## Effects of excluded volume and correlated disorder on the shapes of self-avoiding walks

### Dr Christian von Ferber (Coventry University, UK)

### Abstract

We apply the field theoretical renormalization group to analyze universal
shape properties of long polymer chains modelled as self-avoiding walks in
a correlated environment. Many analytical calculations focus on the
scaling exponents that govern conformational properties of polymer
macromolecules. Here, we consider observables that are related to
universal ratios. These are universal in the sense that given a polymer in
a good solvent, they are independent of the chemical structure of the
macromolecules and the details of the solvent. In particular we focus on
ratios characterising the deviation from the spherical shape as well as
the relation between the end-end distance and the gyration radius of the
polymer. These questions have a long history going back to Kuhn's work on
random walks 1934 with the aim to understand the viscosity of polymer
solutions. Here, we address the question of the influence of excluded
volume and correlated disorder on the shapes acquired by the long flexible
macromolecules. This question may be of relevance for the understanding of
the behavior of macromolecules in colloidal solutions, near microporous
membranes, or even in a biological environment. To this end, we consider a
model of polymers in D dimensions in an environment with structural
obstacles, characterized by a pair correlation function h(r), that decays
with distance r according to a power law: h(r) ~ 1/r^a . We apply the
field-theoretical renormalization group approach and expand in both (4-D)
and (4-a) to estimate ratios that characterise the end-end distance, the
gyration radius and rotationally invariant measures of the deviation from
the spherical shape. [V Blavatska, C von Ferber, and Yu. Holovatch;
Effects of disorder on the shapes of macromolecules, Condensed Matter
Physics (2011)].