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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 5 — Mar. 11, 2013
  • pp: 6578–6583
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Anisotropic conductivity rotates heat fluxes in transient regimes

Sébastien Guenneau and Claude Amra  »View Author Affiliations


Optics Express, Vol. 21, Issue 5, pp. 6578-6583 (2013)
http://dx.doi.org/10.1364/OE.21.006578


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Abstract

We present a finite element analysis of a diffusion problem involving a coated cylinder enabling the rotation of heat fluxes. The coating consists of a heterogeneous anisotropic conductivity deduced from a geometric transformation in the time dependent heat equation. In contrast to thermal cloak and concentrator, specific heat and density are not affected by the transformation in the rotator. Therein, thermal flux diffuses from region of lower temperature to higher temperature, leading to an apparent negative conductivity analogous to what was observed in transformed thermostatics. When a conducting object lies inside the rotator, it appears as if rotated by certain angle to an external observer, what can be seen as a thermal illusion. A structured rotator is finally proposed inspired by earlier designs of thermostatic and microwave rotators.

© 2013 OSA

1. Introduction

Six years ago, it was suggested by Pendry, Schurig and Smith that an object surrounded by a coating consisting of an heterogeneous anisotropic material becomes invisible for electromagnetic waves [1

1. J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780–1782 (2006) [CrossRef] [PubMed] .

]. The theoretical idea based on geometric transformations in the Maxwell’s equations [2

2. D. Van Dantzig, “Electromagnetism, independent of metrical geometry,” Proc. Kon. Ned. Akad. v. Wet. 37, 521–531 (1934).

] has been since then confirmed by some experiments. Leonhardt independently analyzed conformal invisibility devices using the Schrödinger equation, which suppresses the anisotropy of the cloak at the cost of constraining the cloaking to relatively small wavelengths compared to the object to hide [3

3. U. Leonhardt, “Optical Conformal Mapping,” Science 312, 1777–1780 (2006) [CrossRef] [PubMed] .

]. A plasmonic route to invisibility was proposed by Alu and Engheta [4

4. A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 95, 016623 (2005) [CrossRef] .

] but it relies on a specific knowledge of the shape and material properties of the object being concealed. Last, Milton and Nicorovici proposed to cloak a countable set of line sources using anomalous resonance when it lies in the close neighborhood of a cylindrical coating filled with a negative refractive index material [5

5. G. Milton and N. A. P. Nicorovici, “On the cloaking effects associated with anomalous localised resonance,” Proc. R. Soc. London Ser. A 462, 3027–3059 (2006) [CrossRef] .

, 6

6. N. A. P. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, “Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance,” Opt. Express 15, 6314–6323 (2007) [CrossRef] [PubMed] .

].

In the present article, we build upon our recent proposal of cloaking in thermodynamics [7

7. S. Guenneau, C. Amra, and D. Veynante, “Transformation thermodynamics: cloaking and concentrating heat flux, Opt. Express 20, 8207–8218 (2012) [CrossRef] [PubMed] .

], which relies on a covariant formulation of the heat equation [8

8. H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids (Oxford University Press, 1959).

] and study a rotator for heat fluxes, which is a counterpart of the transformed medium Chen and Chan introduced in optics [9

9. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105 (2007) [CrossRef] .

, 10

10. H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and Experimental Realization of a Broadband Transformation Media Field Rotator at Microwave Frequencies,” Phys. Rev. Lett. 102(18), 183903 (2009) [CrossRef] [PubMed] .

]. It is also worthwhile noticing mathematicians working in the field of inverse problems were already aware of the counter-intuitive physics of the anisotropic conducting equation, albeit in a different context: cloaking in electric impedance tomography [11

11. A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas. 24, 413–419 (2003) [CrossRef] [PubMed] .

]. Interestingly, the isomorphism between the anisotropic conductivity and thermostatic equations make it possible to control the pathway of heat flux in a stationary setting, as observed by Fan et al. [12

12. C.Z. Fan, Y. Gao, and J. P. Huang, “Shaped graded materials with an apparent negative thermal conductivity,” Appl. Phys. Lett. 92, 251907 (2008) [CrossRef] .

] (see also [6

6. N. A. P. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, “Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance,” Opt. Express 15, 6314–6323 (2007) [CrossRef] [PubMed] .

] for analogous cloaking in electrostatics) and experimentally validated by Narayano and Sato [13

13. S. Narayana and Y. Sato, “Heat flux manipulation with engineered thermal materials,” Phys. Rev. Lett. 108, 214303 (2012) [CrossRef] [PubMed] .

