Exceptions to Fourier’s law at the macroscale

Significance

Traditional analysis of heat conduction in materials assumes that heat diffuses (Fourier’s law) with exceptions only at the nanoscale. Here, scrutinizing limits of this assumption, we examine translucent materials through which energy also transfers by electromagnetic radiation, and we perform the experiments in a vacuum to avoid air convection. We conclude that the independent pathway of energy transfer by electromagnetic radiation produces macroscopic-scale exceptions to predictions made using Fourier’s law. These empirical findings offer a challenge for theorists and a unique approach to engineer heat management.

Abstract

The usual basis to analyze heat transfer within materials is the equation formulated 200 years ago, Fourier’s law, which is identical mathematically to the mass diffusion equation, Fick’s law. Revisiting this assumption regarding heat transport within translucent materials, performing the experiments in vacuum to avoid air convection, we compare the model predictions to infrared-based measurements with nearly mK temperature resolution. After heat pulses, we find macroscale non-Gaussian tails in the surface temperature profile. At steady state, we find macroscale anomalous hot spots when the sample is topographically rough, and this is validated by using two additional independent methods to measure surface temperature. These discrepancies from Fourier’s law for translucent materials suggest that internal radiation whose mean-free-path is millimeters interacts with defects to produce small heat sources that by secondary emission afford an additional, non-local mode of heat transport. For these polymer and inorganic glass materials, this suggests unique strategies of heat management design.

The basis of heat transfer within materials is generally considered to be the equation formulated 200 years ago, Fourier’s law, which mathematically is identical to the mass diffusion equation, Fick’s law (1, 2). Ironically, although regarding mass transport, the ubiquity of “anomalous” transport has stimulated an intellectually vigorous branch of applied physical science (3–5), regarding heat transport, the 200-y-old ansatz continues to be the paradigm. Recent new understanding of heat transfer dwells almost exclusively on nanometer-scale physics, dimensions smaller than the diffusion mean free path (2, 6). It is reasonable to find violations at the nanoscale but to find violations at macroscopic scales would be startling as this would go beyond standard textbook thinking. Viewed as a problem of energy in transit, this is a quintessential example of the redistribution of energy when a system is forced out of equilibrium, which is one of the most rapidly advancing modern themes of modern condensed matter physics (7), though not yet linked in the minds of researchers to the problem of heat propagation.

Here, we test the limits of the Fourier model by selecting translucent materials (polymers and inorganic glasses) through which energy can also transfer by electromagnetic radiation. Our idea was that because translucent materials are not fully transparent, the mean free path in relevant windows of wavelength might exceed the usual assumption of nanometers or micrometers and become macroscopic. Strips of test material, clamped at each end, were suspended in a homemade vacuum chamber (SI Appendix, Fig. S1) whose sapphire windows allow surface emissivity to be measured by an infrared camera whose sensitivity was assisted by cryogenic cooling. The vacuum level, ≈25 Pa, was selected after finite element simulation (FEA) predicted no significant convection at this pressure (SI Appendix, Fig. S2). The sample was shielded from thermal radiation by a material with ultra-low thermal emissivity ≈0.03 (tinfoil) with efficacy illustrated in SI Appendix, Fig. S3. Using emitted infrared (IR) radiation to detect temperature, the wavelengths in the region 3 to 5 μm take advantage of the detector’s greatest sensitivity. We report the temperature of a surface layer whose thickness for a representative material, polystyrene (SI Appendix, Fig. S4), we quantified to be ≈80 μm. The precaution of vacuum-environment avoids the alternative heat transfer pathway of convection through air (1). Measurements in vacuum are more precise even at constant temperature (SI Appendix, Fig. S5). We confirmed that in isothermal conditions, the camera read the identical sample temperature everywhere, supporting the interpretation that the IR camera measured temperature with fidelity and that the sample thermal emissivity was uniform. Fluctuations from spot to spot in vacuum were ±0.03 K (SI Appendix, Fig. S5).

Results

Heat Pulses.

