Origin of effective non-Fourier heat conduction phenomena in heterogeneous materials

Abstract: Phenomenological models of non-Fourier heat conduction often lack a strict microstructural foundation, leading to ambiguities when modeling complex heterogeneous materials. In this study, we derive a continuum heat equation beyond Fourier's law using spatial volume averaging for a two-component system. We analytically prove that the experimentally observed static and dynamic thermal diffusivity arise directly from the distinct material properties, concluding that heterogeneous media are inherently over-diffusive. The resulting heat equation is thermodynamically compatible, and the microstructural origin allows the calculation of non-Fourier transport coefficients. Furthermore, we demonstrate that finite-sample boundaries introduce higher-order spatial non-localities, thereby explaining the size dependence of over-diffusion. We validate the model against experimental data across metal and carbon foams, rocks, and metal-organic frameworks.

• The paper derives a generalized Jeffreys heat-flux equation from two Fourier-conducting phases using volume averaging, with relaxation times and non-local terms determined by constituent properties and interfacial heat exchange.
• Heterogeneous materials are inherently over-diffusive, with predicted intrinsic diffusivity ratios ranging from 1.27 in air-filled aluminum foam to 16.1 in water-filled aluminum foam, provided boundary conditions reveal the effect.
• The model links finite-sample size to apparent diffusivity, predicts that higher-order spatial terms can contribute 25–35% during early transients, and imposes a second-law bound on the non-local length scale.

Overview and motivation

This paper by R. Kovács addresses a long-standing ambiguity in the modeling of heat conduction in heterogeneous media: phenomenological non-Fourier models (Jeffreys, Guyer–Krumhansl, dual-phase-lag) are routinely fitted to experimental data on foams, rocks, and composites without a microstructural justification. The central contribution is a rigorous derivation of an effective macroscopic heat equation from first principles, using spatial volume averaging over a representative volume element (RVE) applied to a two-component system in which each phase individually obeys Fourier's law. The key result is that the experimentally observed static and dynamic thermal diffusivities emerge analytically from the distinct thermophysical properties of the constituents, and that heterogeneous media are inherently over-diffusive — a strong claim with direct consequences for how flash experiments on porous materials should be interpreted.

The framework is local thermal non-equilibrium (LTNE): two coupled internal energy balances for phases $\alpha$ (solid matrix) and $\beta$ (saturating fluid), each conducting per Fourier's law and exchanging heat through a volumetric interfacial coefficient $H$. The paper departs from classical LTNE by allowing the interfacial exchange to be spatially non-local when scale separation between RVE size $L$ and macroscopic length $L_{\mathrm{mac}}$ breaks down.

Derivation of the effective equation

Starting from microscopic Fourier conduction in both rigid, stationary phases, the author applies Gray's decomposition $$T_{\alpha\mathrm{m}} = T_\alpha + \tilde{T}_{\alpha\mathrm{m}}$$, the spatial averaging theorem, and the standard closure $$\tilde{T}_{\alpha\mathrm{m}} = \mathbf{b}_\alpha \cdot \nabla T_\alpha$$ to obtain phase-averaged fluxes. Summing the phase balances yields the standard mixture energy balance for the heat-capacity-weighted temperature $T$, while subtracting them yields an evolution equation for the inter-phase temperature difference $\delta T = T_\alpha - T_\beta$. Eliminating $\delta T$ produces a constitutive relation for the total macroscopic flux:

$\beta$0

a generalized Jeffreys equation with an additional third-order spatial term. The transport coefficients are fully determined by microstructural quantities:

• Flux relaxation time: $\beta$1,
• Gradient relaxation time: $\beta$2,
• Non-locality coefficient: $\beta$3,

where $\beta$4 and $\beta$5. The non-locality enters through a second-order Taylor expansion of the interfacial temperature difference across the distributed interface, introducing a length scale $\beta$6 tied to pore size — this is the mechanism by which finite-sample boundaries generate higher-order spatial terms and hence size-dependent over-diffusion.

Two structural remarks are important. First, because each phase obeys Fourier's law and porosity is uniform, the macroscopic flux is irrotational; consequently the derived equation cannot recover the Guyer–Krumhansl form with its positive $\beta$7 term, which originates in phonon hydrodynamics with rotational flux. The resemblance is purely mathematical. Second, unlike dual-phase-lag constructions — which are known to violate the Clausius–Duhem inequality — no arbitrary Taylor expansion of a lagged flux law is invoked; the timescales arise constructively from the coupled balances.

