The Young's moduli of three types of carbon allotropes: a molecular mechanics model and a finite-element method

Abstract

The close-form expressions of the Young's moduli and the fracture stresses of cyclicgraphene, graphyne and supergraphene along their armchair and zigzag directions are derived based on a molecular mechanics model. Checking against present finite-element calculations of their Young's moduli shows that the explicit solutions are reasonable. The obtained analytical solutions should be of great help for understanding the mechanical properties of the graphene-like materials.

1. Introduction

When graphene was exfoliated out by Novoselov et al. at first in 2004 [1], it has been widely considered as a very promising material for applications in nanoengineering due to their excellent mechanical and electronic properties [2–5]. Recently, the future application prospects of graphene-like materials have attracted much interest and stimulated numerous studies [6–9]. Although the large homogeneous sheets of other carbon allotropes, such as supergraphene, cyclicgraphene and graphyne have not been reported so far, great efforts have been made to synthesize and assemble a precursor and subunit of graphyne from a new chemical method [10,11]. In particular, their physical and chemical properties such as stability, thermal conductivity and electronic properties have been investigated systematically by density functional theory [12–15].

Despite their importance and the studies of available molecular dynamics (MD) simulations and continuum modelling [16], the link between molecular and continuum descriptions of their mechanical properties is not established yet.

In order to overcome the limitations of the atomistic simulations and continuum modelling, both of the ‘stick-spiral’ model (SSM) [17] and the beam model (using finite-element (FE) method) [18,19] based on the interatomic potentials are effectively developed to characterize the mechanical properties of the three types of carbon allotropes (cyclicgraphene, graphyne and supergraphene) in this paper. Furthermore, the close-form expressions of their Young's moduli and fracture stresses in armchair and zigzag sheets are derived based on the molecular mechanics model. The explicit solutions of the SSM are further validated from current FE calculations based on the beam model.

2. Young's moduli of three types of carbon allotropes in zigzag and armchair sheets based on the molecular mechanics model

The molecular structures for three carbon allotropes are shown in figure 1. The network of cyclicgraphene is composed of C3 and C12 circles. Carbon atoms in C3 circle are interconnected by carbon–carbon bonds and there is an olefinic bond (−C=C−) between C3 circle and C3 circle [12]. The network of graphyne can be formed by connecting each hexagon C6 by linear carbon chain which is formed by inserting acetylenic bond (−C≡C−) into carbon–carbon bond [12]. Supergraphene contains two kinds of chemical bonds, in which the bonds at the apex of the hexagon are more stable than the other bonds [12]. All of the three allotropes consist of two groups via the classification of Heimann et al. [20] based on valence orbital hybridization (N). The cyclicgraphene belong to the system with pure sp² hybridization type (N=2), while the others belong to the mixed sp²+sp¹ hybridization type (1<N<2). The detailed types of the bonds are shown in figure 2.

Figure 1.

Atomic structures for three carbon allotropes of (a) cyclicgraphene (zigzag), (b) graphyne (zigzag) and (c) supergraphene (zigzag) (unit cells are marked with dark parallelograms).

Figure 2.

Bond lengths of three carbon allotopes of (a) cyclicgraphene, (b) graphyne and (c) supergraphene in the energy minimized structures [16].

In the SSM of the molecular mechanics, the total energy, U, at small strains can be expressed as the sum of energies [17]

$$Et=U_{\rho }+U_{\theta }=\frac{1}{2}\sum _i{K}_i{(\mathrm{d}{b}_i)}^2+\frac{1}{2}\sum _j{C}_j{(\mathrm{d}{\theta }_j)}^2,$$ 2.1

where U_ρ and U_θ are energies associated with bond stretching and angle variation, K_i and C_j are the force constants associated with bond stretching and angle variation, db_i and dθ_j are the elongation of bond i and the variance of bond angle j, respectively.

