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A Series Circuit of Thermal Rectifiers: An Effective Way to Enhance Rectification RatioShiqian Hu, Meng An, Nuo Yang,* and Baowen Li*

nonlinear lattice models,[6] graded mass density,[2,7] asym-metric structure,[3–5,9,42] molecular junction,[10] or an interface between two different materials.[11–13,43] Very recently, based on phase change materials,[44,45] the thermal rectification can be enhanced significantly. However, available reports to enhance the thermal rectification ratio only focus on optimizing the performance of single thermal rectifier.

In this letter, a series circuit of TRs is proposed to enhance the thermal rectification. Based on our previous work, the different temperature dependence of thermal conductivity in graphene and graphene phononic crystal (GPnC),[46] we con-struct an asymmetric graphene/GPnC structure (Figure 1a) and found its thermal rectification. Furthermore, inspired by the series effect in electronic circuits, the graphene/GPnC1/GPnC2 structure (Figure 2a) is used to investigate the series effect of thermal rectification. The dependence of the thermal rectification on the geometric size of structure is also dis-cussed in details.

A schematic picture of the asymmetric graphene/GPnC structure is shown in Figure 1a, which is comprised of gra-phene and GPnC with the same length. The size of pores in GPnC is characterized by L1. The neck width (Ln) is fixed as 0.71 nm in all structures studied here. In our simulations, the lattice constant and thickness of graphene are used as 0.142 and 0.334 nm, respectively. The fixed (periodic) boundary condition is adopted along longitudinal (transversal) direction.

In nonequilibrium molecular dynamics simulations, the three layers atoms close to the two ends are coupled with the Langevin heat bath[47] at temperature TL = T0 (1 + Δ) and TR = T0 (1 − Δ), respectively, where T0 is the average tem-perature and Δ is the normalized temperature difference. Therefore, the atoms in the left end are at a higher (lower) temperature in the case of Δ > 0 (Δ < 0).

The bonding interaction between carbon atoms is described by the energy potential of a Morse bond and a har-monic cosine angle, which include both two-body and three-body potential terms.[48] We use the velocity Verlet algorithm to integrate the discretized differential equations of motions, where the time step is set as 0.5 fs. The heat flux transferred across the system is defined as the energy transported along it in unit time.[3] The structure is relaxed long enough (3 ns) such that the system reaches the steady state. Then, both the temperature and heat flux are recorded, which are averaged over 4.5 ns. It is noted that the values of temperature dif-ference (ΔT = |TL − TR|) are consistent when the heat flux (J+/J−) used for calculating the thermal rectification ratio is DOI: 10.1002/smll.201602726

Thermal Rectification

S. HuCenter for Phononics and Thermal Energy ScienceSchool of Physics Science and EngineeringTongji UniversityShanghai 200092, P. R. China

M. An, Prof. N. YangState Key Laboratory of Coal CombustionHuazhong University of Science and Technology (HUST)Wuhan 430074, P. R. ChinaE-mail: nuo@hust.edu.cn

M. An, Prof. N. YangNano Interface Center for Energy (NICE)School of Energy and Power EngineeringHuazhong University of Science and Technology (HUST)Wuhan 430074, P. R. China

Prof. B. LiDepartment of Mechanical EngineeringUniversity of ColoradoBoulder, CO 80309, USAE-mail: Baowen.Li@Colorado.Edu

Phonon, the quantized mode of vibration, is responsible for heat transport in dielectric and semiconducting crystals. The understanding of phonon properties provides a theoretical foundation for thermal devices, such as thermal rectifiers,[1–13] optomechanical crystals,[14,15] thermal cloaking,[16–20] thermo-electrics,[21–25] thermocrystals,[26–28] and phonon nano-capacitor,[29] etc., which could lead to a new technological revolution.

