Phase Transitions and Change of Type in Low-Temperature Heat Propagation

Abstract

Abstract. Classical heat pulse experiments have shown heat to propagate in waves through crystalline materials at temperatures close to absolute zero. With increasing temperature, these waves slow down and finally disappear, to be replaced by diffusive heat propagation. Several features surrounding this phenomenon are examined in this work. The model used switches between an internal parameter (or extended thermodynamics) description and a classical (linear or nonlinear) Fourier law setting. This leads to a hyperbolic-parabolic change of type, which allows wavelike features to appear beneath the transition temperature and diffusion above. We examine the region around and immediately below the transition temperature, where dissipative effects are insignificant. Significantly, these features appear only at certain temperatures below which the materials reach their peak thermal conductivities (at approximately 18.5 K and 4.5 K for NaF and Bi, respectively). No wavelike behavior is found in NaF and Bi at higher temperatures, where only diffusive heat propagation is observed. Further, the speed, U E , at which small amplitude thermal waves propagate is a decreasing function of temperature in the region where the waves can be detected, after which the diffusion process dominates. This hyperbolic region appears separated from the diffusive region by a "critical" temperature, ϑ λ , at which U E = 0 [1]. The aim of this paper is to understand the dynamics of regular solutions having temperatures close to that of the phase transition. We begin, in section 2, by describing a phenomenological onedimensional model which uses an internal variable behaving as an order parameter. In section 3, we will examine properties of the phase transition, and in section 4, we obtain conditions under which this class of solutions remain smooth. Some explicit cases are, finally, examined in section 5.

Mechanics of Microstructured Solids 2

“…Connections between heat conduction and nonlinear hyperbolic type equations have been studied by many authors who were concerned about the physical model: a review on heat waves is due to Joseph and Preziosi [12], subsequent results, also referred to experiments, are given by Saxton and Saxton [18]. However, these models are concerned about some refinements which take into account a richer phenomenology, which originates nonlinearities, with respect to that one of the model here considered.…”

“…In the present paper space derivatives are found making use of the discrete Fourier transform (DFT). Calculations are carried out using SciPy package [13]: for DFT the FFTW [6] library and for ODE solver the F2PY [18] generated Python interface to ODEPACK Fortran code [8] is used.…”

Nonlinear Hyperbolic Equations and Linear Heat Conduction with Memory

“…Connections between heat conduction and nonlinear hyperbolic type equations have been studied by many authors who were concerned about the physical model: a review on heat waves is due to Joseph and Preziosi [14], subsequent results, also referred to experiments, are given by Saxton and Saxton [20]. However, these models are concerned about some refinements of which take into account a richer phenomenology, which originates nonlinearities, with respect to that one of the model here considered.…”

Decay of the 3D Quasilinear Hyperbolic Equations with Nonlinear Damping

“…Recently, the authors in [7] proved convergence rates for the half space problem. Moreover, the authors in [3] and [4] considered phase transition and the effect of damping respectively. When (V) = − with being a given positive constant and ℎ(V) = 1, system (10) reduces to the well-known -system with linear damping.…”

“…[1][2][3][4] and references therein). To improve the model, one may generalize the history dependence of q by modifying (4) or, as was done in [2], by introducing a suitable nonlinear dependence in (7),…”