In the SSM [17], the twisting moment M_j resulting from bond angle variation and the stretching force F_i resulting from bond elongation can be expressed as

$$M_j=C_j\hspace{0.17em}\mathrm{d}{\theta }_j\mathrm{and}{F}_i=K_i\hspace{0.17em}\mathrm{d}{b}_i.$$ 2.2

To obtain the Young's moduli of all the carbon allotropes, each infinite carbon allotrope is considered to overcome the boundary effect from the SSM. As shown in figures 2 and 3, there are eight chemical bond lengths a (−C−C−), b (−C=C−), c (aromatic bond), d(−C−C−), e (−C≡C−), g (analogical aromatic bond), h (−C=C−) and i (−C=C−) [16] and ten bond angles α, β, θ, φ, ω, η, ϕ, γ, δ and ξ. The longitudinal external tensile stress will result in bond elongation da, db, dc, dd, de, dg, dh and di and bond angle variances dα, dβ, dθ, dφ, dω, dη, dϕ, dγ, dδ and dξ. In the model, an elastic stick with an axial stiffness of K is used to address the relationship of external force versus bond length variation of carbon–carbon bond. The spiral spring with a stiffness C is employed to describe the twisting moment resulting from an angular distortion of bond angle. Three types of carbon allotropes subjected to a tensile force along the zigzag direction are shown in figure 3.

Figure 3.

Stress analysis scheme of three carbon allotropes subjected to an external force along the zigzag direction. (a) Cyclicgraphene, (b) graphyne and (c) supergraphene.

A unit cell of graphyne consists of five atomic structure models, which is shown on the left side of figure 3b and the other three atomic structure models is shown on the right side of figure 3b. As shown in figure 2, we denote the force f contributed by bond b, c, e and g along the external tensile direction so that the total force of cyclicgraphene, graphyne and supergraphene on the sheets is 2f, 6f and 2f, respectively. For cyclicgraphene, we define the force along the vertical bond is f₁, while the force along the bond a is f₂. According to the force balance, it can be derived that

$$f_1=f\mathrm{and}{f}_2=0.$$ 2.3

By decomposing the force f into two directions for three carbon allotropes, of which one is along the bond direction and the other perpendicular to the bond, force equilibrium of bond extension leads to

$$\begin{array}{rl}f\sin \left(\frac{\theta }{2}\right)& =K\mathrm{d}b,f_2=K\mathrm{d}a,\\ f\sin \left(\frac{\phi }{2}\right)& =K\mathrm{d}c,f\sin \left(\frac{\eta }{2}\right)=K\mathrm{d}d,f\sin \left(\frac{\eta }{2}\right)=K\mathrm{d}e\\ \mathrm{and}f\sin \left(\frac{\delta }{2}\right)& =K\mathrm{d}g,f\sin \left(\frac{\delta }{2}\right)=K\mathrm{d}h,f\sin \left(\frac{\delta }{2}\right)=K\mathrm{d}i.\end{array}\}$$ 2.4

By dividing bond b, c, e and i into two halves, the twisting moment on the right half by the left half is $(\hspace{0.17em}fb/2)\cos (\theta /2)$, $(\hspace{0.17em}fc/2)\cos (\phi /2)$, $(\hspace{0.17em}f(2d+e)/2)\cos (\eta /2)$ and $(\hspace{0.17em}f(2g+2h+i)/2)\cos (\delta /2)$, respectively. Then four equilibrium equations about bond angle variance can be derived as

$$\begin{array}{rl}\frac{fb}{2}\cos \left(\frac{\theta }{2}\right)& =C\mathrm{d}\theta -C\mathrm{d}\varphi ,\\ \frac{fc}{2}\cos \left(\frac{\phi }{2}\right)& =C\mathrm{d}\phi -C\mathrm{d}\gamma ,\\ \frac{f(2d+e)}{2}\cos \left(\frac{\eta }{2}\right)& =C\mathrm{d}\eta -C\mathrm{d}\omega \\ \mathrm{and}\frac{f(2g+2h+i)}{2}\cos \left(\frac{\delta }{2}\right)& =C\mathrm{d}\delta -C\mathrm{d}\xi .\end{array}\}$$ 2.5

Similarly, we can derive

$$f\left(\frac{b}{2}\cos \left(\frac{\theta }{2}\right)+\frac{a}{2}\cos \left(\frac{\alpha }{2}\right)\right)-f_1\frac{a}{2}\cos \left(\frac{\alpha }{2}\right)=C\mathrm{d}\alpha -C\mathrm{d}\beta .$$ 2.6

