Randomly crosslinked macromolecular systems: Vulcanization transition to and properties of the amorphous solid state

Abstract

As Charles Goodyear discovered in 1839, when he first vulcanised 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 delocalised, is transformed into a solid state, in which a nonzero fraction of macromolecules have spontaneously become localised. This solid state is a most unusual one: localisation 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 vulcanisation 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 vulcanisation 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 idealised situation, which subsequently turns out to be surprisingly relevant. Thus, we are motivated to hypothesise an explicit form for the order parameter in the amorphous solid state that is parametrised in terms of two physical quantities: the fraction of localised monomers, and the statistical distribution of localisation lengths of localised 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 field-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 s...

Universality and its origins at the amorphous solidification transition

“…As discussed in detail in [4], because we are concerned with the transition between liquid and amorphous solid states, both of which have uniform macroscopic density, the one-replica sector order parameter is zero on both sides of the transition. This means that the sought free energy can be expressed in terms of contributions referring to the higher-replica sector order parameter, alone.…”

“…However, the disorder-averaged particle density cannot detect the transition between the liquid and the amorphous solid states, because it is uniform (and has the same value) in both states: a subtler order parameter is needed. As shown earlier, for the specific cases of randomly crosslinked [1,3,4] and end-linked [8] macromolecular systems, the appropriate order parameter is instead:…”

“…Such a characterization weaves in the physical notion that amorphous systems should show a spectrum of possibilities for the behavior of their constituents, and adopts the perspective that it is this spectrum that one should aim to calculate. This hypothesis translates into the following expression for the replica order parameter [3,4]:…”

“…During the last decade there has been an ongoing effort to obtain an ever more detailed understanding of the behavior of randomly cross-linked macromolecular systems near the vulcanization transition [1][2][3][4]. This effort has been built from two ingredients: (i) the Deam-Edwards formulation of the statistical mechanics of polymer networks [5]; and (ii) concepts and techniques employed in the study of spin glasses [6].…”

Two simple approaches to sol–gel transition

“…2 The first theoretical representations of such type transitions have been elaborated in classical works by Flory [5] - [7]. Modern description of the glassy state of the macromolecule networks (see [8] - [10]) is based on pioneering contributions to the theory of soft condensed matter: the Deam-Edwards theory [11] of a cross-linked tangled macromolecule, and the Edwards-Anderson theory [12] of spin glass.…”