Melts of polymeric fractals and D-dimensional manifolds: Saturation vs screening

Abstract

This paper discusses the behavior of melts of polymeric fractals in their dense state. It is shown that melts of polymeric fractals of spectral dimension D behave differently when D exceeds a critical value, i.e., D S ϭ2d/(2ϩd). Here d is the dimension of space. For larger connectivities the fractals are saturated, whereas fractals with smaller connectivity behave similarly to melts of linear polymer chains. In the latter case the polymers interpenetrate each other to a great extent, screen their excluded volume, and retain their ideal shape. This conclusion is drawn by studying screening processes in melts of polymeric manifolds of arbitrary connectivity. It is shown that for systems above the critical spectral dimension a screening length exists, i.e., in other words, a concentration, for which the screening condition can never be satisfied. It is shown that this fact corresponds to saturation. Below the critical connectivity dimension the classical excluded volume screeningcomparable to the case in linear polymer chains-takes place. The condition on the spectral dimension is equivalent to a condition in terms of the Gaussian fractal dimension d f ϭ2D/(2ϪD), i.e., the melt saturates when the embedding space dimension is less or equal than the Gaussian fractal dimension, i.e., dрd f .

On the Mechanism of Hydrodynamic Reinforcement in Elastic Composites

“…The clusters overlap only if their connectivity is not too large. We have worked out criteria for these conditions in. , It was shown that the spectral dimensions should not exceed 6 / 5 for ideal and 4 / 3 for nonideal clusters. Clusters with larger connectivities do no longer overlap significantly.…”

Elasticity of entangled polymer loops: Olympic gels

“…It is well known, that the excluded volume forces are screened out [1,11]. The screening of the excluded volume can also be asserted to the strong interpenetration of linear (= one dimensional connectivity) chains [12]. As a result, the chains behave gaussian in the melt.…”

Microgels and fractal structures at interfaces and surfaces

“…Thus fractals with lower spectral dimensions screen their excluded volume, whereas polymeric fractals with larger spectral dimension saturate. Then they form soft balls, which cannot interpenetrate each other and are well separated from each other [11,12]. This state has an analogon in the case of linear polymers.…”

“…We have shown earlier [11,12], that fractal polymers and gels can interpenetrates each other and screen excluded volume forces, wherever their connectivity is low. In terms of the spectral dimension this is the case, when it is lower than a critical value, i.e.…”