Remote cooling by a novel thermal lens with anisotropic positive thermal conductivity

A novel thermal lens that can achieve a remote cooling effect is designed by transformation thermodynamics. The effective distance between the separate hot source and cold source is shortened by our shelled thermal lens without any negative thermal conductivity. Numerical simulations verify the performance of our thermal lens. Based on the effective medium theory, we also propose a practical way to realize our lens using two-layered isotropic thermal media that are both found in nature. The proposed thermal lens will have potential applications in remote temperature control and in creating other thermal illusions.

Transformation Optics (TO) can be utilized to design special electromagnetic media with pre-designed functions by using a coordinate transformation from the virtual space to the real space [1][2][3] . Many novel optical/electromagnetic devices have been designed by TO, including invisibility cloakings [4][5][6] , novel lenses 7,8 , field rotators 9-11 , concentrators 12 , illusion optical devices 13,14 , and others. Based on the heat equation's form-invariant property under coordinate transformations, transformation thermodynamics (TT) was first proposed by Guenneau, S. et al. 15 . In recent years, manipulating heat flux by a coordinate transformation method has attracted much attention and developed many novel thermal devices, e.g. thermal cloakings [16][17][18] , concentrators for heat collections 19,20 , heat hyper-lenses for heat focusing 21 , heat hoses 22 , heat flux rotators 23 , thermal diodes 24 , heat illusion devices [25][26][27][28][29] , novel thermal lenses 30,31 , devices for a heat directional transmission 32 , etc. In 2016, our group proposed a novel thermal lens based on spatial translations and folding coordinate transformations together to achieve a remote heating/cooling operation on temperature distribution 31 . However, it is still challenging to realize the proposed thermal lens due to the negative thermal conductivity introduced from the spatial folding transformation. In some applications, we need a remote cooling technique. In some applications, we want to achieve a very high temperature in an given region (the target region) while not influencing its neighboring area (i.e. the temperature is low around the target region). We need to use a remote cooling technique to cool down the temperature around the target region but not influencing the hot target region. For example, in laser tumor therapy, we hope normal cells around the targeted tumor cell are not killed by the high temperature produced by the laser. In this case, we need a cooling technique to keep the temperature in normal cells relatively low. The cooling source cannot be set inside the human body, which means that a remote cooling technique is required. Such a remote cooling can also be applied to cool down the neighboring area around an inevitable hot site through a remote cold source. In this study, we use the spatial translation and compression transformations together to achieve a remote cooling effect using a thermal lens with homogeneous positive anisotropic thermal conductivity. Figure 1(a) shows the basic function of our thermal lens. The whole lens is composed of five different regions of media with homogenous anisotropic thermal conductivity (indicated by different labels). The hot source (a red star) and cool source (blue star) are set in the left and right sides of the region V, respectively, within the shelled structure (i.e. our lens). The function of this shelled lens is to reduce the effective spatial distance between the hot source and the cool source, which makes the cool source effectively much closer to the hot source than their real distance. The effective cooling region is the region outside the shelled structure ABCD. We now show how to design the proposed lens using TT. Figure 1(b) and (c) show the corresponding relation between the reference space and the real space. The quantities with and without primes are those in the real space and the reference space, respectively, throughout the paper. Two triangular regions, B 1 A 1 D 1 and B 1 ′ C 1 D 1 ′ , in real space are spatially translated from B 0 A 0 D 0 and B 0 ′ C 0 D 0 ′ , respectively, in the reference space. The thermal materials in these two regions are still the same as the background medium due to the spatial translational transformation. The middle region (i.e. Region V) and the shelled regions (i.e. regions I-IV) are compressed and stretched, respectively, from the real space to the reference space. The whole coordinate transformation can be summarized as:

Methods
As shown in Fig. 1(c), a, b and d determine the geometrical size of our shell in the real space. Note that sgn is the sign function, which is defined by: The thermal conductivity in each region can be calculated with the help of TT 15 . Only the regions I-V in Fig. 1(a) or (c) have thermal conductivity different from the background medium (i.e. other regions without labels) with a constant isotropic thermal conductivity κ 0 . The required thermal conductivity in the region V is given by: The hot source (red star) and cool source (blue star) are set in the background medium. The coordinate transformation between the reference space (b) and the real space (c). In (c), quadrangles  The required thermal conductivity in the other four regions I-IV can be summarized by: x x x y x y y y z z The distance between the two triangular regions B 1 A 1 D 1 and B 1 ′ C 1 D 1 ′ in the real space is d, which is also the thickness of the region V. The function of the whole coordinate transformation (i.e. the effect of the transformation medium) is to shorten the effective distance between these two regions. Actually, the effective distance between these two regions is d 0 if the other regions I-V are filled by our lens described by Eqs (3) and (4). Since the effective distance between the two triangular regions is shortened (e.g. d 0 « d), the cooling effect would be improved if we set a hot source and a cold source at any position in these two triangular regions, separately.
We use D r to indicate the real distance between the hot and cold sources, which includes the distance from the two sources to the edge of region V and the physical thickness of region V (see Fig. 1(a)). Since the function of our lens is to shorten the effective distance between the two triangular regions B 1 A 1 D 1 and B 1 ′ C 1 D 1 ′ (i.e. the thickness of region V), the effective distance between the hot and cold sources are D e = D r − d + d 0 .

