ergy and the grain boundary orientation with respect to theapplied strain.
II. CALCULATIONAL METHODS
Four sets of simulations, each with a different type of initial structure, were considered. The first set consisted of single symmetrical tilt GB’s with 001 and 011 misorien-tation axes with no preexisting flaws. The initial structures of
these GB’s are based on a coincident-site lattice model forgroup-IV materials in which each atom is fourfoldcoordinated.25 To estimate theoretical strengths, the crystalswere strained at a rate of 1% ps along the direction perpen-dicular to the GB plane Fig. 1a. This was accomplishedby moving two regions of atoms 3 Å wide and 10 latticeparameters on either side of the GB away from the interface.During strain, the atomic positions within the end regionswere held constant while the remaining atoms were allowedto move by integrating classical equations of motion usingforces from the analytic potential discussed below. Periodicboundary conditions were maintained within the GB plane,and each system contained approximately 4000 atoms. Thequantities considered in this set of simulations were the
maximum fracture stresses of the GB’s compared to thebulk, and the GB work for fracture, which characterizes theability of the material to adsorb energy without failing.
The structures examined in the second set of simulationsconsisted of crystals in which a surface notch 30 Å longoriented perpendicular to the direction of strain was insertedinto the GB Fig. 1b. Strain was applied to these systemsas described above until a crack started to propagate, afterwhich the coordinates of the atoms in the end regions were
left unchanged. Periodic boundary conditions were appliedalong the z axes only see Fig. 1 for the axis orientation, andthe simulations were performed for several 8018020 Ådiamond samples, each of which contained approximately50 000 atoms. The third set of simulations was identical tothe second except that different GB orientations with respectto the notch and applied strain were examined. These werecarried out so that both transgranular and intergranular crackpropagation could be modeled Fig. 1c.
In the final type of simulation, crack propagation in amore complicated polycrystalline microstructure containingfive 011 tilt GB’s and two triple junctions was simulated.The microstructure used in this simulation, which is illus-
FIG. 1. Illustrations of the simulated systems. a Simulation setup for modeling the fracture strengths of individual GB’s; b system formodeling intergranular crack propagation; c crack propagation at different initial orientations between a notch and GB plane; d crackpropagation in a microstructure with realistic features.
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trated in Fig. 1d, was constructed to mimic that observedexperimentally in CVD diamond.26 All atoms in the structureare four-fold coordinated, and there are no cavities betweenthe grains. To construct this system, two triple junctions hav-ing a common 9 GB were first connected to one anotheralong this GB. The system size was then increased by trans-
lating other sections of the crystal to the open regions. Thefinal system had dimensions 15026020 Å contained120 000 atoms and was periodic in the z direction. Details of the atomic structures of the 93(111)3(111) and2793(211) triple junctions are describedelsewhere.27
A reactive analytic bond-order potential was used tomodel the interatomic forces. This formalism, which is basedon the second moment approximation to the local electronicdensity of states, models the interatomic energy as a sum of repulsive and attractive pair potentials. The attractive pairterms, which model bonding from the valence electrons, aremodulated by analytic bond order functions whose values
depend on coordination, bond angles, radical character of thebond, and an approximation to conjugation effects arisingfrom adjacent unsaturated atoms. The form of the potential issimilar to that in Ref. 28 with two modifications. The first isa slightly different set of functional forms for the pair termsand the bond-order function29 that better describe the elasticproperties of diamond compared to those given in Ref. 28.These properties are listed in Table I along with correspond-ing experimental data. The second modification, describedbelow, is related to the cut-off function used to restrict thepotential to nearest neighbor interactions.
