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6CCVD Research

Dependence of Heat Transport in Solids on Length-Scale, Pressure, and Temperature - Implications for Mechanisms and Thermodynamics

At a Glance

Section titled “At a Glance”
MetadataDetails
Publication Date2021-01-18
JournalMaterials
AuthorsAnne M. Hofmeister
InstitutionsPlanetary Science Institute
Citations10
AnalysisFull AI Review Included

Dependence of Heat Transport in Solids: Analysis for Engineers

Section titled “Dependence of Heat Transport in Solids: Analysis for Engineers”

Executive Summary

Section titled “Executive Summary”

This research fundamentally re-evaluates the mechanism of heat transport in bulk solids, moving beyond traditional phonon-based models by demonstrating the critical role of radiative diffusion (light transport) across diverse materials.

  • Length-Scale Dependence Confirmed: Thermal Diffusivity (D) is not a material constant but depends strongly on sample thickness (L) for bulk solids (0.03 to 10 mm), following the universal relationship D(L) = D∞[1 - exp(-bL)].
  • Radiative Diffusion Mechanism: The observed length dependence and the temperature behavior (D increasing linearly with T at high temperatures) confirm that heat conduction is governed by the diffusion of low-frequency infrared light down the thermal gradient.
  • Universal Temperature Law: A 3-parameter empirical formula, D(T) = F* (298/T)G + HT, accurately fits D(T) data for insulators, metals, and semiconductors above 298 K, providing a unified description of thermal behavior.
  • New Thermodynamic Constraint: A novel thermodynamic identity was derived: the logarithmic pressure derivative of specific heat equals the negative of linear compressibility (dln(cp)/dP = -ÎČL). This relationship is consistent with light diffusion, as compression alters the space occupied by matter but not by light.
  • DAC Data Reliability Issues: High-pressure measurements using ultra-thin diamond anvil cell (DAC) samples (”m scale) are shown to be highly problematic, as ballistic radiative transfer (direct light transmission) dominates over true diffusive heat conduction.

Technical Specifications

Section titled “Technical Specifications”

The following table summarizes key quantitative findings and experimental parameters derived from Laser Flash Analysis (LFA) and high-pressure data analysis.

ParameterValueUnitContext
LFA Sample Thickness (L)0.3 to 15mmRange for bulk material studies
LFA Measurement Accuracy±2%Achieved by contact-free, transient method
Universal D(L) Fit FormD(L) = D∞[1 - exp(-bL)]mm2 s-1Applies to all materials (insulators, metals, Si) at 298 K
Attenuation Parameter (b)6.32 D∞-0.48mm-1Empirical correlation for insulators and metals (R=0.99)
High-T D(T) Fit FormD(T) = F* (298/T)G + HTmm2 s-1Universal 3-parameter fit for T > 298 K
Power Law Exponent (G) Range0.3 to 2DimensionlessVaries based on material structure (e.g., metals G ≈ 0)
Radiative Term Coefficient (H)~0.001K-1Typical value for insulators, controlling high-T D upturn
Bridgman’s Parameter (g)~7.3DimensionlessUniversal value derived from dln(Îș)/dP for diverse solids
DAC Foil Thickness0.1 to 4”mUsed in high-pressure studies; prone to ballistic error
New Thermodynamic Identitydln(cp)/dP = -ÎČL% GPa-1Relates specific heat change to linear compressibility

Key Methodologies

Section titled “Key Methodologies”

The study relies primarily on the transient Laser Flash Analysis (LFA) technique, modified to ensure accurate measurement of diffusive heat transport by eliminating systematic errors.

  1. LFA Setup and Conditions: Experiments utilized a controlled atmosphere furnace, a high-energy pulsed laser, and an IR detector. Samples were thin, flat slabs (0.3 to 15 mm) held contact-free by their edges to prevent thermal losses.
  2. Surface Preparation: Top and bottom surfaces were coated with graphite (less than 1 ”m thick). The bottom coat converts the narrow laser pulse into a broad blackbody spectrum for absorption, while the top coat enhances IR emissions for detection.
  3. Ballistic Transfer Mitigation: Spurious fast (ballistic) radiative transfer, which is enhanced in thin samples, was removed from the raw temperature-time (T-t) curves by applying advanced mathematical models (Cowan and Blumm et al. models) during data fitting.
  4. Data Analysis (D Calculation): Thermal diffusivity (D) was calculated using the time (t1/2) required for the rear surface to reach half its maximum temperature, adjusted for external radiative cooling (Cowan’s parameter φ).
  5. High-Pressure Data Selection: Only reliable low-pressure data (less than 2 GPa) collected on thick, mm-scale samples (piston-cylinder or multi-anvil devices) were used to extract pressure derivatives (dln(Îș)/dP), as these lengths minimize the L-dependence and ballistic errors.
  6. Mechanism Confirmation: The derived D(L,T) and Îș(P) relationships were compared against a radiative diffusion model, which uses the material’s absorption coefficient A(Îœ,T) as the key parameter, confirming that heat transfer is analogous to light diffusion.

