SHAPES OF POLYMERS AND POLYMER NETWORKS
Christian von Ferber
Applied Mathematics Research Centre, Coventry UniversityThe discovery and experimental confirmation by light scattering experiments in the early 1920ies that polymers in solution are well described as long flexible chains of high numbers of (often identical) units has given rise to research in modeling their properties in Mathematics and Physics . Subsequently, models using random walks and self-avoiding walks were developed as well as sophisticated experimental and mathematical methods. Light and X-ray scattering allows to measure the mean extensions of the polymer coils in solution due to the correlations of monomers belonging to the same chain.
In the present lectures we in turn discuss and explore the possibilities of determining the shapes of these polymers coils in solution: e.g. for such random chains we may determine the deviation of the shape of these coils from the symmetric disk like distribution. We will find that indeed coils of such chains display a non-spherical shape which we may measure by defining an “asphericity” parameter. In turn, three dimensional polymers may display deviations from the spherical shape in terms of either an oblate (pancake like) shapes or a prolate (cigar like) shapes. We will find that the prolate shape prevails for linear chains in three dimensions. The way to measure these shapes is via a gyration tensor (similar to the inertia tensor for rigid bodies).
A simplified intuitive version of measuring the shape of a random-walk coil has been proposed earlier by Kuhn .
These lectures intend to introduce various concepts that have been used successfully to tackle the intricate problems of determining polymer shape parameters for polymers with and without self-avoidance.
We start by discussing numerical and analytical methods to determine the shapes of two- and three-dimensional random walks. This research has a long history dating back to R. Koyama  as well as Solc, Gobush and Stockmeyer .
We further explore the impact of excluded volume on the shapes of linear chain-like polymers following the seminal work by Aronovitz and Nelson .
Finally, we will investigate situations involving branched polymers, starting with star-branched polymers, i.e. structures where a number of identical chains are all attached by one end to a central core. Further, we look at other regular branched structures such as 'comb'-like architectures and star-burst dendrimers.
Finally, we demonstrate a semi-numerical methods to calculate the shapes of branched structures building on work by Wei and Eichinger . Applying these methods we investigate random branched networks with different branching statistics such as loop-less networks with different degree distributions.
 H. Staudinger. Uber Polymerisation. Berichte der deutschen chemischen Gesellschaft, 53 (1920) 1073–1085.
 P. Debye. Methods to determine the electrical and geometrical structure of molecules (Nobel Lecture, December 12, 1936); see also P. Debye. Debye function. Ann. d. Phys. 32 (1912) 789.
 W. Kuhn. Ueber die Gestalt fadenformiger Molekule in Losungen. Colloid Zeitschrift, 68 (1934) 2.
 R. Koyama. Excluded volume effect on the radius of gyration of chain polymers. J. Phys. Soc. Jpn. 22 (1967) 973.
 W. Gobush, K. Solc, W.H. Stockmayer. Statistical mechanics of random-flight chains. V Excluded volume expansion and second virial coefficient for linear chains of varying shape. J. Chem. Phys. 60 (1974) 12.
 J.A. Aronovitz, D.R. Nelson. Universal features of polymer shapes. J. Phys. France, 47 (1986) 1445.
 G. Wei, B.E. Eichinger. On shape asymmetry of Gaussian molecules. J. Chem. Phys. 93 (1990) 1430.