Theory for Thermal Bi/Multistability: Nonlinear Thermal Conductivity

Abstract

In this chapter, we theoretically design diffusive bistability (and even multistability) in the macroscopic scale, which has a similar phenomenon but a different mechanism from its microscopic counterpart (Wang et al., Phys. Rev. Lett. 101, 267203 (2008)); the latter has been extensively investigated in the literature, e.g., for building nanometer-scale memory components. By introducing second- and third-order nonlinear terms (opposite in sign) into diffusion coefficient matrices, bistable energy or mass diffusion occurs with two different steady states, identified as “0” and “1”. In particular, we study heat conduction in a two-terminal three-body system. This bistable system exhibits a macro-scale thermal memory effect with tailored nonlinear thermal conductivities. Finite-element simulations confirm the theoretical analysis. Also, we suggest experiments with metamaterials based on shape memory alloys. This framework blazes a trail in constructing intrinsic bistability or multistability in diffusive systems for macroscopic energy or mass management.

1 Opening Remarks

Modern electronic techniques face increasingly prominent heat dissipation problems due to shrinking chip sizes and increasing integration levels [1]. Fortunately, the past decade has witnessed the possibility of manipulating heat transport at nanoscale [2,3,4,5,6,7], which provides a promising method for evolving electron-based computation. Phononics, a microscale interpretation of controlling heat flow to carry and process information, has flourished since then [8]. To date, indispensable elements of phononic computers, including thermal diodes [2], thermal logical gates [3], and thermal memories [4], have been proposed theoretically and experimentally. The thermal memory requests a nonlinear bistable thermal circuit for basic phononic information storage. Two different steady states can be demonstrated as “0” and “1” beyond the same boundary condition, just like the electronic counterpart. Although this concept was proposed in 2008 [4], the studies of thermal bistability (TBIS) devices are still far from being satisfactory (say, compared with existing research on optical bistability), which prohibits its practical applications. This situation is because most studies are executed at a microscopic scale, but the nanofabrication capacity is limited.

Recent progress in TBIS focuses on achieving bistability by introducing nonlinear thermal radiation for forming the negative differential thermal resistance (NDTR) [9,10,11,12,13,14,15], in which the Steffan-Boltzmann’s law deviates. With success in optical bistability [16,17,18,19,20], it is natural to migrate its methods into thermal radiation for TBIS because both optical and thermal-radiation processes can be classified as wave physics. Comprehensively, TBIS is realized mainly in two ways: the radiative phase transition at a specific temperature region [9,10,11,12] and the anomalous radiative phenomenon such as near-field radiation or nonlinear optical resonances [13,14,15]. The switching time between two states has been improved to several hundred $\mu s$ in the laboratory. The nonlinear thermal radiation can be a potential theoretical scheme for achieving TBIS.

However, in a macroscopic diffusive system such as heat conduction, TBIS has never been touched because of the absence of a theoretical framework analogous to its counterpart in wave systems. Nevertheless, heat conduction, a sort of major heat transfer mode described by the diffusion equation [21], can not apply to the method in wave processes because of the distinction of governing equations between diffusive and wave systems [22]. Hence, it is necessary to consider the conduction TBIS due to its ubiquity. On the one hand, thermal conduction still plays a primary role of heat dissipation in traditional electron-based computation. Conduction TBIS devices may well couple thermal and electronic memory. On the other hand, great progress has been made in manipulating macroscopic thermal conduction at will, especially in recent decades, by using the theory of transformation thermotics and thermal metamaterials [23,24,25,26,27,28,29,30,31,32], which may facilitate the design and manufacture of conduction TBIS devices. Here, we establish a bistability theory for treating diffusive systems. We take heat conduction as a classical diffusive system and deduce the non-linear heat-conduction parameters by adopting two theoretical methods. Finite-element simulations confirm it and further demonstrate a practical thermal memory process. We also give a proof-of-principle experimental design by adopting the temperature-trapping theory [32]. The theoretical framework applies to tailoring diffusion coefficient matrices for bistability (and multistability) in diffusion.

