Grain Boundary Energy and Grain Size Dependences of ThermalConductivity of Polycrystalline GrapheneH. K. Liu,†,‡ Y. Lin,*,† and S. N. Luo*,‡

†State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China,Chengdu, Sichuan 610054, P. R. China‡The Peac Institute of Multiscale Sciences, Chengdu, Sichuan 610207, P. R. China

ABSTRACT: We investigate with molecular dynamics simulations thedependences of thermal conductivity (κ) of polycrystalline graphene on grainboundary (GB) energy and grain size. Hexagonal grains and grains with randomshapes and sizes are explored, and their thermal properties and phonon densitiesof states are characterized. It is found that κ decreases exponentially withincreasing GB energy, and decreasing grain size reduces κ. GB-induced phononsoftening and scattering, as well as reduction in the number of heat conductingphonons, contribute to the decrease in thermal conductivity.

■ INTRODUCTION

Graphene is a two-dimensional (2D) carbon film with uniquethermal, mechanical, and electronic properties bearing promisein many applications.1−4 Defect-free graphene’s extraordinaryproperties such as ultrahigh electronic mobility,5,6 superiorthermal conductivity,7,8 and mechanical strength with excep-tional stretchability,9−11 have been demonstrated in exper-imental12−15 and theoretical16−19 investigations. Recently, achemical vapor deposition (CVD) technique was developed togrow large-scale single-layer graphene films20,21 consisting ofrandomly oriented graphene grains. It is shown that the grainboundaries (GBs) result in not only the deterioration ofelectronic22,23 and mechanical properties,24−26 but also thermaltransport properties.27−29 Yazyev and Louie introduced ageneral approach for constructing dislocations in graphenecharacterized by arbitrary Burgers vectors as well as grainboundaries,30 covering a whole range of possible misorientationangles. Bagri et al. studied thermal transport across twin grainboundaries using nonequilibrium molecular dynamics (MD)simulations.27

Previous studies focused mostly on parallel, regular-structured (e.g., repeating pentagon−heptagon pairs)GBs,31−33 which largely represent bicrystals in a sense.However, grains can be of different shapes and sizes and mayform GB triple junctions and other random, complex patternsin reality. A recent independent study examined heatconduction in a special, idealized polycrystalline graphenewith small hexagonal grains and repeating pentagon−heptagondislocations.34 Since GB structure and thus GB energy, grainsize, and grain shape may all contribute to thermal conductivity,here we perform MD simulations of more realistic polycrystal-line structures containing hexagonal grains or grains of random

grain shapes and sizes and explore the effects of the average GBenergy and grain size (in terms of GB fraction) on thermalconductivity of polycrystalline graphene. GBs induce reductionin thermal conductivity, and this reduction is quantified and canbe attributed to phonon softening and scattering.

■ METHODOLOGYDifferent grain patterns have been observed in experiments,including uniform hexagonal graphene flakes grown on thesurface of liquid copper.35 We thus construct polycrystallinegraphene structures with the widely used Voronoi tessellationmethod,36−38 where a set of grain centers is first specified, andfor each center there is a corresponding region consisting of allatoms closer to that center than to any other centers. Theformal definition is

= ∈ | ≤ ≠G A S d A C d A C j i( , ) ( , ) for alli i j (1)

Here S is a space created with a distance function d. A, Gi, andCi denote atoms, grain i, and grain center i, respectively.GBs may form polygons of different shapes due to different

grain centers and different numbers of grains chosen. A unitpolycrystalline configuration with hexagonal grains and tilt GBsis shown in Figure 1(a). Type-1 grain in Figure 1(a) is thereference grain for rotation. The rotation angle is 30° for type2; for types 3 and 4, the rotation angles are ϕ and −ϕ,respectively. All the neighboring grains are of different graintypes. This construction leads to random GBs (in order tomimic reality) and shapes, so the GB structures are with

Received: August 8, 2014Revised: September 25, 2014Published: October 2, 2014

Article

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© 2014 American Chemical Society 24797 dx.doi.org/10.1021/jp508035b | J. Phys. Chem. C 2014, 118, 24797−24802

randomly occurring defects of different types includingvacancies and octagons (Figure 1(c) and Figure 1(d)). Allthe grains in polycrystalline graphene are structurally the sameexcept that neighboring grains are rotated relatively by an angle.The dimensions of the polycrystalline graphene sheet are about60 nm × 70 nm, containing about 160 000 carbon atoms. Weexplore five ϕ values for hexagonal configurations (Table 1). In

addition, a fully random, 10 grain, polycrystalline structure isgenerated with random grain centers and rotation angles(Figure 1(b)). To characterize the bulk GB characteristics of apolycrystalline graphene specimen, one parameter is theaverage GB energy of polycrystalline graphene, EGB, defined as