Commercial Applications

Section titled “Commercial Applications”

The findings have significant implications for thermal management and material selection across several high-performance engineering sectors, particularly those dealing with thin films and extreme environments.

  • Thermal Management in Microelectronics:
    • The strong D(L) dependence means that bulk thermal properties are irrelevant for thin films (L < 1 mm). Accurate thermal modeling of microprocessors, power devices, and thermal interface materials (TIMs) must incorporate the length-scale attenuation parameter (b).
  • High-Temperature Systems (Aerospace/Energy):
    • For systems operating above 500 K, the intrinsic radiative component (HT term) dominates heat transfer. Material selection must prioritize low IR absorption coefficients (A(Îœ,T)) in the relevant frequency range, not just low phonon scattering rates.
  • CVD Diamond and Wide Bandgap Materials (Relevant to 6ccvd.com):
    • Materials with extremely high thermal diffusivity (like diamond, SiC, and GaN) exhibit strong ballistic effects. The LFA methodology developed here is essential for obtaining accurate, intrinsic D values for these materials, which are critical for high-power RF and high-frequency electronics.
  • Thermal Barrier Coatings (TBCs):
    • The D(L) formula provides a quantitative basis for designing TBCs (e.g., Yttrium-Stabilized Zirconia, YSZ) where thickness is a primary design variable for controlling heat flow.
  • High-Pressure Industrial Processes:
    • The new thermodynamic relationship provides engineers with a reliable method to predict changes in heat capacity (cp) under extreme compression, improving the design and safety of high-pressure synthesis and processing equipment.
View Original Abstract

Accurate laser-flash measurements of thermal diffusivity (D) of diverse bulk solids at moderate temperature (T), with thickness L of ~0.03 to 10 mm, reveal that D(T) = D∞(T)[1 − exp(−bL)]. When L is several mm, D∞(T) = FT−G + HT, where F is constant, G is ~1 or 0, and H (for insulators) is ~0.001. The attenuation parameter b = 6.19D∞−0.477 at 298 K for electrical insulators, elements, and alloys. Dimensional analysis confirms that D → 0 as L → 0, which is consistent with heat diffusion, requiring a medium. Thermal conductivity (Îș) behaves similarly, being proportional to D. Attenuation describing heat conduction signifies that light is the diffusing entity in solids. A radiative transfer model with 1 free parameter that represents a simplified absorption coefficient describes the complex form for Îș(T) of solids, including its strong peak at cryogenic temperatures. Three parameters describe Îș with a secondary peak and/or a high-T increase. The strong length dependence and experimental difficulties in diamond anvil studies have yielded problematic transport properties. Reliable low-pressure data on diverse thick samples reveal a new thermodynamic formula for specific heat (∂ln(cP)/∂P = −linear compressibility), which leads to ∂ln(Îș)/∂P = linear compressibility + ∂lnα/∂P, where α is thermal expansivity. These formulae support that heat conduction in solids equals diffusion of light down the thermal gradient, since changing P alters the space occupied by matter, but not by light.

Tech Support

Section titled “Tech Support”

Original Source

Section titled “Original Source”

References

Section titled “References”
  1. 2007 - Thermal conductivity of the Earth [Crossref]
  2. 1952 - Transmission of radiation through glass in tank furnaces
  3. 1965 - The effect of isotopes on lattice heat conduction. I. Lithium fluoride [Crossref]
  4. 1998 - Thermal conductivity and ballistic-phonon transport in the cross-plane direction of superlattices [Crossref]
  5. 2012 - Determining phonon mean free paths from observations of quasiballistic thermal transport [Crossref]
  6. 2015 - Experimental evidence of very long intrinsic phonon mean free path along the c-axis of graphite [Crossref]
  7. 2017 - Isolating lattice from electronic contributions in thermal transport measurements of metals and alloys and a new model [Crossref]
  8. 2014 - Thermal diffusivity of electrical insulators at high temperatures: Evidence for diffusion of phonon-polaritons at infrared frequencies augmenting phonon heat conduction [Crossref]

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