## Anisotropic conductivity rotates heat fluxes in transient regimes |

Optics Express, Vol. 21, Issue 5, pp. 6578-6583 (2013)

http://dx.doi.org/10.1364/OE.21.006578

Acrobat PDF (1754 KB)

### 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

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

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] .

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] .

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] .

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

## 2. Transformed heat equation for an area preserving transform

*p*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

*κ*∇

*u*is encompassed in Eq. (1)).

*p*(

**x**,

*t*) =

*p*

_{0}

*H*(

*t*)

*δ*(

**x**−

**x**

_{0}), 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**=

**x**

_{0}.

*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:

*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] .

**T**=

**J**

^{T}**J**/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.

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

**J**) = 1. This greatly simplifies the control of heat fluxes in transient regimes, compared to what was done in [7

**20**, 8207–8218 (2012) [CrossRef] [PubMed] .

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

*r*=

*R*

_{1}through an angle

*θ*

_{0}, and gradually diminishes the rotation angle through the coronna

*R*

_{1}<

*r*<

*R*

_{2}up to a vanishing rotation angle on the outer boundary of the disc

*r*=

*R*

_{2}. The effect on the metric is shown for 2 angles

*θ*

_{0}in Fig. 1(d) and Fig. 2(d).

**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

*R*

_{1}(interior radius) and

*R*

_{2}(exterior radius). The Jacobian matrix

**J**

_{xx}_{′}is therefore merely the product of three elementary Jacobians :

**J**

_{xx}_{′}=

**J**

_{x}_{θ}**J**

_{θθ}_{′}

**J**

_{θ}_{′}

_{x}_{′}, where

**J**

*=*

_{x}_{θ}*∂*(

*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

**J**

_{θ}_{′}

_{x}_{′}associated with the change to Cartesian co-ordinates (

*x*′,

*y*′) is such that

**J**

_{θ}_{′}

_{x}_{′}=

*∂*(

*r*′,

*θ*′)/

*∂*(

*x*′,

*y*′) = diag(1, 1/

*r*′)

**R**(−

*θ*′), using the fact that

**R**(

*θ*′)

^{−1}=

**R**(−

*θ*′). Finally,

**J**

_{θθ}_{′}=

*∂*(

*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] .

**J**

_{θθ}_{′}) = 1 as the azimuthal stretch preserves the overall volume within the disc

*r*=

*r*′ ≤

*R*

_{2}(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.

**T**through its inverse, and using again the fact that

*θ*=

*θ*′, we obtain

*t*=

*θ*

_{0}

*rf*′(

*r*)/(

*f*(

*R*

_{2}) −

*f*(

*R*

_{1})) =

*θ*

_{0}

*r*/(

*R*

_{2}−

*R*

_{1}).

## 3. Illustrative numerical examples

*π*/4 in the transformation matrix. One can clearly see that the isotherms are tilted through an angle 3

*π*/4 inside the inner disc

*r*′ <

*R*

_{1}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. S. Narayana and Y. Sato, “Heat flux manipulation with engineered thermal materials,” Phys. Rev. Lett. **108**, 214303 (2012) [CrossRef] [PubMed] .

*R*

_{1}<

*r*′ <

*R*

_{2}, so that the rotator is itself invisible thermally for an external observer.

*θ*

_{0}=

*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

*θ*

_{0}=

*π*/2. Finally, we propose a structured design of a rotator for thermodynamics in Fig. 4.

## 4. Conclusion

**20**, 8207–8218 (2012) [CrossRef] [PubMed] .

**90**, 241105 (2007) [CrossRef] .

**92**, 251907 (2008) [CrossRef] .

**108**, 214303 (2012) [CrossRef] [PubMed] .

**108**, 214303 (2012) [CrossRef] [PubMed] .

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

## References and links

1. | J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science |

2. | D. Van Dantzig, “Electromagnetism, independent of metrical geometry,” Proc. Kon. Ned. Akad. v. Wet. |

3. | U. Leonhardt, “Optical Conformal Mapping,” Science |

4. | A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E |

5. | G. Milton and N. A. P. Nicorovici, “On the cloaking effects associated with anomalous localised resonance,” Proc. R. Soc. London Ser. A |

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 |

7. | S. Guenneau, C. Amra, and D. Veynante, “Transformation thermodynamics: cloaking and concentrating heat flux, Opt. Express |

8. | H. S. Carslaw and J. C. Jaeger, |

9. | H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. |

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. |

11. | A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas. |

12. | C.Z. Fan, Y. Gao, and J. P. Huang, “Shaped graded materials with an apparent negative thermal conductivity,” Appl. Phys. Lett. |

13. | S. Narayana and Y. Sato, “Heat flux manipulation with engineered thermal materials,” Phys. Rev. Lett. |

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

- J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science312, 1780–1782 (2006). [CrossRef] [PubMed]
- D. Van Dantzig, “Electromagnetism, independent of metrical geometry,” Proc. Kon. Ned. Akad. v. Wet.37, 521–531 (1934).
- U. Leonhardt, “Optical Conformal Mapping,” Science312, 1777–1780 (2006). [CrossRef] [PubMed]
- A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E95, 016623 (2005). [CrossRef]
- 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]
- 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]
- S. Guenneau, C. Amra, and D. Veynante, “Transformation thermodynamics: cloaking and concentrating heat flux, Opt. Express20, 8207–8218 (2012). [CrossRef] [PubMed]
- H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids (Oxford University Press, 1959).
- H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett.90, 241105 (2007). [CrossRef]
- 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]
- A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas.24, 413–419 (2003). [CrossRef] [PubMed]
- 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]
- S. Narayana and Y. Sato, “Heat flux manipulation with engineered thermal materials,” Phys. Rev. Lett.108, 214303 (2012). [CrossRef] [PubMed]
- 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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