Critical Properties of the Emergent Random Solid in Vulcanization/gelation Transition

160 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2005.The vulcanization/gelation transition is a continuous equilibrium phase transition from a liquid state to a random solid state, controlled by the density of permanent crosslinks between the constituent particles. The emergent random solid state is characterized by nonzero shear rigidity and particle localization (in high enough dimensions) about random positions, where their localization lengths are statistically distributed. Founded upon previous work that had constructed a replicated Landau-Wilson free energy for the transition, and had analyzed the critical region within the liquid state, this Thesis focuses on the nature of fluctuations in the random solid state and the critical behavior of physical quantities detecting it. The first part of this Thesis investigates the Goldstone-type low energy, long wave-length fluctuations associated with the spontaneous breakdown of a (global, continuous) translational symmetry at the transition. These fluctuations are identified with the shear deformations of the emergent random solid, whose shear modulus and elastic free energy are derived utilizing this identification. The impact of such fluctuations on the statistical distribution of localization lengths is ascertained. In the second part of this Thesis, a thorough analysis of the critical region within the random solid state is presented on implementing a Renormalization Group approach. The critical-fluctuation-correction to the mean-field distribution of localization lengths is determined from a perturbative calculation of the Equation of State for the vulcanization/gelation field theory (to lowest order in an expansion in epsilon, i.e., the difference of the upper critical dimension and the spatial dimension). Such a calculation is challenging owing to the nature of translational symmetry breaking in the replicated field theory. The third part of this Thesis deduces the scaling of entropic shear rigidity near the vulcanization/gelation transition. The shear modulus exponent is analyzed within a Renormalization Group approach, and it is shown that the critical exponent can assume two distinct fixed-point values depending on the strength of the excluded-volume interaction between the constituent particles, thereby resolving an old controversy over its value.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD