Carbon nanomaterials for thermal rectification

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5

Tengfei Ma and Yan Wang Department of Mechanical Engineering, University of Nevada, Reno, NV, United States

CHAPTER OUTLINE 5.1 Introduction ................................................................................................................................. 103 5.2 Applications of Thermal Rectification ............................................................................................ 104 5.3 Common Mechanisms of Thermal Rectification in Macroscopic and Nanosystems............................ 106 5.4 Thermal and Mechanical Properties of Carbon Nanomaterials......................................................... 108 5.5 Thermal Rectification Based on Carbon Nanomaterials ................................................................... 110 5.6 Outlook and Conclusion................................................................................................................ 116 References ......................................................................................................................................... 118

5.1 INTRODUCTION Rectification, which can convert an AC signal to a DC signal, is a well-known phenomenon in the electronics field. It can be realized by an electric diode that conducts electric current in one direction while blocks the current in the opposite direction. As shown in Fig. 5.1A, by using an electric rectifier, we can remove the electric current in a certain direction (half-wave rectification). Moreover, as shown in Fig. 5.1B, we can transform an AC signal into DC without losing any information (full-wave rectification) if several diodes are arranged in an appropriate way. Thermal rectification is a phenomenon similar to electric rectification. Conventionally, heat conduction is usually independent to the sign of temperature gradient or the direction of heat current in the linear heat diffusion regime [1]. However, in certain intentionally engineered structures, that is, thermal rectifiers, there is a diode-like heat transfer behavior where the heat current changes in magnitude when the applied temperature bias is reversed in direction, as illustrated in Fig. 5.2. A perfect thermal rectifier would be one that is highly thermal conductive in one direction while insulating in the other. To quantify the degree of thermal rectification, the thermal rectification factor or ratio can be defined as: η5

G1 2 G2 κ1 2 κ2 5 G2 κ2

Carbon Based Nanomaterials for Advanced Thermal and Electrochemical Energy Storage and Conversion. DOI: https://doi.org/10.1016/B978-0-12-814083-3.00005-6 © 2019 Elsevier Inc. All rights reserved.

(5.1)

103

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FIGURE 5.1 Electric rectification. (A) Half-wave rectification, (B) full-wave rectification.

FIGURE 5.2 Illustration of thermal rectification, in which the magnitude of heat flux depends on the direction of heat flow.

where G1 (or κ) is the thermal conductance (or thermal conductivity) in one direction while G
(or κ) is in the reverse direction. According to this definition, we may consider an ideal rectifier as one being a perfect insulator in one direction (G1 5 0) while conducting in the other (G
6¼0). In that case, the rectification factor η is infinite. It is worth mentioning that the above way of defining the degree of thermal rectification is not used unanimously in literature. There are several different ways of quantifying thermal rectification, and readers are advised to note carefully the definition of η when they compare the thermal rectification ratio among different studies.

5.2 APPLICATIONS OF THERMAL RECTIFICATION Thermal rectification can be used in many aspects such as thermal management and thermal signal processing. For instance, Varga et al., used the thermal diode panels for buildings in the cooling season [2]. In their experiment, when the temperature is high enough, the special panel can have high thermal transport coefficient in the forward heat transfer direction because of the phase change of the water and the induced heat convection in the panel. However, in the reverse direction

