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Structure-dependent mechanical properties of extended beta-graphyne
Accepted Manuscript Structure-dependent mechanical properties of extended β-graphyne
Jiaxing Qu, Hongwei Zhang, Jianxin Li, Shexu Zhao, Tienchong Chang PII:
S0008-6223(17)30500-6
DOI:
10.1016/j.carbon.2017.05.051
Reference:
CARBON 12027
To appear in:
Carbon
Received Date:
22 February 2017
Revised Date:
18 April 2017
Accepted Date:
14 May 2017
Please cite this article as: Jiaxing Qu, Hongwei Zhang, Jianxin Li, Shexu Zhao, Tienchong Chang, Structure-dependent mechanical properties of extended β-graphyne, Carbon (2017), doi: 10.1016/j. carbon.2017.05.051
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ACCEPTED MANUSCRIPT Structure-dependent mechanical properties of extended -graphyne Jiaxing Qu1 • Hongwei Zhang1 • Jianxin Li1 • Shexu Zhao1 • Tienchong Chang1,2,*
1State
Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil
Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
2Shanghai
Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of
Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China
Abstract Owing to their remarkable electronic and thermal properties, graphynes have been considered new promising carbon materials after graphene. However, our understanding of their structure-dependent mechanical properties is far from complete. In this paper an analytical molecular mechanics model is proposed to mathematically establish a relationship between the structure and the elastic properties of graphyne. Extensive molecular dynamics simulations are performed for comparison. We show that the elastic properties of graphyne exhibit a strong dependence on its structure. The in-plane stiffness, in-plane shear stiffness and layer modulus decrease with increasing percentage of the acetylenic linkages, while the Poisson’s ratio increases. Further analysis demonstrates that this dependence of structure attributes to the change in bond density. Based on the concept of effective bond density, scaling laws are developed for the elastic properties of graphyne, which demonstrate a useful approach for linking mechanical properties of graphene allotropes to those of graphene.
*
1
Corresponding author. Tel: 86(21)56331208. Email: [email protected].
ACCEPTED MANUSCRIPT
1. Introduction Carbon exhibits versatile flexibility in forming different hybridization states (i.e., sp, sp2, sp3). Apart from some naturally existed ones, such as graphite (sp2hybridize) and diamond (sp3-hybridize), many other carbon allotropes are possible to be constructed by altering the periodic binding motifs in the networks consisting of sp3-, sp2- and sp-hybridized carbon atoms [1-14]. In particular, graphene, a sp2-hybridized carbon allotrope, exhibits extraordinary electronic properties [15,16] such as the quantum Hall effect and giant carrier mobility, as well as excellent optical, thermal, and mechanical performance, making graphene an attractive building material for nano devices. This sp2-hybridized carbon allotrope inspired the combination of carbon allotropes with more hybridized states. With the introduction of sp constituents, Baughman et al. [17] postulated that graphynes can be alternative carbon planar forms constructed by partial or complete replacement of the aromatic bonds of graphene with the acetylenic chains (single- and triple-bond) [18]. Based on different ratio of sp constituents, graphyne can have unlimited geometrical configurations, among which four typical structures, i.e.,
graphyne, graphyne, graphyne and 6, 6, 12-graphyne have attracted much attention recently [17-20]. Graphynes extend some novel properties beyond those of primitive graphene [21-30] and have been proposed to be used as energy storage [31], anode [32], gas separation [33], and water desalination materials [34]. Among graphyne family, 2
ACCEPTED MANUSCRIPT graphyne (Fig. 1) exhibits outstanding properties. Malko et al. [25] predicted that Dirac cones and their associated electronic transport exist in graphyne, just like in graphene. Ouyang et al. [35] found that the zigzag -graphyne nanoribbons possess superior thermoelectric performance with a thermoelectric figure of merit (ZT) of 0.51.5 at room temperature, at least one order higher than that of graphene. Despite these studies, the understanding of mechanical properties of graphyne is very limited. The insertion of acetylenic linkages is detrimental to the mechanical properties of graphyne. For example, the elastic modulus of graphyne is at least one order lower than that of graphene [36]. The deterioration in mechanical properties (fracture strength and Young’s modulus) of different graphynes is attributed to the lower atom density [37]. However, the structure dependent mechanical property of -graphyne is unexplored, especially the analytical relationship between the elastic properties and acetylene repeats (or bond density).
Figure 1. (a) Schematic illustration of geometric structure of -graphyne (b) Bond types in -graphyne, where a, b and c are the lengths of aromatic, single and triple bonds, respectively. The length of carbyne chains is l=[(n+1)b + nc], 3
ACCEPTED MANUSCRIPT where n is the index number. In this paper, based on a molecular mechanics model, we obtain a set of closeform expressions for the elastic properties of -graphyne, which perfectly match results from molecular dynamics simulations. We show that the in-plane stiffness, inplane shear stiffness and layer modulus are decreasing functions while Poisson’s ratio is an increasing function of the number of acetylenic linkages.
2. Analytical Model According to molecular mechanics theory, the force field depends on the relative position of individual atoms. When a structure is deformed, the energy stored in the structure is a sum of several individual energy sources. In the molecular force field, the total potential energy of system, Et, can be expressed as Et U U U U +U vdw U es
(1)
where U, U U and U refer to energies associated with bond stretching, angle variation, inversion and torsion, respectively; Uvdw, Ues are energies associated with van der Waals and electrostatic interactions. In case of different materials and loading conditions, various functional forms may be used for these energy terms, and some secondary energy terms are neglected to obtain the analytical results more directly. In this work, for -graphyne subjected to in-plane loadings at small strains, the energy terms associated with torsion, inversion, 4
ACCEPTED MANUSCRIPT van de Waals and electrostatic interactions are negligible [36, 38-43], and only those with bond stretching and angle variation are predominant in the total potential energy. We use harmonic potentials to characterize the interactions between atoms, which had been proved to be efficient and accurate enough at small strains. Just like other carbon allotropes [36, 38-44], the structure of -graphyne (Fig. 1) can be treated as a stick-spiral system, and a stick-spiral model can be used to obtain the explicit expressions of its elastic properties. In the stick-spiral model, an elastic stick with an axial stiffness of K is used to model the stretching force of the carboncarbon bond and a spiral spring with a stiffness of C to model the moment arising from the variation of the angle. Furthermore, the bending stiffness of stick is assumed to be infinite because the chemical bonds will always remain straight regardless of the applied load. The total potential energy of the system can thus be expressed as a function of the bond elongation and the bond angle variance
Et U U
2 1 1 K i (dri ) 2 C j d j 2 i 2 j
where dri is the elongation of bond i and dj is the variance of bond angle j.
