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Thermal properties of graphene-based polymer composite materials: A molecular dynamics study
Results in Physics 16 (2020) 102974
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Thermal properties of graphene-based polymer composite materials: A molecular dynamics study
T
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Junjie Chen , Baofang Liu, Xuhui Gao Department of Energy and Power Engineering, School of Mechanical and Power Engineering, Henan Polytechnic University, Jiaozuo, Henan, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Graphene Composite materials Thermal conductivity Temperature discontinuity Carbon cross-links Thermal boundary resistance
Interfaces between graphene play a crucial yet poorly understood role in the transport of heat in graphene-based composite materials. The primary focus of this study was on developing a quantitative understanding of the mechanism underlying the heat flow through the structure of a graphene-based polymer matrix composite material. The macroscopic thermal properties of the composite material were studied to provide an effective way to improve thermal conductivity for polymers. Molecular dynamics simulations were performed to understand the thermal resistance of interfaces between graphene and to optimize the condition of the interface for nanoscale thermal reinforcement. A modified Maxwell Garnett effective medium theory was used to predict the thermal conductivity of the composite material. The results indicated that the thermal properties of a graphenebased polymer matrix composite material depends upon the details of the microscopic structure and atomic interactions within the composite material. Interfaces between graphene contribute significantly to the thermal conductivity of the composite material. This is even more critical for composite materials where the lateral size of graphene can affect the thermal performance of the composite material significantly. Low thermal boundary resistance and high thermal conductivity can be achieved by introducing the structure of a carbon cross-link network between graphene. The proper combination of graphene with large size and carbon cross-links with high density can lead to a fifty-fold improvement in overall thermal performance for the polymer matrix composite material. The relative position between graphene can be adjusted for the further optimization of interfacial conditions, but with little success in nanoscale thermal reinforcement. The bulk thermal conductivity of graphene nanoribbons has been found to be about 2570 W/m·K at room temperature. The results can provide a theoretical basis for understanding the applied physics of thermal transport at the nanoscale in graphene-based polymer matrix composite materials.
Introduction Graphene is a two-dimensional monolayer of carbon atoms that possesses remarkable mechanical, electrical, and thermal properties [1,2]. In addition, graphene has a large surface area and can be a safer analog to carbon nanotubes, which makes graphene an attractive candidate for biomedical applications [3,4], conductive textile coatings [5,6], optical elements [7,8], battery electrode materials [9,10], etc. Among different graphene-based materials, polymer-graphene nanocomposites have gained significant attention [11,12] due, at least in part, to their combination of the properties of graphene, such as thermal and electrical conductivity [13,14], thermal stability [15,16], mechanical properties [17,18], and optical properties [19,20], and the flexibility of polymers, including processability into a variety of
material shapes [21,22]. Accordingly, polymer-graphene nanocomposites comprising graphene dispersed in a polymer matrix have been the subject of numerous developments [23,24]. These graphene-containing materials can be made by a variety of techniques such as melt-blending, electrospinning, doping, chemical vapor deposition, and self-assembly [25,26] to yield materials of different shapes and sizes including nanofibers, membranes, and papers [27,28]. Yet, polymer nanocomposites suffer from limitations related to the type of polymers and uneven dispersion of the graphene that can diminish performance attributes of the resultant material [29,30]. Several other problems associated with composite materials include use of expensive carbon nanotubes rather than graphene, material preparation methods that are impractical for largescale commercial production, and processing difficulties [31,32].
⁎ Corresponding author at: Department of Energy and Power Engineering, School of Mechanical and Power Engineering, Henan Polytechnic University, 2000 Century Avenue, Jiaozuo, Henan 454000, China. E-mail address: [email protected] (J. Chen).
https://doi.org/10.1016/j.rinp.2020.102974 Received 20 December 2019; Received in revised form 22 January 2020; Accepted 22 January 2020 Available online 25 January 2020 2211-3797/ © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).
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dT −1 k g=− Jx ⎛ x ⎞ ⎝ dx ⎠
Specific interest lies in the area of polymer composites in which graphene is added to polymers and other compositions to provide thermal conductivity [33,34]. Experiments indicate that graphene exhibits extremely high phonon-dominated thermal conductivity of up to about 1500–2500 W/ m·K at room temperature [35,36]. The extremely high thermal conductivity of graphene has many potential applications in various engineering fields [37,38]. For instance, the use of graphene can effectively reduce thermal boundary resistance by at least one order of magnitude to satisfy the increasing demand for heat dissipation of electronic devices [39,40]. Achievements, however, have fallen short of aspiration. The potential of graphene as a reinforcement material for polymers has not been fully exploited [41,42]. It is of significant importance to optimize the interface between graphene for nanoscale thermal reinforcement [43,44]. However, it remains unclear how the condition of the interface affects the thermal conductivity of graphenebased polymer matrix composite materials. Fortunately, recent studies have demonstrated that chemical means such as functionalization can be effective in improving heat transport properties [45,46], and this is likely to be a major focus in the near term. In this study, the characteristics of thermal transport at the nanoscale were studied to better understand the mechanism underlying the heat flow through the structure of a graphene-based polymer matrix composite material. Molecular dynamics was used to model the transfer of heat through the interface between graphene, and to investigate the characteristics of thermal transport at the nanoscale. A modified Maxwell Garnett effective medium theory was used to predict the macroscopic thermal properties of the composite material. The objective of this study is to determine how the thermal conductivity of a graphene-based polymer matrix composite material varies with the condition of the interface between graphene. Emphasis is placed on improving the thermal conductivity of the composite material through optimizing the condition of the interface between graphene.
(1)
in which kg is the thermal conductivity of the graphene nanoribbon modeled, J is the local heat flux, x is the direction of heat flow, T is temperature, and the last term represents the gradient of temperature between the hot region and one of the cold regions. An implicit assumption in the above description is that thermal conduction is anisotropic. In this context, the heat flux through the structure is parallel to the gradient of temperature between the hot region and the cold regions. The heat flux through the graphene nanoribbon takes the following form
Jx =
1 2tA
∑ transfer
m 2 2 (vhot − vcold ) 2
(2)
in which t represents simulation time, A is the cross-sectional area perpendicular to the direction of heat flow, m is the mass of the exchanged atoms, and vhot and vcold represent the velocities of the hot and cold atoms, respectively. The cross-sectional area perpendicular to the direction of heat flow is defined as
A = wd
(3)
wherein w is the width of the graphene nanoribbon, and d is the thickness of the graphene nanoribbon. The thickness is assumed to be 0.335 nm, which is the interplanar distance of carbon atoms stacked parallel to each other in a graphite crystal. Thermal conductivity is proportional to the mean free path for phonon scattering [51,52], on the basis of the theory of phonon transport in nanostructures. The effective mean free path for phonon scattering can be written as −1 −1 −1 leffective = lphonon − phonon + lphonon − boundary
(4)
in which leffective is the effective mean free path for phonon scattering in nanostructures, lphonon-phonon is the free path associated with bulk phonon-phonon scattering, and lphonon-boundary is the free path associated with phonon-boundary scattering. On the basis of the relation determined by the above equation, the thermal conductivity of the graphene nanoribbon modeled can be expressed as follows:
Computational methods With the significance of understanding the thermal resistance at the interface between adjacent graphene nanoribbons, triple graphene nanoribbons are used as a model system in this study. The characteristics of heat transfer through armchair-edged graphene nanoribbons are studied using molecular dynamics simulations, which are a powerful tool to investigate heat transport properties such as thermal boundary resistance [47,48]. Specifically, the reverse non-equilibrium molecular dynamics approach is employed to determine the thermal conductivity of graphene nanoribbons and the thermal boundary resistance at the interface between adjacent graphene nanoribbons using the MüllerPlathe algorithm [49]. The overall heat transport properties of a system containing graphene nanoribbons can be derived by utilizing molecular dynamics simulations [50]. Heat flux through one individual armchair-edged graphene nanoribbon is depicted schematically in Fig. 1. The structure of the graphene nanoribbon modeled herein is divided into specific slabs by defining hot and cold regions. The hot region is located at the middle of the structure, whereas the cold regions are located at the two ends of the structure. A known heat flux is imposed through the structure of finite length. Heat will flow from the hot region to the cold regions in an attempt to equalize the temperature difference, in accordance with the second law of thermodynamics. The resulting gradient of temperature between the hot region and the cold regions can be determined. The total energy is conserved in the process. When the system reaches a steady state finally, the thermal conductivity can be determined in terms of the imposed heat flux and the resulting gradient of temperature between the hot region and one of the cold regions. Fourier's law of heat conduction is used in the direction of heat flow in its one-dimensional form as follows:
−1 −1 kg−1 ∝ lphonon − phonon + lphonon − boundary
(5)
The above equation predicts that the relationship between thermal conductivity and the free path associated with phonon-boundary scattering can be represented by a straight line. The bulk thermal conductivity of the graphene nanoribbon modeled can therefore be derived from the intercept of the straight line by extrapolation based on an infinite system size. The LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) code [53] is used to simulate the heat transfer process, with the AIREBO (Adaptive Intermolecular Reactive Empirical Bond Order) potential in its second generation form [54]. This potential function can be used to model intermolecular interactions in condensed-phase hydrocarbon systems [55,56]. The non-bonded interactions between graphene nanoribbons, such as van der Waals forces, are modeled with the L-J (Lennard-Jones) potential. Before starting molecular dynamics simulations, the total potential energy of the system is minimized by iteratively adjusting atom coordinates until one of the convergence criteria is satisfied. Thereafter, heat flux is imposed through the system using the Müller-Plathe algorithm. Molecular dynamics simulations are then performed on the relaxed structure, with the velocity-Verlet integrator and a time step of 0.5 fs. After the system reaches a steady state finally, molecular dynamics simulations will continue to be carried out up to 2 ns in order to ensure reliable statistics. Temperature profiles are determined in the direction of heat flow. 2
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Fig. 1. Schematic diagram of heat flux through one individual armchair-edged graphene nanoribbon. It is assumed and indicated that the red region has a higher value of temperature than the blue regions and that the property being transported in the graphene nanoribbon therefore flows from the red region to the blue regions. The arrows indicate the direction of heat transfer between the hot and cold regions.
