Phonon Thermal Transport at Interfaces of a Graphene/Vertically Aligned\\ Carbon Nanotubes/Hexagonal Boron Nitride Sandwiched Heterostructure

Menglin Li(李檬璘), Muhammad Asif Shakoori, Ruipeng Wang(王瑞鹏), and Haipeng Li(李海鹏)$^{\hyperlink{s}{*}}$

doi:10.1088/0256-307X/41/1/016302

⇦Chinese Physics Letters, 2024, Vol. 41, No. 1, Article code 016302⇨ Phonon Thermal Transport at Interfaces of a Graphene/Vertically Aligned Carbon Nanotubes/Hexagonal Boron Nitride Sandwiched Heterostructure Menglin Li (李檬璘), Muhammad Asif Shakoori, Ruipeng Wang (王瑞鹏), and Haipeng Li (李海鹏)* Affiliations School of Materials Science and Physics, China University of Mining and Technology, Xuzhou 221116, China Received 16 October 2023; accepted manuscript online 1 December 2023; published online 9 January 2024 ^*Corresponding author. Email: haipli@cumt.edu.cn Citation Text: Li M L, Shakoori M A, Wang R P et al. 2024 Chin. Phys. Lett. 41 016302 Abstract Molecular dynamics simulation is used to calculate the interfacial thermal resistance of a graphene/carbon nanotubes/hexagonal boron nitride (Gr/CNTs/hBN) sandwiched heterostructure, in which vertically aligned carbon nanotube (VACNT) arrays are covalently bonded to graphene and hexagonal boron nitride layers. We find that the interfacial thermal resistance (ITR) of the Gr/VACNT/hBN sandwiched heterostructure is one to two orders of magnitude smaller than the ITR of a Gr/hBN van der Waals heterostructure with the same plane size. It is observed that covalent bonding effectively enhances the phonon coupling between Gr and hBN layers, resulting in an increase in the overlap factor of phonon density of states between Gr and hBN, thus reducing the ITR of Gr and hBN. In addition, the chirality, size (diameter and length), and packing density of sandwich-layer VACNTs have an important influence on the ITR of the heterostructure. Under the same CNT diameter and length, the ITR of the sandwiched heterostructure with armchair-shaped VACNTs is higher than that of the sandwiched heterostructure with zigzag-shaped VACNTs due to the different chemical bonding of chiral CNTs with Gr and hBN. When the armchair-shaped CNT diameter increases or the length decreases, the ITR of the sandwiched heterostructure tends to decrease. Moreover, the increase in the VACNT packing density also leads to a continuous decrease in the ITR of the sandwiched heterostructure, attributed to the extremely high intrinsic thermal conductivity of CNTs and the increase of out-of-plane heat transfer channels. This work may be helpful for understanding the mechanism for ITR in multilayer vertical heterostructures, and provides theoretical guidance for a new strategy to regulate the interlayer thermal resistance of heterostructures by optimizing the design of sandwich layer thermal interface materials.

