Transport Equation Approach Research Articles

Extended finite similitude and dimensional analysis for scaling

The theory of scaling called finite similitude does not involve dimensional analysis and is founded on a transport-equation approach that is applicable to all of classical physics. It features a countable infinite number of similitude rules and has recently been extended to other types of governing equations (e.g., differential, variational) by the introduction of a scaling space Omega _{beta }, within which all physical quantities are deemed dependent on a single dimensional parameter beta . The theory is presently limited to physical applications but the focus of this paper is its extension to other quantitative-based theories such as finance. This is achieved by connecting it to an extended form of dimensional analysis, where changes in any quantity can be associated with curves projected onto a dimensional Lie group. It is shown in the paper how differential similitude identities arising out of the finite similitude theory are universal in the sense they can be formed and applied to any quantitative-based theory. In order to illustrate its applicability outside physics the Black-Scholes equation for option valuation in finance is considered since this equation is recognised to be similar in form to an equation from thermal physics. It is demonstrated that the theory of finite similitude can be applied to the Black-Scholes equation and more widely can be used to assess observed size effects in portfolio performance.

Collective bulk excitations in the QGP: From the weakly non‐ideal case to the strong correlation limit

AbstractCollective bulk excitation propagating in the medium of a quark‐gluon plasma are studied. The quark‐gluon plasma medium is described with two different models to investigate the effect of binary collisions and shear viscosity to account for damping and non‐ideality. To find the dispersion relation the longitudinal dielectric functions of the quark‐gluon plasma are used within quantum hydrodynamics and transport equation approaches. The results were obtained numerically by finding the roots of the dispersion relations. Both models are compared with the hard thermal loop approximation as the ideal case for both models.

Effect of interfacial area modeling method on interfacial drag predictions for two-fluid model calculations in large diameter pipes

The nuclear industry requires accurate modeling efforts to ensure safety standards are consistently achieved. To that end, the sensitivity of void fraction and interfacial area concentration predictions to changes in the interfacial area concentration modeling methods was evaluated. Several interfacial area concentration correlations and interfacial area transport equation coalescence and breakup models have been compared to the data of Schlegel et al. and Shen et al. using a computational tool developed in MATLAB. Bubble coalescence and breakup empirical constants were optimized for the group void fractions and interfacial area concentrations using the computational tool and the surface response method. The results indicate that the transport equation approach has a higher accuracy for both the void fraction and interfacial area concentration predictions than the correlation approach. The Smith et al. model is shown to have the highest void fraction predictive accuracy. The optimized model is shown to successfully increase accuracy of the interfacial area concentration prediction. This is also the highest accuracy observed for interfacial area concentration predictions.

Effect of four-phonon interaction on phonon thermal conductivity and mean-free-path spectrum of high-temperature phase SnSe

The phonon thermal conductivity and mean-free-path (MFP) spectrum of high-temperature phase SnSe (β-SnSe) are studied using the Boltzmann transport equation and ab initio approaches. The particle picture for phonon transport in β-SnSe is revisited, and the imaginary phonon frequencies caused by the ground-state within conventional density-functional theory are resolved. We show that between 800 and 950 K, the in-plane and cross-plane thermal conductivity has an average decrease of 38% and 19%, respectively, when four-phonon scatterings are considered. This large suppression of phonon transport stems mainly from the strong redistribution scattering process. With both the phonon and electron MFP spectra revealed, a characteristic length of 10 nm is suggested to reduce the in-plane and cross-plane thermal conductivity by 18% and 52%, respectively, via nanostructure engineering without sacrificing the power factor.

Anharmonic effects on lattice dynamics and thermal transport of two-dimensional InTe monolayer

The lattice thermal conductivity (κl) plays a key role in the performance of thermoelectric (TE) materials, where the lower values lead to higher figure of merit values. Two-dimensional (2D) group III-VI monolayers such as InTe are promising materials for TE energy generation owing to their low κl that lead to high TE figure of merit values. In this work, we investigate the influence of the lattice anharmonicity on the lattice thermal conductivity of InTe monolayer. The thermodynamic parameters are calculated by using the self-consistent phonon (SCP) theory. The κl value of the InTe monolayer is obtained to be 0.30 Wm−1 K−1 by using the standard Boltzmann transport equation (BTE) approach, while it is 3.58 Wm−1 K−1 by using SCP + BTE approach. These results confirm the importance of the anharmonic effects on the κl value, where it was found to be significantly higher (91%) using the SCP + BTE approach than that obtained using the standard BTE approach.

