Abstract
The hardness of nanotwinned diamond (ntdiamond) is reported to be more than twice that of the natural diamond, thanks to the fine spaces between twin boundaries (TBs), which block dislocation propagation during deformation. In this work, we explore the effects of additional TBs in ntdiamond using molecular dynamics (MD) calculations and introduce a novel intersectional nanotwinned diamond (intdiamond) template for future laboratory synthesis. The hardness of this intdiamond is predicted by first analyzing individual dislocation slip modes in twinned grains and then calculating the bulk properties based on the Sachs model. Here we show that the hardness of the intdiamond is much higher than that of ntdiamond. The hardening mechanism of intdiamond is attributed to the increased critical resolved shear stress due to the presence of intersectional TBs in ntdiamond; this result is further verified by MD simulations. This work provides a new strategy for designing new superhard materials in experiments.
Introduction
Diamond is the hardest, stiffest and least compressible crystalline materials in the world. Understanding and further improving its hardness is scientifically fascinating and technologically important^{1,2,3}. In the past few decades, numerous efforts have been made in these directions, both experimentally and theoretically^{4,5,6,7,8,9}. It has been demonstrated that the hardness of the diamond can be improved by refining its grain size and/or twin thickness according to the wellknown Hall–Petch effect^{10,11}. For example, nanograined diamond (ngdiamond) with grain sizes of 10–30 nm has been reported as high as 110–140 GPa in Knoop hardness, significantly higher than that of singlecrystal diamond^{4,5,12}. Nanotwinned diamond (ntdiamond) with an average twin thickness (λ) of 5–8 nm, synthesized by compressing onionstructured precursors^{7,8}, is recently reported to possess Vickers hardness of 175–200 GPa, setting a new world record. Can the hardness of ntdiamond be further increased? This is a fundamental scientific question for designing new superhard materials with potentially wideranging implications^{13}.
The recent molecular dynamic (MD) simulations on ntdiamond indicated that the origin of the unprecedented hardness originates from two factors: high lattice frictional stress due to the strong sp^{3} C–C bonding and high athermal stress due to Hall–Petch effect^{14}. Experimental and theoretical studies also show that twin boundaries (TBs) can continuously harden covalent materials with decreasing twin thickness (λ) to very small values (on the order of a few nm)^{7,8,15,16}. Therefore, for covalent materials, a practical strategy to achieve superhardness is to introduce more TBs into the microstructure. Based on this idea, a novel intersectional nanotwinned diamond (intdiamond) model is constructed by inducing intersectional TBs into ntdiamond. After analyzing dislocation slip modes in individual intdiamond grains, the hardness of bulk intdiamond is calculated based on the Sachs model^{17}. We find that the hardness of the intdiamond is much higher than that of the ntdiamond. This result is further verified by MD simulations. This work provides a new strategy for designing new superhard materials in future experiments.
Results
Dislocation slip modes in intdiamond grains
For diamond, dominant dislocations are along the <110> directions slipping in the (111) plane with Burgers vector of \(\frac{1}{2}\) <110> for perfect dislocation and \(\frac{1}{6}\) <112> for glideset partial dislocation^{14,18}. According to the angle between Burgers vector and dislocation line direction, there are six types of dislocations: glideset 0° perfect dislocations, glideset 30° partial dislocations, glideset 60° perfect dislocations, glideset 90° partial dislocations, shuffleset 0° perfect dislocations, and shuffleset 60° perfect dislocations^{14}. Among them, shuffleset 0° perfect dislocation has the lowest critical resolved shear stress (CRSS) for dislocation motion and the lowest barrier strength when reacting with TBs^{14}. Here the CRSS is defined as the threshold stress of dislocation motion, and the barrier strength is defined as the threshold stress of dislocation reaction with the TB when the activation energy reaches zero^{19}. Therefore, at room temperature, the hardness of the diamond is primarily controlled by the behavior of shuffleset 0° perfect dislocations^{16,20}. Therefore, in the present study, the hardness of the intdiamond will be analyzed based on the behavior of shuffleset 0° perfect dislocations.
