection method, to generate off-lattice random walks of specified, though arbitrarily large, thickness in and prove that our method is ergodic on the space of thick walks. The data resulting from our implementation of this method is consistent with the scaling of the squared radius of gyration of random walks, with no thickness constraint. Based on this, we use the data to describe the complex relationship between the presence and nature of knotting and size, thickness and shape of the random walk. We extend the current understanding of excluded volume by expanding the range of analysis of how the squared radius of gyration scales with length and thickness. We also examine the profound effect of thickness on knotting in open chains. We will quantify how thickness effects the size of thick open chains, calculating the growth exponent for squared radius of gyration as a function of thickness. We will also show that for radius , increasing thickness by 0.1 decreases the probability of knot formation by 50% or more.

In our paper, the expression for ht x z 0 t x z 0 zi is incorrect by a factor of 1 2 . Equation (2) should read

A model system of side-group liquid crystalline polymers (SGLCPs) with systematically varied molecular weight (from 78 to 420 kg/mol; PDI ≤ 1.16) and spacer length (8−12 atoms long) was prepared by polymer analogous synthesis. Matching the structure of the mesogenic units to that of the nematic solvent produced excellent solubility, even at molecular weights an order of magnitude greater than in prior literature on SGLCP solutions. Addition of up to 10 wt % polymer did not affect the ordinary and extraordinary refractive indices of the nematic host (4-pentyl-4‘-cyanobiphenyl, 5CB), indicating that the order parameter was not significantly affected by the polymer.

We develop a theory of polymers in a nematic solvent by exploiting an analogy with two-dimensional quantum bosons at zero temperature. We argue that the theory should also describe polymers in an isotropic solvent. The dense phase is analyzed in a Bogoliubovlike approximation, which assumes a broken symmetry in the phase of the boson order parameter. We find a stiffening of the longitudinal fluctuations of the nematic field, calculate the density-density correlation function, and extend the analysis to the case of ferro- and electrorheological fluids. The boson formalism is used to derive a simple hydrodynamic theory which is indistinguishable from the corresponding theory of polymer nematics in an isotropic solvent at long wavelengths. We also use hydrodynamics to discuss the physical meaning of the boson order parameter. A renormalization group treatment in the dilute limit shows that logarithmic corrections to polymer wandering, predicted by de Gennes, are unaffected by interpolymer interactions. A continuously variable Flory exponent appears for polymers embedded in a two-dimensional nematic solvent. We include free polymer ends and hairpin configurations in the theory and show that hairpins are described by an Ising-like symmetry-breaking term in the boson field theory.

A brief review will be given of the current situation in the theory of self-avoiding walks (SAWs). The Domb-Joyce model first introduced in 1972 consists of a random walk on a lattice in which eachN step configuration has a weighting factor Π i=0 N−2 Πj=i+2/N(1−ωδij). Herei andj are the lattice sites occupied by the ith and jth points of the walk. When ω=0 the model reduces to a standard random walk, and when ω=1 it is a self-avoiding walk. The universality hypothesis of critical phenomena will be used to conjecture the behavior of the model as a function ofω for largeN. The implications for the theory of dilute polymer solutions will be indicated.