R Brak

Critical exponents from nonlinear functional equations for partially directed cluster models

We present a method for the derivation of the generating function and computation of critical exponents for several cluster models (staircase, bar-graph, and directed column-convex polygons, as well as partially directed self-avoiding walks), starting with nonlinear functional equations for the generating function. By linearizing these equations, we first give a derivation of the generating functions. The nonlinear equations are further used to compute the thermodynamic critical exponents via a formal perturbation ansatz. Alternatively, taking the continuum limit leads to nonlinear differential equations, from which one can extract the scaling function. We find that all the above models are in the same universality class with exponents γu=-1/2, γi=-1/3, and ϕ=2/3. All models have as their scaling function the logarithmic derivative of the Airy function.

A directed walk model of a long chain polymer in a slit with attractive walls

We present the exact solutions of various directed walk models of polymers confined to a slit and interacting with the walls of the slit via an attractive potential. We consider three geometric constraints on the ends of the polymer and concentrate on the long chain limit. Apart from the general interest in the effect of geometrical confinement, this can be viewed as a two-dimensional model of steric stabilization and sensitized flocculation of colloidal dispersions. We demonstrate that the large width limit admits a phase diagram that is markedly different from the one found in a half-plane geometry, even when the polymer is constrained to be fixed at both ends on one wall. We are not able to find a closed form solution for the free energy for finite width, at all values of the interaction parameters, but we can calculate the asymptotic behaviour for large widths everywhere in the phase plane. This allows us to find the force between the walls induced by the polymer and hence the regions of the plane where either steric stabilization or sensitized flocculation would occur.

A collapse transition in a directed walk model

The authors consider a directed walk model of linear polymers in dilute solution, with an energy associated with the number of near-neighbour contacts in the walk. For this model they can derive an exact expression for the generating function in two variables conjugate to the number of steps and the number of contacts. They discuss the analytic structure of this generating function and identify the transition corresponding to collapse.

New Results for Directed Vesicles and Chains near an Attractive Wall

In this paper we present new exact results for single fully directed walks and fully directed vesicles near an attractive wall. This involves a novel method of solution for these types of problems. The major advantage of this method is that it, unlike many other single-walker methods, generalizes to an arbitrary number of walkers. The method of solution involves solving a set of partial difference equations with a Bethe Ansatz. The solution is expressed as a “constant-term” formula which evaluates to sums of products of binomial coefficients. The vesicle critical temperature is found at which a binding transition takes place, and the asymptotic forms of the associated partition functions are found to have three different entropic exponents depending on whether the temperature is above, below, or at its critical value. The expected number of monomers adsorbed onto the surface is found to become proportional to the vesicle length at temperatures below critical. Scaling functions near the critical point are determined.

Exact solution of the staircase and row-convex polygon perimeter and area generating function

An explicit expression is obtained for the perimeter and area generating function G(y, z)= Sigma n>or=2 Sigma m>or=1 cn,mynZm, where cn,m is the number of row-convex polygons with area m and perimeter n. A similar expression is obtained for the area-perimeter generating function for staircase polygons. Both expressions contain q-series.

A scaling theory of the collapse transition in geometric cluster models of polymers and vesicles

Much effort has been expended in the past decade to calculate numerically the exponents at the collapse transition point in walk, polygon and animal models. The crossover exponent phi has been of special interest and sometimes is assumed to obey the relation 2- alpha =1/ phi with the alpha the canonical (thermodynamic) exponent that characterizes the divergence of the specific heat. The reasons for the validity of this relation are not widely known. The authors present a scaling theory of collapse transitions in such models. The free energy and canonical partition functions have finite-length scaling forms whilst the grand partition function has a tricritical scaling form. The link between the grand and canonical ensembles leads to the above scaling relation. They then comment on the validity of current estimates of the crossover exponent for interacting self-avoiding walks in two dimensions and propose a test involving the scaling relation which may be used to check these values.

Exact solution of the row-convex polygon perimeter generating function

An explicit expression is obtained for the perimeter generating function G(y)= Sigma n>or=2 any2n for row-convex polygons on the square lattice, where an is the number of 2n step row-convex polygons. An asymptotic expression for an approximately A mu nn-3/2 is obtained, where mu =3+22 and A are given. The authors also show that the generating function is an algebraic function and that it satisfies an inhomogeneous linear differential equation of degree three.

Directed animals on two-dimensional lattices

Previously, directed animals on square and triangular lattices have been enumerated by area, and have been found to have simple generating functions, whilst the hexagonal lattice generating function has not been obtained. Directed animals on several new lattices are enumerated, one class of which is solved exactly. Directed animals by bonds (with and without loops) are also enumerated. In each case an asymptotic growth like n-1/2 mu n is observed and precise estimates for mu are given.

A self-interacting partially directed walk subject to a force

We consider a directed walk model of a homopolymer (in two dimensions) which is self-interacting and can undergo a collapse transition, subject to an applied tensile force. We review and interpret all the results already in the literature concerning the case where this force is in the preferred direction of the walk. We consider the force extension curves at different temperatures as well as the critical-force temperature curve. We demonstrate that this model can be analysed rigorously for all key quantities of interest even when there may not be explicit expressions for these quantities available. We show which of the techniques available can be extended to the full model, where the force has components in the preferred direction and the direction perpendicular to this. Whilst the solution of the generating function is available, its analysis is far more complicated and not all the rigorous techniques are available. However, many results can be extracted including the location of the critical point which gives the general critical-force temperature curve. Lastly, we generalize the model to a three-dimensional analogue and show that several key properties can be analysed if the force is restricted to the plane of preferred directions.

On the behaviour of collapsing linear and branched polymers

In this paper, we have reviewed the behaviour of a variety of models of collapse transitions in linear and branched polymers. Both rigorous and numerical results are presented. Some of the work described is still in progress, so the results are incomplete. Nevertheless, the overall picture demonstrates how the association of an attractive fugacity between nearby monomers can give rise to a collapse transition in a wide variety of models. These then qualitatively describe the transitions undergone by collapsing linear and branched polymers.