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Letter pubs.acs.org/NanoLett
Multilayer Graphene Enables Higher Efficiency in Improving Thermal Conductivities of Graphene/Epoxy Composites Xi Shen,† Zhenyu Wang,† Ying Wu,† Xu Liu,† Yan-Bing He,‡ and Jang-Kyo Kim*,† †
Department of Mechanical and Aerospace Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong ‡ Engineering Laboratory for Functionalized Carbon Materials, Graduate School at Shenzhen, Tsinghua University, Shenzhen 518055, China S Supporting Information *
ABSTRACT: The effects of number of graphene layers (n) and size of multilayer graphene sheets on thermal conductivities (TCs) of their epoxy composites are investigated. Molecular dynamics simulations show that the in-plane TCs of graphene sheets and the TCs across the graphene/epoxy interface simultaneously increase with increasing n. However, such higher TCs of multilayer graphene sheets will not translate into higher TCs of bulk composites unless they have large lateral sizes to maintain their aspect ratios comparable to the monolayer counterparts. The benefits of using large, multilayer graphene sheets are confirmed by experiments, showing that the composites made from graphite nanoplatelets (n > 10) with over 30 μm in diameter deliver a TC of ∼1.5 W m−1 K−1 at only 2.8 vol %, consistently higher than those containing monolayer or few-layer graphene at the same graphene loading. Our findings offer a guideline to use cost-effective multilayer graphene as conductive fillers for various thermal management applications. KEYWORDS: Graphene, number of layers, size, composites, thermal conductivity
T
values are far lower than the expectations in view of the extremely high TC of graphene. Although higher TCs can be achieved by increasing graphene contents,12 the concomitant, large increase in viscosity of polymer makes the composites very difficult to process, limiting their real-world applications as TIMs. In addition, if the graphene content exceeds the percolation threshold, the much increased electrical conductivities of composites may cause short circuits when applied as TIMs. Therefore, a relatively low graphene content with a high efficiency in improving TCs of composites is desired. To make the most of the inherently high TC of graphene, previous studies were focused mainly on full exfoliation and uniform dispersion10 of individual graphene sheets in the matrix and tailoring the interface between graphene and the matrix to lower the interfacial resistance.11,19,21−23 While the dimensions of graphene, including the number of layers (n) and the lateral size, often show a significant impact on the electrical and mechanical properties of graphene papers24 and graphene/ polymer composites,25,26 rare attention has been attracted to their effects on TCs of the composites. It is found27−30 that the thermal transport in two-dimensional graphene was greatly affected by its size and n. Because of the long phonon mean free path, the in-plane TC of graphene depends on its length along the heat flow direction with increasing TC as the length increases.27 Although an increase in n led to a reduction in TC
hermal management with fast heat dissipation has become one of the most critical issues in modern electronic devices.1 The heat generated in electronics, optoelectronics, batteries, and so forth needs to be removed efficiently to prevent the malfunction of the device due to overheating.2 Traditional heat removal methods involve using polymer-based composites with thermally conductive fillers, such as silver and nickel, as thermal interface materials (TIMs).2 These materials possess a thermal conductivity (TC) over 1 W m−1 K−1 at the expense of high filler loading and thus heavy weights of the devices. Since the discovery of graphene,3 graphite nanoplatelets (GNPs) were initially considered as alternative fillers to traditional metals.4−7 Later, the pioneering studies on graphene, multilayer graphene, and their composites for thermal management applications8−10 have stimulated a surge of interests to explore graphene and its derivatives as the new generation of conductive fillers to effectively improve the TCs of polymer composites.11−13 A low filler loading of graphene results in an unprecedented improvement in TCs of epoxy composites14,15 and much reduced contact resistance with metal surfaces16 compared with traditional metallic fillers. Apart from reducing the weight of the electronics, the required low filler content lowers the viscosity for easy processing of the composites, a desired manufacturing requirement. While myriad achievements have been made in understanding the thermal transport in graphene17,18 and across the graphene/ polymer interface,19,20 the TCs of composites containing graphene sheets at relatively low filler contents of less than 10 vol % are generally in the range of 1−5 W m−1 K−1. These © XXXX American Chemical Society
Received: February 18, 2016 Revised: May 2, 2016
A
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Figure 1. (a) Three-dimensional model containing a graphene sheet (highlighted in yellow) embedded within an epoxy system for computing the inplane TC of embedded graphene; (b) the temperature gradient in graphene along the heat flux (z-) direction; and (c) TCs of embedded graphene sheets in epoxy as a function of number of layers.
