The mechanical properties of single-crystal and ultrananocrystalline diamond: A theoretical study

Chemical Physics Letters 414 (2005) 351–358 www.elsevier.com/locate/cplett

The mechanical properties of single-crystal and ultrananocrystalline diamond: A theoretical study Jeffrey T. Paci

a,*

, Ted Belytschko b, George C. Schatz

a

a

b

Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3113, USA Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3111, USA Received 1 July 2005; in final form 1 August 2005

Abstract We examine the mechanical properties of single-crystal and ultrananocrystalline diamond (UNCD) by simulating their fracture using semiempirical quantum mechanics and density functional theory. Our results predict a failure strain of 0.13 and a fracture stress of 100 GPa for UNCD, which are 37% and 43%, respectively, that of single-crystal diamond. The YoungÕs modulus of UNCD is E = 1.05 TPa which is only slightly smaller than that of single-crystal diamond (E = 1.09 TPa). The UNCD fracture stress value (rf = 100 GPa) is very large compared to that observed experimentally (rf < 5 GPa). We use Griffith theory to show that this difference is due to defects in UNCD.
2005 Elsevier B.V. All rights reserved.

1. Introduction Thin diamond films composed of extremely small (3– 5 nm) diamond grains and atom-wide grain boundaries (
0.2 nm wide) can now be produced using plasmaenhanced chemical vapor deposition techniques [1–4]. The material, which is called ultrananocrystalline diamond (UNCD), is very strong with mechanical properties (Vickers hardness, HV = 88 GPa [5], and YoungÕs modulus, E = 950 GPa [6]) similar to those of singlecrystal diamond (HV = 100 GPa [7], and E = 1050 GPa [8,9]). The films are also smooth, with a mean surface roughness of
20–40 nm [10,11]. Nitrogen-doping of the plasma can be used to make these films conduct significant amounts of electric current [12–17]. These properties combine to make UNCD an excellent candidate for use in the development of microelectromechanical systems [5,18].

*

Corresponding author. Fax: +1 847 467 4996. E-mail address: [email protected] (J.T. Paci).

0009-2614/$ - see front matter
2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.08.019

We are interested in examining the strain-to-fracture behavior of these UNCD films using semiempirical quantum mechanics and density functional theory. Strain is applied to the diamond cluster, and the positions of the atoms are adjusted to minimize its energy, subject to the constraint provided by the strain. The structure of UNCD films makes a complete examination of their fracture properties difficult. In addition to the grain boundaries which form between adjacent 3–5 nm diamond grains, these films contain other types of defects. Groups of diamond grains tend to form structures which are
100 nm in size. Interfaces form between adjacent groups of diamond grains, producing grain boundaries with structures which are not well understood. In this work, our primary focus will be on the mechanical properties of atom-wide grain boundaries which form between adjacent 3–5 nm diamond grains. References to the term Ôgrain boundaryÕ in the remainder of the Letter are meant to refer to these grain boundaries. The effect of defects larger than these grain boundary structures will, however, also be discussed.

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There are a variety of methods which can be used to approach these types of problems. Empirical bond-order potentials, such as the Tersoff–Brenner potential, have been used successfully to explain features of grain boundary structures [19]. The best higher level results are generally obtained using electronic structure calculations. An example is the semiempirical quantum mechanical method MSINDO [20] we use in this work. This approach is more computationally expensive (by at least an order of magnitude) and can, therefore, only be used on smaller systems (systems containing a few hundred atoms). Better yet are self-consistent density functional theory approaches. The SIESTA code [21] used in this work is one example of such an approach. It is more computationally expensive than MSINDO (by about an order of magnitude) but should provide results which are as quantitatively accurate as any currently available and computationally practical method. Regardless of the method used to perform the simulations of interest, it is necessary to first calculate optimized molecular geometries, and then to calculate their energies. The geometry calculations are very time consuming compared to the energy calculations. This suggests a hybrid approach where a method like MSINDO is used to calculate structures, combined with a method like SIESTA to calculate their energies. Such a hybrid method provides the possibility of getting significantly better energies, for a modest increase in computational cost. We explore the speed and accuracy of this approach for single-crystal and UNC diamond in this work. The remainder of the Letter is organized as follows. In Section 2, the MSINDO and SIESTA methods used in our simulations are described, as are the clusters used to model single-crystal and UNC diamond. The results of simulations performed on these two clusters are the subject of Sections 3 and 4, respectively. In Section 5, we propose a reason for experimentally observed UNCD strengths based on our surface energy calculations for a grain boundary. A summary is presented in Section 6.

