Lattice Boltzmann Modeling of Phonon Heat Conduction in Superlattice Structures

2010 ◽  
Author(s):  
Cristian J. San Marti´n ◽  
Amador M. Guzma´n ◽  
Rodrigo A. Escobar
Keyword(s):  
Thermal Conductivity ◽  
Free Path ◽  
Lattice Boltzmann ◽  
Phonon Dispersion ◽  
Effective Conductivity ◽  
Mean Free Path ◽  
One Dimension ◽  
The Mean ◽  
Superlattice Structures ◽  
Boltzmann Method

The results of temperature prediction and determination of effective thermal conductivity in periodic Si-Ge superlattice in one dimension, at length scale comparable to the mean free path are presented. Classical heat transfer models such as Fourier’s law do not represent what actually happens within electronic devices at these length scales. Phonon-border and phonon-interface scattering effects provide discontinuous jumps in temperature distribution when the mean free path is comparable with the device’s characteristic length, a relation given by the Knudsen number (Kn). For predicting the temperature within the periodic Si-Ge superlattice use is made of the lattice Boltzmann method in one dimension, using Debye’s model in the phonon dispersion relation. The predictions show that as Kn increases, so do the jumps at the borders, the same as at the interfaces. The prediction also shows that the effective conductivity of the Si-Ge superlattice decreases as Kn and the number of layers of material increase, and that keff decreases as the magnitude of p increases, a factor that allows heat flow between one layer and another. Use of gray LBM leads to good approximations of the actual temperature field and thermal conductivity values for the superlattice materials model when the physics of phonons established by Debye’s model is used.

Accuracy of the Lattice-Boltzmann Method

1997 ◽  
Vol 08 (04) ◽  
pp. 747-752 ◽  
Author(s):  
Robert S. Maier ◽  
Robert S. Bernard
Keyword(s):  
Mach Number ◽  
Boundary Conditions ◽  
Lattice Boltzmann Method ◽  
Free Path ◽  
Lattice Boltzmann ◽  
Mean Free Path ◽  
Duct Flow ◽  
Reynolds Number Flow ◽  
The Mean ◽  
Boltzmann Method

The accuracy of the lattice-Boltzmann method (LBM) is moderated by several factors, including Mach number, spatial resolution, boundary conditions, and the lattice mean free path. Results obtained with 3D lattices suggest that the accuracy of certain two-dimensional (2D) flows, such as Poiseuille and Couette flow, persist even when the mean free path between collisions is large, but that of the 3D duct flow deteriorates markedly when the mean free path exceeds the lattice spacing. Accuracy in general decreases with Knudsen number and Mach number, and the product of these two quantities is a useful index for the applicability of LBM to 3D low-Reynolds-number flow. The influence of boundary representations on LBM accuracy is captured by the proposed index, when the accuracy of the prescribed boundary conditions is consistent with that of LBM.

scholarly journals Ballistic Heat Transport in Nanocomposite: The Role of the Shape and Interconnection of Nanoinclusions

Nanomaterials ◽  
2021 ◽  
Vol 11 (8) ◽  
pp. 1982
Author(s):  
Paul Desmarchelier ◽  
Alice Carré ◽  
Konstantinos Termentzidis ◽  
Anne Tanguy
Keyword(s):  
Thermal Conductivity ◽  
Molecular Dynamics ◽  
Molecular Dynamics Simulations ◽  
Free Path ◽  
Mean Free Path ◽  
Strong Increase ◽  
Moderate Increase ◽  
Energy Propagation ◽  
The Mean ◽  
Dynamics Simulations

In this article, the effect on the vibrational and thermal properties of gradually interconnected nanoinclusions embedded in an amorphous silicon matrix is studied using molecular dynamics simulations. The nanoinclusion arrangement ranges from an aligned sphere array to an interconnected mesh of nanowires. Wave-packet simulations scanning different polarizations and frequencies reveal that the interconnection of the nanoinclusions at constant volume fraction induces a strong increase of the mean free path of high frequency phonons, but does not affect the energy diffusivity. The mean free path and energy diffusivity are then used to estimate the thermal conductivity, showing an enhancement of the effective thermal conductivity due to the existence of crystalline structural interconnections. This enhancement is dominated by the ballistic transport of phonons. Equilibrium molecular dynamics simulations confirm the tendency, although less markedly. This leads to the observation that coherent energy propagation with a moderate increase of the thermal conductivity is possible. These findings could be useful for energy harvesting applications, thermal management or for mechanical information processing.

