Structural, Vibrational, and Thermal Properties of Nanocrystalline

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Structural, Vibrational and Thermal Properties of Nanocrystalline Graphene in Atomistic Simulations Konstanze R. Hahn, Claudio Melis, and Luciano Colombo J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b11556 • Publication Date (Web): 13 Jan 2016 Downloaded from http://pubs.acs.org on January 26, 2016

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The Journal of Physical Chemistry

Structural, vibrational and thermal properties of nanocrystalline graphene in atomistic simulations Konstanze R. Hahn,∗ Claudio Melis, and Luciano Colombo Department of Physics, University of Cagliari, Cittadella Universitaria, I-09042 Monserrato (CA), Italy E-mail: [email protected]

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Abstract Two different methods have been applied to create differently structured nanocrystalline graphene samples used in molecular dynamics simulations. In the first method, graphene sheets are generated by grain growth from individual nucleation seeds. The second method applies Voronoi tessellation to define single crystalline domains in the simulation cells. The differently generated nanocrystalline graphene sheets show significant variations in the grain size distribution and the shape of the crystalline domains. Furthermore, out-of-plane corrugation is found to be more pronounced in samples generated by the Voronoi method for small grain sizes (≤15 nm). Marginal differences are observed in the distribution of polygonal rings in the grain boundaries which might result from the geometrical shape of the grain boundaries. Thermal conductivity has been determined using the approach-to-equilibrium molecular dynamics formalism. A lower thermal conductivity is observed in Voronoi-samples for grain sizes between 5 and 15 nm which is attributed to the stronger out-of-plane corrugation.

Introduction Common fabrication methods of monolayer graphene include epitaxial film growth, 1,2 thermal sublimation of silicon carbide 3 and reduction of graphene oxide. 4 Probably the most widely used method for graphene production is chemical vapor deposition (CVD) owing to its simplicity, the relatively low costs and the ability to be used in large-scale production. 5–8 In all methods, defects are introduced in the crystalline structure resulting from limitations of the kinetic properties in the growth process and defects of the substrates. The generation of grain boundaries and defects in such polycrystalline graphene can drastically change mechanical and electronic properties of the system. 9–11 The polycrystalline nature with dimensions down to the micro- and nanoscale is also supposed to be responsible for the wide range of values of the thermal conductivity measured experimentally in graphene (600 to 10000 W/mK 12–15 ). Regarding the effect of environmental parameters, a significant 2 ACS Paragon Plus Environment

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difference in the thermal conductivity has been found for suspended (∼1600 W/mK) 16 and SiO2 supported (600 W/mK) 13 graphene. Furthermore, thermal conductivity of CVD produced graphene has been shown to be reduced by ∼30% when wrinkles are introduced in the structure. 17 Considering the significant effect of graphene morphology on its technologically important properties, detailed investigation of grain boundaries and the polycrystalline structure in graphene is essential to be able to control its electrical, thermal and mechanical properties. Several experimental, 18–22 theoretical 23,24 and combined 25,26 studies have been performed on the investigation of grain boundaries in graphene. Early studies observed tilted grain boundaries using scanning tunneling microscopy (STM). 19 Using transmission electron microscopy (TEM) it has been possible to characterize the pentagon-heptagon pattern at graphene grain boundaries experimentally. 21,27 Using STM, Graphene grown on Ni(111) has been shown to form straight lines of grain boundaries with defect patterns consisting of octagon-heptagon pairs. 28 Combined density functional theory (DFT) and molecular dynamics (MD) simulations showed increased strength of graphene with increasing misorientation angle in symmetric grain boundaries. 29 The increased strength, however, relies rather on the arrangement and density of defects that are created at the interface than on the misorientation angle itself as has been shown by MD simulations. 30 Furthermore, hybrid MD simulations have revealed increased out-of-plane corrugation of carbon atoms with increasing misorientation angles. 23 In another study based on MD simulations, the out-of-plane corrugation has been correlated to the thermal conductivity in such systems, suggesting increased scattering of flexural phonons with enhanced out-of-plane corrugation. 31 So far, comparatively smaller effort has been devoted to the systematic investigation of the effect of size, shape and distribution of grains on graphene properties and, in particular, on its thermal conductivity. Tuning the process conditions, single-crystalline domains of a size in the centimetre-scale have been successfully generated. 32 Typical sizes of so-produced