Fig. 2. Schematic diagram of heat flux through the structure of the triple graphene nanoribbons modeled in this study. There is an overlap between two adjacent graphene nanoribbons, as indicated by the magenta regions.
cross-links is arranged in a two-dimensional honeycomb lattice with its two adjacent graphene nanoribbons. More specifically, the carbon-tocarbon covalent bond length in these carbon cross-links is 0.142 nm, which is the same as that in graphene nanoribbons. The bond angle of three connected carbon atoms in these carbon cross-links is 120 degrees, which is the angle formed between each carbon atom in these carbon cross-links and its two nearest neighboring carbon atoms in two adjacent graphene nanoribbons. The thermal boundary resistance between two adjacent graphene nanoribbons can be written as
The width of the graphene nanoribbon modeled is assumed to be 2.4 nm. The length of the graphene nanoribbon modeled is assumed to be 40 nm, unless otherwise specified in this study. In addition, triple graphene nanoribbons are also considered in this study to simulate the heat transfer process and to determine the thermal resistance at the interface between two adjacent graphene nanoribbons. The triple structure modeled herein is depicted schematically in Fig. 2. In this case, the width of each graphene nanoribbon is 2.4 nm, and the length of each graphene nanoribbon is 40 nm. There is an overlap between two adjacent graphene nanoribbons, with 4 nm in length, unless otherwise stated, and a fixed distance apart from each other. This fixed distance is assumed to be 0.25 nm, unless otherwise specified in this study. For valence bond theory, the interatomic potential of a carbon-tocarbon covalent bond is about two orders of magnitude larger than that of a non-covalent bond between graphene sheets [57,84]. The phonon thermal transport between two adjacent graphene nanoribbons can therefore be enhanced by forming chemical bonds, such as carbon cross-links, at their interface. In this study, the number of carbon crosslinks between two adjacent graphene nanoribbons varies to understand how to achieve the optimum thermal performance for a graphene-based composite material by forming a thermally conductive network. Carbon cross-links are covalent bonds, which can be formed by chemical reactions. Carbon cross-links are usually used to promote a change in the physical properties of carbon-based materials. These carbon cross-links take the form of covalent bonds. Each carbon atom of these carbon
R=
ΔTx Jx
(6)
in which R is the thermal boundary resistance between two adjacent graphene nanoribbons, ΔTx is the difference in temperature in the direction of heat flow, and Jx is the imposed heat flux in the direction of heat flow. The density of cross-links has been found to play a significant role in the mechanical and physical properties of a composite material [58,59]. This parameter is introduced in this study to evaluate the effect of the number of carbon cross-links on the thermal boundary resistance between two adjacent graphene nanoribbons. The surface density of carbon cross-links is defined as
ρ= 3
N lo h
(7)
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in which ρ is the surface density of carbon cross-links, N is the number of carbon cross-links, lo is the length of overlap between two adjacent graphene nanoribbons, h is the distance apart from two adjacent graphene nanoribbons. Effective medium theory can be used to describe the macroscopic properties of a graphene-based composite material. In this study, graphene nanoribbons are assumed to be uniformly dispersed in the matrix material. The macroscopic thermal properties of the graphene-based composite material are characterized by effective thermal conductivity obtained from the parameters of the graphene-based composite material using effective medium theory. More specifically, the thermal conductivity of the graphene-based composite material is determined using the Maxwell Garnett approximation theory corrected by adding a thermal boundary resistance term to the thermal conductivity of the graphene nanoribbon modeled in this study. The thermal conductivity of the graphene-based composite material takes the following form:
kc = k g [3km + 2f (k g − km )]/[(3 − f ) k g + fkm + fRk g km/δ ]
(8) Fig. 3. Equilibrated temperature profiles obtained for one individual armchairedged graphene nanoribbon. The length of the graphene nanoribbon modeled is 40 nm.
in which kc is the thermal conductivity of the graphene-based composite material, km is the thermal conductivity of the matrix material, f is the volume fraction of the graphene nanoribbon modeled, and δ is the thickness of the graphene nanoribbon modeled. The thermal resistance across the triple graphene nanoribbons is comprised of five individual resistances in series. The effective thermal resistance arises as stated in the following expression
lg, l lg, m lg, r lt = + Rl − m + + Rm − r + kt k g, l kg, m kg, r
constant, i.e., the relationship between temperature and the distance along the direction of heat flow is more-or-less linear. However, the thermal conductivity determined by the above method varies depending upon the length of the graphene nanoribbon modeled in this study. The results are presented in Fig. 4, in which the thermal conductivity of the graphene nanoribbon at room temperature is expressed as a function of the length of the graphene nanoribbon. Thermal conductivity predictions are made for various lengths and then a linear extrapolation procedure is performed. The length of the graphene nanoribbon varies in the range from 20 to 200 nm. The results obtained for the specific thermal resistance of the graphene nanoribbon are also presented in Fig. 4; specific thermal resistance is the reciprocal of thermal conductivity. The thermal conductivity of the graphene nanoribbon increases with increasing its length. The main cause of this behavior is that the length of the graphene nanoribbon modeled in this study is much less than the mean free path for phonon scattering in the bulk material. Phonons will be considered the dominate means of thermal conduction within the graphene nanoribbon modeled in this study. The bulk phonon mean free path can be up to 700 nm at room temperature [66,67]. Therefore, in addition to phonon–phonon scattering, phonon-boundary scattering is also of great importance to the thermal conductivity of the graphene nanoribbon. The results presented in Fig. 4 validate the relationship between thermal conductivity and the free path associated with phononboundary scattering, as expressed by equation (5). The methodology for determining thermal conductivity, as described above, can also be applied to three-dimensional systems [68,85]. The y-intercept of the straight line in Fig. 4 refers to the bulk thermal conductivity of the graphene nanoribbon modeled, which is about 2570 W/m·K at room temperature. The results obtained from the model are in good agreement with those determined theoretically [69,70] and experimentally [71,86], which have a wide range of thermal conductivities between 2000 and 5000 W/m·K for single layer graphene at room temperature. From the results presented here, it is clear that size dependence arises through phonon-boundary scattering. The length of a graphene nanoribbon should be long enough to predict the bulk thermal conductivity of the graphene nanoribbon accurately using the linear extrapolation procedure described above.