Fracture simulations provide a stringent test of the reli-ability of interatomic potentials because fracture properties
are not usually included in a fitting database. During crackpropagation, atoms are subjected to different bonding envi-ronments where the high strain near a crack tip results inatomic configurations that are far from their ideal bondlengths and bond angles. In particular, fracture properties aresensitive to the behavior of the interatomic potential near aninflection point. In the scheme of nearest-neighbor inter-atomic potentials, which are typical for describing bondingin covalent materials, the interaction must be cut off beforethe second neighbor distance using a suitable switching func-tion. The form of the switching function need not influencemost of the bulk and surface properties of materials andtherefore is usually chosen arbitrarily rather than fit to somephysical property. However, the switching function is crucial
for describing bond breaking during the fracture process; ar-bitrary switching functions can therefore result in nonphysi-cal behavior in atomistic fracture simulations. In preliminarysimulations, for example, it was found that the influence of aswitching function on the inflection point of the interatomicpotential can result in very high stresses and strains requiredfor diamond fracture.31
In the initial version of the potential,28 a switching func-
tion cuts off the interaction between 1.7 and 2.0 Å. Thisnearest-neighbor bonding model for carbon is well justifiedby the nature of covalent bonding and works well for mostequilibrium structures. However, the fixed switching func-tion approach is problematic as C-C bonds are stretched be-yond 1.7 Å because it significantly influences the forces inthe vicinity of the inflection point 1.85 Å in diamond forthe 111 direction. To avoid this problem in the presentstudy, the cutoff distance was extended far beyond the in-flection point. To preserve the nearest-neighbor character of interactions, a bond list using the original 2 Å cutoff dis-tance was constructed for the initial system that was leftunchanged during the simulations. This ad hoc scheme
solves the cutoff problem while still describing bond break-ing and changes in the chemistry of the bond during cleavagee.g., formation of double, triple, and conjugated bonds.However, its application is restricted to phenomena that in-volve bond breaking and rehybridization, but not new bondformation.
It is well established17 that the area bounded by the curveof stress as a function of the separation between atomicplanes up to the maximum stress should approximately equalthe surface energy . With the modified cutoff procedure,this area calculated for separation of 111 diamond planes isabout 0.85 , which is very reasonable. The calculated maxi-mum tensile strength for bond breaking in the 111 directionis 96 GPa, which is close to the value calculated by Tyson32
using atomic force constants. These tests demonstrate thereliability of the interatomic potential for simulating fracture.
III. RESULTS AND DISCUSSIONS
In subsection A below, analysis of GB cohesion based onenergetic considerations is discussed, followed in subsectionB by the results of molecular dynamic simulations of cleav-age of individual GB’s. The first purpose of these calcula-tions is to determine the extent to which GB strengths cal-
culated using GB cohesion energies via Orovan’s criterion,for example are consistent with those obtained frommolecular-dynamics simulations. The second purpose of these calculations is to determine the types of GB’s havingrelatively high strengths. In the final subsection results on thecrack behavior in the material containing GB’s are presented.Of primary interest is to determine if critical stresses of theintergranular crack propagation calculated with the Griffithcriterion using GB cohesion energies are consistent withthose obtained in the dynamic simulations. The inter- versustransgranular crack propagation in diamond is also dis-cussed.
A. Grain boundary cohesion
The energy required to cleave a brittle material along aGB plane without plastic deformation is defined through therelation17
TABLE I. Properties of diamond given by the analytic bond-order potential, DFT/LDA calculations and experiments.
Property Analytic PotentialDFT/ LDA Experiment
Lattice constant Å 3.566 3.52 3.566a
Bulk modulus Mbar 4.45 4.62 4.44a
Shear modulus
Mbar
5.4 5.2 5.0–5.5
b
C11 Mbar 10.78 11.11 10.81a
C12 Mbar 1.31 1.38 1.25a
C44 Mbar 6.8 5.95 5.79a
aFrom Ref. 30.bFrom Ref. 1.
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where 1 , 2 are energies of the two unreconstructed sur-faces created due to cleavage and E gb is the GB energy ( E gb
is zero for bulk cohesion. In diamond, where plastic defor-mation is negligible and therefore Eq. 1 may be applied, itis expected that calculated GB cohesive energies will give
reasonable estimates for the relative strengths of grains andgrain boundaries.Summarized in Table II and plotted in Fig. 2 are GB
cohesive energies for 001 symmetrical tilt GB’s as a func-tion of misorientation angle . Surface energies were calcu-lated for a few surface orientations and then extrapolatedthrough the entire misorientation range. The GB energies,which are necessary for calculating GB cleavage energies,were evaluated over the entire range of misorientation angleusing a multiscale modeling approach that combines con-tinuum and atomic-level models.33 The relations in Fig. 2awere obtained from density-functional calculations using thelocal-density approximation DFT/LDA. Details of these
calculations are given elsewhere.
33
Plotted in Fig. 2b arethe same curves calculated with the analytic bond-order po-tential described above. Although the cleavage energies areoverall lower for the analytic potential compared to the first-principles DFT/LDA calculations, the relative energies forgrain boundary and bulk cleavage are similar for the twomethods. Cleavage energies of most 001 and 011 tiltGB’s Table II are about 60–75 % of those for the idealbulk crystals with the same orientation. It is also apparentfrom Fig. 2 and Table II that special short-period GB’s 5(120), 5(130) for 001 tilt axes and 9(122) GBfor 011 tilt axes possess higher cleavage energies relativeto GB’s in the nearby misorientation range. Calculated GBcleavage energies will be used in the section B for evaluation
of GB theoretical strengths that then will be compared withthe GB strengths obtained from dynamic simulations.