Experiments on the flow of heat in liquid helium below 0.7 °K

1958 ◽  
Vol 246 (1246) ◽  
pp. 390-405 ◽  
Keyword(s):  
Thermal Conductivity ◽  
Liquid Helium ◽  
Free Path ◽  
Empirical Relation ◽  
Mean Free Path ◽  
Series Of Experiments ◽  
The Mean ◽  
Set Up ◽  
Low Pressures ◽  
Measured Thermal Conductivity

A series of experiments has been performed to study the steady flow of heat in liquid helium in tubes of diameter 0.05 to 1.0 cm at temperatures between 0.25 and 0.7 °K. The results are interpreted in terms of the flow of a gas of phonons, in which the mean free path λ varies with temperature, and may be either greater or less than the diameter of the tube d . When λ ≫ d the flow is limited by the scattering of the phonons at the walls, and the effect of the surface has been studied, but when λ ≪ d viscous flow is set up in which the measured thermal conductivity is increased above that for wall scattering. This behaviour is very similar to that observed in the flow of gases at low pressures, and by applying kinetic theory to the problem it can be shown that the mean free path of the phonons characterizing viscosity can be expressed by the empirical relation λ = 3.8 x 10 -3 T -4.3 cm. This result is inconsistent with the temperature dependence of λ as T -9 predicted theoretically by Landau & Khalatnikov (1949).

scholarly journals The thermal conductivity of carbon dioxide

1927 ◽  
Vol 114 (767) ◽  
pp. 354-366 ◽  
Keyword(s):  
Thermal Conductivity ◽  
Free Path ◽  
Critical Pressure ◽  
Mean Free Path ◽  
Double System ◽  
Wire Method ◽  
The Mean ◽  
The Absolute ◽  
The Many

Of the many experimental determinations of the thermal conductivity of Co 2 which have been made, the absolute values given by the various observers vary from 3·07 × 10 -5 cal. sec. -1 cm. -1 deg. -1 (Winkelman, 1), to 3·39 × 10 -5 cal. sec. -1 cm. -1 deg. -1 (Weber, 2), and generally speaking the experiments were modifications of two principal methods, namely, the electrically heated wire of Schleimacher (3) and the cooling thermometer method. In both of these methods convection losses were present to a degree depending on the dimensions and disposition of the apparatus, and on the pressure of the gas; therefore, in the author’s opinion, the discrepancies amongst various observers are due to the practice of attempting to eliminate these convective losses by diminishing the pressure. Such a procedure is justifiable only if the reduction of pressure is not carried beyond the point at which the mean free path of the molecules becomes comparable with the dimensions of the containing vessel. This is a critical point in the determination of the conductivity of a gas, as the authors’ experiments on Co 2 indicate that the convection becomes negligible only at pressures for which the mean Free Path Effect is such that the significance imposed on the conductivity by Fourier’s law loses its meaning, and below this critical pressure the conductivity varies with the pressure in a manner depending on the dimensions of the vessel containing the gas. In the experiments of Gregory and Archer (4), on the thermal conductivities of air and hydrogen, the use of a double system of electrically-heated wires enabled the authors accurately to identify the critical pressure at which convective losses became negligible. This is an extremely important point in all applications of the hot-wire method to the absolute determination of the conductivities of gases, and alone justifies the procedure of lowering the pressure to eliminate convective losses. Above this critical pressure it is necessary to disentangle the conduction and convection losses, and below, the meaning of conduction loses its ordinary significance.

