Equation Of Phonon Radiative Transfer Research Articles

In-plane thermal conductivity measurements of Si μ-cantilevers using the suspended thermo-reflectance (STR) technique.

The Suspended Thermoreflectance (STR) technique is described in this paper. This optoelectronic measurement tool performs thermal characterization of freestanding micro-/nanoscale materials. STR performs thermal mapping at the submicron level and produces unconstrained thermal conductivity unlike other optical measurement techniques where independent conductivity measurement is not possible due to their reliance on heat capacity. STR works by changing the temperature of a material and collecting the associated change in light reflection from multiple points on the sample surface. Reflection is a function of the material being tested, the wavelength of the probe light, geometry, and the composition of the specimen for transparent and quasi-transparent materials. In this article, Si μ-cantilevers are studied. In addition, a thermal analytical model is developed and incorporated with optical equations to characterize the conductivity of the Si μ-cantilevers. The analytical model is compared with a finite element model to check its applicability in the STR experiment and data analysis. To validate the technique, the thermal conductivity of 2 and 3 µm thick Si μ-cantilevers was determined using STR at a temperature range of 20-350K and compared to simulations using the equation of phonon radiative transfer and literature values.

Anisotropic Klemens model for the thermal conductivity tensor and its size effect

With the constraint from Onsager reciprocity relations, here we generalize the Klemens model for intrinsic phonon-phonon scattering from isotropic to anisotropic. Combined with the anisotropic Debye dispersion, this anisotropic Klemens model leads to analytical expressions for heat transfer along both ab-plane (κab) and c-axis (κc), suitable for both layered and chainlike materials with any anisotropy ratio of the dispersion and the scattering. This combination of dispersion and scattering is further applied to the equation of phonon radiative transfer to model heat conduction across anisotropic thin films. The model is compared to experimental κab and κc of bulk graphite at high temperatures (T≥0.1×max(θD,ab,θD,c), where θD,ab and θD,c are the two Debye temperatures), leading to best-fit scattering parameters that capture the harmonic and anharmonic nature of the intra- and inter-layer bonding of graphite. With no further adjusting of these parameters, the anisotropic scattering model performs at least 9 times better than the isotropic one at explaining the experimental κc of graphite thin films. This size effect is further justified by comparing the accumulation function of κc.

Transient, Sub-Continuum, Heat Conduction in Irregular Geometries

Abstract Phonon transfer in irregular shapes is important for assessing the influence of shape effect on thermal transport characteristics of low-scale films. It becomes critical for evaluating the contribution of the scattering phonons to the phonon intensity distribution inside the film. Hence, the sub-continuum ballistic-diffusive model is incorporated to formulate the phonon transport in an irregular geometry of low-size film adopting the transient, frequency-independent, equation of phonon radiative transfer. The discrete ordinate method is used in the numerical discretization of the governing transport equation. It is demonstrated that the geometric feature of the film influences the phonon intensity distribution within the film material. The transport characteristics obtained from the Fourier and the ballistic-diffusive models are markedly different in their spatial and temporal behavior. This is true when the device sizes are of the same order of magnitude as the mean-free path of the heat carriers.

Entropy Generation Rate for Stationary Ballistic-Diffusive Heat Conduction in a Rectangular Flake

Thermodynamic irreversibility in low dimensional flake is considered and entropy generation rate in two-dimensional thin film is examined, Equation of phonon radiative transfer is solved for two-dimensional and rectangular diamond flake. Volumetric and total entropy generation rate are evaluated incorporating the formulation of thermal radiation heat transfer. The influence of flake aspect ratio (width to height) on the entropy generation rate is explored while keeping the Diamond flake area constant for all aspect ratios considered. The findings reveal that the entropy generation rate increases with increasing aspect ratio for fixed boundary conditions.

ENTROPY GENERATION RATE IN A MICROSCALE THIN FILM

This paper presents a new formulation of the rate of entropy generation in thin films whose thickness is of the order of the mean-free-path or less. In this relation, an expression for the gradient of the equivalent equilibrium temperature is proposed that is a function of the gradient of the phonon intensity at any point inside the thin film. It is shown that the proposed expression reduces to the familiar gradient of the thermodynamic temperature in the diffusive limit. Furthermore, the new formulation is used to compute the entropy generation rate for the case of steady-state, one-dimensional heat transfer in a thin film by first solving the Equation of Phonon Radiative Transfer to determine the phonon intensity. These computations are performed both for the silicon and the diamond thin films, for a range of Knudsen numbers starting from the diffusive limit up until the ballistic limit. It is found that the entropy generation rate attains a peak value at Kn = 0.7 and decreases for other Knudsen numbers when non-equilibrium transport is adopted in the analysis. However, rate of entropy generation increases almost linearly for the equilibrium heating situation. This is true for both the silicon and the diamond thin films.

