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

At a Glance

Metadata	Details
Publication Date	2021-01-18
Journal	Materials
Authors	Anne M. Hofmeister
Institutions	Planetary Science Institute
Citations	10
Analysis	Full AI Review Included

Technical Documentation & Analysis: Advanced Heat Transport in Solids

Reference: Hofmeister, A.M. Dependence of Heat Transport in Solids on Length-Scale, Pressure, and Temperature: Implications for Mechanisms and Thermodynamics. Materials 2021, 14, 449.

Executive Summary

This research provides critical insights into the fundamental mechanisms of heat transport, directly impacting the engineering and selection of high-performance thermal materials, such as MPCVD diamond.

Mechanism Confirmation: The study confirms that heat transport in solids (insulators, semiconductors, and metals) is fundamentally governed by the diffusion of low-frequency infrared (IR) light (radiative diffusion), rather than traditional phonon-phonon scattering models.
Length-Scale Dependence: Thermal diffusivity ($D$) and thermal conductivity ($\kappa$) are proven to be strongly dependent on sample thickness ($L$), following the relationship $D(L,T) = D_{\infty}[1 - \exp(-bL)]$. This necessitates precise control over material dimensions for accurate characterization and application.
High-D Material Requirements: High-diffusivity materials (like MgO, Si, and by extension, diamond) exhibit the strongest length dependence, requiring thin, high-precision samples (sub-mm) to isolate intrinsic properties and model ballistic transfer effects.
Methodological Rigor: Accurate measurements rely on advanced Laser Flash Analysis (LFA) techniques combined with sophisticated models (Cowan, Blumm) to successfully remove spurious ballistic radiative transfer, especially in thin films and high-temperature regimes.
High-Pressure Challenges: High-pressure Diamond Anvil Cell (DAC) studies using ultra-thin metal foils ($\sim 4$ µm) are shown to be highly problematic, with rear-surface heating often resulting from direct laser light penetration rather than heat diffusion, underscoring the difficulty of transport measurement at the micro-scale.
Thermodynamic Link: New thermodynamic formulae are derived, linking the pressure dependence of specific heat to linear compressibility, which supports the model of heat conduction as the diffusion of light.

Technical Specifications

The following hard data points and relationships were extracted from the analysis of diverse bulk solids and thin films:

Parameter	Value	Unit	Context
Universal $D(L,T)$ Fit (Thick $L$)	$D(T) = F T^{-G} + H T$	mm$^{2}$ s$^{-1}$	Applies to thick non-metallic single crystals ($L > 1$ mm).
Attenuation Parameter $b$ (298 K)	$6.19 D_{\infty}^{-0.477}$	mm$^{-1}$	General fit for insulators, elements, and alloys.
Thermal Diffusivity $D_{\infty}$ (MgO)	18.867	mm$^{2}$ s$^{-1}$	Asymptotic value for $L \to \infty$ at 298 K.
Attenuation $b$ (MgO)	1.8394	mm$^{-1}$	Attenuation parameter for MgO at 298 K.
Thermal Diffusivity $D_{\infty}$ (Si)	109.02	mm$^{2}$ s$^{-1}$	Asymptotic value for Si at 298 K.
Attenuation $b$ (Si)	0.64621	mm$^{-1}$	Attenuation parameter for Si at 298 K.
LFA Sample Thickness Range	0.03 to 15	mm	Range of bulk samples studied.
DAC Foil Thickness	$\sim 0.1$ to $\sim 4$	µm	Ultra-thin films used in high-pressure experiments (Fe, Pt, Ir).
LFA Measurement Accuracy	$\pm 2$	%	Achieved accuracy due to quantifying length and time, not heat input.
Radiative Cooling Parameter ($\phi$)	$0$ to $10$	Dimensionless	Used in Cowan’s model to fit $T-t$ curves (Figure 2a).
Bridgman’s Parameter ($g$)	$\sim 7.3$	Dimensionless	Universal value for diverse solids, suggesting a universal heat transfer mechanism.

Key Methodologies

The core experimental technique utilized was Laser Flash Analysis (LFA), supplemented by analysis of high-pressure Diamond Anvil Cell (DAC) data.