First, consider the heat propagation kinetics after heating by laser pulses of modest power. For example, after pulses 0.2 s long with ≈40 μm beam width, the temperature T_r was ≈3 K at 1 s elapsed time, where T_r is the temperature after subtracting the background temperature (Fig. 1A). To avoid non-generic details of the experimental setup, these measurements were initiated after several times the pulse duration. There was no measurable impact on heat propagation of the wavefront of the focused laser beam if one accepts FEA simulations based on Fourier’s law (SI Appendix, Fig. S6). Temperature jump was proportional to laser power (Fig. 1B). Further testing the fidelity of these measurements, we used three independent methods by using a steady-state temperature gradient to enable measurements at many different temperatures. In order of increasing resolution, these were i) a thermocouple placed variably on the sample’s top surface, whose temperature was read as voltage; ii) a coating of thermochromic liquid crystal painted onto the sample, whose temperature was read as color hue; and iii) an infrared camera as described in the previous paragraph, whose temperature was read as calibrated surface emissivity. Agreement is quantitative (Fig. 1C). In the end of this paper, a section discusses this and other reasons to have confidence in the data.

Fig. 1.

Time-dependent temperature after pulse heating. (A) T_r spatial distribution (T_r is temperature after subtracting the background temperature) measured by IR camera at 1 s after a laser beam (focused Gaussian beam with width ≈40 μm, 20 mW) impinges on a strip of polystyrene (PS, 350 kg-mol⁻¹) for 0.2 s. Arrow shows the direction from which the impinges the sample. (B) T_{r_jump} after a heat pulse for 0.2 s, plotted against the laser power at the sample, with other conditions same as in panel A. Error bar is SD from five measurements. Red dashed line is guide to the eye. (C) After heating the sample to various steady-state temperatures above ambient (22 ± 0.1 °C), ΔT measured by IR camera is plotted against temperature measured by thermocouple voltage (ΔT_ThC) and hue of liquid crystal coating (ΔT_LC). (D) Evaluated at a radial distance of 3 mm from the heat source after pulse heating, normalized T_r is plotted against time for PS. (E) Same as panel D but for rubber (polyisioprene with 10% crosslinker). (F) Same as panel D but for quartz (neutral density filter). Red dashed curves in D–F show FEA simulation based on the Fourier law.

Temperature at positions millimeters away from the heat pulse rose too rapidly to be consistent with the Gaussian profiles predicted by Fourier’s law. This cannot be attributed to the chain character of molecules within our polymer samples because quartz (an inorganic glass) displays a qualitatively similar anomaly nor can it be attributed to the glassy state of these materials because the polymer rubbery state also displays it. The anomalous temperature jump at early times after temperature pulse is illustrated for polystyrene glass (Fig. 1D), polyisoprene rubber at room temperature (Fig. 1E), and quartz (a neutral density filter) (Fig. 1F). Observing the same trend for rubber, which is soft and flexible, rules out a trivial thermomechanical explanation by hypothesizing gradient-induced cracks within these materials. Finally, because these research-grade samples of materials were chemically pure, translucent optically, and free of visible optical defects, these findings cannot be easily dismissed on the grounds of sample impurities.

Now we consider temperature spatial distribution at fixed elapsed time, inspired by the many known examples of anomalous diffusive mass transport (3–5). Implementing this, we radially average temperature at fixed distances from the heat pulse and find that Gaussian temperature profiles are recovered with increasing time (Fig. 2A and SI Appendix, Fig. S7). This transition could not reproduced by finite element simulation of a hypothetical surface layer with different heat conductivity than the bulk, however (SI Appendix, Fig. S8). Reasoning that if these materials absorb radiation the distance dependence should be exponential according to the Beer-Lambert law (8), we decompose the observed non-Gaussian profiles into the sum of two components, one Gaussian and the other exponentially decaying. The macroscopically large decay length of the latter—1 mm (Fig. 2) and 1 to 2 mm in other situations (SI Appendix, Fig. S7)—is consistent with our observations of non-Fourier temperature profiles at millimeter length scales. This is why the classical Rosseland approximation to describe heat transfer (1) does not apply—the radiation mean free path in this region of wavelength is so large that it is has become comparable to the sample dimensions.

Fig. 2.

Position-dependent temperature after pulse heating. (A) Normalized temperature profiles in radial direction at elapsed times 1 s and 10 s show a non-Gaussian tail that gradually disappears. (B) Internal radiation contribution plotted against thermal diffusivity of the material for six materials identified in the panel. (C) Non-Gaussian parameter (α) plotted against time for three materials identified in the panel and also for FEA simulation for a sample with these same geometrical dimensions. The graph highlights τ for each material: the time for kurtosis to decay to the value characteristic of Gaussian diffusion. (D) The τ are plotted against thermal diffusivity for six materials identified in the panel. The error bars in B and D are SD from five independent measurements. Lines with slope unity in B and D are guides to the eye.

