Theory Of Heat Transport Research Articles

A Fokker–Planck code for laser plasma interaction in femtosecond-laser shock peening

A Fokker–Planck code is developed to simulate the laser–plasma interaction in the femtosecond-laser shock peening and forming processes. A numerical scheme dealing with high-energy concentration and its resulting steep gradient are presented, and the source code is provided as supplementary material for further usage. The breakdown of the classical heat transport theory is observed when the laser intensity increases. The difference in heat flow between the classical theory and simulation is presented. It is found that the classical heat transport theory overestimates heat flow by orders of magnitude during femtosecond-laser shock peening or forming. As a result, the electron pressure can be underestimated using the classical hydrodynamic code.

Microscopic theory and quantum simulation of atomic heat transport

Quantum simulation methods based on density-functional theory are currently deemed unfit to cope with atomic heat transport within the Green-Kubo formalism, because quantum-mechanical energy densities and currents are inherently ill-defined at the atomic scale. We show that, while this difficulty would also affect classical simulations, thermal conductivity is indeed insensitive to such ill-definedness by virtue of a sort of gauge invariance resulting from energy extensivity and conservation. Based on these findings, we derive an expression for the adiabatic energy flux from density-functional theory, which allows heat transport to be simulated using ab-initio equilibrium molecular dynamics. Our methodology is demonstrated by comparing its predictions with those of classical equilibrium and ab-initio non-equilibrium (M\"uller-Plathe) simulations for a liquid-Argon model, and finally applied to heavy water at ambient conditions.

State space approach for the vibration of nanobeams based on the nonlocal thermoelasticity theory without energy dissipation

In this article, an Euler-Bernoulli beam model based upon nonlocal thermoelasticity theory without energy dissipation is used to study the vibration of a nanobeam subjected to ramp-type heating. Classical continuum theory is inherently size independent, while nonlocal elasticity exhibits size dependence. Among other things, this leads to a new expression for the effective nonlocal bending moment as contrasted to its classical counterpart. The thermal problem is addressed in the context of the Green-Naghdi (GN) theory of heat transport without energy dissipation. The governing partial differential equations are solved in the Laplace transform domain by the state space approach of modern control theory. Inverse of Laplace transforms are computed numerically using Fourier expansion techniques. The effects of nonlocality and ramping time parameters on the lateral vibration, temperature, displacement and bending moment are discussed.

Myth about nano-fluid heat transfer enhancement

The paper examines a few claims about the merit of using nano-particles to enhance the rate of heat transfer. In the literature, a few mechanisms were put forward to explain the magic of enhancing the rate of heat transfer by many folds compared with a clear fluid by adding a few percentages of nano-particles to the working fluid. In this work, a few introduced mechanisms is examined in the light of classical physics of the thermal diffusion and the fluid mechanics. Yet, no one able to break those simple physics; therefore, they should be valid for particles floating in a fluid. The paper for the first time reports the effect of particle alignment with the direction of heat flow (temperature gradient) effects on the effective thermal conductivity, which is significant. Furthermore, forced and natural convections are examined based on the classical theory of heat transport. We demonstrated the reason of why the rate of heat transfer decreases by adding nano-particles for a fluid experiencing natural convection.

Heat transport as torsional responses and Keldysh formalism in a curved spacetime

We revisit a theory of heat transport in the light of a gauge theory of gravity and find the proper heat current with a corresponding gauge field, which yields the natural definitions of the heat magnetization and the Kubo-formula contribution to the thermal conductivity as torsional responses. We also develop a general framework for calculating gravitational responses by combining the Keldysh and Cartan formalisms. By using this framework, we explicitly calculate these two quantities and reproduce the Wiedemann-Franz law for the thermal Hall conductivity in the clean and non-interacting case. Finally, we discuss an effective action for the quantized thermal Hall effect in $(2 + 1)$-D topological superconductors.

Nonlocal theory for heat transport at high frequencies

We develop a nonlocal theory for heat conduction under high-frequency temperature fields and apply the theory to explain reductions of the apparent thermal conductivity observed in recent experiments. Our nonlocal theory is an analytical solution of the Boltzmann transport equation for phonons in a semi-infinite solid, similar to a prior nonlocal theory for heat conduction under a high-temperature gradient but subjected to periodic heating at the surface. The boundary condition of periodic heating, as opposed to prior calculations of heating by a single laser pulse, better mimics time-domain thermoreflectance (TDTR) and broadband frequency-domain thermoreflectance (BB-FDTR) measurements. We find that, except for pure crystals at high frequencies, the effective thermal conductivity derived using the nonlocal theory compares well with calculations of a modified Callaway model that includes an upper limit on the phonon mean-free path at twice the thermal penetration depth. For pure crystals, however, the effective thermal conductivity derived from the out-of-phase calculations are independent of frequency, in agreement with prior TDTR measurements, due to the countereffect of reduced heat flux and diminished relative phase between the heat flux and temperature oscillations at high frequencies. Our results suggest that empirical interpretation of ballistic phonons not contributing to heat conduction is not general and can only be applied to measurements on alloys and not pure crystals, even when a large laser spot size is used in the experiments and the interfacial thermal resistance is negligible.

