# Bragg Mirrors for Thermal Waves

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## Abstract

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## 1. Introduction

## 2. Thermal Waves—Cattaneo-Vernotte Equation

## 3. Thermal Superlattice

## 4. Thermal Bragg Mirrors

#### Tuning of the Thermal Reflectance

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Cattaneo, C. Sulla Conduzione Del Calore; Springer: Berlin/Heidelberg, Germany, 2011; p. 485. [Google Scholar]
- Vernotte, P. Les paradoxes de la théorie continue de lequation de la chaleur. CR Acad. Sci. Paris
**1958**, 246, 3154–3155. [Google Scholar] - Kaminski, W. Hyperbolic Heat Conduction Equation for Materials with a Nonhomogeneous Inner Structure. J. Heat Trans.
**1990**, 112, 555–560. [Google Scholar] [CrossRef] - Cassol, G.O.; Dubljevic, S. Hyperbolicity of the heat equation. IFAC-PapersOnLine
**2019**, 52, 63–67. [Google Scholar] [CrossRef] - Joseph, D.D.; Preziosi, L. Heat waves. Rev. Mod. Phys.
**1989**, 61, 41. [Google Scholar] [CrossRef] - Marín, E.; Vaca-Oyola, L.; Delgado-Vasallo, O. On thermal waves velocity: Some open questions in thermal waves’ physics. Rev. Mex. Fis. E
**2016**, 62, 1–4. [Google Scholar] - Gandolfi, M.; Benetti, G.; Glorieux, C.; Giannetti, C.; Banfi, F. Accessing temperature waves: A dispersion relation perspective. Int. J. Heat Mass Trans.
**2019**, 143, 118553. [Google Scholar] [CrossRef] [Green Version] - Roetzel, W.; Putra, N.; Das, S.K. Experiment and analysis for non-Fourier conduction in materials with non-homogeneous inner structure. Int. J. Therm. Sci.
**2003**, 42, 541–552. [Google Scholar] [CrossRef] - Graßmann, A.; Peters, F. Experimental investigation of heat conduction in wet sand. Heat Mass Transf.
**1999**, 35, 289–294. [Google Scholar] [CrossRef] - Jaunich, M.; Raje, S.; Kim, K.; Mitra, K.; Guo, Z. Bio-heat transfer analysis during short pulse laser irradiation of tissues. Int. J. Heat Mass Transf.
**2008**, 51, 5511–5521. [Google Scholar] [CrossRef] - Mandhukar, A.; Park, Y.; Kim, W.; Sunaryanto, H.J.; Berlin, R.; Chamorro, L.P.; Bentsman, J.; Ostoja-Starzewski, M. Heat conduction in porcine muscle and blood: Experiments and time-fractional telegraph equation model. J. R. Soc. Interface
**2019**, 16, 20190726. [Google Scholar] [CrossRef] [PubMed] - Mitra, K.; Kumar, S.; Vedevarz, A.; Moallemi, M.K. Experimental Evidence of Hyperbolic Heat Conduction in Processed Meat. J. Heat Transfer
**1995**, 117, 568–573. [Google Scholar] [CrossRef] - Xu, F.; Seffen, K.A.; Lu, T.J. Non-Fourier analysis of skin biothermomechanics. Int. J. Heat Mass Transf.
**2008**, 51, 2237–2259. [Google Scholar] [CrossRef] - Kovács, R.; Ván, P. Generalized heat conduction in heat pulse experiments. Int. J. Heat Mass Transf.
**2015**, 83, 613–620. [Google Scholar] [CrossRef] [Green Version] - Ahmadikia, H.; Fazlali, R.; Moradi, A. Analytical solution of the parabolic and hyperbolic heat transfer equations with constant and transient heat flux conditions on skin tissue. Int. J. Heat Mass Transf.
**2012**, 39, 121–130. [Google Scholar] [CrossRef] - Huberman, S.; Duncan, R.A.; Chen, K.; Song, B.; Chiloyan, V.; Ding, Z.; Maznev, A.A.; Chen, G.; Nelson, K.A. Observation of second sound in graphite at temperatures above 100 K. Science
**2019**, 364, 375–379. [Google Scholar] [CrossRef] [Green Version] - Beardo, A.; López-Suárez, M.; Pérez, L.A.; Sendra, L.; Alonso, M.I.; Melis, C.; Bafaluy, J.; Camacho, J.; Colombo, L.; Rurali, R.; et al. Observation of second sound in a rapidly varying temperature field in Ge. Sci. Adv.
