Universal $L^{-3}$ finite-size effects in the viscoelasticity of confined amorphous systems

Abstract

We present a theory of viscoelasticity of amorphous media, which takes into account the effects of confinement along one of three spatial dimensions. The framework is based on the nonaffine extension of lattice dynamics to amorphous systems, or nonaffine response theory. The size effects due to the confinement are taken into account via the nonaffine part of the shear storage modulus G . The nonaffine contribution is written as a sum over modes in k-space. With a rigorous argument based on the analysis of the k-space integral over modes, it is shown that the confinement size L in one spatial dimension, e.g. the z axis, leads to a infrared cut-off for the modes contributing to the nonaffine (softening) correction to the modulus that scales as L −3 . Corrections for finite sample size D in the two perpendicular dimensions scale as ∼ (L/D) 4 , and are negligible for L D. For liquids it is predicted that G ∼ L −3 in agreement with a previous more approximate analysis, whereas for amorphous materials G ∼ G bulk + βL −3 . For the case of liquids, four different experimental systems are shown to be very well described by the L −3 law.

Universal G′ ∼ L–3 Law for the Low-Frequency Shear Modulus of Confined Liquids

“…For the lower limit of the non-affine integral, we used an "infrared" cutoff, by assuming that the liquid is confined along the z direction, such that L ≡ L z ≪ L x , L y ; the lower limit in the non-affine integral over k-space thus reduces to 1/L, as displayed in eq 4. A rigorous estimate of the k-integral in eq 4, including exact prefactors, can be found in ref 39.…”

Deformations, relaxation and broken symmetries in liquids, solids and glasses: a unified topological field theory

“…The theoretical discovery of the finite elasticity of confined liquids has been put forward using the framework of nonaffine elasticity of structurally disordered condensed matter systems [63]. The key point is that confinement along one spatial direction effectively cuts off low-energy eigenmodes of the system which are responsible for the non-affine softening contribution to the shear modulus [81]. The infrared cutoff due to confinement thus make the non-affine contribution effectively smaller, and leads to a finite low-frequency shear modulus in liquids with good wetting to the solid boundaries (the latter boundary condition is required to ensure the propagation of low-frequency phonons).…”