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npj Computational Materials Volume 6, Article Number: 119 (2020) Cite this article
According to reports, the hardness of nano twin diamond (nt-diamond) is more than twice that of natural diamond, thanks to the fine space between twin boundaries (TB), which can prevent dislocation propagation during deformation. In this work, we used molecular dynamics (MD) calculations to explore the impact of additional TBs in nt-diamond, and introduced a new type of crossed nano twin diamond (int-diamond) for future laboratory synthesis template. The hardness of the whole diamond is predicted by first analyzing the slip mode of single dislocations in twin crystal grains, and then calculating the overall characteristics according to the Sachs model. Here, we show that the hardness of int type diamond is much higher than that of nt type diamond. The hardening mechanism of int-diamond is attributable to the increase in critical analytical shear stress caused by the existence of cross TBs in nt-diamond; MD simulations further verify this result. This work provides a new strategy for designing new superhard materials in experiments.
Diamond is the hardest, hardest, and least compressible crystalline material in the world. Understanding and further improving its hardness is of scientific and technological significance1,2,3. In the past few decades, a lot of efforts have been made in these directions, including experiments and theories4,5,6,7,8,9. It has been proved that according to the well-known Hall-Page effect 10, 11, the hardness of diamond can be improved by refining the grain size and/or twin thickness of diamond. For example, it has been reported that nanocrystalline diamond (ng-diamond) with a grain size of 10–30 nm has a Knoop hardness of 110–140 GPa, which is significantly higher than that of single crystal diamond 4, 5, and 12. The average twin thickness (λ) of nano twin diamond (nt-diamond) is 5–8 nm. It is synthesized by compressing the precursor 7,8 of the onion structure. It has recently been reported to have a Vickers hardness of 175–200 GPa, creating New world record. Can the hardness of nt-diamond be further improved? This is the basic scientific problem of designing new superhard materials with potentially wide-ranging impact.
Recent molecular dynamics (MD) simulations of nt-diamond show that the unprecedented hardness is due to two factors: high lattice friction stress caused by strong sp3 CC bonds and high non-thermal stress caused by the Hall-Page effect. Experimental and theoretical studies have also shown that twin boundaries (TB) can continuously harden covalent materials while reducing the twin thickness (λ) to a very small value (approximately a few nanometers) 7,8,15,16. Therefore, for covalent materials, a practical strategy to achieve ultra-hardness is to introduce more TB into the microstructure. Based on this idea, a new type of int-diamond model was constructed by introducing cross TBs into nt-diamond. After analyzing the dislocation slip mode in a single diamond grain, the hardness of the bulk diamond is calculated based on the Sachs model. We found that the hardness of int-diamond is much higher than that of nt-diamond. The MD simulation further verified this result. This work provides a new strategy for designing new superhard materials in future experiments.
For diamond, the main dislocations slide in the (111) plane along the <110> direction, and the Burgers vector is \(\frac{1}{2}\) <110> for perfect dislocations and \(\frac{ 1}{1}{ 6}\) <112> is used for slip part of dislocation 14,18. According to the angle between the Burgers vector and the direction of the dislocation line, there are six types of dislocations: 0° complete dislocation of the sliding sleeve, 30° partial dislocation of the sliding sleeve, 60° complete dislocation of the sliding sleeve, and 90° partial dislocation of the sliding sleeve. , shuffle-set 0° perfect dislocation and shuffle-set 60° perfect dislocation 14. Among them, the shuffle-set 0° perfect dislocation has the lowest critically resolved shear stress (CRSS) of dislocation motion and the lowest barrier strength when reacting with TBs14. Here CRSS is defined as the threshold stress for dislocation movement, and the barrier strength is defined as the threshold stress for dislocation reaction with TB when the activation energy reaches zero. Therefore, at room temperature, the hardness of diamond is mainly controlled by the behavior of shuffling the 0° perfect dislocation16,20. Therefore, in this study, the hardness of the overall diamond will be analyzed based on the behavior of a random group of 0° perfect dislocations.