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(reverse mode), thermal transport can be negligible because no convection occurs and thus the panel behaves as an insulator. The experiment demonstrated that for the reverse mode, the apparent conductivity is only 0.07 W/m K, while the apparent conductivity in the forward direction is three to five times larger, based on the temperature difference. Mohamad et al. implemented thermal rectification/diode into an integrated solar collector
storage tank system [3]. The dimension of the tank is 0.5 m high, 0.78 m long, and 0.5 m wide. The front face has an angle of 37 degrees, which is optimal to receive solar rays perpendicular to the surface at most times. In addition, black matte paint with an absorptivity of B0.96 is painted in the front face, which is also covered with a sheet of glass (thickness of 3-mm and transmissivity of 0.88 [4]). There is a 2.5-cm thick air gap between the glass and painted surface to reduce the heat loss to the ambient. For the other surfaces, the tank is insulated with 5.0-cm thick Styrofoam boards. The thermal diode was built by fixing a sheet of Plexiglas inside the tank (more details of the system can be found in Ref. [5]). The thermal diode is realized by allowing a light plastic strip to rotate clockwise while preventing anticlockwise rotation. Consequently, the flow induced by density difference will lead the water flow in a clockwise direction when the water is heated in the daytime, while in the night time when the water is cooled, the diode will prevent water flow from the channel to the tank. Zhu et al. developed a kind of thermal rectifier that can be used for active heat flow control [5]. It is considered to be the first temperature-gated thermal rectifier devices using VO2 beams. In the beams, the transformation between metallic and insulating phase can be controlled by temperature. Therefore, thermal rectification in this device can be actively turned on and off by changing the temperature, which functions like a thermal gate. When the device is cooled down to below 340K, the mixed phases of metallic and insulating coexist, and a maximum rectification of 28% was observed. However, when the temperature is above 340K, the device becomes completely metallic and thermal rectification disappears. As shown in Fig. 5.3A, when the

FIGURE 5.3 Temperature-gated thermal rectifier. All panels are reproduced from J. Zhu, et al., Temperature-gated thermal rectifier for active heat flow control, Nano Lett. 14 (8) (2014) 4867
4872 with permission.

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rectification is on, thermal transport is dependent on the sign of the temperature gradient. While the temperature of the device (TG) is higher than the critical temperature (TC), the direction of the temperature gradient will not affect the heat transport. Fig. 5.3B shows the microscope image of the device, which is located on suspended membranes for thermal conductance measurement. Fig. 5.3C shows the scanning electron microscopy (SEM) image of asymmetrical VO2 beam.

5.3 COMMON MECHANISMS OF THERMAL RECTIFICATION IN MACROSCOPIC AND NANOSYSTEMS It may seem that thermal rectification violates the second law of thermodynamics, but it indeed exists more commonly than expected. As shown in Fig. 5.4, a difference in the Nusselt number is expected for the natural convection between a hot plate (red) and a cold one (blue), which is well understood in undergraduate-level heat transfer textbooks. This phenomenon is induced by the different flow field governed by gravity and buoyancy [6]. The convection process induced by temperature difference, which is also called Rayleigh
Bernard convection, will occur when the Rayleigh number (RaL), which is defined as [6]: RaL 5

gβðTH 2 TC ÞL3 αν

(5.2)

is larger than a critical value of Racr 5 1708. In the above equation, g is the gravitational constant, β is the volumetric thermal expansion coefficient, TH and TC are the temperature of the hot and cold plate, respectively, α is the thermal diffusivity, and ν is the viscosity of the fluid. If the upper plate is heated, there is no fluid motion and the heat transport is governed only by heat conduction. Therefore, the second case in Fig. 5.4 generally possesses larger heat current compared with the first case for the same temperature difference, if the Rayleigh number is higher than 1708, when there is an onset of temperature-difference-induced fluid motion in the second case. Thermal rectification can also be achieved by thermal strain/warping at interfaces of materials. Different materials can exhibit different temperature dependencies on the thermal or mechanical properties. Therefore, interfaces constructed with different materials might show different contact areas when we change the temperatures at each side of the interface. Both experiments and theoretical calculations have demonstrated thermal rectification at interfaces between two different materials with specific surface roughness. The basic mechanism can be illustrated by Fig. 5.5 [7]. When

FIGURE 5.4 Thermal rectification based on natural convection.