In an intdiamond grain, two different orientation twin boundaries TB_{1} and TB_{2} coexist and interweaved, whose average twin thickness is λ_{1} and λ_{2}, respectively (Fig. 1a)^{21,22}. TB_{1} and TB_{2} segment the grain into domains with four different crystallographic orientations, that is, orientations D_{1}, D_{2}, D_{3}, and D_{4}. Among these different orientations, lattices of D_{1} and D_{2}, D_{2} and D_{3}, D_{3} and D_{4}, and D_{4} and D_{1} are mirror images across TB_{1}, TB_{2}, TB_{1}, and TB_{2}, respectively. Consequently, the slip systems in an intdiamond grain can be expressed by a combination of four Thompson tetrahedra (in Fig. 1b): ABCD, ABCD^{1}, A^{1}BCD and A^{2}BCD^{1}, which correspond to D_{1}, D_{2}, D_{3}, and D_{4}, respectively. The combination of these four Thompson tetrahedra results in 39 slip systems in an intdiamond grain (Table 1). According to dislocation line directions and their slip plane orientations, the slip modes of shuffleset 0° perfect dislocations in intdiamond are divided into four types: slip transfer (ST) mode, confined layer slip (CLS) mode, confined slip transfer mode I (CSTI) and confined slip transfer mode II (CSTII) modes, all of which are schematically plotted in Fig. 1b. For ST mode, the slip planes are parallel to either TB_{1} or TB_{2}, and the respective dislocation lines are located within either TB_{2} or TB_{1}. For CLS mode, the slip planes are parallel to either TB_{1} or TB_{2}, while the respective dislocation lines are nonparallel to TB_{2} or TB_{1}. For CSTI mode, both slip planes and dislocation lines are nonparallel to any TB. For CSTII mode, the slip planes are nonparallel to any TB, while the respective dislocation lines are located within either TB_{1} or TB_{2}. As a result, the interactions of these slip modes with TBs are different. This leads to different CRSSs for the four slip modes. For ST mode, the dislocationTB reaction is characterized by dislocations propagating within the TBs^{23,24}, and the corresponding CRSS is determined by the lattice friction stress and barrier strength of shuffleset 0° perfect dislocations reacting with the TBs^{25}. For CLS mode, the dislocation motion is confined between two TBs and its CRSS can be evaluated by increased dislocation energy due to the dislocation line is left on the two TBs^{23,26}. For CSTI mode, the shuffleset 0° perfect dislocation is confined between two TBs and then become shuffleset 60° perfect dislocation to parallel TB when it reaches TB because the initial dislocation line is nonparallel to TB. Finally, the shuffleset 60° perfect dislocation interaction with TB to propagate through TB, and its CRSS is determined by the barrier strength of shuffleset 60° perfect dislocations reacting with TBs and increased dislocation energy because produced dislocation on two TBs. For CSTII mode, although its dislocation motion is similar to that in CSTI mode, its barrier strength is determined by shuffleset 0° perfect dislocations reacting with the TB. In order to obtain CRSSs for these slip modes, barrier strengths of shuffleset 0° and 60° perfect dislocations reacting with TBs must be calculated first.
Barrier strengths of dislocation reaction with TB
Shuffleset 0° and 60° perfect dislocations reacting with TBs can be considered as a kink nucleation and migration process^{14,27,28} (Fig. 1c and Supplementary Fig. 1). The shearstress dependent activation energy for kink nucleation and migration are calculated (detail in “Methods” section), and the results are plotted in Fig. 1d. With increasing shear stress, the activation energies for shuffleset 0° and 60° perfect dislocations reacting with TB reach zero at shear stresses of 19 and 48 GPa, respectively. These stresses are considered the respective barrier strengths for shuffleset 0° and 60° perfect dislocations. The twin intersecting points can provide the pinning obstacle for the slip of dislocation when the shuffleset dislocation slip along the twin plane. However, in ntdiamond, the shuffleset dislocations slip along twin plane energetically show no advantage over those along other slip planes^{16}. Therefore, the shuffleset dislocation is favor to slip along the slip planes rather than twin plane, and the pinning effect of intersecting points on shuffleset dislocation motion is neglected in this work.
CRSS for ST mode
In ST mode, dislocation motions are blocked by TBs (inset of Fig. 2a)^{29}. According to dislocation pileup theory^{30}, the CRSS (τ_{css}) of this mode is expressed as the following:^{25}
where τ_{0} is lattice frictional stress; G is the shear modulus; b is the magnitude of the Burgers vector; λ is twin thickness of intersectional twin; τ_{TB} is the barrier strength of shuffleset 0° perfect dislocations when reacting with the TB. Both modulus and stress are in GPa, and all length parameters are in nm.