of suspended or freestanding graphene,28 the opposite was true for the TCs of supported29 and encased graphene sheets.30 Such a difference in thermal transport behavior between the suspended and supported graphene sheets lies in the fact that the surrounding material, for example, the substrate or matrix, can damp the phonon energy31 in graphene, lowering the TC when the graphene sheet is supported on or encased in another material. In fact, the TC of graphene supported on a SiO2 substrate was only ∼580 W m−1 K−1 at room temperature,32 almost an order of magnitude lower than the suspended one, ∼ 5000 W m−1 K−1.9 With increasing n, the TC of supported graphene gradually rose to the TC of bulk graphite of ∼1900 W m−1 K−1.30 Such phenomena suggest that the TC of graphene sheets embedded in a matrix would similarly increase with increasing n, and by the same token the TCs of composites may be better improved by reinforcing with multilayer graphene sheets. However, to achieve higher TCs of composites with higher n of graphene, two other factors need to be considered. One important factor besides the inherent TC of filler itself is the thermal conduction across the filler/matrix interface, and its dependence on n is largely unknown. Another factor worthy of close scrutiny is the lateral size of graphene, which often shows significant impacts on mechanical and electrical properties of graphene papers and composites.24−26 Understanding the effect of n on TCs of composites has great implications for the next generation thermal management applications. Multilayer graphene sheets usually consist of stacked graphene layers with n = 2 to 10, while GNPs have larger thicknesses up to 100 nm.33 From the practical perspective, the fabrication process becomes far more complicated and costly when n is reduced to a single layer. Therefore, multilayers are preferable for large-scale production. It follows then that the requirements for using multilayer graphene sheets need to be established to achieve the same or even higher TCs of composites than using monolayer graphene sheets. Herein, we used a combined multiscale modeling and experimental approach to elucidate the effects of n and lateral
size of graphene sheets on TCs of graphene/epoxy composites. The effect of n on thermal conduction across the graphene/ epoxy interface is studied by molecular dynamics simulation (MDS), and the importance of graphene size is probed using an analytical model. Combined with experimental data available in the literature, the TCs of GNPs with different lateral sizes are measured to verify the foregoing modeling results. In-plane TC of Embedded Graphene. The effect of n on TC of graphene embedded in an epoxy matrix is evaluated based on MDS using the program Materials Studio (Accerlys). The Condensed-phase Optimized Molecular Potentials for Atomistic Simulation Studies (COMPASS) force field34 was used to simulate interatomic interactions for the whole system. The models containing graphene sheets embedded in the epoxy system (see Methods for details) for computing TCs of embedded graphene by Reverse Non-Equilibrium Molecular Dynamic (RNEMD) simulations are shown in Figure 1a. The two ends of the simulation domain were maintained at highand low-temperatures (Figure 1b), respectively. The TC of embedded graphene κz is given by J κ z = − dT dz
(1)
where J is the heat flux and dT/dz is the temperature gradient in the graphene sheet (see Supporting Information for details). Similar to the suspended graphene, the TC of embedded graphene may also depend on its length due to the long phonon mean free path.35 Therefore, we used a scaling method to extract the TC of embedded graphene (Figure S1, Supporting Information). The TC of embedded graphene in epoxy in the heat flux (z-) direction initially increased rapidly with increasing n and then slowly when n > 5, converging to the asymptotic value of bulk graphite as n → ∞ (Figure 1c). This trend agrees with the previous observations for the supported and encased graphene with a SiO2 substrate.30,36 The double exponential fitting was B
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Figure 2. (a) Comparison of VPS of carbon atoms in the suspended and embedded graphene sheets. The arrows indicate the recovery of vibrational power when n in the embedded graphene increases from 1 to 5. (b) VPS of carbon atoms in different layers of five-layer graphene. The vibrational power is recovered in the inner layers that are away from the matrix.