2. Simulation methods We use the MSINDO [20] semiempirical self-consistent field molecular orbital program to perform simulations of the fracture of UNCD. This method is based on the intermediate neglect of differential overlap (INDO) approximation. In this approximation, differential overlap between atomic orbitals on the same atom is included in one-center electron-repulsion integrals, but is neglected in two-centered electron-repulsion integrals [22]. As we will show by way of comparison to higher level simulation results and calculations using experimental constants, this quantum mechanical program can be

expected to provide semi-quantitative values for the mechanical properties of UNCD structures. The program allows for the use of periodic boundary conditions (PBCs) in three dimensions using the cyclic cluster model [23]. The essence of this model is the creation of identical environments for translationally equivalent cluster atoms. The availability of PBCs in a reasonably fast code was a key factor in choosing MSINDO. Geometry optimization in MSINDO is performed using a variable metric method, the Broyden–Fletcher– Goldfarb–Shanno algorithm [24,25]. This algorithm requires derivative calculations, one-dimensional subminimization, and N2 storage, where N is the number of grid points used in the numerical representation of the potential energy surface. Density functional theory (DFT) calculations are performed using the Spanish initiative for electronic simulations with thousands of atoms (SIESTA) [21]. It is a self-consistent DFT method which uses numerical atomic orbital basis set functions and norm-conserving pseudo-potentials. Both local density (LDA-LSD) and generalized gradient (GGA) approximations to the exchange-correlation energy are available. Localized linear combinations of the occupied orbitals are used, so computer time and memory scale linearly with system size. We have run SIESTA calculations using both LDA and GGA functionals. The Perdew–Zunger (PZ) functional has been used for our LDA calculations, and the Perdew–Burke–Ernzerhof (PBE) functional for our GGA functional calculations. All calculations were performed using a double zeta plus polarization basis set (DZP), the largest basis set available in SIESTA. Pulay mixing at levels as high as 10 was used, whenever necessary, to facilitate self-consistent field (SCF) convergence. Conjugate gradient (CG) optimization of atomic positions was performed in all of the simulations which involve the relaxation of atomic coordinates. A single k-point was used to sample the Brillouin zone. Simulations of UNCD are performed using clusters containing 208 carbon atoms. Atoms are arranged to produce the so-called R 13 twist (1 0 0) [26] grain boundary [19,27–29]. Shown in Fig. 1, it is composed of 16 planes of 13 atoms. The top four, center eight and bottom four planes have a lattice spacing that corresponds to that of perfect diamond. The top and center sections as well as the center and bottom sections are twisted by 67.4 relative to each other, resulting in grain boundaries. The two grain boundaries, although similar to each other, are not identical. Three-dimensional periodic boundary conditions are used in order to avoid unphysical cluster-edge effects. High energy twist grain boundaries, of which the R 13 twist (1 0 0) grain boundary is an example, are the most common type of grain boundaries in UNCD [28]. In

J.T. Paci et al. / Chemical Physics Letters 414 (2005) 351–358

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structure corresponding to zero strain. Single-crystal diamond simulations are performed on the center eight atom planes of the UNCD cluster, and contain 104 atoms. By fixing the unit cell in directions perpendicular to the applied load, we have considered a state of uniaxial strain. Since the Poisson ratio of diamond is 0.07, the extensional modulus in uniaxial stress is 1.01 of the modulus in uniaxial strain, and we will not distinguish between the two in the following. We do not know whether the fracture stress, rf, would be significantly effected by our assumption of uniaxial strain, as compared to uniaxial stress.