Lattice Boltzmann Modeling of Subcontinuum Energy Transport in Crystalline and Amorphous Microelectronic Devices

2006 ◽  
Vol 128 (2) ◽  
pp. 115-124 ◽  
Author(s):  
Rodrigo Escobar ◽  
Brian Smith ◽  
Cristina Amon
Keyword(s):  
Thin Films ◽  
Thermal Conductivity ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Energy Transport ◽  
Phonon Dispersion ◽  
Numerical Models ◽  
Phonon Transport ◽  
Nonlinear Frequency ◽  
Boltzmann Method

Numerical simulations of time-dependent energy transport in semiconductor thin films are performed using the lattice Boltzmann method applied to phonon transport. The discrete lattice Boltzmann method is derived from the continuous Boltzmann transport equation assuming first gray dispersion and then nonlinear, frequency-dependent phonon dispersion for acoustic and optical phonons. Results indicate that a transition from diffusive to ballistic energy transport is found as the characteristic length of the system becomes comparable to the phonon mean free path. The methodology is used in representative microelectronics applications covering both crystalline and amorphous materials including silicon thin films and nanoporous silica dielectrics. Size-dependent thermal conductivity values are also computed based on steady-state temperature distributions obtained from the numerical models. For each case, reducing feature size into the subcontinuum regime decreases the thermal conductivity when compared to bulk values. Overall, simulations that consider phonon dispersion yield results more consistent with experimental correlations.

Size Effects of Heat Conduction for Silicon Nanograins

2014 ◽  
Vol 633-634 ◽  
pp. 34-37
Author(s):  
Ya Fen Han ◽  
Hai Dong Liu
Keyword(s):  
Thermal Conductivity ◽  
Heat Conduction ◽  
Kinetic Theory ◽  
Free Path ◽  
Size Effects ◽  
Mean Free Path ◽  
Structure Model ◽  
The Mean

The structure model of silicon nanograins was built. And then based the modification of the mean free path of phonons according to the size of nanograins, the expression of thermal conductivity in nanograins was obtained according to the phonon kinetic theory. The dependence of the thermal conductivity of silicon nanograins on size was investigated. The results showed that thermal conductivity of nanograins decrease with the reduction of characteristic sizes when the characteristic sizes of nanograins are comparable to or smaller than the phonon mean free path.

Joule Heating and Thermal Conductivity Determination of Nanoscale Metallic Thin Films and Interconnects

2005 ◽  
Author(s):  
Siva P. Gurrum ◽  
William P. King ◽  
Yogendra K. Joshi ◽  
Koneru Ramakrishna
Keyword(s):  
Thin Films ◽  
Thermal Conductivity ◽  
Free Path ◽  
Joule Heating ◽  
High Performance ◽  
Effective Conductivity ◽  
Mean Free Path ◽  
Current Crowding ◽  
Metal Line ◽  
Thin Metallic Film

Evolution of high performance microprocessors has resulted in a steady decrease in on-chip feature sizes. Increasing requirements on maximum current density are expected to increase interconnect temperature drastically due to Joule heating. As interconnect dimensions approach the electron mean free path range, effective conductivity reduces due to size effects. Thermal characterization of sub-micron interconnects and thin films is thus highly important. This work investigates current crowding and the associated Joule heating near a constriction in a thin metallic film and proposes a novel technique to determine thermal conductivity of thin metallic films and interconnects in the sub-100 nm range. Scanning Joule Expansion Microscopy (SJEM) measures the thermal expansion of the structure whose thickness is comparable to the mean free path of electrons. Numerical solution of heat conduction equation in the frequency space is used to obtain a fit for effective thermal conductivity. A thermal conductivity of ~ 80.0 W/mK provides a best fit to the data. This is about one-third the bulk thermal conductivity of gold, which is 318 W/mK at room temperature. Using Wiedemann-Franz Law a thermal conductivity of 92.0 W/mK is obtained after measuring the electrical resistivity of the metal line. This is close to that obtained through numerical fit.