(9)
in which lt is the total length of the triple graphene nanoribbons, kt is the thermal conductivity of the triple graphene nanoribbons, lg is the length of a graphene nanoribbon in the triple structure, subscripts l, m, and r identify the left graphene nanoribbon, the middle graphene nanoribbon, and the right graphene nanoribbon in the triple structure, Rl-m is the thermal boundary resistance between the left graphene nanoribbon and the middle graphene nanoribbon, and Rm-r is the thermal boundary resistance between the middle graphene nanoribbon and the right graphene nanoribbon. In the simplest case, the three types of graphene nanoribbons are identical, and the overlap between two adjacent graphene nanoribbons has a short length. The above equation can therefore be simplified as
kt = 3k g lg /(3lg + 2k g R)
(10)
The above expression is then used to predict the thermal conductivity of the graphene-based composite material modeled in this study. Results and discussions Thermal conductivity of graphene nanoribbons Equilibrated temperature profiles obtained for one individual armchair-edged graphene nanoribbon are shown in Fig. 3. Temperature is nonlinear within the hot and cold regions, as the change of temperature is not proportional to the change of the distance along the direction of heat flow. The main cause of this nonlinear phenomenon is a finite-size effect arising from the system, since the characteristic scale of the system is comparable with the mean free path of the atoms, as discussed in the literature [60,61]. In this case, a very steep gradient in temperature may induce a significant nonlinear response such that Fourier's law may be no longer applicable. A similar nonlinear phenomenon of the change in temperature has also been noted in the case of carbon nanotubes [62,63]. This indicates that the transport of heat within graphene or carbon nanotubes is not fully diffusive [64,65]. The thermal conductivity of the armchair-edged graphene nanoribbon is determined by the temperature gradient that can be treated as a
Thermal boundary resistance Interfaces between adjacent graphene often contribute significantly to the thermal conductivity of the resulting composite material [72,73]. Understanding the thermal resistance at the interface between two 4
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Fig. 4. Specific thermal resistance and thermal conductivity of one individual armchair-edged graphene nanoribbon at room temperature as a function of the length of the graphene nanoribbon.
decreases with increasing the number of carbon cross-links, which will eventually lead to a reduction in the thermal boundary resistance between two adjacent graphene nanoribbons. The results are presented in Fig. 6, in which the thermal boundary resistance at room temperature is expressed as a function of the surface density of carbon cross-links. Thermal boundary resistance is very sensitive to the condition of the interface between two adjacent graphene nanoribbons. More specifically, thermal boundary resistance decreases with increasing the surface density of carbon cross-links. Carbon cross-links should be present in an amount sufficient to ensure that thermal boundary resistance is reduced effectively. Low thermal boundary resistance is technologically
adjacent graphene nanoribbons is of great significance in the study of its thermal conductivity. The thermal resistance results from the transport of phonons across the interface. The thermal boundary resistance between two adjacent graphene nanoribbons is predicted for the triple graphene nanoribbons depicted schematically in Fig. 2. Equilibrated temperature profiles obtained for the triple graphene nanoribbons are shown in Fig. 5. Discontinuity in temperature does occur at the interface between two adjacent graphene nanoribbons, and there is also significant difference in temperature. The presence of thermal boundary resistance corresponds to a discontinuous temperature across the interface. However, the difference in temperature
Fig. 5. Equilibrated temperature profiles obtained for the triple graphene nanoribbons modeled in this study. The length of each graphene nanoribbon is 40 nm. 5
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Fig. 6. Effect of the surface density of carbon cross-links on the thermal boundary resistance between two adjacent graphene nanoribbons at room temperature. The length of each graphene nanoribbon is 40 nm.
term “interval distance” refers to the distance apart from two adjacent graphene nanoribbons. The results are presented in Fig. 7, in which the thermal boundary resistance at room temperature is expressed as a function of overlap length or interval distance. The overlap length varies from 0 to 8 nm, and the interval distance varies from 0 to 4.8 nm. It is clear that the thermal boundary resistance decreases with increasing overlap length and decreasing interval distance. However,
important for applications for which very high efficiency in heat dissipation is required, which is of great significance to the development of microelectronic devices [74,75]. The effects of overlap length and interval distance on the thermal boundary resistance between two adjacent graphene nanoribbons are evaluated at room temperature. The term “overlap length” refers to the length of overlap between two adjacent graphene nanoribbons. The
Fig. 7. Effects of overlap length and interval distance on the thermal boundary resistance between two adjacent graphene nanoribbons at room temperature. 6
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Fig. 8. Thermal conductivity of a graphene-based polymer matrix composite material at room temperature as a function of the length of graphene nanoribbons. The thermal conductivity of the polymer matrix material is considered in this study to be 0.35 W/m·K at room temperature. There are no carbon cross-links between two adjacent graphene nanoribbons.
high efficiency in heat dissipation is required, even if the graphene nanoribbons use in a polymer matrix composite material is long enough. The thermal conductivity of the polymer matrix composite material is highly dependent upon the content of graphene nanoribbons dispersed in the polymer matrix material, as discussed above. Higher loading levels of such a two-dimensional carbon material may eventually lead to a higher thermal conductivity of the polymer matrix composite material. Unfortunately, there is always a challenge in achieving a highly loaded and substantially uniform dispersion of graphene nanoribbons in the polymer matrix material. There are two main factors contributing to this challenge: the tendency of nanoscale dispersants to aggregate [80,81] and the nature of the surface of the twodimensional carbon material [82,83], such as graphene nanoribbons, used in this study. It is assumed in this study that a substantially uniform dispersion of graphene nanoribbons is achieved in the polymer matrix material, as described in the model above. This assumption is reasonable if the content of graphene nanoribbons in the polymer matrix composite material is low enough. In this context, the content of graphene nanoribbons considered in this study for the polymer matrix composite material is therefore limited to the low level from 2 to 8% by volume. On the other hand, this study does not address the issue of the effect of the width of graphene nanoribbons on the thermal properties of the polymer matrix composite material. However, the resulting effect is similar to that of the length of graphene nanoribbons. The effect of the width of graphene nanoribbons may be even more pronounced due to the much shorter free path associated with phonon-boundary scattering. It is worth noting that the model developed in this study is still applicable to make predictions about the thermal properties of the polymer matrix composite material when there is a change in the width of graphene nanoribbons. The graphene nanoribbons modeled in this study have a well-defined edge structure. If these graphene nanoribbons have a variety of edge functional groups, the model can be modified to account for this factor in order to make accurate predictions about the thermal properties of the polymer matrix composite material. The effects of overlap length and interval distance on the thermal conductivity of the polymer matrix composite material are evaluated at room temperature. The results are presented in Fig. 9, in which the
both overlap length and interval distance have little effect on the thermal boundary resistance between two adjacent graphene nanoribbons, in comparison with the presence of carbon cross-links.
Thermal conductivity of graphene-based composite materials The effect of the length of graphene nanoribbons on the thermal conductivity of a graphene-based polymer matrix composite material is studied. The results are presented in Fig. 8, in which the thermal conductivity of the composite material at room temperature is expressed as a function of the length of graphene nanoribbons. The thermal conductivity of polymers is low, often in the range of from 0.2 to 0.5 W/ m·K at room temperature [76,77]. The mean value may be considered to be an approximation close to the thermal conductivity of common polymers. Therefore, the thermal conductivity of the polymer matrix material is considered in this study to be 0.35 W/m·K at room temperature. The content of graphene nanoribbons varies from 2 to 8% by volume, and there are no carbon cross-links between two adjacent graphene nanoribbons. The results presented in Fig. 8 indicate that the thermal conductivity of the polymer matrix composite material increases with increasing the length of graphene nanoribbons. The length of graphene nanoribbons affects the thermal conductivity of the composite material significantly, especially when the content of graphene nanoribbons is high. It is clear that a high level of graphene content facilitates a very efficient phonon transmission between two adjacent graphene nanoribbons, which will lead to an increase in the thermal conductivity of the composite material. The maximum thermal conductivity presented in Fig. 8 is close to the thermal conductivity of stainless steel, which is a common metal material with moderate thermal conductivity. However, the thermal conductivity of graphene-based polymer matrix composite materials has so far fallen short of that predicted here [78,79], and thus the potential of graphene as a reinforcement material for polymers has not been fully realized. At a low level of graphene content, phonons do not efficiently couple across the interface between two adjacent graphene nanoribbons. Therefore, the thermal conductivity of the composite material is low in all the cases of graphene length studied here, as shown in Fig. 8. It is therefore often necessary to employ a high level of graphene content for thermal management applications for which very 7
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Fig. 9. Effects of overlap length and interval distance on the thermal conductivity of a graphene-based polymer matrix composite material at room temperature. The content of graphene nanoribbons in the resulting composite material is 8% by volume. There are no carbon cross-links between two adjacent graphene nanoribbons.