direction, as shown in Figure 3a. To quantify the TC across the interface, we created a thermal circuit at the interface analogous to an electrical circuit. The total thermal resistance, RT, is comprised of resistance generated by three resistors
used to reflect such a physical trend which was characterized by the decreasing rate of increment with increasing n and also to provide an estimation of the TC of bulk graphite (see Methods for details).36 The extrapolation of the data using the double exponential fitting gave us the TC of bulk graphite, 1854 W m−1 K−1, which is in excellent agreement with experimental values.28,30 The phonon vibrational power spectrum (VPS) is a useful tool to understand the phonon transport behavior. The phonon VPS was derived from the discrete Fourier transform of the velocity autocorrelation function by35 D(ω) =
∫0
RT = R ep / g + R g + R g/ep
where Rep/g is the resistance at the epoxy/graphene interface when the heat flows into graphene, Rg is the resistance generated by graphene, and Rg/ep is the resistance at graphene/ epoxy interface when the heat leaves graphene. The two interface resistances, Rep/g and Rg/ep, are given by
τ
⟨v(0) ·v(t )⟩exp( −iωt )dt
(3)
(2)
R ep/g =
where D(ω) is the phonon VPS at frequency ω; and ⟨v(0)·v(t)⟩ is the correlation function. The velocity is correlated at every 2 fs from time origin with a total integration time τ = 5 ps. The surrounding epoxy matrix suppressed the vibration of carbon atoms in graphene sheets, resulting in a lower vibrational power of embedded graphene compared with suspended one (Figure 2a). With increasing n, the vibrational power was partially recovered due mainly to the presence of inner layers whose vibrations were hardly affected by the surrounding matrix (Figure 2b). This means that an increase in n is potentially beneficial for high TCs of composites because of the rise in TCs of graphene sheets. TC Across Graphene/Epoxy Interface. The TC across the interface is usually much lower than that in the plane direction, one of the major reasons behind the limited TCs of bulk composites. RNEMD simulations were carried out to investigate the thermal conduction across the graphene/epoxy interface. A multilayer graphene sheet was embedded between two epoxy cells in parallel and perpendicular to the heat flux
ΔT1 , J
R g/ep =
ΔT2 J
(4)
where ΔT1 and ΔT2 are the temperature gradients at the two interfaces, respectively, and J is the heat flux in the thickness direction. The interface resistance, RI, is defined as the average of the two resistances:
RI =
R ep/g + R g/ep (5)
2
The interface conductivity, κI, is then given by κI =
⎛ RI ⎞−1 ⎜ ⎟ ⎝t ⎠
(6)
where t is the thickness of the interface. The TC of graphene in the thickness direction, κg, is given by
J κ g = − dT dx
C
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Figure 3. (a) Simulation domain consisting of graphene sandwiched between epoxy for computing TC across the interface and the thermal circuit at the interface. Large temperature drops were observed at the two interfaces. (b) κg, κx, κI, as a function of n. (c) RI between graphene and epoxy as a function of n. (d) Models used to calculate EI. (e) EI as a function of n. The inset shows the correlation between EI and RI.