With G = 540 GPa according to ref. ^{14}, τ_{TB} = 19 GPa as calculated above and τ_{0} = 10.3 GPa (see “Methods” section) and assuming λ_{1} = λ_{2} = λ, twin thicknessdependent CRSS for ST mode is rewritten as the following:
which is plotted in Fig. 2a. This CRSS increases with decreasing twin thickness, and the trend and quantitative values are similar to that of ST mode in ntdiamond^{14}.
CRSS for CLS mode
The CRSS for CLS mode can be calculated on the basis of the virtual work principle (Fig. 2), and it is expressed as: refs. ^{25,31}
where θ is the angle between the slip plane and the twin plane and λ is twin thickness; ν is Possion ratio; ϕ is the angle between the dislocation line and the Burgers vector, α is dislocation core parameter^{23}.
With the corresponding parameters from ref. 14 and materials parameters of the diamond (listed in Supplementary Table 1), the CRSS for CLS mode is expressed as:
Using Eq. 4, the CRSS for CLS mode is calculated and plotted in Fig. 2a. This CRSS increases with decreasing twin thickness, and the values are similar to that of CLS mode in ntdiamond^{14}.
CRSS for CSTI mode
In CSTI mode, dislocation motions are confined by TB_{1} or TB_{2} and blocked by TB_{2} or TB_{1}, respectively. Therefore, the corresponding CRSS is affected by both the Hall–Petch effect and the confined layer slipping effect. To obtain CRSS, both dislocation pileup and CLS models are used (in Fig. 2b), and the CRSS for CSTI mode is expressed as:
Where τ_{0} is lattice frictional stress, τ_{TB} is the barrier strength of shuffleset 60° perfect dislocation reacting with TB, and ν is the Poisson ratio. All other parameters have been defined before.
Based on the parameters of the diamond from ref. ^{14} (listed in Supplementary Table 1), the CRSS of CSTI mode is expressed as:
The resulting CRSS as a function of twin thickness is plotted in Fig. 2b. This CRSS increases with decreasing twin thickness. Because the CRSS is affected by both the Hall–Petch effect and the confined layer slipping effect, it can be considered as a superposition of ST and CLS modes. At the same twin thickness condition, this CRSS is higher than that of ST and CLS modes.
CRSS for CSTII mode
Similar to CSTI mode, the CRSS for CSTII is affected by both the Hall–Petch effect and the confined layer slipping effect; therefore, the CRSS for CSTII can be expressed by Eq. 5. The difference is that in this case, τ_{TB} refers to the barrier strength of shuffleset 0° perfect dislocations reacting with TB. Based on Eq. 5 and the parameters of the diamond, the CRSS for CSTII mode is expressed as:
The CRSS of CSTII mode thus calculated is plotted in Fig. 2b. Due to the lower barrier strength of shuffleset 0° perfect dislocations, the CRSS is smaller than that of CSTI at the same twin thickness. Owing to the combined Hall–Petch and confined layer slipping effects, this CRSS is also higher than that of ST and CLS modes at the same twin thickness.
Hardness of bulk intdiamond based on the Sachs model
The Sachs model is a singleslip system model for mechanical properties of polycrystalline materials, and it is a particularly effective method to investigate the yield strength of polycrystalline materials with anisotropic slip systems^{17}. For intdiamond, dislocations in multiple twin domains change directions and slip planes in such complex manner (as shown in Figs. 1 and 2) that the yield strength cannot be evaluated using a simple Taylor model^{32}. Here, we model the yield strength of bulk intdiamond by considering 6000 grains on the basis of the Sachs model (see “Methods” section). Macroscopic yield strength is defined as the stress level at which 90% of the grains yield. The Vickers hardness is then assumed to be three times the compressive yield strength^{33,34,35,36,37}. The resulting hardness of intdiamond increases with decreasing twin thickness (both λ_{1} and λ_{2}), and is consistently higher than that of ntdiamond with the same twin thickness (λ_{1}) and grain size (Fig. 3b). At a twin thickness of 0.62 nm, intdiamond reaches a hardness of 668 GPa, ~67% higher than that of the ntdiamond (401 GPa). These results indicate that the hardness limit for ntdiamond can be raised further by adding intersectional TBs in ntdiamond. Although the hardness limit of the intdiamond can be improved to 668 GPa, it is still below the theoretical hardness (~810 GPa) of diamond calculated by the 9τ_{theo}, where the τ_{theo} is the theoretical shear strength of diamond.