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polycrystalline grains, however, are in the order of micrometers. 21 Recently, it has been shown how the grain growth and thus the particle shape and size can be controlled by the process temperature in the CVD production process. 33 The shape of crystalline domains can further be controlled by the pyrolysis rate 34 and the concentration of the process gas. 35 Triggering these process conditions, flower-like, hexagonal and round-shaped graphene grains have been produced. 34,35 Applying atomistic simulations it has been shown recently that the mechanical strength of graphene is reduced by 50% when grain boundaries in form of polycrystals are introduced. 6 The thermal conductivity of nanocrystalline graphene generated by grain growth, has been estimated using approach-to-equilibrium molecular dynamics (AEMD) simulations proving a strong dependence of the thermal conductivity on the grain size. 36 Other studies have applied equilibrium MD simulations and the Green-Kubo expression for the investigation of the thermal conductivity in nanocrystalline graphene. 37,38 Polycrystalline graphene samples in these works have been generated by different approaches. Mortazavi et al. generated nanocrystalline samples by grain growth 37 while Liu et al. used Voronoi tessellation. 38 The two studies focused on the investigation of different aspects. A direct comparison of the two is thus problematic. In this work, we have generated nanocrystalline graphene with grain sizes up to 20 nm using two different methods. In the first, nanocrystalline graphene has been generated where grains have been grown by an iterative algorithm (bottom-up). The other one uses the Voronoi tessellation scheme to generate a pattern of crystalline domains which have been cut-out of a single crystalline sample cell (top-down). The structure of the generated simulation domains has been determined based on the particle size distribution, the radius of gyration, the shape of the grains and the out-of-plane buckling of the whole graphene sheet. In addition, the thermal conductivity of the nanocrystalline graphene generated by the two different methods is investigated applying AEMD simulations. 39


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Methods Creation of nanocrystalline graphene In analogy to experiments, generation of nanostructured systems for theoretical investigations can be realized either by bottom-up or top-down methods. In top-down methods, first, homogeneous amorphous or crystalline simulation cells are created which are then modified by scissoring and rearrangement of partially homogeneous sections to create nanostructured materials. In contrast, in bottom-up methods, nanostructures are constructed from single atoms or small building blocks (for example crystalline seeds) and simulation techniques such as MD are used to grow the nanostructures from the initial seeds. Here, simulation cells have been generated using both methods. The grain growth method is a bottom-up approach while the Voronoi method can be regarded as a top-down approach.

Grain growth method For the grain growth method, an algorithm has been developed where, for a given rectangular simulation cell with width Lx and length Lz , a specified number of nucleation sites has been placed at random positions in the cell with the constraint that neighboring seed atoms have a certain distance between each other. The number of nucleation sites determines the dimension of the generated grains. Here, densities from 0.32 to 3.7·10−4 seeds/nm2 have been used to obtain grain sizes from 0.7 to 20 nm. An arbitrary angle between 0 and 60◦ has been assigned to each nucleation site to define the orientation of the crystalline growth of graphene. Under-coordinated atoms on the edge of each grain have been added to a list of possible reactive sites. In an iterative process, the reactive site has been randomly selected from this list to which the next atom has been added. The list of reactive atoms has been updated in each step until no reactive atoms were left. Grain growth has been terminated, i.e. atoms have been eliminated from the list of reactive atoms, when the respective C atom was three-fold coordinated or atoms from



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neighboring cells were closer than 95% of the equilibrium C-C bond length (here: 1.3978 ˚ A). The method has been described in more detail previously. 36 A section of a sample cell is shown in Figure 1a.

Figure 1: Section of nanocrystalline graphene with average radius of gyration of 7.7 nm generated by (a) grain growth and (b) the Voronoi method. Red-colored grain boundary atoms have been identified based on their morphological properties (C atoms of non-hexagonal rings).