Graphene, a 2D structure of a single atomic layer of carbon arranged in a honeycomb lattice,[30] exhibits many intriguing properties. In addition to its unique electronic and optical properties, graphene is a promising material for thermal management of future nanoelectronic devices due to its superior thermal conductivity at room temperature.[5,31–33] In addition, due to the outstanding phonon confinement[34,35] and unique phonon coherent interference effect,[26,36,37] the porous graphene has shown potential applications in many fields, such as gas separation and purification,[38] water desali-nation,[39] ion channel,[40] and DNA sequencing.[41]

Similar to electronic counterpart, the thermal rectifier (TR) also plays a vital role in phononics/thermal circuits.[1] The thermal rectification means that the magnitude of heat flux changes when the temperature gradient is reversed in direction. Massive efforts have been devoted to investigate a high-performance thermal rectifier. Up to now, most theo-retical studies and experimental tests are realized based on

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recorded. All the results presented here are averaged from 12 independent simulations from different initial conditions. The error bar is the standard errors of the results of these simulations.

To show the thermal rectification effect quantitatively, the thermal rectification ratio (Rect) is defined as

ectR rr

= −+

(1)

where r+ (r−) is the thermal resistance of the system when Δ < 0 (> 0).

We define the generalized heat flux transmission function (τr) as[49]

B

Jk Trτ = ∆±

±

(2)

where J+ (J−) is the heat current from the right to the left (the left to the right), corresponding to Δ < 0 (> 0), kB is the Boltzmann constant, and ΔT is the temperature difference. In the ballistic limit, the transmission function is simply equal to 3N (the number of propagating phonon modes). According the definition of thermal resistance (r = ΔT/J), we can easily obtain the Rect as

ectr

rR r

rττ= =−

++−

(3)

First, the thermal rectification effect is studied in an asymmetric graphene/GPnC structure (Figure 1a). The

temperature profiles in the direction from GPnC to graphene and the inverse direction in asymmetric graphene/GPnC structure are both plotted in Figure 1b. We have checked the

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Figure 1. a) Schematic picture of asymmetric graphene/GPnC structure. The length (width) of the GPnC is set as L (W). L1 (size of pores) and Ln (neck width) are used to characterize the GPnC structure. Ln is fixed as 0.71 nm. b) The temperature profiles correspond to the heat flux along the direction from GPnC to graphene and the inverse direction in the asymmetric graphene/GPnC structure. c) The thermal rectification ratio as a function of the average temperature T0 with Δ = 0.5. d) The dependence of thermal rectification ratio on ΔT. The corresponding parameters are L = 50 nm, W = 7 nm, L1 = 0.71 nm, T0 = 1500 K.

Figure 2. Schematic pictures of a) graphene/GPnC1/GPnC2 structure, b) asymmetric graphene/GPnC structure, and c) GPnC1/GPnC2 structure. L = Lgraphene + LGPnC, and PL = Lgraphene/LGPnC. For all the three structures, the widths (W) are set as 7 nm. The fixed (periodic) boundary condition is applied along the transverse (longitudinal) direction. The sizes of pores in the middle (right) region are characterized by L1 (L2).

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overall size effect of rectification ratio in the graphene/GPnC structure. Although there are obvious size dependences of thermal conductivity of graphene[35,50] and GPnC,[46] the results show that the rectification ratio is insensitive to the overall length of the structure (details shown in Figure S1 in the Supporting Information). The thermal rectification ratio is shown in Figure 1c as a function of average temperature (T0). It is found that the heat flux from GPnC region to gra-phene region is higher than that from graphene region to GPnC region when the temperature difference is exchanged in direction and the thermal rectification ratio is insensitive to T0. Therefore, we choose T0 as 1500 K in the following studies. The dependence of thermal rectification ratio on the temperature difference (ΔT) at 1500 K is shown in Figure 1d. It is observed that the thermal rectification ratio increases when ΔT increases from 300 to 2700 K.