5
(2)
ACCEPTED MANUSCRIPT Figure 2. (a) -graphyne (n=1) subject to tension. The dashed box represents the unit cell of graphyne (with length of UA and UZ), while the color blocks mark the representative local structures used in the present analytical model. (b) Force equilibrium of local structures. Elastic properties of 2D isotropic materials can be described by Young’s modulus, Y, and Poisson’s ratio, . Owing to the ambiguous definition of the thickness of monolayer 2D materials, we use the in-plane stiffness, YS, instead of Y in the following discussion. The elastic properties of -graphyne can be obtained directly by analyzing the force-deformation response of its unit cell (Fig. 2a) in the stick-spiral model, similar to the procedure presented by Hou et al. [36, 43] for deriving the analytical expressions for the elastic properties of -graphyne and graphyne. We consider a -graphyne sheet under axial loads along the armchair direction as shown in Fig. 2a. Force equilibrium analysis of local structures (Fig. 2b) in a unit cell (with a length of UA and UZ in the armchair and zigzag directions, respectively) leads to the following governing equations. 2 f 2 K a daA
(3)
f 2 cos K n n 1 dbZ ndcZ = K n lZ
(4)
bZ n 2 2 bZ cZ f 2 sin m (Caa Cab )d Cab d （2Caa Cab )d
(5)
f1 K n n 1 dbA ndcA
(6)
and
6
ACCEPTED MANUSCRIPT
aZ f1 f 2 sin Cab d 2d 2
(7)
f1 f 2 cos K a daZ
(8)
c i cZ i b b + 1 1 Z n i Z Z f 2 sin Cbc di ,1 i n 1 1 4 4 2 2
(9)
In the above equations, Ka is the stick stiffness of aromatic bonds (the subscripts A and Z represent the armchair and zigzag orientations, respectively), Kn is the stiffness of acetylenic chains [7, 43-45], Caa, Cab, and Cbc are the spiral stiffness of bond angle
and i. f1 and f2 are the internal forces. We notice that, due to the unique structure of -graphyne, the unit length UA can be obtained in two ways (see Appendix A). To assure the geometry continuity, the variances of the two expressions for UA should be equal to each other, leading to (see Appendix A) f1 f 2
(10)
Now we have eight Eqs. (3)-(10), and nine unknown parameters daA, dlA, daZ, dlZ, d d, di, f1 and f2. This means that all the other parameters can be expressed as functions of f2. The expressions for strains in the armchair and zigzag direction are
A
Z
7
2 dU A f 1 1 3 1 2 Caa r0 2 UA a K n 4 K a 8Cab Cab 2Caa
2 dU Z f 1 1 21Cab 2 Caa r0 3 r02 2 U Z a 2Kn 4Ka 8Cab Cab 2Caa 8Cbc
(11)
(12)
ACCEPTED MANUSCRIPT where n+1)b+nc+(1/2)a, a(b+c), a2, (b+c){[(n(n+1)(4n−1)/12) (b+c)]−[((2n+1)−(−1)n/8)(b−c)]}, with a=a/r0, b=b/r0, c=c/r0, and r0 the reference aromatic bond length in graphene. The in-plane stiffness YS and Poisson’s ratio can be obtained as YS
2 f1 2 f 2 2 3 a A
2 1 3 1 2 Caa r0 / 3 K n 4 K a 8Cab Cab 2Caa
1 1
2 Z 1 1 21Cab 2 Caa r0 3 r02 8Cab Cab 2Caa 8Cbc A 2Kn 4Ka
(13)
1 1 3 1 2 Caa r02 K n 4 K a 8Cab Cab 2Caa (14)
Although Eqs. (13) and (14) are derived in the armchair direction, they are exactly the expressions for in-plane stiffness and Poisson’s ratio of graphyne along arbitrary direction because of its isotropic characteristic due to six-fold in-plane structural symmetry. According to the isotropic elastic theory, the in-plane shear stiffness of graphyne can be expressed as
GS
2 YS 1 2 1 1Cab 22 Caa r0 3 r02 = / + 2 1 2Cab Cab 2Caa 4Cbc 3 Kn
(15)
Another parameter to describe the macroscopic physical properties, the bulk modulus, is expressed as B=Y/3(1−2) for 3D bulk materials. However, for 2D materials, no out-of-plane deformation will be induced by the in-plane deformation. Consequently, the 2D bulk modulus or layer modulus of 2D isotropic materials can be defined as [36],
L
8
2 YS 1 2 1 1 2 Caa 1Cab r0 3 r02 = / 2 1 Ka 2Cab Cab 2Caa 4Cbc 3 Kn
(16)
ACCEPTED MANUSCRIPT For graphene, we have n=0, = == = = a=K0=K, Cab=Caa=C. Eqs. (13)-(16) can thus be reduced to
YS
8 3KC Kr02 18C
(17)
Kr02 6C Kr02 18C
(18)
2 3KC Kr02 6C
(19)
GS
L
K 2 3
(20)
The first three are identical to the analytical results obtained in the previous works [36, 39, 43], and the last one is newly obtained. It is noted that these equations apply only in the small deformation range and extremely low temperature. For graphynes under relatively large strains and high temperature, the six-fold structural symmetry will be broken, thus the mechanical properties will show anisotropic, i.e., strain-induced-anisotropy, which has been also found in nanotubes [41] and graphene [46].
3. MD Simulation To validate the analytical results, we use MD simulations to explore the elastic properties of graphyne. In the calculations, we investigate ten graphyne sheets with n varying from 1 to 10. The sizes of the investigated graphyne sheets are all 9
ACCEPTED MANUSCRIPT around 20×20 nm. Similar to graphene, the edges of graphyne are denoted armchair or zigzag according to the bond orientations. All MD simulations are performed using the code LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator). The bonding interaction between the carbon atoms is described by REBO [47] potential, which has been widely used to model carbon-carbon interactions. Periodic boundary conditions are applied along the three dimensions (the box length in the out-of-plane direction is 10 nm). All simulations are subjected to a NPT ensemble. The temperature is set as 1 K (in trying to eliminate the temperature effect). The pressure is set to zero (it may experience an oscillating magnitude of several MPa in the in-plane direction). The time step is taken as 0.5 fs. Prior to the loading, the initial model is relaxed until the energy of the system is fully minimized. After reaching the equilibrium states, an engineering strain rate of 0.0005 ps-1 is applied for uniaxial tensile, shear or biaxial deformation. The inplane stiffness YS, in-plane shear stiffness GS and layer modulus L are extracted by the second derivative of the corresponding strain energy at a small strain (within 5%), whereas the Poisson’s ratio is derived from the MD results of GS and L (both of which are calculated at biaxial loading conditions).
4. Results and Discussion Molecular parameters of graphyne used in the present analytical model can be determined based on results from first principle calculations or MD simulations 10
ACCEPTED MANUSCRIPT [21, 48, 49]. Our own MD simulations indicate that the lengths of the aromatic, single and triple bond are respectively 0.140, 0.139, and 0.133 nm, consistent with those reported in the literature [17, 21, 48, 50]. For convenience, as in previous works [36, 43], we take a=a/r0≈1, b=b/r0≈1, c=c/r0≈0.85, with r0=0.142 nm, Cbb=1.42 nN nm, Cbc=0.423 nN nm, and Kn=79.4 nN/l =559/(1.85n+1) nN/nm.
4.1 In-plane stiffness The in-plane stiffness of graphyne predicted by Eq. (13) is shown in Fig. 3. It is seen that, with the introduction of acetylenic chains, the in-plane stiffness of graphyne is substantially reduced in comparison with pure sp2 graphene. For example, when n=5, the in-plane stiffness is about 29 N/m, less than 10% of that of graphene. The longer the acetylenic chains, the lower the in-plane stiffness. This descending trend has been observed also in other graphynes [21, 48-51]. For instance, with n increasing from 1 to 5, the in-plane stiffness of -graphyne decreases from 32 to 0.55 N/m [36], and that of -graphyne decreases from 180 to 60 N/m [43]. The degradation of the stiffness can be considered as a result of the lowering bond density. Molecular structures with lower bond density have a weaker resistance to deformation and therefore a smaller stiffness. Owing to the lack of extensive studies on the mechanical properties of graphyne, the available data for the in-plane stiffness of graphyne in the literature is only for n=1. A value of 87.4 N/m is obtained from MD simulations [49] and 73.1 N/m from first principles (DFT) calculations [51], both of which are in 11
ACCEPTED MANUSCRIPT good agreement with the present analytical prediction, 79.3 N/m, and the present MD calculation, 83.0 N/m.