2 Theoretical Foundation

A diffusive system is usually described by force causing a flux. For example, Fourier’s law $\boldsymbol{J}=-\kappa \nabla T$ implies that the heat flux is induced by a temperature gradient, similar to Ohm’s law $\boldsymbol{I}=-\epsilon \nabla U$ and Fick’s law $\boldsymbol{q}=-D\nabla n$. Generally, the relation between fluxes and forces of a diffusive system can be written as

$$\begin{aligned} Y_i=\sum _{j=1}^{n}K_{ij}X_j, \end{aligned}$$

(18.1)

where i represents the variety of fluxes and j symbolizes different kinds of forces. Considering a simple single-field diffusion, $i=j$, Eq. (18.1) comes into no-coupled transport in the system. If the elements of transport coefficient tensor $K_{ii}$ are constant, the relation between a flux $Y_i$ and a force $X_i$ is linear. However, bistability requests that the system deviates from a linear relation between $Y_i$ and $X_i$. The nonlinearity of elements in the coefficient matrices becomes necessary for getting two or more steady-state solutions in the diffusion equation. Let us take Fourier’s law as an example. Here, nonlinearity in macroscopic heat conduction can owe to the temperature-dependence of thermal conductivity, $\kappa (T)$ [33]. Thus, by engineering $\kappa (T)$ of a thermal circuit, an NDTR [4, 5] will work, which induces anomalous thermal diffusion. This property is essentially important for obtaining macroscopic TBIS.

Fig. 18.1

Adapted from Ref. [34]

a A two-terminal model for thermal bistability. Heat transfers along the x-axis. A and B are two different heat-conduction materials. C is a region for reading out and writing in. $T_h$ and $T_c$ are temperatures of heat baths. $T_0$ is the temperature of region C. b Schematic diagram of heat flow in region A (dotted red line)and B (dashed blue line), and the net flow of region C (solid black line). A and B have different nonlinear thermal conductivities, resulting in three intersections.

Inspired by the model proposed in Ref. [4], we consider a two-terminal three-body heat transport model presented in Fig. 18.1a without loss of generality. In this case, heat flows along the x axis. A and B are two heat-conduction materials with the same sizes denoted as L (length) and S (cross-sectional area). The middle small region C shows a uniform temperature distribution due to a relatively high thermal conductivity. So C is set for extracting state information of the system. We aim to observe two divergent steady temperatures within C under the same boundary condition to achieve TBIS. Both extremes are connected to heat baths. We fix $T_h$ and $T_c$ as the temperature of two heat baths respectively and set $T_0$ as the temperature of region C. According to the continuity of heat flow, $T_0$ has a unique solution under the steady state if A and B are linear heat-conduction materials (namely, their thermal conductivities $\kappa _A$ and $\kappa _B$ are temperature-independent constants). The heat flows $J_A$ and $J_B$ running through A and B are linear monotonic functions of $T_0$, which can be verified by $J_A=\kappa _A(T_h-T_0)/L$ and $J_B=\kappa _B(T_0-T_c)/L$. Their changes concerning $T_0$ are two straight lines with one intersection point [$J_A(T_0)=J_B(T_0)$], which refers to the unique heat-conduction steady state. However, steady states can increase if A and B are two nonlinear heat-conduction materials (their thermal conductivities depend on the temperature). Here we denote $\kappa _A(T)$ and $\kappa _B(T)$ as their thermal conductivities, respectively

$$\begin{aligned} \kappa _A(T) = \kappa _{A0}+ \sum _{m} \chi _{Am} T^{m}, \end{aligned}$$

(18.2)

$$\begin{aligned} \kappa _B(T) = \kappa _{B0}+ \sum _{n} \chi _{Bn} T^{n}, \end{aligned}$$

(18.3)

where m and n are positive integers. The linear relation between heat flow $J_A$ (or $J_B$) and $T_0$ deviates. As illustrated in Fig. 18.1b, more than one intersection point of $J_A$ and $J_B$. That is to say, thermal bistability or multistability phenomena can appear due to nonlinear heat conduction.