Results
Numerical simulations are given for a hot point source (375 K) and a cold point source (275 K) separated by a fixed distance D r = 12 cm with and without our lens, respectively, in Fig. 2(a) and (d). The temperature distribution outside our lens is much lower than the case in which the lens is removed, which means that the cooling effect is greatly improved by introducing our lens.
We could expect that if d 0 is smaller, the effective spatial distance between the hot source and the cold source is smaller (D e = D r − d + d 0 ), which indicates a better cooling effect. As shown in Fig. 2, the cooling effect improves as d 0 becomes smaller while keeping the physical distance between the hot source and the cold source D r unchanged.
Another interesting aspect is that, if the real distance between the hot source and the cold source is unchanged, and only the parameter d in Eq. (4), which determines the thermal media of our thermal lens, is changed (other parameters a, b, d 0 do not change), the performance of our thermal lens also changes. Considering the effective distance between the hot and cold sources are D e = D r − d + d 0 , and thus the larger d, the better cooling effect (see Fig. 3). Note that d cannot be too much larger, given that the geometrical size of the whole lens is fixed (i.e. d/2 + a < b). Experiment design. Since thermal conductivity in each region of the whole lens is positive, it can be realized by naturally-available materials. We design a feasible thermal lens with layered isotropic thermal materials based on the effective medium theory. The background medium is chosen to be thermal epoxy with a constant thermal conductivity of 3.4 W/(m·K). The size of the lens is chosen as d = 20 cm, d 0 = 1 cm, a = 5 cm, b = 20 cm. We use two layered isotropic thermal media to achieve an effective anisotropic thermal conductivity: where κ A and κ B are thermal conductivities of two isotropic media. κ \\ and κ ⊥ are the effective anisotropic thermal conductivities along and orthogonal to the direction of interface between two isotropic media, respectively. f A and f B are the filling factor of the medium A and the medium B, respectively, which satisfy f A + f B = 1.
For the region V in the middle of our device, we choose copper with thermal conductivity of 394 W/(m·K) as the medium A. κ \\ and κ ⊥ are determined by parameters d and d 0 (i.e. the real distance and the effective distance between the hot source and the cold source, respectively), and calculated by Eq. (3): κ \\ = 68 and κ ⊥ = 0.17. The required filling factor of the medium A and the thermal conductivity of the medium B are two unknown quantities which can be solved from Eq. (5): f A = 17.2% and κ B = 0.141 W/(m·K). We can use PVC with thermal conductivity of 0.14 W/(m·K) as the medium B.
For the Regions I, II, III and IV, we only need to design two layered-media in Region I, and media in other regions can be obtained by the symmetry transformation due to the symmetry of the whole lens with respect to the x′ and y′ axes (see the structure in Fig. 4(a)). The direction of the principal axis of the anisotropic medium in Region I can be obtained by diagonalizing the matrix in Eq. (4) 33 : Figure 2. The finite element method (FEM) simulation results. We plot the temperature distribution outside our thermal lens (colored white): (a)-(c) with our thermal lens, and (d) without our thermal lens. The outer boundary is set by the room temperature (295 K). We set one hot point source with fixed temperature 375 K and a cold point source with fixed temperature 275 K inside our thermal lens (marked by red and blue stars, respectively). The background medium is chosen as a thermal epoxy with a conductivity of 3.4 W/(m·K), a mass density of 3.1e 3 kg/m 3 , and a thermal capacity of 550 J/(kg·K). The conductivity of our thermal lens is given by Eqs (3) and (4)   χ-γ system is the local principal axis system, in which the anisotropic thermal conductivity in Eq. (4) can be expressed by a diagonal matrix. For the thermal lens with the same parameters as in Fig. 2, the angle between  the x′ -axis and χ-axis is θ = 71[deg], and the two principal values of thermal conductivity are 1.6 W/(m·K) and 2.9 W/(m·K) (i.e. κ \\ = 2.9 and κ ⊥ = 1.6). We choose bismuth, with thermal conductivity of nearly 8 W/mK, as medium A. By solving Eq. (5), we can obtain the required thermal conductivity of the medium B and the filling factor: κ B = 1.275 W/(m·K) and f A = 24.2%. Thus, fused silica with thermal conductivity of 1.32 W/(m·K) can be used as medium B 34 . The structure of the lens is shown in Fig. 4(a). Numerical simulation results of the lens composed of a two-layered medium are given in Fig. 4(d), which shows good cooling effect compared to the case where the lens is removed in Fig. 4(b).
We should note that the purpose of our design is to cool down the region outside the shell ABCD in Fig. 1(c) by our lens. We only plot the temperature distribution outside the shell in all our simulations (e.g. in Figs 2, 3 and 4). If we add the temperature distribution of other regions (e.g. inside our lens), we have to rescale the colorbar of the figures and consequently it will be hard to see the difference of the temperature distribution outside the shell for the cases with and without the lens from the figures.

Conclusion
By combining spatial translations and compression transformations, we designed a novel thermal lens for a remote cooling function. The hot source and the cold source can be separated by a certain distance while keeping a good cooling effect by applying our thermal lens around them. Based on the effective medium theory, we also design a specific lens composed of a layered isotropic thermal medium that is available in nature. Numerical simulations verify the function of the proposed lens.