Thermodynamic admissibility

Thermodynamic compatibility is not postulated but inherited from the microscale. The volume-averaged entropy production, evaluated near equilibrium, reduces to a positive quadratic form whose conductivity matrix must be positive definite. Sylvester's criterion applied to that matrix yields

$\beta$8

which is equivalent to $\beta$9 and is strictly stronger than the condition $H$0. The non-local term in $H$1 contributes negatively to entropy production, so it must be bounded — a concrete, physically interpretable stability constraint absent from phenomenological treatments. This result directly links the sign and magnitude of the $H$2 term to second-law compliance.

Model reduction and term magnitudes

Order-of-magnitude analysis for a typical rock heat-pulse experiment ($H$3 mm, $H$4 s, $H$5) splits the response into two regimes:

Regime	Characteristic scales	Role of $H$6
Early transient ($H$7)	$H$8 mm	~25–35% of leading terms
Late transient ($H$9)	$L$0, $L$1 s	<4%, negligible

The early-time estimate uses the thermal penetration depth rather than sample thickness, avoiding systematic underestimation of higher-order derivatives. The implication is significant: the third-order term, which is invisible in extended irreversible thermodynamics and internal-variable formulations, may be materially relevant during fast transients, and prior experimental evaluations omitting it warrant re-examination.

Relation to the two-temperature model

The $L$2-representation of the derived equation is structurally identical to that obtained by eliminating phase temperatures from the classical 2T model — both yield a damped wave-type equation with a biharmonic correction. The differences lie at the level of the constitutive structure: the present formulation carries the timescale ratio $L$3 explicitly, avoids the physically ambiguous partitioning of boundary heat fluxes between phases (e.g., in a flashed metal foam), and admits systematic non-local extensions via $L$4. Neither model supports true wave propagation despite their mathematical wave-like structure — a limitation the author states plainly.

Over-diffusion and scale dependence

Defining the over-diffusion coefficient $L$5, the derivation gives

$L$6

with $L$7 always. Hence $L$8 identically: any two-component medium with distinct constituent properties is inherently over-diffusive under local exchange. Fourier resonance ($L$9) requires the fine-tuned condition $L_{\mathrm{mac}}$0, which the author treats as mathematically possible but not physically generic. Note also that observing over-diffusion requires appropriate boundary conditions; the coefficient characterizes material timescales only.

For sub-RVE samples, projecting onto the first decaying eigenmode of the finite domain maps $L_{\mathrm{mac}}$1, producing an apparent, thickness-dependent coefficient:

$L_{\mathrm{mac}}$2

which explains the size dependence reported in room-temperature experiments on rocks. Since typical volumetric exchange coefficients are $L_{\mathrm{mac}}$3 W/(m³ K), the correction matters mainly for thin samples — precisely the regime accessible to laser-flash apparatus.

Experimental validation

Five case studies quantify the framework against published data:

Material	$L_{\mathrm{mac}}$4 (W/m K)	$L_{\mathrm{mac}}$5	$L_{\mathrm{mac}}$6 (intrinsic)	$L_{\mathrm{mac}}$7 at thinnest sample
Air-filled Al foam	6.675	2.0 ms	1.27	1.22 (2 mm)
Water-filled Al foam	6.847	18.1 ms	16.1	15.5 (2 mm)
Air-filled carbon foam	10.02	—	1.64	1.51 (2 mm)
Water-saturated limestone	2.12	0.5 s	1.56	1.25 (2 mm)
HKUST-1/RGO pellet	0.42	0.45 s	3.15	1.81 (0.7 mm)

The limestone value agrees with prior measurements, and the MOF/RGO case is particularly instructive: pure MOF pellets behave diffusively, while over-diffusion emerges only in composites — identifying the active two-phase pair as MOF-matrix/RGO rather than solid/air. For the 0.68–1.15 mm pellets actually tested, the predicted apparent coefficient drops substantially below the intrinsic value, quantitatively explaining why small samples suppress the full over-diffusive response. The water-filled aluminum foam prediction ($L_{\mathrm{mac}}$8) has not been experimentally verified; the author flags this as an open test, noting that $L_{\mathrm{mac}}$9 ms is negligible for industrial exchangers but potentially relevant to space applications.