Voronoi method First the cell dimensions and randomly chosen seed sites have been defined. The simulations cells have been divided into regions based on the seed positions using the Voronoi tessellation method. The Voronoi formalism defines a region Rp associated with the seed sp according to Rp = {x ∈ R2 | δ (x, sp ) ≤ δ (x, sj ) for all j 6= p},

(1)

where x is the position in the simulation cell, sj the center of the other seeds and δ is the euclidean distance. Each of the so-defined regions has been cut-out from a template cell of crystalline graphene and rotated by a randomly assigned angle. A section of a sample cell is shown in Figure 1b. At least two different configurations have been created for each sample dimension and grain size. The average radius of gyration (definition see below) has been changed from 1 nm to 20 nm using samples with a cell length of 200 nm. 6 ACS Paragon Plus Environment

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Creation of simulation cells by the two methods described above only considers geometrical aspects. Interatomic interactions are neglected. Furthermore, simulation cells are created only in two dimensions, omitting corrugation in out-of-plane direction. To account for the latter, sample cells have been relaxed during MD simulations at elevated temperatures as described below.

Morphology analysis The grain size in nanocrystalline graphene has been characterized by the number of atoms per grain, by a hypothetical radius assuming a circular shape and by the radius of gyration. From the number of atoms nC the area AG which is occupied by the grain can be calculated according to 3√ 3 · d2C · nC , 4

AG =

(2)

where dC is the equilibrium bond length between C atoms in the crystalline cell. Assuming a circular shape of the grains, the ideal radius is calculated from the area of the grain q (rid = AπG ). Furthermore, the grain size has been characterized by the radius of gyration rG,j . The radius of gyration includes information on the moment of inertia and gives thus an indication of the shape of the grains. It has been determined for each seed j according to

2 rG,j

PnC,j =

i

(~ri,j − ~rCM,j )2 , nC,j

(3)

where ~ri,j is the position of atom i in grain j and rCM,j and nC,j are the center of mass and the number of atoms in the grain, respectively. To get more information of the shape of the grains, a shape factor has been defined which represents the ratio of the longest to the shortest distance of peripheral C atoms to the center of mass of the grain. Grain boundary atoms have been identified by their structural properties. All C atoms that do not belong to a hexagonal ring have been selected by this method. The distribution 7 ACS Paragon Plus Environment


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of polygonal rings in the entire simulation cell has been determined by iteratively following the nearest and next-nearest neighbor positions of each atom.

AEMD simulations Velocity rescaling with a time step of 0.5 fs has been applied for the equilibration process of the flat (2D) and unrelaxed simulation cells. It has been initiated at a temperature of 300 K for 10 ps. Subsequently, the cells have been heated to a temperature of 4000 K with a heating rate of 105 K/ps (in 35 ps) and relaxed at this temperature for 255 ps. The cells have been cooled back to 300 K with a cooling rate of 10 K/ps (in 370 ps). This procedure has been followed by a canonical run for 200 ps. Molecular dynamics simulations have been performed using the lammps code 40,41 and covalent interactions between carbon atoms have been described by the second-generation reactive empirical bond order (REBO) potential which has previously been shown to give reasonable results for graphene-based systems. 36,42,43 The thermal conductivity has been determined based on the AEMD methodology. In this formalism, the simulation cell is firstly divided into two regions with equal length Lz /2. One of these two compartments is equilibrated at a high temperature (Th =400 K), the other compartment at a low temperature (Tc =200 K) by velocity rescaling. This creates an initial step-like function of the temperature along the sample length in z-direction. 39 Next, the evolution of the average temperature in the hot (Th ) and cold (Tc ) reservoir is recorded during a transient regime towards equilibrium of microcanonical evolution. Based on Fourier’s theorem of thermal transport and the given step-like initial temperature profile, P α2n κ ¯t the evolution of the temperature gradient (∆T = Th − Tc ) follows ∆T (t) = ∞ , n=1 Cn e where κ ¯=

κ ρcv

is the thermal diffusivity with the density ρ of the material and its specific

heat cv . This expression is fitted to the temperature gradient obtained from the simulations to determine κ. More details on the AEMD methodology can be found elsewhere. 39