**2021**, 7, eabg4677. [Google Scholar] [CrossRef] [PubMed] - Farhat, M.; Guenneau, S.; Chen, P.Y.; Alù, A.; Salama, K.N. Scattering Cancellation-Based Cloaking for the Maxwell-Cattaneo Heat Waves. Phys. Rev. Appl.
**2019**, 11, 044089. [Google Scholar] [CrossRef] [Green Version] - Chen, A.L.; Li, Z.Y.; Ma, T.X.; Li, X.S.; Wang, Y.S. Heat reduction by thermal wave crystals. Int. J. Heat Mass Transf.
**2018**, 121, 215–222. [Google Scholar] [CrossRef] [Green Version] - Gandolfi, M.; Giannetti, C.; Banfi, F. Temperonic crystal: A superlattice for temperature waves in graphene. Phys. Rev. Lett.
**2020**, 125, 265901. [Google Scholar] [CrossRef] - Ordóñez-Miranda, J.; Alvarado-Gil, J. Effective thermal properties of layered systems under the parabolic and hyperbolic heat conduction models using pulsed heat sources. J. Heat Transf.
**2011**, 133. [Google Scholar] [CrossRef] - Vázquez, F.; Ván, P.; Kovács, R. Ballistic-Diffusive Model for Heat Transport in Superlattices and the Minimum Effective Heat Conductivity. Entropy
**2020**, 22, 167. [Google Scholar] [CrossRef] [Green Version] - Estrada-Wiese, D.; del Río, J.A.; de la Mora, M.B. Heat transfer in photonic mirrors. J. Mater. Sci. Mater. Electron.
**2014**, 25, 4348–4355. [Google Scholar] [CrossRef] [Green Version] - Borca-Tasciuc, T.; Liu, W.; Liu, J.; Zeng, T.; Song, D.W.; Moore, C.D.; Chen, G.; Wang, K.L.; Goorsky, M.S.; Radetic, T.; et al. Thermal conductivity of symmetrically strained Si/Ge superlattices. Superlattices Microstruct.
**2000**, 28, 199–206. [Google Scholar] [CrossRef] - Chen, G.; Neagu, M. Thermal conductivity and heat transfer in superlattices. Appl. Phys. Lett.
**1997**, 71, 2761–2763. [Google Scholar] [CrossRef] - Esquivel-Sirvent, R.; Cocoletzi, G. Band structure for the propagation of elastic waves in superlattices. J. Acoust. Soc. Am.
**1994**, 95, 86–90. [Google Scholar] [CrossRef] - El Boudouti, E.H.; Djafari-Rouhani, B. Acoustic waves in finite superlattices. Phys. Rev. B
**1994**, 49, 4586–4592. [Google Scholar] [CrossRef] [PubMed] - De la Cruz, G.G. Bulk and surface plasmons in graphene finite superlattices. Superlattices Microstruct.
**2019**, 125, 315–321. [Google Scholar] [CrossRef] - Jena, S.; Tokas, R.B.; Thakur, S.; Udupa, D.V. Tunable mirrors and filters in 1D photonic crystals containing polymers. Phys. E Low-Dimens. Syst. Nanostructures
**2019**, 114, 113627. [Google Scholar] [CrossRef] - Al-sheqefi, F.; Belhadj, W. Photonic band gap characteristics of one-dimensional graphene-dielectric periodic structures. Superlattices Microstruct.
**2015**, 88, 127–138. [Google Scholar] [CrossRef] - Kone, I.; Domingue, F.; Reinhardt, A.; Jacquinot, H.; Borel, M.; Gorisse, M.; Parat, G.; Casset, F.; Pellissier-Tanon, D.; Carpentier, J.; et al. Guided acoustic wave resonators using an acoustic Bragg mirror. Appl. Phys. Lett.
**2010**, 96, 223504. [Google Scholar] [CrossRef] - Herrera, L. Causal heat conduction contravening the fading memory paradigm. Entropy
**2019**, 21, 950. [Google Scholar] [CrossRef] [Green Version] - Barletta, A.; Zanchini, E. Hyperbolic heat conduction and local equilibrium: A second law analysis. Int. J. Heat Mass Transf.
**1997**, 40, 1007–1016. [Google Scholar] [CrossRef] - Özısık, M.N. Heat Conduction; Wiley-Interscience Publication: Hoboken, NJ, USA, 1993. [Google Scholar]
- Bockh, P.V.; Wetzel, T. Heat Transfer, 3rd ed.; Springer: Berlin/Heidelberg, Germany, 1993; Volume 4. [Google Scholar]
