### Transformational Thermodynamics: Cloaks to Keep You Cool

**Sébastien Guenneau (left) and Claude Amra (right)**

**Authors: Sebastien Guenneau and Claude Amra**

**Affiliation: Institut Fresnel, Centre National Recherche Scientifique, Aix-Marseille University**

**and Ecole Centrale Marseille, France**

In 2006, two papers -- published in the same issue of 'Science' -- revolutionized the world of classical optics with the concept of transformation optics (TO). One was by Ulf Leonhardt of University of St Andrews, Scotland, UK [1] and the other was by John Pendry of Imperial College, London, UK and David Schurig and David Smith of Duke University, USA [2]. The concept of transformation optics allows coordinate changes in an isotropic homogeneous dielectric medium that can lead to a anisotropic heterogeneous metamaterial described by tensors of permittivity and permeability, a fact foreseen twenty years ago in two visionary papers on computational electromagnetic [3,4].

The conceptual breakthrough in 2006 was to note that one can blow-up a point into a finite region using a change of coordinates known from mathematicians working on inverse problems [5], in order to conceal this region from electromagnetic waves, and to demonstrate the feasibility of an invisibility cloak for microwaves [6], thereby bringing perhaps the most stunning electromagnetic paradigm of all times into reality!

However, similar changes of coordinates can also be applied to other wave equations, such as linear water waves propagating at the surface of a fluid [7], pressure waves propagating in a fluid [8], coupled pressure and shear waves propagating in a solid material [9,10], or flexural waves in thin elastic plates [11,12]. Such metamaterials designed using transformation acoustics (TA) could be used to protect regions from tsunamis (in general from ocean waves) or earthquakes (especially from surface elastic waves known as Rayleigh waves) on a larger scale! They could also be used to improve sound in opera theaters or to hide submarines from sonars (silence cloak).

But that’s not the end of the invisible story, as one can also leave the world of TO and TA and enter the brave new world of transformation thermodynamics (TT, or T

^{2}), whereby one now wishes to control diffusion processes, such as heat. Some precursory theoretical and numerical studies on the conduction equation in anisotropic media [5,13,14] have shown that one can control the diffusive heat flow in new ways in the static limit, a fact experimentally demonstrated this year [15].

**Figure 1: [Click on the image to view high resolution version] Numerical simulation showing the distribution of temperature in a region heated from the left (temperature=100 degrees Celsius, red color). The temperature gradually decreases away from the source, until it reaches a temperature of 0 degree Celsius on the left hand side (blue color). Importantly, one sees that the temperature vanishes inside the inner disc of the thermal cloak (annulus containing an anisotropic heterogeneous conductivity). This thermal protection is achieved by curving the isothermal curves (black lines).**

Our group at the Fresnel Institute in Marseille and the Ecole Centrale in Paris has shown, under the umbrella of the French National Center for Scientific Research (CNRS), that TT is a valid concept: one can control the flow of heat when time flows [16], and it is enough to use concentric layers with isotropic homogeneous conductivity to design invisibility cloaks (to protect a region from heat, see figure 1) and concentrators (to enhance heat exchange in a region). This opens unprecedented routes towards heat insulators -- for instance, for Green houses and also for heat harvesting in photovoltaics.

**References:**

**[1]**U. Leonhardt, “Optical Conformal Mapping”, Science 312, 1777 (2006). Abstract.

**[2]**J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields”, Science, 312, 1780 (2006). Abstract.

**[3]**A. Nicolet, J.F. Remacle, B. Meys, A. Genon and W. Legros, “Transformation methods in computational electromagnetic“, Journal of Applied Physics, 75, 6036-6038 (1994). Abstract.

**[4]**A.J. Ward and J.B. Pendry, “Refraction and geometry in Maxwell’s equations“, Journal of Modern Optics, 43, 773-793 (1996). Abstract.

**[5]**A Greenleaf, M Lassas and G Uhlmann, “On nonuniqueness for Calderon’s inverse problem“, Mathematical Research Letters, 10, 685–693 (2003). Full Article.

**[6]**D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies”, Science, 314, 977 (2006). Abstract.

**[7]**M. Farhat, S. Enoch, S. Guenneau, and A. B. Movchan, “Broadband Cylindrical Acoustic Cloak for Linear Surface Waves in Fluid”, Physical Review Letters, 101, 134501 (2008). Abstract.

**[8]**S. Zhang, C. Xia, and N. Fang, “Broadband Acoustic Cloak for Ultrasound Waves”, Physical Review Letters, 106, 024301 (2011). Abstract.

**[9]**G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant from”, New Journal of Physics, 8, 248 (2006). Abstract.

**[10]**M. Brun, S. Guenneau and A.B. Movchan, "Achieving control of in-plane elastic waves". Applied Physics Letters, 94, 061903 (2009). Abstract.

**[11]**M. Farhat, S. Guenneau, S. Enoch, and A. B. Movchan, “Ultrabroadband Elastic Cloaking in thin Plates”, Physical Review Letters, 103, 024301 (2009). Abstract.

**[12]**N. Stenger, M. Wilhelm, and M. Wegener, “Experiments on Elastic Cloaking in Thin Plates”, Physical Review Letters, 108, 014301 (2012). Full Article. 2Physics Article.

**[13]**C. Z. Fan, Y. Gao, and J. P. Huang, “Shaped graded materials with an apparent negative thermal conductivity", Applied Physics Letters, 92, 251907 (2008). Abstract.

**[14]**T. Chen, C. N. Weng, and J. S. Chen, “Cloak for curvilinearly anisotropic media in conduction", Applied Physics Letters, 93, 114103 (2008). Abstract.

**[15]**Supradeep Narayan and Yuki Sato, “Heat flux manipulation with engineered thermal materials", Physical Review Letters 108, 214303 (2012). Abstract.

**[16]**S. Guenneau, C. Amra, and D. Veynante, ‘’Transformation thermodynamics: cloaking and concentrating heat flux’’, Optics Express, 20, 8207 (2012). Abstract.

Labels: Complex System 2, Invisibility Cloak 2, Metamaterial 2

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