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Pincus relations for randomly branched polymers
Randomly branched polymer chains (or trees) are a classical subject of polymer physics with connections to the theory of magnetic systems, percolation and critical phenomena. More recently, the model has been reconsidered for RNA, supercoiled DNA and the crumpling of topologically constrained polymers. While solvable in the ideal case, little is known exactly about randomly branched polymers with volume interactions. In the first part of the presentation, I will review the Flory theory for interacting trees for good solvent, θ-solutions and melts and compare its predictions to a wide range of available analytical and numerical results [1]. Even though the predictions are surprisingly good, the approach is inherently limited. In the second part of the presentation, I use a combination of scaling arguments and computer simulations to analyse distribution functions for a wide variety of quantities characterising the tree connectivities and conformations. We observe [2] a generalised Kramers relation for the branch weight distributions and find that distributions of contour and spatial distances are of Redner-des Cloizeaux type, q(x) = C|x|^θ exp[−(K|x|)^t]. We propose a coherent framework, including generalised Fisher-Pincus relations, relating most of the RdC exponents to each other and to the contact and Flory exponents for interacting trees.
[1] Flory theory of randomly branched polymers, Ralf Everaers, Alexander Y. Grosberg, Michael Rubinstein, and Angelo Rosa, Soft Matter, in press
[2] Beyond Flory theory : Distribution functions for interacting lattice trees, A. Rosa and R. Everaers, Phys. Rev. E, in press