DOI:10.1088/0256-307X/41/1/016302 © 2024 Chinese Physics Society Article Text With the development of electronic devices such as mobile phones and chips, showing the trend of integration, miniaturization and high frequency, the heat flow density of electronic devices has increased significantly. The problem of heat dissipation is a crucial factor that limits the performance of electronic products. Therefore, reducing interfacial thermal resistance (ITR) has become a significant approach to enhancing performance. In recent years, graphene (Gr) has attracted the attention of researchers because of its unique physical properties such as high carrier mobility, excellent thermal conductivity, and mechanical properties. Gr-based materials can significantly improve the heat dissipation performance of nanodevices and raise the energy conversion efficiency of electronic devices. Therefore, Gr-based two-dimensional materials are expected to become good thermal interface materials for the next-generation micro–nano electronic devices,[1] providing new solutions to the current problems related to heat dissipation for high-power electronic devices. A new type of two-dimensional heterostructures made of two or more two-dimensional materials fixed vertically usually has unusual physical properties.[2,3] Among them, hexagonal boron nitride (hBN) is an appropriate candidate for producing a heterostructure with Gr. This is not only due to its analogous honeycomb structure to Gr but also because it has small lattice mismatches with Gr. In a vertical van der Waals (vdW) heterostructure composed of Gr and hBN, the hBN is generally used as a promising dielectric substrate for Gr-based nano-electronic devices. Recently, Gr/hBN vdW heterostructures have also aroused great interest due to their unique Moiré pattern. Compared with Gr deposited on silica, the mobility of Gr on an hBN substrate shows a significant improvement by three times.[4,5] Singh and Kumar studied the influence of the interface on the mechanical properties of a Gr/hBN vdW heterostructure.[6] Compared with single-layer hBN, heat transport in a Gr/hBN vdW heterostructure is significantly enhanced.[7,8] Chen et al.[9] experimentally measured the Seebeck coefficient of Gr/hBN/Gr sandwich vdW heterostructures to be $-$99.3 µV$\cdot$K$^{-1}$, indicating that Gr/hBN heterostructures have potential for applications in the field of thermoelectric power. However, improvement in the thermoelectric $ZT$ value of Gr/hBN heterostructures requires an effective reduction in thermal conductivity. The interfacial thermal conductance (ITC) or ITR (= 1/ITC), a crucial aspect for Gr/hBN heterostructures used in thermal interface materials, has also been extensively investigated experimentally and computationally. The experimentally measured ITC of the Gr/hBN interface is approximately 7.4 MW$\cdot$m$^{-2}$$\cdot$K$^{-1}$,[10] consistent with the theoretical values.[11] Ren et al.[12] observed the rotation-angle dependence of interfacial thermal transport across a Gr/hBN heterostructure. Zou and Cao[13] reported that the interlayer coupling between Gr and hBN is too weak to change the phonon group velocities of the Gr layer but reduces the relaxation time of the out-of-plane acoustic (ZA) phonon modes due to breaking of the symmetry-based selection rule. Chen et al.[14] found that the interlayer coupling strength has a great influence on the coupling between out-of-plane and in-plane phonons in Gr/hBN vdW heterostructures. Our previous theoretical study also found that enhanced interlayer coupling is conducive to reducing the interlayer ITR of a Gr/silicene vdW heterostructure.[15] Therefore, research of tunable phonon thermal transport in Gr/hBN heterostructures is a key scientific issue for Gr/hBN heterostructure applications.[5] Although the above-mentioned studies have advanced our understanding of thermal transport in Gr/hBN vdW heterostructures, ITR in heterostructures containing both strong covalent bonds and weak vdW interactions has yet to be further explored.[14] Compared with the large number of reported Gr heat transport studies, there are relatively few studies investigating phonon heat transport of carbon nanotubes (CNTs) and their composite structures. The theoretical value of CNT axial thermal conductivity can be as high as 3000 W$\cdot$m$^{-1}$$\cdot$K$^{-1}$[16] and CNTs have a very low thermal expansion coefficient ($10^{-6}$ K$^{-1}$). Therefore, CNTs are considered to be potential thermal conductive fillers for increasing the thermal conductivity of matrix materials such as polymers and metals.[17-19] For instance, CNTs arranged vertically between two solid surfaces can form a good contact with the solid surface under reasonable pressure. Vertically aligned CNTs (VACNTs), with advantages of high thermal conductivity, thermal stability and flexibility, are expected to solve the problem of chip heat dissipation as excellent thermal management materials.[20,21] Yu et al.[22] used chemical vapor deposition to vertically grow VACNT arrays on copper sheets coated with silicon carbide to achieve high orientation of CNTs along the vertical direction of the copper sheet. The high contact thermal resistance between CNTs and metal interfaces is an urgent problem to be solved before applications. However, there are few reports on the regulation of heat transfer in VACNT-based heterostructures. In this Letter, VACNTs are introduced into a Gr/hBN vdW heterostructure to form Gr/VACNT/hBN sandwiched heterostructures via covalent bonding of a sandwich layer between Gr and hBN layers. The molecular dynamics (MD) simulation method is used to simulate the room-temperature interfacial heat transport properties of Gr/VACNT/hBN sandwiched heterostructures, and the effects of diameter, length, chirality, and number of CNTs on ITR are discussed. Analysis of the phonon density of state (PDOS) of each layer of the sandwiched heterostructure reveals that the sandwich layer covalent bonding mechanism induces a decrease in ITR of the heterostructure. The findings of this study are of great guiding significance for applications of carbon-based thermal interface materials. Model and Method of Calculations. Figure 1 schematically shows the Gr/VACNT/hBN heterostructure. The sandwich-layered VACNTs are covalently bonded to the upper Gr and the lower hBN to form a Gr/VACNT/hBN sandwiched heterostructure. In this study, the optimized Tersoff potential[18,19] developed by Lindsay and Broido is used to describe the interatomic interaction among C, N, and B atoms of the Gr layer and the hBN layer. The optimized Tersoff potential can simulate the phonon properties and heat transport properties of Gr and hBN nanostructures.[23-26] The interlayer interaction between Gr and the hBN vdW heterostructure is expressed by the Lennard–Jones (LJ) potential, $V(r_{ij})=4\varepsilon [{({\frac{\sigma }{r_{ij}}})^{12}-({\frac{\sigma}{r_{ij}}})^{6}}]$, where $\varepsilon$ is the depth of the potential well, $\sigma$ is the distance at which the interparticle potential is zero, and $r_{ij}$ is the distance between atoms $i$ and $j$. Here, the parameters of the LJ potential are $\varepsilon_{\scriptscriptstyle{\rm C-B}} = 3.294$ meV, $\sigma_{\scriptscriptstyle{\rm C-B}} = 3.411$ Å, $\varepsilon_{\scriptscriptstyle{\rm C-N}} = 4.068$ meV, and $\sigma_{\scriptscriptstyle{\rm C-N}} = 3.367$ Å.[27,28]