Another important issue is the relative bulk cleavage en-ergies of different low-index planes in diamond, which hasbeen widely discussed in the literature on mechanical prop-erties of diamond.1,2 Our DFT/LDA calculations as well asthose carried out by Kern and Hafner34 using similar tech-
niques predict that for diamond the 110 surface is energeti-cally more stable than the unreconstructed 111 (11) sur-face for both relaxed and unrelaxed structures. This result iscontrary to earlier reports on the mechanical properties of diamond.1 The earlier calculations of surface energies werebased on the evaluation of the energy of broken bonds perunit area of different surfaces. However, this does not ac-count for the strong bonding for carbon that leads to sig-nificant reduction of the energy of the 110 surface atomsdue to the formation of -bonded chains along the surface.Chain formation on the 110 surface is compatible with thegeometry of the underlying lattice and does not require anysurface reconstruction. Surface energies of 110 and unre-
constructed 111 (1
1) diamond surfaces calculated withinthe DFT/LDA approach are 5.6 and 6.6 J/m2, respectively.To further illustrate the chemistry of the various surfaces,
ball-and-stick models are given at the bottom of Fig. 2athat indicate the different bonding and defect types associ-ated with atoms on surfaces within each misorientation inter-val. Atoms on the 100 surface corresponding 90° pos-sess two dangling bonds each. The 130 surface 53.13°contains two types of atoms on which there are either one ortwo dangling bonds. The 120 surface 36.87° containsatoms with two dangling bonds each and atoms forming -bonded chains. Atoms on the 110 surface 0° form -bonded chains only. All free surfaces at angles intermedi-ate between these delimiting angles contain a mix of the
TABLE II. Grain boundary theoretical strength properties, calculated from molecular dynamics simula-tions for 001 and 011 symmetrical tilt GB’s STGB. The ratio W GB / W 111 is the relative work forfracture. Cohesive energies were calculated both with DFT/LDA method and using the bond order potentialBOP
bonding types of the delimiting surfaces. As a result, thecurve of the dependence of surface energy on the misorien-tation angle consists of three straight lines with slightly dif-ferent slopes Fig. 2. In summary, the ‘‘ -bonding chemis-try’’ should be taken into account when analyzing energeticcharacteristics of covalent materials with pronounced
bonding.
B. Dynamic simulation of theoretical strength
Below are discussed theoretical fracture characteristicsi.e., those for a sample with no preexisting cracks obtainedfrom molecular dynamics simulations for bulk diamond withlow-index orientations and selected 001 and 011 sym-metrical tilt GB’s.
1. Ideal crystal strength
In the simulations using the slow straining method withthe bond-order interatomic potential, the maximum tensilestress for the 111-oriented diamond is 96 GPa and the cor-responding strain is 15.6%. For the frozen separation tech-nique the maximum strength and strain are 5% and 20%,respectively, higher than those obtained in dynamic simula-tions. A variety of theoretical tensile strength values for111-oriented diamond have been reported; these valuesrange from 200 GPa Ref. 35 using Orovan’s criterion and106 GPa estimated assuming Morse-type interatomicinteractions32 to 53 GPa Ref. 36 including third-orderelastic coefficients. Thus, our value for maximum tensilestress for the 111-oriented diamond is closer to that re-ported by Tyson.32
It is well established experimentally that cleavage of dia-mond results in predominantly 111 planes.1,2,37 Explana-tions of the preference for 111 cleavage were based on the
consideration that most preferred cleavage would be that in-volving the least cleavage energy.1 According to estimates of cleavage energies for diamond by Field,1 cleavage should bealong 111 planes. Another explanation involves the crite-rion of the least applied stress for cleavage along particularplanes.2 According to calculations by Whitlock and Ruoff,36
the stress required to produce tensile fracture is minimal forthe 111 direction although for the 011 orientation re-ported the stress value is only slightly higher. In addition tothe least fracture stress and least cleavage energy criteria,discussed above, further considerations regarding preferredcleavage planes may be developed. First, if critical stressesfor cleavage for particular planes are close, the criterion of
the least work for fracture might be involved for furtheranalysis. The work for fracture characterizes the ability of the material to adsorb energy without fracture. In general, thework for fracture might exceed the cleavage energy sinceadditional mechanisms of energy dissipation beyond purebond breaking along the fracture plane during loading mayappear. For example, the possibility of significant bondbending at loading along the 001 direction exists and for011 direction, too, although less pronounced because atthe given geometries the bonds make an angle with the axisof the applied strain. As a result for 001 and 011 orien-tations more strain may be relaxed by changing of bondangles as compared to the 111 crystal orientation. Anotherconsideration regarding preference planes for cleavage in
FIG. 2. Cleavage energies of 001 symmetrical tilt GB’s in diamond. a Results from DFT/LDA calculations; b values calculated withthe bond-order potential. For comparison, the bulk crystal cleavage energies are also shown.
