Development of High-Performance Solid-State Thermal Diodes Using Unusual Behavior of Thermal Conductivity Observed for Ag₂Ch (Ch = S, Se, Te)

Abstract

Thermal diodes are a kind of new device which controls the direction and magnitude of heat flow. The performance of thermal diode is evaluated by thermal rectification ratio (TRR) defined as |J_large|/|J_small|. Solid-state thermal diodes are made with two materials possessing different temperature dependence of thermal conductivity. Their performance is improved by the significant variation of thermal conductivity with temperature. In this study, therefore, we employed Ag₂Ch (Ch = S, Se, Te) because these materials are characterized by a drastic change in thermal conductivity due to a structural phase transition in the temperature range of 373 < T < 473 K. We prepared Ag₂Ch samples by means of self-propagating high-temperature synthesis method. The phases involved in the samples were identified using powder X-ray diffraction. We measured temperature dependence of thermal conductivity for prepared samples by means of laser flash method, and consequently observed 200–300% change in their thermal conductivity. The thermal diode consisting of Ag₂S and Ag₂Te was designed using the measured thermal conductivity. We experimentally confirmed that the developed thermal diode showed TRR = 2.1 ± 0.1 when it was placed between two heat reservoirs kept at T_H = 473 K and T_L = 405 K.

 

This Paper was Originally Published in Japanese in J. Thermoelec. Soc. Jpn. 16 (2019) 3–7.

1. Introduction

Recently, heat flow control technologies attract significant attention for their potential to construct the society of low energy consumption. Once heat flow control technologies are established, they allow us to utilize abundant waste heat existing anywhere by efficiently transporting that to places demand on heat. We introduce here a thermal diode, in which the magnitude of heat flow strongly depends on its directions. The thermal diode, for their characteristic, leads to effective control of heat flow and be considered as one of the key technologies for managing the unused heat.

In a steady-state, heat flow density J_Q in a solid is described by the Fourier’s law.   

\begin{equation} \boldsymbol{J}_{\mathbf{Q}} = -\kappa\nabla T \end{equation} (1)

Equation (1) simply explains that J_Q is proportional to a temperature gradient −∇T and proportional coefficient κ, which is known as thermal conductivity. Therefore, making the direction dependence of |J_Q|, we need to employ special mechanisms leading to direction dependence on κ.

Several mechanisms leading to thermal rectification were proposed.¹⁾ By seriously considering reproducibility and practicality of these, we focused on a thermal rectification mechanism brought about by a composite consisting of two different materials showing opposite temperature-dependent thermal conductivity.^2,3) As shown in Fig. 1, one of the materials, here we tentatively call it as A, shows increasing thermal conductivity with increasing temperature, while the other B shows decreasing. When the materials A and B located at the high- and low-temperature side respectively, both materials simultaneously show high thermal conductivity, and hence we observe a large heat flow J_large from A to B. When the temperature of heat reservoirs is reversed, both materials turn out to possess lower thermal conductivity, and the heat flow also becomes small (J_small). The thermal rectification effect is thus obtainable.

Fig. 1

Schematic illustration of a thermal diode consisting of materials A and B showing different temperature dependence of thermal conductivity.^2,3) Here, l represents the total length of thermal diode and x is the length ratio of A to l.

The performance of thermal diode is generally evaluated by thermal rectification ratio (TRR) defined as follows   

\begin{equation} \mathit{TRR} = |\boldsymbol{J}_{\text{large}}|/|\boldsymbol{J}_{\text{small}}| \end{equation} (2)

For obtaining large TRR, we need to employ materials possessing the remarkable temperature dependence of thermal conductivity. Unfortunately, however, solid materials generally possess less significant temperature-dependent at high temperatures above room temperature. Therefore, it is of great difficulty to realize an effective thermal rectification in the temperature range where abundant waste heat exists.

