Temperature dependence of boundary conditions in phonon hydrodynamics of smooth and rough nanowires

Abstract

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

Heat waves and phonon–wall collisions in nanowires

“…The functions C (T ) and a (T ) have been inferred in the paper by Sellitto et al (2010b) for silicon nanowires (see equations (2.6) and (2.7) therein) by a comparison with the experimental data for the steady-state effective thermal conductivity both in the case of rough-walled silicon nanowires (Hochbaum et al 2008;Martin et al 2009) and in the case of smooth-walled nanowires (Li et al 2003) for different radii.…”

“…The wall parameters C and a are given, too. They follow from equations (2.3) once the functions C (T ) and a (T ), inferred in Sellitto et al (2010b), have been taken into account. In particular, the superposed r means that we are referring to rough walls (with a typical peak height D = 3 nm and an average separation between neighbouring peaks D = 6 nm), whereas the superposed s refers to smooth walls.…”

Non-local effects in radial heat transport in silicon thin layers and graphene sheets

“…The choice of this geometry takes us beyond the usual one-dimensional situations, in which some of the non-local effects vanish identically. In fact, in one-dimensional heat transport in nanowires and thin layers, the non-local effects are also present through the variation of the heat flux inside the system, but their effects may be embedded in size-dependent thermal conductivity (Alvarez et al 2009(Alvarez et al , 2011Sellitto et al 2010b). Although this approach is very useful for practical applications, it slightly obscures some essential points of non-local transport.…”

“…However, the effect of the radius will be further considered later, in figure 4. We have taken two different values for T 0 , namely, T 0 = 150 K and T 0 = 100 K. The corresponding wall-accommodation parameters C = 0.36 and C = 0.46 have been taken from Sellitto et al (2010b) and refer to a smooth wall, so that the effective thermal conductivity at T 0 = 150 K and T 0 = 100 K are, respectively, l eff = 23.2 Wm −1 K −1 and l eff = 18.8 Wm −1 K −1 , according to equation (3.1). As it is possible to observe, the temperature behaviour arising from equation (3.2) (solid lines in figure 3) shows a hump; namely, when r < , the temperature increases with radius.…”

Constitutive equations for heat conduction in nanosystems and nonequilibrium processes: an overview

“…Then, the nonlocal term in Eq. (5) may be more important than the heat flux itself [32][33][34][35], and that equation reduces to…”