Fig. 10. Effect of the surface density of carbon cross-links on the thermal conductivity of a graphene-based polymer matrix composite material at room temperature. The content of graphene nanoribbons in the resulting composite material is 8% by volume.
discussed above. In addition, it is expected that the thermal conductivity has strong dependence on the thermal boundary resistance. In this context, the content of graphene nanoribbons in the resulting composite material is assumed to be 8% by volume for the sake of simplicity.
thermal conductivity at room temperature is expressed as a function of overlap length or interval distance. There are no carbon cross-links between two adjacent graphene nanoribbons. Both overlap length and interval distance have been found to have little effect on the thermal boundary resistance between two adjacent graphene nanoribbons, as 8
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between graphene. The length of graphene plays an important role in the thermal conductivity of the composite material, especially when the content of graphene is high. Both thermal boundary resistance and thermal conductivity are very sensitive to the presence of carbon crosslinks between graphene. A fifty-fold improvement in overall thermal performance can be achieved for the polymer matrix composite material by properly combining graphene with large size and carbon crosslinks with high density. In the absence of carbon cross-links, phonons do not efficiently couple across the interface between graphene. Both overlap length and interval distance have little effect on the thermal boundary resistance between graphene and the thermal conductivity of the composite material. The results could offer valuable guidance for improving the thermal properties of graphene-based polymer matrix composite materials. This study does not address the issues of the stability of the resulting polymer matrix composite material nor the ability of graphene nanoribbons to be dispersed in the matrix material. Graphene nanoribbons to be effectively dispersed in the matrix material is of importance to the thermal conductivity of the polymer matrix composite material. A substantially uniform dispersion would be of great utility in permitting exploitation of the unique thermal properties of the polymer matrix composite material. Therefore, the development of a molecular dynamics modeling technology with taking into account the dispersion of graphene nanoribbons in the matrix material is of significant technical and commercial importance. Further study is needed to determine the effect of the degree of dispersion on the thermal properties of graphenebased polymer composite materials by performing molecular dynamics simulations for the nanostructure with a complex geometry.
The results presented in Fig. 9 indicate that the thermal conductivity of the composite material increases with increasing overlap length and decreasing interval distance. However, both overlap length and interval distance have little effect on the thermal conductivity of the composite material, in comparison with the length of graphene nanoribbons. The main cause of this behavior is that overlap length and interval distance do not affect the thermal boundary resistance between two adjacent graphene nanoribbons effectively, as discussed above. Consequently, a high degree of overlap will not lead to a significant increase in the thermal conductivity of the composite material. The thermal conductivity of the composite material increases with deceasing interval distance. However, the reduction of interval distance cannot lead to significant improvement in the thermal conductivity of the composite material. These graphene nanoribbons are essentially discontinuities each other. Consequently, the absence of carbon cross-links makes it more difficult for phonons to transmit across the interface between two adjacent graphene nanoribbons, which will eventually lead to lower thermal conductivity of the composite material. While graphene nanoribbons do exhibit high thermal conductivity, their effectiveness is limited by how well phonons can be transmitted across the interface between two adjacent graphene nanoribbons. To understand the importance of the interface, the effect of the surface density of carbon cross-links on the thermal conductivity of the composite material at room temperature is investigated. The results are presented in Fig. 10, in which the thermal conductivity at room temperature is expressed as a function of the surface density of carbon cross-links. The content of graphene nanoribbons in the resulting composite material is 8% by volume. It is clear from Fig. 10 that thermal conductivity is very sensitive to the condition of the interface between two adjacent graphene nanoribbons. The surface density of carbon cross-links affects the thermal conductivity of the composite material significantly. More specifically, the thermal conductivity of the composite material increases with increasing the surface density of carbon cross-links. In this case, the interface dominates the overall thermal conductivity of the composite material. High surface density of carbon cross-links allows for significant heat flow through the interface. Consequently, the key to reducing thermal boundary resistance and to improving thermal conductivity is the use of carbon cross-links between two adjacent graphene nanoribbons. What is needed is the ability to combine long graphene nanoribbons with the high surface density of carbon crosslinks, which reduce the effect of thermal boundary resistance such that maximum thermal performance can be realized. For example, a fiftyfold improvement in overall thermal performance can be achieved for the polymer matrix composite material by properly combining graphene with large size and carbon cross-links with high density.
CRediT authorship contribution statement Junjie Chen: Conceptualization, Methodology, Software, Formal analysis, Investigation, Resources, Writing - original draft, Writing review & editing, Supervision, Project administration, Funding acquisition. Baofang Liu: Methodology, Software, Formal analysis, Investigation. Xuhui Gao: Methodology, Software, Formal analysis, Investigation. Acknowledgement This work was supported by the National Natural Science Foundation of China (No. 51506048). References [1] Geim AK, Novoselov KS. The rise of graphene. Nat Mater 2007;6(3):183–91. [2] Geim AK. Graphene: Status and prospects. Science 2009;324(5934):1530–4. [3] Yang Y, Asiri AM, Tang Z, Du D, Lin Y. Graphene based materials for biomedical applications. Mater Today 2013;16(10):365–73. [4] Bitounis D, Ali-Boucetta H, Hong BH, Min D-H, Kostarelos K. Prospects and challenges of graphene in biomedical applications. Adv Mater 2013;25(16):2258–68. [5] Shateri-Khalilabad M, Yazdanshenas ME. Fabricating electroconductive cotton textiles using graphene. Carbohydr Polym 2013;96(1):190–5. [6] Mizerska U, Fortuniak W, Makowski T, Svyntkivska M, Piorkowska E, Kowalczyk D, et al. Electrically conductive and hydrophobic rGO-containing organosilicon coating of cotton fabric. Progress in Organic Coatings, Volume. Article Number 2019;137:105312. [7] Liu M, Yin X, Ulin-Avila E, Geng B, Zentgraf T, Ju L, et al. A graphene-based broadband optical modulator. Nature 2011;474(7349):64–7. [8] Liu M, Yin X, Zhang X. Double-layer graphene optical modulator. Nano Lett 2012;12(3):1482–5. [9] Mohanapriya K, Jha Neetu. Hierarchically hybrid nanostructure of carbon nanoparticles decorated graphene sheets as an efficient electrode material for supercapacitors, aqueous Al-ion battery and capacitive deionization. Electrochim Acta 2019;324:134870. https://doi.org/10.1016/j.electacta.2019.134870. [10] Dada OJ. Higher capacity utilization and rate performance of lead acid battery electrodes using graphene additives. J Storage Mater 2019;23:579–89. [11] Singh V, Joung D, Zhai L, Das S, Khondaker SI. Sudipta Seal, Graphene based materials: Past, present and future. Prog Mater Sci 2011;56(8):1178–271. [12] Stankovich S, Dikin DA, Dommett GHB, Kohlhaas KM, Zimney EJ, Stach EA, et al. Graphene-based composite materials. Nature 2006;442(7100):282–6. [13] Balandin AA, Ghosh S, Bao W, Calizo I, Teweldebrhan D, Miao F, et al. Superior thermal conductivity of single-layer graphene. Nano Lett 2008;8(3):902–7.