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Results and Discussion Structure of nanograins Figure 1 shows a section of the nanograins in graphene sheets generated by the two different methods (grain growth and Voronoi) described above. Illustration of grain boundary atoms indicates that the boundaries in graphene sheets generated by the grain growth are curved (serpent-like) while the boundaries in Voronoi-generated sheets form rather straight lines. Recently, straight grain boundaries have been generated experimentally in graphene grown on Ni(111). 28 Such grain boundaries have been found to consist mostly of a combination of pentagonal and octagonal C-rings. A detailed analysis of the structural properties of the grains in graphene sheets generated by the two methods has been done based on the number of atoms in each grain, the radius of gyration and a shape factor which gives an indication of the shape of grains. From the number of atoms, the area (AG ) of each grain can be calculated. From this and assuming a perfect circular shape of the grains, it is possible to calculate the ”ideal” radius of the grains (as described above). The radius of gyration (Eq. 3) has been used here to describe the grain size distribution. In nanocrystalline graphene generated with the grain growth method, grain size distribution follows a normal distribution (Figure 2a) according to (x−µr )2 1 e− 2σ2 . f (x) = √ 2πσ

(4)

The spread of grain sizes in cells generated by the Voronoi method is remarkably larger and the approximation by a normal distribution is arguable (Figure 2b). The distribution of grain sizes is thus analyzed by the standard deviation σgsd of the arithmetic mean. To better characterize the behavior of the distribution with increasing radius of gyration, the standard deviation has been expressed with respect to arithmetic mean radius of gyration



0.7

a

c

voronoi growth

0.6 0.5

σgsd /rG [-]

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0.4 0.3 0.2

b

0.1 5

10

15

20

rG [nm]

Figure 2: Grain size distribution in nanocrystalline graphene, generated by (a) grain growth and (b) the Voronoi method in cells with rG =2.5 nm. (c) Relative standard deviation of grain size in grain-growth (triangles) and Voronoi (diamonds) samples as a function of the average dimension of the grains (rG ). rG according to s 1 σgsd = rG rG

Pn

i=1

(x − rG )2 . n

In the grown cells, the relative standard deviation (

σgsd ) rG

(5) fluctuates between 0.16 and

0.35 with an average value of 0.25 (Figure 2c). Both the average value and the fluctuations are significantly larger in cells generated by the Voronoi method where relative standard deviations are found between 0.32 and 0.66. This demonstrates a remarkable variety in the grain dimensions in simulations cells that have been generated by the Voronoi method. The structure of nanocrystalline grains in the generated graphene sheets has been further characterized by their shape. A shape factor has been defined which represents the ratio between the longest and the shortest distance in a grain from its center of mass (Figure 3). Generation of nanocrystalline graphene using the grain growth method results in grains that are nearly circular. The average value of the shape factor has been calculated to be 1.7±0.1. In contrast, the Voronoi method yields nanocrystalline graphene samples with a great variety 10 ACS Paragon Plus Environment

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of grain shapes (Figure 3). The shape factor fluctuates significantly around the average value of 2.5±0.6. 4 voronoi growth

shape factor [-]

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shape factor =

3

2

1 5

10

15

20

rG [nm]

Figure 3: Shape factor in nanocrystalline graphene with varying average radius of gyration rG . It is apparent that the shape of grains generated by the Voronoi method (green) is less homogeneous than systems generated by the grain growth method (red). The colored lines and regions indicate the average value of the shape factor and its standard deviation, respectively. The shape factor is defined as the ratio between the longest and the shortest distance of atoms from the center of mass of the grain they belong to (right panel).

Both the grain size distribution and the behavior of the shape factor demonstrate large irregularities and a notable diversity in the morphology of the crystalline grains generated by the Voronoi method. Generation of nanocrystalline graphene by grain growth on the other hand leads to cells with a homogeneous distribution of grain sizes and shapes that are respectively close to a circular shape. It is proposed that the two different generation methods presented here can be applied and further adapted to generate graphene sheets with different properties similar to experimentally produced graphene.