Up to now, large thermal rectification ratio exceeding two was experimentally achieved with the mechanism described above by three different groups. The reported values of TRR and the temperature range were (a) TRR = 2.6 (T_H = 317 K, T_L = 280 K),⁴⁾ (b) TRR = 2 (T_H = 318 K, T_L = 298 K),⁵⁾ and (c) TRR = 2.2 (T_H = 900 K, T_L = 300 K).⁶⁾

In the case of (a) and (b), solid-liquid transition in polymers was used to make a sudden change in thermal conductivity. This strategy allows the authors to obtain the large TRR value with the relatively small temperature difference of ΔT = T_H − T_L < 40 K. However, the utilize of liquid requires a special vessel and complicated device structure.

In the case of (c), the authors employed aluminum-based quasicrystal that shows a drastic increase in electronic thermal conductivity with increasing temperature and also employed CuGuTe₂ characterized by an unusual decrease in lattice thermal conductivity with increasing temperature most likely due to the strong anharmonic lattice vibration. The large TRR exceeding two was obtained in the device composed solely of solid-state materials. Their thermal diode, however, required a large temperature difference (ΔT = 600 K) for the maximum performance and did not work fully in the temperature range of abundant waste heat: T < 473 K.

Accordingly, all the thermal diodes described above had several problems that prevent us from utilizing them in practical applications. For developing practical thermal diodes working in the temperature range from 300 to 473 K, where a vast amount of waste heat is available, we need to employ solid materials possessing thermal conductivity varying drastically in this temperature range. In this study, therefore, we employed silver chalcogenides Ag₂Ch (Ch = S, Se, Te) as the component materials of the thermal diode. Silver chalcogenides exhibit a structural phase transition at T_x = 373–473 K, and their high-temperature phases are widely known as superionic conductor.⁷⁾ Notably, a drastic variation of thermal conductivity was reported in the vicinity of T_x.^8,9) We consider that the utilize of this drastic change should enable us to develop a new solid-state thermal diode possessing large TRR with a minor temperature difference below 473 K.

To develop a thermal diode possessing large TRR with a small temperature difference using Ag₂Ch, we prepared high-quality single-phase samples, measured their physical properties, and evaluated the performance of fabricated thermal diode.

2. Experimental Method

Powders of Ag (99.9%), S (99.99%), Se (99.9%), and Te (99.9%) were weighted out to become stoichiometry compositions of Ag₂Ch. The weighted powders were mixed and compressed into a cylindrical shape at room temperature. For a phase formation, then the compressed samples were subjected to a self-propagating high-temperature synthesis (SHS) method in a vacuum atmosphere. The obtained samples were crushed into powder again and sintered into dense pellets of 10 mm in diameter using hot pressing at 623–923 K for 10 min under the uniaxial pressure of 40 MPa to conduct an accurate physical property measurement.

Phases involved in the samples were identified by X-ray diffraction (XRD) using Cu-Kα radiation (40 mA, 40 kV) in BRUKER, NEW D8 ADVANCE.

The temperature dependence of thermal conductivity κ(T) was calculated from thermal diffusivity α(T), specific heat Cp(T), and sample density d using the following formula.   

\begin{equation} \kappa(T) = \alpha(T) \times Cp(T) \times d \end{equation} (3)

We used a laser flash method in NETZSCH, LFA457 for measuring α(T) and Cp(T), and d were estimated utilizing an Archimedes method at room temperature. The samples in a cylindrical shape of 10 mm in diameter and 1.0 mm in thickness were used for the measurements.