3. Single-crystal diamond

Fig. 1. The cluster used to model UNCD. This cluster is under zero strain. The horizontal arrows indicate the locations of the two grain boundaries (GB). The vertical arrows indicate the direction of applied strain.

order to investigate their strength, we strain the clusters in the (1 0 0) direction, the direction normal to these grain boundaries. The structure is 1.5 · 0.9 · 0.9 nm when not under strain. This means that the density of grain boundaries in our UNCD model is
50 times the density of grain boundaries in real UNCD. Although we would like to examine larger clusters which would make it possible to do simulations with more realistic grain boundary densities, it is not computationally practical to do so using the same high level of theory. UNCD cluster simulations are performed by starting from a geometry close to the unstrained minimum energy structure. In MSINDO, estimates for the structure are produced by setting the spacing between all planes equal to the spacing between planes in perfect diamond. For the SIESTA calculations, the initial structure was produced by first doing a CG optimization on the center eight atom planes, including relaxation of the unit cell size. The full UNCD structure was then built from these atomic coordinates. In both MSINDO and SIESTA calculations, the unit cell size is then fixed, and the position of all atoms is re-optimized. From the resulting geometry, planes of atoms are moved apart or pushed together using a uniform increase or decrease in plane spacing. The unit cell is adjusted by a length equal to the total applied strain (only in the direction in which strain is imposed), and the position of all atoms optimized. Additional strain is applied by starting from the optimized structure of the previous strain step. The structure corresponding to the minimum of the energy versus strain curve created by this procedure is defined as the

The energy versus strain curves resulting from straining the single-crystal diamond cluster shown in Fig. 2 are presented in Fig. 3a. Results produced using MSINDO and SIESTA are shown. In Fig. 3a, the unstrained (lowest energy) structure is assigned an energy of zero, and other energies reported relative to this value. No negative strain values are shown in the figure but a strain-free initial state was found by placing the cluster under sufficient compression to accurately determine the location of the energy minimum. The strains encountered in these simulations are sufficiently large so that generally the strain and stress measures must be carefully defined. However, here we are dealing with a state of uniaxial strain and a constant cross-sectional area, so most of these complications can be avoided.

Fig. 2. The cluster used to model single-crystal diamond. This cluster is under zero strain. The arrows indicate the direction in which strain is applied.

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a

b

’MSINDO’ ’PBE//MSINDO’ ’PZ’ ’PBE’

2.5

350

300

250

Stress (GPa)

2

Energy (eV/atom)

’MSINDO’ ’PBE//MSINDO’ ’PZ’ ’PBE’

1.5

1

200

150

100 0.5 50

0

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0

0.05

0.1

0.15

Strain

0.2

0.25

0.3

0.35

0.4

Strain

Fig. 3. The energy (a) and stress (b) versus strain curves resulting from straining the single-crystal diamond cluster shown in Fig. 2. Results produced using MSINDO (+), PBE//MSINDO (asterisks), PZ (empty squares), and PBE (filled squares).

We will use as a measure of strain, the stretch, defined by
¼

L  L0 DL  ; L0 L0

ð1Þ

where L and L0 are the current and initial (unstrained) lengths, and DL is the elongation. The work-conjugate stress is the first Piola–Kirchhoff stress (transpose of the nominal stress), P, which for a uniform stress state is given by [30] F P¼ ; A0

A0 EL0 2
; 2

ð6Þ

where U is the total energy. Energy points corresponding to |
| < 0.03 are used to fit the numerical data to determine E. Stresses are calculated by rðiÞ ¼