Out-of-plane corrugation Recently, MD simulations have been applied to study the effect of out-of-plane buckling at grain boundaries in graphene on the phonon transport properties. 31 A larger degree of out-of-plane buckling has been shown to increase the scattering of long-wavelength phonons 11 ACS Paragon Plus Environment


in the grain boundary possibly leading to a decrease in thermal conductivity. In the latter study, the out-of-plane buckling has been analyzed directly at the grain boundary. Here, the corrugation of atoms in the entire simulation cell has been considered for the analysis of the out-of-plane buckling. In the present simulations, the generated graphene sheets are initially perfectly flat. During the equilibration process, the graphene sheets become corrugated in out-of-plane direction (Figure 4b). The corrugation has been defined as the dislocation of C atoms perpendicular to the 2D graphene plane (average position of all atoms). From the arithmetic mean and the standard deviation, a standard normal distribution can be extrapolated (Figure 4a).

a

growth voronoi

b

5 nm

population

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c 14 nm

20 nm -1

-0.5

0

0.5

1

out-of-plane corrugation [nm]

Figure 4: (a) Corrugation in out-of-plane direction of nanocrystalline graphene sheets in simulation cells with rG =5, 14 and 20 nm. (b) Section of a corrugated nanocrystalline graphene sheet (rG =10 nm generated by grain growth) where boundary atoms are indicated in red. (c) Corrugation of all atoms as a function of their position in out-of-plane direction showing a sinusoidal deformation (rG =14 nm generated by grain growth).

In some simulation cells with larger grain sizes (rG > 8 nm), a bimodal distribution has been observed. This results from a sinusoidal deformation of the graphene sheet which is periodic in the cell length Lz . To eliminate the effects of this long-period deformation, the


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Table 1: Out-of-plane corrugation defined as the FWHM value of the distribution of atom positions in out-of-plane direction for samples shown in Figure 4a.

rG [nm] 5 14 20

growth FWHM [nm] κ [W/mK] 0.085 0.181 0.257

84.9 151.8 151.7

Voronoi FWHM [nm] κ [W/mK] 0.259 0.346 0.356

86.1 129.2 146.2

dislocation of C atoms has been calculated with respect to this sinusoidal function which has been fitted to the out-of-plane position of all atoms in the simulation cell (Figure 4c). From the standard deviation the full-width half maximum (FWHM) has been calculated and is compared in Table 1 for the two generation methods and for different grain sizes. For grown cells it increases from 0.09 to 0.26 nm with growing grain size from 5 to 20 nm indicating enhanced buckling in nanocrystalline graphene with larger grain sizes. Graphene samples generated by the Voronoi method show similar effects. However, the buckling is less affected by the grain size. The FWHM value increases from 0.26 to 0.36 nm for increasing cell size from 5 to 20 nm. For small grain sizes (≤ 14 nm), buckling is significantly larger in the Voronoi samples than in the samples with grown cells (Figure 4a, Table 1). In addition, the out-of-plane corrugation has been quantified by the root-mean-square displacement (RMS) from the average position of all atoms or from the sinusoidal function in cases where long-period deformation was observed. Figure 5 shows the RMS as a function of the grain size. At all grain sizes, corrugation in the Voronoi samples is more pronounced with respect to the grain growth samples. Furthermore it is observed that corrugation tends to increase with increasing grain size in particular in the grain growth samples. The higher corrugation in Voronoi samples can possibly result from the fact that grains are tailored to the desired nanosize with the Voronoi method, imposing artificial constraints (C atoms respectively close or far from each other) in the boundary regions rather than naturally grown from nucleation seeds as in the grain growth method. 13 ACS Paragon Plus Environment


1.6 1.4 1.2 RMS [nm]

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1 0.8 0.6 growth voronoi

0.4 0.2 0

5

10

15

20

rG [nm]

Figure 5: Root-mean-square displacement of C atoms from the plane of the graphene sheet as a function of the grain size (rG ) in Voronoi (diamonds) and grain growth (triangles) samples.