Conclusions The characteristics of nanoscale heat transport in a graphene-based polymer matrix composite material were studied to determine how the thermal conductivity of the composite material varies with the condition of the interface between graphene. Molecular dynamics simulations were performed and a modified Maxwell Garnett effective medium theory was used to understand the basic mechanism of the thermal resistance across the interface, and to optimize the condition of the interface for nanoscale thermal reinforcement. The results indicated that the bulk thermal conductivity of graphene nanoribbons is about 2570 W/m·K at room temperature. The thermal conductivity of a graphene-based polymer matrix composite material depends strongly upon the details of the microscopic structure and atomic interactions within the composite material, especially the condition of the interface between graphene. The thermal conductivity of the composite material can be improved by increasing the length of graphene, the surface density of carbon cross-links, and the length of overlap between graphene, or by decreasing the interval distance 9
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Results in Physics journal homepage: www.elsevier.com/locate/rinp
Thermal properties of graphene-based polymer composite materials: A molecular dynamics study
T
⁎
Junjie Chen , Baofang Liu, Xuhui Gao Department of Energy and Power Engineering, School of Mechanical and Power Engineering, Henan Polytechnic University, Jiaozuo, Henan, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Graphene Composite materials Thermal conductivity Temperature discontinuity Carbon cross-links Thermal boundary resistance
Interfaces between graphene play a crucial yet poorly understood role in the transport of heat in graphene-based composite materials. The primary focus of this study was on developing a quantitative understanding of the mechanism underlying the heat flow through the structure of a graphene-based polymer matrix composite material. The macroscopic thermal properties of the composite material were studied to provide an effective way to improve thermal conductivity for polymers. Molecular dynamics simulations were performed to understand the thermal resistance of interfaces between graphene and to optimize the condition of the interface for nanoscale thermal reinforcement. A modified Maxwell Garnett effective medium theory was used to predict the thermal conductivity of the composite material. The results indicated that the thermal properties of a graphenebased polymer matrix composite material depends upon the details of the microscopic structure and atomic interactions within the composite material. Interfaces between graphene contribute significantly to the thermal conductivity of the composite material. This is even more critical for composite materials where the lateral size of graphene can affect the thermal performance of the composite material significantly. Low thermal boundary resistance and high thermal conductivity can be achieved by introducing the structure of a carbon cross-link network between graphene. The proper combination of graphene with large size and carbon cross-links with high density can lead to a fifty-fold improvement in overall thermal performance for the polymer matrix composite material. The relative position between graphene can be adjusted for the further optimization of interfacial conditions, but with little success in nanoscale thermal reinforcement. The bulk thermal conductivity of graphene nanoribbons has been found to be about 2570 W/m·K at room temperature. The results can provide a theoretical basis for understanding the applied physics of thermal transport at the nanoscale in graphene-based polymer matrix composite materials.
Introduction Graphene is a two-dimensional monolayer of carbon atoms that possesses remarkable mechanical, electrical, and thermal properties [1,2]. In addition, graphene has a large surface area and can be a safer analog to carbon nanotubes, which makes graphene an attractive candidate for biomedical applications [3,4], conductive textile coatings [5,6], optical elements [7,8], battery electrode materials [9,10], etc. Among different graphene-based materials, polymer-graphene nanocomposites have gained significant attention [11,12] due, at least in part, to their combination of the properties of graphene, such as thermal and electrical conductivity [13,14], thermal stability [15,16], mechanical properties [17,18], and optical properties [19,20], and the flexibility of polymers, including processability into a variety of
material shapes [21,22]. Accordingly, polymer-graphene nanocomposites comprising graphene dispersed in a polymer matrix have been the subject of numerous developments [23,24]. These graphene-containing materials can be made by a variety of techniques such as melt-blending, electrospinning, doping, chemical vapor deposition, and self-assembly [25,26] to yield materials of different shapes and sizes including nanofibers, membranes, and papers [27,28]. Yet, polymer nanocomposites suffer from limitations related to the type of polymers and uneven dispersion of the graphene that can diminish performance attributes of the resultant material [29,30]. Several other problems associated with composite materials include use of expensive carbon nanotubes rather than graphene, material preparation methods that are impractical for largescale commercial production, and processing difficulties [31,32].
⁎ Corresponding author at: Department of Energy and Power Engineering, School of Mechanical and Power Engineering, Henan Polytechnic University, 2000 Century Avenue, Jiaozuo, Henan 454000, China. E-mail address: [email protected] (J. Chen).
https://doi.org/10.1016/j.rinp.2020.102974 Received 20 December 2019; Received in revised form 22 January 2020; Accepted 22 January 2020 Available online 25 January 2020 2211-3797/ © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).
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dT −1 k g=− Jx ⎛ x ⎞ ⎝ dx ⎠
Specific interest lies in the area of polymer composites in which graphene is added to polymers and other compositions to provide thermal conductivity [33,34]. Experiments indicate that graphene exhibits extremely high phonon-dominated thermal conductivity of up to about 1500–2500 W/ m·K at room temperature [35,36]. The extremely high thermal conductivity of graphene has many potential applications in various engineering fields [37,38]. For instance, the use of graphene can effectively reduce thermal boundary resistance by at least one order of magnitude to satisfy the increasing demand for heat dissipation of electronic devices [39,40]. Achievements, however, have fallen short of aspiration. The potential of graphene as a reinforcement material for polymers has not been fully exploited [41,42]. It is of significant importance to optimize the interface between graphene for nanoscale thermal reinforcement [43,44]. However, it remains unclear how the condition of the interface affects the thermal conductivity of graphenebased polymer matrix composite materials. Fortunately, recent studies have demonstrated that chemical means such as functionalization can be effective in improving heat transport properties [45,46], and this is likely to be a major focus in the near term. In this study, the characteristics of thermal transport at the nanoscale were studied to better understand the mechanism underlying the heat flow through the structure of a graphene-based polymer matrix composite material. Molecular dynamics was used to model the transfer of heat through the interface between graphene, and to investigate the characteristics of thermal transport at the nanoscale. A modified Maxwell Garnett effective medium theory was used to predict the macroscopic thermal properties of the composite material. The objective of this study is to determine how the thermal conductivity of a graphene-based polymer matrix composite material varies with the condition of the interface between graphene. Emphasis is placed on improving the thermal conductivity of the composite material through optimizing the condition of the interface between graphene.
(1)
in which kg is the thermal conductivity of the graphene nanoribbon modeled, J is the local heat flux, x is the direction of heat flow, T is temperature, and the last term represents the gradient of temperature between the hot region and one of the cold regions. An implicit assumption in the above description is that thermal conduction is anisotropic. In this context, the heat flux through the structure is parallel to the gradient of temperature between the hot region and the cold regions. The heat flux through the graphene nanoribbon takes the following form
Jx =
1 2tA
∑ transfer
m 2 2 (vhot − vcold ) 2
(2)
in which t represents simulation time, A is the cross-sectional area perpendicular to the direction of heat flow, m is the mass of the exchanged atoms, and vhot and vcold represent the velocities of the hot and cold atoms, respectively. The cross-sectional area perpendicular to the direction of heat flow is defined as
A = wd
(3)
wherein w is the width of the graphene nanoribbon, and d is the thickness of the graphene nanoribbon. The thickness is assumed to be 0.335 nm, which is the interplanar distance of carbon atoms stacked parallel to each other in a graphite crystal. Thermal conductivity is proportional to the mean free path for phonon scattering [51,52], on the basis of the theory of phonon transport in nanostructures. The effective mean free path for phonon scattering can be written as −1 −1 −1 leffective = lphonon − phonon + lphonon − boundary
(4)
in which leffective is the effective mean free path for phonon scattering in nanostructures, lphonon-phonon is the free path associated with bulk phonon-phonon scattering, and lphonon-boundary is the free path associated with phonon-boundary scattering. On the basis of the relation determined by the above equation, the thermal conductivity of the graphene nanoribbon modeled can be expressed as follows:
Computational methods With the significance of understanding the thermal resistance at the interface between adjacent graphene nanoribbons, triple graphene nanoribbons are used as a model system in this study. The characteristics of heat transfer through armchair-edged graphene nanoribbons are studied using molecular dynamics simulations, which are a powerful tool to investigate heat transport properties such as thermal boundary resistance [47,48]. Specifically, the reverse non-equilibrium molecular dynamics approach is employed to determine the thermal conductivity of graphene nanoribbons and the thermal boundary resistance at the interface between adjacent graphene nanoribbons using the MüllerPlathe algorithm [49]. The overall heat transport properties of a system containing graphene nanoribbons can be derived by utilizing molecular dynamics simulations [50]. Heat flux through one individual armchair-edged graphene nanoribbon is depicted schematically in Fig. 1. The structure of the graphene nanoribbon modeled herein is divided into specific slabs by defining hot and cold regions. The hot region is located at the middle of the structure, whereas the cold regions are located at the two ends of the structure. A known heat flux is imposed through the structure of finite length. Heat will flow from the hot region to the cold regions in an attempt to equalize the temperature difference, in accordance with the second law of thermodynamics. The resulting gradient of temperature between the hot region and the cold regions can be determined. The total energy is conserved in the process. When the system reaches a steady state finally, the thermal conductivity can be determined in terms of the imposed heat flux and the resulting gradient of temperature between the hot region and one of the cold regions. Fourier's law of heat conduction is used in the direction of heat flow in its one-dimensional form as follows:
−1 −1 kg−1 ∝ lphonon − phonon + lphonon − boundary
(5)
The above equation predicts that the relationship between thermal conductivity and the free path associated with phonon-boundary scattering can be represented by a straight line. The bulk thermal conductivity of the graphene nanoribbon modeled can therefore be derived from the intercept of the straight line by extrapolation based on an infinite system size. The LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) code [53] is used to simulate the heat transfer process, with the AIREBO (Adaptive Intermolecular Reactive Empirical Bond Order) potential in its second generation form [54]. This potential function can be used to model intermolecular interactions in condensed-phase hydrocarbon systems [55,56]. The non-bonded interactions between graphene nanoribbons, such as van der Waals forces, are modeled with the L-J (Lennard-Jones) potential. Before starting molecular dynamics simulations, the total potential energy of the system is minimized by iteratively adjusting atom coordinates until one of the convergence criteria is satisfied. Thereafter, heat flux is imposed through the system using the Müller-Plathe algorithm. Molecular dynamics simulations are then performed on the relaxed structure, with the velocity-Verlet integrator and a time step of 0.5 fs. After the system reaches a steady state finally, molecular dynamics simulations will continue to be carried out up to 2 ns in order to ensure reliable statistics. Temperature profiles are determined in the direction of heat flow. 2
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Fig. 1. Schematic diagram of heat flux through one individual armchair-edged graphene nanoribbon. It is assumed and indicated that the red region has a higher value of temperature than the blue regions and that the property being transported in the graphene nanoribbon therefore flows from the red region to the blue regions. The arrows indicate the direction of heat transfer between the hot and cold regions.