Firehose and Mirror Instabilities in a Collisionless Shearing Plasma

Hybrid-kinetic numerical simulations of firehose and mirror instabilities in a collisionless plasma are performed in which pressure anisotropy is driven as the magnetic field is changed by a persistent linear shear $S$. For a decreasing field, it is found that mostly oblique firehose fluctuations grow at ion Larmor scales and saturate with energies $\ensuremath{\propto}{S}^{1/2}$; the pressure anisotropy is pinned at the stability threshold by particle scattering off microscale fluctuations. In contrast, nonlinear mirror fluctuations are large compared to the ion Larmor scale and grow secularly in time; marginality is maintained by an increasing population of resonant particles trapped in magnetic mirrors. After one shear time, saturated order-unity magnetic mirrors are formed and particles scatter off their sharp edges. Both instabilities drive sub-ion-Larmor--scale fluctuations, which appear to be kinetic-Alfv\'en-wave turbulence. Our results impact theories of momentum and heat transport in astrophysical and space plasmas, in which the stretching of a magnetic field by shear is a generic process.

Fractional-order theory of heat transport in rigid bodies

The non-local model of heat transfer, used to describe the deviations of the temperature field from the well-known prediction of Fourier/Cattaneo models experienced in complex media, is framed in the context of fractional-order calculus. It has been assumed (Borino et al., 2011 [53], Mongioví and Zingales, 2013 [54]) that thermal energy transport is due to two phenomena: (i) A short-range heat flux ruled by a local transport equation; (ii) A long-range thermal energy transfer proportional to a distance-decaying function, to the relative temperature and to the product of the interacting masses. The distance-decaying function is assumed in the functional class of the power-law decay of the distance yielding a novel temperature equation in terms of α-order Marchaud fractional-order derivative (0⩽α⩽1). Thermodynamical consistency of the model is provided in the context of Clausius–Plank inequality. The effects induced by the boundary conditions on the temperature field are investigated for diffusive as well as ballistic local heat flux. Deviations of the temperature field from the linear distributions in the neighborhood of the thermostated zones of small-scale conductors are qualitatively predicted by the used fractional-order heat transport model, as shown by means of molecular dynamics simulations.

Scaling theory of heat transport in quasi-one-dimensional disordered harmonic chains

We introduce a variant of the banded random matrix ensemble and show, using detailed numerical analysis and theoretical arguments, that the phonon heat current in disordered quasi-one-dimensional lattices obeys a one-parameter scaling law. The resulting β function indicates that an anomalous Fourier law is applicable in the diffusive regime, while in the localization regime the heat current decays exponentially with the sample size. Our approach opens a new way to investigate the effects of Anderson localization in heat conduction based on the powerful ideas of scaling theory.

Heat and spin transport in a cold atomic Fermi gas

Motivated by recent experiments measuring the spin transport in ultracold unitary atomic Fermi gases [Sommer et al., Nature (London) 472, 201 (2011); Sommer et al., New J. Phys. 13, 055009 (2011)], we explore the theory of spin and heat transport in a three-dimensional spin-polarized atomic Fermi gas. We develop estimates of spin and thermal diffusivities and discuss magnetocaloric effects, namely the the spin Seebeck and spin Peltier effects. We estimate these transport coefficients using a Boltzmann kinetic equation in the classical regime and present experimentally accessible signatures of the spin Seebeck effect. We study an exactly solvable model that illustrates the role of momentum-dependent scattering in the magnetocaloric effects.

A Continuum Approach to Thermomass Theory

A continuum approach to the thermomass theory for nonlinear heat transport is developed and its compatibility with the general framework of continuum thermodynamics is investigated. The heat flux is supposed to depend on the absolute temperature together with a vectorial internal variable, which is proportional to the drift velocity of the heat carriers. A generalized heat-transport equation, which is capable to bring Fourier, Maxwell–Cattaneo–Vernotte and thermomass-theory equations as special cases is derived. Propagation of heat waves along a nonequilibrium steady state is analyzed as well.

