Boli Zhou, UT-Austin

"Scale-dependent elasticity as a probe of the amorphous solid state"

Abstract: The vulcanization transition is the name given to the continuous phase transition -- exhibited by rubber -- from the liquid state to the amorphous solid state, driven by increasing the density of the random chemical cross-links that permanently connect the constituent particles at random. In the liquid state, the particles are totally delocalized, but in the amorphous solid state the cross-linking causes a fraction of them to become localized in space. Although the mean positions of the localized particles are random, showing no periodicity, the single-particle static density at different positions in space are correlated. Thus, one can design an order parameter capable of distinguishing between the two states: liquid and amorphous solid. On the other hand, the presence of quenched disorder hints the applicability of replica statistical mechanics. Accordingly, one should be able devise a kind of replica order parameter governed by an associated replica Landau theory that is stringently constrained in its form by symmetry and length-scale arguments. Analysis of such a Landau theory indicates that the amorphous solid state is characterized by the fraction of localized particles and the statistical distribution of particle localization lengths. Moreover, the amorphous solid state spontaneously breaks a key symmetry -- that of the relative translations of the replicas -- and we use the corresponding Goldstone fields to develop the elasticity theory for the amorphous solid state. Peculiar to this setting is the presence of continuously distributed localization lengths that diverge as the transition to the liquid state is approached. This phenomenon brings a special significance to the scale-dependent elasticity as a probe of the distribution of localization lengths. We compute this scale-dependent elasticity and show its relationship with the distribution of localization lengths.