Converting the patterns of local heat flux via thermal illusion device

Converting the patterns of local heat flux via thermal illusion device N. Q. Zhu,1 X. Y. Shen,1,a and J. P. Huang1,2,b 1Department of Physics, State Key Laboratory of Surface Physics, and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Fudan University, Shanghai, 200433, China 2Collaborative Innovation Center of Advanced Microstructure, Nanjing University, Nanjing, 210093, China


I. INTRODUCTION
In 2008, Fan et al. 1 started to adopt the coordinate transformation approach to propose a class of thermal metamaterials working as thermal cloaking, which makes heat flow around an "invisible" region and eventually returns to its original pathway as if the "invisible region" does not exist.This cloaking originates from the fact that the thermal conduction equation remains form-invariant under coordinate transformation, and it helps to pave a new way to steer heat flux.As a result, much attention has been paid by theorists and experimentalists, [2][3][4][5][6][7][8] and there come out a lot of thermal metamaterials with novel thermal properties beyond cloaking, 6,[10][11][12][13] such as concentrators (which are used to enhance the temperature gradient in a specific region), 3,14 inverters (which allow heat to apparently flow from a colder to a warmer region without violating the second law of thermodynamics), 1,15 rotators (which can rotate the flow of heat as if the heat comes from a different direction), 3,15 and camouflage. 16ontrolling the flow of heat enlightens us to design an illusion device for thermal conduction.8][19] This concept has been extended to other fields (say, acoustics [20][21][22] ) where wave equations dominate.Obviously, an illusion of thermal conduction not only offers a different way for controlling heat conduction but also has extensive applications in misleading the detectors of temperature distribution signatures.Just as our recent work shows, a thermal illusion device based on the thermal conduction equation can be created by adopting a complementary layer. 23However, the thermal conductivity of the complementary layer should be negative.For complying with the second law of thermodynamics, one must apply external work on the system. 10,13Inspired by the former work, 23 in order to build thermal illusion devices without using negative-conductivity materials, here we attempt to propose a new kind of thermal illusion device.To this end, we shall change the flow direction of local heat flux inside the device whereas the pattern of heat flux outside the device keeps unchanged (as if the device does not exist).That is, a phenomenon of thermal illusion is created through the thermal illusion device.In order to check whether the device works or not, we shall resort to two-dimensional finite-element simulations.

A. Coordinate transformation approach
Considering a typical thermal conduction process, heat flux is proportional to a temperature gradient.The thermal conduction equation can then be written as where ρ and C are the density and heat capacity respectively and T represents temperature evolving with time t at each point X = (x, y) in the space.In equation ( 1), κ is thermal conductivity, and Q is a heat source.For a steady state process, the distribution of temperature T is independent of time t, and thus the first term in equation ( 1) vanishes.Throughout this work, we suppose there is no heat source, Q = 0. Therefore, equation ( 1) can be reduced to ∇ • (−κ∇T) = 0. Upon a change of variable X = (x, y) → X = (x , y ) described by a Jacobian transformation matrix J, this equation, ∇ • (−κ∇T) = 0, takes the following form, where J T is the transposed matrix of J, and det(J) is the determinant of J. Thus the new thermal conductivity in the transformed space X can be expressed as

B. Transformation 1
The coordinate transformation is constructed in two-dimensional Cartesian coordinate systems and schematically presented in Figure 1.Suppose the radius of the sector is a, the length of OM is b and the side length of the square is 2c.
Similarly, we use X = (x, y) to represent an arbitrary point in the original space, and X = (x , y ) to denote the corresponding point in the transformed space.According to the geometrical relation that r = x 2 + y 2 and r = x 2 + y 2 , we can derive the mapping transformation in Regions I and II, Clearly, equation ( 4) is used to geometrically compress the triangle OCD (original space) to the sector O AB (transformed space).Accordingly the vertical lines in the original space is distorted to arcs in the transformed space.If we fill Regions I and II with appropriate anisotropic thermal conductivities developed by transformation norms, 1 parallel heat flux propagating through these regions can be converged.Equation (4) can then be easily derived as Therefore, we are able to obtain the anisotropic thermal conductivity tensor κ 1 of the illusion device, All To illustrate this conductivity tensor is positive and provide practicable parameters for experiment, 24 we can diagonalize it by rotating the principle axis by an appropriate angle α.This angle can be calculated by using the relation: where κ x y , κ x x , κ y y are the elements of the tensor κ 1 .As a result, we obtain the diagonalized tensor as .