Efficiency improvement of discrete-ordinates method for interfacial phonon transport by Gauss-Legendre integral for frequency domain

The thermal conduction through interfaces plays an important role for optimization and new designs in micro/nano energy system. However, the available knowledge and understanding on the physical picture of phonon transport at interfaces are still insufficient. The discrete-ordinates method (DOM) has been recently applied to solve the equation of phonon radiative transfer (EPRT) but it still suffers from expensive computational cost, especially with interfaces of different materials. In the present work, we introduce the Gauss-Legendre quadrature for frequency domain to improve the efficiency of discrete-ordinates method between heterogeneous materials over the uniform discretization scheme. The present scheme is verified for cross-plane interfacial phonon transport, with comparisons with both theoretical solution and kinetic-type Monte Carlo method. One to two orders of magnitude improvements on CPU time have been obtained numerically.

Numerical study of 3-D microscale heat transfer of a thin diamond slab under fix and moving laser heating

Laser heating is one of the most practical operations in the field of solid circuit production and thin condensed film treatment. The correct prediction of the heat propagation and flux into the micro/nanothin slab under laser heating has high practical importance. Many theoretical and numerical investigations have been performed for analysis of micro/nanoheat conduction based on one or 2-D approximations. For moving laser heating of thin films, with asymmetric paths, the one or 2-D analysis cannot be applied. The most appropriate equation for micro/ nanoheat transfer is the Boltzmann transport equation which predicts the phonon transport, precisely. In the present work, the 3-D microscale heat conduction of a diamond thin slab under fix or moving laser heating at very small time scales has been studied. Hence, the transient 3-D integro-differential equation of phonon radiative transfer has been derived from the Boltzmann equation transport and solved numerically to find the heat flux and temperature of thin slab. Regarding the boundary and interface scattering and the finite relaxation time in the equation of phonon radiative transfer, leads to more precise prediction than conventional Fourier law, especially for moving laser heating.

Heat transfer modelling in the Si–Ge nanoparticle composites by numerical solution of the equation of phonon radiative transfer

Heat transfer modelling in the Si–Ge nanoparticle composites by numerical solution of the equation of phonon radiative transfer

Finite element analysis of transient ballistic–diffusive phonon heat transport in two-dimensional domains

While sub-continuum heat conduction becomes more important as the size of micro/nanodevices keeps shrinking under the mean free path of heat carriers, its computation still remains challenging to the general engineering community due to the lack of easily accessible numerical simulation tools. To address this challenge, this article reports the finite element analysis (FEA) of transient ballistic-diffusive phonon heat transport in a two-dimensional domain using a commercial package (COMSOL Multiphysics). The Boltzmann transport equation under the gray relaxation-time approximation was numerically solved by discretizing the angular domain with the discrete ordinate method (DOM) and the spatial domain with the FEA. The DOM-FEA method was validated by comparing the results with different benchmark studies, such as the equation of phonon radiative transfer, the ballistic-diffusive equation, and the finite difference method of the phonon Boltzmann transport equation. The calculation of phonon heat transport for a 2-D square slab reveals that heat conduction becomes more ballistic with temperature jumps at boundaries as Knudsen number (Kn) increases. The ballistic nature also significantly affects transient thermal behaviors at high Kn numbers. The obtained results clearly demonstrate the capability of the DOM-FEA as a promising engineering tool for calculating sub-continuum phonon heat transport. (C) 2014 Elsevier Ltd. All rights reserved. (Less)

C213 非対称形状Si薄膜の熱伝導

The size effect on thermal transport property due to phonon boundary scattering was studied as controlling heat conduction and "thermal rectifier". We discussed on the thermal rectification effect from the calculation and measurement of asymmetric structured Si membranes. Heat conduction was calculated by using radiation calculation of ANSYS Fluent based on Boltzmann transport theory which is development of equation of phonon radiative transfer. And we measured temperature distribution of micro porous Si membrane by using infrared thermometry system which enabled microscopic measurement.

Entropy Generation in Thin Films Evaluated From Phonon Radiative Transport

One of the approaches for micro/nanoscale heat transfer in semiconductors and dielectric materials is to use the Boltzmann transport equation, which reduces to the equation of phonon radiative transfer under the relaxation time approximation. Transfer and generation of entropy are processes inherently associated with thermal energy transport, yet little has been done to analyze entropy generation in solids at length scales comparable with or smaller than the mean free path of heat carriers. This work extends the concept of radiation entropy in a participating medium to phonon radiation, thus, providing a method to evaluate entropy generation at both large and small length scales. The conventional formula for entropy generation in heat diffusion can be derived under the local equilibrium assumption. Furthermore, the phonon brightness temperature is introduced to describe the nature of nonequilibrium heat conduction. A diamond film is used as a numerical example to illustrate the distribution of entropy generation at the walls and inside the film at low temperatures. A fundamental knowledge of the entropy generation processes provides a thermodynamic understanding of heat transport in solid microstructures; this is particularly important for the performance evaluation of thermal systems and microdevices.

Brownian Motion Description of Heat Conduction by Phonons

A non-Markovian partial differential equation, rooted in the theory of Brownian motion, is proposed for describing heat conduction by phonons. Although a finite speed of propagation is a built-in feature of the equation, it does not give rise to an inauthentic wave front that results from the application of Cattaneo's equation. Even a simplified, analytically tractable version of the equation yields results close to those found by solving, through more elaborate means, the equation of phonon radiative transfer.