=−

EE E

hLGBpl pr

(2)

where Epl is the total energy of the polycrystalline graphene; Epris the total energy of the corresponding pristine graphene withthe same number of atoms; h is the nominal graphiteinterplanar distance (3.35 Å); and L is the total length ofGBs. Free surface contributions to the total energy areconsidered in the calculation. For comparing bulk thermalconductivities, the average GB energy is more appropriate thanindividual GB energies since there are numerous different GBsin a polycrystalline specimen. The structural and defect

characteristics of pristine and polycrystalline configurationsexplored in this work are summarized in Table 1. The totalnumbers of defects in different hexagonal-grained configu-rations are nearly the same owing to their same GB fractions.The random-grained polycrystalline graphene is more defectivegiven its larger GB fraction. The pentagon and heptagon defecttypes are dominant among different defects; the fractions ofthese two defect types in the hexagonal-grained structures aresimilar, indicating that the difference in their GB energies ismostly due to other defects.GB energy is a most useful and widely used method in

characterizing GBs. We also characterize polycrystallinegraphene with selected area electron diffraction (SAED)simulation.39 The scattered intensity I(k) is

= *I

F FN

kk k

( )( ) ( )

(3)

with the structure factor

∑ π= ·=

F f ik k r( ) exp(2 )j

N

j j1 (4)

Here k is the reciprocal space vector; r is the position of anatom in the real space; f is the atomic scattering factor forelectrons;40 and N is the number of atoms in the selected area/region. For a given wavelength λ, the diffraction angle θ and kare related via Bragg’s law41

θλ

= = | |d

k2 sin 1

hkl (5)

where d represents the d-spacing between (hkl) planes. We useλ = 0.025 Å (200 keV)40 in SAED simulations. Figure 2 shows

the SAED patterns: with increasing GB energy, the diffractionspots become broadened, and the Debye−Sherrer ringsbecomes less sparse showing increased randomness and“disordering”.In addition, the grain size effects are explored by scaling

down the grain sizes (or the area of a graphene sheet) listed inTable 1, while keeping the GB configurations and thus, EGB, thesame. A convenient parameter to characterize the average grainsize is GB fraction, ρGB = L/S, where S is the area of thegraphene sheet. For hexagonal grains, the grain size is reduced

Figure 1. Atomic configurations of polycrystalline graphene withhexagonal grains (a) and with grains of random shapes and sizes (b).Some structure details at the GB triple junctions in (a) are shown in(c) and (d). Color-coding refers to different grains (e.g., 1−4).

Table 1. Structural and Defect Characteristics of the Pristineand Polycrystalline Graphene Configurationsa

ϕ system size EGB (J/m2) Npent Nhept Noct Nother Ntotal

pristine 160000 0.00 0 0 0 0 05° 158422 4.60 109 111 61 44 32510° 158416 1.42 122 131 38 21 31215° 158429 2.11 123 129 41 21 31420° 158425 2.94 118 123 49 23 31325° 158412 3.85 112 117 59 29 317random 158183 6.52 156 167 112 94 529

aHere N denotes defect number; subscripts pent, hept, oct, and otherrefer to pentagon defects, heptagon defects, octagon defects, and othertypes of defects, respectively.

Figure 2. Calculated SAED patterns of the pristine graphene, thehexagonal-grained polycrystalline graphene, and the random-grainedpolycrystalline graphene simulated in our work. The scale bar refers tothe relative electron intensity. The electron energy is 200 keV.

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from about 30 to 10 and 5 nm; these values correspond to ρGB= 0.03, 0.09, and 0.18, respectively. For different polycrystallinegraphene specimens, their bulk thermal conductivities are ofdirect interest, so it is more suitable to use the parameters forcharacterizing their bulk properties, EGB and ρGB, than those forindividual GBs.The large-scale atomic/molecular massively parallel simulator

(LAMMPS) is used in our MD simulations.42 The atomicinteractions are defined by a widely used, optimized Tersoffpotential,43 which has been shown to improve the accuracy ofthermal calculations.28,44−46 Thermal conductivity is calculatedwith the Green−Kubo method47,48

∫κ =Ω

⟨ ⟩∞

k TJ J t t

1(0) ( ) di i

B2 0 (6)

where Ω = Sh is the system volume; kB is the Boltzmannconstant; T is temperature; Ji is the ith component of the heatflux (i = x, y); and t is time. The term within the angle bracketsdenotes the heat flux autocorrelation function (HFACF).The polycrystalline graphene configurations are relaxed with

the conjugate gradient method and then equilibrated with theconstant−volume−temperature ensemble and a Nose-Hooverthermostat at 300 K for 1 ns, followed by 0.5 ns thermalizationwith the microcanonical (NVE) ensemble. Periodic boundaryconditions are applied in the in-plane directions, and the out-of-plane dimension of the supercell is sufficiently large.27 Thenheat flux is recorded for autocorrelation during long NVE runs.Since the simulations are performed at discrete time steps of Δt= 0.5 fs, eq 6 can be rewritten as49