Fig. 2. Schematic diagram of heat flux through the structure of the triple graphene nanoribbons modeled in this study. There is an overlap between two adjacent graphene nanoribbons, as indicated by the magenta regions.
cross-links is arranged in a two-dimensional honeycomb lattice with its two adjacent graphene nanoribbons. More specifically, the carbon-tocarbon covalent bond length in these carbon cross-links is 0.142 nm, which is the same as that in graphene nanoribbons. The bond angle of three connected carbon atoms in these carbon cross-links is 120 degrees, which is the angle formed between each carbon atom in these carbon cross-links and its two nearest neighboring carbon atoms in two adjacent graphene nanoribbons. The thermal boundary resistance between two adjacent graphene nanoribbons can be written as
The width of the graphene nanoribbon modeled is assumed to be 2.4 nm. The length of the graphene nanoribbon modeled is assumed to be 40 nm, unless otherwise specified in this study. In addition, triple graphene nanoribbons are also considered in this study to simulate the heat transfer process and to determine the thermal resistance at the interface between two adjacent graphene nanoribbons. The triple structure modeled herein is depicted schematically in Fig. 2. In this case, the width of each graphene nanoribbon is 2.4 nm, and the length of each graphene nanoribbon is 40 nm. There is an overlap between two adjacent graphene nanoribbons, with 4 nm in length, unless otherwise stated, and a fixed distance apart from each other. This fixed distance is assumed to be 0.25 nm, unless otherwise specified in this study. For valence bond theory, the interatomic potential of a carbon-tocarbon covalent bond is about two orders of magnitude larger than that of a non-covalent bond between graphene sheets [57,84]. The phonon thermal transport between two adjacent graphene nanoribbons can therefore be enhanced by forming chemical bonds, such as carbon cross-links, at their interface. In this study, the number of carbon crosslinks between two adjacent graphene nanoribbons varies to understand how to achieve the optimum thermal performance for a graphene-based composite material by forming a thermally conductive network. Carbon cross-links are covalent bonds, which can be formed by chemical reactions. Carbon cross-links are usually used to promote a change in the physical properties of carbon-based materials. These carbon cross-links take the form of covalent bonds. Each carbon atom of these carbon
R=
ΔTx Jx
(6)
in which R is the thermal boundary resistance between two adjacent graphene nanoribbons, ΔTx is the difference in temperature in the direction of heat flow, and Jx is the imposed heat flux in the direction of heat flow. The density of cross-links has been found to play a significant role in the mechanical and physical properties of a composite material [58,59]. This parameter is introduced in this study to evaluate the effect of the number of carbon cross-links on the thermal boundary resistance between two adjacent graphene nanoribbons. The surface density of carbon cross-links is defined as
ρ= 3
N lo h
(7)
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in which ρ is the surface density of carbon cross-links, N is the number of carbon cross-links, lo is the length of overlap between two adjacent graphene nanoribbons, h is the distance apart from two adjacent graphene nanoribbons. Effective medium theory can be used to describe the macroscopic properties of a graphene-based composite material. In this study, graphene nanoribbons are assumed to be uniformly dispersed in the matrix material. The macroscopic thermal properties of the graphene-based composite material are characterized by effective thermal conductivity obtained from the parameters of the graphene-based composite material using effective medium theory. More specifically, the thermal conductivity of the graphene-based composite material is determined using the Maxwell Garnett approximation theory corrected by adding a thermal boundary resistance term to the thermal conductivity of the graphene nanoribbon modeled in this study. The thermal conductivity of the graphene-based composite material takes the following form:
kc = k g [3km + 2f (k g − km )]/[(3 − f ) k g + fkm + fRk g km/δ ]
(8) Fig. 3. Equilibrated temperature profiles obtained for one individual armchairedged graphene nanoribbon. The length of the graphene nanoribbon modeled is 40 nm.
in which kc is the thermal conductivity of the graphene-based composite material, km is the thermal conductivity of the matrix material, f is the volume fraction of the graphene nanoribbon modeled, and δ is the thickness of the graphene nanoribbon modeled. The thermal resistance across the triple graphene nanoribbons is comprised of five individual resistances in series. The effective thermal resistance arises as stated in the following expression
lg, l lg, m lg, r lt = + Rl − m + + Rm − r + kt k g, l kg, m kg, r
constant, i.e., the relationship between temperature and the distance along the direction of heat flow is more-or-less linear. However, the thermal conductivity determined by the above method varies depending upon the length of the graphene nanoribbon modeled in this study. The results are presented in Fig. 4, in which the thermal conductivity of the graphene nanoribbon at room temperature is expressed as a function of the length of the graphene nanoribbon. Thermal conductivity predictions are made for various lengths and then a linear extrapolation procedure is performed. The length of the graphene nanoribbon varies in the range from 20 to 200 nm. The results obtained for the specific thermal resistance of the graphene nanoribbon are also presented in Fig. 4; specific thermal resistance is the reciprocal of thermal conductivity. The thermal conductivity of the graphene nanoribbon increases with increasing its length. The main cause of this behavior is that the length of the graphene nanoribbon modeled in this study is much less than the mean free path for phonon scattering in the bulk material. Phonons will be considered the dominate means of thermal conduction within the graphene nanoribbon modeled in this study. The bulk phonon mean free path can be up to 700 nm at room temperature [66,67]. Therefore, in addition to phonon–phonon scattering, phonon-boundary scattering is also of great importance to the thermal conductivity of the graphene nanoribbon. The results presented in Fig. 4 validate the relationship between thermal conductivity and the free path associated with phononboundary scattering, as expressed by equation (5). The methodology for determining thermal conductivity, as described above, can also be applied to three-dimensional systems [68,85]. The y-intercept of the straight line in Fig. 4 refers to the bulk thermal conductivity of the graphene nanoribbon modeled, which is about 2570 W/m·K at room temperature. The results obtained from the model are in good agreement with those determined theoretically [69,70] and experimentally [71,86], which have a wide range of thermal conductivities between 2000 and 5000 W/m·K for single layer graphene at room temperature. From the results presented here, it is clear that size dependence arises through phonon-boundary scattering. The length of a graphene nanoribbon should be long enough to predict the bulk thermal conductivity of the graphene nanoribbon accurately using the linear extrapolation procedure described above.
