Article pubs.acs.org/JPCB
Increasing the Thermal Conductivity of Graphene-Polyamide-6,6 Nanocomposites by Surface-Grafted Polymer Chains: Calculation with Molecular Dynamics and Effective-Medium Approximation Yangyang Gao†,# and Florian Müller-Plathe*,† †
Eduard-Zintl-Institut für Anorganische und Physikalische Chemie and Center of Smart Interfaces,Technische Universität Darmstadt, Alarich-Weiss-Str.4, 64287 Darmstadt, Germany S Supporting Information *
ABSTRACT: By employing reverse nonequilibrium molecular dynamics simulations in a full atomistic resolution, the effect of surface-grafted chains on the thermal conductivity of graphene-polyamide-6.6 (PA) nanocomposites has been investigated. The interfacial thermal conductivity perpendicular to the graphene plane is proportional to the grafting density, while it first increases and then saturates with the grafting length. Meanwhile, the intrinsic in-plane thermal conductivity of graphene drops sharply as the grafting density increases. The maximum overall thermal conductivity of nanocomposites appears at an intermediate grafting density because of these two competing effects. The thermal conductivity of the composite parallel to the graphene plane increases with the grafting density and grafting length which is attributed to better interfacial coupling between graphene and PA. There exists an optimal balance between grafting density and grafting length to obtain the highest interfacial and parallel thermal conductivity. Two empirical formulas are suggested, which quantitatively account for the effects of grafting length and density on the interfacial and parallel thermal conductivity. Combined with effective medium approximation, for ungrafted graphene in random orientation, the model overestimates the thermal conductivity at low graphene volume fraction (f < 10%) compared with experiments, while it underestimates it at high graphene volume fraction ( f > 10%). For unoriented grafted graphene, the model matches the experimental results well. In short, this work provides some valuable guides to obtain the nanocomposites with high thermal conductivity by grafting chain on the surface of graphene. the intrinsic thermal conductivity of CNTs17,18 or graphene.19−21 Until now, the net effect of covalent functionalization on the thermal transport properties of CNT or graphene polymer composites is still unclear. There is a large scatter in the reported experimental thermal conductivities of composites: By functionalizing graphene, the thermal conductivity of nanocomposites can be increased by 28-fold over a pure epoxy resin at 20 wt %.22 Some work,23,24 however, reported that moderate chemical modifications (chemical treatment for 0.5 h) are beneficial to improve the thermal conductivity of the CNT composites. Gojny et al.25 and Gulotty et al.26 even find that nonfunctionalized CNTs led to a larger improvement of thermal conductivity of polymer composites compared with functionalized CNTs. The functionalization of graphene and CNTs on the thermal conductivity behaves in a similar way because of similar chemical properties of CNTs and graphene. In experiments, it is very difficult to control factors such as the dispersion and purity of graphene, and the processing conditions, which can affect the thermal conductivity of nanocomposites.27,28In molecular dynamics simulations, end-
1. INTRODUCTION Due to the weak van der Waals interactions between chains and the randomness in molecular arrangement, bulk polymers have very low thermal conductivities of 0.1−1 W/m·K at room temperature.1 On the other hand, carbon nanotubes (CNTs) and graphene have supreme thermal conductivities (up to 5000 W/m·K).2−5 Thus, they have been proposed as filler in polymer nanocomposites to improve the thermal transport. Unfortunately, the reported increase (0.5 h) reduces the thermal conductivity. The trend of experiments is consistent with that of our simulation. Conversely, the thermal conductivity Λrandom/λPA varies only from 15.2 to 17.8 when the grafting length increases from L = 6 to L = 28 at f = 10%. This means the grafting density dominates the effect on the thermal conductivity. We turn to the second case where the graphene platelets are oriented along one direction. In the EMA, eq 8, the thermal conductivity Λ∥/λPA along the orientation direction is proportional to the volume fraction of graphene. This indicates that the energy is conducted independently within the polymer and within the graphene. For carbon nanotubes,34 the simulated and experimental parallel conductivities are found to be much smaller than expected on the basis of simple additivity. This is because heat transfers above proportion within the PA matrix due to the bad coupling between the ungrafted carbon nanotubes and interfacial PA. For the grafted graphene-filled system, the grafted PA chains improve the interfacial coupling and more heat can flow through the graphene. Thus, the parallel thermal conductivity λ∥ increases with the grafting density and grafting length in Figures 3a and 4b. However, in Figure S14, the thermal conductivity Λ∥/λPA decreases for the grafted systems because of the decreased thermal conductivity of grafted graphene. In the MD simulations, the coupling is taken into consideration; while the EMA model omits it. So on one hand, the model overestimates the parallel thermal conductivity; on the other hand, the trend with the grafting density is contrary to the MD results.