(Figure 3c), which is in agreement with the experimental37 and simulation results.38 The interface resistance decreased gradually with increasing n, and became nearly saturated at above approximately seven layers. The interface resistance between two surfaces is closely related to the interface adhesion energy between the two materials.39,40 A higher interface adhesion energy generally leads to a lower interface resistance. A model containing epoxy and graphene was built to calculate the interface adhesion energy, as shown in Figure 3d. Periodic boundary conditions were applied in the x- and y-directions, and a vacuum slab of 10 nm was applied in the z-direction. It is noted that the graphene sheets, regardless of n, remained relatively flat during the simulation. Thinner graphene sheets may be more flexible than thicker multilayer graphene and therefore tend to form wrinkles, crumples, and folding within the matrix.41 Although such flexibility of graphene sheets may affect their coupling with the matrix, it was not captured in the MDS due to the nanometer size of graphene sheets and periodic boundary conditions, which was smaller than the typical size of wrinkles in real composites. Nonetheless, we are more interested in understanding the intrinsic effect of n on TCs across graphene sheets. As such, the interface adhesion energy was determined by the van der Waals (vdW) interactions between them by
where dT/dx is the temperature gradient within the graphene sheets. The TC across the interface/graphene/interface, κx, is given by κx =
⎛ RT ⎞−1 ⎜ ⎟ ⎝ d ⎠
(8)
where d is the total thickness of graphene and two interfaces. For monolayer graphene, the TC of interface/graphene/ interface, κx, was only 0.051 W m−1 K−1 (Figure 3b), almost 1 order of magnitude lower than that of epoxy. The TC increased to 0.35 W m−1 K−1 by nearly 6 folds when n was 13. It exceeded that of epoxy (∼0.2 W m−1 K−1) when n was 7, becoming no longer a factor limiting the TC of composites. The increasing κx with n was due to the simultaneously increased TC of graphene in the thickness direction, κg, and interface conductivity, κI. As n increased from 1 to 13, κI increased marginally from 0.050 to 0.064 W m−1 K−1 by 30%, while κg increased significantly to ∼1 W m−1 K−1. This means that the increment of κx was mainly due to the increasing κg. However, it is also worth noting that κx was only about one-third of κg, meaning that the presence of interface greatly limited the TC of interface/graphene/ interface. The extremely low κI ( 10 and large lateral sizes, rather than monolayer or few-layer graphene with n = 2 to 5 for real-world thermal management applications. Comparison with Experimental Results. To verify the above theoretical findings, experiments were carried out to measure the TCs of GNP/epoxy composites with GNPs having different lateral sizes and similar thicknesses, which are compared with results for other graphene sheets containing monolayer or few-layer graphene taken from literature. GNPs with different sizes were fabricated by changing the sonicating time of expanded graphite (EG) in acetone45 (see Methods for details). Two different groups of GNPs, designated as GNP-8 and GNP-20, were obtained with sonication durations of 8 and 20 h, respectively. The mean sizes of two groups were 34.4 and 1.7 μm, while the thicknesses were ∼21 and ∼15 nm for GNP8 and GNP-20, respectively (Figure S2, Supporting Information). The aspect ratios calculated from these dimensions were 114 and 1645 for GNP-20 and GNP-8, respectively. The GNPs were mixed with an epoxy matrix to fabricate composites (see Supporting Information for details). The dispersion of GNPs in the epoxy matrix is an important factor affecting various properties of composites. The scanning electron microscope (SEM) images of the composites (Figure S3, Supporting Information) indicate uniform dispersion of GNPs regardless of their sizes, suggesting that the agglomeration of GNPs would not be a factor affecting the comparison
(9)