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diamond suggests that there can be planes of the preferentiallocation of microdefects, which can serve as nuclei for mi-crocracks. For example, it is known that 111 planes inCVD diamond contain a large amount of twins and stackingfaults.26 Finally, the most important characteristic defining aplane of preferred cleavage would be a critical stress formicrocrack propagation, assuming that in-grown microcrackembryos are approximately of equal sizes in the possible
cleavage planes. This case is discussed in section C. 1 of thepresent paper based on the dynamic simulation of microc-racks using the bond order potential. However, to completethe analysis of the reason for very marked preference of 111 cleavage in diamond, further first-principles calcula-tions are required to obtain least-critical stress values as wellas the least work for fracture for low-index orientations fromfracture dynamic simulations.
Based on the predicted DFT-LDA calculations of the bulkcohesive energies for the low-index orientations in diamondTable II, it is evident that the least cleavage energy crite-rion does not explain the preferable cleavage along 111planes since the cohesive energy for 011 planes is 17%
lower. The dynamic simulations with the bond-order poten-tial demonstrate that the critical fracture stress for 011-oriented diamond is 20% higher than for the 111 orienta-tion even though the energy of the 011 surface is lowerthan that of the 111 surface. This is related to the fact thatthe potential predicts that -bonded chains, which signifi-cantly decrease the surface energy, form primarily only afterthe interplane separation reaches the distance correspondingto the maximum stress. As a result, before the maximumstress is reached, the bonds behave as if they are purelysingle bonds. While physically plausible, accurate first-principles calculations of the fracture dynamics are requiredto further characterize simultaneous bond rupture and -bonded chain formation during cleavage of a 011 ori-ented diamond crystal. Thus, the least critical tensile stressobtained with the bond-order potential, in principle, explainspreferable cleavage for 111 planes as compared to 011planes. Dynamic simulations with the bond-order potentialalso demonstrate that the work for fracture is 30% lower for111 planes in comparison with 011 planes, and 60%lower than that for 001 planes Table II. Thus, the leastwork for fracture for the 111 plane is also consistent withexperimental observations.
It should also be noted that first-principles calculationssuggest that the lowest energy among the low-index faces isfor the reconstructed 111 surface34 involving formation of seven-five member rings on the surface. If this surface recon-
struction occurs simultaneously with bond breaking alongthe 111 surface, it could significantly decrease the work forfracture for the 111 plane. First principles calculations34
also demonstrate that among low-index hydrogenated sur-faces the 111 surface is more energetically stable. Thusatomic hydrogen can also, in principle, decrease the work forfracture for 111 planes.
2. Individual grain boundary strengths
Some typical stress-strain curves obtained from dynamicsimulations of ideal diamond and systems containing GB’sare illustrated in Fig. 3. The stress-strain curves for the short-period special 5130 GB and the long-period
41450 GB are shown in the figure. The fracture stressesand strains for the bicrystals are significantly lower thanthose for the ideal crystal.
Theoretical strength properties of different GB’s with001 and 011 misorientation axes estimated from the dy-namic simulations are summarized in Table II and Fig. 4.Critical fracture stresses of samples with GB’s are about30–60 % lower than those for ideal diamond Fig. 4, TableII. The work for fracture for various types of GB’s is 40–80 % lower depending on the GB type than that for a 111ideal diamond orientation Table II.
The theoretical fracture stress of a crystal according to theOrovan criterion,38 depends on the cleavage energy ,Young’s modulus E , and the interplanar spacing a0 in theunstressed state of the planes perpendicular to the tensile axisthrough the relation
max E / a01/2. 2
This expression assumes that the energy required to breakthe bonds is provided by the stored elastic energy in theregion nearby the fracture plane. Although it is well estab-lished that the magnitude of the ideal breaking strength isgenerally overestimated by Orovan’s relation by up to a fac-
FIG. 3. Stress-strain curves for a 111-oriented ideal diamondsample and for samples containing 5130 and 41450
GB’s.
