Generalization of Fourier’s Law into Viscous Heat Equations
At a Glance
Section titled “At a Glance”| Metadata | Details |
|---|---|
| Publication Date | 2020-01-28 |
| Journal | Physical Review X |
| Authors | Michele Simoncelli, Nicola Marzari, Andrea Cepellotti, Michele Simoncelli, Nicola Marzari |
| Institutions | Laboratoire de Chimie Théorique, École Polytechnique Fédérale de Lausanne |
| Citations | 49 |
| Analysis | Full AI Review Included |
Executive Summary
Section titled “Executive Summary”This analysis details the generalization of Fourier’s law into a set of “Viscous Heat Equations,” providing a rigorous mesoscopic framework for modeling thermal transport in dielectric crystals across diffusive (Fourier) and hydrodynamic (second sound, Poiseuille flow) regimes.
- Generalized Heat Transport: Derived two novel coupled partial differential equations (PDEs) for temperature and phonon drift velocity, representing the thermal counterpart of the Navier-Stokes equations for fluids.
- Thermal Viscosity Defined: Introduced thermal viscosity (µ) as a fundamental transport coefficient, determined microscopically by the contributions of even-parity relaxons (collective phonon excitations). This complements thermal conductivity, which is determined by odd-parity relaxons.
- Fourier Deviation Number (FDN): Defined a dimensionless parameter (FDN) that quantifies the deviation from standard Fourier’s law, accurately predicting the temperature and size window where hydrodynamic effects emerge.
- Graphite Validation: The model’s predictions for the hydrodynamic window in graphite (60-85 K, 10-20 µm) show excellent agreement with recent pioneering experimental observations of second sound.
- Diamond Prediction: Predicts that hydrodynamic behavior (non-diffusive heat flow) can be observed in micrometer-sized diamond crystals at non-cryogenic, room temperatures (300 K).
- Computational Efficiency: The mesoscopic viscous heat equations offer a computationally simpler alternative to solving the full microscopic Boltzmann Transport Equation (LBTE) while retaining accuracy in the hydrodynamic regime.
Technical Specifications
Section titled “Technical Specifications”Key transport coefficients and characteristic parameters for bulk materials at 300 K, derived from first-principles calculations:
| Parameter | Value | Unit | Context |
|---|---|---|---|
| Thermal Viscosity (µiiii) | 0.003368 | Pa·s | Bulk Graphite (in-plane) |
| Thermal Viscosity (µiiii) | 0.002107 | Pa·s | Bulk Diamond |
| Thermal Viscosity (µiiii) | 0.000774 | Pa·s | Bulk Silicon |
| Thermal Conductivity (κii) | 1993.79 | W/mK | Bulk Graphite (in-plane) |
| Thermal Conductivity (κii) | 2774.99 | W/mK | Bulk Diamond |
| Thermal Conductivity (κii) | 145.71 | W/mK | Bulk Silicon |
| Graphite Hydrodynamic Window | 60 - 85 | K | Temperature range for maximum FDN (10 µm sample). |
| Diamond Hydrodynamic Prediction | Room Temperature | N/A | Predicted for micrometer-sized crystals (size >= 1 µm). |
| Specific Heat (C) (Graphite) | 6.513 | J/cm3K | Bulk, 300 K |
| Specific Momentum (A) (Graphite) | 4.965 x 10-3 | J·s/cm3 | Bulk, 300 K |
Key Methodologies
Section titled “Key Methodologies”The viscous heat equations and associated parameters were derived using a combination of first-principles calculations and advanced kinetic theory:
- First-Principles Input: Second- and third-order interatomic force constants were calculated using the Quantum ESPRESSO distribution (LDA functional) to accurately model structural, vibrational, and anharmonic properties of graphite, diamond, and silicon.
- Relaxons and LBTE Solution: The linearized Boltzmann Transport Equation (LBTE) for phonons was solved exactly via diagonalization of the scattering matrix, yielding relaxons (eigenvectors).