where ET is the total energy of the whole system; Eg is the energy of graphene without interacting with epoxy; and Eep is the energy of epoxy without interacting with graphene. The interface adhesion energy consistently increased with increasing n, as shown in Figure 3e, analogous to TC of embedded graphene (Figure 1c). This observation can be explained by the transparency of monolayer graphene to vdW interaction42,43 where the vdW force was transmitted through the monolayer graphene. In other words, the noncontacting layers in the middle of graphene sheet can impose an attractive force to epoxy molecules through the vdW interactions. When the number of graphene layer increased, more internal layers were able to interact with epoxy molecules, giving rise to a higher EI. The rate of increment started to slow down when n > 7. This observation signifies that the vdW forces imposed by graphene layers beyond approximately seven layers away from epoxy molecules were partially screened by the multilayer graphene in between because they gradually became an opaque barrier to vdW forces with increasing n.42,43 The increasing EI led to a lower RI with increasing n. EI and RI exhibited an almost inverse linear relationship (see inset of Figure 3e). TCs of Bulk Composites. Although increasing n is shown to improve the TCs of embedded graphene both in the plane direction and across the interface, it is more important to verify the same for TCs of bulk composites. From the practical application point of view, multilayer graphene sheets are preferred to monolayer ones due to their easiness in large-scale production. The change in lateral size of graphene also affects TCs of composites. Therefore, we adopted an analytical model based on the effective medium theory for particulate composites44 to predict TCs of bulk composites containing graphene with different lateral and thickness dimensions. We also evaluated the feasibility of using multilayer graphene sheets to replace monolayer ones for better enhancement of TCs of composites. The accuracy of the original model was improved (see Methods for details) by replacing the constant bulk TC of graphene, κp, and the interface resistance, RI, with the thickness dependent values, κp (n) and RI (n), obtained from MDS (Figures 1c and 3c), respectively. In addition, the individual graphene sheets were considered as having a disc shape with an aspect ratio of α (= D/h), where D is the diameter and h is the thickness. The improved model is given in eq 17. Parametric studies were performed for graphene sheets with different D E
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Figure 5. (a) TCs of composites containing two groups of GNPs as a function of filler content and comparison with those containing monolayer or few-layer graphene reported in literature (Yu et al.6 and Shahil et al.10); (b) comparison of TCEFs of graphene sheets with different n and aspect ratios between the analytical model (solid lines) and experimental data (unfilled symbols). Experimental data: blue star for current study; red circle;6 blue diamond;7 red triangle;10 black right-pointing triangle.11
well with the predictions with the exception of GNP-8 with a very high aspect ratio of 1645, for which the prediction overestimated the TC. GNPs with such an extremely high aspect ratio are vulnerable to folding and wrinkles (Figure S4, Supporting Information), detrimental to the TCs of GNPs themselves and their composites.13,46 Unable to take into account such an effect, however, the analytical model tended to overestimate the TC of GNP-8. Nevertheless, the general trend of increasing TCEF with increasing aspect ratio for graphene sheets of similar thicknesses (i.e., symbols with the same colors) agreed with predictions. For graphene sheets with similar aspect ratios of ∼200, GNPs (blue diamond) showed the highest TCEF while monolayer graphene (black triangle) was the least efficient among three types of graphene, which is in excellent agreement with the predictions. These experimental findings verified our theory based on the multiscale modeling, signifying the importance of large lateral size in achieving high TCs of composites with multilayer graphene. To demonstrate the effectiveness of our GNP/epoxy composites for practical TIM applications, GNP/epoxy composites were sandwiched between two thin copper (Cu) foils, and the TCs of such sandwich structures were measured using the same laser flash technique (see Supporting Information for details). The “effective” or “apparent” TC47 includes the contribution from the contact resistance between TIMs and connecting surfaces. The apparent TCs of composites containing 2.8 vol % GNP-20 and GNP-8 were measured to be 0.62 and 1.29 W/mK, respectively, which were smaller than their respective “intrinsic” TCs due to the existence of thermal contact resistance (Figure S5, Supporting Information). Similar