Ballistic-diffusive Heat Conduction Research Articles

A Case Study of Non-Fourier Heat Conduction Using Internal Variables and GENERIC

Abstract Applying simultaneously the methodology of non-equilibrium thermodynamics with internal variables (NET-IV) and the framework of General Equation for the Non-Equilibrium Reversible–Irreversible Coupling (GENERIC), we demonstrate that, in heat conduction theories, entropy current multipliers can be interpreted as relaxed state variables. Fourier’s law and its various extensions—the Maxwell–Cattaneo–Vernotte, Guyer–Krumhansl, Jeffreys type, Ginzburg–Landau (Allen–Cahn) type and ballistic–diffusive heat conduction equations—are derived in both formulations. Along these lines, a comparison of NET-IV and GENERIC is also performed. Our results may pave the way for microscopic/multiscale understanding of beyond-Fourier heat conduction and open new ways for numerical simulations of heat conduction problems.

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.

Ballistic-Diffusive Heat Conduction in Thin Films by Phonon Monte Carlo Method: Gray Medium Approximation Versus Phonon Dispersion

Abstract The gray medium approximation treating all phonons with an averaged and representative mean-free-path (MFP) is an often used method in analyzing ballistic-diffusive heat conduction at nanoscale. However, whether there exists a reasonable value of the average MFP which effectively represents the entire spectrum of modal MFPs remains unclear. In this paper, phonon Monte Carlo (MC) method is employed to study the effects of the gray medium approximation on ballistic-diffusive heat conduction in silicon films by comparing with dispersion MC simulations. Four typical ways for calculating the average MFP with gray medium approximation are investigated. Three of them are based on the weighted average of the modal MFPs, and the remaining one is based on the weighted average of the reciprocals of the modal MFPs. The first three methods are found to be good at predicting effective thermal conductivity and heat flux distribution, but have difficulties in temperature profile, while the last one performs better for temperature profile than effective thermal conductivity and heat flux distribution. Therefore, none of the average MFPs can accurately characterize all the features of ballistic-diffusive heat conduction for the gray medium approximation. Phonon dispersion has to be considered for the accurate thermal analyses and modeling of ballistic-diffusive heat transport. Our work could be helpful for further understanding of phonon dispersion and more careful use of the gray medium approximation.

Topology optimization of the volume-to-point heat conduction problem at micro- and nano-scale

The volume-to-point (VP) heat conduction problem is one of the fundamental problems of cooling for electronic devices. The existed reports about the VP problem are mainly based on the Fourier’s law which works well at the macroscopic scale. However, the length scale of modern electronic devices has reduced to micro- and nano-scale, at which optimization methods that are capable of dealing with the non-Fourier heat conduction are desired now. In this paper, phonon Boltzmann transport equation (BTE) and solid isotropic material with penalization (SIMP) method are coupled to develop a topology optimization method for ballistic-diffusive heat conduction. Phonon BTE is transformed into equation of phonon radiative transport, which is solved by the discrete ordinate method. To realize the topology optimization, SIMP method is adopted to penalize the phonon extinction coefficient, which equals to the reciprocal of phonon mean-free-path, and an explicit constraint on the global gradient of the nominal material density is used to ensure the solutions being well-posed and mesh-independent. By using the developed topology optimization method, it is found that the optimal material distributions for the VP problem in ballistic-diffusive heat conduction significantly deviate from the traditional tree-like structure obtained in diffusive heat conduction, and the results vary with the Knudsen number (<i>Kn</i>). This is related to the different coefficient interpolation ways in the SIMP method and phonon ballistic transport. When <i>Kn</i> → 0, instead of converging to the conventional tree-like structure which fully stretches into the interior zone, the new method gradually produces the result obtained by the topology optimization which interpolates the reciprocal of the thermal conductivity in diffusive heat conduction. As <i>Kn</i> increases, the high thermal-conductive filling materials show a trend to gather around the low-temperature boundary, and there are more thick and strong trunk structures, less tiny and thin branch structures in the optimized material distributions. In addition, the ratio of the optimized average temperature to the value of the uniform material distribution <inline-formula><tex-math id="M1">\begin{document}$\left( {T_{{\rm{ave}},{\rm{opt}}}^{\rm{*}}/T_{{\rm{ave}},{\rm{uni}}}^{\rm{*}}} \right)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20190923_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20190923_M1.png"/></alternatives></inline-formula> also increases. The dependence of the topology optimization results on <i>Kn</i> can be attributed to the size effect of the thermal conductivity caused by phonon ballistic transport. In the diffusive heat conduction, filling materials with different length scales have the same efficiency to build high thermal-conductive channels. However, with ballistic effect enhancing, size effect makes the effective thermal conductivities of the branch structure lower than those of the trunk structure, as the former is smaller than the latter. As a result, the branch structures are less efficient compared with the trunk structures in terms of building high thermal-conductive channels, and the optimal material distributions have more trunk structures and fewer branch structures. When the ballistic effect becomes significant enough, say at <i>Kn</i> = 0.1, the topology optimization gets a dough-like material distribution in which branches merge into trunks. The proposed topology optimization method have the potential to provide guidance in designing nanoscale electronic devices for improving the heat dissipation capability.

