BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

Exact evaluation of the mean-square fluctuation of the position vector of a crosslinking point in the Gaussian network Deguchi, Tetsuo


The Gaussian network plays a central role in the study on the fundamental elastic behavior of various polymer networks such as rubbers and gels [1, 2]. Here we remark that many bio-materials are made of gels. Recently, a new method has been introduced for generating an ensemble of random conformations of graph-shaped polymers in terms of topologically constrained Gaussian random walks (TCRW) or Gaussian random graph embeddings [3]. It is one of the key properties of TCRW that the probability distribution function of the bond vectors in polymer conformations of TCRW is composed of the normal distributions with unit variance. In this talk we critically study Floryâ s approximate expression for the mean square fluctuation of the end-to-end vector r around its average value \(r\) with functionality \(f\) [4] $$â ¨2 â ¨(r â â ¨râ ©) â ©2 2Nb f$$ Here N is the number of the Kuhn segments in the network subchain connecting a crosslinking point to another one. We express the fluctuation â ¨(Î r)2â © in terms of resistance distances, and evaluate it rigorously. We argue that Floryâ s expression should be valid if the functionality f is very large, based on the numerical experiments of large random graphs with functionality f, i.e., regular graphs with functionality f. We also discuss the results of Ref. [5]. The results of the present talk should be important not only in materials science but also in applications of biomaterials.

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