Fig. 1. Schematic diagram of the Gr/VACNT/hBN sandwiched heterostructure: (a) top view, (b) front view.

The MD simulations are performed using the LAMMPS software.[29] The simulation time step is set to 0.1 fs and the periodic boundary conditions applied in the $xy$ plane. First, at a temperature of 300 K, the heterostructure is relaxed successively by 0.5 ns in the isothermal–isobaric ensemble, canonical ensemble and microcanonical ensemble, respectively, and then the system is verified to reach an equilibrium state. Next, the heat dissipation method[30,31] is used to calculate the ITR for the fully relaxed system. The specific calculation process of ITR is as follows: Gr in the upper layer is quickly heated to 500 K during 50.0 fs, while hBN in the lower layer is kept at 300 K. After 50.0 fs, the thermostat is removed and the heat is transmitted from the upper Gr layer through the sandwich layer of VACNTs to the lower hBN layer. After 800.0 ps, the Gr and hBN layers reach a new equilibrium state, as shown in Fig. 2(a). The change in energy dissipation of the upper Gr over time is expressed as \begin{align} \frac{\partial E_{t} }{\partial t}=\frac{A}{R}({T_{\rm Gr} -T_{\rm hBN}}), \tag {1} \end{align} where $R$ is the ITR, $A$ is the area of the heterostructure interface, $T_{\rm Gr}-T_{\rm hBN}$ is the temperature difference between Gr and hBN layers, and $E_{t}$ is the energy of the Gr layer. The energy $E_{t}$ after integration becomes \begin{align} E_{t} =E_{0} +\frac{A}{R}\int_0^t {({T_{\rm Gr} -T_{\rm hBN}})} dt, \tag {2} \end{align} where $E_{0}$ is the initial energy dissipated by Gr as heat, as shown in Fig. 2(b), and the ITR ($R$) is obtained by linear fitting. In order to verify the reliability and accuracy of this method, we selected a Gr/hBN vdW heterostructure with an interface area of 98.0 Å $\times $ 100.9 Å with an interlayer layers of 3.4 Å.[32,33] The calculated ITR of the Gr/hBN vdW heterostructure is $R=(1.86 \pm 0.08) \times 10^{-7}$ m$^{2}$$\cdot$K$\cdot$W$^{-1}$, which is consistent with the ITR of the a Gr/hBN vdW heterostructure of the same size reported in the literature ($R =1.5\times 10^{-7}$ m$^{2}$$\cdot$K$\cdot$W$^{-1}$[34]).

Fig. 2. (a) Evolution of Gr temperature ($T_{\rm Gr}$), hBN temperature ($T_{\rm hBN}$) and Gr energy ($E_{\rm Gr}$) with time $t$ during the heat dissipation process. (b) Linear relationship between Gr energy ($E_{\rm Gr}$) and time integration of temperature difference ($T_{\rm Gr}-T_{\rm hBN}$). Here, it is assumed that heat flows from Gr to hBN.