Radial ballistic-diffusive heat conduction in nanoscale

ABSTRACTHeat conduction in radius direction is of great importance to the use of two-dimensional materials and experiments. In this paper, radial ballistic-diffusive heat conduction in nanoscale is investigated by the phonon Monte Carlo (MC) method and phonon Boltzmann transport equation. We find that owing to the two-dimensional nature, the radial heat transport is dominated by two parameters, including the Knudsen number (Kn) and the radius ratio of the two concentric boundaries, the former of which is defined as the ratio of the phonon mean-free-path to the distance of the two boundaries. Compared with the one-dimensional cases, radial ballistic transport not only leads to boundary temperature jumps and the size effect of the effective thermal conductivity, but also results in a nonlinear temperature profile in logarithm radius coordinate, a difference of the inner and outer boundary temperature jumps, a stronger size effect, and a nonuniform local thermal conductivity within the system. When the value of Kn is far less than one, diffusive transport predominates and the effect of the radius ratio is negligible. Whereas, when Kn is comparable to or larger than one, the intensity of ballistic transport compared to diffusive transport will be increased significantly as the radius ratio decreases. In addition, the models for the temperature profile and the effective thermal conductivity are derived by an interpolation of the limit solutions and modification of the previous model, respectively. The good agreements with the phonon MC simulations demonstrate their validity.

A hybrid phonon Monte Carlo-diffusion method for ballistic-diffusive heat conduction in nano- and micro- structures

The existing phonon Monte Carlo (MC) for ballistic-diffusive heat conduction are limited to small and simple structures owing to the huge time cost following with the increasing scale. This article presents a new hybrid phonon Monte Carlo-diffusion method for ballistic-diffusive heat conduction, which successfully characterizes the ballistic effect with significantly reducing the computational cost. Based on the idea that the phonon-boundary scattering mainly affects the regions adjacent to the boundaries when the system is considerably large, the whole system is divided into three zones: the boundary MC zone and the middle diffusion zone, between which is the overlap zone. By using an alternating method and setting virtual phonon bath or specular reflection as the boundary condition for the MC zones, the results of the phonon tracing MC and diffusion equation can be coupled and converge at the overlap zone. To verify, the cross-plane and in-plane film heat conduction, where slip boundary conditions are the major characteristics of the ballistic-diffusive regime, are simulated by the hybrid method as well as standard phonon tracing MC which works as a benchmark. It is found that the hybrid method can accurately predict the distributions of temperature and heat flux in the system with nearly the same precision as the phonon tracing MC while the computation time can reduce up to 90%, validating its potential use for larger and more complex structures.