In an internal diamond grain, two different orientation twin boundaries TB1 and TB2 coexist and interweave, and their average twin thickness is λ1 and λ2 (Figure 1a) 21, 22. TB1 and TB2 divide the crystal grains into domains with four different crystal orientations, namely orientations D1, D2, D3, and D4. In these different orientations, the lattices of D1 and D2, D2 and D3, D3 and D4, D4 and D1 are mirror images across TB1, TB2, TB1, and TB2, respectively. Therefore, the slip system in the overall diamond grain can be represented by the combination of four Thompson tetrahedra (Figure 1b): ABCD, ABCD1, A1BCD, and A2BCD1, corresponding to D1, D2, D3, and D4, respectively. The combination of these four Thompson tetrahedrons produced 39 slip systems in a monolithic diamond grain (Table 1). According to the direction of the dislocation line and the orientation of the slip surface, the slip mode of the shuffle-set 0° perfect dislocation inside the diamond is divided into four types: slip transfer (ST) mode, restricted layer slip (CLS) mode, restricted slip Transfer mode I (CST-I) and restricted sliding transfer mode II (CST-II) modes, all of these modes are schematically drawn in Figure 1b. For ST mode, the slip surface is parallel to TB1 or TB2, and the corresponding dislocation line is located in TB2 or TB1. For CLS mode, the slip surface is parallel to TB1 or TB2, and the corresponding dislocation line is not parallel to TB2 or TB1. For the CST-I mode, neither the slip surface nor the dislocation line is parallel to any TB. For CST-II mode, the slip surface is not parallel to any TB, and the corresponding dislocation line is located in TB1 or TB2. Therefore, the interaction between these slip patterns and TB is different. This causes the CRSS of the four slip modes to be different. For ST mode, the dislocation-TB reaction is characterized by dislocations propagating in TBs23,24, and the corresponding CRSS is determined by the lattice friction stress and barrier strength of a random group of 0°perfect dislocations reacting with TBs25. For the CLS mode, the dislocation movement is limited between two TBs, and since the dislocation line remains on the two TBs, its CRSS23,26 can be evaluated by the increased dislocation energy. For CST-I mode, the shuffle-set 0° perfect dislocation is limited between two TBs, and then when it reaches TB, it becomes shuffle-set 60° perfect dislocation parallel TB, because the initial dislocation line is not parallel to TB . Finally, the shuffle-set 60° perfect dislocation interacts with the TB to propagate through the TB, and its CRSS is increased by the barrier strength of the shuffle-set 60° perfect dislocation reacting with the TBs and the increase due to the generation of dislocations on the two TBs. The dislocation energy is determined. For the CST-II mode, although its dislocation movement is similar to that of the CST-I mode, its barrier strength is determined by the perfect dislocation of the shuffle-set 0° reacting with TB. In order to obtain the CRSS of these slip modes, the barrier strength of the perfect dislocations of 0° and 60° of the shuffle-set that react with TBs must first be calculated.
A schematic diagram of the structure of an int-diamond. The lattices of D1 and D2, D2 and D3, D3 and D4, D4 and D1 are mirror images of the double boundaries TB1, TB2, TB1 and TB2, respectively. GB is the grain boundary. b. Four dislocation slip modes inside diamond: slip transfer mode (ST), restricted layer slip mode (CLS), restricted slip transfer mode I (CST-I) and restricted slip transfer mode II (CST-II). c A schematic diagram of the perfect dislocation of a shuffle-set screw BD and the reaction with TB when it reaches TB. The d shuffle-set screw perfectly dislocations the stress-dependent activation energy of the reaction between BD and AD and TB.
The reaction of Shuffle-set 0° and 60° perfect dislocations with TBs can be considered as kink nucleation and migration processes 14, 27, 28 (Figure 1c and Supplementary Figure 1). Calculate the shear stress-related activation energy for kink nucleation and migration (detailed information is in the "Methods" section), and the results are plotted in Figure 1d. With the increase of shear stress, the activation energy of the perfect dislocations of shuffle-set 0° and 60° and TB reached zero at shear stresses of 19 and 48 GPa, respectively. These stresses are considered to be the respective barrier strengths of the perfect dislocations of shuffle-set 0° and 60°. The double intersection point can provide a pinning obstacle for the dislocation slip when the shuffle-set dislocation slips along the twin plane. However, in nt-diamond, shuffle-set dislocations sliding along two planes show no energy advantage over dislocations along other sliding planes. Therefore, shuffle-set dislocations are conducive to sliding along the slip plane rather than the twin plane. This paper ignores the pinning effect of the intersection point on the movement of the shuffle-set dislocation.
In ST mode, dislocation movement is prevented by TB (inset of Figure 2a)29. According to the dislocation stacking theory 30, the CRSS (τcss) of this mode is expressed as follows: 25
Where τ0 is the lattice friction stress; G is the shear modulus; b is the size of the Burgers vector; λ is the twin thickness of the cross twins; τTB is the barrier strength when the perfect dislocation reacts with TB at shuffle-set 0° . The modulus and stress are in GPa, and all length parameters are in nm.
a Calculated CRSS for slip transfer (ST) mode and confined layer slip (CLS) mode. The dislocation accumulation model and the virtual work principle are used in ST and CLS modes, respectively. b Calculated CRSS for restricted sliding transfer mode I (CST-I) and restricted sliding transfer mode II (CST-II). For the CST-I and CST-II models, it can be considered as a superposition of ST and CLS models. The illustration is a schematic diagram of the perfect dislocation AB, AD, BC and BD reaction of a random set screw with twin boundaries TB1 and TB2. GB is the grain boundary.
According to the reference, G = 540 GPa. In Figure 14, τTB = 19 GPa, calculated as above, τ0 = 10.3 GPa (refer to the "Methods" section) and assuming λ1 = λ2 = λ, the double-thickness-related CRSS of the ST mode is rewritten as follows:
It is plotted in Figure 2a. The CRSS increases with the decrease of the twin thickness, and the trend and quantitative value are similar to the ST mode in nt-diamond14.