From geometrical relationships along the zigzag direction, it can be shown that

$$\begin{array}{rl}\alpha +2\beta & =2\pi ,\theta +2\varphi =2\pi ,\\ \phi +2\gamma & =2\pi ,\eta +2\omega =2\pi \\ \mathrm{and}\delta +2\xi & =2\pi .\end{array}\}$$ 2.7

By differentiating both sides of equation (2.7), it is derived that

$$\begin{array}{rl}\mathrm{d}\alpha +2\mathrm{d}\beta & =0,\mathrm{d}\theta +2\mathrm{d}\varphi =0,\\ \mathrm{d}\phi +2\mathrm{d}\gamma & =0,\mathrm{d}\eta +2\mathrm{d}\omega =0\\ \mathrm{and}\mathrm{d}\delta +2\mathrm{d}\xi & =0.\end{array}\}$$ 2.8

The axial strains of cyclicgraphene, graphyne and supergraphene along the zigzag direction are defined as ε₁, ε₂ and ε₃. Using equations (2.3)–(2.8), together with the relation θ=ϕ=2π/3, α = π/3, β=5π/6, φ=γ=η=ω=2π/3 and δ=ξ=2π/3, the axial strains along the zigzag direction can be defined as

$$\begin{array}{rl}{\epsilon }_1& =\frac{\mathrm{d}(b\sin (\theta /2)+2a\sin (\alpha /2))}{b\sin (\theta /2)+2a\sin (\alpha /2)}=\frac{(3f/4K)+(b^{2}f/24C)+\left(\sqrt{3}abf/12C\right)}{\left(\sqrt{3}/2\right)b+a},\\ {\epsilon }_2& =\frac{\mathrm{d}(2c\sin (\phi /2)+(2d+e)\sin (\eta /2))}{2c\sin (\phi /2)+(2d+e)\sin (\eta /2)}=\frac{(15f/2K)+(f{c}^2/6C)+(f{(2d+e)}^2/12C)}{\sqrt{3}(2c+2d+e)}\\ \mathrm{and}{\epsilon }_3& =\frac{\mathrm{d}((2g+2h+i)\sin (\delta /2))}{(2g+2h+i)\sin (\delta /2)}=\frac{(15f/2K)+(f{(2g+2h+i)}^2/12C)}{\sqrt{3}(2g+2h+i)}.\end{array}\}$$ 2.9

We define the width of unit cell perpendicular to the tensile direction as L₁, L₂ and L₃. From geometrical relationships along the zigzag, it can be shown that

$$\begin{array}{rl}{L}_1& =4\left(\frac{b}{2}+a\cos \left(\frac{\alpha }{2}\right)+\frac{b}{2}\cos \frac{\theta }{2}\right),\\ L_2& =3\left(c+2c\cos \left(\frac{\psi }{2}\right)+2d+e\right)\\ \mathrm{and}{L}_3& =3\hspace{0.17em}(2g+2h+i).\end{array}\}$$ 2.10

When a sheet with the width L of unit cell perpendicular to the tension direction is subjected to a tensile force F, Young's modulus Y can be defined as

$$Y=\frac{F}{Lt\epsilon },$$ 2.11

where t is the thickness of the sheet, and F=2f, 6f and 2f for cyclicgraphene, graphyne and supergraphene, respectively. The thicknesses t of cyclicgraphene, graphyne and supergraphene are 3.47, 3.46 and 3.64 Å, respectively [16].

Substituting equations (2.9) and (2.10) into equation (2.11), together with the thickness t₁, t₂ and t₃, the Young's moduli Y ₁, Y ₂ and Y ₃ of cyclicgraphene, graphyne and supergraphene can be derived as

$$\begin{array}{rl}{Y}_1& =\frac{1}{\sqrt{3}{t}_1((3/4K)+(b^2/24C)+\left(\sqrt{3}ab/12C\right))},\\ Y_2& =\frac{2\sqrt{3}}{t_2((15/2K)+(c^2/6C)+({(2d+e)}^2/12C))}\\ \mathrm{and}{Y}_3& =\frac{2}{\sqrt{3}{t}_3((15/2K)+({(2g+2h+i)}^2/12C))}.\end{array}\}$$ 2.12

In the previous literature [16], bond lengths can be derived as shown in figure 2. When the data of K=742 nN nm⁻¹, C=1.42 nN nm⁻¹ [17], a=1.59 Å, b=1.30 Å, c=1.40 Å, d=1.39 Å, e=1.33 Å, g=1.40 Å, h=1.34 Å, i=1.34 Å, t₁=3.47 Å, t₂=3.46 Åand t₃=3.64 Åare given [16], the Young's moduli of cyclicgraphene, graphyne and supergraphene are derived as 461.19, 448.53 and 84.81 GPa, respectively.

