From random to self-avoiding walks

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 $1$.

Polymers in random media: An introduction

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.

Dendrite growth in solid polymer electrolytes has been frequently analyzed since it was the critical issue that limited its applications. For the liquid electrolyte systems, the dilute solution theory and the classic Nernst-Planck equation have been proven to be useful tools for analysis of ion transport dynamics and especially the dendrite initiation at Sand’s time. However, characterization of the Sand’s time in solid polymer electrolyte systems is challenging and also seldomly performed. From the experimental perspective, operando observations have been done, but the true local current density was different from the geometrical average current density since there was always heterogeneous current distribution for millimeter-scale electrolytes. From the theoretical perspective, the dendrite initiation time followed the trend predicted by Sand’s equation, but discrepancies were found such as dendrites observed at under-limiting currents or the order-of-magnitude extended Sand’s time compared to theoretical predictions. Here, we use transparent microcapillary cells for solid polymer electrolyte systems to overcome these challenges. These specialized cells allow direct operando optical observations, while the micron-scale cross-sectional area minimizes the discrepancy between true local and averaged geometrical current densities. Comparison between operando image and voltage response during constant current polarization shows that, unlike liquid electrolyte cases, the dendrite initiation for solid polymer electrolytes doesn’t always trigger a voltage spike. We also derived transport parameters from the measured Sand’s time, limiting current density, and conductivity and the cross-validated transport parameters were used to calculate theoretical Sand’s time that can be compared with the experimental values. Using the analytical solution from the Nernst-Planck equation and numerical calculations using Newman’s concentrated solution model and COMSOL Multiphysics, the predicted Sand’s time for the dilute and concentrated solution theory both matched closely with our experimental values. This work demonstrates that while the polarization process and the onset of lithium dendritic growths in solid polymer electrolytes can be still accurately predicted by the dilute solution theory, it may not always result in voltage responses similar to that of the liquid electrolyte cases. It’s also suggested that ensuring the homogenous distribution of lithium flux and avoiding the localized overlimiting current density is the key to realizing the dendrite-free polymer electrolytes.

Bead-spring chain model for the dynamics of dilute polymer solutions: Part 2. Comparisons with experimental data

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.

Recent studies suggest that cosolute mixtures may exert significant non-additive effects upon protein stability. The corresponding liquid-vapor interfaces may provide useful insight into these non-additive effects. Accordingly, in this work, we relate the interfacial properties of dilute multicomponent solutions to the interactions between solutes. We first derive a simple model for the surface excess of solutes in terms of thermodynamic observables. We then develop a lattice-based statistical mechanical perturbation theory to derive these observables from microscopic interactions. Rather than adopting a random mixing approximation, this dilute solution theory (DST) exactly treats solute-solute interactions to lowest order in perturbation theory. Although it cannot treat concentrated solutions, Monte Carlo (MC) simulations demonstrate that DST describes the interactions in dilute solutions with much greater accuracy than regular solution theory. Importantly, DST emphasizes a fundamental distinction between the "intrinsic" and "effective" preferences of solutes for interfaces. DST predicts that three classes of solutes can be distinguished by their intrinsic preference for interfaces. While the surface preference of strong depletants is relatively insensitive to interactions, the surface preference of strong surfactants can be modulated by interactions at the interface. Moreover, DST predicts that the surface preference of weak depletants and weak surfactants can be qualitatively inverted by interactions in the bulk. We also demonstrate that DST can be extended to treat surface polarization effects and to model experimental data. MC simulations validate the accuracy of DST predictions for lattice systems that correspond to molar concentrations.

Abstract

The binary cluster integral β for polymer segments is evaluated on the basis of the very crude smoothed-potential cell model, regarding it as equivalent to the second virial coefficient for a binary liquid mixture of small molecules. For polystyrene and polyisobutylene in various solvents at various temperatures, it is shown that the model can interpret qualitatively the behavior of β in terms of intermolecular forces. However, the calculated values of β are about one order of magnitude greater than the observed values, which are one order of magnitude smaller than the values expected from second virial coefficients observed for small molecules. This indicates that interactions between polymer segments in dilute solution are different from those between small molecules. It is pointed out that the difference arises from the fact that polymer segments bound in the chain have fewer degrees of freedom, translational and rotational, than free small molecules.

Test of Current Viscosity Theories of Dilute Polymer Solutions

Monte Carlo simulation of a confined random-walk chain

The prediction of conformal invariance for the statistics of self-avoiding random walks restricted to both semi-infinite and wedge-shaped geometries are tested by extrapolating exact enumerations. Close agreement is found, both for the angular distribution of the end-to-end vector, and for the dependence of the critical exponent gamma 2 on the opening angle of the wedge.

The t dependence of the average of the end-to-end distance (Rt2) of t-step random walks on self-avoiding walks is studied by small cell real-space renormalisation. The results are in qualitative agreement with earlier Monte Carlo data.