Radiative Resistance at The Nano-scale: Thermal Barrier

In recent years, many studies have been conducted on radiative heat transfer between objects with separation distances less than the thermal wavelength Polder and Van Hove (1971); Pendry (1999); Rousseau et al. (2009); Volokitin and Persson (2007); Zare et al. (2019). Because the radiative flux at this scale violates Stephen Boltzmann’s law, heat flux management by controlling geometrical parameters and system-specific features has attracted much attention. Dependence on parameters is an interesting part of radiative heat transfer in two-body systems. These parameters can be either internal, such as size, shape, orientation, distance and material composition of objects Ben-Abdallah et al. (2015, 2009); Singer et al. (2015); Xu et al. (2019); Nikbakht (2018), or an external parameter such as magnetic field in magneto-optical systems, thermal boundary conditions, the properties of surrounding media, or an electric field in metallic material systemsOtt et al. (2019); Abraham Ekeroth et al. (2017); Latella and Ben-Abdallah (2017); Ben-Abdallah (2016); Latella et al. (2018a); Volokitin (2019, 2020); Volokitin and Persson (2020). The quantitative form of the radiative heat flux can change as parameters are varied. In particular, the heat flux can be enhanced or decreased, or the net direction can change. As we move up from two-body to three-body systems, the radiative heat transfer can be tuned by changing the parameters, but theoretically and more recently it has been shown experimentally that three-body systems can provide the possibility to enhance radiative heat transfer over two-body counterparts Ben-Abdallah et al. (2011); Nikbakht (2014); Dong et al. (2018); Thompson et al. (2020); Kan et al. (2019); Messina and Antezza (2014); Dong et al. (2017a, a); Messina et al. (2018). There has been a large amount of literature seeking to improve thermal rectification, thermal switching and thermal splitting by controlling various material and structural parametersGu et al. (2015); Ramezan Choubdar and Nikbakht (2016); Song et al. (2019). As the number of objects in the system increases, the many-body effects become very significant Nikbakht (2017); Pérez-Rodríguez et al. (2019); Dong et al. (2017b); Ben-Abdallah (2019); Tervo et al. (2019); Song et al. (2020) and as expected it influences the dynamics of temperature Nikbakht (2015); Latella et al. (2020). On the other hand, the dynamics and the steady-state radiative heat flux may exhibit sensitive dependence on parameters, initial conditions and also on the thermal boundary conditions Latella et al. (2019); Dyakov et al. (2014); Reina et al. (2020). Recent theoretical work on the thermal bistability has highlighted the importance of an external heat flux on the thermal switching in near-field radiative heat transfer Reina et al. (2020). It has been previously established that the many-body parallel planar systems can provides distinctive properties for significant enhancement of near-field heat transport Latella et al. (2018b); Iizuka and Fan (2018), and the geometrical properties as well as the initial condition for temperatures can have a remarkable effect on the temperature evolution Latella et al. (2017).

The transfer of large amounts of energy between system components in the near-field regime results in a very strong temperature coupling at these scales. However, finding ways to minimize radiation heat transfer in systems that require thermal insulation is particularly important. In conductive heat transfer this isolation is mainly carried out using multi-layer structures Tien and Cunnington (1973); Koebel et al. (2012); Ong (2017). The temperature profile in such structures does not show a monotonic trend across an interface between different materials. Instead, there is a temperature difference athwart the boundaries. Similarly, a system of planar objects can resist radiation heat flux and it is expected to cause discontinuities in temperature profile. In other words, the components of the system provide a thermal resistance that must be considered in thermal design or analysis.

