We use phonon hydrodynamics with a surface slip flow as a simplified macroscopic model accounting for a reduction in lateral thermal conductivity in nanosystems. For high Knudsen numbers, the corresponding effective thermal conductivity decreases linearly with the radius or the width, in contrast with the quadratic dependence predicted by usual phonon hydrodynamics. The linear dependence is accounted for by the surface slip flow. The difference in the expressions for the surface tangential flow in the hydrodynamic and the diffusive regimes is commented on and the influence of boundary conditions on the form of the effective thermal conductivity is explored

Phonon hydrodynamics is used to analyze the influence of porosity and of pore size on reduction in thermal conductivity in porous silicon, with respect to crystalline silicon. The expressions predict that the thermal conductivity is lower for higher porosity and for smaller pore radius, as a consequence of phonon ballistic effects. The theoretical results describe experimental data better than the assumption that they only depend on porosity.

In the analysis of nanosystems, the phonon-wall interaction must be incorporated to the usual description of phonon hydrodynamics, as surface effects become comparable to bulk effects in these systems. In the present paper, we analyze the temperature dependence of two phenomenological coefficients describing the specular and diffusive collisions, on one side, and backscattering collisions, on the other side, in silicon nanowires. Furthermore, we also propose for them a qualitative microscopic interpretation. This dependence is important because it strongly influences the temperature dependence of the effective thermal conductivity of nanosystems

A heat-transport equation incorporating nonlocal and nonlinear contributions of the heat flux is derived in\ud the framework of weakly nonlocal nonequilibrium thermodynamics. The motivation for these terms arises from\ud applications to nanosystems, where strong gradients are found, due to the small distance over which changes\ud in temperature and heat flux take place. This equation generalizes to the nonlinear domain previous equations\ud used in the context of phonon hydrodynamics. Compatibility with second law of thermodynamics is investigated\ud and a comparison with the thermomass model of heat transport is carried out. The analogy between the\ud equations describing the heat flow problem and the hydrodynamic equations is shown and the stability of the\ud heat flow is analyzed in a special case

The influence of weakly nonlocal effects on the speed of second sound along or against the direction of a nonvanishing average heat flow is explored in a formalism based on a dynamical nonequilibrium temperature. The restrictions from the second law of thermodynamics on the nonlocal evolution equation of this temperature are studied from a gradient extension of classical Liu procedure, and the corresponding form of the entropy is derived. Stability requirements obtained from the second differential of the entropy are seen to restrict a nonlinear dependence of the thermal conductivity on the temperature gradient

A dynamical nonequilibrium temperature has been proposed to describe relaxational equations for the heat\ud flux. This temperature provides an alternative description to the Maxwell-Cattaneo equation. In the linear\ud regime and in bulk systems both descriptions are equivalent but this is not so when nonlinear effects are\ud included. Here we explore the influence of nonlinear terms on the phase speed of heat waves in nonequilibrium\ud steady states in both theoretical models and we show that their predictions are different. This could allow to\ud explore which description is more suitable, when experiments on these situations will become available.\ud Furthermore, we have analyzed a nonlinear and nonlocal constitutive equation for the heat flux and we have\ud shown its analogy with the Navier-Stokes equation in the regime of phonon hydrodynamics in nanosystems.\ud This analogy allows one to define a dimensionless number for heat flow, analogous to the Reynolds number,\ud and to predict a critical heat flux where nonlinear effects could become dominant

The restrictions placed by the second law of thermodynamics on the boundary conditions have a special interest in nanosystems, where the Knudsen boundary layer, whose width is of the order of the mean-free path of heat carriers, becomes comparable to the size of the whole system. Here we explore second-order boundary conditions and show that the constraints of the classical irreversible thermodynamics are too restrictive, and that other formalisms going beyond local-equilibrium approach yield more realistic constraints for hydrodynamic phonon flow along nanowires. Furthermore, our analysis suggests a transition to zero thermal conductivity for very thin nanowires due to phonon backscattering. © 2010 American Institute of Physics

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