U ði þ 1Þ  U ðiÞ ½DLði þ 1Þ  DLðiÞA0

ð7Þ

and the strains by

F ¼ P. A0

ð3Þ

The response of the diamond is path-independent, so it is hyperelastic and its response can be described by a potential U so that

dU dU r¼P ¼ ¼ L0 ; ð4Þ d
dL
is equivwhere the last equality follows from Eq. (1). U alent to the specific internal energy. If we assume that the stress is linearly related to the strain near the unstrained state, then near the unstrained state r ¼ E
¼ E


A0 L0 ¼ U ¼U

ð2Þ

where A0 is the initial cross-sectional area and F is the applied force. Since the cross-sectional area is unchanged during deformation, the area is unchanged so the nominal stress equals the physical (Cauchy) stress, r, i.e. r¼

where E is YoungÕs modulus. The energy relation (Eq. (4)) can then be integrated to give

ðL  L0 Þ ; L

ð5Þ


ðiÞ ¼

DLði þ 1Þ þ DLðiÞ ; 2L

ð8Þ

where, for example, A0 = 8.57 · 1019 m2 for the MSINDO structures, and i = 1, 2, . . . ,N, where N is the number of data points on a given energy versus length curve. Some of the energy versus strain data is discontinuous. This is due to abrupt changes in atomic structure which occur at large strains. Examination of the structures indicates that the discontinuities result from small but abrupt shifts in the position of one or several atoms as strain is increased. The ÔMSINDOÕ result shown in Fig. 3a was produced by straining the cluster in MSINDO, as described in Section 2. The large decrease in energy observed for all curves as strain is increased beyond
0.3 is due to relaxation which takes place after the diamond fractures. A parabolic fit to the small-strain portion of the ÔMSINDOÕ curve produces a YoungÕs modulus value of 1.56 TPa. This value is higher than the known value for single-crystal diamond (E = 1.05 TPa [8,9]). The rf

J.T. Paci et al. / Chemical Physics Letters 414 (2005) 351–358

shown in Fig. 3b (rf = 277 GPa) is also high, as a higher level calculation (DFT) result of rf = 225 GPa has been reported [31]. In this work, Telling et al. apply strain to a cuboid unit cell (containing 8 carbon atoms). Although not reported explicitly, we assume they strained the crystal using an algorithm in which strain was applied gradually. The errors in E and rf are a shortcoming of MSINDO; we have observed similar over-predictions of YoungÕs modulus and fracture stress for the semiempirical method PM3 when applied to carbon nanotubes [32]. Optimized geometry calculations are very time consuming compared to the calculation of the energies of the resulting structures. This fact suggests the use of MSINDO to calculate structures, and a higher level method (in this case SIESTA) to calculate their energies. This hybrid approach, in which the PBE functional was used, produced the data for the curves labeled ÔPBE// MSINDOÕ in Fig. 3a,b. The structure with the lowest energy is assigned an energy of zero, and other energies reported relative to this value. Values of E = 1.01 TPa and rf = 219 GPa were obtained using PBE//MSINDO, which agree very well with diamond. Two additional curves are shown in Fig. 3a,b. These are the results of straining the single-crystal diamond structure using SIESTA. Use of the PZ functional leads to a prediction of E = 1.09 TPa and rf = 239 GPa, while the PBE functional gives E = 1.09 TPa and rf = 233 GPa. The PBE results are probably better, as the LDA often leads to over-binding. This appears to not be very important here as the two functionals produce nearly the same results. Both full-strain SIESTA results compare favorably with the experimental value of E = 1.05 TPa, and the other DFT fracture stress result reported earlier (rf = 225 GPa) [31]. In these PZ and PBE simulations, fracture occurs violently. All of the original diamond structure disappears and the cluster rearranges itself to form an amorphous carbon. In the MSINDO simulations, fracture occurs cleanly between atomic planes five and six. A comparison between PBE//MSINDO and fullstrain SIESTA results suggest that the PBE//MSINDO method is useful. It predicted E and rf values which are within 8% and 6%, respectively, of our PBE fullstrain values. This hybrid method appears, therefore, to be a practical alternative to a full-strain calculation using SIESTA, as it is nearly as accurate and much faster, as we will show later. The fact that the structures formed after fracture using PZ or PBE versus MSINDO are qualitatively different, is a concern. Failure strains for the diamond shown in Fig. 2 have also been calculated. As discussed above, PBE results are probably the most reliable. This method predicts a value of
f = 0.35, which is the same as the result reported in the Telling et al. DFT study [31]. Values pro-