A. Calculations of Net Heat Flow

We define $J_0=J_B-J_A$ as the net heat flow from the region C. $J_0=0$ is the necessary condition that a steady-state system should satisfy. In a TBIS system, $J_0(T_0)=0$ has three real solutions. These three points are candidates of steady points. But the point where $\partial J_0/\partial T_0<0$ should be excluded because it is an unstable equilibrium point. Then, which steady state will the system come into? This depends on the initial conditions. As shown in Fig. 18.1b, a cubic function (rather than a quadratic function) can construct a bistable system perfectly. Thus, we can speculate that the index terms in Eqs. (18.2) and (18.3) should be kept up to the second terms. This means $m=n=2$. Accordingly, Eqs. (18.2) and (18.3) can be reduced as

$$\begin{aligned} \kappa _A(T) = \kappa _{A0}+ \chi _{A1} T+\chi _{A2} T^{2}, \end{aligned}$$

(18.4)

$$\begin{aligned} \kappa _B(T) = \kappa _{B0}+ \chi _{B1} T+\chi _{B2} T^{2}. \end{aligned}$$

(18.5)

Distinctly, two factors influence such a TBIS system. When one factor dominates, the system will become state I (on) and vice versa into state II (off). For TBIS, the dominating factors depend on the temperature-evolution direction. For example, if relaxing from a low-temperature state, factor I dominates, and the system will enter state I. On the contrary, an initial high-temperature state will conclude in another final state. So $\chi _{A1}$ and $\chi _{A2}$ (or $\chi _{B1}$ and $\chi _{B2}$) are inferred to have opposite signs. Based on the above analysis, we are in a position to calculate the thermal conductivity parameters for a TBIS system.

The nonlinear thermal conductivity values show position-dependent (one-to-one mapping to position x) in a steady state. But the heat flows $J_A$ and $J_B$ are independent of x due to the heat flow conservation. $J_A$ can be written as $J_A=\kappa _{eA}S\langle \nabla T_A\rangle$, where $\kappa _{eA}$ is the effective thermal conductivity and $\langle \nabla T_A\rangle$ is the corresponding average temperature gradient of A. So is $J_B$. Then we can derive $J_A$ and $J_B$ as

$$\begin{aligned} J_A = \kappa _{eA}S \langle \nabla T_A\rangle = \frac{\kappa _{eA}(T_h-T_0)S}{L}, \end{aligned}$$

(18.6)

$$\begin{aligned} J_B= \kappa _{eB}S \langle \nabla T_B \rangle = \frac{\kappa _{eB}(T_0-T_c)S}{L}. \end{aligned}$$

(18.7)

To conclude $J_A$ and $J_B$, the effective thermal conductivities should be deduced. For simplicity, we assume B is a linear heat-conduction material ($\chi _{Bn}=0$), and only hold A’s nonlinearity. This simplification will not affect the cubic relation between net heat flow $J_0$ and $T_0$. Then $\kappa _{eA}$ and $\kappa _{eB}$ can be written as

$$\begin{aligned} \kappa _{eA} = \dfrac{\int _{T_0}^{T_h}\kappa _A(T)}{T_h-T_0} = \frac{\kappa _{A0}T_h+ \dfrac{1}{2}\chi _{A1}T_h^{2} +\dfrac{1}{3}\chi _{A2}T_h^{3} -\left( \kappa _{A0}T_0h+ \dfrac{1}{2}\chi _{A1}T_0^{2}+\dfrac{1}{3}\chi _{A2}T_0^{3}\right) }{T_h-T_0}, \end{aligned}$$

(18.8)

$$\begin{aligned} \kappa _{eB} = \kappa _{B0}. \end{aligned}$$

(18.9)

Substituting Eqs. (18.8) and (18.9) into Eqs. (18.6) and (18.7), we get

$$\begin{aligned} J_A = -\frac{S}{L}\left[ \frac{1}{3}\chi _{A2}T_0^{3} + \frac{1}{2}\chi _{A1}T_0^{2} + \kappa _{A0}T_0 - \left( \frac{1}{3}\chi _{A2}T_h^{3} + \frac{1}{2}\chi _{A1}T_h^{2} + \kappa _{A0}T_h\right) \right] , \end{aligned}$$

(18.10)

$$\begin{aligned} J_B = \frac{S}{L}(\kappa _{B0}T_0-\kappa _{B0}T_c). \end{aligned}$$