Slip Boundary Conditions in Ballistic–Diffusive Heat Transport in Nanostructures

ABSTRACTBallistic–diffusive heat conduction, which is predominantly affected by boundaries and interfaces, will occur in nanostructures whose characteristic lengths are comparable to the phonon mean free path (MFP). Here, we demonstrated that interactions between phonons and boundaries (or interfaces) could lead to two kinds of slip boundary conditions in the ballistic–diffusive regime: boundary temperature jump and boundary heat flux slip. The phonon Boltzmann transport equation (BTE) with relaxation time approximation and the phonon tracing Monte Carlo (MC) method were used to investigate these two slip boundary conditions for the ballistic–diffusive heat conduction in nanofilms on a substrate. For cross-plane heat conduction where the boundary temperature jump is the dominant non-Fourier phenomenon, ballistic transport causes the temperature jumps and thus introduces a ballistic thermal resistance. Importantly, when considering the interface effect, the corresponding model was derived based on the phonon BTE and verified by comparing with the MC simulations. In addition, an interface–ballistic coupling effect was identified, which indicates inapplicability of the standard thermal resistance analysis. In contrast, for the in-plane case that is controlled by boundary heat flux slip, both phonon boundary scattering and perturbation of the phonon distribution function induced by the interface can cause heat flux slip, leading to a variation in in-plane thermal resistance. In addition, a model beyond the Fuchs-Sondheimer formula, which can address both the boundary scattering and the interface effects, was derived based on the phonon BTE. The good agreements with the MC simulations indicate its validity.

Ballistic-diffusive heat conduction in multiply-constrained nanostructures

Ballistic-diffusive heat conduction in multiply-constrained nanostructures is theoretically studied based on the phonon Boltzmann transport equation. The results show that different constraints influence the thermal transport in different ways. In the direction parallel to the heat flow, the phonon ballistic transport can cause temperature jumps at the boundaries in contact with the phonon baths. In contrast, for lateral constraint, the heat flux is reduced near the boundaries due to phonon-boundary scattering. A thermal conductivity model for multiply-constrained nanostructures is then derived from the phonon Boltzmann transport equation. The influences of different constraints are combined on the basis of Matthiessen's rule. The model accurately characterizes the thermal conductivities of various typical nanostructures, including nanofilms (in-plane and cross-plane) and finite length nanowires of various cross-sectional shapes (e.g. circular and rectangular). The model predictions also agree well with Monte Carlo simulations and experimental data for silicon nanofilms and nanowires.

The effective thermal conductivity of ballistic–diffusive heat conduction in nanostructures with internal heat source

In nanostructures whose characteristic lengths are comparable to the phonon mean free path, the ballistic–diffusive heat conduction leads to the size effect, geometry dependence and anisotropy of the effective thermal conductivity. In the present work, we have studied the effective thermal conductivity of the ballistic–diffusive heat conduction in nanostructures (including nanofilms and nanowires) with internal heat source using Monte Carlo simulation and Boltzmann transport equation. It is found that the effective thermal conductivity of nanostructures with internal heat source is significantly lower than that with temperature difference, though it still increases with the increasing characteristic length. The models for the effective thermal conductivity and the temperature distribution of the cross-plane heat conduction in the nanofilms with internal heat source are directly derived from the phonon Boltzmann transport equation, and the comparisons with the Monte Carlo simulations well confirm their validities. As for the effective thermal conductivity of the in-plane nanofilms and nanowires with internal heat source, referring to the Matthiessen’s rule, the models are in the form of keff/kbulk=1/(1+αKn), with the parameter α obtained by the best fitting with the Monte Carlo simulations. Moreover, the diffusive heat conduction equation with the effective thermal conductivity can well characterize the temperature distributions in the in-plane nanofilms and long nanowires, while it fails in the short nanowires due to the influence of the axial constraints.

Molecular dynamics simulation of the thermophysical properties and phase change behaviors of aluminum nanoparticles