In-plane stiffness (N/m)
100 Analytical model MD simulation Empirical equation DFT by A.R. Puigdollers et al. MD by Zhang et al.
80 60 40 20 0
2
4
6
8
10
Number of acetylenic linkages
Figure 3. In-plane stiffness of -graphyne as a function of the number of acetylenic linkages.
4.2 Poisson’s ratio Different from the in-plane stiffness, the Poisson’s ratio of graphyne is insensitive to the length of acetylenic chains, as shown in Fig. 4. In magnitude, Poisson’s ratio derived from the analytical model increases from 0.49 to 0.56 as the number of acetylenic linkages varies from 1 to 10. The value of Poisson’s ratio for n=1 is comparable to the existing result from first principles calculations [51]. We note that the present analytical results are obtained without taking the effect of temperature into consideration. When thermal fluctuations exist, Poisson’s ratio of a two-dimensional material can be significantly modified by the thermally induced outof-plane deformations. The present analytical expression for Poisson’s ratio is useful. 12
ACCEPTED MANUSCRIPT First, it is applicable when the out-of-plane deformation of the graphyne layer is restricted (e.g., when it is adhered to a substrate). Second, the analytical expressions of other properties, e.g., the in-plane shear stiffness and layer modulus, can be derived based on the expression of the Poisson’s ratio, as shown by Eqs. (15) and (16).
1.0 Analytical model MD simulation Empirical equation DFT by A.R. Puigdollers et al.
Poisson's ratio
0.8 0.6 0.4 0.2 0.0
2
4
6
8
10
Number of acetylenic linkages
Figure 4. Poisson’s ratio as a function of the number of acetylenic linkages.
4.3 In-plane shear stiffness and layer modulus The in-plane shear stiffness and the layer modulus of graphyne are shown in Fig. 5. We see that both the shear stiffness and layer modulus decrease with increasing the percentage of acetylenic linkages, similar to the in-plane stiffness. With the index number n increasing from 1 to 10, the shear stiffness decreases from 26 to 5 N/m, and the layer modulus from 78 to 18 N/m. By comparison, with n increasing from 1 to 10, the shear stiffness of -graphyne decreases from 9.3 to 0.14 N/m [36], and that of -graphyne decreases from 76 to 12 N/m [43]. The results from MD calculations are also shown in Fig. 5 for comparison. It is 13
ACCEPTED MANUSCRIPT seen that the analytically predicted results for the shear stiffness and the layer modulus are in excellent agreement with those from MD simulations. In contrast, the analytically predicted in-plane stiffness is slightly larger than the MD calculated one. This might be due to the fact that the shear stiffness and layer modulus are calculated in MD simulations under biaxial loading conditions which may limit the possible outof-plane deformation of the graphyne sheet and therefore the assumption of pure 2D configuration in the analytical model can be well met. However, the in-plane stiffness is calculated under a uniaxial stress which provides less constraint to the out-of-plane deformation and thus a larger deviation of the calculated in-plane stiffness from the analytically predicted one. a
b Analytical model MD simulation Empirical equation
20
Layer Modulus (N/m)
Shear stiffness (N/m)
30
10
0
2
4
6
8
10
Number of acetylenic linkages
100 Analytical model MD simulation Empirical equation
75
50
25
0
2
4
6
8
10
Number of acetylenic linkages
Figure 5. Dependence of the in-plane shear stiffness (a) and the layer modulus (b) on the number of acetylenic linkages.
5. Scaling Laws The introduction of acetylenic chains causes an obvious chain-length dependent mechanical properties of graphynes. In other words, the mechanical properties of 14
ACCEPTED MANUSCRIPT graphynes depends significantly on the length of acetylenic chains. This is intrinsically attributed to the change in bond density. Previous studies have reported that the bond density plays an important role in determining the Young’s modulus and Poisson’s ratio of the carbon allotropes [37, 43]. Sun et al. [37] evaluated the mechanical properties of 11 different graphene allotropes and showed that the elastic modulus increases with the increasing bond density. Cranford and Buehler [50], using a simple spring-network model, showed that the in-plane stiffness of -graphyne is approximately inversely proportional to the number n of acetylenic linkages. Hou et al. [43] showed that the in-plane stiffness of -graphyne is proportional to the effective bond density rather than the real bond density. The concept of the effective bond density is originated from the analytical results obtained by the stick-spiral model, and can thus more accurately capture the decreasing trend of the scale effect on the mechanical properties of graphynes. In this section, we will establish the scaling laws for the elastic properties of graphyne based on the concept of the effective bond density, i.e., obtain empirical equations of how mechanical properties of -graphyne scale with the index number n (i.e., the length of acetylenic chains). The layer modulus of a 2D material represents its ability of resisting deformation under a hydrostatic stress/stain condition. When a 2D material is upon an in-plane hydrostatic stress, its all bonds will equally share the in-plane stress. In this sense, it is expected that the layer modulus is proportional to the real bond density
bond, hence we assume
15
ACCEPTED MANUSCRIPT L
1 bond 2
(21)
where is a coefficient. This linear relation is confirmed by MD calculations, as shown in Fig. 6. However, the in-plane stiffness of -graphyne is not a linear function of the real bond density (see Fig. 6), as has been observed for -graphyne too [43]. We thus define the effective bond density, eff, on which the in-plane stiffness of a 2D material is proportionally dependent, i.e., YS eff
(22)
For 2D isotropic materials, we further have
1
YS 1 eff 2L bond
(23)
It is seen that the scaling laws for the elastic properties of graphynes can be obtained once the effective bond density is determined.
100
Stiffness (N/m)
80
In-plane sitffness Layer modulus
60 40 20 0
0.1
0.2
0.3
0.4 -2
Real bond density (Å )
16
0.5
ACCEPTED MANUSCRIPT Figure 6. MD results for the in-plane stiffness and layer modulus as functions of the real bond density. The layer modulus is almost linearly dependent on the real bond density, but the in-plane stiffness is not.