(18.11)

Defining shape factor $\Gamma =S/L$, then $J_0$ can be expressed as

$$\begin{aligned} \begin{aligned} J_0&= J_B-J_A \\&= \Gamma \left[ \frac{1}{3}\chi _{A2}T_0^{3} + \frac{1}{2}\chi _{A1}T_0^{2} + (\kappa _{A0}+\kappa _{B0})T_0 - \left( \frac{1}{3}\chi _{A2}T_h^{3} + \frac{1}{2}\chi _{A1}T_h^{2} + \kappa _{A0}T_h + \kappa _{B0}T_c\right) \right] . \end{aligned} \end{aligned}$$

(18.12)

Equations (18.4) and (18.5) describe the nonlinear heat conduction. Generally, it is hard to solve the nonlinear heat conduction differential equation. So we may adopt an effective-thermal-conductivity approximation to avoid the nonlinear terms above. In addition, Kirchhoff’s transformation provides another way to make the nonlinear equation linearization [35]. As it works well in one-dimensional heat conduction problems, we can get exact solutions to temperature distributions in our model. Then comparing the two results, we can verify the above approximation results.

Let us still consider the nonlinear heat conduction in region A and assume region B has a linear thermal conductivity. Under a steady state, heat conduction in region A can be described as

$$\begin{aligned} \frac{\partial }{\partial x}\left[ \kappa _A(T)\frac{\partial T}{\partial x}\right] =0. \end{aligned}$$

(18.13)

Here, we define a new variable U, which has the same unit as a temperature,

$$\begin{aligned} U=U(T)=\int _{T_{ref}}^{T}\frac{\kappa _A(T')}{\kappa _A(T_{ref})}dT', \end{aligned}$$

(18.14)

where $T_{ref}$ is an arbitrary reference temperature. And Eq. (18.13) can be transformed as

$$\begin{aligned} \frac{\partial }{\partial x}\left[ \kappa _A(T)\frac{\partial T}{\partial U}\frac{\partial U}{\partial x}\right] =0. \end{aligned}$$

(18.15)

Combing Eqs. (18.14) and (18.15), we can get a heat-conduction equation with U,

$$\begin{aligned} \frac{\partial ^{2}U}{\partial x^{2}}=0. \end{aligned}$$

(18.16)

If we take $T_{ref}=0$ K, the variable U and corresponding upper and lower bounds can be deduced as

$$\begin{aligned} U(T)=\dfrac{\int _{0}^{T}\left( \kappa _{A0}+\chi _{A1}T' +\chi _{A2}T'^{2}\right) dT'}{\kappa _{A0}} =\dfrac{\kappa _{A0}T+\frac{1}{2}\chi _{A1}T^{2} +\frac{1}{3}\chi _{A2}T^{3}}{\kappa _{A0}}, \end{aligned}$$

(18.17)

and

$$\begin{aligned} \left\{ \begin{array}{lllll} U_1=\dfrac{\kappa _{A0}T_h+\frac{1}{2}\chi _{A1}T_h^{2} +\frac{1}{3}\chi _{A2}T_h^{3}}{\kappa _{A0}}~(x=a),\\ \\ U_2=\dfrac{\kappa _{A0}T_0+\frac{1}{2}\chi _{A1}T_0^{2} +\frac{1}{3}\chi _{A2}T_0^{3}}{\kappa _{A0}}~(x=b).\\ \end{array} \right. \end{aligned}$$

(18.18)

Combing Eqs. (18.16) and (18.18) together, we can solve the expression of U as

$$\begin{aligned} U(x) = \frac{U_2-U_1}{L}x+U_1, \end{aligned}$$

(18.19)

which indicates the value of U at each position. It is easy to migrate U(x) back to T(x). Thus, by means of the intermediate variable U, we can find the relation between T and x as

$$\begin{aligned} U=\dfrac{\kappa _{A0}T+\frac{1}{2}\chi _{A1}T^{2} +\frac{1}{3}\chi _{A2}T^{3}}{\kappa _{A0}}= \frac{U_2-U_1}{L}x+U_1. \end{aligned}$$