FIG. 4. Fracture stresses and strains of 001 tilt GB’s in dia-mond. a Values obtained from molecular-dynamics simulationsopen circles and fracture stresses evaluated from the Orovan cri-terion solid circles.
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tor of two,35 the Orovan criterion has nonetheless been usedto investigate the relationship between maximum stress andcleavage energies of GB’s. Fracture stresses of GB’s calcu-
lated with the Orovan criterion are given in Fig. 4. The quan-titative difference between the results of atomic simulationsand the analytic approach is not surprising. However, as canbe seen from Fig. 4, there is little qualitative correlation be-tween maximum stresses calculated from Orovan’s criterionand those obtained from dynamic simulations. This is due tothe specific mechanism of fracture initiation within a GB. Atthe critical local stress which is several times higher than theapplied stress, nucleation of microcracks occur within thedislocation cores at the GB that are initially under tensionFigs. 5a and 5b. This is followed by bond breakingalong the interface Figs. 5c and 5d. Thus Orovan’s cri-terion cannot be used to accurately estimate theoretical
strengths for GB’s in diamond because of the nonuniformityof the cleavage energy distribution along the interface.
Structures of the surfaces after cleavage are illustrated inFig. 6. Fracture surfaces for the 5013 GB are rougherthan those of higher GB’s. This indicates a higher work forfracture for the 5013 GB in accordance with results inTable I. Thus, critical local stresses rather than GB cleavageenergies define theoretical strengths of different types of GB’s. Evidently, the maximum local stress depends on theintrinsic stresses in the vicinity of a GB that is enhanced byexternal stress when load is applied. Additional analysis isrequired to establish the correlation between the critical localstresses, applied load, and GB structure that defines the in-
trinsic GB stress.It can be concluded that the relative theoretical strength of a GB is determined by its type. For example, these simula-tions have shown that certain short-period 001 symmetricaltilt GB, namely the 1 0° and 90°, 5012 36.87°, and 5013 53.13°, possess about a 30%higher critical stress and 30% higher work for fracture thanGB’s in the nearby misorientation range Table II, Fig. 4.Fracture strength is also higher for the special 9122 GBthan for other two GB’s 27 and 81 with 011 tiltaxes, which were also studied Table II. This result is con-sistent with physical properties of special GB’s in metals andceramics that can be significantly different from those forother GB’s in the nearby misorientation range.15,39
C. Crack propagation
1. Ideal crystals
As it was outlined above, the 111 plane is the experi-mentally preferred cleavage plane in diamond,1 althoughother cleavage facets have been also observed,37 particularlythe 110 plane. Because diamond is a brittle material, it isexpected that the Griffith criterion can predict relatively ac-curate critical stresses for bulk cleavage for samples with
pre-existing cracks. Prior estimates of surface energies1
fromsimple bond-scission analyses suggest that the 111 surfacehas the lowest surface energy followed by the 110 and100 surfaces, respectively. Based on this cleavage energyranking for low-index facets, from the Griffith criterion itfollows that the critical stress of a crack propagation is mini-mal for the 111 plane, what corresponds to the experimen-tal observations. However, as it was discussed above, simplebond-scission evaluations are inconsistent with the surfaceenergies obtained from the DFT-LDA calculations as well asusing the bond-order potential see Table II. The criticalstresses calculated from the Griffith equation using surfaceenergies and elastic properties from the analytic potential are
reported in Table III. These values predict a 23% highercritical stress for crack propagation along the 111 planecompared to the 110 plane. This trend is maintained withsurface energies taken from the DFT/LDA calculations seeTable III.
To explore the reason for the discrepancy between theexperimentally observed cleavage planes and the predictedcritical stresses from the Griffith criterion in Table III, simu-lations of crack propagation within 111, 110, and 100planes were performed. The model system for this set of calculations is illustrated in Fig. 1b. The simulationsyielded critical stresses for the 110 and 100 surfaces thatare 4% and 15% higher, respectively, than that for the 111surface. This is consistent with experiment, but inconsistent
FIG. 5. Close up view of fracture initiation and propagationwithin the 25340 GB in diamond.
FIG. 6. Surface structures after cleavage. a Cleavage along the1497100 GB; b cleavage along the 5013 GB; c cleav-age along the 27255 GB.