- Ostoja-Starzewski, M. A derivation of the Maxwell–Cattaneo equation from the free energy and dissipation potentials. Int. J. Eng. Sci.
**2009**, 47, 807–810. [Google Scholar] [CrossRef] - Thidé, B. Electromagnetic Field Theory; Upsilon Books: Uppsala, Sweden, 2004. [Google Scholar]
- Camacho de la Rosa, A.; Becerril, D.; Gómez-Farfán, G.; Esquivel-Sirvent, R. Time-Harmonic Photothermal Heating by Nanoparticles in a Non-Fourier Medium. J. Phys. Chem. C
**2021**, 125, 22856–22862. [Google Scholar] [CrossRef] - Berto, P.; Mohamed, M.S.A.; Rigneault, H.; Baffou, G. Time-harmonic optical heating of plasmonic nanoparticles. Phys. Rev. B
**2014**, 90, 035439. [Google Scholar] [CrossRef] [Green Version] - Green, D.R. Thermal Surface Impedance for Plane Heat Waves in Layered Materials. J. Appl. Phys.
**1966**, 37, 3095–3099. [Google Scholar] [CrossRef] - Li, B.c.; Zhang, S.y. The effect of interface resistances on thermal wave propagation in multi-layered samples. J. Phys. D Appl. Phys.
**1997**, 30, 1447. [Google Scholar] [CrossRef] - Pérez-Álvarez, R.; García-Moliner, F. Transfer Matrix, Green Function and Related Techniques: Tools for the Study of Multilayer Heterostructures; Publicacions de la Universitat Jaume I: Castellón, Spain, 2004. [Google Scholar]
- Patel, H.A.; Garde, S.; Keblinski, P. Thermal resistance of nanoscopic liquid-liquid interfaces: Dependence on chemistry and molecular architecture. Nano Lett.
**2005**, 5, 2225–2231. [Google Scholar] [CrossRef] [PubMed] - Hasan, M.R.; Vo, T.Q.; Kim, B. Manipulating thermal resistance at the solid–fluid interface through monolayer deposition. RSC Adv.
**2019**, 9, 4948–4956. [Google Scholar] [CrossRef] [Green Version] - Knobel, R. An Introduction to the Mathematical Theory of Waves; American Mathematical Society: Providence, RI, USA, 2000; Volume 3. [Google Scholar]
- Ranut, P.; Nobile, E. On the effective thermal conductivity of metal foams. J. Phys. Conf. Ser.
**2014**, 547, 012021. [Google Scholar] [CrossRef] - Kundu, B. Exact analysis for propagation of heat in a biological tissue subject to different surface conditions for therapeutic applications. Appl. Math. Comput.
**2016**, 285, 204–216. [Google Scholar] [CrossRef] - Craciunescu, O.I.; Howle, L.E.; Clegg, S.T. Experimental evaluation of the thermal properties of two tissue equivalent phantom materials. Int. J. Hypertherm.
**1999**, 15, 509–518. [Google Scholar] - Sánchez, A.; Porta, A.V.; Orozco, S. Photonic band-gap and defect modes of a one-dimensional photonic crystal under localized compression. J. Appl. Phys.
**2017**, 121, 173101. [Google Scholar] [CrossRef] - Scalora, M.; Bloemer, M.J.; Pethel, A.S.; Dowling, J.P.; Bowden, C.M.; Manka, A.S. Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures. J. Appl. Phys.
**1998**, 83, 2377–2383. [Google Scholar] [CrossRef] - Kushwaha, M.S.; Halevi, P. Band-gap engineering in periodic elastic composites. Appl. Phys. Lett.
**1994**, 64, 1085–1087. [Google Scholar] [CrossRef] - Esquivel-Sirvent, R.; Noguez, C. Theory of the acoustic signature of topological and morphological defects in SiC/porous SiC laminated ceramics. J. Appl. Phys.
**1997**, 82, 3618–3620. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Finite layered system made of consecutive layers with different thermal properties. Each layer is characterized by its thickness d, thermal speed $\upsilon $ and the thermal diffusivity $\alpha $. The system is bounded by a substrate and outer medium. (