The fractions of different slip modes occurring in the yielded intdiamond grains are statistically analyzed, and the results are plotted in the insert of Fig. 3a. The fraction of grains yielded by ST mode slips increases whereas the fraction of grains yielded by CLS and CSTII mode slips decreases with decreasing twin thickness. Slips by CSTI mode are difficult to activate due to the high CRSSs, therefore the fraction of grains yielded by CSTI mode slips is essentially zero within the twin thickness range studied here. Hence, the hardness of bulk intdiamond is primarily due to slips in the ST, CLS, and CSTII modes with twin thicknesses up to 10 nm. As the CRSS of CSTII slip is higher than that of slip mode in ntdiamond, the hardness of intdiamond is higher than that of ntdiamond^{14}.
Verification of intdiamond hardness by MD simulation
To further confirm the calculated results by Sachs model, the yield strength of polycrystalline intdiamond is studied by using MD simulation. The calculated stressstrain curve of the intdiamond is plotted in Fig. 4. The yield strength is equal to 165 GPa for intdiamond, 154 and 161 GPa for ntdiamond at twin thickness of 5.5 and 1.2 nm, and 140 GPa for ngdiamond, respectively. Although the strain rate (5 × 10^{8} s^{−1}) in MD simulation is higher than that of the experiment, these results qualitatively confirm that the yield strength of the intdiamond is larger than that of the ntdiamond, and further confirm the simulated results by our Sachs model.
Applicability of the intdiamond idea to synthesis
Huang et al.^{7} and Tao et al.^{8} have shown that, with properly selected precursory materials and under carefully controlled synthesis conditions, nano twins can be consistently introduced in ngdiamond. TEM observations show that in ntdiamond, a significant portion of the grains contains intersectional nanotwins, forming tweedlike pattern characteristic of the intdiamond (cf. Fig, 2 in ref. ^{7} and cf. Supplementary Fig. 2 in ref. ^{8}). These observations suggest that intdiamond is readily manufacturable. The challenge is how to produce intdiamond consistently in bulk samples. Based on previous experimental results, two potentially important parameters to be grain size of the onion carbon precursors (highly deformed graphene layers in precursors increase the chance of TB formation in the diamond formation) and pressure and temperature conditions (high nucleation rates for diamond formation also increase the chance of producing TBs). Therefore, predeformation onion carbon precursors by uniaxial compression, large shear deformation^{38}, and improvement of the synthesis pressure are a feasible method to form intdiamond in the experiment.
Discussion
We have examined the mechanical properties of diamond with a novel microstructure by introducing intersectional twin boundaries in ngdiamond. A total of 39 slip systems in four slip modes of this designer diamond (intdiamond) are systematically analyzed. Based on the critical resolved shear stress of the four modes, we calculate the hardness of bulk intdiamond using the Sachs model and show that intdiamond is much harder than ntdiamond. The hardening mechanism of intdiamond is attributed to the intersectional TBs, which block dislocations motions, resulting in increased CRSS. These results are further confirmed by direct polycrystalline MD simulations. This work provides a new strategy for designing new superhard materials in experiments.
Methods
Barrier strength
The dislocation reaction with TB is a process of kink formation and migration, and it is schematically plotted in Supplementary Fig. 1. To simulate this process, a cuboid diamond twin structure model was first built. In this model, their x, y, and z axis are along diamond matrix’s _{.}\([11\bar2]\), \([\bar 110]\), and [111] directions, the dimensions of x, y, and z axis are 20.9, 2.5, and 13.8 nm, respectively, and contains about 120000 carbon atoms. Next, a series of kinked shuffleset 0° and 60° perfect dislocation with different kink pair widths was introduced in the twin plane of the cuboid diamond twin structure model by using dislocation displacement field method (in Supplementary Fig. 1)^{39,40}.
MD simulations were then performed by using LAMMPS program^{41}, and C–C bonding interactions were described by LCBOP potential^{42}. Periodic boundary condition was only imposed along the y direction and the free surface was imposed in x and z directions. All these constructed structures were relaxed via energy minimization under different shear stress conditions. After relaxation, kink width dependent system energies were obtained and the maximum excess energy can be considered as kink formation energy (2E_{f}) at given shear stress. At the same time, a kink migration energy (E_{m}) were calculated by using NEB method^{43}. Finally, the activation energy Q of dislocation reaction with TB is obtained according to Q = 2E_{f} + E_{m}^{27}. The shear stressdependent activation energy for dislocation reaction with TB is plotted in Fig. 1d. As shown in Fig. 1d, when the activation energy of dislocation reaction with TB reaches zero, the corresponding shear stress can be considered as the barrier strength for a dislocation reacting with TB.