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FIGURE 5.5 Change in thermal contact area due to strain or warping.

heat flows from the top to the bottom (which means material A is maintained at a higher temperature), the contact area will decrease due to the bending deformation of material B, and, therefore, the thermal conductivity will decrease. However, when the heat flow from the bottom to the top (which means material B is at a higher temperature), the surface of material B will be flatten by thermal expansion, and the thermal conductivity will thus increase. As early as 1955, Barzelay et al. measured the thermal contact resistance of the interface between aluminum and stainless steel [8]. Different external pressures were loaded on the structure in their experiment. The thermal resistance was found to decrease dramatically when the pressure was increased in the low-pressure range; then it stopped decreasing, and saturated at a constant value in the higher-pressure range. They also found that the thermal contact resistance varied with the heat flow direction. The thermal resistance was lower when heat flowed from aluminum to stainless steel. In addition, the degree of rectification increased with increasing pressure in the ranges of 0
400 psi. In this pressure range, the boundary conductance changed slightly with pressure in the direction of stainless steel to aluminum while it increased almost linearly in the opposite direction. So far, thermal rectification has been observed experimentally or predicted theoretically in various structures. The earliest designs of thermal rectifications are based on macroscopic mechanisms (as discussed in Ref. [7]), while, recently, various nanoengineered structures, for example, asymmetric structures or structures with lattice defects (as discussed in Refs. [7,9], and later in this chapter), have been demonstrated or predicted to be promising thermal rectifiers. Generally speaking, at least one type of linearity and asymmetry must be introduced to the system to generate thermal rectification. For example, combining two or more materials with different temperature dependence of the thermal conductivity could lead to thermal rectification even at the macroscopic scale. Go et al. has analytically proven that “the necessary condition for thermal rectification (in bulk materials) is that the thermal conductivity of the material or structure be a function of both space and temperature, and that this function is not separable” [10]. Even though a local thermal conductivity is not well-defined in nanosized materials, the above

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conclusion derived for macroscopic systems also seemingly works well in such small-scale systems, for example, asymmetrically defected graphene nanoribbons (GNRs) [11]. The existence of thermal rectification (conduction only) in a pure, nondefective, single material has been a puzzle since the discovery of a few GNR-based thermal rectifiers [12,13]. Based on the conventional heat diffusion picture, in which heat conduction is governed by Fourier’s law, thermal rectification should not exist in a macroscopic homogeneous material [1]. Thermal rectification in nanosized structures is thus caused by other mechanisms. In Ref. [1], three interrelated mechanisms were revealed to be responsible for thermal rectification in asymmetric GNRs, namely, asymmetric phonon spectra overlap, inseparable dependence of thermal conductivity on temperature and space, and phonon edge (or boundary for 3D materials) localization, which will be discussed in more detail later in this chapter. Carbon-based thermal rectifiers will be emphasized in this chapter owing to their high flexibility, fracture toughness, high strength, and high-temperature stability. We will also demonstrate that the abundance of carbon elements, the existence of various carbon allotropes, and the versatility in ways of engineering carbon-based structures render them outstanding candidates for thermal rectifiers.

5.4 THERMAL AND MECHANICAL PROPERTIES OF CARBON NANOMATERIALS Various materials and structures have been proposed to be promising candidates of thermal rectifier. Carbon nanomaterials (CNMs) are one of the most popular and appropriate choices for designing thermal rectification devices due to their special physical and chemical properties. Generally speaking, CNMs, such as graphene, carbon nanotube (CNT), and GNRs, typically possess high and tunable thermal conductivity. This, in conjugate with the low thermal expansion coefficient, render them suitable to be fabricated into thermal rectifiers. Graphene is one of the most attractive CNMs because of its unique mechanical and thermal properties. Balandin et al. obtained superior thermal conductivity of single-layer graphene experimentally [14]. Fig. 5.6 shows their experiment set up, in which laser was used to induce a heat wave in the graphene layer. The measured thermal conductivity at room temperature is in the range of B4840 to B5300 W/m K. Seol et al. also reported extremely high thermal conductivity in the range of 3000
5000 W/m K for 2D suspended graphene in their experiment [15]. Based on 2D graphene, different graphitic forms, for example, 0D buckyballs, quasi-1D CNT and GNRs, and 3D graphite can be formed, as shown in Fig. 5.7 [16]. These carbon allotropes also possess interesting thermal and mechanical properties. For example, CNTs are good conductors of phonons, which are the primary carriers of heat for nonmetallic materials. Therefore, CNTs can have very high thermal conductivity. At room temperature, molecular dynamics simulations of CNTs have predicted the thermal conductivity to be up to 6000 W/m K [17,18]. However, theoretical models have predicted that the thermal conductivity of CNT can be divergent with its length [19,20]. Recent experiments reported the thermal conductivity of millimeter-long CNT can be 8640 W/m K [21].

