Metathermotics: Nonlinear thermal responses of core-shell metamaterials

Shuai Yang, Liujun Xu, and Jiping Huang

Phys. Rev. E 99, 042144 – Published 29 April 2019

Abstract

Thermal metamaterials based on core-shell structures have aroused wide research interest, e.g., in thermal cloaks. However, almost all the relevant studies only discuss linear materials whose thermal conductivities are temperature-independent constants. Nonlinear materials (whose thermal conductivities depend on temperatures) have seldom been touched; however, they are important in practical applications. This situation largely results from the lack of a general theoretical framework for handling such nonlinear problems. Here we study the nonlinear responses of thermal metamaterials with a core-shell structure in two or three dimensions. By calculating the effective thermal conductivity, we derive the nonlinear modulation of a nonlinear core. Furthermore, we reveal two thermal coupling conditions, under which this nonlinear modulation can be efficiently manipulated. In particular, we reveal the phenomenon of nonlinearity enhancement. Then this theory helps us to design a kind of intelligent thermal transparency devices, which can respond to the direction of thermal fields. The theoretical results and finite-element simulations agree well with each other. This work not only offers a different mechanism to achieve nonlinearity modulation and enhancement in thermotics, but also suggests potential applications in thermal management, including illusion.

Received 2 February 2019

DOI:https://doi.org/10.1103/PhysRevE.99.042144

Physics Subject Headings (PhySH)

Heat transfer Thermal properties

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Shuai Yang, Liujun Xu, and Jiping Huang^*

Department of Physics, State Key Laboratory of Surface Physics, and Key Laboratory of Micro and Nano Photonic Structures (MOE), Fudan University, Shanghai 200433, China

^*jphuang@fudan.edu.cn

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Vol. 99, Iss. 4 — April 2019

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Images

Figure 1
Schematic diagrams of (a) two-dimensional or (b) three-dimensional core-shell structure in Cartesian coordinate system; (b) only shows one eighth of the structure for the sake of clarity. The core or shell radius is $r_{1}$ or $r_{2}$ ; the core, shell, and matrix have thermal conductivities of $κ_{c} (T)$ , ${\overset{\leftrightarrow}{κ}}_{s}$ , and $κ_{m}$ , respectively. Other details can be found in the text.
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Figure 2
Two-dimensional simulation results. (a1)–(c1) show the nonlinear modulation $η$ ( $= χ_{e} / χ_{c}$ ) under the thermal coupling conditions determined by Eqs. (10), (11) with three variables: (a1) $κ_{θ θ} / κ_{r r}$ , (b1) $p_{1}$ , and (c1) $p_{2}$ . The solid lines in (a1)–(c1) are calculated from Eq. (12), and the symbols are obtained from finite-element simulations. The solid lines in (a2)–(c2) are calculated from Eq. (15) under the thermal coupling conditions determined by Eqs. (13), (14). Other parameters: (a1), (a2) $p_{1} = 0.25$ and $p_{2} = π / 16$ ; (b1), (b2) $κ_{θ θ} / κ_{r r} = 2$ and $p_{2} = π / 16$ ; (c1), (c2) $κ_{θ θ} / κ_{r r} = 2$ and $p_{1} = 0.25$ ; (a1)–(c1) $κ_{c} (T) = 400 + 0.05 T W / (m K)$ , $κ_{s} = - \sqrt{κ_{r r} κ_{θ θ}} = - 400 W / (m K)$ is the effective scalar thermal conductivity, and $κ_{m} = 400 W / (m K)$ ; (a2)–(c2) $κ_{c} (T) = 400 + 0.05 T W / (m K)$ , $κ_{s} = \sqrt{κ_{r r} κ_{θ θ}} = 400 W / (m K)$ , and $κ_{m} = 400 W / (m K)$ .
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Figure 3
Three-dimensional simulation results. The solid lines in (a1)–(c1) [or (a2)–(c2)] are calculated from Eq. (21) [or Eq. (24)], and the symbols are obtained from finite-element simulations. Other parameters are the same as those in Fig. 2 except for (a1), (a2) $p_{1} = 0.125$ and $p_{2} = π / 48$ , (b1), (b2) $κ_{θ θ} / κ_{r r} = 2$ and $p_{2} = π / 48$ , and (c1), (c2) $κ_{θ θ} / κ_{r r} = 2$ and $p_{1} = 0.125$ .
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Figure 4
Switchable thermal transparency device: (a) is the schematic diagram based on the nonlinear core and linear shell and (b), (c) are corresponding finite-element simulation results. (b) is in open state and (c) is in close state. (d) is the schematic diagram based on the linear core and nonlinear shell and (e), (f) are corresponding finite-element simulation results. (e) is in open state and (f) is in close state. Other parameters in (a)–(f): $κ_{c a} = 400 + 70 \times (T - 293) W / (m K)$ , $κ_{c b} = 400 - 70 \times (T - 293) W / (m K)$ , $κ_{s} = 100 W / (m K)$ , $κ_{c} = 20 W / (m K)$ , $κ_{s a} = 400 + 8 \times (T - 293) W / (m K)$ , $κ_{s b} = 400 - 8 \times (T - 293) W / (m K)$ , and $κ_{m} = 200 W / (m K)$ . Here $κ_{c a}$ , $κ_{c b}$ , $κ_{s a}$ , and $κ_{s b}$ are temperature dependent, which could, in principle, be designed by using shape memory alloys according to the method proposed in Refs. [28, 31].
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Figure 5
Replacing (a) apparent negative thermal conductivity with (b) line sources and (c) point sources. The simulation box of (a)–(c) is $16 \times 16$ cm. The radii of shell $r_{s}$ and core $r_{c}$ are 3.2 and 1.6 cm, respectively. The thermal conductivities of background material and core in (a)–(c) are $50 W / (m K)$ . The thermal conductivities of the shells in (a) and (b), (c) are $- 50 W / (m K)$ and $20 W / (m K)$ , respectively. Line sources are applied on the two boundaries of the shell in (b). The temperatures of line sources obey $T = 293 - 250 r_{c, s}^{2} x / (x^{2} + y^{2})$ K, where $(x, y)$ represent the Cartesian coordinates whose origin locates in the center of the simulation box. Point sources with radius 0.1 cm are applied on the two boundaries of the shell in (c). The temperatures of point sources can be calculated with $T = 293 - 250 r_{c, s}^{2} x / (x^{2} + y^{2})$ K according to the source positions.
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