]. However, unlike for [12

12. C.Z. Fan, Y. Gao, and J. P. Huang, “Shaped graded materials with an apparent negative thermal conductivity,” Appl. Phys. Lett. 92, 251907 (2008) [CrossRef] .

, 13

13. S. Narayana and Y. Sato, “Heat flux manipulation with engineered thermal materials,” Phys. Rev. Lett. 108, 214303 (2012) [CrossRef] [PubMed] .

], the object and rotator which we study here lie well within the intense near field of the heat source, and time plays an essential role, as in experiments on thermodynamics cloaking [14

14. R. Schittny, M. Kadic, S. Guenneau, and M. Wegener, “Experiments on transformation thermodynamics: Molding the flow of heat,” arXiv:1210.2810

] from Wegener’s group.

2. Transformed heat equation for an area preserving transform

We consider the two-dimensional diffusion equation in a bounded cylindrical domain Ω with a source p
ρ(x)c(x)ut=(κ(x)u)+p(x,t),
(1)
where u represents the distribution of temperature evolving with time t > 0, at each point x = (x, y) in Ω. Moreover, κ is the thermal conductivity (W.m−1.K−1 i.e. watt per meter kelvin in SI units), ρ is the density (kg.m−3 i.e. kilogram per cubic meter in SI units) and c the specific heat (or thermal) capacity (J.K−1.kg−1 i.e. joule per kilogram kelvin in SI units). It is customary to let κ go in front of the spatial derivatives when the medium is homogeneous. However, here we consider a heterogeneous medium, hence the spatial derivatives of κ are derivatives taken in distributional sense, thereby ensuring continuity of the heat flux κu is encompassed in Eq. (1)).

In this article, we further consider a source with a time step (Heaviside) variation and a singular (Dirac) spatial variation, that is: p(x, t) = p0H(t)δ(xx0), with H the Heaviside function and Delta the Dirac distribution. This means that the source term is constant throughout time t > 0, while it is spatially localized on the line x = x0.

In the sequel, we adopt a covariant approach of the heat equation. For this, let us consider a map from a co-ordinate system {x′, y′} to the co-ordinate system {x, y} given by the transformation characterized by x(x′, y′) and y(x′, y′). Note that it is the transformed domain and co-ordinate system that are mapped onto the initial domain with Cartesian coordinates, and not the opposite. This change of co-ordinates is characterized by the transformation of the differentials through the Jacobian:
(dxdy)=Jxx(dxdy),withJxx=(x,y)(x,y).
(2)

The only thing to do in the transformed coordinates is to replace the materials (often homogeneous and isotropic) by equivalent ones that are inhomogeneous (their thermal characteristics are no longer piecewise constant but merely depend on x′, y′ co-ordinates) and anisotropic ones (tensorial nature) whose properties are given by a new conductivity in Eq. (1) [7

7. S. Guenneau, C. Amra, and D. Veynante, “Transformation thermodynamics: cloaking and concentrating heat flux, Opt. Express 20, 8207–8218 (2012) [CrossRef] [PubMed] .

]
κ__=J1κJtdet(J)=κJ1JTdet(J)=κT1,
(3)
which is matrix-valued (anisotropic and hetereogeneous). Here, T = JTJ/det(J) is a representation of the metric tensor, which is associated with a distorted mesh, see Fig. 1(d) for the case of a rotator. Moreover, the product of density by heat capacity in the left hand side of Eq. (1) is now multiplied by the determinant of the Jacobian matrix J of the transformation.

Fig. 1 Normalized temperature field for a source located on top i.e. along the line x0 = (x, 4.10−4m) which diffuses heat in a medium with diffusivity κ/() = 1.10−5m2.s−1 containing a rotator of center (0, 0) and radii R1 = 5.10−5m and R2 = 3.10−4m, and rotation angle θ0 = 3π/4. The temperature is normalized throughout time on the upper side of the cell. Snapshots of temperature distribution at t = 0.005s (a), t = 0.01s (b) and t = 0.1s (c); (d) The mesh formed by streamlines and isothermal values in the long time regime t ≥ 0.1s illustrates the deformation of the transformed thermal space.

The associated time-dependent transformed heat equation takes the form [7

7. S. Guenneau, C. Amra, and D. Veynante, “Transformation thermodynamics: cloaking and concentrating heat flux, Opt. Express 20, 8207–8218 (2012) [CrossRef] [PubMed] .