In solid state physics, phonons are the quantized form of lattice vibration, and the thermal conduction of crystals is mainly determined by the interaction of phonons.[35] PDOS is an important parameter that reflects the thermal properties of crystals at the microscopic level. The degree of overlap of the PDOS of layered materials determines the phonon transmission behavior of the interface. Experimentally, the PDOS can be measured using inelastic neutron diffraction technology. Based on MD simulations, PDOS can be obtained by Fourier transform of the velocity autocorrelation function as[36] \begin{align} D(\omega)=\int_0^\tau {\varGamma ({t})} \exp ({-i\omega {t}})dt, \tag {3} \end{align} where $\omega$ is the frequency, $D(\omega)$ is the frequency of the PDOS, and $\varGamma$(t) is the velocity autocorrelation function written as $\varGamma (t)=\langle {v(t)\cdot v(0)}\rangle$. The ITR between heterostructure interfaces is closely related to the PDOS overlap factor $S$ of the materials on both sides of the interface,[37] \begin{align} S=\frac{\int_0^{+\infty } {D_{1}(\omega)D_{2}(\omega)} d\omega }{\int_0^{+\infty } {D_{1}(\omega)} d\omega \int_0^{+\infty } {D_{2} (\omega)} d\omega}, \tag {4} \end{align} where the $D_{1}$ and $D_{2}$ are the PDOSs of the materials on both sides of the interface. The larger the value of the overlap factor $S$, the stronger the phonon coupling between the two layers, and thus the more conducive it is to phonon transport. Effect of VACNT Diameter on the ITR of the Heterostructure. In the Gr/VACNT/hBN sandwiched heterostructure the area of the Gr and hBN layers is 119.35 Å $\times $ 118.08 Å, and the thickness of the sandwich layer is 50.42 Å. We select 11 kinds of armchair CNTs: the chirality index ($m$,$m$) increases from (20,20) to (70,70), and the corresponding nanotube diameter $d$ increases from 27.12 Å to 94.92 Å. Figure 3 shows the relationship between the diameter of a single armchair-shaped CNT and the ITR of the Gr and hBN layers. It can be seen that the diameter of CNT (20,20) is $d=27.12$ Å, and the corresponding ITR ($R=32.16 \pm 1.11$ m$^{2}\cdot$K$\cdot$GW$^{-1}$) of sandwiched heterostructure is one order of magnitude smaller than the ITR of the Gr/hBN vdW heterostructure ($R=186 \pm 8$ m$^{2}\cdot$K$\cdot$GW$^{-1}$). Obviously, the covalent bonding of VACNTs effectively increases the phonon coupling between the Gr and hBN layers, which is conducive to enhancing the interlayer interface heat conduction. As the diameter $d$ of the armchair-shaped CNTs ($m$,$m$) increases from 27.12 Å to 94.92 Å, the ITR of the sandwiched heterostructure gradually decreases to 28% due to the increase in the number of CNT atoms bonded to the Gr or hBN layers.

Fig. 3. Effect of armchair CNT diameter ($d$) on the ITR ($R$) of a Gr/VACNT/hBN sandwiched heterostructure, with ($m$,$m$) showing the chiral index of armchair nanotubes.