For three types of carbon allotropes subjected to a tensile force along the armchair direction, their molecular structures are shown in figure 4.

Figure 4.

Stress analysis scheme of unit cells of armchair sheets. (a) Cyclicgraphene, (b) graphyne and (c) supergraphene.

For armchair sheets, the similar calculations can be obtained

$$\begin{array}{rl}{\epsilon }_1& =\frac{(17f/2K)+(f{b}^2/4C)+\left(\sqrt{3}fab/6C\right)}{2\sqrt{3}a+3b},\\ {\epsilon }_2& =\frac{(45f/2K)+(f{c}^2/2C)+(f{(2d+e)}^2/4C)}{3(2c+2d+e)}\\ \mathrm{and}{\epsilon }_3& =\frac{(15f/2K)+(f{(2f+2g+h)}^2/12C)}{\sqrt{3}(2f+2g+h)}.\end{array}\}$$ 2.13

and

$$\begin{array}{rl}{Y}_1& =\frac{2\sqrt{3}}{t_1((17/2K)+(b^2/4C)+(\sqrt{3}ab/6C))},\\ Y_2& =\frac{6\sqrt{3}}{t_2((45/2K)+(c^2/2C)+({(2d+e)}^2/4C))}\\ \mathrm{and}{Y}_3& =\frac{2}{\sqrt{3}{t}_3((15/2K)+({(2f+2g+h)}^2/12C))}.\end{array}\}$$ 2.14

From above available data, the Young's moduli of cyclicgraphene, graphyne and supergraphene are derived as 535.77, 448.53 and 84.81 GPa, respectively.

3. Young's moduli of three types of carbon allotropes along the zigzag and armchair directions based on finite-element method

The molecular mechanics-based space-frame structure model was proposed and was successfully used to calculate the equivalent macro mechanical properties of carbon nano-materials such as carbon nanotube [18,19,21,22], graphene [23] and Boron Nitride [24]. The FE beam structures of the three typical cyclicgraphene, graphyne and supergraphene sheets can be easily built from the coordinates of the MD structures [16]. In order to reduce the boundary effect on their Young's moduli, the three typical rectangular sheets with the sizes of 111.35×94.73 Å , 125.67×105.71 Å and 124.03×103.64 Å are modelled [7] (figure 5). The Young's moduli of the beams and their circular cross-section radii are determined as [23],

$$E_{\mathrm{beam}}=\frac{K_r^{2}r}{4\pi K_{\theta }},$$ 3.1

and

$$d=4\sqrt{\frac{K_{\theta }}{K_r}},$$ 3.2

where K_r=742 nN nm⁻¹, K_θ=1.42 nN nm [17] and r is the bond length (in §2, there are seven chemical bond lengths a, b, c, d, e, g, h and i in the three typical sheets). As the equivalent Young's modulus of graphene nanoribbon is almost independent on Poisson's ratio v of the beam [25], here we choose v=0.1 of the beam in all the following FE calculations.

Figure 5.

The finite-element structures of (a) cyclicgraphene, (b) graphyne and (c) supergraphene.

All the present FE calculations are performed using the commercial ANSYS 14.0 package with 2-node BEAM 188 element. One end part of each FE model is fixed and the other end part is given by a displacement of 0.1 Åalong the armchair and zigzag directions (figure 6), respectively. The relationship of reaction force versus displacement at the end part along the armchair and zigzag directions of the three structures are captured in the simulations, as shown in figure 7.

Figure 6.

One end part is fixed and the other end part is subjected to a displacement of 0.1 Åalong the zigzag direction of cyclicgraphene sheet.

Figure 7.

Reaction force–displacement curves along different directions of three carbon allotropes based on the finite-element calculations.

Based on the obtained results of reaction force versus displacement, the equivalent Young's modulus could be calculated by equation (3.3) as follows:

$$E=\frac{F/Wt}{\mathrm{\Delta }L/L},$$ 3.3

where ΔL is the displacement at the end part of the sheet, F is the reaction force, W and L are the width and the length of the sheet, respectively.