355

Table 1 Summary of the mechanical properties of single-crystal diamond, and UNCD Cluster

E (TPa)


f

rf (GPa)

Single-crystal (MSINDO) Single-crystal (PBE//MSINDO) Single-crystal (PZ) Single-crystal (PBE) UNCD (MSINDO) UNCD (PBE//MSINDO) UNCD (PBE)

1.56 1.01 1.09 1.09 1.53 0.955 1.05

0.34 0.37 0.36 0.35 0.14 0.16 0.13

277 219 239 233 163 116 100

duced using our other calculation methods are all within 6% of this PBE result. YoungÕs modulus, fracture stress and failure strain values for single-crystal diamond are summarized in Table 1.

4. UNCD The energy versus strain curves resulting from straining the pure UNCD cluster shown in Fig. 1 are presented in Fig. 4a. Results produced using the MSINDO, PBE//MSINDO and PBE methods are shown. This cluster is strained using a step size of ˚ , and YoungÕs modulus values calculated DL = 0.1 A using parabolic fits to points for which |
| < 0.03. The MSINDO simulations yield E = 1.53 TPa and rf = 163 GPa (see Fig. 4b). This E value is slightly smaller than the analogous single-crystal diamond result (E = 1.56 TPa), and seems unrealistically high based on the known YoungÕs modulus of single-crystal diamond and our experience with the YoungÕs modulus for single-crystal diamond, as predicted using MSINDO. Similarly, the fracture stress value seems high. As in the case of single-crystal diamond, we have used PBE to calculate the energy of the MSINDO structures. The results are also presented in Fig. 4a,b and are labeled ÔPBE//MSINDOÕ. The YoungÕs modulus and fracture stress values associated with these curves are E = 0.955 TPa and rf = 116 GPa, respectively. As the PBE//MSINDO method, when used for single-crystal diamond, produced results which were very close to: (1) full-strain PBE calculations; (2) the results of other DFT calculations, and (3) the experimental value for E, it is reasonable to expect that this hybrid method should give at least reasonably quantitative accuracy for UNCD. To further investigate this, we have performed fullstrain calculations on the UNCD cluster, using PBE. The results, also shown in Fig. 4a,b, provide values of E = 1.05 TPa and rf = 100 GPa. These full-strain PBE results are probably the most quantitatively accurate of the three methods used in the UNCD simulations.

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J.T. Paci et al. / Chemical Physics Letters 414 (2005) 351–358

a

b 200

’MSINDO’ ’PBE//MSINDO’ ’PBE’

’MSINDO’ ’PBE//MSINDO’ ’PBE’

0.5 150

Stress (GPa)

Energy (eV/atom)

0.4

0.3

100

0.2 50 0.1

0

0 0

0.05

0.1

0.15

0.2

Strain

0

0.05

0.1

0.15

0.2

Strain

Fig. 4. The energy (a) and stress (b) versus strain curves resulting from straining the UNCD cluster shown in Fig. 1. Results produced using MSINDO (+), PBE//MSINDO (asterisks), and PBE (filled squares).