Consistency of the phenomenological theories of wave-type heat transport with the hydrodynamics of a phonon gas

Various phenomenological theories of wave-type heat transport, which can be interpreted as the models of an isotropic rigid heat conductor with an internal vector state variable, have been proposed in the literature with the objective to describe the second sound propagation in dielectric crystals. The aim of this paper is to analyze the relation between these phenomenological approaches and the phonon gas hydrodynamics. The four-moment phonon gas hydrodynamics based on the maximum entropy closure of the moment equations with nonlinear isotropic phonon dispersion relation is considered for this purpose. We reformulate the equations of this hydrodynamics in terms of energy and quasi-momentum as the primitive fields and subsequently demonstrate that, from the macroscopic point of view, they can be understood as describing the reference model of an isotropic rigid heat conductor with quasi-momentum playing the role of the internal vector state variable. This model is determined by the entropy function and the additional scalar potential, but if the finite domain of phonon wave vectors is approximated by the whole space, the additional potential can be expressed in terms of the entropy function and its first derivatives. Then the transformation of primitive fields and the expansion of thermodynamic potentials in powers of the square of quasi-momentum enable us to compare the reference model with the models proposed earlier in the literature. It is shown that the previous models require some subtle modifications in order to achieve full consistency with phonon gas hydrodynamics.

Theory of heat transport of normal liquid 3He in aerogel

The introduction of liquid 3He into a silica aerogel provides us with a model system in which to study the effects of disorder on the properties of a strongly correlated Fermi liquid. The transport of heat, mass and spin exhibits cross-over behavior from a high temperature regime, where inelastic scattering dominates, to a low temperature regime dominated by elastic scattering off the aerogel. We report exact and approximate solutions to the Boltzmann–Landau transport equation for the thermal conductivity of liquid 3He, including elastic scattering of quasi-particles by the aerogel and inelastic quasi-particle collisions. These results provide quantitative predictions on the transport properties of liquid 3He in aerogels over a wide range of pressures, temperatures and aerogel densities. In particular, we obtain a scaling function, F(T/T⋆), for the normalized thermal conductivity, , in terms of a reduced temperature, T/T⋆, where T⋆ is a cross-over temperature defined by the elastic and inelastic collision rates. Theoretical results are compared with the available experimental data for the thermal conductivity.

Thermal properties of char obtained by pyrolysis: A molecular dynamics simulation study

The thermal conductivity of pyrolytic char obtained by ultrahigh temperature decomposition of polyethylene specimen via molecular dynamics simulations is investigated as a function of temperature and microstructural characteristics. We find that the simulated thermal conductivity dependence on the average coordination number is modified by formation of graphene-like microtopological features in carbonaceous char. The dependence of thermal conductivity on temperature and average coordination number is explained in terms of an analytical model, based on the Einstein’s theory of heat transport. The deviations due to the formation of graphene sheet-like units are taken into consideration by introducing corresponding corrections in the elastic properties of char.

Advanced simulation of electron heat transport in fusion plasmas

Electron transport in burning plasmas is more important since fusion products first heat electrons. First-principles simulations of electron turbulence are much more challenging due to the multi-scale dynamics of the electron turbulence, and have been made possible by close collaborations between plasma physicists and computational scientists. The GTC simulations of collisionless trapped electron mode (CTEM) turbulence show that the electron heat transport exhibits a gradual transition from Bohm to gyroBohm scaling when the device size is increased. The deviation from the gyroBohm scaling can be induced by large turbulence eddies, turbulence spreading, and non-diffusive transport processes. Analysis of radial correlation function shows that CTEM turbulence eddies are predominantly microscopic but with a significant tail in the mesoscale. A comprehensive analysis of kinetic and fluid time scales shows that zonal flow shearing is the dominant decorrelation mechanism. The mesoscale eddies result from a dynamical process of linear streamers breaking by zonal flows and merging of microscopic eddies. The radial profile of the electron heat conductivity only follows the profile of fluctuation intensity on a global scale, whereas the ion transport tracks more sensitively the local fluctuation intensity. This suggests the existence of a nondiffusive component in the electron heat flux, which arises from the ballistic radial E X B drift of trapped electrons due to a combination of the presence of mesoscale eddies and the weak de-tuning of the toroidal precessional resonance that drives the CTEM instability. On the other hand, the ion radial excursion is not affected by the mesoscale eddies due to a parallel decorrelation, which is not operational for the trapped electrons because of a bounce averaging process associated with the electron fast motion along magnetic field lines. The presence of the nondiffusive component raises question on the applicability of the usual quasilinear theory for the CTEM electron transport. This is in contrast to the good agreement between the quasilinear transport theory and simulation results of the electron heat transport in electron temperature gradient (ETG) turbulence, which is regulated by a wave-particle decorrelation. Therefore, the transport in the CTEM turbulence is a fluid-like eddy mixing process even though the linear CTEM instability is driven by a kinetic resonance. In contrast, a kinetic process dominates the transport in the ETG turbulence, which is characterized by macroscopic streamers.