The Young's moduli of cyclicgraphene, graphyne and supergraphene along their armchair and zigzag directions are derived based on the SSM and FE calculations in table 1. The FE results agree well with those of SSM for cyclicgraphene and graphyne, in which the maximum relative error is around 17%. However, the difference between the FE results (beam model) and SSM is more than 40% for supergraphene. In our previous work [25], the difference of the two methods (SSM and beam model) is clarified in detail, in which the SSM overestimates and the beam model underestimates the mechanical properties in narrow graphene nanoribbons under in-plane bending condition by comparison with MD simulations (the maximum difference can be up to 300%) [25]. In figures 3c and 4c, 20 bonds bear the in-plane bending even if the supergraphene is subjected to a uniaxial tension, which is the main reason to lead to the large difference. The Young's moduli of graphyne (170 N m⁻¹ (armchair) and 224 N m⁻¹ (zigzag)) were obtained from MD simulations in previous work [7], while the Young's moduli of graphyne from our FE calculations are 159.5 N m⁻¹ (armchair) and 158 N m⁻¹ (zigzag) (The Young's modulus is the product of the conventional Young's modulus with the thickness), respectively. We stressed that the previous MD results were obtained using REAXFF potential, in which the functions of the REAXFF potential are completely different with present harmonic potential (equation (2.1)). The comparison between previous MD results with present SSM or FE results is not suitable. Therefore, we only compare the FE results with our molecular mechanics model as the parameters of the FE model are obtained from the present harmonic potential.

Table 1.

TheYoung's moduli of cyclicgrphene, grapheme and supergraphene along their armchair and zigzag directions using SSM and FE calculations (the superscript characters of z and a represent zigzag and armchair sheets, respectively).

Young's modulus (GPa)	cyclicgraphene	graphyne	supergraphene	graphene
FE	438.51a	461.18a	50.80a	1040a
	447.70z	457.20z	56.85z	1042z
SSM	461.19a	448.53a	84.81a	1060a
	535.77z	448.53z	84.81z	1060z

4. Fracture stresses of three carbon allotropes

To obtain the relationship between the fracture stresses of different carbon allotropes and the tensile strength of different bonds, the bond strengths of carbon allotropes must be well studied. If a chemical bond is more unsaturated than other chemical bonds, it will break more easily. For cyclicgraphene, the olefinic bond is more fragile than the single bond (−C−C−) when they are subjected to a same force. Similarly, the acetylenic bond breaks more easily than others of graphyne, and the olefinic bond is the most fragile bond for supergraphene.

As shown in figure 1, each sheet can be formed by a number of unit cells. For example, supergraphene is constructed from some regular hexagonal supercell. Along the zigzag direction, the vertical carbon–carbon bond (bond a, figure 2) and slanting olefinic bond (bond b, figure 2) are most probably to break for cyclicgraphene, while slanting acetylenic bond and olefinic bond are most fragile for graphyne and supergraphene, respectively. For cyclicgraphene, we assume that the vertical carbon–carbon bond break earlier than slanting olefinic bond.

We define fracture force of the most fragile bond as f₀, and then the tensile strength of a bond can be defined as

$$\sigma =\frac{4f_0}{\pi d^2}.$$ 4.1

When the number of the unit cell is equal to n, the total external force F₁, F₂ and F₃ of cyclicgraphene, graphyne and supergraphene can be derived as

$$\begin{array}{rl}{F}_1& =(2n+2)f_0,\\ F_2& =(6n+2)f_0\sin \left(\frac{\eta }{2}\right)\\ \mathrm{and}{F}_3& =(2n+2)f_0\sin \left(\frac{\delta }{2}\right).\end{array}\}$$ 4.2

The fracture stress of three carbon allotropes can be defined as

$${\sigma }_{\mathrm{s}}=\frac{F}{Lt}.$$ 4.3

As each sheet consists of many unit cells (figure 5), the width L₁, L₂ and L₃ of cyclicgraphene, graphyne and supergraphene can be defined as

$$\begin{array}{rl}{L}_1& =(n+1)\left(2a\cos \left(\frac{\alpha }{2}\right)+b+2b\cos \left(\frac{\theta }{2}\right)\right)+n\left(2a\cos \left(\frac{\alpha }{2}\right)+b\right),\\ L_2& =(n+1)\left(4c\cos \left(\frac{\phi }{2}\right)+c+2(2d+e)\right)+n\left(2c\cos \left(\frac{\phi }{2}\right)+2c+2d+e\right)\\ \mathrm{and}{L}_3& =(n+1)\left(1+2\cos \left(\frac{\delta }{2}\right)\right)(2g+2h+i)+n(2g+2h+i).\end{array}\}$$ 4.4