Microscale Thermal Characterization for Two Adjacent Dielectric Thin Films

Heat conduction in two adjacent layers of dielectric thin e lms are analyzed by solving an equation of phonon radiative transfer. In the present work, the physical model is derived with an equilibrium temperature at the imaginary black interface, so that the two layers can be linked. Moreover, the effective mean free path can be corrected by the phonon scattering rate on grain boundaries of imperfect interface. Chemical-vapor-deposited diamond layer on silicon is chosen as the structure of the test case. The studies found that the interface is a critical issue in microscale thermal characteristics; for example, a temperature jump at the interface characterizes the effective thermal conductivity on diamond e lm and ine uences the temperature level on the silicon substrates. Energy transmission to the other layer increases the temperature jump at the interface and lowers the temperature level on the silicon substrate without changing the phonon mean free path. The disorder interface with various grain structures increases the phonon scattering rate and shortens the effective mean free path signie cantly; therefore, the temperature gradient across the diamond e lm increases and the normalized temperature jump at the interface is decreased substantially. Nomenclature C = heat capacity d = grain size I = phonon intensity k = thermal conductivity N = number of atoms q = heat e ux T = temperature t = time U = internal energy V = phonon group velocity a = energy transmission coefe cient h = grain boundary scattering strength u = Debye temperature L = phonon mean free path m = cosine of the angle between the phonon propagation velocity and the x direction t = relaxation time V = solid angle Subscripts D = defects or at Debye temperature eff = effective quantity G = grain GB = grain boundary i = medium i j = medium j S = at imaginary interface U = Umklapp process v = spectral quantity

Transient ballistic and diffusive phonon heat transport in thin films

Ballistic and diffusive phonon transport under small time and spatial scales are important in fast-switching electronic devices and pulsed-laser processing of materials. The Fourier law represents only diffusive transport and yields an infinite speed for heat waves. Although the hyperbolic heat equation involves a finite heat wave speed, it cannot model ballistic phonon transport in short spatial scales, which under steady state follows the Casimir limit of phonon radiation. An equation of phonon radiative transfer (EPRT) is developed which shows the correct limiting behavior for both purely ballistic and diffusive transport. The solution of the EPRT for diamond thin films not only produces wall temperature jumps under ballistic transport but shows markedly different transient response from that of the Fourier law and the hyperbolic heat equation even for predominantly diffusive transport. For sudden temperature rise at one film boundary, the results show that the Fourier law and the hyperbolic heat equation can significantly over- or underpredict the boundary heat flux at time scales smaller than the phonon relaxation times.

Thermal conductivities of quantum well structures

This work analyzes the size and the boundary effects of a gallium arsenide- (GaAs) based quantum well (QW) structure on the thermal conductivity of the well material. Calculations show that the order of phonon mean free path (MFP) is equal to or even longer than the typical dimension of the well (-200 A or less). Holland's model is applied to match the thermal conductivity data of bulk GaAs from 2 to above 600 K. The equation of phonon radiative transfer (EPRT) developed from the Boltzmann transport equation is then introduced for the heat transport in the QW structure. Boundary conditions are built from the diffuse phonon mismatch theory, and approximate solutions are obtained for the cases of heat flow perpendicular and parallel to the well. Results show that the thermal conductivity of the quantum well can be one order-of-magnitude lower than that of its corresponding bulk form at room temperature. The size and boundary effects also cause anisotropy of the thermal conductivity, even though the unit cell of GaAs is cubic.

Microscale Heat Conduction in Dielectric Thin Films

Heat conduction in dielectric thin films is a critical issue in the design of electronic devices and packages. Depending on the material properties, there exists a range of film thickness where the Fourier law, used for macroscale heat conduction, cannot be applied. This paper shows that in this microscale regime, heat transport by lattice vibrations or phonons can be analyzed as a radiative transfer problem. Based on Boltzmann transport theory, an equation of phonon radiative transfer (EPRT) is developed. In the acoustically thick limit, ξL ≫ 1, or the macroscale regime, where the film thickness is much larger than the phonon-scattering mean free path, the EPRT reduces to the Fourier law. In the acoustically thin limit, ξL ≪ 1, the EPRT yields the blackbody radiation law q = σ (T14 − T24) at temperatures below the Debye temperature, where q is the heat flux and T1 and T2 are temperatures at the film boundaries. For transient heat conduction, the EPRT suggests that a heat pulse is transported as a wave, which becomes attenuated in the film due to phonon scattering. It is also shown that the hyperbolic heat equation can be derived from the EPRT only in the acoustically thick limit. The EPRT is then used to study heat transport in diamond thin films in wide range of acoustical thicknesses spanning the thin and the thick regimes. The heat flux follows the relation q = 4σT3ΔT/(3ξL/4 + 1) as derived in the modified diffusion approximation for photon radiative transfer. The thermal conductivity, as currently predicted by kinetic theory, causes the Fourier law to overpredict the heat flux by 33 percent when ξL ≪ 1, by 133 percent when ξL = 1, and by about 10 percent when ξL increases to 10. To use the Fourier law in both ballistic and diffusive transport regimes, a simple expression for an effective thermal conductivity is developed.