Lattice Boltzmann Simulation of High-Diffusivity Problems With Application to Energy Transport in a High-Temperature Solar Thermochemical Reactor

2013 ◽  
Author(s):  
Like Li ◽  
Abhishek Singh ◽  
Nicholas AuYeung ◽  
Renwei Mei ◽  
Jörg Petrasch ◽  
...  
Keyword(s):  
Diffusion Coefficient ◽  
High Temperature ◽  
Free Path ◽  
Analytical Solutions ◽  
Lattice Boltzmann ◽  
Energy Transport ◽  
Mean Free Path ◽  
Numerical Errors ◽  
The Mean ◽  
Solar Thermochemical

We describe a general technique for rescaling the high-diffusivity convection diffusion equation (CDE) when it is simulated with the lattice Boltzmann equation (LBE) method. The macroscopic CDE is recovered from the kinetic-based LBE when the mean free path of the particles is much smaller than the lattice grid size. As the relaxation time and the mean free path are proportional to the diffusion coefficient of the CDE in LBE models, direct use of a large diffusion coefficient would lead to large numerical errors in LBE simulations. To improve accuracy, we rescale the CDE by choosing a large time scale and a moderate relaxation coefficient so that the characteristic Fourier number for the diffusion process remains the same. The rescaled LBE model is first validated with two numerical tests for which analytical solutions are available: the transient heat conduction in a semi-infinite solid and that inside a circle. The comparison between the LBE results and analytical solutions shows that the numerical errors are greatly reduced when the high diffusion coefficient is rescaled down. It is then applied in a diffusion-radiation coupled model to simulate the energy transport in a high-temperature solar thermochemical reactor for hydrogen production. Rescaling of the solar flux boundary conditions and the chemical reaction source terms due to the rescaling of the diffusion coefficient is also discussed and the simulation results will be used to optimize the cavity-reactor design.

Experimentelle und theoretische Untersuchungen des isothermen und des adiabatischen Hall-Effektes an Se

1953 ◽  
Vol 8 (8) ◽  
pp. 453-459
Author(s):  
Rolf Diestel
Keyword(s):  
Thermal Conductivity ◽  
Theoretical Result ◽  
Free Path ◽  
Hall Coefficient ◽  
Mean Free Path ◽  
Type Semiconductor ◽  
The Mean ◽  
The Difference ◽  
P Type

Supposing an arbitrary dependence of the mean free path on energy, the isothermal and the adiabatic Hall coefficients are calculated for a p-type semiconductor by means of the theory of electrons. The difference between the Hall coefficients calculated in this way decrease considerably with increasing thermal conductivity of the lattice. Even for substances with very small thermal conductivity (i. e. Se etc.), the difference amounts to only about 1% in relation to the isothermal Hall coefficient. This theoretical result is proved by measurements on Se; the measured difference slightly exceeds the limit of error (5,2±4,1)%.

Estimation of Anisotropic Thermal Conductivity in Nanoscale Confined Semiconductors via Lattice Boltzmann Method

2005 ◽  
Author(s):  
Sartaj S. Ghai ◽  
Woo Tae Kim ◽  
Cristina H. Amon ◽  
Myung S. Jhon
Keyword(s):  
Thermal Conductivity ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Transport Model ◽  
Mean Free Path ◽  
Lattice Boltzmann Model ◽  
Semiconductor Film ◽  
Anisotropic Thermal Conductivity ◽  
Transient Thermal ◽  
Boltzmann Method

A novel transient thermal transport model based on lattice Boltzmann method is developed to capture the sub-continuum effects including anisotropic thermal behavior of solids at nanoscale. Rigorous boundary condition treatment is incorporated via ghost boundary formulation. These sub-continuum effects deviate significantly from the bulk behavior and can not be accurately captured by the continuum based models such as Fourier equation. We observed that as the thickness of the semiconductor film is decreased to the scale of its carrier’s mean free path, the thermal conductivity of the film reduces drastically from its bulk value and starts to show anisotropic behavior. In addition, a temperature jump, which does not exist at the bulk conditions, is observed at the interfaces. These sub-continuum effects are successfully captured by the lattice Boltzmann model and simple equations have been developed to accurately estimate these effects using the film geometry and properties.

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