In Refs. [38,39] the authors reported that the effect of CNT chirality on the thermal conductivity of a single-wall CNT of the same diameter is less than 20%. However, the influence of CNT chirality on the ITR of Gr/VACNT/hBN sandwiched heterostructure is unclear. We select two chiral CNTs with a length of 50.42 Å and a diameter of 40.68 Å, namely an armchair-shaped CNT (30,30) and zigzag CNT (52,0), to form a Gr/VACNT/hBN sandwiched heterostructure, and calculate their ITR. It is found that the ITR ($R=20.04 \pm 1.74$ m$^{2}\cdot$K$\cdot$GW$^{-1}$) of the armchair-shaped CNT (30,30) sandwiched vertical heterostructure is about 25% higher than that of the zigzag CNT (52,0) sandwiched vertical heterostructure ($R=15.98 \pm 2.28$ m$^{2}\cdot$K$\cdot$GW$^{-1}$). Therefore, the chirality of the CNT has an important influence on the ITR of the Gr/VACNT/hBN sandwiched heterostructure. The reason may be that the axial atomic arrangement and edge atomic distribution are different in armchair and zigzag CNTs, resulting in different chemical bonding of chiral CNTs with Gr and hBN to influence phonon transport between the interfaces of the upper and lower layers. Effect of VACNT Length and Packing Density on the ITR of a Heterostructure. We choose a Gr and hBN layer with an area 60 Å $\times$ 60 Å and study the influence of VACNT length on the ITR of the Gr/VACNT/hBN sandwiched heterostructure. Within the error range of calculation, the direction of applied heat flow has little effect on the ITR of the heterostructure. As shown in Fig. 4, for an applied heat flow directed from the Gr layer to the hBN layer, as the VACNT length increases from 10.0 Å to 90.0 Å, the ITR of the Gr/VACNT/hBN sandwiched heterostructure increases from $R=9.23 \pm 1.31$ m$^{2}\cdot$K$\cdot$GW$^{-1}$ to $R=35.70 \pm 2.11$ m$^{2}\cdot$K$\cdot$GW$^{-1}$. This shows that the length of the VACNT has a significant impact on the ITR of the heterostructure, and an increase in the length of the sandwich VACNT is not conducive to heat conduction of the interlayer interface. This is because an increase in the thickness of the sandwich layer will increase the length and conduction time of the heat transfer path as well as the ITR.[40] It should be noted here that the calculated ITR between Gr and hBN layers in heterostructures containing both strong covalent bonds and weak vdW interactions is different from the size dependence of longitudinal phonon transport in nanoscale CNTs.[41] As was reported, when the length of CNTs is very short and similar to the mean free path of phonons (about 80 nm at 300 K[38]), phonon transport is dominated by ballistic transport, and thus phonon thermal conductivity increases linearly with the increase in length of CNTs.

Fig. 4. Effect of CNT (5,5) length ($L$) on the ITR of a Gr/VACNT/hBN sandwiched heterostructure. The interface area between Gr and hBN is 60 Å $\times$ 60 Å and the diameter of CNT (5,5) $d=6.78$ Å.

In the sandwiched heterostructures we insert 1–5 VACNTs (5,5) with a diameter of 6.78 Å and length of 50.42 Å to study the influence of VACNT orderly packing density on the ITR of the sandwiched heterostructure. The calculation results are shown in Fig. 5. With increase in the number of VACNTs in the intermediate layer, the ITR of the sandwiched heterostructure continues to decrease. Compared with the sandwiched heterostructure with one VACNT, the ITR of the sandwiched heterostructure with five VACNTs is reduced by about 77%. This is due to the high intrinsic thermal conductivity of CNTs and the increase in out-of-plane heat transfer channels. Therefore, increasing the VACNT packing density of the intermediate layer is conducive to improving interfacial heat transport from the Gr layer to the hBN layer, which is of great significance for designs of interfacial thermal management materials.

Fig. 5. Effect of VACNT orderly packing density ($\sigma$) on the ITR of Gr/VACNT/hBN sandwiched heterostructures. The insets show the top views of the sandwiched heterostructure. The interface area between Gr and hBN is 60 Å $\times$ 60 Å and the diameter of CNT (5,5) $d=6.78$ Å.

Fig. 6. ITR of Gr/VACNT/hBN sandwiched heterostructures with four and five randomly arranged CNTs. The interface area between Gr and hBN is 60 Å $\times$ 60 Å and the diameter of CNT (5,5) $d=6.78$ Å.