To determine the effective bond density, we take graphene structure as a reference configuration, and define the relative effective bond density in -graphyne as
eff
eff ge eff
(24)
ge where eff is the effective bond density in graphene. The relative real bond density in
-graphyne is bond
bond 3 ge bond 4n 3
(25)
ge where bond is the real bond density in graphene. For simplicity a≈b≈c≈1 is used in
deriving Eq. (25) and will be used in the following to determine the effective bond density. If we recall Eq. (A6), we find that f1/f2 decreases drastically with n, e.g., from 0.5 for n=0 (graphene) to 0.001 for n=5. This means that, when the graphyne sheet subjects to uniaxial tension along armchair direction, the applied force will mainly deliver along the path shown in Fig. 7, further indicating that the effective bond number of acetylenic chains in the zigzag direction should be less than their real bond number. We see also that, the force delivery path is not a straight line but a 17
ACCEPTED MANUSCRIPT kinked one, which will lead to a lower in-plane stiffness due to a smaller deformation resistance. The deformation resistance is obviously associated with the kink offset distance. Here we introduce a structural parameter d1/d (in which d1 and d are defined in Fig.7) to account for the effect of the kinked force delivery path on the inplane stiffness of graphyne. For -graphyne,
3 4n 3
(26)
We find that, the relative effective bond density defined by the following Eq. (27) can well capture the scale effect on the mechanical properties, i.e.
eff bond
1 ( ) 2 3 1 9 2 2 1 ( / bond ) (4n 3) 2 2( 4n 3)
(27)
Figure 7. The main force transfer path (colored yellow) in graphyne under an armchair direction tension. The elastic properties of -graphyne can thus be empirically expressed as (derivation can be found in Appendix B) 18
ACCEPTED MANUSCRIPT
YS
1
ge 3 1 9 Y 2 S (4n 3) 2 2( 4n 3)
9
1
9
(28)
ge 2 2 2 2(4 n 3) 2 2(4 n 3 )
GS
3GSge 1 3 9 9 (4n 3) 2 2(4n 3) 2 2 2(4n 3) 2
L
3 Lge 4n 3
(29)
(30)
(31)
The results predicted by the above empirical Eqs. (28)-(31) are shown in Figs. 3-5. It is seen that these equations can well reproduce the MD simulation data, further confirming that the mechanical properties of -graphyne are closely related to the effective bond density. Owing to the physical basis of the relative effective bond density, the scaling laws presented here can establish direct connections between the mechanical properties of -graphyne and pristine graphene, and have demonstrated a useful approach for linking mechanical properties of graphene allotropes to those of graphene by determining the relative effective bond density of graphene allotropes.
6. Conclusions Based on a stick-spiral model, we have presented an analytical model for
graphyne, and obtained close-form expressions for its elastic properties. Based on these expressions, the structure dependent elastic properties of graphyne are investigated. It is found that the in-plane stiffness, in-plane shear stiffness, and layer 19
ACCEPTED MANUSCRIPT modulus of graphyne significantly decreases, while the Poisson’s ratio slightly increases, with the increasing length of acetylenic chains. These analytical predictions are in excellent agreements with MD simulations. We develop also the scaling laws for the elastic properties of graphyne by determining the relative effective bond density. These scaling laws establish direct connections between the mechanical properties of -graphyne and graphene. Although the present scaling laws are obtained for -graphyne, the approach developed here can be applied to other graphene allotropes. The present findings provide a fundamental understanding of the mechanical properties of graphyne, and a useful approach for linking mechanical properties of graphene allotropes to those of graphene in an efficient way.
Acknowledgements The authors acknowledge the financial supports from the NSF of China (Grant No. 11425209) and Shanghai Pujiang Program (Grant No. 13PJD016). The MD simulations were carried out on the computing platform of the International Center for Applied Mechanics in Energy Engineering (ICAMEE), Shanghai University.
20
ACCEPTED MANUSCRIPT Appendix A The lengths of the unit cell can be expressed as functions of bond lengths and bond angles U A 2 n 1 bA 2ncA 2aZ cos
(A1)
n k 2aA 4 bZ cos bZ cZ cos i ) 2aZ cos k 1 i 1 n k U Z 4 bZ sin ( bZ cZ sin( i )) 2aZ sin k 1 i 1
(A2)
The variances of UA and UZ can thus be obtained as dU A
2 f1 f1 f 2 32 Caa ( f1 f 2 ) + Kn 2Ka 4Cab (2Cab Caa )
(A3)
3(21Cab 2 Caa )( f1 f 2 )ro 2 3 f 2 3 ro 2 9f f f 2 1 2 + 2Ka Kn 4Cab (2Cab Caa ) 2Cbc
dU Z
3 f1 f 2 21Cab 2 Caa r02 3 f 2 3 r02 3 f2 3( f1 f 2 ) Kn 2K a 4Cab Cab 2Caa 2Cbc
(A4)
To assure the geometry continuity, the two expressions of dUA in Eq. (A3) should be equal to each other, leading to f1 f 2
(A5)
with 2 1 1 3 2 Caa 1Cab r0 4 Caa Cab Cab 2Ka Kn
21
5 3 2 Caa 1Cab ro2 33 r02 1 4 Caa Cab Cab 2Cbc 2Ka 2Kn
(A6)
ACCEPTED MANUSCRIPT Appendix B When the relative effective bond density is defined in Eq. (27), the elastic properties of -graphyne can be derived as YS eff YSge
1
ge 3 1 9 YS (4n 3) 2 2(4n 3) 2
ge eff eff eff 1 1 ge 9 9 1 ge 2 2 bond bond bond 2 2(4 n 3) 2 2 (4 n 3)
(B1) (B2)
3 1 9 2 2(4n 3) 2 (4 n 3) Y S GS 2GSge (1 ge ) 2(1 ) 3 1 ge 9 9 2 2 2 2 2(4n 3) 2 2(4n 3)
3GSge 1 3 9 9 2 2 (4n 3) 2 2(4n 3) 2 2(4n 3) L bond Lge
22
3 Lge 4n 3
(B3) (B4)
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ACCEPTED MANUSCRIPT [22] J. Zhou, K. Lv, Q. Wang, X. S. Chen, Q. Sun and P. Jena, Electronic structures and bonding of graphyne sheet and its BN analog, J. Chem. Phys 134(17) (2011) 174701. [23] B. G. Kim and H. J. Choi, Graphyne: Hexagonal network of carbon with versatile Dirac cones, Phys. Rev. B 86(11) (2012) 115435. [24] Y. Li, L. Xu, H. Liu and Y. Li, Graphdiyne and graphyne: From theoretical predictions to practical construction, Chem. Soc. Rev. 43(8) (2014) 2572-2586. [25] D. Malko, C. Neiss, F. Vines and A. Gorling, Competition for graphene: Graphynes with direction-dependent Dirac cones, Phys. Rev. Lett. 108(8) (2012) 086804. [26] H. Sevinçli and C. Sevik, Electronic, phononic, and thermoelectric properties of graphyne sheets, Appl. Phys. Lett. 105(22) (2014) 223108. [27] G. Wang, M. Si, A. Kumar and R. Pandey, Strain engineering of Dirac cones in graphyne, Appl. Phys. Lett. 104(21) (2014) 213107. [28] W. Wu, W. Guo and X. C. Zeng, Intrinsic electronic and transport properties of graphyne sheets and nanoribbons, Nanoscale 5(19) (2013) 9264-9276. [29] P. H. Jiang, H. J. Liu, L. Cheng, D. D. Fan, J. Zhang, J. Wei, Thermoelectric properties of γgraphyne from first-principles calculations, Carbon 113 (2017) 108-113. [30] S. Ma, M. Zhang, L. Z. Sun and K. W. Zhang, High-temperature behavior of monolayer graphyne and graphdiyne, Carbon 99 (2016) 547-555. [31] S. Chattaraj, K. Srinivasu, S. Mondal and S. K. Ghosh, Hydrogen trapping ability of the pyridinelithium(+) (1:1) complex, J. Phys. Chem. A 119(12) (2015) 3056-3063. [32] B. Jang, J. Koo, M. Park, H. Lee, J. Nam, Y. Kwon, et al., Graphdiyne as a high-capacity lithium ion battery anode material, Appl. Phys. Lett. 103(26) (2013) 263904. 25