Lattice friction stress
For shuffleset 0° perfect dislocation slip in diamond, it also can be considered as a process of kink formation and migration, and the schematic for this process is plotted in insert of Supplementary Fig. 2. To simulate this process, a diamond structure model was built first. In this mode, its x, y, and z axis are along the \([11\bar 2]\), \([\bar 110]\) and [111] direction and with dimensions of 20.9, 2.5, and 13.8 nm, respectively. Then, a series of kinked shuffleset 0° perfect dislocation with different kink pair widths were introduced in the slip plane located at the center of the diamond structure model. On the basis of these models, the shear stress dependent kink formation and migration energy is obtained by adding shear stress to the diamond structure model by using the method as described above (methods section of Barrier strength). Finally, the shear stressdependent activation energy of shuffleset 0° perfect dislocation slip in diamond is plotted in Supplementary Fig. 2. When the activation energy of the dislocation slip reaches zero, the corresponding stress is the lattice friction stress.
intdiamond hardness by using Sachs model
Sachs model is an effective method to investigate the yield strength for polycrystalline materials with anisotropic slip system. In Sachs model, the yield strength of each grain can be expressed as:
where the \(\sigma _n^m\) represents the yield strength of mth slip system in nth grain and it can be expressed as following:
where the \(\tau _m^{{\mathrm{CRSS}}}\) is the CRSS of mth slip system; \(\mu _n^m\) is the Schmid factor of mth slip system in nth grain.
In this work, a polycrystalline model with 6000 random orientations grains is considered. The yield strength for each grain can be obtained by using Eqs. 8 and 9. On the basis of these critical yield strength, we determine whether the grain yielded under a given uniaxial stress condition. As shown in Fig. 3a, the fraction of yielded grains increased with increasing uniaxial stress, when the fraction of yielded gain reaching 90%, the corresponding stress can be considered as the yield strength for this polycrystalline material. Further its hardness can be obtained by tripling its yield strength^{33,34,35}.
MD simulation method for yield strength calculation
In this work, atomic models for intdiamond, ntdiamond, and ngdiamond were constructed by using Voronoi polyhedron method^{44}. As shown in Supplementary Fig. 3, each model contains 20 grains with an average grain size of 16.23 nm. For intdiamond model, the twin boundaries TB_{1} have two types: Σ3(111) and Σ27(115) and twin boundary TB_{2} is Σ3(111). In this structure model, the fraction of twin boundary Σ3(111) is ~75%, and the fraction of twin boundary Σ3(111) can be further improved to asymptotically approach unity^{45}. The twin thickness is 5.5 nm for TB_{1}, and 1.2 nm for TB_{2}. In the ntdiamond model, the twin thickness is 5.5 and 1.2 nm.
The MD simulations were then performed on these atomic models by using the popular LAMMPS code^{41}, and atomic configurations were visualized and analyzed by using the OVITO package^{46}. In this MD simulation, the C–C bonding interactions were described by Tersoff potential^{47}, and the isothermal–isobaric (NPT) scheme was used^{48}. The timestep is set as 0.001 ps, relaxation time is 200 ps. After structure optimization under 300 K and ambient pressure, the compressive deformation is applied along x direction under a constant strain rate of 5 × 10^{8} s^{−1} with a total true strain of 0.3, and the corresponding stressstrain curves were recorded. The maximum stress in the recorded stressstrain curves can be considered as the corresponding yield strength.
Data availability
The authors declare that the data supporting the findings of this study are available within the paper and its Supplementary Information files.
Code availability
All atomic simulations were performed by using the open source LAMMPS code^{41}.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (NSFC, Grant Numbers 51925105, 51771165 and 51525205).
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B.W. conceived the project. J.X. performed all calculations. J.X and B.W. analyzed the calculated results. J.X., B.W., B.X., X.Z., Y.W., and Y.T. cowrote the paper. All authors discussed the results and commented on the paper.
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Xiao, J., Wen, B., Xu, B. et al. Intersectional nanotwinned diamondthe hardest polycrystalline diamond by design. npj Comput Mater 6, 119 (2020). https://doi.org/10.1038/s41524020003873
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