In these simulations, the UNCD fractures at both grain boundaries. Some cross-linked strings of atoms remain attach to the top and middle, and middle and bottom sections of single-crystal diamond. The three single-crystal diamond sections remain mostly intact. Fracture occurs in the MSINDO simulations in qualitatively the same way, except that only the bottom grain boundary breaks. Failure strains for the UNCD shown in Fig. 1 have also been calculated. The PBE results are probably the most reliable, with the method predicting
f = 0.13. Values produced using the other methods are all within 23% of this result. The failure strains as predicted by the three computational methods as well as E and rf values, are also summarized in Table 1. Although the most accurate, the full-strain PBE calculations are very time-consuming, requiring well over a month of computer time on a 2 GHz AMD Opteron processor. For these calculations, one strain step must be completed before the next can be started. The speed and accuracy of the PBE//MSINDO method suggests its usefulness for other calculations on these types of systems. The MSINDO component of this calculation took about 5 days on the Opteron processor. It, like the full-strain PBE calculation, requires one strain step to be completed before the next can be started. The PBE component of the PBE//MSINDO calculation only requires the performance of single-point energy calculations. Although each takes about 12 hours, they can be done simultaneously, and as soon as a given MSINDO structure is available. These hybrid calculations produce E and rf values which are 9% lower and 16% higher, respectively, than the corresponding full-strain PBE cal-

culations. Depending on the application, errors of this size may be acceptable, considering the orderof-magnitude savings in computer time the PBE// MSINDO method provides as compared to a full-strain calculation using PBE. Such hybrid simulations have obvious advantages over simply using MSINDO which overestimates E and rf by 46% and 63%, respectively, as compared to the full-strain PBE results. This especially, considering that PBE//MSINDO simulations only requires
10% more time to perform than the corresponding MSINDO simulations.

5. The mechanical properties of UNCD The full-strain PBE calculations give E = 1.05 TPa,
f = 0.13, and rf = 100 GPa, as shown in Table 1. These values suggest UNCD has a YoungÕs modulus only slightly smaller than that of single-crystal diamond, and a failure strain and fracture stress which are 37% and 43%, respectively, of the values for single-crystal diamond. This UNCD value for E agrees well with the experimental value of E = 950 GPa [6]. The fracture stress values discussed to this point have all been theoretical rf values. Indentation testing is the most often used experimental method for determining the strength of single-crystal diamond. Using a large set of such data, Field and Pickles [33] estimate rf = 4.0 GPa for singlecrystal diamond. Thus, in the case of single-crystal diamond, the theoretical rf is very large compared to experiment-based predictions for rf. From this value of rf, Field and Pickles inferred that the diamond contained sharpended defects with a length of approximately 1000 nm.

J.T. Paci et al. / Chemical Physics Letters 414 (2005) 351–358

 rf ¼

pEc 2cð1  m2 Þ

1=2 ;

ð9Þ

14 ’single-crystal’ ’UNCD’ ’UNCD experiment’

12

10

s f (GPa)

In [34], Espinosa et al. report the observation of defects which are on this length scale in their UNCD samples. They used membrane deflection experiments and found rf
1–5 GPa for their UNCD. We next examine the scale of defects expected in UNCD based on our calculations by Griffith theory [35]. The Griffith equation for a penny-shaped crack of radius c gives the fracture stress as [35]