The temperature dependence of electrical resistivity ρ(T) was measured by the conventional four-probe method. Electronic thermal conductivity κ_el(T) was roughly calculated from the measured ρ(T) using the Wiedemann-Franz law.   

\begin{equation} \kappa_{\text{el}}(T) = LT/\rho(T) \end{equation} (4)

Here L = 2.44 × 10⁸ WΩ K⁻² is the constant known as Lorenz number. Lattice thermal conductivity κ_lat(T) was calculated simply using the following formula.   

\begin{equation} \kappa_{\text{lat}}(T) = \kappa(T) - \kappa_{\text{el}}(T) \end{equation} (5)

The thermal resistance of samples varies not only with κ but also with sample dimensions. In the thermal diode shown in Fig. 1, therefore, the value of TRR is determined by the temperature dependence of κ and the length ratio x = L_A/(L_A + L_B) of two materials. Using a method proposed by Takeuchi et al.,¹⁰⁾ TRR values were estimated from a temperature-dependent thermal conductivity of two materials, a temperature of heat reservoirs, and a length ratio of two materials, assuming no heat leak from the side of the materials. After determining the temperature of two heat reservoirs and length ratio x for maximum TRR value, we made thermal diode consisting of two samples with the determined length ratio x.

For evaluating TRR, we measured heat flow through thermal diode using the self-made setup in which one was evaluated by the steady-state method (See Fig. 2). Thermal grease was applied to all interfaces between the heaters, titanium rods, and samples, and then the whole rods were pressed at 1 MPa for good thermal contact. The heaters placed on both sides of the rods enable us to change the direction of heat flow through the thermal diode under the same condition of thermal contact. The heat flow measurements were performed in a vacuum atmosphere for removing a convection effect. To reduce radiation loss of heat, we additionally used aluminum sheets for covering the titanium rods and the sample. Heat flows through thermal diode were calculated from the Fourier’s law using the cross-section area and temperature gradient in the rod of low-temperature side, and the thermal conductivity of titanium.¹¹⁾

Fig. 2

Schematic illustration of our heat flow measurement system.

3. Results and Discussions

Figure 3 displays the XRD patterns of synthesized powder samples (Ag₂Ch_meas.) together with the calculated¹²⁾ patterns (Ag₂Ch_calc.) from the reported crystal structure of Ag₂S,¹³⁾ Ag₂Se,¹⁴⁾ and Ag₂Te.¹⁵⁾ The measured XRD patterns were in good agreement with the calculated ones. Therefore, our samples were safely identified free from secondary phases. Notably, by using the SHS method, Ag₂Ch single-phase samples could be synthesized within 30 min from the weighing of starting powders.

Fig. 3

Measured XRD patterns of Ag₂Ch (Ch = S, Se, Te). Calculated XRD patterns from the reported crystal structure^13–15) are also plotted.

Figure 4 shows the temperature dependence of thermal conductivity for prepared bulk samples. All the samples possessed a drastic change (200–300%) in thermal conductivity during the phase transition. Its reproducibility was confirmed among several samples made in some different batches. We should emphasize that these silver chalcogenides showed an extremely low thermal conductivity about less than 1.5 Wm⁻¹K⁻¹ and a peak in the vicinity of phase transition. We also realized that the temperature dependence of thermal conductivities varies with the type of chalcogenide elements.

Fig. 4

Temperature dependence of the thermal conductivity measured for Ag₂Ch (Ch = S, Se, Te).

Figure 5 shows the temperature dependence of thermal diffusivity and specific heat used to calculate the thermal conductivity of samples #1 in Fig. 4. The peak of specific heat indicates the structural phase transition of Ag₂Ch. Notably, the peak was also clearly observed in thermal diffusivity for Ag₂Se and Ag₂Te, and it resulted in a larger κ(T) peak. The presence of the peak in κ(T) is a controversial issue because this behavior is unusual, and the data was measured with one of the dynamical techniques, i.e. laser flash method. From these experimental facts, however, we consider the peak of thermal conductivity observed during phase transformation in Ag₂Ch is not an artifact produced by dynamic measurement but a physical characteristic. Despite the difficulty of steady-state measurements to observe such a rapid change of thermal conductivity in a narrow temperature range, we are trying to prove the existence of the peak in κ(T) experimentally and report it in another paper.

Fig. 5

Temperature dependence of the (a) thermal diffusivity and (b) specific heat measured for Ag₂Ch (Ch = S, Se, Te).

