Statistical mechanics of the amorphous solid state of randomly crosslinked macromolecules

In 1839 Charles Goodyear discovered the vulcanization of rubber: by heating natural rubber (which is a liquid), together with sulfur, he obtained an elastic solid material. This liquid-solid transformation was later found to entail the random crosslinking of the linear macromolecules forming natural rubber, until the crosslink density is high enough that an equilibrium phase transition--the vulcanization transition--from the liquid to the amorphous solid state occurs. In this thesis a semi-microscopic theory of the vulcanization transition is developed, and the emergent amorphous solid state is studied in detail, close to the transition. In the first part of this work a derivation is presented of the free energy functional for a system of randomly crosslinked linear macromolecules, starting from a semi-microscopic model. Stationary points of the free energy are obtained that represent, respectively, the liquid and the amorphous solid state. A continuous phase transition between the two states is found to occur. It is shown that in the liquid state all monomers are delocalized, but in the amorphous solid state, some fraction (called the "gel fraction") of the monomers become localized near random mean positions, and thermally fluctuate about these positions with random r.m.s. displacements (called "localization lengths"). The distribution of localization lengths is computed, and it is shown that both this distribution and the order parameter exhibit simple scaling properties near the transition. Both the gel fraction and the typical inverse localization length are found to vanish continuously at the transition. In the second part of this work the stability of the amorphous solid state is analyzed for a class of systems undergoing liquid-amorphous-solid phase transitions driven by the effect of random constraints. This class of systems includes vulcanized macromolecular systems, as well as others, including endlinked ones. This stability analysis is performed within two different formulations: one involving a Landau theory that is common to all systems in the class, and another involving the semi-microscopic theory for randomly crosslinked macromolecules (discussed in the first part of this work). The results are the same in the two formulations. The stability matrix is obtained for fluctuations around the stationary point corresponding to the amorphous solid state. All the eigenvalues of the stability matrix are shown to be non-negative near the transition. In fact, they are all positive, except for a zero mode associated with the spontaneously broken continuous translational symmetry of the system. Therefore the amorphous solid state is found to be stable. Signatures of the transition to the amorphous solid state include not only the random localization of a fraction of the particles but also the emergence of a nonzero static shear modulus. In the third part of this work, a semi-microscopic statistical-mechanical theory of the latter signature is presented that accounts for both thermal fluctuations and quenched disorder. It is found that the shear modulus grows continuously from zero at the transition, and does so with the classical exponent, i.e., with the third power of the excess crosslink density. It is also found that, quite surprisingly, the external stresses do not spoil the spherical symmetry of the localization clouds of the particles near the transition.U of I Onlythesi