With the development of energy storage technology, phase change materials which can be used to store thermal energy have received much attention in recent years. The nano-metallic materials are universally used as phase change materials due to their many desirable thermophysical properites. In this paper, the molecular dynamics simulation method is adopted to simulate the variations of melting point, density and phonon thermal conductivity of the nano aluminum with grain size ranging from 0.8 nm to 3.2 nm. The variations of density, specific heat capacity and phonon thermal conductivity with temperature of aluminum nanoparticles at a grain size of 1.6 nm are also studied. By using the embedded-atom potential, the thermophysical properties and phase change behaviors of aluminum nanoparticles are stimulated. The phase transition temperature of aluminum nanoparticles is studied based on the energy-temperature curve and the specific heat capacity-temperature curve. The surface energy theory and the size effect theory are applied to the analysis of the variation of the melting point of the aluminum nanoparticles, and the results show that the melting point increases as grain size augments, and it increases slowly when its grain size is between 2.2 nm and 3.2 nm but still holds the trend of increase. In order to obtain accurate thermal conductivity, the Green-Kubo method is adopted to calculate the phonon thermal conductivity of aluminum nanoparticle. As the grain size of aluminum nanoparticles increases, its density monotonically decreases, and the thermal conductivity monotonically increases linearly, which is in line with the theory of phonon. Similarly, with the increase of temperature, the density and thermal conductivity of aluminum nanoparticles of 1.6 nm in grain size both decrease. Moreover, the density of aluminum nanoparticle is generally lower than that of its bulk material. The study also shows that the heat transfer manner of aluminum nanoparticle is based on ballistic-diffusive heat conduction instead of the traditional diffusive heat conduction when it is in a nanoscale. The simulation studies the thermophysical properties of nanoparticles from the atomic perspective, and is of significance for guiding the design of the phase change materials based on the aluminum nanoparticles for thermal energy storage.

Phonon ballistic-diffusive heat conduction in silicon nanofilms by Monte Carlo simulations

Abstract An efficient Monte Carlo (MC) method on the basis of introducing a model of phonon scattering processes is proposed to simulate the ballistic-diffusive heat conduction in silicon nanofilms. The calculated thermal conductivity of nanofilms agrees with the experimental data, which is indicative of the validity of our simulations. The boundary temperature jump caused by the effects of phonon ballistic transport is observed by the MC technique. It is found that the boundary temperature jump increases with the Knudsen number (Kn). Theoretical models for predicting the boundary temperature jumps are also derived from the phonon Boltzmann transport equation. Model 1 is derived based on the acoustically thin approximation (Kn ≫ 1), whereas model 2 is obtained by the diffusive approximation (Kn ≪ 1). Furthermore, we derive model 3 in the intermediate region by an empirical way i.e. averaging model 1 and model 2. The theoretical models agree well with the MC simulations in different regions.

On an instability exhibited by the ballistic-diffusive heat conduction model of Xu and Hu

We show that the constitutive relation for the thermal flux proposed by Xu & Hu (2011) admits an unconditional instability. We also highlight the difference between mathematical models containing delay and those that include relaxation effects.

A transient ballistic–diffusive heat conduction model for heat pulse propagation in nonmetallic crystals

A hybrid phonon gas model was developed to describe heat pulse propagation in dielectric crystals. In this model, heat energy is described as a mixture of longitudinal and transverse phonon gases. The novel aspects of this model include: (1) the distinction between longitudinal and transverse phonon excitations is taken into account; (2) transitions between ballistic and second sound transport, ballistic and diffusive transport, and second sound and diffusive transport are successfully captured. For the first time, benchmark cases of heat-pulse experiments in NaF at low temperatures in ballistic, ballistic–diffusive, and diffusive regions have been completely reconstructed in numerical simulations, which demonstrate the validity of the new model. It was elucidated how heat pulses are transmitted by longitudinal, ballistic transverse and dispersive transverse phonons at low temperatures. The numerical results not only yield new insight in physics of transient ballistic–diffusive heat conduction with effects from different phonon scattering processes, but also provide numerical tools to study similar transient ballistic–diffusive heat conduction in nanoelectronics and modern optoelectronics.

A Hybrid Phonon Gas Model for Transient Ballistic-Diffusive Heat Transport

We present a continuum hybrid phonon gas model to describe transient ballistic-diffusive heat transport. In this model, heat energy is carried by a mixture of longitudinal and transverse phonon gases so that the distinction between longitudinal and transverse phonon excitations is taken into account. This new model is validated by the successful reconstruction of benchmark cases of heat-pulse experiments in NaF, which have never been completely reconstructed before. It is elucidated how thermal pulses are transmitted by longitudinal and transverse phonon gases. This model not only helps us yield new insight in transient ballistic-diffusive heat conduction mechanisms but also provides numerical tools to study transient ballistic-diffusive heat conduction in nanoelectronic and modern optoelectronics.