The temperature dependence of electrical resistivity for Ag₂Ch is given in Fig. 6. The electronic thermal conductivity was roughly calculated using Wiedemann-Franz law, and the lattice thermal conductivity was estimated by subtracting the electronic thermal conductivity from the measured ones. From the analysis, we consider that the electronic thermal conductivity contributes to the difference in the absolute value of thermal conductivity between the low- and high-temperature phases. The absence of a significant drop in the electrical resistivity at the phase transition suggests the lattice thermal conductivity contributes to the peak in the measured thermal conductivity. We also emphasize here that the low- and high-temperature phases of Ag₂Ch possessed extremely low lattice thermal conductivity less than 0.5 Wm⁻¹K⁻¹. This extremely low lattice thermal conductivity in low- and high-temperature phases are most likely due to the significant anharmonic lattice vibration and the ionic conduction, respectively.

Fig. 6

Temperature dependence of the electrical resistivity measured for Ag₂Ch (Ch = S, Se, Te).

From the temperature dependence of thermal conductivity shown in Fig. 4, a large thermal rectification effect was naturally expected in the thermal diode consisting of Ag₂S and Ag₂Te. We calculated TRR as a function of both the temperature of heat reservoirs and the length ratio x of two materials. As a typical example, the TRR calculation under the condition of (T_H, T_L) = (473 K, 405 K) was plotted as a function of x in Fig. 7. From the calculation, we predicted that TRR = 2.3 as the maximum value would be obtainable with the length ratio $L_{\text{Ag}_{2}\text{S}}:L_{\text{Ag}_{2}\text{Te}} = 56:44$.

Fig. 7

Thermal rectification ratio (TRR) of the thermal diode consisting of Ag₂S and Ag₂Te plotted as a function of the length ratio $x = L_{\text{Ag}_{2}\text{S}}/(L_{\text{Ag}_{2}\text{S}} + L_{\text{Ag}_{2}\text{Te}})$.

The thermal diode consisting of Ag₂S and Ag₂Te in a cylindrical shape of 10 mm in diameter was prepared with a thickness of 2.23 mm and 1.76 mm, respectively, to become $L_{\text{Ag}_{2}\text{S}}:L_{\text{Ag}_{2}\text{Te}} = 56:44$. The thermal diode was imposed between the two heat reservoirs T_H = 473 K and T_L = 405 K, and the direction dependence of the heat flow was measured. When Ag₂S and Ag₂Te placed at high- and low-temperature side, respectively, we obtained J_large, and the opposite condition provided us with J_small. The values of |J_large| and |J_small| were plotted as a function of heating time in Fig. 8. The solid-lines were obtained from the experiment, and the dashed-lines were calculated from the temperature dependence of thermal conductivity.

Fig. 8

Heat flow measured for the prepared thermal diode consisting of Ag₂S and Ag₂Te plotted as a function of the heating time. Solid and dashed lines indicate measured and calculated ones, respectively.

The magnitudes of heat flow were kept constant over the plotted period. This fact indicates that the experimental data were obtained under steady heat flow conditions. The large difference between |J_large| and |J_small| means that a significant thermal rectification effect is successfully achieved. We should emphasize here that the obtained thermal rectification ratio was TRR_meas. = 2.1 ± 0.1, and the composite consisting of Ag₂S/Ag₂Te is the first bulk thermal diode showing TRR > 2 in the temperature range of 373–473 K. The measured value (TRR_meas.) was slightly smaller than the calculated one (TRR_calc.). This difference would be caused by a radiative heat loss or a migration of Ag ions under the temperature gradient.

We are considering applications of a thermal diode in a heat exchanger using water. In such a case, at least, T_L should be less than the boiling temperature of the water. For Ag₂Ch, partial element substitutions in Ch site cause a significant shift of phase transition temperature.¹⁶⁾ This characteristic allows us to precisely control the working temperature of the silver chalcogenides thermal diode. We are trying to develop a thermal diode working at such a temperature range involving 373 K.