(9)
in which lt is the total length of the triple graphene nanoribbons, kt is the thermal conductivity of the triple graphene nanoribbons, lg is the length of a graphene nanoribbon in the triple structure, subscripts l, m, and r identify the left graphene nanoribbon, the middle graphene nanoribbon, and the right graphene nanoribbon in the triple structure, Rl-m is the thermal boundary resistance between the left graphene nanoribbon and the middle graphene nanoribbon, and Rm-r is the thermal boundary resistance between the middle graphene nanoribbon and the right graphene nanoribbon. In the simplest case, the three types of graphene nanoribbons are identical, and the overlap between two adjacent graphene nanoribbons has a short length. The above equation can therefore be simplified as
kt = 3k g lg /(3lg + 2k g R)
(10)
The above expression is then used to predict the thermal conductivity of the graphene-based composite material modeled in this study. Results and discussions Thermal conductivity of graphene nanoribbons Equilibrated temperature profiles obtained for one individual armchair-edged graphene nanoribbon are shown in Fig. 3. Temperature is nonlinear within the hot and cold regions, as the change of temperature is not proportional to the change of the distance along the direction of heat flow. The main cause of this nonlinear phenomenon is a finite-size effect arising from the system, since the characteristic scale of the system is comparable with the mean free path of the atoms, as discussed in the literature [60,61]. In this case, a very steep gradient in temperature may induce a significant nonlinear response such that Fourier's law may be no longer applicable. A similar nonlinear phenomenon of the change in temperature has also been noted in the case of carbon nanotubes [62,63]. This indicates that the transport of heat within graphene or carbon nanotubes is not fully diffusive [64,65]. The thermal conductivity of the armchair-edged graphene nanoribbon is determined by the temperature gradient that can be treated as a
Thermal boundary resistance Interfaces between adjacent graphene often contribute significantly to the thermal conductivity of the resulting composite material [72,73]. Understanding the thermal resistance at the interface between two 4
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Fig. 4. Specific thermal resistance and thermal conductivity of one individual armchair-edged graphene nanoribbon at room temperature as a function of the length of the graphene nanoribbon.
decreases with increasing the number of carbon cross-links, which will eventually lead to a reduction in the thermal boundary resistance between two adjacent graphene nanoribbons. The results are presented in Fig. 6, in which the thermal boundary resistance at room temperature is expressed as a function of the surface density of carbon cross-links. Thermal boundary resistance is very sensitive to the condition of the interface between two adjacent graphene nanoribbons. More specifically, thermal boundary resistance decreases with increasing the surface density of carbon cross-links. Carbon cross-links should be present in an amount sufficient to ensure that thermal boundary resistance is reduced effectively. Low thermal boundary resistance is technologically
adjacent graphene nanoribbons is of great significance in the study of its thermal conductivity. The thermal resistance results from the transport of phonons across the interface. The thermal boundary resistance between two adjacent graphene nanoribbons is predicted for the triple graphene nanoribbons depicted schematically in Fig. 2. Equilibrated temperature profiles obtained for the triple graphene nanoribbons are shown in Fig. 5. Discontinuity in temperature does occur at the interface between two adjacent graphene nanoribbons, and there is also significant difference in temperature. The presence of thermal boundary resistance corresponds to a discontinuous temperature across the interface. However, the difference in temperature
Fig. 5. Equilibrated temperature profiles obtained for the triple graphene nanoribbons modeled in this study. The length of each graphene nanoribbon is 40 nm. 5
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Fig. 6. Effect of the surface density of carbon cross-links on the thermal boundary resistance between two adjacent graphene nanoribbons at room temperature. The length of each graphene nanoribbon is 40 nm.
term “interval distance” refers to the distance apart from two adjacent graphene nanoribbons. The results are presented in Fig. 7, in which the thermal boundary resistance at room temperature is expressed as a function of overlap length or interval distance. The overlap length varies from 0 to 8 nm, and the interval distance varies from 0 to 4.8 nm. It is clear that the thermal boundary resistance decreases with increasing overlap length and decreasing interval distance. However,
important for applications for which very high efficiency in heat dissipation is required, which is of great significance to the development of microelectronic devices [74,75]. The effects of overlap length and interval distance on the thermal boundary resistance between two adjacent graphene nanoribbons are evaluated at room temperature. The term “overlap length” refers to the length of overlap between two adjacent graphene nanoribbons. The
Fig. 7. Effects of overlap length and interval distance on the thermal boundary resistance between two adjacent graphene nanoribbons at room temperature. 6
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Fig. 8. Thermal conductivity of a graphene-based polymer matrix composite material at room temperature as a function of the length of graphene nanoribbons. The thermal conductivity of the polymer matrix material is considered in this study to be 0.35 W/m·K at room temperature. There are no carbon cross-links between two adjacent graphene nanoribbons.
high efficiency in heat dissipation is required, even if the graphene nanoribbons use in a polymer matrix composite material is long enough. The thermal conductivity of the polymer matrix composite material is highly dependent upon the content of graphene nanoribbons dispersed in the polymer matrix material, as discussed above. Higher loading levels of such a two-dimensional carbon material may eventually lead to a higher thermal conductivity of the polymer matrix composite material. Unfortunately, there is always a challenge in achieving a highly loaded and substantially uniform dispersion of graphene nanoribbons in the polymer matrix material. There are two main factors contributing to this challenge: the tendency of nanoscale dispersants to aggregate [80,81] and the nature of the surface of the twodimensional carbon material [82,83], such as graphene nanoribbons, used in this study. It is assumed in this study that a substantially uniform dispersion of graphene nanoribbons is achieved in the polymer matrix material, as described in the model above. This assumption is reasonable if the content of graphene nanoribbons in the polymer matrix composite material is low enough. In this context, the content of graphene nanoribbons considered in this study for the polymer matrix composite material is therefore limited to the low level from 2 to 8% by volume. On the other hand, this study does not address the issue of the effect of the width of graphene nanoribbons on the thermal properties of the polymer matrix composite material. However, the resulting effect is similar to that of the length of graphene nanoribbons. The effect of the width of graphene nanoribbons may be even more pronounced due to the much shorter free path associated with phonon-boundary scattering. It is worth noting that the model developed in this study is still applicable to make predictions about the thermal properties of the polymer matrix composite material when there is a change in the width of graphene nanoribbons. The graphene nanoribbons modeled in this study have a well-defined edge structure. If these graphene nanoribbons have a variety of edge functional groups, the model can be modified to account for this factor in order to make accurate predictions about the thermal properties of the polymer matrix composite material. The effects of overlap length and interval distance on the thermal conductivity of the polymer matrix composite material are evaluated at room temperature. The results are presented in Fig. 9, in which the
both overlap length and interval distance have little effect on the thermal boundary resistance between two adjacent graphene nanoribbons, in comparison with the presence of carbon cross-links.
Thermal conductivity of graphene-based composite materials The effect of the length of graphene nanoribbons on the thermal conductivity of a graphene-based polymer matrix composite material is studied. The results are presented in Fig. 8, in which the thermal conductivity of the composite material at room temperature is expressed as a function of the length of graphene nanoribbons. The thermal conductivity of polymers is low, often in the range of from 0.2 to 0.5 W/ m·K at room temperature [76,77]. The mean value may be considered to be an approximation close to the thermal conductivity of common polymers. Therefore, the thermal conductivity of the polymer matrix material is considered in this study to be 0.35 W/m·K at room temperature. The content of graphene nanoribbons varies from 2 to 8% by volume, and there are no carbon cross-links between two adjacent graphene nanoribbons. The results presented in Fig. 8 indicate that the thermal conductivity of the polymer matrix composite material increases with increasing the length of graphene nanoribbons. The length of graphene nanoribbons affects the thermal conductivity of the composite material significantly, especially when the content of graphene nanoribbons is high. It is clear that a high level of graphene content facilitates a very efficient phonon transmission between two adjacent graphene nanoribbons, which will lead to an increase in the thermal conductivity of the composite material. The maximum thermal conductivity presented in Fig. 8 is close to the thermal conductivity of stainless steel, which is a common metal material with moderate thermal conductivity. However, the thermal conductivity of graphene-based polymer matrix composite materials has so far fallen short of that predicted here [78,79], and thus the potential of graphene as a reinforcement material for polymers has not been fully realized. At a low level of graphene content, phonons do not efficiently couple across the interface between two adjacent graphene nanoribbons. Therefore, the thermal conductivity of the composite material is low in all the cases of graphene length studied here, as shown in Fig. 8. It is therefore often necessary to employ a high level of graphene content for thermal management applications for which very 7
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Fig. 9. Effects of overlap length and interval distance on the thermal conductivity of a graphene-based polymer matrix composite material at room temperature. The content of graphene nanoribbons in the resulting composite material is 8% by volume. There are no carbon cross-links between two adjacent graphene nanoribbons.