The number of VACNTs prepared in the experiment is enormous, and there may be defects and disordered arrangements locally. We simulate the effect of VACNT random distribution on ITR under the same packing density. We construct ten random packing configurations for four and five CNTs in heterostructures. The ITR values of these CNTs are shown in Fig. 6. It is observed that the average values of ITR are $7.45 \pm 0.69$ m$^{2}\cdot$K$\cdot$GW$^{-1}$ for four CNTs and $5.31 \pm 0.37$ m$^{2}\cdot$K$\cdot$GW$^{-1}$ for five CNTs with random arrangements. This indicates that the influence of VACNT packing arrangement on ITR is less than 10% of the statistical average, and the fluctuation decreases with increase in VACNT packing density. Therefore, the increase in the number of heat dissipation channels bonded between interfaces is the main factor contributing to the decrease in ITR. Phonon Density of States. Usually, small differences in the PDOS of the interfacial material will have a great effect on the ITR of the heterostructure. Therefore, heat transfer in the heterostructure depends on the degree of matching of the PDOS of different materials. According to Eq. (4), the overlap factor $S$ between the layers in the Gr/VACNT/hBN sandwiched heterostructure is calculated as shown in Fig. 7. In the Gr/VACNT/hBN sandwiched heterostructure, the PDOS of Gr is similar to the PDOS of CNTs in the frequency band 0–60 THz, and the corresponding PDOS overlap factor $S$(Gr–CNT) = 0.172. However, the PDOS of hBN and CNTs is very different, and the corresponding PDOS overlap factor $S$(hBN–CNT) = 0.0222, indicating that the phonon coupling between hBN and the CNT interface is weak. Therefore, reducing the ITR of hBN and CNT is the key to improving the out-of-plane heat transport of the Gr/VACNT/hBN sandwiched heterostructure. In addition, the overlap factor of Gr and hBN in the Gr/VACNT/hBN sandwiched heterostructure $S$(Gr/$\cdots$/hBN) = 0.0226; however, in the Gr/hBN vdW heterostructure, the overlap factor of Gr and hBN, $S$(Gr/hBN) = 0.0175 is about 20% smaller than that of $S$(Gr/$\cdots$/hBN), which shows that the introduction of VACNTs effectively enhances the phonon coupling between Gr and hBN layers and thus reduces the ITR. We reported in our previous work that ITR decreases with increase in the interface coupling strength in vdW heterostructures.[15] Compared with vdW bonding, the covalent bonding in Gr-based thermal interface materials can significantly reduce the thermal contact resistance. This indicates that stronger interatomic interactions are more effective for phonon transport across the interfaces.[11]

Fig. 7. Phonon density of states (PDOS) and overlap factor ($S$) of Gr, CNT, and hBN in the Gr/VACNT/hBN sandwich structure: (a) $S$(Gr/hBN) = 0.0175, (b) $S$(Gr–CNT) = 0.172, (c) $S$(hBN–CNT) = 0.0222, (d) $S$(Gr/$\cdots$/hBN) = 0.0226.

In summary, we have calculated the ITR of Gr/VACNT/hBN sandwiched heterostructures using MD simulation. The effects of the size (diameter and length), chirality, and packing density of VACNTs on the interfacial heat transport of the heterostructure are studied. It is found that the ITR of the Gr/VACNT/hBN sandwiched heterostructure is 1–2 orders of magnitude smaller than the ITR of a Gr/CNT vdW heterostructure of the same size, which shows that introducing a VACNT sandwich layer effectively enhances the phonon coupling between Gr and hBN layers, resulting in an increase in the PDOS overlap factor between Gr and hBN and thereby reducing the interlayer ITR of Gr and hBN. In addition, the diameter, length, chirality, and packing density of sandwich CNTs have an important influence on the ITR of the heterostructure. For the same diameter and length of CNT, the ITR of a sandwiched heterostructure with armchair-shaped VACNTs is higher than that with zigzag-shaped VACNTs, which is due to the different axial atomic arrangement and edge atomic distribution under different chiralities. For armchair-shaped CNTs, as the diameter of the CNT increases or the length of the nanotubes decreases, and the ITR of the sandwiched heterostructure tends to decrease. Moreover, the increase in the packing density of intermediate-layer CNTs also leads to a continuous decrease in the ITR of the heterostructure, which is due to the extremely high intrinsic thermal conductivity of CNTs and an increase in heat transfer channels outside the heterostructure surface. Therefore, increasing the packing density of CNTs in the intermediate layer is conducive to improving interfacial heat transfer from the Gr layer to the hBN layer. This is of great significance for the design of interfacial heat management materials. The present results may be helpful for understanding the interfacial heat transport mechanism of multilayer vertical heterostructures and provide theoretical guidance for optimizing the thermal management of micro–nano electronic devices and for using interfaces to regulate nanoscale heat transport. Acknowledgments. This work was supported by the Fundamental Research Funds for the Central Universities of China (Grant No. 2019ZDPY16). We are also grateful for the support of the funding for the key discipline of physics at the China University of Mining and Technology and also the support of the Shenzhen Yuliang Technology Co., Ltd. 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