Besides, considering the practical application of the device reported in this work, we should pay attention to the destruction of bulk thermal diodes due to the volume change during the phase transition. This serious problem, however, would be eliminated by the metal-like ductile nature reported for Ag₂S.¹⁷⁾ We consider that this feature enables us to fabricate a device that is tolerant of strain caused by phase transition and thermal expansion.

4. Summary

For developing a solid-state thermal diode showing a large thermal rectification effect with a small temperature difference, in this study, we fabricated and evaluated the thermal diode consisting of silver chalcogenides Ag₂Ch (Ch = S, Se, Te). Single-phase Ag₂Ch samples were prepared using the SHS method, and detailed temperature-dependent physical property measurements were conducted. We clarified that all Ag₂Ch samples show the drastic change (200–300%) in thermal conductivity at the phase transition and the different thermal conductivity behavior depending on the chalcogen element. Also, we revealed the contribution of the lattice and electronic thermal conductivity to the unusual behavior of thermal conductivity. Consequently, we succeeded in developing the solid-state thermal diode possessing TRR > 2 composed of Ag₂S and Ag₂Te when it is placed between two heat reservoirs kept at T_H = 473 K and T_L = 405 K.

REFERENCES

• 1)   G.  Wehmeyer,  T.  Yabuki,  C.  Monachon,  J.  Wu and  C.  Dames: Appl. Phys. Rev. 4 (2017) 041304.
• 2)   M.  Peyrard: Europhys. Lett. 76 (2006) 49–55.
• 3)   B.  Hu,  D.  He,  L.  Yang and  Y.  Zhang: Phys. Rev. E 74 (2006) 060201.
• 4)   A.L.  Cottrill,  S.  Wang,  A.T.  Liu,  W.J.  Wang and  M.S.  Strano: Adv. Energy Mater. 8 (2018) 1702692.
• 5)   E.  Pallecchi,  Z.  Chen,  G.E.  Fernandes,  Y.  Wan,  J.H.  Kim and  J.  Xu: Mater. Horiz. 2 (2015) 125–129.
• 6)   T.  Takeuchi: Sci. Technol. Adv. Mater. 15 (2014) 064801.
• 7)   H.  Okazaki: J. Japan Inst. Met. Mater. 16 (1977) 689–697.
• 8)   H.  Chen,  Z.  Yue,  D.  Ren,  H.  Zeng,  T.  Wei,  K.  Zhao,  R.  Yang,  P.  Qiu,  L.  Chen and  X.  Shi: Adv. Mater. 31 (2018) 1806518.
• 9)   D.  Jung,  K.  Kurosaki,  Y.  Ohishi,  H.  Muta and  S.  Yamanaka: Mater. Trans. 53 (2012) 1216–1219.
• 10)   T.  Takeuchi,  H.  Goto,  R.S.  Nakayama,  Y.I.  Terazawa,  K.  Ogawa,  A.  Yamamoto,  T.  Itoh and  M.  Mikami: J. Appl. Phys. 111 (2012) 093517.
• 11)   C.Y.  Ho,  R.W.  Powell and  P.E.  Liley: J. Phys. Chem. Ref. Data 1 (1972) 279.
• 12)   F.  Izumi and  K.  Momma: Solid State Phenom. 130 (2007) 15–20.
• 13)   R.  Sadanaga and  S.  Sueno: Mineral. J. 5 (1967) 124–143.
• 14)   J.  Yu and  H.  Yun: Acta Crystallogr. Sec. E 67 (2011) i24.
• 15)   A.  van der Lee and  J.L.  Boer: Acta Crystallogr. Sec. C 49 (1993) 1444–1446.
• 16)   S.  Miyatani: J. Phys. Soc. Jpn. 15 (1960) 1586–1595.
• 17)   X.  Shi,  H.  Chen,  F.  Hao,  R.  Liu,  T.  Wang,  P.  Qiu,  U.  Burkhardt,  Y.  Grin and  L.  Chen: Nat. Mater. 17 (2018) 421–426.