Classical modeling of radiation transfer in materials would not predict this because in those models for optically thick materials, meaning that the mean free path of photons is much smaller than the characteristic length scale of the medium, the heat flux relation for radiative transfer has the same form as the Fourier law and only modifies the material’s effective thermal diffusivity (1). The materials studied here are more translucent than those described by models of this kind. If this radiation took the form of black body radiation, it would be dominated by wavelengths in the visible, infrared, and longer range of wavelengths (SI Appendix, Fig. S4A), so translucency of these samples is fundamental. Moreover, radiation propagates more rapidly than diffusion, so it contributes proportionately more at locations where diffusive heat flux has not yet arrived after a heat pulse. At the short time of 1 s, contribution by the former can amount to nearly 50% of the total and is larger, the lower the material’s thermal diffusivity (Fig. 2B). It is reasonable to recover the expected Gaussian temperature profiles when heat arrives later by the usual diffusive mechanisms.

We calculated the non-Gaussian parameter α, plotted against time in Fig. 2C, which should equal zero for a random walk (the Fourier law). For this assessment, we extracted the kurtosis (9) from data generated from the temperature distribution (SI Appendix, Fig. S9) and subtracted the baseline three; slight deviations of the simulated α from 0 at early and late times can be attributed to pulse memory and sample boundary effects (10), respectively. As thermal diffusivity of the material increases, the magnitude of this anomaly decreases (Fig. 2D).

Steady-state Temperature Gradients.

Steady-state temperature gradients also revealed anomalies. Experimentally, heating tape was used to clamp one end of the sample with care taken to shield both the clamp and the heater from transmitting thermal radiation through the vacuum, and the other end was attached to another shielded metal clamp at room temperature. Consider first a sample that contains a dimple-shaped topographical depression (Fig. 3A). After subtracting the background thermal gradient as explained in SI Appendix, Figs. S10 and S11, we analyze deviations from the background temperature gradient along the length direction and compare to FEA simulation (Fig. 3B) after selecting a suitable FEA mesh size (SI Appendix, Fig. S12), where T_r is the temperature after subtracting the background thermal gradient. Surprisingly, the surface temperatures on the two edges of the dimple are not symmetric: spots of anomalously higher temperature (≈0.5 K) appear not only after the leading interface on the side of higher temperature but also at the trailing interface on the side of cooler temperature (Fig. 3B). This disagrees with the FEA simulations, which predict symmetric deviations at these positions. This is so even when the FEA simulations incorporate surface radiation loss and surface-to-surface radiation (SI Appendix, Fig. S13) and even when one postulates a hypothetical surface layer with different thermal conductivity than the bulk (SI Appendix, Fig. S14). In the literature, the Fourier law has been modified to produce non-Brownian diffusion (11, 12) but our FEA simulations of the solutions of parabolic and hyperbolic equations did not predict the observed anomalous hot spots (SI Appendix, Fig. S15), which is expected because at steady state, there is no time dependence. The possible wavelike character of heat has also been considered in the literature (13–15) so we set up pairs of side-by-side depressions, but this experiment gave a null result: No interference patterns were observed (SI Appendix, Fig. S16). Our observations contrast with the hot-cold symmetry that is built into the random walk Fourier model.

Fig. 3.

Temperature at and near depressions in steady-state thermal gradients. (A) Schematic sketch of a strip with depression radius 2.5 mm. (B) Temperature profile in length direction after background subtraction for temperature gradient 1 K-mm⁻¹. Top: FEA simulation; red lines, guides to the eye, highlight regions of higher temperature. Middle and Bottom: two independently prepared samples (PS, 350 k g-mol⁻¹). The shaded region shows fluctuations from five repeat measurements. (C) Temperature rise T_{r-hot spot} measured at the hotter side, plotted against thermal gradient. (D) T_{r-hot spot} plotted against depression depth for radius 2.5 mm and gradient 1 K-mm⁻¹. (E) T_{r-hot spot} plotted against depression radius for depth 1 mm and gradient 1 K-mm⁻¹. In C–E, lines guide the eye and error bars are SD from five measurements using independently prepared samples. (F) Schematic sketch of a strip containing an asymmetric cylindrical-symmetry depression. (G) Temperature profile in length direction after background subtraction for gradient 1 K-mm⁻¹. Top: FEA simulation. Bottom: temperature profile in length direction for experiments in panels H and I. (H) Color temperature map for panel F when sample is positioned with lower depression curvature on hotter side. (I) Same as panel H except lower depression curvature is on colder side.

