Statistical Mechanics of Vulcanisation and the Spontaneous Emergence of Static-Density Fluctuations

We present a statistical-mechanical analysis of a system of randomly cross-linked macromolecules, focusing on the emergence of random static-density fluctuations as the solidification transition is approached. In mean-field theory the problem can be formulated in terms of a single macromolecule in an effective random potential. The distribution of the random potential, which must be calculated self-consistently, determines all moments of the static-density fluctuations. We find that the solidification transition occurs when an infinite network of cross-linked macromolecules is formed, and that a sequence of nonlinear density response functions diverges simultaneously at this transition.