C. Transformation 2
Transformation 1 mentioned above deals with the steady-state thermal conduction and converges the parallel heat flux on the vertex of the triangle, namely, from parallel patterns to nonparallel patterns.For the sake of completeness, here we want to attempt an inverse behavior, i.e., from non-parallel patterns to parallel patterns.To proceed, we assume a temperature field associated with an extremely small ring located at the centre where heat flow diverges from it, and we aim to design a device converting the dispersed heat flux (non-parallel patterns) into parallel patterns.
A proper approach to achieve this goal is to make the following transformation that is illustrated in Figure 2. Suppose the radius of the small circle is a, the side length of the square is 2b and the radius of the big circle is √ 2b.We extend the sector O AB (original space) to the triangle OCD (transformed space).The geometrical relation of the transformation means that each point in an arc located in the original space will be mapped to a vertical segment in the transformed space.As a result, when the divergent heat flux reaches the device, they will change to a parallel pattern due to the distorted coordinates.
Also, we use X = (x, y) and X = (x , y ) to respectively represent an arbitrary point in the original space and the corresponding point in the transformed space.We obtain the mapping transformation in Regions I and III, The thermal conductivity κ 2 for Regions I and III can be derived from equation (3), With the help of the rotation matrix, we can derive the thermal conductivity κ 2 for Regions II and IV, Also, the values of the components of κ 2 and κ 2 do not have any relation with either a or b and are only given by the coordinates of the points.Incidentally, there exist det(κ 2 ) = 1 and det(κ 2 ) = 1.That is, the two tensors are both positive, which can also be confirmed by using the same diagonalization method as we adopted at the end of Section II B (for Transformation 1).Since Transformation 2 is just a reverse operation of Transformation 1, the diagonalized conductivity tensor for Transformation 2 would be same as or similar to what we derived for Transformation 1.As a result, the diagonalized conductivity tensor of κ 2 is , and that of κ 2 is It should be noted that there is no ambiguity or singularity in the transformation.There has been a relationship between (x, y) and (x, y).From Eq. ( 5) and ( 7), we can derive: −1 ≤ y/x = y /x ≤ 1. Obviously, by using this relationship it can be proved that the parameters of the tensors are all limited.Therefore, there is no singularity in the transformation.

III. RESULTS
We perform finite element simulations based on commercial software COMSOL Multiphysics (http://www.comsol.com) to check whether the device works indeed.

B. Transformation 2
In Figure 5(a), we show the temperature distribution of a small circle settled in a host medium.Both of them are filled up with isotropic thermal conductivity κ 2 , which is set as 5 W/ (m•K).

IV. CONCLUSIONS
Since all the positive conductivity tensors we derived above are symmetrical matrices, which can be diagonalized by rotating the principle axis by an appropriate angle.The new simplified conductivity matrices can be easily used to implement the device with effective medium theories. 25hus, it should be more convenient to build such a thermal illusion device.
In a word, by adopting the coordinate transformation approach of heat conduction, we have designed a thermal illusion device composed of materials with positive conductivity tensors.The device is able to convert the parallel pattern of heat flux into non-parallel pattern (Figure 3) and vice versa (Figure 5).Meanwhile, the heat flux (or temperature distribution) outside the device keeps the same as if the device does not exist.The two-dimensional finite-element simulations have been used to confirm the desired effects.This work not only gives an approach to control heat flux, but also provides a new method to create thermal illusion facilities without adopting negative thermal conductivities.

FIG. 1 .
FIG. 1. Schematic diagram of Transformation 1.The radius of the sector is a, the length of OM is b, and the side length of the square is 2c.
FIG. 2. Schematic diagram of Transformation 2. The radius of the inner (small) circle is a, the side length of the square is 2b, and the radius of the outer (big) circle is √ 2b.

Figure 3 (FIG. 4 .
Figure 3(a) illustrates the temperature distribution in the original space, where the thermal conductivity κ 1 is 1 W/ (m•K).The temperature of the left boundary is set to be 273.15K while the right to be 373.15K.The upper and lower boundaries are set to be thermal insulation.The white arrows representing the heat flux are parallel.

FIG. 5 .
FIG. 5. Results of two-dimensional finite-element simulations.(a) The temperature distribution for a material with conductivity κ 2 = 5 W/ (m•K).The parameters are set as a = 0.50 m and b = 0.50 m.An extremely small ring is located at the centre with temperature (T ) at its edge, T = 493.15K; the boundary condition is set as T = 293.15K.The white arrows are diverging from the centre.(b) Simulation result of Transformation 2. Both the parameters and the boundary condition are the same as those in (a).The conductivities of materials filled in the circle (whose boundary is a white circle) are κ 2 and κ 2 , given by equation (8) and equation (9), and the thermal conductivity in the rest space is κ 2 = 5 W/ (m•K).The conditions of the extremely small ring and the boundary are same as those in (a).Within the region of the white circle, the heat flux represented by white arrows is distorted to a parallel pattern.