ACCEPTED MANUSCRIPT [33] H. Zhang, Y. Luo, X. Feng, L. Zhao and M. Zhang, Flexible band gap tuning of hexagonal boron nitride sheets interconnected by acetylenic bonds, Phys. Chem. Chem. Phys. 17(31) (2015) 20376-20381. [34] M. Xue, H. Qiu and W. Guo, Exceptionally fast water desalination at complete salt rejection by pristine graphyne monolayers, Nanotechnology 24(50) (2013) 505720. [35] T. Ouyang and M. Hu, Thermal transport and thermoelectric properties of beta-graphyne nanostructures, Nanotechnology 25(24) (2014) 245401. [36] J. Hou, Z. Yin, Y. Zhang and T. Chang, Structure dependent elastic properties of supergraphene, Acta. Mech. Sinica-PRC (2016) 1-6. [37] H. Sun, S. Mukherjee, M. Daly, A. Krishnan, M. H. Karigerasi and C. V. Singh, New insights into the structure-nonlinear mechanical property relations for graphene allotropes, Carbon 110 (2016) 443-457. [38] T. Chang, A molecular based anisotropic shell model for single-walled carbon nanotubes, J. Mech. Phys. Solids 58(9) (2010) 1422-1433. [39] T. Chang and H. Gao, Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model, J. Mech. Phys. Solids 51(6) (2003) 1059-1074. [40] T. Chang, J. Geng and X. Guo, Chirality- and size-dependent elastic properties of single-walled carbon nanotubes, Appl. Phys. Lett. 87(25) (2005) 251929. [41] J. Geng and T. Chang, Nonlinear stick-spiral model for predicting mechanical behavior of single-walled carbon nanotubes, Phys. Rev. B 74(24) (2006) 245428. [42] X. Guo, J. Geng and T. Chang, Prediction of chirality- and size-dependent elastic properties of single-walled carbon nanotubes via a molecular mechanics model, P. Roy. Soc. A-Math. Phy. 26
ACCEPTED MANUSCRIPT 462(2072) (2006) 2523-2540. [43] J. Hou, Z. Yin, Y. Zhang and T. Chang, An analytical molecular mechanics model for elastic properties of graphyne-n, J. Appl. Mech. 82(9) (2015) 094501. [44] A. K. Nair, S. W. Cranford and M. J. Buehler, The minimal nanowire: Mechanical properties of carbyne, EPL-Europhys.Lett. 95(1) (2011) 16002. [45] A. J. Kocsis, N. A. R. Yedama and S. W. Cranford, Confinement and controlling the effective compressive stiffness of carbyne, Nanotechnology 25(33) (2014) 335709. [46] T. Ma, B. Li and T. Chang, Chirality- and curvature-dependent bending stiffness of single layer graphene, Appl. Phys. Lett. 99(20) (2011) 201901.
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27
Jiaxing Qu, Hongwei Zhang, Jianxin Li, Shexu Zhao, Tienchong Chang PII:
S0008-6223(17)30500-6
DOI:
10.1016/j.carbon.2017.05.051
Reference:
CARBON 12027
To appear in:
Carbon
Received Date:
22 February 2017
Revised Date:
18 April 2017
Accepted Date:
14 May 2017
Please cite this article as: Jiaxing Qu, Hongwei Zhang, Jianxin Li, Shexu Zhao, Tienchong Chang, Structure-dependent mechanical properties of extended β-graphyne, Carbon (2017), doi: 10.1016/j. carbon.2017.05.051
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ACCEPTED MANUSCRIPT Structure-dependent mechanical properties of extended -graphyne Jiaxing Qu1 • Hongwei Zhang1 • Jianxin Li1 • Shexu Zhao1 • Tienchong Chang1,2,*
1State
Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil
Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
2Shanghai
Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of
Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China
Abstract Owing to their remarkable electronic and thermal properties, graphynes have been considered new promising carbon materials after graphene. However, our understanding of their structure-dependent mechanical properties is far from complete. In this paper an analytical molecular mechanics model is proposed to mathematically establish a relationship between the structure and the elastic properties of graphyne. Extensive molecular dynamics simulations are performed for comparison. We show that the elastic properties of graphyne exhibit a strong dependence on its structure. The in-plane stiffness, in-plane shear stiffness and layer modulus decrease with increasing percentage of the acetylenic linkages, while the Poisson’s ratio increases. Further analysis demonstrates that this dependence of structure attributes to the change in bond density. Based on the concept of effective bond density, scaling laws are developed for the elastic properties of graphyne, which demonstrate a useful approach for linking mechanical properties of graphene allotropes to those of graphene.
*
1
Corresponding author. Tel: 86(21)56331208. Email: [email protected].
ACCEPTED MANUSCRIPT
1. Introduction Carbon exhibits versatile flexibility in forming different hybridization states (i.e., sp, sp2, sp3). Apart from some naturally existed ones, such as graphite (sp2hybridize) and diamond (sp3-hybridize), many other carbon allotropes are possible to be constructed by altering the periodic binding motifs in the networks consisting of sp3-, sp2- and sp-hybridized carbon atoms [1-14]. In particular, graphene, a sp2-hybridized carbon allotrope, exhibits extraordinary electronic properties [15,16] such as the quantum Hall effect and giant carrier mobility, as well as excellent optical, thermal, and mechanical performance, making graphene an attractive building material for nano devices. This sp2-hybridized carbon allotrope inspired the combination of carbon allotropes with more hybridized states. With the introduction of sp constituents, Baughman et al. [17] postulated that graphynes can be alternative carbon planar forms constructed by partial or complete replacement of the aromatic bonds of graphene with the acetylenic chains (single- and triple-bond) [18]. Based on different ratio of sp constituents, graphyne can have unlimited geometrical configurations, among which four typical structures, i.e.,
graphyne, graphyne, graphyne and 6, 6, 12-graphyne have attracted much attention recently [17-20]. Graphynes extend some novel properties beyond those of primitive graphene [21-30] and have been proposed to be used as energy storage [31], anode [32], gas separation [33], and water desalination materials [34]. Among graphyne family, 2
ACCEPTED MANUSCRIPT graphyne (Fig. 1) exhibits outstanding properties. Malko et al. [25] predicted that Dirac cones and their associated electronic transport exist in graphyne, just like in graphene. Ouyang et al. [35] found that the zigzag -graphyne nanoribbons possess superior thermoelectric performance with a thermoelectric figure of merit (ZT) of 0.51.5 at room temperature, at least one order higher than that of graphene. Despite these studies, the understanding of mechanical properties of graphyne is very limited. The insertion of acetylenic linkages is detrimental to the mechanical properties of graphyne. For example, the elastic modulus of graphyne is at least one order lower than that of graphene [36]. The deterioration in mechanical properties (fracture strength and Young’s modulus) of different graphynes is attributed to the lower atom density [37]. However, the structure dependent mechanical property of -graphyne is unexplored, especially the analytical relationship between the elastic properties and acetylene repeats (or bond density).
Figure 1. (a) Schematic illustration of geometric structure of -graphyne (b) Bond types in -graphyne, where a, b and c are the lengths of aromatic, single and triple bonds, respectively. The length of carbyne chains is l=[(n+1)b + nc], 3
ACCEPTED MANUSCRIPT where n is the index number. In this paper, based on a molecular mechanics model, we obtain a set of closeform expressions for the elastic properties of -graphyne, which perfectly match results from molecular dynamics simulations. We show that the in-plane stiffness, inplane shear stiffness and layer modulus are decreasing functions while Poisson’s ratio is an increasing function of the number of acetylenic linkages.