This trend was confirmed for several other amorphous materials and also a semi-crystalline material that is opaque at visible wavelengths but translucent in the infrared (SI Appendix, Fig. S17). We quantified the respective influences of the applied temperature gradient (Fig. 3C), depression depth (Fig. 3D), and depression radius (Fig. 3E) from raw data illustrated in SI Appendix, Figs. S18 and S19. Altogether, these findings indicate larger anomalous heating at interfaces, the larger the temperature gradient. Physically, it is reasonable to observe this trend of more anomaly when these quantities increase because the area of geometric depression increases with depth and radius. The breakdown of symmetry in the hot–cold directions also holds for other geometrical patterns of surface topography (SI Appendix, Fig. S20). Seeking to increase the anomaly, we fabricated asymmetrically curved cylindrical depressions, gently curved and more sharply curved near the two ends, as sketched in Fig. 3F. The line temperature distribution after removing the background gradient is compared to FEA simulation in Fig. 3G. Encouragingly, i) this produces at the leading interface a broad slab-shaped region of elevated temperature, which contrasts with the narrow hot spot that results from a circularly shaped depression; ii) width of the slab is controlled by adjusting the local curvature, as illustrated by switching the direction of heat propagation (Fig. 3I); iii) the larger the interface curvature at the leading interface, the more elevated the relative temperature in the slab region. The control parameter is interface curvature normal to the direction of energy transfer. However, when samples were made less translucent by blending 10% carbon black into polyisoprene, this sharply reduces the hot spot (SI Appendix, Fig. S21).

Remarks about Selection of Infrared (IR) Imaging to Evaluate Temperature.

Infrared (IR) imaging outperforms other methods in terms of response time and spatial resolution. We have shown quantitative agreement between three independent methods to measure temperature—placing a thermocouple on the sample's top surface to measure temperature as indicted by voltage, using thermochromic liquid crystal coating on the sample to measure temperature as indicated by color hue and utilizing an IR camera with calibrated surface emissivity to measure temperature as indicated by thermal emissivity (Fig. 1C). We emphasize that under isothermal conditions, our IR imaging shows the same temperature everywhere on our samples, which shows the validity of our assumption that the thermal emissivity is the same regardless of whether the sample is uniform or structured in its topography. The conditions of these experiments are in the high emissivity range, >0.92.

We now consider possible limitations. First, while it is true that we probe surface (not bulk) temperature, this is not considered problematic because the data cannot be modeled using a two-layer model of thermal conductivity (SI Appendix, Figs. S8 and S14). While it is true that every sample absorbs some infrared emitted in principle, this is not considered problematic when one considers the high infrared transmissivity of the materials we studied (SI Appendix, Fig. S4). We quantified the depth of near-surface material probed, 80 micrometers for these translucent samples (SI Appendix, Fig. S4).

As an operational matter, the angle of incidence from the sample into the detector is not considered to be problematic. Within a vacuum chamber, our samples are positioned parallel to a sapphire window through which surface emissivity passes and is detected ≈0.5 m away by the IR camera equipped with a telescopic lens. As the area of the camera detector, 100 mm × 56 mm, is much larger than the measured sample area, 20 mm × 15 mm or 20 mm × 20 mm, we obtain consistent results regardless of how we shift sample position within the imaging area. While it is true that imperfect radiation shielding could heat the sample, the large metal vacuum chamber (stainless steel) did not heat up during measurements and heaters were shielded from the sample using tinfoil whose thermal emissivity (≈0.03) is exceptionally low and verified from measurement (SI Appendix, Fig. S3) the efficacy of this shielding.

We considered the possibility that scattered IR radiation might be detected by the IR detector and misinterpreted as heat, but by experimental design, the IR detector could not detect scattering of the deep-blue (405 nm) laser used for heat pulse experiments and scattered thermal radiation would be below the bounds of experimental resolution for Rayleigh and Mie scattering (SI Appendix, Fig. S22). In SI Appendix, Fig. S22, one sees that thermal radiation scattered into the IR detector, i.e., at angle 90° to the sample, would be <0.01% of the incident intensity when we considered a range of putative scattering centers with dimensions in the range 4 nm to 4 μm. This conclusion is consistent with noticing that temperature T sensed by the IR detector scales weakly with radiance I, T ≈ I_rad^0.1 (SI Appendix, Fig. S4B). Finally, we note that the IR measurements made at steady-state were checked independently using measurements based on a thermocouple and on a thermochromic liquid crystal, and these measurements yielded consistent temperature results though the thermocouple and liquid crystal methods are based on different physical principles than IR imaging. Taking all these factors into account, we conclude that these concerns are not problematic. In a future experiment, it would be desirable to quantify infrared scattering by illuminating the sample with an infrared laser whose wavelength falls within the detection range of the IR imaging camera.