Grain boundary atoms The concentration of boundary atoms can have a significant effect on the thermal transport in nanocrystalline graphene. At a specific grain size, it gives an indication on the boundary thickness. As described above grain boundary atoms have been identified as atoms that do not belong to a hexagonal ring. In addition, the distribution of rings in the grain boundaries has been characterized (Figure 6c for rG =5, 14 and 20 nm). The most abundant non-hexagonal rings are pentagons and heptagons for both generation methods (Figure 6c) in agreement with numerous studies discussing pentagon-heptagon patterns of grain boundaries in polycrystalline graphene. 21,23,24,27,44 Nevertheless, a small difference between the fraction of pentagons and heptagons is notable with a slightly smaller number of heptagons in all structures. This is compensated by octagonal and higher-numbered rings that are additionally observed in grain boundaries of polycrystalline graphene. 6,28 The distribution of polygons in nanocrystalline graphene is similar for both generation methods. For larger grains (rG ≥14 nm), however, a slight increase of octagonal rings is observed in


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a

growth

c

0.5

voronoi growth

0.4

5 nm

0.3 0.2 0.1 0 0.4

14 nm

0.3 0.2

b

voronoi

0.1

population

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0 0.4

20 nm

0.3 0.2 0.1

4

5

7

8

9

0

ring size

Figure 6: Section of nanocrystalline simulations cells generated by (a) grain growth and (b) the Voronoi method showing the boundary atoms (in red). (c) Distribution of polygonal rings in simulation cells with rG of 5, 14 and 20 nm. samples generated by the Voronoi method. This could possibly result from the shape of the grain boundary which rather forms a straight line in the Voronoi-samples but is more curved and serpent-like in the grown samples. Interestingly enough, the existence of pentagonoctagon patterns in straight grain boundaries of polycrystalline graphene has been proven experimentally. 28 For a rough estimation of the boundary thickness, the nanograins have been approximated by a circular shape. With this assumption, the boundary thickness can be approximated from the concentration of boundary atoms cGB according to

cGB =

t AGB =2 , AG rid


(6)


where AGB is the area of the grain boundary and AG is the area of the entire grain. Using the data of both generation methods, a boundary thickness of 1.94 ˚ A is estimated (Figure 7). This is about 20% larger than the spacial extension of one atomic layer in the simulation set-up used here (1.6 ˚ A) indicating an average boundary thickness of approximately 1.2 atomic layers. As shown before, in particular the shape of grains generated by the Voronoi method are rather long than circular (Figure 3). For non-circular shapes, the value of t overestimates the actual boundary thickness. In fact, t is found to be slightly higher (1.96 ˚ A) if only datapoints from the Voronoi method are used than using datapoints only of the grain growth (1.93 ˚ A). However, the effect is marginal. It is thus proposed that the boundary thickness in differently created graphene sheets is similar. 50 growth voronoi 40

cGB [%]

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30

rid

t

20

10

0 0

5

10

15

20

25

30

rid [nm] Figure 7: Concentration of grain boundary atoms cGB as a function of the ideal radius rid . Assuming ideal circular shape of the grains, the boundary thickness t can be estimated from cGB = r2·t and resulted in t = 1.9 ˚ A. id

Recently, a combined experimental and theoretical study discovered bimodal phonon scattering graphene grain boundaries resulting from boundary roughness scattering and scattering in the disordered region of the grain boundary. 45 Boundary roughness scattering has 16 ACS Paragon Plus Environment

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been found to dominate the thermal resistance for small mismatch angles with grain boundaries up to 0.25 nm. With the estimated grain boundary thickness of 0.19 nm of the samples created here, boundary roughness scattering is supposed to be the dominating scattering mechanism.

Vibrational density of states The vibrational density of states (VDOS) has been calculated as the Fourier transform of the autocorrelation function of the atomic velocities collected during a canonical simulation at 300 K for 10 ps. Figure 8 shows the VDOS for the two polycrystalline systems compared to crystalline graphene. The acoustic phonon spectrum (0-40 THz, Figure 8b) remains unaffected by the changes grain sizes from 5 to 20 nm. In addition, no difference is observed between cells generated by the voronoi or the grain growth method. Considering that the main contributors to thermal conductivity are acoustic phonons, similar thermal transport properties are thus expected for the two nanocrystalline systems. Optical phonons, on the other hand, are affected both by the generation method and the grain size. For small crystalline domains (rG =5 nm), a red shift of the main peak is observed (Figure 8c). The main peak at 53 THz in crystalline graphene can be attributed to the so-called G peak which results from bond stretching of sp2 atoms in rings and chains. 46 It has been shown previously that strain alters the position of the G peak in 2D graphene sheets. 47–50 In polycrystalline graphene both tensile and shear strain lead to phonon softening which is reflected in a red shift of the vibrational G peak. In the polycrystalline graphene samples generated here the atom density is slightly lower compared to single crystalline graphene (39.40 atoms/nm2 ) indicating tensile strain in the polycrystalline samples. The density in samples generated by the Voronoi method is marginally lower and similar for all grain sizes (39.39 atoms/nm2 ). In samples generated by the grain growth method, on the other hand, the density is significantly lower in particular in samples with small grains. The average density in samples with grain sizes up to rG =5 nm 17 ACS Paragon Plus Environment