to the intrinsic TCs, the composites containing GNP-8 showed much higher apparent TC than those containing GNP-20, indicating that the performance of composites maintained in the practical setup. The extracted thermal contact resistances with the copper foil for both composites were similar, that is, 11.9 ± 3.6 and 13.4 ± 2.4 mm2 K/W for the composites with GNP-20 and GNP-8, respectively. These values are comparable to those of commercial TIMs,48 few-layer graphene/polymer composites47 and carbon nanotube bucky papers49 (Table S1, Supporting Information). In addition, a comparison of the apparent TC ratio of composite to matrix, κA/κm,A, was made among composites containing graphene sheets with different n (Table S1, Supporting Information). Our GNP-8/epoxy composites showed a κA/κm,A ratio of ∼6.4 at 2.8 vol %, which was higher than that of graphene-multilayer graphene/thermal grease (κA/ κm ∼ 2.4 at 2 vol %)10 and few-layer graphene/thermal grease
between TCs of composites containing different groups of GNPs. The TCs of the composites containing GNPs with different lateral sizes and filler contents measured by the laser flash method are shown in Figure 5a and compared with those containing monolayer or few layer graphene sheets taken from the literature.6,10 The dimension of these composites are listed in Table 1. The TCs of composites in general increased with Table 1. Experimental Data Used in Figure 5 for Graphene Sheets with Different Dimensions reference monolayer graphene11 Few-layer graphene6 graphene−multilayer graphene10 GNPs7 GNP-20 (current work) GNP-8 (current work)
thickness [nm]
size [μm]
aspect ratio
TCEF [%]
1 1.7 1.45
0.2 0.35 0.6
200 206 414
27.5 120 230
20 14.9 20.9
3.9 1.7 34.4
195 114 1645
158 96 244
increasing graphene content and aspect ratio. Although GNP-8 and GNP-20 had similarly larger thicknesses than the other monolayer or few-layer graphene sheets taken for comparison, the composites containing GNP-8 showed the highest TCs while those with GNP-20 exhibited the lowest TCs among all, for a given graphene volume fraction, contingent on their aspect ratio. To further evaluate the efficiency of graphene sheets with different lateral and thickness dimensions in improving the TCs of composites, the thermal conductivity enhancement factor (TCEF)10 is used, which is defined as the enhancement in TC per 1 vol % of fillers κ − κm × 100% TCEF = c κm (10) where κc is the TC of composite containing 1 vol % of fillers, and κm is the TC of polymer matrix. Figure 5b presents the TCEFs of composites predicted based on eqs 17 and 10, which are compared with the experimental data given in Table 1. As expected, the prediction shows that the TCEF increased with increasing aspect ratio. With the same aspect ratio, the TCEF of graphene sheets with a larger n was always higher than those with a smaller n. The experimental data were categorized into three groups based on the thickness of graphene sheets: namely, (i) monolayer graphene11 (n = 1, black symbols); (ii) few-layer graphene6,10 (n = 2 to 5, red symbols); and (iii) GNPs7 (n > 10, blue symbols). The experimental data agreed F
DOI: 10.1021/acs.nanolett.6b00722 Nano Lett. XXXX, XXX, XXX−XXX
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Nano Letters composites (κA/κm ∼ 2 at 3.4 vol %)47 at similar graphene contents. The above comparisons corroborate the effectiveness of composites containing thick graphene sheets (n > 10) with large lateral sizes for practical TIM applications. Multilayer graphene can also be used for effective thermal management of Li-ion batteries by incorporating them in the phase change material, for example, paraffin wax.50 In conclusion, we investigated the possibility of using multilayer graphene sheets instead of monolayer ones to more efficiently improve the TCs of epoxy composites based on combined multiscale modeling and experimental approach. The MDSs show that both TCs in the plane direction and across the graphene/epoxy interface increased with increasing n, which is the foundation for using multilayer graphene sheets. However, the analytical model reveals that the higher n in multilayer graphene sheets without proportionally increasing their lateral sizes resulted in lower TCs of bulk composites. When the multilayer graphene sheets had the same aspect ratios as the monolayer ones, the former always outperformed in improving TCs of composites. To verify the findings from the above multiscale