(18.20)

Taking the derivative of Eq. (18.20) with respect to x in region A, we get $\dfrac{\partial T}{\partial x}\bigg |_A$,

$$\begin{aligned} \dfrac{\partial T}{\partial x}\bigg |_A= \frac{\kappa _{A0}(U_2-U_1)}{\kappa _A(T)L}. \end{aligned}$$

(18.21)

Then, the net outflow of heat from C can be written as

$$\begin{aligned} \begin{aligned}&J_0^{*}=J_B^{*}-J_A^{*}= \kappa _{B0}\frac{T_0-T_c}{L}S+\kappa _A(T)\dfrac{\partial T}{\partial x}\bigg |_A S \\&=\Gamma \left[ \frac{1}{3}\chi _{A2}T_0^{3} + \frac{1}{2}\chi _{A1}T_0^{2} + (\kappa _{A0}+\kappa _{B0})T_0 - \left( \frac{1}{3}\chi _{A2}T_h^{3} + \frac{1}{2}\chi _{A1}T_h^{2} + \kappa _{A0}T_h + \kappa _{B0}T_c\right) \right] , \end{aligned} \end{aligned}$$

(18.22)

which echoes with Eq. (18.12). It is definite that a nonlinear one-dimensional heat conduction process can be simplified by executing the space averaging of $\kappa (T)$, which makes a detour around the nonlinear terms. This will facilitate the disposal of nonlinear-heat-conduction case.

B. Tailoring Nonlinear-thermal-Conductivities Coefficients

We can see $J_0$ satisfies a cubic relation with $T_0$. Now we construct another cubic function $J_0'(T_0)$ with three zero points $T_{01}$, $T_{02}$, $T_{03}$ (suppose $T_c<T_{01}<T_{03}<T_{02}<T_h$). $J_0'$ can be written as

$$\begin{aligned} \begin{aligned} J_0'&= \alpha [(T_0-T_{01})(T_0-T_{02})(T_0-T_{03})] \\&=\alpha [T_0^{3}-(T_{01}+T_{02}+T_{03})T_0^{2}+ (T_{01}T_{02}+T_{01}T_{03}+T_{02}T_{03})T_0-T_{01}T_{02}T_{03}]. \end{aligned} \end{aligned}$$

(18.23)

$\alpha$ is the pre-coefficient with a unit J/K. $T_{01}$ and $T_{02}$ are the two designed stable temperatures of region C. By comparing the coefficient and constant terms of Eqs. (18.12) and (18.23), we acquire a set of equations

$$\begin{aligned} \left\{ \begin{array}{lllll} \frac{1}{3}\Gamma \chi _{A2}=\alpha ,\\ \\ \frac{1}{2}\Gamma \chi _{A1}=-\alpha (T_{01}+T_{02}+T_{03}),\\ \\ \Gamma (\kappa _{A0}+\kappa _{B0})=\alpha (T_{01}T_{02}+T_{01}T_{03}+T_{02}T_{03}),\\ \\ -\Gamma \left( \frac{1}{3}\chi _{A2}T_h^{3} + \frac{1}{2}\chi _{A1}T_h^{2} + \kappa _{A0}T_h + \kappa _{B0}T_c\right) =-\alpha T_{01}T_{02}T_{03}.\\ \end{array} \right. \end{aligned}$$

(18.24)

Then we achieve

$$\begin{aligned} \begin{array}{lllll} \kappa _{A0}=\dfrac{\alpha }{\Gamma }\left[ \dfrac{-T_h^{3} +(T_{01}+T_{02}+T_{03})T_h^{2} -(T_{01}T_{02}+T_{01}T_{03}+T_{02}T_{03})T_c +T_{01}T_{02}T_{03}}{T_h-T_c}\right] ,\\ \\ \kappa _{B0}=\dfrac{\alpha }{\Gamma }\left[ \dfrac{T_h^{3} -(T_{01}+T_{02}+T_{03})T_h^{2} +(T_{01}T_{02}+T_{01}T_{03}+T_{02}T_{03})T_h -T_{01}T_{02}T_{03}}{T_h-T_c}\right] ,\\ \\ \chi _{A1}=-\dfrac{2\alpha }{\Gamma }(T_{01}+T_{02}+T_{03}),\\ \\ \chi _{A2}= \dfrac{3\alpha }{\Gamma }.\\ \end{array} \end{aligned}$$