In this article, we have investigated the radiative thermal current and steady state temperature profile in a parallel planar system which is subjected to an external temperature gradient and transfer heat in the form of radiation. Using a simple example it is shown that the steady-state radiative thermal thermal current can be tuned by engineering the intrinsic properties of each layer. Moreover, inspired by the idea of Kapitza resistance and interfacial thermal resistance Pollack (1969); Kapitsa and Haar (1986); Maiti et al. (1997), we have introduced a radiative thermal resistance in parallel planar objects and demonstrate the possibility of extreme radiative thermal isolation. We have introduced radiative thermal barrier and the linear and non-linear resistances are calculated for barriers with different materials. It is shown that both thermal current and temperature profile in steady-state regime depend strongly on the width and the composition of the barrier. The numerical results indicate that for a given composite system, due to the existence of a thermal barrier, the radiative thermal resistance depends not only on the width of the barrier but also on the separation distances. While we have used Silicon Carbide (SiC) and hexagonal Boron Nitride (hBN) as typical materials for slabs, the proposed formalism is general and can be applied to any planar system with arbitrary parameters (materials, widths, vacuum gap distances). Moreover, the proposed quantities (thermal resistance and radiative current) can be used to analyze the results of many studies in the field of radiative heat transfer and improve our understanding of the subjects in this field.

The structure of the paper is as follow. The formalism is developed in Sec. II. In Sec. III we start with a simple prototypical example of a thermal barrier. We have computed the temperature profile in an array of Silicon Carbide (SiC) slabs when a hexagonal Boron Nitride (hBN) or SiC slab is embedded in the middle of the system Sec. III.1. The radiative thermal resistances are calculated in Sec. III.2. The effect of the barrier thickness on the temperature profile and thermal resistances are investigated in Sec. III.3. In Sec. III.4 the thermal resistance of the barrier compared for near-field and far-field regime. Finally, this study is summarized in Sec. IV.

The schematic of the system under consideration is shown in Fig. 1. It consists of N parallel slabs with thicknesses $\Delta_{i}$ separated by vacuum gaps of width $\delta$. The slabs are along the z-axis at positions $z_{i}$. The system is in thermal baths (environment) from left and right at fixed temperatures $T_{L}\equiv T_{0}$ and $T_{R}\equiv T_{N+1}$, respectively. Moreover, the first ($i=1$) and the last ($i=N$) slabs are connected to reservoirs with fixed temperatures $T_{1}$ and $T_{N}$, respectively. Based on the boundary condition of the system, i.e. $(T_{L},T_{1},T_{N},T_{R})$, the radiative heat transfer that takes place along the system, drives the system from an initial non-equilibrium state $[T_{L},T_{1},T_{2}(0),\cdots,T_{N-1}(0),T_{N},T_{R}]$ to a non-equilibrium steady states configuration $[T_{L},T_{1},T_{2}^{eq},\cdots,T_{N-1}^{eq},T_{N},T_{R}]$, in which each slab is in local thermal equilibrium and there is no time variation of physical variables. The temporal behaviour of each slab is governed by equation:

$$\rho_{i}c_{i}\Delta_{i}\dfrac{\partial T_{i}}{\partial t}=\varphi_{i}(T_{L},T_{1},\cdots,T_{N},T_{R})~{}~{}i=2,\cdots,N-1$$

(1)

where $\Delta_{i}$ is the thickness of the slab with mass density $\rho_{i}$ and heat capacity $c_{i}$. For a given thermal boundary condition and initial condition for temperatures, the net radiative heat flux per unit surface received by the i-th slab is given by Latella et al. (2017):

$$\varphi_{i}=\sum_{j=0}^{N+1}\left[\int_{0}^{{\infty}}\frac{d\omega}{2\pi}\int_{0}^{{\infty}}\dfrac{dk}{2\pi}k\sum_{p=\{S,P\}}\Theta(\omega,T_{j})\mathcal{T}^{j,i}(\omega,k,p)\right],$$

(2)

for $i=2,\cdots,N-1$. In this relation, the second summation runs over two physical polarization states of the radiating field, i.e. S and P polarizations respectively. Moreover, $\Theta(\omega,T_{j})=\hbar\omega/(e^{\hbar\omega/k_{B}T_{j}}-1)$ denotes the mean energy of Plank oscillator at temperature $T_{j}$. The contribution of each slab in the heat flux is given by the transmission coefficient $\mathcal{T}^{j,i}=\mathcal{T}^{j,i}(\omega,k,p)$, which depends on the geometrical and intrinsic features of the system (see the appendix for more detail). The solution of Eq. (1) can be visualized as a trajectory in a (N-2)-Dimensional phase space. However, we are only interested in the long-time behavior of the system, i.e. the steady-state temperature profile that the system is able to reach as $t\rightarrow\infty$. Since the right-hand-side of Eq. (1) does not depend on $t$ explicitly, the system is autonomous and we only need to find fixed point of the system Strogatz (2000). The steady state temperature profile of Eq. (1) is defined by the fact that the net energy flux on each slab vanishes, i.e. $\varphi_{i}(\textbf{T}^{*})=0$ for $i=2,\cdots,N-1$, where $\textbf{T}^{*}\equiv(T_{L},T_{1},T^{eq}_{2},\cdots,T_{N-1}^{eq},T_{N},T_{R})$ is the fixed point of the system in phase space. For further investigation of the steady-state of the system, we summarized and extend Eq. (2) to cover all system components, including slabs, environments and reservoirs. Hence