]:
ρ(x)c(x)det(J)ut=(κT1u)+det(J)p(x,t).
(4)
Here, we would like to consider a geometric transform which is area preserving i.e. such that det(J) = 1. This greatly simplifies the control of heat fluxes in transient regimes, compared to what was done in [7

7. S. Guenneau, C. Amra, and D. Veynante, “Transformation thermodynamics: cloaking and concentrating heat flux, Opt. Express 20, 8207–8218 (2012) [CrossRef] [PubMed] .

]. For this, let us now consider the following transform [9

9. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105 (2007) [CrossRef] .

]
r=r,θ=θ+θ0,r<R1,r=r,θ=θ,r>R2,r=r,θ=θ+θ0f(R2)f(r)f(R2)f(R1),R1<r<R2,
(5)
which rotates the inner disc r = R1 through an angle θ0, and gradually diminishes the rotation angle through the coronna R1 < r < R2 up to a vanishing rotation angle on the outer boundary of the disc r = R2. The effect on the metric is shown for 2 angles θ0 in Fig. 1(d) and Fig. 2(d).

Fig. 2 Normalized temperature field for a source located on top i.e. along the line x0 = (x, 4.10−4m) which diffuses heat in a medium with diffusivity κ/() = 1.10−5m2.s−1 containing a rotator of center (0, 0) and radii R1 = 1.10−4m and R2 = 3.10−4m, and rotation angle θ0 = π/2. The temperature is normalized throughout time on the upper side of the cell. Snapshots of temperature distribution at t = 0.005s (a), t = 0.01s (b) and t = 0.1s (c); (d) The mesh formed by streamlines and isothermal values in the long time regime t ≥ 0.1s illustrates the deformation of the transformed thermal space.

It remains to compute the transformation matrix T associated with this transformation and to express it in the Cartesian co-ordinates (x′, y′). As a result, we shall obtain the anisotropic conductivity characterizing the thermic metamaterial in the cylindrical rotator defined by the radii R1 (interior radius) and R2 (exterior radius). The Jacobian matrix Jxx is therefore merely the product of three elementary Jacobians : Jxx = JxθJθθJθx, where Jxθ = (x, y)/(r, θ) = R(θ)diag(1, r) is the Jacobian associated with the change to cylindrical co-ordinates (r, θ), which involves the rotation (unimodular) matrix R(θ). In the same way, the Jacobian Jθx associated with the change to Cartesian co-ordinates (x′, y′) is such that Jθx = (r′, θ′)/(x′, y′) = diag(1, 1/r′)R(−θ′), using the fact that R(θ′)−1 = R(−θ′). Finally, Jθθ = (r, θ)/(r′, θ′) is the azimuthally stretched cylindrical co-ordinates as proposed in [9

9. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105 (2007) [CrossRef] .

]. Importantly, det(Jθθ) = 1 as the azimuthal stretch preserves the overall volume within the disc r = r′ ≤ R2 (unitary transform). As a first consequence, the transformation does not affect the product of heat capacity and density, which do not play a particular role in the design of thermic rotator.

Altogether, the material properties of the thermic rotator are described by the transformation matrix T through its inverse, and using again the fact that θ = θ′, we obtain T1=Jxx1JxxTdet(Jxx)=R(θ)1Jθθ1R(θ)1R(θ)TJθθTR(θ)T=R(θ)Jθθ1JθθTR(θ) that we give explicitly in Cartesian coordinates:
T1=((T1)11(T1)12(T1)21(T1)22),
(6)
where
(T1)11=12tcos(θ)sin(θ)+t2cos2(θ),(T1)22=1+2tcos(θ)sin(θ)+t2sin2(θ),(T1)12=(T1)21=t2cos(θ)sin(θ)t(cos2(θ)sin2(θ)),
(7)
and t = θ0rf′(r)/(f(R2) − f (R1)) = θ0r/(R2R1).

3. Illustrative numerical examples

The anisotropic heterogeneous conductivity in the rotator should be now inserted in the transformed heat equation, which we want to solve numerically. The finite element method is the tool of choice, and is implemented in the commercial package COMSOL MULTIPHYSICS.

In Fig. 1, we show the result of considering a rotation angle 3π/4 in the transformation matrix. One can clearly see that the isotherms are tilted through an angle 3π/4 inside the inner disc r′ < R1 compared to the isotherms generated by the heat source outside the rotator: This leads to an apparent negative conductivity, whereby heat diffuses from regions of low to high temperature, a counterintuitive effect already encountered in thermostatic contexts [12

12. C.Z. Fan, Y. Gao, and J. P. Huang, “Shaped graded materials with an apparent negative thermal conductivity,” Appl. Phys. Lett. 92, 251907 (2008) [CrossRef] .

