RANDOMLY CROSS-LINKED MACROMOLECULAR SYSTEMS - VULCANIZATION TRANSITION TO AND PROPERTIES OF THE AMORPHOUS SOLID-STATE

As Charles Goodyear discovered in 1839, when he first vulcanized rubbe r, a macromolecular liquid is transformed into a solid when a sufficie nt density of permanent crosslinks is introduced at random. At this co ntinuous 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 equilib rium amorphous solid state: it is properly viewed as a solid state tha t bears the same relationship to the liquid and crystalline states as the spin glass state of certain magnetic systems bears to the paramagn etic 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 macromolecu les. Our primary focus is on the equilibrium properties of such system s, especially in the regime of Goodyear's vulcanization transition. Th is approach rests firmly on techniques from the statistical mechanics of disordered systems pioneered by Edwards and co-workers in the conte xt of macromolecular systems, and by Edwards and Anderson in the conte xt of magnetic systems. We begin with a review of the semi-microscopic formulation of the statistical mechanics of randomly crosslinked macr omolecular systems due to Edwards and co-workers, in particular discus sing 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 intuit ion, we examine the order parameter in an idealized situation, which s ubsequently turns out to be surprisingly relevant. Thus, we are motiva ted to hypothesize an explicit form for the order parameter in the amo rphous solid state that is parametrized in terms of two physical quant ities: the fraction of localized monomers, and the statistical distrib ution of localization lengths of localized monomers. Next, we review t he symmetry properties of the system itself, the liquid state and the amorphous solid state, and discuss connections with scattering experim ents. Then, we review a representation of the statistical mechanics of randomly crosslinked macromolecular systems from which the quenched d isorder has been eliminated via an application of the replica techniqu e. We transform the statistical mechanics into a held-theoretic repres entation, which exhibits a close connection with the order parameter, and analyse this representation at the saddle-point level. This analys is reveals that sufficient crosslinking causes an instability of the l iquid state, this state giving way to the amorphous solid state. To ad dress the properties of the amorphous solid state itself, we solve the self-consistent equation for the order parameter by adopting the hypo thesis discussed earlier. Hence, we find that the vulcanization transi tion is marked by the appearance of a non-zero fraction of localized m onomers, which we compute, the dependence of this fraction on the cros slink density indicating a connection with random graph theory and per colation. We also compute the distribution of localization lengths tha t characterizes the ordered state, which we find to be expressible in terms of a universal scaling function of a single variable, at least i n the vicinity of the transition. Finally, we analyse the consequences of incorporating a certain specific class of correlations associated with the excluded-volume interaction.