$$\varphi_{i}=F_{i,i}+\sum_{j\neq i}F_{j,i}+F^{ext}_{i},$$

(3)

for $i=0,1,\cdots,N,N+1$. In second term the value of index $j$ runs from $0(L)$ to $N+1(R)$ (including the reservoirs and the external environment). Here $F_{i,i}\leqslant 0$ is a radiative cooling of the i-th component which could be a slab, a reservoir or an environment. Moreover, $F_{j,i}\geqslant 0$ is the power transferred from the j-th component to the i-th one. Finally, an external amount of heat ($F^{ext}$) should be transfer from, or to, heat reservoirs and environments in order to keep them in constant temperatures. It should be emphasized that $F_{i}^{ext}=0$ for slabs with varying temperature, i.e. $i=2,\cdots,N-1$ to match Eq (2).

Suppose the left environment absorbs the rate of heat $F_{L}^{ext}$ and first slab absorbs the rate of heat $F_{1}^{ext}$. According to the conservation of energy $\sum_{i=0}^{N+1}\varphi_{i}=0$, it is easy to show that $F_{L}^{ext}(t\rightarrow\infty)+F_{1}^{ext}(t\rightarrow\infty)=-[F_{R}^{ext}(t\rightarrow\infty)+F_{N}^{ext}(t\rightarrow\infty)]$, which can be referred to as the left-right symmetry at steady state. Specifically, we can say now that the same amount of heat which is given to the left environment and reservoir at a time is equal to the one that taken from the right environment and reservoir, or visa versa. By setting $\varphi_{i}=0$ in Eq (3) for $i=0,1$ and using Eq. (2), it is straightforward to show that

	$\displaystyle F_{1}^{ext}$	$\displaystyle=$	$\displaystyle-\sum_{j=0}^{N+1}\int_{0}^{{\infty}}\frac{d\omega}{2\pi}\int_{0}^{{\infty}}\dfrac{dk}{2\pi}k\sum_{p}\Theta(\omega,T_{j})\mathcal{T}^{j,1},$		(4)
	$\displaystyle F_{L}^{ext}$	$\displaystyle=$	$\displaystyle-\sum_{j=0}^{N}\int_{0}^{{\infty}}\frac{d\omega}{2\pi}\int_{0}^{{\infty}}\dfrac{dk}{2\pi}k\sum_{p}\Theta_{j,R}\mathcal{T}^{L,j},$		(5)

with $\Theta_{j,R}=\Theta(\omega,T_{j})-\Theta(\omega,T_{R})$. It is clear that these external powers are time-dependent during the initial stage of the dynamics of temperatures in the system. However, they eventually approach the steady-state values as the system reaches local thermal equilibrium. In addition, we know that due to the temperature difference caused by the boundary conditions, the radiative thermal current flows along the z direction either to the left or to the right. Hence, we define the net current flow along the system as:

$$J^{eq}=F^{ext}_{L}+F^{ext}_{1}\equiv-(F^{ext}_{R}+F^{ext}_{N}),$$

(6)

which remains constant in the steady-state regime. Here, $J^{eq}$ represents the radiant energy passing through the system in local thermal equilibrium. It is important to emphasize that for the system under consideration, the transmission probabilities do not depend on temperature as in phase changed materials, which implies that the power dissipated in each slab ($\varphi_{i}$) is a continues function of temperatures and the off-diagonal elements in the Jacobian matrix of the system ($\frac{\partial\varphi_{i}}{\partial T_{j}},i\neq j$) have constant sign, independent of the system’s state.