2. Analytical Model According to molecular mechanics theory, the force field depends on the relative position of individual atoms. When a structure is deformed, the energy stored in the structure is a sum of several individual energy sources. In the molecular force field, the total potential energy of system, Et, can be expressed as Et U U U U +U vdw U es
(1)
where U, U U and U refer to energies associated with bond stretching, angle variation, inversion and torsion, respectively; Uvdw, Ues are energies associated with van der Waals and electrostatic interactions. In case of different materials and loading conditions, various functional forms may be used for these energy terms, and some secondary energy terms are neglected to obtain the analytical results more directly. In this work, for -graphyne subjected to in-plane loadings at small strains, the energy terms associated with torsion, inversion, 4
ACCEPTED MANUSCRIPT van de Waals and electrostatic interactions are negligible [36, 38-43], and only those with bond stretching and angle variation are predominant in the total potential energy. We use harmonic potentials to characterize the interactions between atoms, which had been proved to be efficient and accurate enough at small strains. Just like other carbon allotropes [36, 38-44], the structure of -graphyne (Fig. 1) can be treated as a stick-spiral system, and a stick-spiral model can be used to obtain the explicit expressions of its elastic properties. In the stick-spiral model, an elastic stick with an axial stiffness of K is used to model the stretching force of the carboncarbon bond and a spiral spring with a stiffness of C to model the moment arising from the variation of the angle. Furthermore, the bending stiffness of stick is assumed to be infinite because the chemical bonds will always remain straight regardless of the applied load. The total potential energy of the system can thus be expressed as a function of the bond elongation and the bond angle variance
Et U U
2 1 1 K i (dri ) 2 C j d j 2 i 2 j
where dri is the elongation of bond i and dj is the variance of bond angle j.
5
(2)
ACCEPTED MANUSCRIPT Figure 2. (a) -graphyne (n=1) subject to tension. The dashed box represents the unit cell of graphyne (with length of UA and UZ), while the color blocks mark the representative local structures used in the present analytical model. (b) Force equilibrium of local structures. Elastic properties of 2D isotropic materials can be described by Young’s modulus, Y, and Poisson’s ratio, . Owing to the ambiguous definition of the thickness of monolayer 2D materials, we use the in-plane stiffness, YS, instead of Y in the following discussion. The elastic properties of -graphyne can be obtained directly by analyzing the force-deformation response of its unit cell (Fig. 2a) in the stick-spiral model, similar to the procedure presented by Hou et al. [36, 43] for deriving the analytical expressions for the elastic properties of -graphyne and graphyne. We consider a -graphyne sheet under axial loads along the armchair direction as shown in Fig. 2a. Force equilibrium analysis of local structures (Fig. 2b) in a unit cell (with a length of UA and UZ in the armchair and zigzag directions, respectively) leads to the following governing equations. 2 f 2 K a daA
(3)
f 2 cos K n n 1 dbZ ndcZ = K n lZ
(4)
bZ n 2 2 bZ cZ f 2 sin m (Caa Cab )d Cab d （2Caa Cab )d
(5)
f1 K n n 1 dbA ndcA
(6)
and
6
ACCEPTED MANUSCRIPT
aZ f1 f 2 sin Cab d 2d 2
(7)
f1 f 2 cos K a daZ
(8)
c i cZ i b b + 1 1 Z n i Z Z f 2 sin Cbc di ,1 i n 1 1 4 4 2 2
(9)
In the above equations, Ka is the stick stiffness of aromatic bonds (the subscripts A and Z represent the armchair and zigzag orientations, respectively), Kn is the stiffness of acetylenic chains [7, 43-45], Caa, Cab, and Cbc are the spiral stiffness of bond angle
and i. f1 and f2 are the internal forces. We notice that, due to the unique structure of -graphyne, the unit length UA can be obtained in two ways (see Appendix A). To assure the geometry continuity, the variances of the two expressions for UA should be equal to each other, leading to (see Appendix A) f1 f 2
(10)
Now we have eight Eqs. (3)-(10), and nine unknown parameters daA, dlA, daZ, dlZ, d d, di, f1 and f2. This means that all the other parameters can be expressed as functions of f2. The expressions for strains in the armchair and zigzag direction are
A
Z
7
2 dU A f 1 1 3 1 2 Caa r0 2 UA a K n 4 K a 8Cab Cab 2Caa
2 dU Z f 1 1 21Cab 2 Caa r0 3 r02 2 U Z a 2Kn 4Ka 8Cab Cab 2Caa 8Cbc
(11)
(12)
ACCEPTED MANUSCRIPT where n+1)b+nc+(1/2)a, a(b+c), a2, (b+c){[(n(n+1)(4n−1)/12) (b+c)]−[((2n+1)−(−1)n/8)(b−c)]}, with a=a/r0, b=b/r0, c=c/r0, and r0 the reference aromatic bond length in graphene. The in-plane stiffness YS and Poisson’s ratio can be obtained as YS
2 f1 2 f 2 2 3 a A
2 1 3 1 2 Caa r0 / 3 K n 4 K a 8Cab Cab 2Caa
1 1
2 Z 1 1 21Cab 2 Caa r0 3 r02 8Cab Cab 2Caa 8Cbc A 2Kn 4Ka
(13)
1 1 3 1 2 Caa r02 K n 4 K a 8Cab Cab 2Caa (14)
Although Eqs. (13) and (14) are derived in the armchair direction, they are exactly the expressions for in-plane stiffness and Poisson’s ratio of graphyne along arbitrary direction because of its isotropic characteristic due to six-fold in-plane structural symmetry. According to the isotropic elastic theory, the in-plane shear stiffness of graphyne can be expressed as
GS
2 YS 1 2 1 1Cab 22 Caa r0 3 r02 = / + 2 1 2Cab Cab 2Caa 4Cbc 3 Kn
(15)
Another parameter to describe the macroscopic physical properties, the bulk modulus, is expressed as B=Y/3(1−2) for 3D bulk materials. However, for 2D materials, no out-of-plane deformation will be induced by the in-plane deformation. Consequently, the 2D bulk modulus or layer modulus of 2D isotropic materials can be defined as [36],
L
8
2 YS 1 2 1 1 2 Caa 1Cab r0 3 r02 = / 2 1 Ka 2Cab Cab 2Caa 4Cbc 3 Kn
(16)
ACCEPTED MANUSCRIPT For graphene, we have n=0, = == = = a=K0=K, Cab=Caa=C. Eqs. (13)-(16) can thus be reduced to
YS
8 3KC Kr02 18C
(17)
Kr02 6C Kr02 18C
(18)
2 3KC Kr02 6C
(19)
GS
L
K 2 3
(20)
The first three are identical to the analytical results obtained in the previous works [36, 39, 43], and the last one is newly obtained. It is noted that these equations apply only in the small deformation range and extremely low temperature. For graphynes under relatively large strains and high temperature, the six-fold structural symmetry will be broken, thus the mechanical properties will show anisotropic, i.e., strain-induced-anisotropy, which has been also found in nanotubes [41] and graphene [46].