Revisiting Mott's model for the thermopower of PdxAg1−x alloys with support of density functional theory

Electronic properties of palladium-silver alloys have been studied for almost a century as a prototype of a transition-metal alloy with entangled $s$ and $d$ bands. We review the literature starting from Mott's model of 1935 and discuss from a quantitative point of view a series of common assumptions stemming from that seminal paper. We also show that spectral functions obtained by unfolding the band structures of a sample of supercells can be used for the ab initio computation of the thermopower of such systems within a Boltzmann transport equation approach.

A transport equation approach for deep neural networks with quenched random weights

We consider a multi-layer Sherrington–Kirkpatrick spin-glass as a model for deep restricted Boltzmann machines with quenched random weights and solve for its free energy in the thermodynamic limit by means of Guerra’s interpolating techniques under the RS and 1RSB ansatz. In particular, we recover the expression already known for the replica-symmetric case. Further, we drop the restriction constraint by introducing intra-layer connections among spins and we show that the resulting system can be mapped into a modular Hopfield network, which is also addressed via the same techniques up to the first step of replica symmetry breaking.

Validation of transition modeling techniques for a simplified fuselage configuration

A Reynolds-averaged Navier-Stokes solver has been equipped with two different transition prediction and modeling techniques for automatic transition prediction on general three-dimensional geometrical configurations. The first technique is a streamline-based approach that applies an automated local linear stability code, the eN-method and a simplified two-N-factor strategy and is applicable with any kind of turbulence model. The second technique is a transition transport equation approach and uses the γ-Reθ,t model. The standard γ-Reθ,t model has been extended by a modeling approach that captures transition due to crossflow instabilities. The extended model can be applied to three-dimensional configuration exhibiting crossflow transition whereas the original γ-Reθ,t model only captures streamwise transition mechanisms. The paper focuses on a three-dimensional inclined 6:1 prolate spheroid which can be considered as a simplified fuselage configuration. For a number of years this configuration has become a standard basic three-dimensional test case for transition prediction approaches and has been used in a number of international workshops and research activities. The focus of the paper is the validation of the two transition modeling approaches for six different angles of attack using two different turbulence models, a standard two-equation eddy-viscosity model and a differential Reynolds-stress model. The simulation results from the two approaches have been compared against each other and to the available experimental data. The comparisons highlight as many qualitative and quantitative similarities and agreements as they reveal quantitative differences and deviations of specific details.

The effect of finite-temperature and anharmonic lattice dynamics on the thermal conductivity of ZrS2 monolayer: self-consistent phonon calculations

Two-dimensional (2D) ZrS2 monolayer (ML) has emerged as a promising candidate for thermoelectric (TE) device applications due to its high TE figure of merit, which is mainly contributed by its inherently low lattice thermal conductivity (LTC). This work investigates the effect of the lattice anharmonicity driven by the temperature-dependent phonon dispersions on the thermal transport of ZrS2 ML. The calculations are based on the self-consistent phonon (SCP) theory to calculate the thermodynamic parameters along with the LTC. The higher-order (quartic) force constants were extracted by using an efficient compressive sensing lattice dynamics technique, which estimates the necessary data based on the emerging machine learning program as an alternative of computationally expensive density functional theory calculations. Resolve of the degeneracy and hardening of the vibrational frequencies of low-energy optical modes were predicted upon including the quartic anharmonicity. As compared to the conventional Boltzmann transport equation (BTE) approach, the LTC of the optimized ZrS2 ML unit cell within SCP + BTE approach is found to be significantly enhanced (e.g., by 21% at 300 K). This enhancement is due to the relatively lower value of phonon linewidth contributed by the anharmonic frequency renormalization included in the SCP theory. Mainly, the conventional BTE approach neglects the temperature dependence of the phonon frequencies due to the consideration of harmonic lattice dynamics and treats the normal process of three-phonon scattering incorrectly due to the use of quasi-particle lifetimes. These limitations are addressed in this work within the SCP + BTE approach, which signifies the validity and accuracy of this approach.