∑ ∑κ = ΔΩ

− +=

−

=

−tk T

N m J n J n m( ) ( ) ( )m

M

n

N m

i iB

21

1

1 (7)

where Ji(m + n) is the ith component of the heat flux at timestep n + m. The total number of integration steps M is set to besmaller than the total number of MD steps, N. The Green−Kubo method requires long time simulations to obtainconverged κ,49 and our simulation durations are 12 ns. EachHFACF is calculated every 120 ps to obtain a convergedthermal conductivity value. There are 100 conductivity valuescalculated for a 12 ns run, and the final κ value is obtained byaveraging the last 70 values.

■ RESULTS AND DISCUSSIONFigure 3 shows a typical HFACF (inset), thermal conductivitiesof consecutive, 100, 120 ps periods, and the average over thelast 70 periods, for pristine graphene containing 160 000 atoms.κ is about 2430 ± 237 W/mK, consistent with previouslyreported values for pristine graphene predicted with the Tersoffpotential23,27 and with other calculations and experiments(2000−5000 W/mK).8,50 Thermal conductivities for pristinegraphene and hexagonal- (5 configurations) and random-grained (1 configuration) polycrystalline graphene, with threedifferent grain sizes or GB fractions, are obtained from theGreen−Kubo method. Defects affect thermal transport at theatomistic scale, and the ensemble thermal transport propertiesare expected to be significantly influenced by GBs inpolycrystalline graphene.51 For given GB configurations orEGB, κ decreases with increasing GB fractions or decreasinggrain sizes, partly owing to increasing phonon scattering at GBs(Figure 4). Mortazavi reported similar results in random-grained polycrystalline graphene with grain size less than 5 nm(grain size is up to 25 nm in this work).52 For a given grain size

or GB fraction ρGB, κ decreases rapidly with increasing EGB, e.g.,for EGB < 3.1 J/m2, and it becomes “saturated” when the GBenergy is bigger than EGB = 6 J/m2 (Figure 5). However,thermal conductivity decreases with decreaing grain size lessrapidly (Figure 4). Thus, at sufficient high GB energies, thermalconductivity is more sensitive to grain size than GB energy.GBs impede thermal transport, consistent with a previous study

Figure 3. Thermal conductivity of pristine graphene as a function oftime during consecutive, 100, 120 ps periods (dashed curves). The redcurve represents the average of the 100 runs. Inset: integration ofHFACF over time.

Figure 4. Normalized thermal conductivity, κ = κ/κ0, of hexagonal-grained graphene as a function of GB fraction. κ0 refers to pristinegraphene. Numbers in the legend denote EGB.

Figure 5. κ as a function of EGB, for pristine graphene andpolycrystalline graphene with hexagonal grains and grains of randomshapes and sizes. The solid curves denote exponential fittings, andnumbers in the legend denote ρGB.

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suggesting that the phonon transmission is weakened byincreased GB energy in a twin boundary system.28 Thesimulation results in Figure 5 can be fitted with an exponentialfunction

κ κε

= + − ∞

⎧⎨⎩⎫⎬⎭a

Eexp GB

(8)

where the normalized thermal conductivity κ = κ/κ0; κ∞, a, andε are fitting parameters. κ∞ is the asymptotic value of thermalconductivity, and ε is the characteristic energy reflecting theoverall effect of different defects at GBs. ε can be regarded asphonon mobility, analogous to the Matthiessen’s rule53

ε ε ε ε ε= + + + + ···1 1 1 1 1

vacancies pentagons heptagons octagons

(9)

Here subscripts denote contributions of various defects to theinverse mobility. In other words, defects at GBs reduce theeffective mobility and thus thermal conductivity.κ∞ decreases with increasing GB fraction (or decreasing grain

size), and it is the opposite for ε. κ∞ is 0.24, 0.20, and 0.17, andε is 0.60, 0.87, and 1.17 J/m2, for ρGB = 0.03, 0.09, and 0.18,respectively. The values suggest that phonon scattering at GBsincreases with increasing GB fraction or decreasing grain sizes.Similarly, Khitun et al. reported that the in-plane thermalconductivity of a quantum-dot superlattice54,55 decreases withincreasing fraction of quantum dots. They described thephonon transport inside a quantum dot within a continuumapproximation and attributed the reduction to acoustic phononscattering. The quantum dots were of different sizes and shapes,analogous to GBs in our cases. Compared to the hexagonal-grained graphene, κ for the random-grained graphene with thesame system sizes is lower (below the fitted curves, Figure 5)due to the higher GB fractions. Thus, increasing GB energy andGB fraction both reduce thermal transport capability. However,for high EGB, the former becomes less effective, and GB fractionplays a more important role since the reduction in κ is morerapid and becomes saturated with increasing EGB, compared todecreasing grain size (cf. Figures 3 and 4).Thermal conductivity is closely related to phonons, in

particular acoustic phonons. We calculate phonon density ofstates (pDOS) for pristine and polycrystalline graphene (Figure6(a)) from additional NVE simulations, where the atomicvelocities are recorded every 2.5 fs for 100 ps, by the Fouriertransformation of time-dependent velocity autocorrelationfunction (VCAF)

