Andreev-Lifshitz hydrodynamics applied to an ordinary solid under pressure

Abstract

We have applied the Andreev-Lifshitz hydrodynamic theory of supersolids to an ordinary solid. This theory includes an internal pressure P , distinct from the applied pressure Pa and the stress tensor λ ik . Under uniform static Pa, we have λ ik = (P −Pa)δ ik . For Pa = 0, Maxwell relations imply that P ∼ P 2 a . The theory also permits vacancy diffusion but treats vacancies as conserved. It gives three sets of propagating elastic modes; it also gives two diffusive modes, one largely of entropy density and one largely of vacancy density (or, more generally, defect density). For the vacancy diffusion mode (or, equivalently, the lattice diffusion mode) the vacancies behave like a fluid within the solid, with the deviations of internal pressure associated with density changes nearly canceling the deviations of stress associated with strain. We briefly consider pressurization experiments in solid 4 He at low temperatures in light of this lattice diffusion mode, which for small Pa has diffusion constant DL ∼ P 2 a . The general principles of the theory -that both volume and strain should be included as thermodynamic variables, with the result that both P and λ ik appear -should apply to all solids under pressure, especially near the solid-liquid transition. The lattice diffusion mode provides an additional degree of freedom that may permit surfaces with different surface treatments to generate different responses in the bulk.

Andreev-Lifshitz supersolid hydrodynamics including the diffusive mode

“…A previous work studied the lattice diffusion mode for a normal solid having distinct velocities associated with momentum (e.g., the normal fluid velocity) and lattice elasticity. 23 The motivation was to consider that the time-delay in the pressurization experiments of Ref. 22 might be due to that mode, under the assumption that the sample studied is not supersolid.…”

“…If σ rather than s were held constant, c 0 would be the sound velocity in an ordinary (non-super) liquid, and c 1 would be the sound velocity in an ordinary solid. 23 Ref. 29 shows that for P a ≪ K we have c 2 1 ≫ c 2 ≫ c 2 0 and strain w (0) ll ≪ 1.…”

“…We have recently studied the lattice diffusion mode of a normal solid (see Ref. 23), based on equations obtained by eliminating the superfluid velocity v s from the theory of Andreev and Lifshitz. 1 We obtained both the diffusion constant and the eigenmode structure (by which we mean the ratios of the deviations from equilibrium of various thermodynamic quantities) for a solid under an externally applied pressure P a (necessary to solidify 4 He, even at T = 0).…”

“…Ref. 23 finds for a solid, by thermodynamic considerations, the dependence of P on P a . The consequences of distinct P , λ ik , and P a had not previously been considered.…”

“…Ref. 23 calculates the effect of P a on the propagating elastic and diffusive modes of an ordinary solid. Ref.…”