modeling, we prepared two different groups of GNPs with different lateral sizes by controlling the sonication time. Compared with the composites made from monolayer or few-layer graphene sheets with similar aspect ratios, the thick GNPs with n > 10 showed the highest efficiency in improving TCs of composites, which is in excellent agreement with the predictions. The GNPs with a mean lateral size of 34 μm and a thickness of ∼20.9 nm achieved the highest efficiency with a TCEF of 244% among all those reported in the literature with minimum effort in the preparation of graphene sheets. This study offers a new perspective in using multilayer graphene sheets as fillers in composites requiring high TCs for thermal management applications because their mass production has a tremendous benefit of simplicity and minimal efforts compared to producing monolayer or few-layer graphene sheets. Methods. MDS. The models of graphene sheets were built similarly to our previous work.51 The model of the epoxy matrix consisted of molecules of diglycidyl ether of bisphenol F (DGEBF) resin and Triethylenetetramine (TETA) hardener. DGEBF and TETA molecules were first packed into a cell with a predefined density of 0.9 g cm−3 using the Amorphous Cell module of Materials Studio. The simulation cell represented an amorphous mixture of two reactive species prior to crosslinking with an experimental stoichiometric ratio.52 Graphene sheets with different numbers of layers were then embedded between two blocks of epoxy to construct the models of composites for thermal transport in the two perpendicular directions (Figures 1a and 3a). The structure was then minimized for 5000 steps followed by a constant pressure and temperature (NPT) run for 500 ps at 500 K to allow for the filler surface to be completely wetted by the polymer molecules. Subsequently, the cross-linking process52 was carried out to simulate the curing of epoxy composites (Figure S6, Supporting Information). The final structure after cross-linking was cooled down to 300 K at a cooling rate of 1 K ps−1. The dimensions of the final model in Figures 1a and 3a were 2 nm × 7 nm × 60 nm and 15 nm × 3 nm × 3 nm, respectively, with periodic boundary conditions along all directions, except the heat flux direction for which fixed boundaries were applied. The TC of embedded graphene in the plane direction and that across the interface were calculated using the RNEMD method53 which has been widely employed to obtain the TC of
graphene.35,54 The model (Figure 1a) was divided into several slabs along the graphene plane direction or z-axis. The heat flux, J, was created by adding a certain amount of energy into the graphene sheet in the first slab and extracting the same amount from the graphene sheet in the last slab every 100 fs. The simulation was carried out with the constant energy (NVE) ensemble to ensure the total energy of the system was conserved. A steady temperature gradient (Figure 1b), dT/dz, was generated in the graphene sheet after 1 ns due to the heat flux. The TC of embedded graphene, κz, is given by Fourier’s law: J κ z = − dT (11)
dz
Similar to the suspended graphene, the TC of embedded graphene may also depend on its length due to the long phonon mean free path. Therefore, a scaling method was used to extract the TC of embedded graphene as described in our previous work.35 The TCs of embedded graphene with different numbers of layers were calculated. All the results obtained from MDS were ensemble averaged over the last 500 ps of the production run in the NVE ensemble. Improved Analytical Model. The TCs of composites were predicted using an analytical model that was improved from the effective medium theory for particulate composites.44 The individual graphene sheets were assumed disc-shaped with a diameter of D and a thickness of h and were randomly dispersed in the epoxy matrix. Using other shapes, such as rectangular shapes, may not affect the final results (see Supporting Information S7 for details). The effective TC of composites, κeff, predicted by the effective medium theory is given by κeff = k(κ m , κ p , RI , f , P)
(12)
