Z. Alexandrowicz

Monte Carlo of Chains with Excluded Volume: a Way to Evade Sample Attrition

A new method for the generation of long self‐avoiding walks on lattice is described: Short self‐avoiding walks—say of length N = 50—are generated by direct Monte Carlo method, then linked in pairs to form “dimers.” Each dimer is tested for intersections between its two halves—those passing the test giving a sample of self‐avoiding walks N = 100, which is dimerized in turn, etc. In this manner the repeated checking for intersections formed with only a few steps (short loops) is substantially avoided, while precisely such intersections are responsible for the heavy attrition with the direct Monte Carlo method. Thus the attrition accompanying the dimerization is quite insignificant even for very large N. With the help of this method walks N = 50 × 27 = 6400 were generated on the 4‐choice cubic lattice, for which the expansion coefficient of the end to end distance is α = 2.2. (The limit reached by others is N ≃ 2000 on the tetrahedral lattice, corresponding to only α ≃ 1.6, while α = 2.2 would require a leng...

Recoil growth: An efficient simulation method for multi-polymer systems

We present a new Monte Carlo scheme for the efficient simulation of multi-polymer systems. The method permits chains to be inserted into the system using a biased growth technique. The growth proceeds via the use of a retractable feeler, which probes possible pathways ahead of the growing chain. By recoiling from traps and excessively dense regions, the growth process yields high success rates for both chain construction and acceptance. Extensive tests of the method using self-avoiding walks on a cubic lattice show that for long chains and at high densities it is considerably more efficient than configurational bias Monte Carlo, of which it may be considered a generalization.

Monte Carlo of Chains with Excluded Volume: Distribution of Intersegmental Distances

A Monte Carlo study of non‐self‐intersecting chains on a four‐choice cubic lattice is undertaken with the help of a recently described “dimerization” method. This method permits to avoid the difficulty of “sample attrition,” enabling the construction of relatively long chains. Certain improvements of the method are described and its validity is substantiated. The distribution and average values of the end to end, as well as of various intrachain distances, are determined for chain lengths varying from N = 32 to 8192. The results indicate that the basic assumptions of several theories on excluded volume need to be revised.

Effect of Excluded Volume on Polyelectrolytes in Salt Solutions

A recently developed theory of excluded volume, relating α to z ∝ M½βL−3 by means of an integral equation, is applied to polyelectrolytes. The line describing α(M½) is the same for all polymers; the dependence of α on the polyelectrolyte parameters (ionic strength and charge density) requires, however, further elaboration. Thus, the excluded volume β is summarily identified with the high‐potential shell coaxially surrounding the charged filament, as treated in a polyelectrolyte theory presented some time ago. Distinction is made between two cases: (a) When the shell radius is smaller than the link length L characterizing the polymer and (b) when the converse holds—so that the length of free hinged links becomes determined by the shell radius. The predicted dependence of α on the molecular weight and on the ionic strength is found to agree satisfactorily with the extensive viscosity data reported by Takahashi and Nagasawa. Additional aspects of investigating α(z) are also discussed.

Analytical Treatment of Excluded Volume. III. Alternative Calculations of the Contacts Density in Chains

The excluded volume effect is expressed in terms of certain probabilities for segment contacts in the real chain, the derivation following a method recently described by Fujita et al. It is then assumed that these probabilities can be calculated from the Gaussian expressions for an “equivalent ideal chain.” On this basis an integral equation is derived for α(z), the asymptotic result being α5 ∼ z. While Fixmans theory [A], the authors previous theory [B], and the present theory [C] express the segment contacts in different ways, they have the common property of preserving the chain connectedness and of assuming contacts probability to be Gaussian. Dissimilar results are reached with the three theories and the reasons for the discrepancy are examined. It appears that the results obtained with the present theory [C], similar to [A] but in contradistinction to [B], do not agree with the original assumption of the contacts probability being Gaussian.

Stochastic Model for Chains with Excluded Volume

A chain consisting of n links with excluded volume (EV) interactions is described by a multilevel hierarchy of segments of increasing length, joined together in succession. Thus the second level segments, t=2, consist of s links, the third consist of s second segments (or s2 links), etc., t=1, 2, ···T, until the chain length is reached with the longest T segment, n=sT. The configurations for this model chain are constructed with a stochastic process which links the segments together, head to tail, with a probability that favors their orientation in the forward direction. Such a partly oriented linkage on one hand, decreases the multiplicity of the chain configurations but, on the other, increases the Boltzmann factor due to EV interactions. The linking probability at each level is, therefore, determined with a variation principle that maximizes the product of these two quantities. An approximate analytical solution shows, that the chain expansion coefficient (accumulated over all T levels) obeys the Fixma...

Simulation of polymers with rebound selection

A crossover from dilute to semidilute regimes of lattice chains has been simulated with the help of a novel method. A Monte Carlo step tries to replace an existing chain by a new chain constructed in biased, excluded volume avoiding steps. Reserve bifurcations are allocated at small intervals of steps and after each interval a selective lottery restores an effectively nonbiased probability distribution. If a construction fails the selection at an nth step, it falls back to n−1, n−2, until it “rebounds” from an unutilized bifurcation. Selection on the go combined with a freedom to rebound, make the construction self-corrective; the principle may have a broad applicability.

Polymers simulated with an improved “rebound selection”

An improved “rebound selection” simulation constructs a polymer chain in biased excluded volume avoiding steps. A construction that in midcourse accumulates a low bias weight tends to be arrested by a selection. Conversely a high weight construction sprouts reserve bifurcation and if subsequently it is arrested, it falls back and rebounds from such bifurcations. The simulation is applied to a condensation transition of thermal 2d lattice chains, and to a dilute–semidilute transition of athermal chains. Semidilute 2d chains do not mix, 3d ones mix as blobs.

Do “grow and exchange” simulations of polymers obey detailed balance?

Efficient simulations grow a polymer chain in a sequence of steps that occasionally branches out, seeking optimal pathways. The fraction of optimal pathways from among all possible ones has to be evaluated, in order to determine a new chain’s ensemble weight and hence its detailed balance acceptance into a multi-chain system. The total number of all possible pathways, however, is huge and has to be sampled. The consequent use of a sampled detailed balance may be questioned. Here it is shown to be exact (at least in the athermal case).

Alternative Approach to the Cluster Theory of Imperfect Gases and Condensation

The expansion of the density and pressure in terms of power series of activity, with cluster integrals as coefficients, is derived with the help of a new method. Rather than expanding at once the entire configurational partition function (3N‐dimensional volume) the expansion here is applied to a quantity constituting the logarithmic increment of the partition function per molecule (three‐dimensional volume). The expansion of this quantity is then recursively used to derive the thermodynamic functions for the system. Formulated in terms of “mathematical” clusters the method leads to a series expansion that, so long as such series converge, remains equivalent to the famous result of Mayers theory. However, as the series become divergent (which presumably corresponds to condensation) the expansion cannot be anymore reduced to Mayers series. When formulated in terms of “physical” clusters the present method permits to derive an expansion that duly allows for physical cluster integrals being mutually exclusi...