Extensions: biological tissue and radiation

Applying the same upscaling to perfused tissue (with blood advection neglected — an admitted oversimplification) yields a perfusion-augmented over-diffusion coefficient $$T_{\alpha\mathrm{m}} = T_\alpha + \tilde{T}_{\alpha\mathrm{m}}$$0 and modified apparent properties $$T_{\alpha\mathrm{m}} = T_\alpha + \tilde{T}_{\alpha\mathrm{m}}$$1, $$T_{\alpha\mathrm{m}} = T_\alpha + \tilde{T}_{\alpha\mathrm{m}}$$2. Two clinical estimates follow: cancellous bone under radiofrequency ablation gives $$T_{\alpha\mathrm{m}} = T_\alpha + \tilde{T}_{\alpha\mathrm{m}}$$3, implying the initial thermal front propagates more than three times faster than Pennes/Fourier predictions — directly relevant to protecting the spinal cord during vertebral tumor ablation. Dermis under pulsed laser treatment gives $$T_{\alpha\mathrm{m}} = T_\alpha + \tilde{T}_{\alpha\mathrm{m}}$$4, suggesting collateral damage risk when pulse durations approach the relaxation time. These numbers rest on estimated effective phase parameters, not component-wise measurements, so they should be read as order-of-magnitude indications.

Radiation is incorporated via a linearized radiative exchange coefficient $$T_{\alpha\mathrm{m}} = T_\alpha + \tilde{T}_{\alpha\mathrm{m}}$$5, augmenting $$T_{\alpha\mathrm{m}} = T_\alpha + \tilde{T}_{\alpha\mathrm{m}}$$6 and $$T_{\alpha\mathrm{m}} = T_\alpha + \tilde{T}_{\alpha\mathrm{m}}$$7. Notably, radiation does not alter the intrinsic $$T_{\alpha\mathrm{m}} = T_\alpha + \tilde{T}_{\alpha\mathrm{m}}$$8 but increases $$T_{\alpha\mathrm{m}} = T_\alpha + \tilde{T}_{\alpha\mathrm{m}}$$9 toward its bulk limit by suppressing the finite-boundary correction — offering an explanation for sharp initial transients in highly porous carbon foams that conduction-only models miss.

Limitations and open questions

Several assumptions bound the results. The framework requires scale separation $$\tilde{T}_{\alpha\mathrm{m}} = \mathbf{b}_\alpha \cdot \nabla T_\alpha$$0 and spatially uniform porosity; the non-local extension is motivated precisely where this separation degrades, creating some tension in strongly heterogeneous syntactic foams. The microstructure is assumed periodic, isotropic, and rigid, with constant properties and no thermal expansion; anisotropic or evolving microstructures fall outside the current closure. The derived equation cannot describe wave propagation or rotational heat flux, so it cannot be mapped onto genuine phonon-hydrodynamic regimes. The equivalence of Guyer–Krumhansl and Jeffreys equations holds only under restrictive conditions (curl-free flux, constant coefficients, restricted sources). Component-wise thermophysical parameters for the MOF/RGO composite and biological tissues are estimates, and the water-filled foam prediction awaits experimental confirmation. Whether the magnitude of $$\tilde{T}_{\alpha\mathrm{m}} = \mathbf{b}_\alpha \cdot \nabla T_\alpha$$1 can be independently confirmed by re-evaluating existing flash datasets remains an open question the author explicitly raises.

Conclusion

The paper establishes that effective non-Fourier heat conduction in heterogeneous materials is not a phenomenological artifact but a rigorous consequence of upscaling two Fourier-conducting, thermally asymmetric phases. It delivers a thermodynamically admissible generalized Jeffreys equation with microstructurally determined coefficients, proves inherent over-diffusion ($$\tilde{T}_{\alpha\mathrm{m}} = \mathbf{b}_\alpha \cdot \nabla T_\alpha$$2), derives a second-law upper bound on the non-local length scale, and explains sample-size dependence of measured over-diffusion through boundary-induced higher-order non-localities. Agreement across metal and carbon foams, saturated limestone, and MOF/RGO composites supports the framework as a constructive alternative to both phenomenological non-Fourier models and ambiguous two-temperature formulations, with concrete implications for transient thermal therapies.