As a result, the system of equations $\varphi_{i}=0$ have only one fixed point and since the system is non-conservative, the fixed point is stable. On the other hand, the steady-state and so the equilibrium thermal current do not depend on the choice of initial condition of the system, i.e. [$T_{2}(0),\cdots,T_{N-1}(0)$]. Using Eqs. (4) to (6), the steady-state thermal current passing through the system can be expressed as

$$J^{eq}=\int_{0}^{{\infty}}\frac{d\omega}{2\pi}\mathcal{C}^{eq}(\omega),$$

(7)

where the monochromatic thermal current coefficient is defined as

$$\mathcal{C}^{eq}(\omega)=\int_{0}^{{\infty}}\dfrac{dk}{2\pi}k\mathcal{S}^{eq}(\omega,k),$$

(8)

where $\mathcal{S}^{eq}(\omega,k)=\mathcal{S}^{eq}_{P}(\omega,k)+\mathcal{S}^{eq}_{S}(\omega,k)$ with

$$\mathcal{S}^{eq}_{p}(\omega,k)=\sum_{j=0}^{N}\Theta_{R,j}^{eq}\left[\mathcal{T}^{L,j}(\omega,k,p)+\mathcal{T}^{j,1}(\omega,k,p)\right].$$

(9)

Here, $\Theta_{R,j}^{eq}=\Theta(\omega,T_{R})-\Theta(\omega,T_{j}^{eq})$ and $\mathcal{S}^{eq}(\omega,k)$ can be interpreted as the steady-state dispersion relation of the thermal current passing through the system. Moreover, $\mathcal{S}^{eq}_{S}(\omega,k)$ and $\mathcal{S}^{eq}_{P}(\omega,k)$ are the dispersion relations of the thermal current for the s- and p-polarized modes. For the special case of $T_{L}=T_{R}$, Eq. (9) reduces to

$$\mathcal{S}^{eq}_{p}(\omega,k)=-\sum_{j=0}^{N+1}\Theta(\omega,T^{eq}_{j})\left[\mathcal{T}^{L,j}(\omega,k,p)+\mathcal{T}^{j,1}(\omega,k,p)\right].$$

(10)

It should be emphasize that, $\mathcal{S}$ depends not only on the choice of materials and system geometrical properties, but also on the steady-state temperature profile. For very large separation distances, where the far-field interaction dominates, the thermal current and so the resistances could mainly decided by the coupling of the middle slabs with the thermal baths (depending on the boundary conditions and materials) rather than the reservoirs temperature gradient. For the sake of simplicity we take $T_{L}=T_{R}=300$ K in our calculations. It is also easy to show that in the absence of thermal baths ($T_{L}=T_{R}=0$ K) we have $\mathcal{S}^{eq}_{p}(\omega,k)=-\sum_{j=1}^{N}\Theta(\omega,T^{eq}_{j})\mathcal{T}^{j,1}(\omega,k,p)$, which for the special case of $N=2$, Eq. (7) reduces to the well known formula for the heat flux exchanged between two parallel slabs Ben-Abdallah et al. (2009).

There is an electrical analogy with radiative heat transfer that can be used to exploited in, see Fig. 1. From this perspective the radiative heat flux is equivalent to the electric current and each slab is a pure resistance to radiative heat flux. Hence, we can define slab resistance as

$$R_{i}=\dfrac{\Delta T_{i}}{J_{i}},$$

(11)

where by definition, $\Delta T_{i}=(T_{i-1}-T_{i+1})/2$ is the temperature difference at the position of i-th slab and $J_{i}$ is the net radiative thermal current passing through it. It is plausible that the thermal current and so the temperatures are function of times. However, when the system reaches steady-state, they depend only on the temperature of the two reservoirs and environments, the slab properties and their separation distances. Since the thermal current has only one path to take in the system under consideration, it is the same through all slab at the steady-state regime, i.e. $J_{i}\rightarrow J^{eq}$. It also seems reasonable to postulate that thermal resistances are independent of environment and reservoirs temperatures in linear regime where $T_{L}\sim T_{1}\sim T_{N}\sim T_{R}$. To this end, a more useful quantity to work with is the linear resistance of slabs

$$R_{i}^{eq}=\lim_{J\rightarrow 0}\dfrac{d(\Delta T_{i})}{dJ}{\Big{|}}_{eq}.$$

(12)

