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

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=2d/(2+d). 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 screening—comparable 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=2D/(2−D), i.e., the melt saturates when the embedding space dimension is less or equal than the Gaussian fractal dimension, i.e., d≤d_f.