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 | Lawrence Berkeley National Laboratory, Laboratoire de Chimie Théorique |
| Citations | 49 |
| Analysis | Full AI Review Included |
Technical Documentation & Analysis: Viscous Heat Transport in MPCVD Diamond
Section titled “Technical Documentation & Analysis: Viscous Heat Transport in MPCVD Diamond”Executive Summary
Section titled “Executive Summary”This research introduces a rigorous framework generalizing Fourier’s law into coupled Viscous Heat Equations (VHE), derived from the linearized Boltzmann Transport Equation (LBTE). This methodology is critical for accurately modeling heat transport in the hydrodynamic regime, where traditional diffusive models fail.
Key findings and implications for 6CCVD’s clients include:
- Thermal Viscosity Defined: The VHE framework introduces thermal viscosity ($\mu$) as a complementary material property to thermal conductivity ($\kappa$), determined by the even-parity relaxon spectrum.
- Diamond Hydrodynamics at Room Temperature: The study predicts that hydrodynamic thermal transport (e.g., second sound, Poiseuille flow) can emerge in micrometer-sized diamond crystals (SCD/PCD) at room temperature (~300 K).
- Critical Length Scales: Hydrodynamic effects are maximized in diamond samples with characteristic lengths ($l_{TOT}$) of 1 µm to 10 µm, requiring precise material fabrication.
- Fourier Deviation Number (FDN): A dimensionless parameter (FDN) is introduced to quantify the deviation from Fourier’s law, providing a low-computational-cost method to identify the hydrodynamic temperature and size window.
- 6CCVD Material Relevance: Experimental validation of these predictions requires ultra-high quality, precisely dimensioned Single Crystal Diamond (SCD) and Polycrystalline Diamond (PCD) materials, which are core offerings of 6CCVD.
Technical Specifications
Section titled “Technical Specifications”The paper provides detailed material parameters derived from first-principles calculations (LBTE) for Single Crystal Diamond (SCD). The following data points, extracted from Table II and Figure 1b (300 K), define the material properties essential for modeling VHE in the hydrodynamic regime:
| Parameter | Value (300 K) | Unit | Context |
|---|---|---|---|
| Thermal Conductivity ($\kappa$) | ~2775 | W / (m K) | Estimated from Fig 1b inset (Isotropic) |
| Thermal Viscosity Component ($\mu^{iiii}$) | 5.725e-03 | Pa·s | Bulk, Isotropic component |
| Specific Heat ($C$) | 2.878e+00 | J / (cm3 K) | Specific heat component |
| Momentum Dissipation Rate ($D_U$) | 2.689e+01 | ns-1 | Inverse timescale |
| Specific Momentum ($A$) | 2.072e-03 | J / (cm3 K) | Specific momentum component |
| Hydrodynamic Window (Size) | 1 - 10 | µm | Predicted largest FDN range |
| Polishing Requirement (SCD) | Ra < 1 | nm | Required for minimizing boundary scattering |
Key Methodologies
Section titled “Key Methodologies”The theoretical framework relies on advanced computational physics to derive macroscopic transport equations from microscopic phonon dynamics.
- Linearized Boltzmann Transport Equation (LBTE): The foundation for describing phonon wavepacket propagation and scattering in dielectric crystals.
- Relaxons and Parity: The LBTE scattering matrix is diagonalized using relaxons (eigenvectors).
- Odd relaxons determine thermal conductivity ($\kappa$).
- Even relaxons determine thermal viscosity ($\mu$).
- Coarse-Graining Derivation: The microscopic LBTE is coarse-grained into two coupled mesoscopic VHEs for the local temperature ($T$) and phonon drift velocity ($u$).
- First-Principles Parameterization: Material parameters ($\kappa$, $\mu$, $C$, $D_U$) are computed from first-principles (DFT/LDA) using the D3Q code, ensuring accuracy without empirical fitting.
- Finite-Size Effects: Boundary scattering effects, crucial for micrometer-sized samples, are approximated using Matthiessen’s rule, combining bulk properties with a ballistic limit for a characteristic sample size ($L_s$).
- Fourier Deviation Number (FDN): A dimensionless parameter is calculated based on the ratio of momentum dissipation ($\pi_3$) and viscous coupling ($\pi_1, \pi_2$) to quickly map the temperature and size window where hydrodynamic effects dominate ($\text{FDN} \ge 0.1$).
6CCVD Solutions & Capabilities
Section titled “6CCVD Solutions & Capabilities”The research explicitly identifies high-quality diamond in the micrometer regime as the ideal platform for experimentally validating the VHE framework and observing room-temperature phonon hydrodynamics. 6CCVD is uniquely positioned to supply the necessary materials and engineering support.
Applicable Materials
Section titled “Applicable Materials”To replicate or extend the hydrodynamic transport research in diamond, 6CCVD recommends the following materials:
- Optical Grade Single Crystal Diamond (SCD): Essential for achieving the lowest possible Umklapp scattering (high intrinsic quality) required for the hydrodynamic regime. SCD offers superior purity and crystal perfection, minimizing momentum dissipation ($D_U$).
- High-Quality Polycrystalline Diamond (PCD): For studies requiring larger lateral dimensions (up to 125mm) or specific grain boundary effects (which influence $L_s$ in the Matthiessen approximation). 6CCVD PCD offers high thermal performance suitable for mesoscopic transport studies.
Customization Potential
Section titled “Customization Potential”Experimental validation of micrometer-scale hydrodynamic phenomena requires materials with extreme precision in dimension and surface quality, directly matching 6CCVD’s core capabilities:
| Research Requirement | 6CCVD Capability | Technical Specification |
|---|---|---|
| Micrometer Thickness | Precise thickness control | SCD/PCD wafers from 0.1 µm to 500 µm |
| Custom Device Geometry | Advanced laser cutting/shaping | Custom plates/wafers up to 125 mm (PCD) |
| Ultra-Smooth Surfaces | High-precision polishing | Ra < 1 nm (SCD), Ra < 5 nm (PCD) |
| Boundary Control | Custom metalization | Internal capability for Au, Pt, Pd, Ti, W, Cu layers for thermal boundary engineering or device integration. |
Engineering Support
Section titled “Engineering Support”The VHE framework and FDN analysis rely on complex material parameters ($\kappa$, $\mu$, $D_U$, $C$) that are highly sensitive to temperature and size. 6CCVD’s in-house team of PhD material scientists specializes in the growth and characterization of MPCVD diamond.
We offer comprehensive engineering support for projects focused on:
- Material Selection: Assisting researchers in selecting the optimal SCD or PCD grade to minimize Umklapp scattering and maximize the hydrodynamic window.
- Dimensioning: Consulting on characteristic length scales ($L_{TOT}$) and geometry required to achieve a target FDN for experimental observation of second sound or Poiseuille flow.
- Device Integration: Providing metalization and polishing services necessary for integrating diamond into complex, micrometer-sized phononic devices.
For custom specifications or material consultation, visit 6ccvd.com or contact our engineering team directly.
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