(18.25)

We can see $\chi _{A1}$ and $\chi _{A2}$ have opposite signs definitely, which echo the inference above. Therefore, a bistable system features that two kinds of factors compete in evolution from a non-equilibrium state to an equilibrium state. $T_{01}$ and $T_{02}$ are representations of two different states, while $T_{03}$ can not exist in a steady state. Equation (18.25) provides guidance in designing nonlinear parameters of heat-conduction objects to realize TBIS. We can calculate the coefficients according to the pre-set zero-point temperatures ($T_{01}$, $T_{02}$, and $T_{03}$), the temperatures of heat baths, and two factors $\Gamma$ and $\alpha$.

This method allows a diffusive system to exhibit bistable states by engineering nonlinear transport coefficients. This intrinsic bistability depends on two competitive factors, reflected by two nonlinear terms with opposite signs. We prove that the second- and third-order nonlinearity of transport coefficients makes bistability effects valid. If the nonlinearity orders are higher, multistability can come to appear. And the switching time depends on the diffusion velocity of heat or mass.

Fig. 18.2

Adapted from Ref. [34]

Analysis of the bistability and NDTR based on the analytical model discussed in the text. a Heat flow in region A (dotted red line), B (dashed blue line), and net flow in region C (solid black line) versus $T_0$ of the system. Here B is a linear heat-conduction material, and the $J_B$ curve is a straight line. $J_A$ and $J_B$ have three intersections. b thermal conductivities of A (dotted red line) and B (dashed blue line) versus $T_0$. The effective thermal conductivity of A is also shown with a solid red line by an integral average of $T_0$. The NDTR region is shadowed in yellow, containing two stable temperature points.

3 Numerical Analysis and Simulation

We draw the graphs to illustrate our methods for tailoring nonlinear thermal conductivities. On basis of the model shown in Fig. 18.1a, we set $T_{01}= 330$ K, $T_{02}= 370$ K, and $T_{03}= 350$ K. The heat and cold baths are fixed at 400 K and 300 K, respectively. Two factors are set as $\alpha =0.001$ J/K and $\Gamma =1$ m. The substitution of these parameters into Eq. (18.25) yields $\kappa _{A0}=366.05$ J/(m K), $\kappa _{B0}=1.05$ J/(m K), $\chi _{A1}=-2.1$ J/(m K$^{2})$, and $\chi _{A2}=0.003$ J/(m K$^{3})$. The curves of $J_A$, $J_B$, and $J_0$ versus $T_0$ are shown in Fig. 18.2a. Three intersections emerge, corresponding to the pre-set parameters $T_{01}$, $T_{02}$, and $T_{03}$. In Fig. 18.2b, the thermal conductivities of A and B versus temperature are depicted. We can see that $\kappa _A(T)$ has negative values in a certain temperature region. This value is calculated as (328.02 K, 371.98 K), which refers to the NDTR region (see the yellow-shadowed region in Fig. 18.2). The region contains two stable temperatures, confirming that the NDTR induces the desired TBIS. These two graphs accord with our expected results as sketched in Fig. 18.1b.

Fig. 18.3

Adapted from Ref. [34]

Net flow $J_0$ versus $T_0$ for different small-shift coefficients. a TBIS behavior for different linear coefficients $\kappa _{A0}$. b TBIS behavior for different linear coefficients $\kappa _{B0}$. c TBIS behavior for different second-order coefficients $\chi _{A1}$. d TBIS behavior for different third-order coefficients $\chi _{A2}$.