∫ωπ

υ υ= ⟨ ⟩ ω∞

g t e t( )1

2(0) ( ) di t

0 (10)

Here ω is angular frequency; υ is velocity; and the term withthe angle brackets represents VACF. Since graphene is 2D, onlyin-plane, longitudinal/transverse phonons are considered.56

Figure 6(a) shows two main peaks in pDOS located around24 and 53 THz (the latter is also referred to as the G-band).Our results on pristine graphene are in accord with previouscalculations.51,57 The G-band of pristine graphene is located at52.2 THz and that of polycrystalline graphene at 50.9 THz.Therefore, the polycrystalline graphene shows a 2% red shift inpDOS compared to the pristine graphene, indicating phononsoftening and a reduction of phonon group velocities (υg).Thermal conductivity at a given temperature T is related to υg,heat capacity C, and phonon mean free path λ as53

∫κ ω υ λ ω ω=∞

C T T13

( , ) ( , )d0

g (11)

Thus, GB-induced reduction in υg contributes to the decreasein κ compared to pristine graphene. However, the difference inred shifts of the pDOS is not pronounced among all thepolycrystalline graphene. For further discussion, we introducethe integrated pDOS (Figure 6(b)) and the following equationfor determining thermal conductivity by polarization modes(longitudinal acoustic, LA, and transverse acoustic, TA,modes)28

∫κ τ ω ω ω ωω ω= ℏ

ω ⎛⎝⎜

⎞⎠⎟g

qf

T12

( ) ( )dd

d ( )d

d0

2

(12)

where q is wave vector, τ is relaxation time, and ℏ is thereduced Planck constant; the 1/2 factor is due to the 2Dnature. For scattering caused by defects, the relaxation time isrelated to the scattering cross-section σ and defect density ηas58

τησυ

= 1

g (13)

The integrated pDOS is shown in Figure 6(b). There are∼10% more phonons in the pristine graphene than thepolycrystalline graphene over the whole frequency range and3% more phonons in the hexagonal-grained graphene than therandom-grained graphene. The variations in the number ofphonons are consistent with those in κ. Meanwhile, increasingGB energy or GB fraction in polycrystalline graphene yieldsmore defects and grain boundaries (Table 1). From eq 10, thelarger scattering cross section σ and defect density η lead to thedecrease in phonon relaxation time τ and thus κ according to eq9.Thus, an increase in η and σ in polycrystalline graphene leads

to a decrease in τ and thus κ, even though there is no obviousdifference in υg. Furthermore, the phonon number reduction ofthe g(ω) peaks, in particular the acoustic phonons, decreasesthe number of heat-conducting phonons and thus heatconduction. In our simulations of polycrystalline graphene,the main contribution of the in-plane pDOS comes from theLA and TA modes around 24 THz (Figure 6(a)). The inset to

Figure 6. (a) Phonon DOS of the pristine and polycrystallinegraphene. The inset shows the details of the first pDOS peak. (b) Theintegrated pDOS. The inset shows integration of the first pDOS peak.

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Figure 6(a) (Figure 6(b)) shows that a higher pDOS peak(more phonon number) corresponds to a lower GB energy or asmaller GB fraction at the same frequency and, along with anincreased relaxation time τ, leads to higher conductivity as seenfrom eq 9.

■ CONCLUSIONS

We have investigated the thermal conductivity of polycrystallinegraphene, including its GB energy and grain size dependences.The configurations are generated by Voronoi tessellation withhexagonal and random-shaped grains, mimicking those in realapplications. The ensemble thermal conductivity is calculatedwith the Green−Kubo method, and its relations with GB size(GB fraction) and GB energy are quantified. Our resultsindicate that the thermal conductivity decreases with increasingGB fraction or GB energy. At sufficient high GB energies,thermal conductivity is more sensitive to grain size than GBenergy. Phonon densities of states are computed for pristineand polycrystalline graphene. Phonon softening and scatteringand reduction in the number of heat-conducting phononscontribute to decreasing heat conductivity with increasing GBenergy or decreasing GB size.

■ AUTHOR INFORMATION

Corresponding Authors*E-mail: linyuan@uestc.edu.cn*E-mail: sluo@pims.ac.cn.

NotesThe authors declare no competing financial interest.

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The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp508035b | J. Phys. Chem. C 2014, 118, 24797−2480224802