Monte Carlo simulation of phonon ballistic diffusive heat conduction in silicon nanofilm

A Monte Carlo (MC) method is proposed by establishing a new model of phonon scattering processes with random sampling according to a scattering probability function. The MC scheme is used to simulate steady and transient ballistic-diffusive heat conduction in silicon nanofilm. In the MC simulations, we trace the phonon bundles that emit into media from the boundaries, and obtain the temperature profiles through statistics of the distribution of phonon bundles. It is found that the size effect of phonon transport leads to a boundary temperature jump which increases with the Knudsen number increasing. The thermal conductivity of the silicon nanofilm is calculated and the results suggest that nanofilm thermal conductivity increases with film thickness increasing, which is in good agreement with the experimental data as well as the results from the theoretical model. The temperature profiles vary with time in the transient simulations, which shows that the heat wave is related to not only time scale but also spatial scale. When the spatial scale becomes smaller, the ballistic transport is more dominant, which leads to stronger heat waves.

A ballistic-diffusive heat conduction model extracted from Boltzmann transport equation

A ballistic-diffusive heat conduction model is derived from the Boltzmann transport equation by a coarse-graining approach developed in the present study. By taking into account of the lagging effect, this model avoids the infinite heat propagation speed implied by the classical Fourier law. By expressing the heat conductivity as a function of the Knudsen number, it accounts for the size effect of the nanoscale heat conduction. The variation of the obtained effective heat conductivity with respect to the characteristic length shows an agreement with the experimental results for thin silicon films and nanowires in the nanoscale regime.

SPH Numerical Modeling for Ballistic-Diffusive Heat Conduction

The classic Fourier heat diffusion theory, in general, does not apply to micro/nanoscale or ultrafast thermal processes, where the mean free path of the heat carriers is comparable to the system size and/or typical times are comparable to the carrier relaxation time. For these processes, the ballistic-diffusive equation is a good alternative. In this context, the smoothed particle hydrodynamics (SPH) method is further developed to solve the one-dimensional ballistic-diffusive equation, by using a boundary treatment technique based on the normalization property of the SPH smoothing kernel. The results indicate that the SPH is a reliable and accurate method capable of yielding correct temperature and heat flux representations in the ballistic-diffusive heat conduction regime.

Ballistic-diffusive heat conduction at nanoscale: GENERIC approach

Heat conduction in nanosize systems has to be studied in settings involving microscopic details that are not seen in the classical Fourier theory. G. Chen has suggested [Phys. Rev. Lett. 86 (2001) 2297] a combination of the Cattaneo setting in which the velocity of the heat propagation is finite and the kinetic theory setting in which phonons are seen as heat carriers. In this Letter we show that if the Cattaneo and the kinetic theories are combined in a way that preserves the structure expressing their compatibility with thermodynamics (GENERIC structure) then both the Cattaneo and the kinetic equations become modified. The modified Cattaneo equations involve the term introduced by Chen and, in addition, new terms that are nonlinear in quantities that disappear at equilibrium. The kinetic equation is modified by new terms involving gradients of the heat flux and the local temperature.

Simulation of Nanoscale Multidimensional Transient Heat Conduction Problems Using Ballistic-Diffusive Equations and Phonon Boltzmann Equation

Heat conduction in micro- and nanoscale and in ultrafast processes may deviate from the predictions of the Fourier law, due to boundary and interface scattering, the ballistic nature of the transport, and the finite relaxation time of heat carriers. The transient ballistic-diffusive heat conduction equations (BDE) were developed as an approximation to the phonon Boltzmann equation (BTE) for nanoscale heat conduction problems. In this paper, we further develop BDE for multidimensional heat conduction, including nanoscale heat source term and different boundary conditions, and compare the simulation results with those obtained from the phonon BTE and the Fourier law. The numerical solution strategies for multidimensional nanoscale heat conduction using BDE are presented. Several two-dimensional cases are simulated and compared to the results of the transient phonon BTE and the Fourier heat conduction theory. The transient BTE is solved using the discrete ordinates method with a two Gauss-Legendre quadratures. Special attention has been paid to the boundary conditions. Compared to the cases without internal heat generation, the difference between the BTE and BDE is larger for the case studied with internal heat generation due to the nature of the ballistic-diffusive approximation, but the results from BDE are still significantly better than those from the Fourier law. Thus we conclude that BDE captures the characteristics of the phonon BTE with much shorter computational time.