- Coefficient Definition: Thermal conductivity (κ) was linked to odd relaxons, while the newly defined thermal viscosity (µ) was linked to even relaxons, highlighting their complementary roles in energy and crystal momentum transport.
- Coarse-Graining and PDE Derivation: The microscopic LBTE was coarse-grained by projecting it onto the subspaces spanned by the conserved quantities (energy and crystal momentum), resulting in two coupled mesoscopic PDEs for temperature (T) and drift velocity (u).
- Boundary Conditions: Numerical simulations imposed fixed temperature (Dirichlet) and zero total heat flux (adiabatic) boundaries, along with a “no-slip” condition (zero drift velocity) on all surfaces.
- Size Effects Approximation: Finite-size and grain-size effects (e.g., for 10 µm polycrystalline graphite) were incorporated using Matthiessen’s rule, combining bulk transport coefficients with their ballistic limits.
Commercial Applications
Section titled “Commercial Applications”This advanced thermal modeling framework is crucial for engineering high-performance materials and devices where traditional Fourier modeling fails.
- High-Performance Thermal Management: Essential for designing efficient heat spreaders and sinks using materials like synthetic diamond (CVD diamond) and graphite, especially in high-power density microchips and 5G/RF electronics.
- Phonon Engineering and Metamaterials: Enables the predictive design of thermal devices (e.g., thermal rectifiers, thermal transistors) that rely on non-diffusive, coherent heat transport phenomena (second sound).
- Micro- and Nanoscale Devices: Critical for modeling heat flow in devices where characteristic lengths are comparable to phonon mean free paths (e.g., carbon nanotubes, graphene interconnects, and micrometer-sized diamond sensors).
- Cryogenic and Quantum Technology: Provides the necessary accuracy to model thermal stability and dissipation in quantum computing and sensing components that operate at extremely low temperatures, where hydrodynamic phonon flow is often dominant.
- Advanced Material Synthesis (CVD): The ability to predict hydrodynamic behavior based on grain size and temperature aids in optimizing the synthesis parameters (e.g., in CVD diamond growth) to achieve specific thermal properties.
View Original Abstract
Heat conduction in dielectric crystals originates from the propagation of\natomic vibrations, whose microscopic dynamics is well described by the\nlinearized phonon Boltzmann transport equation. Recently, it was shown that\nthermal conductivity can be resolved exactly and in a closed form as a sum over\nrelaxons, $\mathit{i.e.}$ collective phonon excitations that are the\neigenvectors of Boltzmann equation’s scattering matrix [Cepellotti and Marzari,\nPRX $\mathbf{6}$ (2016)]. Relaxons have a well-defined parity, and only odd\nrelaxons contribute to the thermal conductivity. Here, we show that the\ncomplementary set of even relaxons determines another quantity --- the thermal\nviscosity --- that enters into the description of heat transport, and is\nespecially relevant in the hydrodynamic regime, where dissipation of crystal\nmomentum by Umklapp scattering phases out. We also show how the thermal\nconductivity and viscosity parametrize two novel viscous heat equations --- two\ncoupled equations for the temperature and drift-velocity fields --- which\nrepresent the thermal counterpart of the Navier-Stokes equations of\nhydrodynamics in the linear, laminar regime. These viscous heat equations are\nderived from a coarse-graining of the linearized Boltzmann transport equation\nfor phonons, and encompass both limits of Fourier’s law and of second sound,\ntaking place, respectively, in the regimes of strong or weak momentum\ndissipation. Last, we introduce the Fourier deviation number as a descriptor\nthat captures the deviations from Fourier’s law due to hydrodynamic effects. We\nshowcase these findings in a test case of a complex-shaped device made of\ngraphite, obtaining a remarkable agreement with the very recent experimental\ndemonstration of hydrodynamic transport in this material. The present findings\nalso suggest that hydrodynamic behavior can appear at room temperature in\nmicrometer-sized diamond crystals.\n
Tech Support
Section titled “Tech Support”Original Source
Section titled “Original Source”References
Section titled “References”- 1955 - Quantum Theory of Solids
- 1960 - Electrons and Phonons: The Theory of Transport Phenomena in Solids