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FIGURE 5.6 Schematic of the experiment set up [9]. Figure is reproduced from Y. Wang, A.K. Vallabhaneni, B. Qiu and X.L. Ruan, Two-dimensional thermal transport in graphene: a review of numerical modeling studies, Nanoscale Microscale Thermophys. Eng. 18 (2), April 2014, 155
182 with permission.

FIGURE 5.7 Different graphitic forms based on graphene [16]. Figure is reproduced from A.K. Geim and K.S. Novoselov, The rise of graphene, Nat. Mater. 6 (3), 2007, 183 with permission.

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Carbon nanotubes also possess superior mechanical properties. The calculated Young’s modulus of CNTs is almost 1 TPa, regardless of their types (single- or multiwalled) and diameters. Poncheral et al. measured a Young’s modulus between 0.7 and 1.3 TPa by using electromechanical resonant vibrations [22]. Wong et al. directly measured the Young’s modulus in 1997 [23]. They used the atomic force microscopy (AFM) to measure the stiffness constant of nanotubes and obtained an average value of 1.28 TPa. Moreover, they also successfully conducted the first mechanical strength measurement and obtained an average bending strength of 14 GPa for nanotubes. Yu et al. conducted a stress
strain experiment on arc-multiwalled nanotubes (arc-MWNT) with an electron microscope, and obtained a Young’s modulus of 0.27
0.95 TPa [24]. They also found that fracture of MWNT can sustain a strain up to 12% without fracture.

5.5 THERMAL RECTIFICATION BASED ON CARBON NANOMATERIALS Owing to the superior properties of CNMs, extensive studies have been conducted to investigate thermal rectification in CNMs. In 2006, Chang et al. successfully realized thermal rectification in the laboratory by modifying nanotubes [both CNTs and boron nitride nanotubes (BNNTs)] [25]. According to previous studies [20,26], one-dimensional thermal transport can be unusual compared with the bulk materials in that Fourier’s law might be invalid. The extraordinary nonlinear thermal effect can help to build thermal rectification structures. Nanotubes, which are quasi-1D, are therefore suitable materials for investigating thermal rectification. For pristine nanotubes with uniform mass distribution, the thermal transport is symmetric; but by adding or doping amorphous C9H16Pt particles, the engineered CNTs and BNNTs possessing nonuniform axial mass distribution (Fig. 5.8) with thermal rectification, can be observed.

FIGURE 5.8 Schematic description of depositing amorphous C9H16Pt (red dots) on a nanotube [25]. Figure is reproduced from C. Chang, D. Okawa, A. Majumdar and A. Zettl, Solid-state thermal rectifier, Science 314 (5802), 2006, 1121
1124 with permission.