357

8

6

4

where c is the fracture surface energy and m is the Poisson ratio. Once rf is exceeded, the crack grows, and the material fails catastrophically. This equation is well suited for estimating rf for brittle solids like diamond. First, consider such a crack in single-crystal diamond. Using E = 1.09 TPa, c = 5.3 J m2 [33] and 2c = 100 nm, one obtains rf = 13 GPa. Similarly, one can calculate that a crack of length 2c = 1130 nm will result in rf = 4.0 GPa. Similar calculations can be performed for a UNCD grain boundary. The fracture surface energy was obtained by taking the difference between the energy of the unstrained UNCD and the energy of the structure broken cleanly at the top grain boundary, as calculated using PBE. The top grain boundary is broken by increasing the separation between the fourth and fifth atomic planes by 0.9 nm relative to their position in the unstrained structure. The unit cell is increased in size by the same amount. The resulting structure is allowed to relax, and then its energy is calculated. After the relaxation, the fourth and fifth atomic planes are more than 1 nm apart. This procedure gives c = 2.6 J m2 which is 49% of the value for single-crystal diamond. There is some uncertainty associated with this surface energy estimate as its value depends on the structural details of our UNCD cluster. According to Eq. (9), a 2c = 100 nm crack in a UNCD grain boundary should then result in a fracture stress rf = 9.3 GPa. Note that although we are now considering a grain boundary which is much longer than that which exists between any pair of 3–5 nm diamond grains, we approximate its behavior using the same grain boundary structure. As mentioned, Espinosa et al. [34] reported fracture stresses for UNCD of rf
1–5 GPa. Fig. 5 shows the Griffith fracture stress based on the inter-granular surface energy computed here and E = 1.05 TPa, for c = 50–600 nm. A bar has also been drawn on the figure. This bar represents the range of fracture stresses measured by Espinosa et al. [34], and has been placed at c = 300 nm as this is the radius of the defect they show, as an example, in Fig. 13 of Ref. [34]. It can be seen that the Griffith fracture stress curve for UNCD passes through (rf = 3.8 GPa) the 1–5 GPa range reported by Espinosa et al. [34]. For comparison, the Griffith frac-

2

0 100

200

300

400

500

600

c (nm) Fig. 5. Griffith fracture stress as a function of crack length, c. Results for single-crystal diamond (solid line) and UNCD (dotted line) are shown. The range of experimentally measured UNCD fracture stress values [34] is represented by the vertical bar at c = 300 nm.

ture stress curve for single-crystal diamond is also included in Fig. 5. Thus, the experimental values and these calculations suggest that UNCD contains defects of significant size. The defects appear to be in the range reported by Field and Pickles [33] for single-crystal diamond, but the fracture stresses are lower because of the significantly lower surface energy of the inter-granular surface. Although the Griffith formula is only approximate because it neglects the non-linearity of the material and lattice trapping effects, the results are nevertheless consistent with the observations. It is clear that the experimental strength of UNCD falls far below the intact crystal or the small granular structure we have studied here and can only be explained by the presence of defects.

6. Summary In this study, we examined the fracture properties of single-crystal and ultrananocrystalline diamond (UNCD) using the semiempirical quantum mechanics method, MSINDO, and the density functional theory methods, LDA and GGA. We found that pure UNCD has a theoretical fracture stress of 100 GPa, and failure strain of 0.13, which are 43% and 37% of the respective single-crystal diamond values. The calculated value of YoungÕs modulus for UNCD (E = 1.05 TPa) is only slightly smaller than that of single-crystal diamond (E = 1.09 TPa). Three approaches were used in this work. In one, MSINDO was used to calculate both the structure and

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energy of the clusters as they were strained to failure. In the second, MSINDO was used to calculate structures as a function of strain, and PBE used to calculate energies of the structures. In the third approach, PBE was used, as MSINDO was in the first approach, to calculate both energies and structures as a function of strain. The third approach is the most accurate but is very computationally time-consuming. The second approach is only modestly more computationally expensive than the first but predicts YoungÕs modulus, failure strain and fracture stress values which are within 8%, 6% and 6%, respectively, of the full-strain PBE values for single-crystal diamond, and 9%, 23% and 16%, respectively, for UNCD. MSINDO alone was much less accurate. Experimentally observed fracture stress values for single-crystal diamond and UNCD are approximately 1/50th of the corresponding theoretical values. We have shown that simple grain boundaries cannot be responsible for this decrease in strength and that Griffith theory, based on the calculated surface energy, suggests defects on the scale of 500 nm; defects of this scale have been observed experimentally.

Acknowledgments We gratefully acknowledge the grant support from the National Science Foundation (Grant CMS 500304472) and NASA University Research, Engineering and Technology Institute on Bio Inspired Materials (BIMat) under Award No. NCC-1-02037.

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