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with the Griffith evaluation using the surface energies. De-tailed analysis of the dynamics revealed that this inconsis-
tency is a result of ‘‘ -bonding chemistry’’ of bond breakingin the 110 plane. The structure of the crack tip for a crackin the 110 plane is illustrated in Fig. 7. Each step of crackpropagation requires breaking bonds between two rows of atoms; the numbers 1 and 2 indicate these rows in Fig. 7.The structure of the resulting surface is such that -bondedchains can form between atoms in the rows denoted by thenumbers 2 and 3 in Fig. 7. This produces relatively low-energy surfaces created during cleavage compared to cleav-age within the 111 and 100 planes. However, the poten-
tial predicts that bond rupture must be almost completedbefore significant energy stabilization can be realized from -bond formation. This leads to the similar critical stress forcrack propagation for this orientation relative to other low-index planes despite the other planes having higher surfaceenergies. This effect, while qualitatively captured by themany-body features of the analytic potential, should be con-firmed by first-principles-based dynamic simulations.
Two directions of crack propagation, 001 and 011,have been also modeled for a crack inserted along a 110plane. It was found that the critical stress for crack propaga-tion in the 001 direction is 11% lower than that in the 011
direction. These two directions are not equivalent; the dis-tance between arrays of broken bonds at each step of crack
front propagation is different, as well as the relative orienta-tion of the crack front and arrays of atoms forming -bondedchains. Thus, simulations predict that within the same cleav-age plane there can be directions with different resistances tocrack propagation.
In summary, the study of bulk cleavage along low-indexfacets in diamond revealed that formal use of Orovan’s cri-terion for the theoretical strength and the Griffith criterionfor the critical stress of crack propagation give a ranking of the strengths for low-index facets that is inconsistent with theexperimental observations. In particular, from these evalua-tions it follows that 011 planes should be the planes of preferred cleavage rather than 111 planes. This is due to thefact that the surface energy, used in the Orovan’s and Griffith
TABLE III. Critical stresses for intergranular crack propagation of an initial crack 30 Å long obtainedfrom molecular dynamic simulations with the bond order potential ( MD ) and calculated from the Griffithcriterion using cleavage energies obtained with the bond-order potential ( G( BO P)) and DFT-LDA calcula-tions ( G( LD A)); toughness calculated from BOP cleavage energies K c(2 E )1/2, toughness from dynamicsimulations K MD MD ( l)1/2
criteria, is lowest for the 011 planes in diamond as con-firmed by the DFT-LDA calculations as well as using thebond-order potential. However, dynamic simulations, par-ticularly with the bond-order potential, give a ranking of strengths of low-index facets that is consistent with experi-ment. As discussed above, this is due to the ‘‘ -bondingchemistry’’ contribution to the failure mechanisms for cova-lent materials with strong bonding. In general, can be con-cluded that for the evaluation of the strength of covalentmaterials along orientations where simple bond-scissionanalysis can not be applied as along the 011 orientation indiamond, the bond strengths should be obtained from thesimulations rather than using criteria involving surface ener-gies. Fracture modeling of diamond samples with low-indexorientations using a tight-binding approach is currently inprogress.
2. Individual grain boundaries
To explore the dynamics of intergranular crack propaga-
tion, a series of simulations were carried out in which a crackwas inserted into a grain boundary and the system wasstrained in the direction perpendicular to the notch until thecrack began to propagate. Two sets of simulations were run.In the first, which included a total of nine GB’s at variousmisorientation angles, the notch was initially placed com-pletely within the GB plane. An example of one of thesesimulations is illustrated in Fig. 1b. In the second set of simulations Fig. 1c, the angle of inclination of the GBplane relative to the notch plane was varied from approxi-mately 10° to 80°, and 4–5 different configurations for eachGB were modeled. The grain with a preexisting notch wasoriented relative to the applied strain so that crack propaga-
tion was initiated in either in the 011 or 001 planes.An example of one of the simulations in which the strain
was applied perpendicular to the GB plane is illustrated inFig. 1b. The cracks propagated within the GB plane in allsystems studied except the 9122 GB for which thecrack deviated to the 111 plane inclined to the 122 planeat 15°. In each case the cleavage surfaces were flat andformed without debris. Critical stresses for intergranularpropagation of an initial crack 30 Å long obtained from thesimulations are illustrated in Fig. 8 and summarized in TableIII. For comparison, critical stresses obtained from the Grif-fith criterion using GB cleavage energies and Young’smoduli calculated with the bond-order potential from TableII are also presented in Fig. 8. It is evident from the figurethat the dependence of critical stress on misorientation issimilar for both approaches the maximum difference isabout 20%, especially given that a variety of additional fac-tors may influence the simulation, including system size,nonlinearity of interatomic interactions, and different tip ra-dii. This suggests that GB cleavage energy is a major param-eter defining GB resistance to crack propagation.