Heat Transport and Temperature Gradient in Silicon-on-Insulator Wafer during Flash Lamp Annealing Process

The temperature gradient formed in a silicon-on-insulator (SOI) wafer during a flash lamp annealing process is calculated on the basis of the heat transport theory. The temperature of SOI wafer, having a 0.04-µm-thick active layer and a 0.1-µm-thick buried oxide (BOX) layer, is calculated. Within 1000 µs, the active layer surface reaches the maximum temperature higher than 1473 K. Because most of the heat is transported by conduction, a very large temperature gradient, such as 3×107 K/m, is formed in the BOX layer owing to its very small heat conductivity. The radiant heat flux through the BOX layer is estimated to be significantly smaller than the conduction heat flux. From a series of calculations taking into account various thicknesses of the active layer and the BOX layer, a BOX layer thicker than 1 µm can significantly increase the active layer temperature.

Vibrational modes and low-temperature thermal properties of graphene and carbon nanotubes: Minimal force-constant model

We present a phenomenological force-constant model developed for the description of lattice dynamics of $s{p}^{2}$ hybridized carbon networks. Within this model approach, we introduce a set of parameters to calculate the phonon dispersion of graphene by fitting the ab initio dispersion. Vibrational modes of carbon nanotubes are obtained by folding the two-dimensional (2D) dispersion of graphene and applying special corrections for the low-frequency modes. Particular attention is paid to the exact dispersion law of the acoustic modes, which determine the low-frequency thermal properties and reveal quantum size effects in carbon nanotubes. On the basis of the resulting phonon spectra, we calculate the specific heat and the thermal conductance for several achiral nanotubes of different diameters. Through the temperature dependence of the specific heat, we demonstrate that phonon spectra of carbon nanotubes show one-dimensional behavior and that the phonon sub-bands are quantized at low temperatures. Consequently, we prove the quantization of the phonon thermal conductance by means of an analysis based on the Landauer theory of heat transport.

Edge stability and transport control with resonant magnetic perturbations in collisionless tokamak plasmas

A critical issue for fusion-plasma research is the erosion of the first wall of the experimental device due to impulsive heating from repetitive edge magneto-hydrodynamic instabilities known as 'edge-localized modes' (ELMs). Here, we show that the addition of small resonant magnetic field perturbations completely eliminates ELMs while maintaining a steady-state high-confinement (H-mode) plasma. These perturbations induce a chaotic behaviour in the magnetic field lines, which reduces the edge pressure gradient below the ELM instability threshold. The pressure gradient reduction results from a reduction in the particle content of the plasma, rather than an increase in the electron thermal transport. This is inconsistent with the predictions of stochastic electron heat transport theory. These results provide a first experimental test of stochastic transport theory in a highly rotating, hot, collisionless plasma and demonstrate a promising solution to the critical issue of controlling edge instabilities in fusion-plasma devices.

Theory of heat transport in a magnetized high-temperature plasma

The transport of charged particles across a strong magnetic field with a small random component is studied in the double diffusion approximation. It is shown that the density of the particles whose initial distribution is stretched along the field satisfies a subdiffusion equation with fractional derivatives. A more general initial particle distribution is also considered, and the applicability of the solutions obtained is discussed.

Thermoelastic relaxation in elastic structures, with applications to thin plates

A new result enables direct calculation of thermoelastic damping in vibrating elastic solids. The mechanism for energy loss is thermal diffusion caused by inhomogeneous deformation, flexure in thin plates. The general result is combined with the Kirchhoff assumption to obtain a new equation for the flexural vibration of thin plates incorporating thermoelastic loss as a damping term. The thermal relaxation loss is inhomogeneous and depends upon the local state of vibrating flexure, specifically, the principal curvatures at a given point on the plate. Thermal loss is zero at points where the principal curvatures are equal and opposite, that is, saddle shaped or pure anticlastic deformation. Conversely, loss is maximal at points where the curvatures are equal, that is, synclastic or spherical flexure. The influence of modal curvature on the thermoelastic damping is described through a modal participation factor. The effect of transverse thermal diffusion on plane wave propagation is also examined. It is shown that transverse diffusion effects are always small provided the plate thickness is far greater than the thermal phonon mean free path, a requirement for the validity of the classical theory of heat transport. These results generalize Zener's theory of thermoelastic loss in beams and are useful in predicting mode widths in micro- and nano-electromechanics systems oscillators.