Exploring the Applications of Fractional Calculus: Hierarchically-Built-Polymers
Par A. Blumen, University of Freiburg, Germany
le mardi 4 Novembre à 14h, Salle des séminaires 3e étage bâtiment A4
Expressions involving fractional derivatives have a long history in the theory and modelling of viscoelastic materials. Here polymers are very challenging, since they may show vast viscoelastic domains, in which “anomalous” behaviours occur. The classical way to model such behaviours is based on the theory of generalized Gaussian structures (GGS) and is naturally related to fractional generalized Langevin equations of non-Markovian nature .
Exemplarily, we take the stiffness of polymers into account and extend the GGS formalism to semiflexible tree-like structures; among them are dendrimers and regular hyperbranched structures [2, 3]. Semiflexibility leads to restrictions on the bonds’ orientations, due to constraints on the bonds’ lengths and on the angles between bonds close to each other. Now, it turns out that the structure of the potential energy for semiflexible treelike polymers (STP) is very simple, given that its corresponding matrix is sparse. This allows us to readily determine the mechanical and dielectric relaxation of several STP, such as stars, dendrimers, dendrimers built from stars and Vicsek fractals.
Of particular importance in applications is the non-Markovian character of these models in the study of chemical reactions: Thus, in a very recent work we investigated the cyclization of chains to rings .
 Gurtovenko. A.A. and Blumen, A. 2005 Adv. Polym. Sci. 182, 171.
 Dolgushev, M. and Blumen, A. 2009 J. Chem. Phys. 131, 044905.
 Dolgushev, M. and Blumen, A. 2013 J. Chem. Phys. 138, 204902.
 Dolgushev, M., Guérin, T., Blumen A., Bénichou, O. and Voituriez, R., 2014 J. Chem. Phys. 141, 014901.