A Theoretical Investigation of the Stability, Electronic, Optical, and Thermoelectric Properties of BaLiP

The density functional theory and post density functional theory techniques are used to investigate the properties of BaLiP for potential applications in photovoltaic and thermoelectric (TE) applications. Density functional calculations of the cohesive energy, phonon‐dispersion, and elastic constant studies establish structural, dynamical, and mechanical stability of the compound. Density functional calculations show that BaLiP is a direct bandgap semiconductor while optical properties at the Bethe–Salpeter equation level of approximation suggest that BaLiP is anisotropic with the optical bandgaps of and eV for in‐plane and out‐of‐plane absorption, respectively. A maximum absorption coefficient of for in‐plane and for out‐of‐plane absorption in the visible range at and eV is found, respectively. Lattice thermal conductivity is computed using a Boltzmann transport equation approach. The calculations reveal that the average lattice thermal conductivity is at room temperature. The figure of merit, which is the main criterion for estimating the TE potential of a material, is estimated to have a maximum of at 1000 K at a hole carrier concentration of . This study shows that BaLiP has some promise as a candidate for applications in electronics, optoelectronics, and TE applications.

Large lattice thermal conductivity, interplay between phonon-phonon, phonon-electron, and phonon-isotope scatterings, and electrical transport in molybdenum from first principles

We describe an ab initio phonon Boltzmann transport equation (BTE) approach accounting for phonon-electron scattering in addition to the well-established phonon-phonon and isotope scatterings. The phonon BTE is linearized and can be exactly solved beyond the relaxation time approximation (RTA). We use this approach to study the lattice thermal conductivity (${\ensuremath{\kappa}}_{\text{ph}}$) of molybdenum (Mo). ${\ensuremath{\kappa}}_{\text{ph}}$ of Mo is found to possess several anomalous features: (1) like in another group VI element tungsten (W), ${\ensuremath{\kappa}}_{\text{ph}}$, with a large value of 37 W ${\mathrm{m}}^{\ensuremath{-}1}$ ${\mathrm{K}}^{\ensuremath{-}1}$ at room temperature, follows weak temperature dependence due to interplay between phonon-phonon (ph-ph), phonon-electron (ph-el), and phonon-isotope (isotope) scatterings; and (2) compared with W, though Mo is much lighter in mass, Mo has a smaller ${\ensuremath{\kappa}}_{\text{ph}}$. This is attributed to weaker interatomic bonding, larger isotope mixture, and larger density of states at Fermi level in Mo. In isotopically pure samples, ${\ensuremath{\kappa}}_{\text{ph}}$ increases from 37 to 48 W ${\mathrm{m}}^{\ensuremath{-}1}$ ${\mathrm{K}}^{\ensuremath{-}1}$ at room temperature. Considering the similarity of the phonon dispersion, our work suggests that chromium should also have a large ${\ensuremath{\kappa}}_{\text{ph}}$, which, rather than the complexity of the electronic band structure argued in the literature, accounts for the significant deviation of measured Lorenz number $L$ from the expected Sommerfeld value. The electrical conductivity ($\ensuremath{\sigma}$) and electronic thermal conductivity (${\ensuremath{\kappa}}_{\text{e}}$) of Mo are also calculated by using an ab initio electron BTE approach. $\ensuremath{\sigma}$ and the total thermal conductivity ($\ensuremath{\kappa}$) agree with the experimental data reasonably. These results demonstrate that the ab initio calculations can quantify the lattice and electronic contributions to $\ensuremath{\kappa}$. We also look into the cumulative $\ensuremath{\sigma}$ and ${\ensuremath{\kappa}}_{\text{ph}}$ with respect to electron and phonon mean free paths (MFPs), respectively, in order to reveal the size effect in Mo. The MFPs of electrons contributing to conductivity range from 5 to 22 nm, whereas the MFPs of phonons primarily distribute between 5 and 73 nm with more than 80% contribution to ${\ensuremath{\kappa}}_{\text{ph}}$. This suggests that a reduced Lorenz number can be observed in Mo nanostructures when the relevant size goes below $\ensuremath{\sim}70$ nm.

Thermal conductivity of silicon at elevated temperature: Role of four-phonon scattering and electronic heat conduction

While using first-principles-based Boltzmann transport equation approach to predict the thermal conductivity of crystalline semiconductor materials has been a routine, the validity of the approach is seldom tested for high-temperature conditions. Most previous studies only focused on the phononic contribution, and neglected the electronic part. In this paper, we present a Boltzmann transport equation study on high-temperature thermal conduction in bulk silicon by considering the effects of both phonons and electrons. Both polar and bipolar contributions to the electronic thermal conductivity are calculated from first-principles. More than 25% of heat is shown to be conducted by electrons at 1500 K. The computed total thermal conductivity of silicon faithfully reproduces the measured data. The approach presented in this paper is expected to be applied to other high-temperature functional materials, and the results could serve as benchmarks and help to explain the high-temperature phonon and electron transport phenomena.

