Beyond Flory theory: Distribution functions for interacting lattice trees

Abstract

While Flory theories [1][2][3][4][5] provide an extremely useful framework for understanding the behavior of interacting, randomly branching polymers, the approach is inherently limited. Here we use a combination of scaling arguments and computer simulations to go beyond a Gaussian description. We analyse distributions functions for a wide variety of quantities characterising the tree connectivities and conformations for the four different statistical ensembles, which we have studied numerically in Refs. [6,7]: (a) ideal randomly branching polymers, (b) 2d and 3d melts of interacting randomly branching polymers, (c) 3d self-avoiding trees with annealed connectivity and (d) 3d self-avoiding trees with quenched ideal connectivity. In particular, we investigate the distributions (i) pN (n) of the weight, n, of branches cut from trees of mass N by severing randomly chosen bonds; (ii) pN (l) of the contour distances, l, between monomers; (iii) pN ( r) of spatial distances, r, between monomers, and (iv) pN ( r|l) of the end-to-end distance of paths of length l. Data for different tree sizes superimpose, when expressed as functions of suitably rescaled observables x = r/ r 2 (N ) or x = l/ l(N ) . In particular, we observe a generalised Kramers relation for the branch weight distributions (i) and find that all the other distributions (ii-iv) are of Redner-des Cloizeaux type, q( x) = C |x| θ exp −(K|x|) t . We propose a coherent framework, including generalised FisherPincus relations, relating most of the RdC exponents to each other and to the contact and Flory exponents for interacting trees.

Influence of Branching on the Configurational and Dynamical Properties of Entangled Polymer Melts

“…Recent simulation studies have suggested that other molecular architectures than linear chains may have mass scaling exponents governing polymer size (e.g., Rnormalg), less than ν=1/2. For randomly branched polymers in their melt state, ν has been proposed to be exactly 1/3 [55,56,57,58], indicating that these polymers also form rather ‘compact’ structures in the melt state [59]. Ring polymers in the melt have been predicted to exhibit this same type of asymptotic scaling [52], strongly suggesting that these polymers belong to the randomly branched polymer universality class when they are in the melt state.…”

Flory theory of randomly branched polymers

“…A forthcoming article 76 discusses how to go beyond Flory theory and the Gaussian approximation by analyzing the distribution functions characterising the tree conformations and connectivity.…”

Computer simulations of melts of randomly branching polymers

“…This delicate balance might well have been destroyed for trees, where the Flory energy needs to be simultaneously minimized with respect to L and R. In two forthcoming publications, we will generalise Flory theory to trees of finite size and extensibility 34 and we will analyse the distribution functions for the quantities, whose mean behaviour we have explored above and in Ref. 17 in an attempt to go beyond Flory theory 35 .…”