Sample Preparation: Samples were prepared as small, thin slabs (0.3 mm to 15 mm thick) with highly parallel faces to ensure one-dimensional heat flow.
Surface Coating: Graphite coatings (<1 µm thickness) were applied to the front surface (for laser absorption and blackbody spectrum generation) and the rear surface (to enhance emissions detection by the IR detector).
LFA Measurement: A short, high-energy laser pulse was applied to the front surface. The temperature rise ($T$) over time ($t$) on the rear surface was recorded remotely using an IR detector.
Adiabatic Modeling (Parker): Initial analysis used the adiabatic model $D = 0.138785 (L^{2} / t_{1/2})$, where $t_{1/2}$ is the time to reach half the maximum temperature.
External Cooling Correction (Cowan): The $T-t$ curves were fitted using Cowan’s model (Equation 16) to account for external radiative heat loss to the surroundings, parameterized by the cooling parameter $\phi$.
Internal Ballistic Correction (Blumm/Hahn): For semi-transparent materials, an extended model (Equation 20) was applied to decouple the slow diffusive heat flux ($I_{dif}$) from the fast, spurious ballistic radiative transfer flux ($I_{bal}$), ensuring the measured $D$ reflects true diffusion.
High-Pressure Analysis: DAC experiments utilized ultra-thin metal foils (e.g., Ir, Fe, Pt) embedded in insulators (e.g., MgO). Heating was achieved via continuous wave or pulsed lasers, with temperature derived from fitting visible/IR emissions.

6CCVD Solutions & Capabilities

The findings of Hofmeister (2021) underscore the critical role of material purity, precise geometry, and controlled surface interactions in accurately characterizing and utilizing high-performance thermal materials. Diamond, possessing the highest known thermal diffusivity, is the ultimate material for extending this research. 6CCVD is uniquely positioned to supply the necessary specialized diamond products.

Applicable Materials for Advanced Thermal Transport Studies

To replicate or extend the high-D transport studies discussed (e.g., surpassing the performance of Si and MgO), 6CCVD recommends the following MPCVD diamond materials:

Material Grade	Application Focus	6CCVD Advantage
Optical Grade Single Crystal Diamond (SCD)	Fundamental physics studies, high-D characterization, DAC experiments.	Highest purity, lowest defect density, enabling intrinsic $D$ measurements far exceeding Si or MgO. Ideal for studying the radiative diffusion mechanism.
High Purity Polycrystalline Diamond (PCD)	Large-area thermal management, bulk $L$-dependence studies, high-power electronics.	Wafers up to 125 mm in diameter, allowing for large $L$ measurements necessary to define the asymptotic $D_{\infty}$ parameter (Equation 22).
Boron-Doped Diamond (BDD)	Studies involving electronic transport (as discussed for metals) or electrochemical applications.	Customizable doping levels to tune electrical conductivity, enabling simultaneous study of thermal and electrical transport properties under pressure.

Customization Potential for Research Replication

The paper highlights the necessity of precise thickness control ($L$) and specific surface treatments (coatings) to isolate diffusive transport and model ballistic effects. 6CCVD offers full customization to meet these stringent requirements:

Precision Thickness Control: We provide SCD and PCD plates/wafers with thickness control from 0.1 µm up to 500 µm (wafers) and 10 mm (substrates). This range covers both the bulk LFA regime (mm scale) and the ultra-thin film regime (µm scale) relevant to DAC studies.
Ultra-Low Roughness Polishing: Accurate LFA relies on parallel ray geometry. Our polishing capabilities ensure exceptional surface quality:
- SCD: Roughness Ra < 1 nm.
- PCD: Roughness Ra < 5 nm (for inch-size wafers).
Custom Metalization Services: The paper utilized thin metal foils (Ir, Pt, Fe) and graphite coatings. 6CCVD offers in-house metalization capabilities (Au, Pt, Pd, Ti, W, Cu) to:
- Replicate the thin metal layers required for high-pressure DAC experiments.
- Provide highly controlled, uniform blackbody absorbing/emitting layers for LFA, replacing or supplementing traditional graphite coatings.

Engineering Support

The complex interplay between length-scale physics, radiative diffusion, and thermodynamic properties requires specialized knowledge. 6CCVD’s in-house PhD team provides authoritative professional support:

Material Selection Consultation: Assistance in selecting the optimal diamond grade (SCD vs. PCD) and orientation for Thermal Transport Studies and High-Pressure Physics projects.
Design Optimization: Guidance on specifying precise thickness ($L$) and surface preparation (polishing, metalization) to minimize experimental artifacts (e.g., ballistic transfer) and ensure data validity according to LFA models.
Global Logistics: Reliable global shipping (DDU default, DDP available) ensures timely delivery of custom diamond components worldwide.

For custom specifications or material consultation, visit 6ccvd.com or contact our engineering team directly.

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 (κ) 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 κ(T) of solids, including its strong peak at cryogenic temperatures. Three parameters describe κ 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(κ)/∂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

Original Source

DOI: https://doi.org/10.3390/ma14020449

References

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