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The results showed that the measured thermal conductivity was 305 W/m K and the rectification level was 2% for CNT at room temperature. For BNNT, the respective thermal rectifications for different size of structures were 7%, 4%, and 3%. The structure will show higher thermal transport performance when heat flowed from C9H16Pt doped region to the other side. For the first time, Hu et al. observed thermal rectification in asymmetric GNRs using molecular dynamics simulations in 2009 [12]. Fig. 5.9A and B shows the rectangular (symmetric) GNR and triangular (asymmetric) GNR structures studied in their research. Nonequilibrium molecular dynamics (NEMD) were used to obtain the thermal conductivity. As shown in Fig. 5.9A and B, ´ the atoms represented by triangles worked as the hot or cold bath with the Nose
Hoover thermostat to control the temperature. Thermal rectification can be observed in the asymmetric triangular structure after reversing the two heat baths. As we can see in Fig. 5.9C, the thermal conductivity of the case in which heat flows from the left (narrower) side to the right (wider) side is lower than the reverse case for the triangular GNR. In contrast, thermal conductivity is independent to the heat flow direction for rectangular GNR, as shown in Fig. 5.9D. The thermal rectification factor, defined as ðκW-N 2 κN-W Þ=κW-N , can be as large as 120% at T 5 180K, although it will decrease apparently with the increase of temperature, as shown in the inset of Fig. 5.9C. Similarly, thermal rectification in asymmetric GNRs was also reported in Yang et al.’s study at a similar time [13]. They observed the dependence of heat flux on the direction of heat flow in asymmetric GNRs from molecular dynamics simulations. Fig. 5.10A shows two different GNR structures they studied: trapezia shaped GNR (TGR) and two-segment-GNR (RGR) with different widths. In their simulation, a hot bath and a cold bath are assigned to the top and bottom with temperature of Ttop 5 T0 (1
Δ) and Tbot 5 T0 (1 1 Δ) to establish a temperature gradient. Therefore, Δ . 0 means the bottom has a higher temperature. The simulation results show that the rectification factor, which is defined similarly as Hu et al.’s work, can be as high as 350% in the TGR, as shown in Fig. 5.10B. This suggests that the heat flux prefers to run along the direction of decreasing width, which was explained qualitatively by the match/mismatch of the phonon spectra between the atomic layers at the two ends. Although this kind of asymmetric nanostructure can realize thermal rectification, it may diminish with larger size. In 2014, Wang et al. [1], using NEMD molecular dynamics simulations, revealed that thermal rectification is significant in nanosized asymmetric GNRs, because of the phonon lateral confinement phenomenon, while its effect diminishes when lateral dimension of the system enlarges. Two different GNR structures (trapezoidal and T-shaped GNRs) were simulated in their research as shown in Fig. 5.11A. In this study, the thermal rectification ratio, or factor (η) is defined as: η5

κforward 2 κbackward κforward

(5.3)

where κforward and κbackward represent the thermal conductivity of two different directions, which are indicated by the arrows in Fig. 5.11A. η can be B40% when the width is relatively low, as shown in Fig. 5.11B. More importantly, η decreases dramatically with the increasing of width (w) but more moderately with length (L), which indicates that the small lateral size is necessary for thermal rectification to occur in asymmetric GNRs.

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FIGURE 5.9 (A, B) Atomic structure of graphene nanoribbons (GNRs). (C, D) Thermal conductivity of (C) asymmetric and (D) symmetric GNRs. The inset of (C) shows the thermal rectification factor η vs temperature. All panels are reproduced from J. Hu, X. Ruan and Y.P. Chen, Thermal conductivity and thermal rectification in graphene nanoribbons: a molecular dynamics study, Nano Lett. 9 (7), 2009, 2730
2735 with permission.

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FIGURE 5.10 (A) Atomic structure of asymmetric GNR. (B) Variation of heat flux and rectification with different Δ. All panels are reproduced from N. Yang, G. Zhang and B. Li, Thermal rectification in asymmetric graphene ribbons, Appl. Phys. Lett. 95 (3), 2009, 033107 with permission.

