Probing the distribution of localization lengths in amorphous solids via wavelength-dependent elasticity

The amorphous solid state exhibited, for example, by randomly crosslinked macromolecular systems has two distinguishing features: (i) it arises via a continuous transition controlled by the crosslink density; and (ii) as a result of the intrinsic randomness, the state is described by a distribution of single-particle localization lengths. Owing to the continuity of the transition, in its vicinity the localization-length distribution is concentrated predominantly at intermediate lengthscales, i.e., lengthscales larger than microscopic but not truly macroscopic. We report on the development of an elasticity theory for the amorphous solid state that is valid not only in the limit of long-wavelength strains but also for strains at wavelengths corresponding to intermediate lengthscales. The corresponding wavelength-dependent shear modulus is sensitive to the distribution of localization lengths, diminishing monotonically with decreasing lengthscale -- a physical reflection of the idea that elasticity at a given lengthscale is primarily supported by particles localized on that or shorter lengthscales. The dependence of the shear modulus on wavelength therefore provides an experimental pathway to probing the distribution of localization lengths.