, 13

13. S. Narayana and Y. Sato, “Heat flux manipulation with engineered thermal materials,” Phys. Rev. Lett. 108, 214303 (2012) [CrossRef] [PubMed] .

]. We note that this apparent negative conductivity effect is markedly enhanced at short times: Thermal exchanges are enhanced in the transient regime. Importantly, heat fluxes are smoothly rotated within the region R1 < r′ < R2, so that the rotator is itself invisible thermally for an external observer.

In Fig. 2 we consider a rotation angle θ0 = π/2 which results in a smaller anisotropy along the azimuthal angle, and a slightly less enhanced apparent negative conductivity and thermal exchanges. In Fig. 3, we consider a conducting object inside the inner disc, and we show that it appears as if rotated by an angle π/2 to an external observer when we take θ0 = π/2. Finally, we propose a structured design of a rotator for thermodynamics in Fig. 4.

Fig. 3 Normalized temperature field for a source located on top i.e. along the line x0 = (x, 4.10−4m) which diffuses heat in a medium with diffusivity κ/() = 1.10−5m2.s−1 containing a rotator of center (0, 0) and radii R1 = 2.10−4m and R2 = 3.10−4m, and rotation angle θ0 = π/2. The temperature is normalized throughout time on the upper side of the cell. Snapshots of temperature distribution at t = 0.005s (a), (b) and t = 0.1s (c), (d); The rectangular object has a diffusivity one hundred times smaller than that of the surrounding medium and is rotated by an angle π/2 in (b) and (d) compared to (a) and (c); The temperature field in (a),(b) and (c),(d) is exactly the same oustide the rotator.
Fig. 4 Normalized temperature field for a source located on the left-hand side i.e. along the line x0 = (−7.10−4m, y) which diffuses heat in a medium with diffusivity κ/() = 1.10−5m2.s−1 containing a structured ring of center (0, 0) and radii R1 = 2.10−4m and R2 = 4.10−4m consisting of four rows with 50 insulating sectors (diffusivity one hundred times smaller than that of the surrounding medium) in each row; Rows are slightly rotated with respect to one another (through an angle θ = π/20), and separated by three thin highly conducting wires (diffusivity 100 times that of surrounding medium) directed along the azimuthal angle. The temperature is normalized throughout time on the left-hand side of the cell. Snapshots of temperature distribution at t = 0.005s (a) and t = 0.02s (b); White curves represent the trajectories of heat fluxes. The achieved rotation of heat flux is π/4.

4. Conclusion

We used the equivalence between a geometric transformation and a change of characteristics of a metamaterial [7

7. S. Guenneau, C. Amra, and D. Veynante, “Transformation thermodynamics: cloaking and concentrating heat flux, Opt. Express 20, 8207–8218 (2012) [CrossRef] [PubMed] .

] (anisotropic conductivity) to compute the thermal response of the rotator proposed in [9

9. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105 (2007) [CrossRef] .

] in the context of optics. Importantly, the rotator design is based upon an area preserving transform, which greatly simplifies the transformed heat equation in transient regime. We proved that the rotator works even if the heat source is very close to the cloak (a fraction of thermal isovalues away), and the apparent negative thermal conductivity [12

12. C.Z. Fan, Y. Gao, and J. P. Huang, “Shaped graded materials with an apparent negative thermal conductivity,” Appl. Phys. Lett. 92, 251907 (2008) [CrossRef] .

, 13

13. S. Narayana and Y. Sato, “Heat flux manipulation with engineered thermal materials,” Phys. Rev. Lett. 108, 214303 (2012) [CrossRef] [PubMed] .

] is markedly enhanced at short times. The rotator enhances exchanges by smoothly rotating heat fluxes. We observed some mirage effect whereby a conducting object located inside the rotator seems to be tilted in accordance with the involved geometric transformation. Finally, a structured rotator was proposed, following designs for thermostatics [13

13. S. Narayana and Y. Sato, “Heat flux manipulation with engineered thermal materials,” Phys. Rev. Lett. 108, 214303 (2012) [CrossRef] [PubMed] .

] and microwaves [10

10. H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and Experimental Realization of a Broadband Transformation Media Field Rotator at Microwave Frequencies,” Phys. Rev. Lett. 102(18), 183903 (2009) [CrossRef] [PubMed] .

].

Acknowledgments

S.G. is thankful for an ERC funding through ANAMORPHISM project. The authors are also grateful for stimulating discussions with M. Zerrad.