In the system under consideration, we have used $T_{L}=T_{R}=300$ K, since the contribution of environments in the thermal current is small compared to that of reservoirs (i.e. $|F^{ext}_{L}|\ll|F^{ext}_{1}|$ especially for small separation distances), the derivative evaluated at thermal equilibrium in the limits of $T_{1}\sim T_{N}$. The total thermal resistance of the system could be calculated by simply adding up the resistance values of the individual resistors, i.e.

$$R_{total}^{eq}=\sum_{i=1}^{N}R_{i}^{eq}\equiv\lim_{\Delta T\rightarrow 0}\frac{\Delta T}{J^{eq}},$$

(13)

with $\Delta T=T_{1}-T_{N}$. We could also define a thermal boundary resistance, similar to the Kapitza resistanceKapitsa and Haar (1986), as the ratio of the temperature variation at the interface to the heat current across it. As a result, like a series circuit, the resistance of each slab would be the addition of its interface resistances

$$R_{i}^{eq}=R_{il}^{eq}+R_{ir}^{eq},$$

(14)

where $R_{il}^{eq}=(T_{i-1}^{eq}-T_{i}^{eq})/2J_{i}^{eq}$ is the resistance of the left interface of the slab and $R_{ir}^{eq}=(T_{i}^{eq}-T_{i+1}^{eq})/2J_{i}^{eq}$ is the right part. It should be emphasizes that the interface resistances $R_{il}^{eq}$ and $R_{ir}^{eq}$ are equivalent of the Kpitza resistance between interfaces, and as long as the thickness of a layer is not zero (or it is sandwiched between different media), the layer persist against the incident thermal current and has a resistance $R_{i}^{eq}\neq 0$.

Based on the framework built above, we can calculate the steady-state temperature profile of a typical radiative thermal barrier. The system that we consider here consists of 15 slabs, which is a hBN or a SiC slab with thickness $\Delta_{8}=\Delta_{barrier}$ sandwiched between 14 identical SiC slabs. The separation between slabs are equal, i.e. $\delta$, and the system is positioned in an environment with constant temperature $T_{L}=T_{R}=300$ K. The thermal gradient is applied by maintaining the two reservoir slabs at constant but different temperatures. For the non-linear regime, the temperature of hot reservoir and cold reservoir are fixed at $T_{1}=400$ K and $T_{15}=300$ K, respectively. However, we have used $T_{1}\sim T_{15}=300$ K to calculate resistances in linear regime. For the complex dielectric function of SiC and hBN, we used the Lorentz-Drude model:

$$\epsilon(\omega)=\epsilon_{\infty}\dfrac{\omega_{L}^{2}-\omega^{2}-i\Gamma\omega}{\omega_{T}^{2}-\omega^{2}-i\Gamma\omega},$$

(15)

where the parameters for silicon carbide (SiC) are as follow: $\epsilon_{\infty}=6.7$ is the high frequency dielectric constant, $\omega_{L}=1.83\times 10^{14}$ rad/s is the longitudinal optical frequency, $\omega_{T}=1.49\times 10^{14}$ rad/s is the transverse optical frequency, and $\Gamma=1.0\times 10^{12}$ rad/s is damping coefficient. While for hexagonal Boron Nitride (hBN) these constants are, $\epsilon_{\infty}=4.9$, $\omega_{L}=3.03\times 10^{14}$ rad/s, $\omega_{T}=2.57\times 10^{14}$ rad/s, and $\Gamma=1.0\times 10^{12}$ rad/s.

In summary, we have used an electrical circuit approach to introduce heat resistance for the components of systems that transmit energy through radiation at the nanoscale. For this purpose we proposed a method for calculating the steady-state radiative current in parallel planar systems. This method can be a useful guide for understanding and optimizing the thermal performance of nanoscale systems. The simulation results indicate that the temperature profile in a parallel planar system, exhibits fantastic changing characteristic around the barriers. We have shown the thermal insulation occurs due to the presence of thermal barrier and the temperature does not show a monotonic trend across the barrier, instead, there is a temperature difference athwart the barrier.