As Charles Goodyear discovered in 1839, when he first vulcanized rubber, a macromolecular liquid is transformed into a solid when a sufficient density of permanent crosslinks is introduced at random. At this continuous equilibrium phase transition, the liquid state, in which all macromolecules are delocalized, is transformed into a solid state, in which a non-zero fraction of macromolecules have spontaneously become localized. This solid state is a most unusual one: localization occurs about mean positions that are distributed homogeneously and randomly, and to an extent that varies randomly from monomer to monomer. Thus, the solid state emerging at the vulcanization transition is an equilibrium amorphous solid state: it is properly viewed as a solid state that bears the same relationship to the liquid and crystalline states as the spin glass state of certain magnetic systems bears to the paramagnetic and ferromagnetic states, in the sense that, like the spin glass state, it is diagnosed by a subtle order parameter. In this article we give a detailed exposition of a theoretical approach to the physical properties of systems of randomly, permanently crosslinked macromolecules. Our primary focus is on the equilibrium properties of such systems, especially in the regime of Goodyear's vulcanization transition. This approach rests firmly on techniques from the statistical mechanics of disordered systems pioneered by Edwards and co-workers in the context of macromolecular systems, and by Edwards and Anderson in the context of magnetic systems. We begin with a review of the semi-microscopic formulation of the statistical mechanics of randomly crosslinked macromolecular systems due to Edwards and co-workers, in particular discussing the role of crosslinks as quenched random variables. Then we turn to the issue of order parameters, and review a version capable, inter alia, of diagnosing the amorphous solid state. To develop some intuition, we examine the order parameter in an idealized situation, which subsequently turns out to be surprisingly relevant. Thus, we are motivated to hypothesize an explicit form for the order parameter in the amorphous solid state that is parametrized in terms of two physical quantities: the fraction of localized monomers, and the statistical distribution of localization lengths of localized monomers. Next, we review the symmetry properties of the system itself, the liquid state and the amorphous solid state, and discuss connections with scattering experiments. Then, we review a representation of the statistical mechanics of randomly crosslinked macromolecular systems from which the quenched disorder has been eliminated via an application of the replica technique. We transform the statistical mechanics into a held-theoretic representation, which exhibits a close connection with the order parameter, and analyse this representation at the saddle-point level. This analysis reveals that sufficient crosslinking causes an instability of the liquid state, this state giving way to the amorphous solid state. To address the properties of the amorphous solid state itself, we solve the self-consistent equation for the order parameter by adopting the hypothesis discussed earlier. Hence, we find that the vulcanization transition is marked by the appearance of a non-zero fraction of localized monomers, which we compute, the dependence of this fraction on the crosslink density indicating a connection with random graph theory and percolation. We also compute the distribution of localization lengths that characterizes the ordered state, which we find to be expressible in terms of a universal scaling function of a single variable, at least in the vicinity of the transition. Finally, we analyse the consequences of incorporating a certain specific class of correlations associated with the excluded-volume interaction.