where κm and κp are the bulk TCs of matrix and graphene, respectively; RI is the interface resistance between graphene and matrix as defined in eq 5; f is the volume fraction of graphene; and P represents geometric and dimensional properties, including lateral size, thickness, and orientation of graphene sheets. Here, we modified the TC of graphene, κp, and the interface resistance, RI, by introducing the thickness-dependent properties as obtained from MDS. As discussed in Figures 1c and 3c, κp and RI are both functions of n. By fitting the data from MDS (Figure 1c), the thickness-dependent TCs of embedded graphene sheets, κp(n), can be obtained:36 κ p(n) = A + B(1 − e(−n / C)) + D(1 − e(−n / E))
(13)
where A, B, C, D, E are five parameters to be fitted by the data obtained from MDS. The thickness-dependent interface resistance, RI(n), was obtained similarly by fitting the data in Figure 3c. The resultant κp(n) and RI(n) were then used to obtain the equivalent TC of composites along the i-axis (i = x, y, and z), κci (n)44 κ p(n)
κic(n) = 1+
(1 + 2p)RI(n)Liκ p(n) h
(14)
where Li (i = x, y and z) is a geometric factor related to the particle shape, p = h/D, and is given by G
DOI: 10.1021/acs.nanolett.6b00722 Nano Lett. XXXX, XXX, XXX−XXX
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Nano Letters Ly = Lz =
p2 2
2(p − 1)
+
p 2 3/2
2(1 − p )
cos−1 p
■
(15)
Lx = 1 − 2Ly
(16)
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
■
κeff (n) = κm
(
κzc(n) − κm Ly c m + Lz(κz (n) − κm)
3 − f 2κ
+
κxc(n) − κm (1 κm + Lx(κxc(n) − κm) κxc(n) − κm L κm + Lx(κxc(n) − κm) x
ACKNOWLEDGMENTS The project was supported by the Research Grants Council (Project Codes: 613811, 16203415) and the Innovation and Technology Commission (Project Code: ITS/141/12) of Hong Kong SAR. X.S. and Z.W. were recipients of the Hong Kong Ph.D. Fellowship. X.S. acknowledges the Tsai Best Student Paper Award given at the 20th International Conference on Composite Materials where part of this paper was presented.
⎤ − Lx)⎥⎦
) (17)
Experimental Details. GNPs with different sizes were produced using natural graphite (NG, supplied by Asbury Graphite Mills) as the raw material similar to our previous work.45 NG and sulfuric acid (1g/30 mL) were mixed and stirred at 200 rpm in a round-bottom flask for 30 min. Then, nitric acid with a volume at 1:3 of sulfuric acid was slowly poured into the mixture, and the new mixture was stirred for 24 h at room temperature. The graphite intercalation compound (GIC) was then obtained by washing the mixture using deionized water and drying in an oven at 60 °C for 24 h. The dried GIC was heated in an oven at 1050 °C for 30 s so that the GIC expanded explosively due to the thermal shock. The EG was then sonicated in acetone (2 mg mL−1) to get exfoliated GNPs. The sizes of GNPs were controlled by the sonication time. Two groups of GNPs, designated as GNP-8 and GNP-20, were produced with sonication durations of 8 and 20 h, respectively. The acetone was then evaporated at 60 °C in an oven overnight to obtained GNP powders. GNPs were added into the epoxy matrix to fabricate composites with different GNP contents (see Supporting Information for details). The size distributions of two groups of GNPs were measured by the Particle Size Analyzer (COULTER LS230). The morphologies of GNPs and GNPs/epoxy composites were characterized on a SEM (JEOL-6390) at an accelerating voltage of 20 kV. The thicknesses of GNPs were measured by the atomic force microscopy (AFM, Scanning Probe MicroscopeNanoScope, Digital Instrument) operating in a tapping mode. The TCs were measured using the laser flash technique. The TC, κ, is given by κ = αCpρ
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(18)
where α is the thermal diffusivity measured by the laser flash instrument (LFA-457, NETZSCH) according to the specification, ASTM E1461; CP is the heat capacity measured using a differential scanning calorimeter (DSC, TA-Q1000); and ρ is the density of the sample.
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The effective TC of composites, κeff(n), containing randomly dispersed graphene sheets therefore is given by (see Supporting Information S8 for details) κ c(n) − κ ⎡ 3 + f ⎢⎣2 κ + Lz (κ c(n)m− κ ) (1 − Lz) + m z z m
SEM images of GNPs; effect of the shape of GNPs; detailed deviation of the analytical model. (PDF)
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.6b00722. Calculation of heat flux and temperature; TC of embedded graphene using scaling method; sizes and thicknesses of GNPs; preparation of GNPs/epoxy composites; apparent TCs and thermal contact resistance measurement; cross-linking process of epoxy using MDS; H
DOI: 10.1021/acs.nanolett.6b00722 Nano Lett. XXXX, XXX, XXX−XXX
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