C E Soteros

The statistical mechanics of random copolymers

Random copolymers are polymers with two or more types of monomer where the monomer sequence is determined by some random process. Once determined, the sequence is fixed so random copolymers are an example of a system with quenched randomness. We review the statistical mechanics of random copolymers, focusing on self-avoiding walk models where there are two types of monomers, A and B, which are randomly distributed along the polymer chain. Theoretical, approximate and numerical results are reviewed for models of the random copolymer adsorption, localization and collapse phase transitions. We concentrate on what is known about the existence of phase transitions, the Morita approximation, and results about self-averaging. We also discuss, in less detail, the replica trick and numerical methods including Monte Carlo methods, exact enumeration and transfer-matrix methods. Important open problems are identified throughout and highlighted in the conclusions.

Statistics of lattice animals

The authors investigate the large-n behaviour of the number of lattice animals with n vertices having alpha cycles per vertex. They prove concavity and continuity properties for the corresponding growth constant and, in particular, show that lattice trees are exponentially scarce in the set of lattice animals. They also consider the corresponding generating function and prove a number of theorems which bound it and set other limits on its possible behaviour.

The free energy of a collapsing branched polymer

The authors consider a number of related lattice models of branched polymers in dilute solution in which the polymer is modelled as a tree or as an animal. In order to model the effect of the thermodynamic properties of changing the temperature, or the quality of the solvent, they consider counting cycles in animals and near-neighbour contacts in both animals and trees. They show that the free energies of these models have common features and derive rigorous upper and lower bounds on the temperature dependence of the free energies. Finally, they derive series data for several of these models and compare their estimates of the limiting free energy with the rigorous bounds.

Polygons and stars in a slit geometry

The authors consider self-avoiding polygons, uniform 3-stars and uniform 4-stars, weakly embeddable in the square lattice and confined between two parallel lines, y=0 and y=L. They show rigorously that the connective constants of these three structures (provided that the corresponding limits exist) are all strictly less than the connective constant kappa (L) of self-avoiding walks, with the same geometrical constraint. They also derive lower bounds on the connective constants of uniform 3-stars and 4-stars. These bounds appear to be strong, at least for small L.

Critical exponents for lattice animals with fixed cyclomatic index

The authors derive an inequality between the number of trees and the number of lattice animals with exactly c cycles, an(c), for all positive c. If they assume that an(c) approximately n- theta c lambda nc, n to infinity , c fixed, they use this to show that theta c= theta o-c where theta 0 is the corresponding exponent for trees.

Stretched polygons in a lattice tube

We examine the topological entanglements of polygons confined to a lattice tube and under the influence of an external tensile force f. The existence of the limiting free energy for these so-called stretched polygons is proved and then, using transfer matrix arguments, a pattern theorem for stretched polygons is proved. Note that the tube constraint allows us to prove a pattern theorem for any arbitrary value of f, while without the tube constraint it has so far only been proved for large values of f. The stretched polygon pattern theorem is used first to show that the average span per edge of a randomly chosen n-edge stretched polygon approaches a positive value, non-decreasing in f, as n → ∞. We then show that the knotting probability of an n-edge stretched polygon confined to a tube goes to one exponentially as n → ∞. Thus as n → ∞ when polygons are influenced by a force f, no matter its strength or direction, topological entanglements, as defined by knotting, occur with high probability.

Statistics of collapsing lattice animals

We consider several lattice models of branched polymers in dilute solution in which the polymer molecule is modelled as a tree or animal. In general the thermodynamic properties of the polymer are determined by monomer-monomer and monomer-solvent interactions. We examine a two-variable model which incorporates both types of interaction and discuss its relationship to other models which have previously been investigated. In particular, we discuss the collapse transition in these models.

Localization of a random copolymer at an interface: an exact enumeration study

We consider a self-avoiding walk on the simple cubic lattice, as a model of localization of a random copolymer at an interface between two immiscible liquids. The vertices of the walk are coloured A or B randomly and independently. The two liquid phases are represented by the two half-spaces z > 0 and z < 0, and the plane z = 0 corresponds to the interface between the two liquids. The energy depends on the numbers of A-vertices with positive z-coordinate and B-vertices with negative z-coordinate. In addition there is a vertex–interface interaction, irrespective of the colour of the vertex. We use exact enumeration and series analysis techniques to investigate the form of the phase diagram and how it changes as the magnitude of the vertex–interface interaction changes.

Contacts in self-avoiding walks and polygons

We prove several results concerning the numbers of n-edge self-avoiding polygons and walks in the lattice Zd which had previously been conjectured on the basis of numerical results. If the number of n-edge self-avoiding polygons (walks) with k contacts is pn(k) (cn(k)) then we prove that κ0 ≡ limn→∞ n-1 log pn(k) = limn→∞ n-1 log cn(k) exists for all fixed k and is independent of k. For polygons in Z2, we prove that there exist two positive functions B1 and B2, independent of n but depending on k, such that B1nkpn(0) ≤ pn(k) ≤ B2nkpn(0) for fixed k and n large. Also, provided the limit exists, we prove that 0 < limn→∞ kn/n < 1. In addition, we consider the number of polygons with a density of contacts, i.e. k = αn, and show that the corresponding connective constant, κ(α), exists and is a concave function of α. For d = 2, we prove that limα→0+ κ(α) = κ0 and the right derivative of κ(α) at α = 0 is infinite.

Self-avoiding polygons and walks in slits

A polymer in a confined geometry may be modeled by a self-avoiding walk or a self-avoiding polygon confined between two parallel walls. In two dimensions, this model involves self-avoiding walks or self-avoiding polygons in the square lattice between two parallel confining lines. Interactions of the polymer with the confining walls are introduced by energy terms associated with edges in the walk or polygon which are at or near the confining lines. We use transfer-matrix methods to investigate the forces between the walk or polygon and the confining lines, as well as to investigate the effects of the confining slits width and of the energy terms on the thermodynamic properties of the walks or polygons in several models. The phase diagram found for the self-avoiding walk models is qualitatively similar to the phase diagram of a directed walk model confined between two parallel lines, as was previously conjectured. However, the phase diagram of one of our polygon models is found to be significantly different and we present numerical data to support this. For that particular model we prove that, for any finite values of the energy terms, there are an infinite number of slit widths where a polygon will induce a steric repulsion between the confining lines.