When the coefficients of nonlinear thermal conductivities have slight variations, will TBIS be broken? Here, we give a $1\%$ value shift to four parameters respectively ($\kappa _{A0}$, $\kappa _{B0}$, $\chi _{A1}$, and $\chi _{A2}$ are increased by $1\%$, respectively). According to the comparisons in Fig. 18.3, the small shift of $\kappa _{B0}$ cannot affect the TBIS, which can be interpreted by the steady heat flow in the system kept almost unchanged. While the thermal conductivity of A varies slightly, TBIS will not exist anymore. So we can conclude that the TBIS of heat conduction is parameter-sensitive. This strict limitation makes it hard to observe the TBIS phenomenon in practical heat-conduction materials. But we can carefully tailor an intrinsic TBIS with pre-designed zero-point temperatures.

Fig. 18.4

Adapted from Ref. [34]

Finite-element simulations of TBIS. a Transient simulation results beyond fixed heat baths’ temperatures. After 0.004 s, the system becomes stable with two different $T_0$ due to the different initial temperatures. b Temperature distribution along x-axis. The left end of the model is set as the origin point ($x=0$). Theoretical heat flows and thermal conductivities of the model are compared with the simulation results in vignettes.

We perform finite-element simulations based on the commercial software COMSOL Multiphysics. We build a model with 9 cm in length and 1 cm in width. Heat conducts along the x-axis. The thermostat region is placed in the center with $\kappa _{C}=1000$ J/(m K). We give 400 K, 500 K, and 600 K as three pre-designed zero points for the thermal conductivities of the left and right parts. $\Gamma$ is 1/4 m according to the model’s geometry. $\alpha$ is arbitrary, and here we take it as 0.0001 J/K. Thus, we can calculate that $\kappa _{A0}=290$ J/(m K), $\kappa _{B0}=6$ J/(m K), $\chi _{A1}=-1.2$ J/(m K$^{2})$, and $\chi _{A2}=0.0012$ J/(m K$^{3})$. The density and specific heat of all materials are set as 10 kg/m$^{3}$ and 10 J/(kg K). Boundary conditions are fixed at 700 K (left) and 300 K (right). Then, we give 300 K and 700 K as initial surface temperatures, see Fig. 18.4a. After the temperature evolution within 0.004 s, the system comes into stable states. However, the final temperatures of C are different according to different initial temperatures, representing two different stable states. The initial temperature of 300 K induces 398.46 K (stable state I) in C, while 700 K leads to 600.12 K (stable state II). The states of C depend on the initial surface temperatures. We fetch the final-state-temperature data of the model along the x-axis and curve it in Fig. 18.4b. State I and II have two different platform temperatures in region C (4$\sim$5 cm). In addition, we plot the theoretical results of heat flow and thermal conductivities versus $T_0$ as inset diagrams in Fig. 18.4b. Both 400 K and 600 K are the pre-set stable values for designing the thermal conductivity parameters. And the simulation results well confirm the theoretical values.

Then, we demonstrate an overall thermal memory process with the designed conduction TBIS in Fig. 18.5a, which is based on the simulation results above. Firstly, we initialize the model by a temperature-writer in 300 K as an initial temperature. After 0.004 s, the system will become steady, and we read out $T_0$ in region C by a temperature-reader. It is 398.46 K now. And then we write in another temperature as 700 K. After 0.004 s, a steady temperature of 600.12 K can be read out. This model’s switching time is 0.004 s, which depends on each part’s density and specific heat. So these two parameters should be considered and optimized when devices are in practical application. This memory process makes the conduction TBIS practicable in fabricating macroscopic thermal memory components.

Fig. 18.5

Adapted from Ref. [34]

a A demonstration of the thermal memory process with the model we design. Four stages are displayed initialization, reading-out, writing-in, and reading-out. b An experimental design based on the temperature-trapping theory. Two stages with different types of SMA are arrayed. The central temperatures depend on the SMA stages’ critical temperatures.