Figure 6. Thermal conductivity of nanocomposites for random orientation of graphene particles Λrandom/λPA as a function of the grafting density. (a1 (graphene length) = 40 μm, f (graphene volume fraction) = 4%.)
predicts it to decrease for gd > 1.56%. There exists an optimum grafting density because of the two competing effects: Grafting increases the thermal coupling of graphene to the polymer matrix but decreases the intrinsic in-plane thermal conductivity of graphene. In addition, for the low grafting density and small graphene particles, we find that the thermal conductivity Λrandom/λPA is below 1.0 (Figure S13) which means that adding small graphene particles actually has an adverse effect on the thermal conductivity compared with the pure polymer. This is because of the low interfacial thermal conductance and the fact that the inherent thermal conductivity of graphene is smaller for small flakes.58−60 So, the thermal conductivity Λrandom/λPA even decreases with the graphene size at gd = 0.00%. To focus on the systems filled with micron (μm) sized graphene used in experiments,61,62 we calculate Λrandom/λPA for different grafting densities and volume fractions f for an assumed graphene size of a1 = 40 μm (Figure 7). At all volume fractions studied here, 1342
DOI: 10.1021/acs.jpcb.5b08398 J. Phys. Chem. B 2016, 120, 1336−1346
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The Journal of Physical Chemistry B For the third case, it is useful to simplify eq 12. Because of λg ≫ λPA, the thermal conductivity Λ⊥/λPA = λg/[λg − f(λg − λPA − αλg)] = λg/[λg − f(λg − αλg)] = 1/[1 − f(1 − α)]. It indicates that the thermal conductivity Λ⊥/λPA is independent of the thermal conductivity and the size of graphene. The calculated Λ⊥/λPA perpendicular to the orientation direction for different grafting densities and different volume fractions is shown in Figure 8. It is found that the thermal conductivity Λ⊥/λPA is less
Figure 9. Comparison of thermal conductivity of nanocomposites for random orientation of graphene particles Λrandom/λPA between experiments11,12,22,61,62,67,68 and model32 for the (a) ungrafted and (b) grafted gaphene (blue and purple curves are calculated via model).
volume fraction (10%). At high volume fraction of graphene, a percolating graphene network can significantly improve the thermal conductivity, which obeys the adjusted critical power law.61 However, the EMA model itself does not take the filler network into consideration.73 In the Supporting Information, we show that the EMA model also mathematically cannot show an upward curvature with the graphene volume fraction, for the parameters ranges typical for graphene−polymer nanocomposites. For grafted graphene, the calculated values match the experiment much better (Figure 9b). This is because of the very uniform dispersion of grafted graphene in the polymer matrix, in experiment and model alike, which does not allow the formation of a network with graphene particles touching each other.