Thermal and mechanical properties of U3Si2: A combined ab-initio and molecular dynamics study

Triuranium disilicide (U3Si2) has been identified as an accident resistant alternative to uranium dioxide (UO2) based fuels. While experimental studies of U3Si2 are not new, only few studies on the thermal conductivity has been reported. Moreover, theoretical studies of the thermal properties of U3Si2 are scarce. We found that inaccuracies in experimental data were a source of discrepancies in recently reported theoretical results, as such need to be addressed. In this work, the structural, thermal, and mechanical properties of, the non-magnetic and magnetic phases, of U3Si2 were investigated using Hubbard corrected density functional theory (DFT+U) and molecular dynamics (MD) simulations. The lattice and electronic thermal properties were determined based on Boltzmann transport equation (BTE) approaches. A considerable anisotropy was determined the lattice and electronic contributions to the thermal conductivity from DFT+U/BTE results, in which the electronic contribution is predominant. A relative good agreement between the lattice contributions to the thermal conductivity, predicted from DFT+U and MD computations, was associated to similarities in the acoustic phonon modes. The present computational results significantly enhance the understanding of the thermal properties of U3Si2. The good agreement between DFT+U/BTE results with experimental data confirm the accident tolerant characteristic of U3Si2, but also further demonstrating the accuracy of the DFT(+U)/BTE approach for the investigation of thermal properties of advanced nuclear fuels.

Erratum: Transport equation approach to calculations of Hadamard Green functions and noncoincident DeWitt coefficients [Phys. Rev. D 84 , 104039 (2011)

Received 10 January 2020DOI:https://doi.org/10.1103/PhysRevD.101.029901© 2020 American Physical Society

Parameter-free model to estimate thermal conductivity in nanostructured materials

Achieving low thermal conductivity and good electrical properties is a crucial condition for thermal energy harvesting materials. Nanostructuring offers a very powerful tool to address both requirements: in nanostructured materials, boundaries preferentially scatter phonons compared to electrons. The computational screening for low-thermal-conductivity nanostructures is typically limited to materials with simple crystal structures, such as silicon, because of the complexity arising from modeling branch- and wave-vector-dependent nanoscale heat transport. The phonon mean-free-path (MFP) dependent Boltzmann transport equation (MFP-BTE) approach is a model that overcomes this limitation. To illustrate this, we analyze thermal transport in 75 nanoporous half-Heusler compounds for different pore sizes. Our calculations demonstrate that, in most cases, the optimization of thermal transport in nanostructures should take into account both bulk thermal properties and geometry-dependent size effects, two aspects that are typically engineered separately. To enable efficient calculations within this paradigm we derive a model, based on the ``gray'' formulation of the BTE, that can decouple the influence of the geometry and the material on the effective thermal conductivity with relatively little loss in accuracy compared to the MFP-BTE. Our study motivates the need for a holistic approach to engineering thermal transport and provides a method for high-throughput low-thermal conductivity materials discovery.

Tailoring phononic, electronic, and thermoelectric properties of orthorhombic GeSe through hydrostatic pressure

In this paper, we systematically investigate the effect of hydrostatic pressure on the phononic and electronic transport properties of orthorhombic p-type GeSe using first-principles based Boltzmann transport equation approach. It is found that the lattice thermal conductivities along the a and c directions increase with pressure, whereas it experiences a decrease along the b direction. This anomalous pressure dependent lattice thermal conductivity is attributed to the combined effect of enhanced phonon group velocity and reduced phonon lifetime. Additionally, the optical phonon branches have remarkable contributions to the total lattice thermal conductivity. The electronic transport calculations indicate that the Seebeck coefficient undergoes a sign change from p-type to n-type along the a direction under pressure, and a dramatic enhancement of the power factor is observed due to the boost of electrical conductivity. The predicted ZT values along the a, b, and c directions are 1.54, 1.09, and 1.01 at 700 K and 8 GPa, respectively, which are about 14, 7.3, and 1.9 times higher than those at zero pressure at experimental carrier concentration of ~1018 cm−3. Our study is expected to provide a guide for further optimization of the thermal and charge transport properties through hydrostatic pressure.