4 Experimental Suggestion

The temperature-trapping theory [32] inspires us to design a proof-of-principle experiment. This theory implies a thermostat region in the center of a spatially symmetric structure within shape memory alloys (SMAs). The thermostat’s temperature depends entirely on the critical temperature of SMAs. Here, we improve this structure and design a two-stage SMAs device to achieve TBIS, as shown in Fig. 18.5b. Two pairs of SMAs are arrayed on both sides, in white and gray, forming two-stage thermal switches. Different types of SMAs are applied in each pair. In particular, these two stages have different critical temperatures $T_1$ and $T_2$, where $T_1<T_2$. In detail, the white stage on the left levels below $T_1$ and bends above $T_1$, while the right one shows the same $T_1$ but inverse deformation. A similar rule works on the gray stage. Heat and cold baths are fixed on both sides with $T_h>T_2$ and $T_c<T_1$. When the whole device is initialized under a low temperature on the left and a high temperature on the right, all the SMAs get straight. The outer stage bends, and the heat flow cannot run into the inner layer when coming to the steady state. The thermostat’s temperature approaches $T_1$. When the initial condition reverses, all stages bend. They will not be level at steady state as $T_h>T_2$ and $T_c<T_1$. This process induces another steady state that $T_2$ is the final temperature of the thermostat. As the SMAs are commercially available, assembling such a two-stage structure is feasible. But the thermal contact resistances may affect the experimental results, which should be considered further.

5 Discussion

For the temperature-dependent thermal conductivity of A depicted in Fig. 18.1a, the third-order nonlinearity is just a necessary condition. We can find that $|\kappa _{A0}|\sim |\chi _{A1}T|\sim |\chi _{A2}T^2|$ is another parameter requirement. Fortunately, these extraordinary thermal properties were proved to emerge in some bulk nonmetallic solids [36]. For example, the thermal conductivity of bulk $\mathrm ZrO_2$ is $4.00-8.72\times 10^{-3}T+1.28\times 10^{-5}T^{2} -5.82\times 10^{-9}T^{3}$[W/(m K)], which agrees qualitatively with the conduction TBIS requires at 10$^3$ K level. It can be applied as material A in our model combing with a common material B. By solving the inverse solutions of Eq. (18.24), namely working out $\alpha$, $T_{01}$, $T_{02}$, and $T_{03}$, one can estimate the experimental bistable temperature for such a structure composed of a nonlinear bulk heat-conduction material plus a common material. Thus, the observation of conduction TBIS in natural materials is practically probable. Besides, using the composite effect of nonlinear heat transfer [37], the fabrication of a conduction TBIS device with composite materials is possible. In this case, the nonlinear thermal conductivities can be well-tailored if adjusting the fraction or configuration of composite bulk components [38]. For example, a core-shell structure [33] and a particle-embedded-in-host structure [39] may be candidates. So we also suggest the composite manufacture method as material A in fabricating the device for application scenarios.

We have established a theoretical framework for achieving bistability in diffusive heat systems. We prove that the TBIS phenomenon exists not only in wave processes (say, nonlinear thermal radiation) but also can be realized in heat-conduction systems. Second- and third-order nonlinearity of thermal conductivity can induce a bistable thermal circuit. When the nonlinearity orders go higher, multistability can be observed as well. We have also given numerical calculation results and show the parameter-sensitive TBIS in heat conduction. Besides, a completed thermal memory process is demonstrated with four stages as an evident consequence. Except for thermal memories, a thermal switch is another possible application. As the designed experiment implies, the switch is initial-temperature-forced and can barrier or allow heat flows due to distinguishable thermal conductivities. As waste heat is dissipated mainly by the diffusive process in traditional computers, conduction TBIS devices can thus be coupled with electronic devices, facilitating thermal calculation based on existing electric calculation.

6 Conclusion

We have introduced an approach to designing macroscopic bistability by taking the heat conduction process as a typical case. Due to the form-similarity of governing equations, this method is applicable in other diffusive systems, such as direct current or particle diffusion systems. Bistability or multistability can be realized by carefully tailoring spatial asymmetry and nonlinearity of diffusive parameters. This method helps generate a significant physical phenomenon in the macroscopic diffusive process and is a potential tool in macroscopic energy or mass management.

7 Exercise and Solution

Exercise

• 1. Discuss the differences between a negative differential thermal resistance and a negative thermal conductivity.

Solution

• 1. A negative differential thermal resistance means that the heat flux decreases when the temperature difference increases, which naturally does not violate the second law of thermodynamics.

  A negative thermal conductivity means heat can spontaneously transport from low to high temperatures, which is usually impossible due to the second law of thermodynamics. However, it can still be effectively realized if an external source is applied.