Fig. 10. Effect of the surface density of carbon cross-links on the thermal conductivity of a graphene-based polymer matrix composite material at room temperature. The content of graphene nanoribbons in the resulting composite material is 8% by volume.
discussed above. In addition, it is expected that the thermal conductivity has strong dependence on the thermal boundary resistance. In this context, the content of graphene nanoribbons in the resulting composite material is assumed to be 8% by volume for the sake of simplicity.
thermal conductivity at room temperature is expressed as a function of overlap length or interval distance. There are no carbon cross-links between two adjacent graphene nanoribbons. Both overlap length and interval distance have been found to have little effect on the thermal boundary resistance between two adjacent graphene nanoribbons, as 8
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between graphene. The length of graphene plays an important role in the thermal conductivity of the composite material, especially when the content of graphene is high. Both thermal boundary resistance and thermal conductivity are very sensitive to the presence of carbon crosslinks between graphene. A fifty-fold improvement in overall thermal performance can be achieved for the polymer matrix composite material by properly combining graphene with large size and carbon crosslinks with high density. In the absence of carbon cross-links, phonons do not efficiently couple across the interface between graphene. Both overlap length and interval distance have little effect on the thermal boundary resistance between graphene and the thermal conductivity of the composite material. The results could offer valuable guidance for improving the thermal properties of graphene-based polymer matrix composite materials. This study does not address the issues of the stability of the resulting polymer matrix composite material nor the ability of graphene nanoribbons to be dispersed in the matrix material. Graphene nanoribbons to be effectively dispersed in the matrix material is of importance to the thermal conductivity of the polymer matrix composite material. A substantially uniform dispersion would be of great utility in permitting exploitation of the unique thermal properties of the polymer matrix composite material. Therefore, the development of a molecular dynamics modeling technology with taking into account the dispersion of graphene nanoribbons in the matrix material is of significant technical and commercial importance. Further study is needed to determine the effect of the degree of dispersion on the thermal properties of graphenebased polymer composite materials by performing molecular dynamics simulations for the nanostructure with a complex geometry.
The results presented in Fig. 9 indicate that the thermal conductivity of the composite material increases with increasing overlap length and decreasing interval distance. However, both overlap length and interval distance have little effect on the thermal conductivity of the composite material, in comparison with the length of graphene nanoribbons. The main cause of this behavior is that overlap length and interval distance do not affect the thermal boundary resistance between two adjacent graphene nanoribbons effectively, as discussed above. Consequently, a high degree of overlap will not lead to a significant increase in the thermal conductivity of the composite material. The thermal conductivity of the composite material increases with deceasing interval distance. However, the reduction of interval distance cannot lead to significant improvement in the thermal conductivity of the composite material. These graphene nanoribbons are essentially discontinuities each other. Consequently, the absence of carbon cross-links makes it more difficult for phonons to transmit across the interface between two adjacent graphene nanoribbons, which will eventually lead to lower thermal conductivity of the composite material. While graphene nanoribbons do exhibit high thermal conductivity, their effectiveness is limited by how well phonons can be transmitted across the interface between two adjacent graphene nanoribbons. To understand the importance of the interface, the effect of the surface density of carbon cross-links on the thermal conductivity of the composite material at room temperature is investigated. The results are presented in Fig. 10, in which the thermal conductivity at room temperature is expressed as a function of the surface density of carbon cross-links. The content of graphene nanoribbons in the resulting composite material is 8% by volume. It is clear from Fig. 10 that thermal conductivity is very sensitive to the condition of the interface between two adjacent graphene nanoribbons. The surface density of carbon cross-links affects the thermal conductivity of the composite material significantly. More specifically, the thermal conductivity of the composite material increases with increasing the surface density of carbon cross-links. In this case, the interface dominates the overall thermal conductivity of the composite material. High surface density of carbon cross-links allows for significant heat flow through the interface. Consequently, the key to reducing thermal boundary resistance and to improving thermal conductivity is the use of carbon cross-links between two adjacent graphene nanoribbons. What is needed is the ability to combine long graphene nanoribbons with the high surface density of carbon crosslinks, which reduce the effect of thermal boundary resistance such that maximum thermal performance can be realized. For example, a fiftyfold improvement in overall thermal performance can be achieved for the polymer matrix composite material by properly combining graphene with large size and carbon cross-links with high density.
CRediT authorship contribution statement Junjie Chen: Conceptualization, Methodology, Software, Formal analysis, Investigation, Resources, Writing - original draft, Writing review & editing, Supervision, Project administration, Funding acquisition. Baofang Liu: Methodology, Software, Formal analysis, Investigation. Xuhui Gao: Methodology, Software, Formal analysis, Investigation. Acknowledgement This work was supported by the National Natural Science Foundation of China (No. 51506048). References [1] Geim AK, Novoselov KS. The rise of graphene. Nat Mater 2007;6(3):183–91. [2] Geim AK. Graphene: Status and prospects. Science 2009;324(5934):1530–4. [3] Yang Y, Asiri AM, Tang Z, Du D, Lin Y. Graphene based materials for biomedical applications. Mater Today 2013;16(10):365–73. [4] Bitounis D, Ali-Boucetta H, Hong BH, Min D-H, Kostarelos K. Prospects and challenges of graphene in biomedical applications. Adv Mater 2013;25(16):2258–68. [5] Shateri-Khalilabad M, Yazdanshenas ME. Fabricating electroconductive cotton textiles using graphene. Carbohydr Polym 2013;96(1):190–5. [6] Mizerska U, Fortuniak W, Makowski T, Svyntkivska M, Piorkowska E, Kowalczyk D, et al. Electrically conductive and hydrophobic rGO-containing organosilicon coating of cotton fabric. Progress in Organic Coatings, Volume. Article Number 2019;137:105312. [7] Liu M, Yin X, Ulin-Avila E, Geng B, Zentgraf T, Ju L, et al. A graphene-based broadband optical modulator. Nature 2011;474(7349):64–7. [8] Liu M, Yin X, Zhang X. Double-layer graphene optical modulator. Nano Lett 2012;12(3):1482–5. [9] Mohanapriya K, Jha Neetu. Hierarchically hybrid nanostructure of carbon nanoparticles decorated graphene sheets as an efficient electrode material for supercapacitors, aqueous Al-ion battery and capacitive deionization. Electrochim Acta 2019;324:134870. https://doi.org/10.1016/j.electacta.2019.134870. [10] Dada OJ. Higher capacity utilization and rate performance of lead acid battery electrodes using graphene additives. J Storage Mater 2019;23:579–89. [11] Singh V, Joung D, Zhai L, Das S, Khondaker SI. Sudipta Seal, Graphene based materials: Past, present and future. Prog Mater Sci 2011;56(8):1178–271. [12] Stankovich S, Dikin DA, Dommett GHB, Kohlhaas KM, Zimney EJ, Stach EA, et al. Graphene-based composite materials. Nature 2006;442(7100):282–6. [13] Balandin AA, Ghosh S, Bao W, Calizo I, Teweldebrhan D, Miao F, et al. Superior thermal conductivity of single-layer graphene. Nano Lett 2008;8(3):902–7.
Conclusions The characteristics of nanoscale heat transport in a graphene-based polymer matrix composite material were studied to determine how the thermal conductivity of the composite material varies with the condition of the interface between graphene. Molecular dynamics simulations were performed and a modified Maxwell Garnett effective medium theory was used to understand the basic mechanism of the thermal resistance across the interface, and to optimize the condition of the interface for nanoscale thermal reinforcement. The results indicated that the bulk thermal conductivity of graphene nanoribbons is about 2570 W/m·K at room temperature. The thermal conductivity of a graphene-based polymer matrix composite material depends strongly upon the details of the microscopic structure and atomic interactions within the composite material, especially the condition of the interface between graphene. The thermal conductivity of the composite material can be improved by increasing the length of graphene, the surface density of carbon cross-links, and the length of overlap between graphene, or by decreasing the interval distance 9
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