Wang et al. proposed three interrelated mechanisms for explaining thermal rectification in asymmetric GNRs and its dependence on lateral size. First, the phonon spectra overlap can be different before and after switching the two thermostats in the NEMD simulations and therefore, different heat transfer behavior will arise. However, when the width increases to macroscopic size, the local phonon spectra will only depend on the temperature, and the phonon spectra overlap will be the same before and after switching the thermostats. As a result, thermal rectification diminishes. Second, the thermal conductivity is dependent on both temperature and space. The phonon lateral confinement effect will induce the dependence of the phonon mean free path and the density of states on the width; thus the thermal conductivity will be dependent on the width. Therefore, when the width of the asymmetric GNR increase to bulk size, the thermal conductivity only depends on

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FIGURE 5.11 (A) Geometrical structure of trapezoidal GNR and T-shaped GNR with definition of length (L) and width (w). The red arrows define the forward direction. (B) Calculated TR ratio (factor) versus length or width of the structure. Rectangular symbols represent the TR ratio of T-shaped GNR versus total length; cross symbols show the TR ratio of T-shaped GNR versus width, and circles is the TR ratio of trapezoidal GNR versus length. All panels are reproduced from Y. Wang, A. Vallabhaneni, J. Hu, B. Qiu, Y.P. Chen and X. Ruan, Phonon lateral confinement enables thermal rectification in asymmetric single-material nanostructures, Nano Lett. 14 (2), 2014, 592
596 with permission.

the temperature so the thermal rectification disappears. Third, phonons can be localized at the edges, and the strength of localization depends on local temperature. The asymmetric localization profile causes thermal rectification in asymmetric GNRs. However, when the GNR width increases, the edge scattering and localization of phonons become less significant and the thermal rectification vanishes. Similarly, Wu et al. [27] observed thermal rectification in single-walled carbon nanohorns (SWNH), which are composed of a horn-shaped, capclosed CNT modified from single-layer graphene. NEMD simulations were carried out in this research and the results suggest that SWNHs are efficient thermal rectifiers. Moreover, they also observed the tunable force constants along the SWNHs’ axes in the stretched structures. Furthermore, the thermal rectification can be enhanced in deformed SWNHs. Based on their simulations, the thermal rectification does not change monotonically with the variation of tensile strain, but a similar trend for different tensile cases can be observed. In particular, thermal rectification is enhanced by small strain while it is reduced by large strain cases. This phenomenon results from the asymmetry of the structure. When a tensile strain is applied, the side with a large radius can hardly be stretched and, as a result, the narrower end can have a larger local strain. Therefore, the interatomic force constants do not change with the same rate along the axial direction [27]. Such force constant gradient is claimed to cause an asymmetric phonon spectra and lead to thermal rectification. Generally speaking, many proposed or fabricated thermal rectifiers have an asymmetric shape, mass density, or an interface between dissimilar materials. However, the manufacture of such delicate structures demands a sophisticated patterning process that limits their stability. The utilizing of interfaces can significantly reduce the effective thermal conductivity, which limits the application of this design. A nanodevice made of a single material, requiring minimum fabrication efforts, and

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FIGURE 5.12 (A) Schematic description of different types of defect including single-vacancy (remove one carbon atom), di-vacancy (remove two carbon atoms), substitutional Si (substitute one carbon atom with silicon atom) and Stone
Wales. (B) Simulation domain for molecular dynamics simulations of pdGNRs. All panels are reproduced from Y. Wang, S. Chen and X. Ruan, Tunable thermal rectification in graphene nanoribbons through defect engineering: a molecular dynamics study, Appl. Phys. Lett. 100 (16), 2012, 163101 with permission.