References and links

1.

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780–1782 (2006) [CrossRef] [PubMed] .

2.

D. Van Dantzig, “Electromagnetism, independent of metrical geometry,” Proc. Kon. Ned. Akad. v. Wet. 37, 521–531 (1934).

3.

U. Leonhardt, “Optical Conformal Mapping,” Science 312, 1777–1780 (2006) [CrossRef] [PubMed] .

4.

A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 95, 016623 (2005) [CrossRef] .

5.

G. Milton and N. A. P. Nicorovici, “On the cloaking effects associated with anomalous localised resonance,” Proc. R. Soc. London Ser. A 462, 3027–3059 (2006) [CrossRef] .

6.

N. A. P. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, “Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance,” Opt. Express 15, 6314–6323 (2007) [CrossRef] [PubMed] .

7.

S. Guenneau, C. Amra, and D. Veynante, “Transformation thermodynamics: cloaking and concentrating heat flux, Opt. Express 20, 8207–8218 (2012) [CrossRef] [PubMed] .

8.

H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids (Oxford University Press, 1959).

9.

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105 (2007) [CrossRef] .

10.

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and Experimental Realization of a Broadband Transformation Media Field Rotator at Microwave Frequencies,” Phys. Rev. Lett. 102(18), 183903 (2009) [CrossRef] [PubMed] .

11.

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas. 24, 413–419 (2003) [CrossRef] [PubMed] .

12.

C.Z. Fan, Y. Gao, and J. P. Huang, “Shaped graded materials with an apparent negative thermal conductivity,” Appl. Phys. Lett. 92, 251907 (2008) [CrossRef] .

13.

S. Narayana and Y. Sato, “Heat flux manipulation with engineered thermal materials,” Phys. Rev. Lett. 108, 214303 (2012) [CrossRef] [PubMed] .

14.

R. Schittny, M. Kadic, S. Guenneau, and M. Wegener, “Experiments on transformation thermodynamics: Molding the flow of heat,” arXiv:1210.2810

OCIS Codes
(000.6850) General : Thermodynamics
(350.6830) Other areas of optics : Thermal lensing
(160.3918) Materials : Metamaterials
(230.3205) Optical devices : Invisibility cloaks

ToC Category:
Metamaterials

History
Original Manuscript: December 13, 2012
Manuscript Accepted: January 31, 2013
Published: March 8, 2013

Citation
Sébastien Guenneau and Claude Amra, "Anisotropic conductivity rotates heat fluxes in transient regimes," Opt. Express 21, 6578-6583 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-6578


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References

  1. J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science312, 1780–1782 (2006). [CrossRef] [PubMed]
  2. D. Van Dantzig, “Electromagnetism, independent of metrical geometry,” Proc. Kon. Ned. Akad. v. Wet.37, 521–531 (1934).
  3. U. Leonhardt, “Optical Conformal Mapping,” Science312, 1777–1780 (2006). [CrossRef] [PubMed]
  4. A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E95, 016623 (2005). [CrossRef]
  5. G. Milton and N. A. P. Nicorovici, “On the cloaking effects associated with anomalous localised resonance,” Proc. R. Soc. London Ser. A462, 3027–3059 (2006). [CrossRef]
  6. N. A. P. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, “Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance,” Opt. Express15, 6314–6323 (2007). [CrossRef] [PubMed]
  7. S. Guenneau, C. Amra, and D. Veynante, “Transformation thermodynamics: cloaking and concentrating heat flux, Opt. Express20, 8207–8218 (2012). [CrossRef] [PubMed]
  8. H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids (Oxford University Press, 1959).
  9. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett.90, 241105 (2007). [CrossRef]
  10. H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and Experimental Realization of a Broadband Transformation Media Field Rotator at Microwave Frequencies,” Phys. Rev. Lett.102(18), 183903 (2009). [CrossRef] [PubMed]
  11. A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas.24, 413–419 (2003). [CrossRef] [PubMed]
  12. C.Z. Fan, Y. Gao, and J. P. Huang, “Shaped graded materials with an apparent negative thermal conductivity,” Appl. Phys. Lett.92, 251907 (2008). [CrossRef]
  13. S. Narayana and Y. Sato, “Heat flux manipulation with engineered thermal materials,” Phys. Rev. Lett.108, 214303 (2012). [CrossRef] [PubMed]
  14. R. Schittny, M. Kadic, S. Guenneau, and M. Wegener, “Experiments on transformation thermodynamics: Molding the flow of heat,” arXiv:1210.2810

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