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The equilibrium amorphous solid state -- formed, e.g., by adequately randomlycrosslinking the constituents of a macromolecular fluid -- is a heterogeneousstate characterized by a universal distribution of particle localizationlengths. Near to the crosslink-density-controlled continuousamorphous-solidification transition, this distribution obeys a scaling form: ithas a single peak at a lengthscale that diverges (along with the width of thedistribution) as the transition is approached. The modulus controllingmacroscale elastic shear deformations of the amorphous solid does not depend onthe distribution of localization lengths. However, it is natural to anticipatethat for deformations at progressively shorter lengthscales -- mesoscaledeformations -- the effective modulus exhibits a scale-dependence, softening asthe deformation lengthscale is reduced. This is because an increasing fractionof the localized particles are, in effect, liquid-like at the deformationlengthscale, and therefore less effective at contributing to the elasticresponse. In this paper, the relationship between the distribution oflocalization lengths and the scale-dependent elastic shear modulus is explored,and it is shown, within the setting of a replica mean-field theory, that theeffective modulus does indeed exhibit scale-dependent softening. Through thissoftening, mesoscale elasticity provides a probe of the heterogeneity of thestate as characterized by the distribution of localization lengths. Inparticular, the response to short-lengthscale elastic deformations is shown toshed light on the asymptotics of the universal localization-length distributionat short localization lengths.

Scale-dependent elasticity as a probe of universal heterogeneity in equilibrium amorphous solids