possessing high thermal conductivity, is a great benefit to this field. Motivated by this demand, Wang et al. proposed asymmetrically defected GNR as a promising thermal rectifier, and systematically studied the effect of various structural parameters on thermal rectification ratio [11]. Specifically, the authors investigated pristine-defected GNR (pdGNR) by adding certain defects, that is, single-vacancy, di-vacancy, substitutional silicon, and Stone
Wales defect, to a perfect GNR. Fig. 5.12 shows the different kinds of defects that were studied in their work. To quantify thermal rectification, they define thermal rectification in the same way as Eq. (5.1), with “ 1 ” denoting the case when the thermostat in the pristine side is maintained at a higher temperature and that in the defected side is maintained at a lower temperature, and vice versa for “
”. It was found that the rectification ratio can be as high as 70%. This work also provided a general view of how temperature difference, structure size (length), and defect concentration (α) can affect the thermal rectification, as shown in Fig. 5.13. Recently, Wang et al. [28] conducted an experimental study on thermal rectification in suspended monolayer graphene. The schematic of the structure is shown in Fig. 5.14A, in which defects in the basal plane of the GNR were generated by ion beam irradiation. The SEM image of the pristine and defect-engineered graphene samples are shown in Fig. 5.14C
E, respectively. The experimental results of three samples are plotted in Fig. 5.14B. Solid and open symbols represent the thermal conductivities of different directions. We can obviously see that no thermal rectification can be detected (the difference is less than 2%) for the pristine graphene, that is, before the defect engineering. However, after the defect engineering, the thermal conductivities of different directions show an apparent difference. The thermal conductivity in the direction of the red arrows shown in

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FIGURE 5.13 The thermal rectification ratio as a function of temperature difference, normalized length, and defect concentration, respectively. All panels are reproduced from Y. Wang, S. Chen and X. Ruan, Tunable thermal rectification in graphene nanoribbons through defect engineering: a molecular dynamics study, Appl. Phys. Lett. 100 (16), 2012, 163101 with permission.

Fig. 5.14C
E are larger than that in the opposite direction. The thermal rectification factors (η) of the engineered graphene (defined as η 5 jλF 2 λB j=λB , where λF and λB are the thermal conductivity of forward and backward direction) is 28, 26, 25% for samples 1, #2, and #3, respectively.

5.6 OUTLOOK AND CONCLUSION Based on the discussions in this chapter, it is obvious that thermal rectification requires a certain form of asymmetry in the structure, either in its geometry, composition, defect concentration, or

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FIGURE 5.14 (A) The schematic diagram of the structure in the experiment. A graphene ribbon suspended between two Au pads working as heater and thermometer. (B) Experimental results of the three sample shown in C, D, and E. (C
E) show three samples in the experiments. Each column shows the pristine graphene with larger scale, the pristine graphene with smaller scale, and the engineered graphene, respectively from left to right. All panels are reproduced from H. Wang, S. Hu, K. Takahashi, X. Zhang, H. Takamatsu and J. Chen, Experimental study of thermal rectification in suspended monolayer graphene, Nat. Commun., 8, 2017, 15843 with permission.

material types. However, it is also evident that the thermal rectification factor of most designs is still not satisfactory for practical applications. Moreover, the delicate nanostructures proposed in most theoretical studies are challenging to fabricate in laboratories, leaving a huge gap between experiments and theories. Thermal rectification solely caused by an asymmetry in geometry is certainly of fundamental importance to heat transfer physics. Due to the phonon lateral confinement mechanism proposed in Ref. [1], the size of the structure might also need to be considered to achieve useful thermal rectifications. The effect of lateral phonon localization will diminish when the size of the lateral dimension of the structure increases to the macroscopic limit, and thermal rectification would thus disappear even with asymmetric structures. This mechanism poses a strict limit to the size of thermal rectifiers based on the idea of asymmetric geometry to micro-/nanoscale. Moreover, a geometric asymmetry alone might be insufficient for achieving significant

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thermal rectification for practical applications. As discussed in this chapter, an asymmetry in defect concentration, mass gradient, or strain field can also induce significant thermal rectification. Therefore, instead of just using asymmetry structures, novel designs coupling two or more of the mechanisms above synergistically could lead to practical thermal rectifiers.

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