Critical stresses for crack propagation within GB’s areabout 30–40 % lower than those for an ideal crystal exceptfor the 5012 and 5013 special GB’s, where themaximum stresses for crack propagation are close to thestresses for crack propagation in ideal samples Table III.
Critical stresses obtained in dynamic simulations for thesespecial GB’s exceed those calculated from the Griffith crite-rion. In addition, it was found that the critical stress for crackpropagation along the 5013 GB with a zigzag arrange-ment of dislocation cores is about 30% lower than that forthe structure with a straight arrangement of dislocationcores.40 The difference in GB resistance to crack propagationfor these two models likely originates from the different ori-entation of bonds within these GB’s relative to the notch. Inthe GB with the zigzag arrangement of structural units, thebonds are almost perpendicular to the direction of crackpropagation, while in the GB with the straight arrangementof structural units bonds are more elongated along the crack
plane and therefore are more resistant to bond rupture.The second series of simulations discussed in this section
were carried out to explore the possibility of a planar crackchanging its path as it propagates from one grain to another.This can result in the absorption of additional energy and aresulting toughness of the polycrystalline system that ishigher than single crystals. The simulation setup for this setof calculations is illustrated in Fig. 1c. Four GB’s, 41,5, 27, and 9, have been studied.
In general, when a crack reaches a GB, it can propagatewithin the GB intergranular fracture, or penetrate into thesecond grain transgranular fracture. Within the secondgrain, the crack can keep moving in the initial direction of propagation or deviate into the easier cleavage plane. Theseevents depend on the GB cleavage energy, relative bulk co-hesive energies of the first and second grains, and the incli-nation angle of the GB relative to the initial crack propaga-tion plane.17 To maintain crack propagation as the cracktransverses the boundary and deviates to the preferred cleav-age plane, the applied load must be increased because thereis a change of a pure mode I crack propagation to a mixedmode. This increases the toughness of the material. If thecrack deflects onto a weak GB, the net resistance toughnessincreases due to an increase in the actual fracture surfacearea.
Figures 9 and 10 show representative snapshots of cleav-
FIG. 8. Critical stresses for crack propagation within 001 tiltGB’s obtained from molecular dynamics simulations solid circles
and calculated from the Griffith criterion open circles. For the
5(013) GB critical stresses for a crack propagation within twoGB models with straight s and zigzag z arrangement of structuralunits are shown.
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age surfaces after crack propagation in systems with 41and 5001 tilt GB’s. Unless an angle between a primarycrack plane and a GB is small Figs. 9a, 9b, and 10a,the crack tends to cross the GB. A simple analysis of thecondition for intergranular crack propagation can be carried
out following Lawn.17 The condition for a crack initially inpure mode I to deflect into a GB in a mixed mode is17
G / G0 R0 GB / R0 G1 , 3
where is the misorientation angle between the initialcrack and GB planes, R0GB is the work of adhesion of the GB, R0(G1) is the bulk cleavage energy of the graincontaining the initial crack propagation, and G ( ) and G (0 )are mechanical-energy-release rates that depend on the anglebetween the crack and the direction of applied load. Values
of G( )/ G(0) and R0GB / R0(011) for several 001 tiltGB’s as functions of GB misorientation angle are plottedin Fig. 11 for a crack initially propagating in an 011 plane.The value of R0(011) the cleavage energy of 011 planes
was calculated by the Griffith formula using the criticalstress obtained from a molecular-dynamics simulation of crack propagation within a 011 plane. The ratioG( )/ G(0), calculated for an isotropic case, was taken fromreference.17 If R0GB / R0(011) exceeds G( )/ G(0) for agiven GB Fig. 11, intergranular crack propagation as thecrack reaches the GB is expected. The results of moleculardynamic simulations in Figs. 9 and 10 correspond reasonablywell to these rough estimations. In crystals containing 011
tilt GB’s it has been observed that cracks tend to exhibittransgranular propagation mainly along 111 planes. Thusthe molecular dynamic simulation results indicate predomi-nantly a transgranular mode of fracture in polycrystalline
diamond.
3. Complex microstructures
Simulated crack propagation in a system containing a net-work of 011 GB’s is illustrated in Fig. 12. Among the GB’s
FIG. 9. Illustration of the fracture surfaces resulting from propa-gation of a crack inclined at different angles to 41(450)GB plane. a 0°, G( )/ G(0 )1, R0GB / R0GB1; b
12.7°, G( )/ G(0 )0.97, R0GB / R0(011)0.43; c
57.7°, G( )/ G(0 )0.61, R0GB / R0(001)0.32; d
77.3°, G( )/ G(0)0.4, R 0GB / R0(011)0.43.