Discussion

Although not modeled in the engineering literature regarding radiative energy (16–18), one knows that the decay of excited vibrational and electronic states after illumination dissipates in part nonradiatively as heat in the local environment (19–23). Though the materials studied here are translucent, they do absorb radiation, and this should manifest as heat. Inherent inhomogeneity of amorphous materials can be one reason for absorption (24–26). Temperature gradients may enhance this. Interfaces are also known to absorb radiation (27, 28). Defects of these kinds would produce a distribution of small heat sources throughout the material, which via secondary emission become a coupled system, thereby providing additional modes of non-local heat transport from temperature jumps at the junction between defects and their environment. Sparsely distributed, these heat sources would allow internally transmitted radiation to propagate significant distances without absorption, as we observe.

We emphasize that our selection of translucent samples distinguishes the materials studied here from those usually considered in the field of heat transfer. Because our samples were translucent, not optically thick, and our heat pulse measurements decay over millimeters (Fig. 2A and SI Appendix, Fig. S7), the mean free path over a relevant range of wavelength is comparable to the sample dimensions. This why in our heat-pulse experiments, the data show heating faster than can be attributed to diffusion (Fig. 1 D–F), indicating that radiation contributes significantly to heat flux during early times after a heat pulse, though the relative contribution of radiation diminishes as diffusion becomes dominant at later times. These materials studied here do not meet the conditions assumed by the so-called Rosseland approximation to estimate radiative heat transfer in optically thick media where the mean free path is smaller than the sample dimension.

When developing relevant theoretical models, we anticipate the role of two basic parameters: first, the mean free path of internal radiation (this, in turn, will depend on the absorption as a function of wavelength), and second, the dimension of the material through which radiation propagates. Their joint action determines the extent to which a material is translucent.

The significance on the practical side is to offer a strategy for designing possible new macroscale heat transfer devices—a matter of topical interest (1, 2, 6, 17, 28).

The significance on the fundamental side is that acceptance of the Fourier law at the macroscale is based on its empirical success, not on first-principles derivation. Here, presenting data that possess more significant figures than appear to have been reported previously, we have identified a pathway of heat conductivity in translucent materials that is not describable using the Fourier law. It is not surprising that these discrepancies were not noticed sooner, as a fact of life is that engineers are accustomed to designing systems that have tolerance. Deeper understanding can benefit by comparison to data with the additional significant figures one can now obtain using modern technology, as we do here.

Materials and Methods

Experiments.

Vacuum chamber.

Vacuum is produced using a turbo pump (Pfeizer Vacuum HiPace® 700, USA). The vacuum chamber includes an optical window made of a 10-mm thick sapphire plate coated with a 2 to 5 μm antireflective film. This window allows for acquisition of radiation signals due to its low IR absorption, which is less than 3% in the 3 to 5 μm range. Because exigencies of inlet–outlet connections limited the achievable vacuum level, we worked at 25 Pa pressure.

Infrared (IR) camera.

The IR camera (FLIR A8300sc, USA) has a large pixel count of 1,280 × 720, with a pixel size of approximately 78 μm and depth of focus of 9 cm. The camera operates within the detection spectra range 3 to 5 μm. Video images are typically collected at 30 frames per second and analyzed using MATLAB codes developed in-house. The large collection area of the IR camera, 100 mm × 56 mm, significantly exceeds the sample area (20 mm × 15 mm or 20 mm × 20 mm). The camera was located outside the vacuum chamber ≈0.5m from the sample. Images were taken using a telescope lens.

Heat pulses.

A focused laser beam from Coherent (OBIS LX 405 nm, USA, P_max = 200 mW) was focused onto the sample with beam width 40 µm.

Steady-state thermal gradients.

To generate a steady temperature gradient on the film sample, heating foil from Thorlabs (USA) was employed. Radiation emitted by the heating foil was shielded using tinfoil sourced from Thomasnet (USA). Temperature was fixed at the two long ends of rectangular strips with dimensions 20 mm × 15 mm × 3 mm. For example, one boundary is set to 313.15 K and the other to 293.15 K to produce the gradient 1 K-mm⁻¹.

Polystyrene glass.