3. MD Simulation To validate the analytical results, we use MD simulations to explore the elastic properties of graphyne. In the calculations, we investigate ten graphyne sheets with n varying from 1 to 10. The sizes of the investigated graphyne sheets are all 9
ACCEPTED MANUSCRIPT around 20×20 nm. Similar to graphene, the edges of graphyne are denoted armchair or zigzag according to the bond orientations. All MD simulations are performed using the code LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator). The bonding interaction between the carbon atoms is described by REBO [47] potential, which has been widely used to model carbon-carbon interactions. Periodic boundary conditions are applied along the three dimensions (the box length in the out-of-plane direction is 10 nm). All simulations are subjected to a NPT ensemble. The temperature is set as 1 K (in trying to eliminate the temperature effect). The pressure is set to zero (it may experience an oscillating magnitude of several MPa in the in-plane direction). The time step is taken as 0.5 fs. Prior to the loading, the initial model is relaxed until the energy of the system is fully minimized. After reaching the equilibrium states, an engineering strain rate of 0.0005 ps-1 is applied for uniaxial tensile, shear or biaxial deformation. The inplane stiffness YS, in-plane shear stiffness GS and layer modulus L are extracted by the second derivative of the corresponding strain energy at a small strain (within 5%), whereas the Poisson’s ratio is derived from the MD results of GS and L (both of which are calculated at biaxial loading conditions).
4. Results and Discussion Molecular parameters of graphyne used in the present analytical model can be determined based on results from first principle calculations or MD simulations 10
ACCEPTED MANUSCRIPT [21, 48, 49]. Our own MD simulations indicate that the lengths of the aromatic, single and triple bond are respectively 0.140, 0.139, and 0.133 nm, consistent with those reported in the literature [17, 21, 48, 50]. For convenience, as in previous works [36, 43], we take a=a/r0≈1, b=b/r0≈1, c=c/r0≈0.85, with r0=0.142 nm, Cbb=1.42 nN nm, Cbc=0.423 nN nm, and Kn=79.4 nN/l =559/(1.85n+1) nN/nm.
4.1 In-plane stiffness The in-plane stiffness of graphyne predicted by Eq. (13) is shown in Fig. 3. It is seen that, with the introduction of acetylenic chains, the in-plane stiffness of graphyne is substantially reduced in comparison with pure sp2 graphene. For example, when n=5, the in-plane stiffness is about 29 N/m, less than 10% of that of graphene. The longer the acetylenic chains, the lower the in-plane stiffness. This descending trend has been observed also in other graphynes [21, 48-51]. For instance, with n increasing from 1 to 5, the in-plane stiffness of -graphyne decreases from 32 to 0.55 N/m [36], and that of -graphyne decreases from 180 to 60 N/m [43]. The degradation of the stiffness can be considered as a result of the lowering bond density. Molecular structures with lower bond density have a weaker resistance to deformation and therefore a smaller stiffness. Owing to the lack of extensive studies on the mechanical properties of graphyne, the available data for the in-plane stiffness of graphyne in the literature is only for n=1. A value of 87.4 N/m is obtained from MD simulations [49] and 73.1 N/m from first principles (DFT) calculations [51], both of which are in 11
ACCEPTED MANUSCRIPT good agreement with the present analytical prediction, 79.3 N/m, and the present MD calculation, 83.0 N/m.
In-plane stiffness (N/m)
100 Analytical model MD simulation Empirical equation DFT by A.R. Puigdollers et al. MD by Zhang et al.
80 60 40 20 0
2
4
6
8
10
Number of acetylenic linkages
Figure 3. In-plane stiffness of -graphyne as a function of the number of acetylenic linkages.
4.2 Poisson’s ratio Different from the in-plane stiffness, the Poisson’s ratio of graphyne is insensitive to the length of acetylenic chains, as shown in Fig. 4. In magnitude, Poisson’s ratio derived from the analytical model increases from 0.49 to 0.56 as the number of acetylenic linkages varies from 1 to 10. The value of Poisson’s ratio for n=1 is comparable to the existing result from first principles calculations [51]. We note that the present analytical results are obtained without taking the effect of temperature into consideration. When thermal fluctuations exist, Poisson’s ratio of a two-dimensional material can be significantly modified by the thermally induced outof-plane deformations. The present analytical expression for Poisson’s ratio is useful. 12
ACCEPTED MANUSCRIPT First, it is applicable when the out-of-plane deformation of the graphyne layer is restricted (e.g., when it is adhered to a substrate). Second, the analytical expressions of other properties, e.g., the in-plane shear stiffness and layer modulus, can be derived based on the expression of the Poisson’s ratio, as shown by Eqs. (15) and (16).
1.0 Analytical model MD simulation Empirical equation DFT by A.R. Puigdollers et al.
Poisson's ratio
0.8 0.6 0.4 0.2 0.0
2
4
6
8
10
Number of acetylenic linkages
Figure 4. Poisson’s ratio as a function of the number of acetylenic linkages.
4.3 In-plane shear stiffness and layer modulus The in-plane shear stiffness and the layer modulus of graphyne are shown in Fig. 5. We see that both the shear stiffness and layer modulus decrease with increasing the percentage of acetylenic linkages, similar to the in-plane stiffness. With the index number n increasing from 1 to 10, the shear stiffness decreases from 26 to 5 N/m, and the layer modulus from 78 to 18 N/m. By comparison, with n increasing from 1 to 10, the shear stiffness of -graphyne decreases from 9.3 to 0.14 N/m [36], and that of -graphyne decreases from 76 to 12 N/m [43]. The results from MD calculations are also shown in Fig. 5 for comparison. It is 13
ACCEPTED MANUSCRIPT seen that the analytically predicted results for the shear stiffness and the layer modulus are in excellent agreement with those from MD simulations. In contrast, the analytically predicted in-plane stiffness is slightly larger than the MD calculated one. This might be due to the fact that the shear stiffness and layer modulus are calculated in MD simulations under biaxial loading conditions which may limit the possible outof-plane deformation of the graphyne sheet and therefore the assumption of pure 2D configuration in the analytical model can be well met. However, the in-plane stiffness is calculated under a uniaxial stress which provides less constraint to the out-of-plane deformation and thus a larger deviation of the calculated in-plane stiffness from the analytically predicted one. a
b Analytical model MD simulation Empirical equation
20
Layer Modulus (N/m)
Shear stiffness (N/m)
30
10
0
2
4
6
8
10
Number of acetylenic linkages
100 Analytical model MD simulation Empirical equation
75
50
25
0
2
4
6
8
10
Number of acetylenic linkages
Figure 5. Dependence of the in-plane shear stiffness (a) and the layer modulus (b) on the number of acetylenic linkages.
5. Scaling Laws The introduction of acetylenic chains causes an obvious chain-length dependent mechanical properties of graphynes. In other words, the mechanical properties of 14
ACCEPTED MANUSCRIPT graphynes depends significantly on the length of acetylenic chains. This is intrinsically attributed to the change in bond density. Previous studies have reported that the bond density plays an important role in determining the Young’s modulus and Poisson’s ratio of the carbon allotropes [37, 43]. Sun et al. [37] evaluated the mechanical properties of 11 different graphene allotropes and showed that the elastic modulus increases with the increasing bond density. Cranford and Buehler [50], using a simple spring-network model, showed that the in-plane stiffness of -graphyne is approximately inversely proportional to the number n of acetylenic linkages. Hou et al. [43] showed that the in-plane stiffness of -graphyne is proportional to the effective bond density rather than the real bond density. The concept of the effective bond density is originated from the analytical results obtained by the stick-spiral model, and can thus more accurately capture the decreasing trend of the scale effect on the mechanical properties of graphynes. In this section, we will establish the scaling laws for the elastic properties of graphyne based on the concept of the effective bond density, i.e., obtain empirical equations of how mechanical properties of -graphyne scale with the index number n (i.e., the length of acetylenic chains). The layer modulus of a 2D material represents its ability of resisting deformation under a hydrostatic stress/stain condition. When a 2D material is upon an in-plane hydrostatic stress, its all bonds will equally share the in-plane stress. In this sense, it is expected that the layer modulus is proportional to the real bond density
bond, hence we assume
15
ACCEPTED MANUSCRIPT L
1 bond 2
(21)
where is a coefficient. This linear relation is confirmed by MD calculations, as shown in Fig. 6. However, the in-plane stiffness of -graphyne is not a linear function of the real bond density (see Fig. 6), as has been observed for -graphyne too [43]. We thus define the effective bond density, eff, on which the in-plane stiffness of a 2D material is proportionally dependent, i.e., YS eff
(22)
For 2D isotropic materials, we further have
1
YS 1 eff 2L bond
(23)
It is seen that the scaling laws for the elastic properties of graphynes can be obtained once the effective bond density is determined.