Thermal transport properties of single-layer black phosphorus from extensive molecular dynamics simulations

We compute the anisotropic in-plane thermal conductivity of suspended single-layer black phosphorus (SLBP) using three molecular dynamics (MD) based methods, including the equilibrium MD method, the nonequilibrium MD (NEMD) method, and the homogeneous NEMD (HNEMD) method. Two existing parameterizations of the Stillinger–Weber (SW) potential for SLBP are used. Consistent results are obtained for all the three methods and conflicting results from previous MD simulations are critically assessed. Among the three methods, the HNEMD method is the most and the NEMD method the least efficient. The thermal conductivity values from our MD simulations are about an order of magnitude larger than the most recent predictions obtained using the Boltzmann transport equation approach considering long-range interactions in density functional theory calculations, suggesting that the short-range SW potential might be inadequate for describing the phonon anharmonicity in SLBP.

Diffusive Phonons in Nongray Nanostructures

Nanostructured semiconducting materials are promising candidates for thermoelectrics (TEs) due to their potential to suppress phonon transport while preserving electrical properties. Modeling phonon-boundary scattering in complex geometries is crucial for predicting materials with high conversion efficiency. However, the simultaneous presence of ballistic and diffusive phonons challenges the development of models that are both accurate and computationally tractable. Using the recently developed first-principles Boltzmann transport equation (BTE) approach, we investigate diffusive phonons in nanomaterials with wide mean-free-path (MFP) distributions. First, we derive the short MFP limit of the suppression function, showing that it does not necessarily recover the value predicted by standard diffusive transport, challenging previous assumptions. Second, we identify a Robin type boundary condition describing diffuse surfaces within Fourier's law, extending the validity of diffusive heat transport in terms of Knudsen numbers. Finally, we use this result to develop a hybrid Fourier/BTE approach to model realistic materials, obtaining good agreement with experiments. These results provide insight on thermal transport in materials that are within experimental reach and open opportunities for large-scale screening of nanostructured TE materials.

Investigation of heat transport across Ge/Si interface using an enhanced ballistic-diffusive model

In this paper, we investigate the heat transport across GermaniumSilicon interface using an enhanced ballistic-diffusive equation (EBDE), where we introduce the temperature jump boundary condition coupled with a thermal boundary resistance (TBR) and an effective thermal conductivity model. This paper focuses on the thermal transport of sub-23 nm Ge/Si thin films. The present model is aimed to describe the ballistic-diffusive phonon transport across Ge/Si interface. We have found that the temperature jump occurs in the interface due to phonon-boundary interactions. In addition, the interfacial heat transport is influenced by the surface roughness effect. The prediction of the suggested EBDE model are in good accordance with analytical method reported in the literature. The proposed model shows excellent agreement with the phonon Boltzmann transport equation (BTE) approach and Monte Carlo simulation (MC). Further, the analytic model for the effective thermal conductivity (ETC) is in strong agreement with experimentally based approach and also the theoretical model.

Thermal conductivity of hexagonal Si, Ge, and Si1-xGex alloys from first-principles

Hexagonal Si and Ge with a lonsdaleite crystal structure are allotropes of silicon and germanium that have recently been synthesized. These materials as well as their alloys are promising candidates for novel applications in optoelectronics. In this paper, we systematically study the phonon transport and thermal conductivity of hexagonal Si, Ge, and their alloys by using the first-principle-based Peierls-Boltzmann transport equation approach. Both three-phonon and four-phonon scatterings are taken into account in the calculations as the phonon scattering mechanisms. The thermal conductivity anisotropy of these materials is identified. While the thermal conductivity parallel to the hexagonal plane for hexagonal Si and Ge is found to be larger than that perpendicular to the hexagonal plane, alloying effectively tunes the thermal conductivity anisotropy by suppressing the thermal conductivity contributions from the middle-frequency phonons. The importance of four-phonon scatterings is assessed by comparing the results with the calculations without including four-phonon scatterings. We find that four-phonon scatterings cannot be ignored in hexagonal Si and Ge as the thermal conductivity would be overestimated by around 10% (40%) at 300 K (900) K. In addition, the phonon mean free path distribution of hexagonal Si, Ge, and their alloys is also discussed.