**b**) Semi-infinite system. The substrate is removed and the layered system extends indefinitely.

**Figure 2.**Reflectance R between the medium labeled material 1 and a half space made of Al (

**a**), material 2 (

**b**). The properties of the different materials are given in Table 1. As explained in the text, the reflectance for thermal waves is $R=\left|r\right|$ (solid) rather than $R={\left|r\right|}^{2}$ (dashed) as in other wave phenomena. For metals that have a very small relaxation time, they behave as perfect reflectors.

**Figure 3.**Penetration length $\delta $ (in mm) as a function of frequency for the materials shown in Table 1. The dotted lines indicate the asymptotic value given by $\delta =2\alpha /v$.

**Figure 4.**For a system with unit cells made of alternating layers of material 1 and material 2 (see Table 1) and both of thickness $d=100$ $\mathsf{\mu}$m, we present in (

**a**–

**c**) the reflectance $R=\left|r\right|$ as a function of frequency for different numbers of unit cells n in a solid line; the dashed line represents the semi-infinite periodic case. In (

**d**), the heat flux into the system at $x=0$ is shown as function of frequency and n. The flux is normalized to the incident flux ${q}_{0}$. The stop-band is seen even for a few number of unit cells. The dotted lines indicate the position of the first stop-band for the semi-infinite crystal.

**Figure 5.**Normal heat flux in a contour plot for a system with unit cells made of alternating layers of material 1 and material 2 assuming that they have relaxation time equal to zero, which corresponds to the Fourier law case.

**Figure 6.**(

**a**–

**c**) Reflectance and normalized (

**d**) heat flux of a system with unit cell m1-m2-Al-m2 (1-2-Al-2). Layers of Material 1 and 2 have a thickness of $d=100\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m while for that of Al ${d}_{al}=1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m.

Material | Thermal Conductivity $\mathit{\kappa}$ (Wm${}^{-1}$ K${}^{-1}$) | Specific Heat Capacity ${\mathit{c}}_{\mathit{V}}$ (J kg${}^{-1}$ K${}^{-1}$) | Mass Density $\mathit{\rho}$ (Kg m${}^{-3}$) | Response Time $\mathit{\tau}$ (s) |
---|---|---|---|---|

epidermis (m1) | 0.235 | 3600 | 1500 | 1 |

dermis (m2) | 0.445 | 3300 | 1116 | 20 |

Al | 237 | 921 | 2707 | $2\times {10}^{-10}$ |

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**MDPI and ACS Style**

Camacho de la Rosa, A.; Becerril, D.; Gómez-Farfán, M.G.; Esquivel-Sirvent, R.
Bragg Mirrors for Thermal Waves. *Energies* **2021**, *14*, 7452.
https://doi.org/10.3390/en14227452

**AMA Style**

Camacho de la Rosa A, Becerril D, Gómez-Farfán MG, Esquivel-Sirvent R.
Bragg Mirrors for Thermal Waves. *Energies*. 2021; 14(22):7452.
https://doi.org/10.3390/en14227452

**Chicago/Turabian Style**

Camacho de la Rosa, Angela, David Becerril, María Guadalupe Gómez-Farfán, and Raúl Esquivel-Sirvent.
2021. "Bragg Mirrors for Thermal Waves" *Energies* 14, no. 22: 7452.
https://doi.org/10.3390/en14227452