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is 38.18 atoms/nm2 while for larger grains (rG >5 nm) it is with 39.23 atoms/nm2 similar to the density in samples generated by the Voronoi method. This explains the red shift of the G peak in the polycrystalline graphene samples which is more prominent in simulation cells generated by grain growth where the main peak appears at 52 THz in samples with rG =5 nm. In graphene sheets generated by the Voronoi method the peak appears at 52.7 THz. With increasing grain size, the red shift is reduced and the peaks in the differently generated cells move closer together. Furthermore, two additional peaks can be distinguished at higher and lower frequencies with respect to the main G band. They become more pronounced at larger grain sizes (rG =14 and 20 nm). Previously, two G band features have been observed in the Raman spectrum of graphene nanoribbons and their intensity ratio has been shown to be dependent on the light polarization. 51 The dependency results from the optical anisotropy of graphene. For the samples studied here, this can possibly be attributed to the different orientations of neighboring crystalline grains.

Thermal conductivity Grain size effects on the thermal conductivity in nanocrystalline graphene have been investigated in simulation cells with a sample length Lz =200 nm. Estimation of the thermal conductivity by MD simulations at a certain cell length Lz in principle underestimates the actual bulk thermal conductivity if Lz is much smaller than the maximum mean free path of phonons. To reliably estimate the bulk thermal conductivity several sample lengths would therefore have to be simulated from which the bulk thermal conductivity can then be estimated. This can be computationally very expensive. For cell sizes where Lz is larger than the largest phonon mean free path, however, the length effect is much less dramatic or, even, at all negligible. 52 The average mean free path of phonons in nanocrystalline graphene with grain sizes of 1 and 2.5 nm has been found to be 30 and 33 nm, respectively. 36 Therefore, in the interest of computational cost and time, simulations have been limited to only one spe18 ACS Paragon Plus Environment

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cific cell length and the resulting thermal conductivity is referred to the thermal conductivity of single crystalline graphene at the identical sample length Lz =200 nm (262 W/mK). Figure 9 shows the relative thermal conductivity of nanocrystalline graphene generated by grain growth and the Voronoi method as a function of the grain size (rG ). It increases continuously with increasing grain size for both methods. Despite the identical general trend, small differences in the thermal conductivity are observed for grain sizes between 5 and 15 nm. In this region, nanocrystalline graphene with grains generated by the grain growth method shows a higher thermal conductivity than the Voronoi samples. A possible explanation for this difference is the higher degree of corrugation that is observed for grain sizes ≤14 nm in samples generated by the Voronoi method (Figure 4a). Previously, MD simulations have evidenced enhanced scattering of long-wavelength flexural phonons at grain boundaries with larger out-of-plane buckling. 31 The out-of-plane corrugation is observed to be even larger at smaller grain sizes in the Voronoi-produced samples (Figure 4a, rG = 5 nm, Table 1). In nanocrystalline materials, however, the phonon spectrum is reduced to the dimension of the crystalline domains, and it is expected that the dimension of the crystalline domains for small grain sizes (≤ 5 nm) is the limiting factor for the thermal transport. Therefore, no difference in the thermal conductivity is observed between the two generation methods for small grains. The effect of increased phonon scattering of long-wavelength phonons at corrugated grain boundaries is only observed at larger grain sizes (≥5 nm). Grain boundaries can be regarded as a connection in series of resistances. 36,37 This argumentation leads to a rational function to describe the κ versus rG behavior according to κc,∞ κ (rG ) = . κc,∞ κc 1 + RGB 2rG