In the following, we give a discussion on the mechanism of rectification in asymmetric graphene/GPnC structure, based on the difference of thermal resistance between GPnC and graphene. Compared with the graphene, the thermal resistance of GPnC has a less sensitive dependence on the temperature.[46] The difference between the total thermal resistances of the two directions originates mainly from the resistance of graphene region. When the heat transfers from GPnC to graphene, the graphene region has a lower temper-ature (≈800 K, Figure 1b), corresponding to a lower thermal resistance. For the inverse direction, from graphene to GPnC, the graphene region has a higher temperature (≈2000 K, Figure 1b), corresponding to a higher thermal resistance. The smaller thermal resistance from GPnC to graphene means the heat flux prefers from GPnC to graphene.

Further, inspired by the series effect in electronic rec-tifiers, the basic element can be composed into complex structure to enhance the rectification ratio. We proposed a three section structure, graphene/GPnC1/GPnC2 (shown in Figure 2a), where the size of pores in the GPnC1 (GPnC2) region is characterized by L1 (L2) and fixed as 0.71 nm (1.55 nm). The GPnC1 and GPnC2 have the same length. In the three section structure, the length of the graphene and GPnC are set as Lgraphene and LGPnC ( GPnC 1L + GPnC 2L ), respectively.

The results of thermal rectification ratio of graphene/GPnC1/GPnC2 versus ΔT are shown in Figure 3a (triangles). Similar to asymmetric graphene/GPnC, the heat flux from GPnC2/GPnC1 region to graphene region is also higher and the value of thermal rectification ratio increases with the ΔT increase from 300 to 2700 K. Meanwhile, the interesting finding is that the thermal rectification ratio of graphene/GPnC1/GPnC2 is enhanced significantly compared with that of asymmetric graphene/GPnC. For a comparison with gra-phene/GPnC1/GPnC2, the structure of graphene/GPnC2/GPnC1 is also studied. The thermal rectification ratio results versus ΔT are shown in Figure 3b (triangles). The surprising finding is that, instead of the enhancement, the value of Rect is reduced.

To interpret this enhancement (reduction) of the thermal rectification ratio by the series effect in three section struc-tures, the thermal rectification ratio of the graphene/GPnC1, TR1, (shown in Figure 2b) and the GPnC1/GPnC2, TR2,

(shown in Figure 2c) are studied. It is found that in Figure 1c, the thermal rectification ratio is not sensitive to T0. How-ever, it depends on ΔT (shown in Figure 1d). To simplify the analysis, we assumed the ratio as a function of ΔT and independent on T0. In order to show the series effect and comparison directly, the same values of ΔT are used in simu-lation for graphene/GPnC1 or GPnC1/GPnC2 as the corre-sponding part in graphene/GPnC1/GPnC2.

The thermal rectification ratio results of graphene/GPnC1 and GPnC1/GPnC2 structure are shown in Figure 3 (circles and squares). For the GPnC1/GPnC2 structure, it is found that the heat flux prefers from GPnC2 to GPnC1. When three section structures are arranged as graphene/GPnC1/GPnC2 (graphene/GPnC2/GPnC1), these two TRs are linked on the same (inverse) thermal rectification direction, which is responsible for the thermal rectification enhancement (reduction).

For the series of two thermal rectifiers, TR1 and TR2, we make a very rough but intuitive assumption that they are independent and not interact with each other. Combined with the Equation (3), we can easily arrive at the constitutive thermal rectification ratio of the series rectifiers

τ ττ τ

= = =−

+

+ +

− −ectse 12

12

1, 2,

1, 2,ect1

ect2R

rr

R Rr r

r r (4)

where ( )ect1

ect2R R is the rectification ratio of TR1 (TR2).

There are two sets of results for three section structures. The first one from series effect prediction by Equation (4) (inverse triangles in Figure 3) based on MD simulation of graphene/GPnC1 and GPnC1/GPnC2. And the other one from MD simulation of three section structures (triangles in

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Figure 3. Rectification ratio versus ΔT in the a) graphene/GPnC1/GPnC2 structure and b) graphene/GPnC2/GPnC1 structure corresponding to upright triangles. Inverse triangles are series effect prediction through Equation (4). Squares (circles) are the rectification ratio in the (a) graphene/GPnC1 (GPnC1/GPnC2) and (b) graphene/GPnC2 (GPnC2/GPnC1) structure, in which the values of ΔT are kept the same as the corresponding part in graphene/GPnC1/GPnC2 and graphene/GPnC2/GPnC1 structure. The parameters are L = 50 nm, W = 7 nm, Ln = 0.71 nm, PL = 1, T0 = 1500 K, L1 = 0.71 nm, and L2 = 1.55 nm. The error bar is standard errors of 12 MD simulations with different initial conditions.