Substituting equations (4.1), (4.4), (3.2) and (4.3) into (4.2) gives the fracture stresses σ_s1, σ_s2 and σ_s3 along the zigzag direction as follows:

$$\begin{array}{rl}{\sigma }_{s1}& =\frac{2\pi K\theta \sigma }{t_1{K}_r(2a\cos (\alpha /2)+b+2b\cos (\theta /2)+(n/n+1)(2a\cos (\alpha /2)+b))},\\ {\sigma }_{s2}& =\frac{2\pi K\theta \sigma \sin (\eta /2)}{t_2{K}_r((n+1)/(3n+1)(4c\cos (\phi /2)+c+2(2d+e))+(n/(3n+1))(2c\cos (\phi /2)+2c+2d+e))}\\ \mathrm{and}{\sigma }_{\mathrm{s}3}& =\frac{2\pi K\theta \sigma \sin (\delta /2)}{t_3{K}_r((1+2\cos (\delta /2))(2g+2h+i)+(n/(n+1))(2g+2h+i))}.\end{array}\}$$ 4.5

However, if the slanting olefinic bond is the first to break for cyclicgraphene, then the total external force F₁ of cyclicgraphene should be expressed as

$$F_1=(2n+2)f_0\sin \left(\frac{\theta }{2}\right).$$ 4.6

The fracture stresses σ_s1 of cyclicgraphene should be expressed as

$${\sigma }_{s1}=\frac{2\pi K\theta \sigma \sin (\theta /2)}{t{K}_r(2a\cos (\alpha /2)+b+2b\cos (\theta /2)+(n/(n+1))(2a\cos (\alpha /2)+b))}.$$ 4.7

Similarly, the fracture stresses along the armchair direction of three carbon allotropes can be expressed as

$$\begin{array}{rl}{\sigma }_{s1}& =\frac{\pi K\theta \sigma }{t_1{K}_r(2a\sin (\alpha /2)+a+2b\sin (\theta /2))}\frac{n+2}{n+1},\\ {\sigma }_{s2}& =\frac{\pi K\theta \sigma }{t_2{K}_r(2(2d+e)\sin (\eta /2)+4c\sin (\phi /2))}\frac{n+2}{n+1}\\ \mathrm{and}{\sigma }_{s3}& =\frac{\pi K\theta \sigma }{2t_3{K}_r(2g+2h+i)\sin (\delta /2)}\frac{n+2}{n+1}.\end{array}\}$$ 4.8

From above analysis, the size-dependent fracture stresses of three carbon allotropes subjected to a tensile force along the zigzag and armchair directions are obtained in equations (4.5) and (4.7) based on the SSM, respectively.

5. Conclusion

In summary, the close-form expressions of the Young's moduli and the fracture stresses of cyclicgraphene, graphyne and supergraphene along their armchair and zigzag directions are derived based on a molecular mechanics model. Checking against present FE calculations of their Young's moduli shows that the explicit solutions are reasonable. The obtained analytical solutions should be of great help for understanding the mechanical properties of the graphene-like materials.

Authors' contributions

B.Z. and J.S. performed all the calculations, interpreted the initial results and wrote the initial manuscript. P.Y. and J.Z. interpreted the final results and edited the final manuscript. All authors gave final approval for publication.

Competing interests

We declare we have no competing interests.

Funding

The support provided by the National Natural Science Foundation of China (grant nos. 11572140 and 11302084), the Programs of Innovation and Entrepreneurship of Jiangsu Province, the Fundamental Research Funds for the Central Universities (grant no. JUSRP11529, JG2015059), the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (NUAA) (grant no. MCMS-0416G01), the Open Fund of Key Laboratory for Intelligent Nano Materials and Devices of the Ministry of Education (NUAA) (grant no. INMD-2015M01) is kindly acknowledged.