Polymer Physics: 25 Years of the Edwards Model

The following sections are included:

• IDENTIFICATION OF PARTICIPANTS IN THE PHOTOGRAPH

• PREFACE

• CONTENTS

Since this meeting is concerned with 25 years of the Edwards model, readers may be interested in the genesis of the model and some comments on its present status. This talk is a purely personal one and does not claim to be a real history, it is just what happened to me.

The statistical mechanics of polymers (self-avoiding walks) are described. A direct method of renormalization is explained for the continuum Edwards model in space dimension d = 4 − ε. This is generalized to self-avoiding crumpled manifolds (i.e. tethered membranes) of internal dimension D (0 ≤ D < 2). In two dimensions the critical properties of polymers are related to conformal invariance. Finally the statistical mechanics of polymers embedded in a fluctuating membrane is briefly described via its link to conformal invariance in presence of “2D quantum gravity”.

We study the critical exponents of self avoiding walks on a family of Sierpinski-type fractals. The members of the family are characterized by an integer b. For large values of b, both the Hausdorff and the spectral dimensions of the fractals tend to 2 from below. We use finite size scaling theory to determine the first two terms in the asymptotic expansions of size exponent ν and the susceptibility exponent γ for large b. The results are compared to predictions of phenomenological theories such as Flory's, and to the ε-expansion techniques.

We discuss the philosophy and application of renormalisation group techniques applied to study the dynamics of dilute polymer solutions. The methodology of this approach is illustrated by means of two detailed examples, viz., (i) the calculation of the dynamic structure factor for a Gaussian polymer chain with hydrodynamic interactions; and (ii) the calculation of the mean-square end-to-end distance for a Gaussian polymer chain in a systematic flow field.

Much has been accomplished by simulations in polymer science since the formulation of the Edwards model [1] and the rate of publication on simulations of macromolecules is still increasing.

The present article contains three lectures. In the first one a brief survey of some recent developments in polymer simulation techniques is presented, in the second lecture the current status of the problem of polymers in random media is summerized, and the last lecture is concerned with self-avoiding flexible surfaces, a most recent development in statistical mechanics.

Statistics of self-avoiding walks (SAWs) on percolating lattices is reviewed. The phase diagram, giving the variation of SAW connectivity constant or the (tricritical) Θ-point variation with lattice dilution, and the critical behaviour, specifically the size exponents of the SAW on the percolation fractals, are discussed. Fractal effects (of the random lattice) on the excluded volume statistics of SAWs are identified.

There are important problems which require the path which emerges physically not only to move but to lose or gain some part of its length. Two notable examples arise in oxbow lakes, and the deposition of feathery material. These problems are described and an outline offered for a solution.

A simple random walk in space if given a dynamical life is described by the Rouse model. The two extensions required to represent the movement of a macromolecule in solution are interactions, and hydrodynamic effects. It is argued that these can be incorporated into equations which then show, as a bonus, that the dynamics fully honours the topological integrity of the molecules. These ideas can be extended to suspensions of liquid drops, for which exact, topologically correct equations can be written down.

The structure of the transport coefficients in the approach to the glass transition is characterized by an essential singularity in their behaviour as functions of the temperature, variously known as the Vogel-Fulcher or Dolittle or Williams-Landel-Ferry laws. This behaviour can be derived directly in certain simple models notably a glass of liquid crystal polymers where mean field theory may be applied. In this paper a new class of closed expressions for tansport coefficients are derived and it is shown that they can be used to derive the Vogel-Fulcher law.

The following sections are included:

• LIST OF SPEAKERS

• LIST OF PARTICIPANTS

• OFFICIALS

• LIST OF CONTRIBUTED TALKS