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Figure 3). The two sets of results agree well with each other as shown in Figure 3. It demonstrates that, through a subtle arrangement of GPnCs, the thermal rectification ratio can be enhanced with series effect strategy. Furthermore, in order to verify our results, we calculated another structure of gra-phene/GPnC1/GPnC2 whose L2 is set as 2.41 nm. The results show that the thermal rectification ratio of graphene/GPnC1/GPnC2 is also enhanced by series effect. And the MD results and theoretical predictions match well (details are shown in Figure S2 in the Supporting Information).

The series effect could be extended to a graded GPnC structure whose pore sizes are continuously tapered. The performance may be maximized in the graded structure if the series effect works. Although there are a few studies on the rectification effect on graded structure,[2,4,5] it is still not clear on its relationship with the series effect which deserves to explore further. Moreover, the series effect could work in other rectifier systems[10–13] and would improve the perfor-mance of rectifier. However, considering the relative high operating temperature difference in our system, a system with applicable temperature difference is needed to develop in future.

Besides the series effect, the size effect should be another important factor of nanoscale devices. The ratio of Lgraphene/LGPnC is defined as PL. Then, the effect of length ratio PL in asymmetric graphene/GPnC on the rectification is investigated. When PL is changed from 1/3 to 3, it is found that the thermal rectification ratio increases with PL increase (shown in Figure 4a). The value of PL is restricted by the overall length (L) and the length of GPnC region (LGPnC). The hardware limits the overall length of system simulated. It should be pointed out that the value of PL can be further increased by increasing L. As mentioned above, the differ-ence of the total thermal resistances come mainly from the

resistance of graphene region. For graphene, the thermal conductivity (κG) increases with its length (L), as κG ~ logL, which has been shown in recent experiment study.[50] As the length ratio (PL) increases, due to size effect, the graphene region will have a larger thermal conductivity and smaller resistance. As shown in Figure 4b, the heat flux increase as PL increases, which results in a larger difference of resistance and thermal rectification ratio.

As both the series effect and size effect are all effective methods to enhance the thermal rectification ratio, we fur-ther calculate the thermal rectification of graphene/GPnC1/GPnC2 structures, whose PLs are set as 3 (the optimized value in size effect). The results show that the combination of size effect and series effect have more significant effect on thermal rectification ratio than anyone of those alone (details are shown in Figure S3 in the Supporting Information).

In summary, using nonequilibrium molecular dynamics simulations, we studied thermal rectification in both two section and three section asymmetric graphene/GPnC struc-tures (Figure 2). The series effect is demonstrated by the consistence between the results of theoretical prediction and that of MD simulations (Figure 3). It is found that both the series effect and size effect are effective strategies to enhance the thermal rectification ratio. The series model advances previous thermal rectifier designs. More recently, the large scale synthesis[51] of porous graphene sheet with pore size ranging from atomic precision (<1 nm) to nanoscaled dimen-sion (≈1–500 nm) has been demonstrated experimentally, therefore, the high-performance thermal rectifier proposed in this paper could be realized experimentally in a foreseeable future.

Supporting Information

Supporting Information is available from the Wiley Online Library or from the author.

Acknowledgements

This project was supported in part by the grants from the National Natural Science Foundation of China: 11334007 and 51576076. The authors are grateful to Junichiro Shiomi for useful discussions. The authors thank the National Supercomputing Center in Tianjin (NSCC-TJ) for providing assistance in computations.

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Received: August 17, 2016Revised: October 14, 2016Published online:

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