To produce each sample, 1 g or 1.25 g polystyrene (Polymer Source, Canada, M_n = 345.5 k, M_w/M_n = 1.07; M = 35 k, M_w/M_n = 1.18) is molded by a silicone rubber-based soft mold (Mold Star) in a vacuum oven at 240 °C for 5 h, then naturally cooled in vacuum overnight. Final dimensions are 20 mm × 15 mm × 3 mm or 20 mm × 20 mm × 3 mm.

Crosslinked rubber.

To produce each sample, 1g polyisoprene (Polymer Source, Canada, M_n = 1,000 k, M_w/M_n = 1.05) is mixed in 50 mL toluene solution containing crosslinking agent (dicumyl peroxide, Sigma-Aldrich) at concentrations of 2, 5 or 10%. The specimens are dried in a PTFE mold in a fume hood then baked in a vacuum oven at 150 °C for 2 h and finally naturally cooled in the vacuum overnight. Unreacted crosslinkers and residual molecules of low molar mass are extracted for 72 h by immersion in toluene (100 mL) at room temperature, exchanging the toluene 24 times. Final dimensions are ~20 mm × 15 mm × 3 mm.

Quartz and PTFE.

Heating of pure quartz by laser pulses (405 nm) was too small to detect using IR imaging, so to increase light absorption in laser pulse experiments, we studied quartz in the form of an absorptive neutral density filter (Thorlabs, USA, optical density 1.0) with dimensions 25.4 mm diameter and 1 mm thickness. For steady-state measurements, we studied a plate of pure quartz (MAKE IT, NC-100, South Korea, 99.99% siloxane) with dimensions 20 mm × 15 mm × 3 mm. The depression in this plate was produced by drilling (3 mm diameter, 1 mm depth) with water cooling. The PTFE and PMMA were obtained from our university’s machine shop (Maker Lab, UNIST, South Korea) and depressions were machined by them.

Thermochromic liquid crystal coatings.

(Advanced Thermal Solutions, Inc., USA, 20 ~ 40 °C, temperature accuracy ± 0.1 °C). As recommended by the company, black backing ink was sprayed to the surface of the specimens using an air sprayer, dried in air, and then the liquid crystal was applied in the same way. Black backing ink and air sprayer were provided by the company.

Thermocouples.

Thermocouples (Fluke, 54 II B, USA, temperature resolution 0.1 K) were placed on the top surface.

FEA Simulations.

Software.

Finite element analysis (FEA) simulations were performed using the “Heat Transfer in Solids” module of COMSOL Multiphysics Version 5.4 software. Sample dimensions: same as in experiments, for example, 20 mm × 15 mm × 3 mm. Mesh size: the option of “physics-controlled extremely fine mesh” with d_min = 0.003 mm, in which the average mesh size of 0.1 mm was set by user and the local mesh was controlled by the software based on the local geometry.

Materials Parameters.

For FEA simulations of polystyrene, we use thermal conductivity κ = 0.28 W-m⁻¹ K⁻¹), density ρ = 1,050 kg-m⁻³, and heat capacity c_P = 1,200 J-kg⁻¹ K⁻¹. For polyisoprene, we use with thermal conductivity κ = 0.15 W-m⁻¹ K⁻¹), density ρ = 910 kg-m⁻³, and heat capacity c_P = 440 J-kg⁻¹ K⁻¹. For quartz, we use thermal conductivity κ = 2.52 W-m⁻¹ K⁻¹), density ρ = 2,620 kg-m⁻³, and heat capacity c_P = 733 J-kg⁻¹ K⁻¹.

Radiation and convection losses.

Thermal radiation loss to the ambient was simulated as $\epsilon \sigma \left({\mathrm{T}}^4-{\mathrm{T}}_{\mathrm{a}\mathrm{m}\mathrm{b}}^4\right)$, and thermal convection loss as $\mathrm{h}\left(\mathrm{T}-{\mathrm{T}}_{\mathrm{a}\mathrm{m}\mathrm{b}}\right)$, where ε, σ, T_amb, and h are the thermal emissivity, Stefan–Boltzmann constant, ambient temperature, and pressure-dependent heat transfer coefficient, respectively. Ambient pressure affects thermal convection.

Pulse heating.

A laser pulse heated the center of a sample with dimensions 20 mm × 20 mm × 3 mm; both a Gaussian beam and a top-hat beam with power 20 mW were considered, each with width 40 μm. The simulation time was 100 s with a time interval of 0.01 s. The room temperature was set as 293.15 K.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

All study data are included in the article and/or SI Appendix.