Figure 8. Perpendicular thermal conductivity for highly oriented graphene particles Λ⊥/λPA as a function of (a) grafting density and (b) graphene volume fraction f. (a1 (graphene length) = 40 μm.)
than or around 1.0. This is reasonable because the heat must completely transfer from the polymer matrix into graphene, then out of graphene into the polymer matrix again. The interfacial resistance is large, and so the graphene platelets become barriers to the heat flux. More specifically, the thermal conductivity Λ⊥/λPA increases with grafting density, since it improves the interfacial conductance; however, it decreases with the volume fraction of graphene, because there are more interfaces at high volume fraction f. By employing the random walk simulations of thermal walkers63 for thermal conductivity, when the graphene particles are perpendicular to the heat flux, a value for Λ⊥/λPA of around 1.0 has been obtained at f = 1% for an interfacial conductance (95 MW/m2 K). This is consistent with our Λ⊥/λPA for gd = 6.25% (90 MW/m2 K) in Figure 8. However, in the experiment,64,65 the thermal conductivity Λ⊥/ λPA was around 1.3−1.4. The difference probably arises because in experiment the fillers are not perfectly aligned along the orientation direction.66 Finally, we compare in Figure 9 the trend of Λrandom/λPA from the EMA model 32 with the experimental results.11,12,22,61,62,67,68 In these experiments, the range of graphene size is between 10 and 60 μm. For ungrafted graphene in Figure 9a, the thermal conductivity Λrandom/λPA from simulation is larger than that from experiments at low 1343
DOI: 10.1021/acs.jpcb.5b08398 J. Phys. Chem. B 2016, 120, 1336−1346
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The Journal of Physical Chemistry B Notes
5. CONCLUSIONS In the present contribution, we have adopted reverse nonequilibrium molecular dynamics simulations to investigate the effect of surface-grafted chains on the thermal conductivity of graphene-polyamide-6.6 (PA) nanocomposites. The interfacial thermal conductivity perpendicular to graphene plane increases continuously with the surface density of grafted chains. It initially increases and then saturates with the length of grafted chains. The thermal conductivity of the polymer matrix is reduced in an interphase of about 2 nm thickness independent of the grafting state. On the other hand, the surface functionalization breaks the conjugate planar structure of graphene and reduces the intrinsic mean free path length of phonons. Thus, the in-plane thermal conductivity of graphene drops as the grafting density increases. The thermal conductivity parallel to graphene plane of composite, however, increases with both grafting density and length. There exists an optimum grafting density for improving the thermal conductivity of nanocomposites because of these two competing effects. The grafting density dominates the interfacial and parallel thermal conductivity, except at excessive grafting densities, where a new interface between grafted and free polymer appears. The effect of grafting density and length on the interfacial and parallel thermal conductivity can be fitted by two empirical formulas. The conductivities obtained from molecular dynamics simulation can be input into an effective medium approximation (EMA), which allows the extrapolation to micrometersized graphene and the calculation of the macroscopic thermal conductivity of nanocomposites. The model always predicts the order of the experimental values correctly. For ungrafted random oriented graphene flakes, however, it overestimates the thermal conductivity compared with experiments at low graphene volume fraction ( f < 10%), while it underestimates the thermal conductivity at high graphene volume fraction ( f > 10%), since it does not allow for the formation of percolating heat-conducting graphene networks. Grafted graphene, in contrast, does not form networks because of the uniform dispersion of graphene in the polymer matrix; therefore, the model matches the experimental results well. The comparison between experiment and EMA model is not as good for orientated graphene, as in experiments the graphene orientation is not perfect. In summary, the results obtained are important for applications of graphene as fillers for improving thermal conductivity of composites.
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The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work is supported by program “Sino-foreign combined postgraduate training” by Beijing Municipal Education Commission and Beijing University of Chemical Technology. Parts of funds are supported by the National Basic Research Program of China 2015CB654700(2015CB654704). Thank Professor Liqun Zhang in Beijing University of Chemical Technology for his continued encouragement and support. Fruitful discussions with Dr. Mohammad Reza Gharib-Zahedi are gratefully appreciated.
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ASSOCIATED CONTENT
S Supporting Information *
This material is available free of charge via the Internet at http://pubs.acs.org/. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.5b08398. Additional figures and tables (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Telephone number: +49-6151-16 6523/4. Present Address #
Yangyang Gao is on leave from Beijing University of Chemical Technology, Beijing, People’s Republic of China. 1344
DOI: 10.1021/acs.jpcb.5b08398 J. Phys. Chem. B 2016, 120, 1336−1346
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