FIG. 10. Illustration of the fracture surfaces resulting frompropagation of a crack inclined at different angles to
5(130) Z GB plane. a 8.1°, G( )/ G(0 )0.95, R0GB / R0(001)0.75; b 36.9°, G( )/ G(0 )0.83, R0GB / R0(011)1; c 53.1°, G( )/ G(0 )0.6, R0GB / R0(011)1; d 81.9°, G( )/ G(0)0.35, R0GB / R0(011)0.75.
FIG. 11. Relative mechanical-energy release rate G( )/ G(0 )open circles and relative crack resistance energy R 0GB / R0(011)solid circles as functions of GB misorientation angle . Calcula-tions are carried out for a preexisting crack inserted within a 011
plane. Every GB is inclined to a preexisting crack at angle
min ;90° .
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in the system, the 27 and 3211 GB’s possess thelowest cohesive energy and are therefore considered weak.In addition to a crack propagating from a notch, anothercrack originated at the intersection of the 27 GB with afree surface top part of Fig. 12 as applied strain was in-creased up to 6%. This ‘top’ crack intersected the triple junc-tion and continued to propagate along the 9 GB. Thecrack originating in the notch deviated to a 111 plane, in-tersected the 3111 GB, and continued to propagatealong a 111 plane until the two cracks coalesced alongnother 111 plane Fig. 12. Thus, both trans- and inter-granular modes of crack propagation were observed in thisset of calculations.
IV. CONCLUSIONS
Fracture strengths and resistances to crack propagation for001 and 011 symmetrical tilt GB’s in diamond have beenstudied via molecular dynamic simulations using a bond-order analytic potential. GB cleavage energies have been cal-
culated with both density-functional theory and the bond-order potential.
It was found that the theoretical fracture stress of indi-vidual GB’s is defined primarily by GB type rather than byGB cleavage energies. In particular, special GB’s possesshigher theoretical fracture stresses than GB’s in the nearbymisorientation range. The mechanism of interface failure isnot that implied by the Orovan criterion, which assumes uni-form distribution of the stored energy along a GB plane andtherefore simultaneous breaking of all bonds along an inter-face. Atomistic simulations demonstrated that failure is ini-tiated within the dislocation cores at the interface when criti-cal local stresses are attained, and then propagated fromthese points along the interface.
Critical stresses for crack propagation within a GB ob-tained from dynamic simulations were consistent with thosecalculated from the Griffith criterion. Toughnesses of specialGB’s was about twice that of other GB’s. This is in agree-ment with other studies and experiments on metals andceramics.1,11
It was found that the chemistry of covalent materials maysignificantly contribute to crack propagation and therefore tofailure mechanisms. In particular, the formation of -bondedchains after crack propagation along 110 surfaces signifi-cantly reduces surface energies of 110 plane as comparedto other low index planes, although the various low-indexsurfaces have similar stresses necessary to break the bonds.To evaluate strength properties of covalent materials alongthe orientations where -bonding reconstruction is signifi-cant, the formal using of the Orovan’s or Griffith criterionsgive incorrect strength properties. For these orientationsstrength properties should be obtained through, for example,dynamic simulations.
Crack propagation has been also studied in systems con-taining GB’s of different types with different initial orienta-tions relative to the notch. In most cases transgranular crackpropagation was observed. From the balance of mechanicalenergy release rate and relative crack resistance of a GB anda grain, it was possible to make rough predictions of theintergranular versus transgranular crack propagation depend-ing on GB type.
ACKNOWLEDGMENTS
Priya Vashishta, Marshall Stoneham, and Michael Fren-klach are thanked for helpful discussions. O.A.S. andD.W.B. were supported by the Office of Naval Researchthrough Contract No. N00014-95-1-0270. A.O. and X.S.were supported by the Air Force Office of ScientificResearch Grant No. F 9620-98-1-0086, USC-LSU Multi-disciplinary University Research Initiative Grant No. F49620-95-1-0452, and the National Science FoundationGrant No. DMR-9711903. Part of the simulations wereperformed on parallel machines in the Concurrent Comput-ing Laboratory for Materials Simulations CCLMS at Loui-siana State University. L.H.Y. was supported by the U.S.Department of Energy under Contract No. W-7405-ENG-48at LLNL.
FIG. 12. Illustration of the fracture surfaces resulting frompropagation of a crack in a realistic microstructure. Different initialconfigurations of a notch relative to the system are explored in thesimulation sets.
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