(7)

The parameters κc,∞ and RGB have been optimized separately for the two generation methods. As a result of the lower thermal conductivity for grain sizes between 5 and 15 nm in 19 ACS Paragon Plus Environment


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Table 2: Parameters fitted to Eq. 7 and 8 for the thermal conductivity in nanocrystalline graphene as a function of rG and cGB , respectively.

generation f (rG ) f (cGB ) method κc,∞ [-] RGB2κc,∞ [nm] κc,∞ [-] RGBdκl c,∞ [-] growth voronoi both

0.925 0.783 0.858

9.29 7.67 8.52

0.976 0.867 0.937

33.3 29.1 31.9

the Voronoi samples, the relative ideal thermal conductivity in crystalline graphene (κc,∞ ) has been determined to be lower (0.78) than for cells generated by grain growth (0.92). With both methods and the assumptions of Eq. 7, the relative thermal conductivity should converge to 1 for infinitely large grains. For grown grains, the optimized value (0.92) is reasonably close. In Eq. 7, a stepwise uniform thermal conductivity is assumed. With the enhanced scattering of flexural phonons at corrugated grain boundaries, however, additional scattering effects are present which might lead to a non-linear behavior of the thermal conductivity. Therefore a lower value of κc,∞ is found for the Voronoi-patterned graphene cells. The rough estimation of the thermal boundary resistance by the optimized parameter RGB resulted in 7.7·10−11 m2 K/W and 8.1 ·10−11 m2 K/W for the grain growth and the Voronoi method, respectively. This result suggests that the topology of grains only marginally affects thermal resistance properties. The thermal conductivity has furthermore been analyzed with respect to the concentration of boundary atoms. With a similar argumentation as for the grain size, the thermal conductivity as a function of the boundary atom concentration can be described according to κ κc,∞ (cGB ) = GB κc 1 + κc,∞d·R cGB l

(8)

with the boundary thickness dl of 1.2 atomic layers of graphene (0.19 nm). The behavior of κ versus cGB is similar for both methods without any notable difference. Therefore, the


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parameters κc,∞ and RGB have been fitted to all datapoints resulting in 0.94 and 2.5 ·10−11 m2 K/W, respectively. Using the concentration of grain boundary atoms as variable for the thermal conductivity according to Eq. 8 approximates the grain size from the concentration of boundary atoms and for a fixed boundary thickness. It can give a rather good estimation of κ but the rough approximation of the grain size leads to a notable underestimation of the thermal resistance RGB .

Conclusions Two different methods have been used to generate nanocrystalline graphene sheets. In the bottom-up approach, nanocrystalline graphene grains have been grown until the entire simulation cell was filled with atoms. In the top-down approach, nanocrystalline graphene has been created using Voronoi tessellation to cut-out crystalline domains and rearrange them. The systems have been analyzed in detail for structural features including the particle size distribution, the shape of the nanograins, the out-of-plane-buckling and the distribution of non-hexagonal rings in the grain boundaries. The size distribution of grains in nanocrystalline graphene generated by grain growth can be described by a normal distribution. In graphene sheets generated by the Voronoi method, the grain size distribution is significantly larger indicating an inhomogeneous distribution of grain sizes. Remarkable differences are also observed in the shape of grains. The grain growth method generates grains with a homogeneous, nearly circular shape while grains generated by the Voronoi method show significant variation in the shape factor (ratio between longest and shortest radius in a grain). The concentration of grain boundary atoms as a function of the grain size has been observed to be similar for both generation methods indicating a similar boundary thickness. Analysis of the polygons in the grain boundaries revealed a similar number of pentagons and heptagons. However, the number of heptagons is slightly lower. This is a result of



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octagons and rings with more members additionally present in the grain boundaries. Samples generated by the Voronoi method with grain sizes larger than 10 nm indicate slightly more octagons. This could result from the shape of the grain boundaries which form rather straight lines. The VDOS is almost identical for acoustic phonons (

Structural, Vibrational, and Thermal Properties of Nanocrystalline

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