100
Stiffness (N/m)
80
In-plane sitffness Layer modulus
60 40 20 0
0.1
0.2
0.3
0.4 -2
Real bond density (Å )
16
0.5
ACCEPTED MANUSCRIPT Figure 6. MD results for the in-plane stiffness and layer modulus as functions of the real bond density. The layer modulus is almost linearly dependent on the real bond density, but the in-plane stiffness is not.
To determine the effective bond density, we take graphene structure as a reference configuration, and define the relative effective bond density in -graphyne as
eff
eff ge eff
(24)
ge where eff is the effective bond density in graphene. The relative real bond density in
-graphyne is bond
bond 3 ge bond 4n 3
(25)
ge where bond is the real bond density in graphene. For simplicity a≈b≈c≈1 is used in
deriving Eq. (25) and will be used in the following to determine the effective bond density. If we recall Eq. (A6), we find that f1/f2 decreases drastically with n, e.g., from 0.5 for n=0 (graphene) to 0.001 for n=5. This means that, when the graphyne sheet subjects to uniaxial tension along armchair direction, the applied force will mainly deliver along the path shown in Fig. 7, further indicating that the effective bond number of acetylenic chains in the zigzag direction should be less than their real bond number. We see also that, the force delivery path is not a straight line but a 17
ACCEPTED MANUSCRIPT kinked one, which will lead to a lower in-plane stiffness due to a smaller deformation resistance. The deformation resistance is obviously associated with the kink offset distance. Here we introduce a structural parameter d1/d (in which d1 and d are defined in Fig.7) to account for the effect of the kinked force delivery path on the inplane stiffness of graphyne. For -graphyne,
3 4n 3
(26)
We find that, the relative effective bond density defined by the following Eq. (27) can well capture the scale effect on the mechanical properties, i.e.
eff bond
1 ( ) 2 3 1 9 2 2 1 ( / bond ) (4n 3) 2 2( 4n 3)
(27)
Figure 7. The main force transfer path (colored yellow) in graphyne under an armchair direction tension. The elastic properties of -graphyne can thus be empirically expressed as (derivation can be found in Appendix B) 18
ACCEPTED MANUSCRIPT
YS
1
ge 3 1 9 Y 2 S (4n 3) 2 2( 4n 3)
9
1
9
(28)
ge 2 2 2 2(4 n 3) 2 2(4 n 3 )
GS
3GSge 1 3 9 9 (4n 3) 2 2(4n 3) 2 2 2(4n 3) 2
L
3 Lge 4n 3
(29)
(30)
(31)
The results predicted by the above empirical Eqs. (28)-(31) are shown in Figs. 3-5. It is seen that these equations can well reproduce the MD simulation data, further confirming that the mechanical properties of -graphyne are closely related to the effective bond density. Owing to the physical basis of the relative effective bond density, the scaling laws presented here can establish direct connections between the mechanical properties of -graphyne and pristine graphene, and have demonstrated a useful approach for linking mechanical properties of graphene allotropes to those of graphene by determining the relative effective bond density of graphene allotropes.
6. Conclusions Based on a stick-spiral model, we have presented an analytical model for
graphyne, and obtained close-form expressions for its elastic properties. Based on these expressions, the structure dependent elastic properties of graphyne are investigated. It is found that the in-plane stiffness, in-plane shear stiffness, and layer 19
ACCEPTED MANUSCRIPT modulus of graphyne significantly decreases, while the Poisson’s ratio slightly increases, with the increasing length of acetylenic chains. These analytical predictions are in excellent agreements with MD simulations. We develop also the scaling laws for the elastic properties of graphyne by determining the relative effective bond density. These scaling laws establish direct connections between the mechanical properties of -graphyne and graphene. Although the present scaling laws are obtained for -graphyne, the approach developed here can be applied to other graphene allotropes. The present findings provide a fundamental understanding of the mechanical properties of graphyne, and a useful approach for linking mechanical properties of graphene allotropes to those of graphene in an efficient way.
Acknowledgements The authors acknowledge the financial supports from the NSF of China (Grant No. 11425209) and Shanghai Pujiang Program (Grant No. 13PJD016). The MD simulations were carried out on the computing platform of the International Center for Applied Mechanics in Energy Engineering (ICAMEE), Shanghai University.
20
ACCEPTED MANUSCRIPT Appendix A The lengths of the unit cell can be expressed as functions of bond lengths and bond angles U A 2 n 1 bA 2ncA 2aZ cos
(A1)
n k 2aA 4 bZ cos bZ cZ cos i ) 2aZ cos k 1 i 1 n k U Z 4 bZ sin ( bZ cZ sin( i )) 2aZ sin k 1 i 1
(A2)
The variances of UA and UZ can thus be obtained as dU A
2 f1 f1 f 2 32 Caa ( f1 f 2 ) + Kn 2Ka 4Cab (2Cab Caa )
(A3)
3(21Cab 2 Caa )( f1 f 2 )ro 2 3 f 2 3 ro 2 9f f f 2 1 2 + 2Ka Kn 4Cab (2Cab Caa ) 2Cbc
dU Z
3 f1 f 2 21Cab 2 Caa r02 3 f 2 3 r02 3 f2 3( f1 f 2 ) Kn 2K a 4Cab Cab 2Caa 2Cbc
(A4)
To assure the geometry continuity, the two expressions of dUA in Eq. (A3) should be equal to each other, leading to f1 f 2
(A5)
with 2 1 1 3 2 Caa 1Cab r0 4 Caa Cab Cab 2Ka Kn
21
5 3 2 Caa 1Cab ro2 33 r02 1 4 Caa Cab Cab 2Cbc 2Ka 2Kn
(A6)
ACCEPTED MANUSCRIPT Appendix B When the relative effective bond density is defined in Eq. (27), the elastic properties of -graphyne can be derived as YS eff YSge
1
ge 3 1 9 YS (4n 3) 2 2(4n 3) 2
ge eff eff eff 1 1 ge 9 9 1 ge 2 2 bond bond bond 2 2(4 n 3) 2 2 (4 n 3)
(B1) (B2)
3 1 9 2 2(4n 3) 2 (4 n 3) Y S GS 2GSge (1 ge ) 2(1 ) 3 1 ge 9 9 2 2 2 2 2(4n 3) 2 2(4n 3)
3GSge 1 3 9 9 2 2 (4n 3) 2 2(4n 3) 2 2(4n 3) L bond Lge
22
3 Lge 4n 3
(B3) (B4)
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