CRSS in CLS mode can be calculated according to the virtual working principle (Figure 2), expressed as: refs. 25,31
Where θ is the angle between the slip surface and the twin plane, λ is the thickness of the twin; ν is the occupancy; ϕ is the angle between the dislocation line and the Burgers vector, and α is the core parameter of the dislocation23.
Use the corresponding parameters from the reference. 14 and diamond material parameters (listed in Supplementary Table 1), CRSS in CLS mode is expressed as:
Use equations. In Figure 4, the CRSS of the CLS mode is calculated and plotted in Figure 2a. The CRSS increases as the twin thickness decreases, and its value is similar to the CLS mode in nt-diamond14.
In CST-I mode, the dislocation movement is restricted by TB1 or TB2 and blocked by TB2 or TB1, respectively. Therefore, the corresponding CRSS is affected by the Hall-Page effect and the restricted layer sliding effect. In order to obtain CRSS, the dislocation accumulation and CLS model are used (Figure 2b). The CRSS of the CST-I mode is expressed as:
Where τ0 is the lattice friction stress, τTB is the barrier strength of the shuffle-set 60° perfect dislocation reacting with TB, and ν is the Poisson's ratio. All other parameters have been defined before.
Based on the diamond parameters in the reference. 14 (listed in Supplementary Table 1), the CRSS of CST-I mode is expressed as:
The resulting CRSS as a function of twin thickness is plotted in Figure 2b. The CRSS increases as the thickness of the twins decreases. Since CRSS is affected by the Hall-Page effect and the confinement layer slip effect, it can be regarded as a superposition of ST and CLS modes. Under the same twin thickness, the CRSS is higher than the ST and CLS modes.
Similar to the CST-I mode, the CRSS of CST-II is affected by the Hall-Page effect and the confinement layer slip effect; therefore, the CRSS of CST-II can be expressed by an equation. 5. The difference is that in this case, τTB refers to the barrier strength of the shuffle-set 0° perfect dislocation that reacts with TB. Based on the equation. 5 and diamond parameters, CRSS in CST-II mode is expressed as:
The thus calculated CRSS of the CST-II mode is plotted in Figure 2b. Due to the low barrier strength of shuffle-set 0° perfect dislocations, CRSS is smaller than CST-I under the same twin thickness. Due to the combination of Hall-Page and restricted layer slip effects, the CRSS is also higher than the ST and CLS modes under the same twin thickness.
The Sachs model is a single-slip system model for the mechanical properties of polycrystalline materials. It is a particularly effective method for studying the yield strength of polycrystalline materials with anisotropic slip systems. For bulk diamond, dislocations in multiple twin domains change direction and slip plane in such a complicated way (as shown in Figure 1 and Figure 2) that a simple Taylor model cannot be used to evaluate the yield strength. Here, we simulate the yield strength of bulk diamond by considering 6000 grains based on the Sachs model (see the "Methods" section). The macroscopic yield strength is defined as the stress level at which 90% of the grains yield. Then assume that the Vickers hardness is three times the compressive yield strength 33, 34, 35, 36, 37. The hardness of the whole diamond increases with the decrease of twin thickness (λ1 and λ2), and is always higher than that of nt diamond with the same twin thickness (λ1) and grain size (Figure 3b). At a twin thickness of 0.62 nm, the hardness of int diamond reaches 668 GPa, which is about 67% higher than that of nt diamond (401 GPa). These results indicate that by adding cross TB to nt-diamond, the hardness limit of nt-diamond can be further increased. Although the hardness limit of solid diamond can be increased to 668 GPa, it is still lower than the theoretical hardness of diamond calculated by 9τtheo (~810 GPa), where τtheo is the theoretical shear strength of diamond.
a Fraction of yielded grains as a function of uniaxial stress. When the proportion of yielding grains reaches 90%, the corresponding uniaxial stress is defined as the yield stress σy. The numbers inserted are the statistical scores of the different slip patterns that occur in the diamond grains produced. b Comparing with ng-diamond and nt-diamond, the calculated hardness of int-diamond. The numbers inserted are the double thickness-related hardness of int-diamond and nt-diamond calculated by the Sachs model, and the comparison of experimental measurement data 7,8. Htheo is the theoretical hardness of diamond calculated from 9τtheo, where τtheo is the theoretical shear strength of diamond.
The scores of different slip modes in the overall diamond grains produced are statistically analyzed, and the results are plotted in the inset of Fig. 3a. The ratio of grains produced by ST mode slip increases, while the ratio of grains produced by CLS and CST-II mode slip decreases with the decrease of twin thickness. Due to the high CRSS, the slip of the CST-I mode is difficult to activate, so in the twin thickness range studied here, the proportion of grains produced by the slip of the CST-I mode is basically zero. Therefore, the hardness of bulk diamond is mainly due to the slip in ST, CLS and CST-II modes with twin thicknesses up to 10 nm. Since the CRSS of CST-II slip in nt-diamond is higher than that of slip mode, the hardness of int-diamond is higher than that of nt-diamond14.
In order to further confirm the calculation results of the Sachs model, MD simulation was used to study the yield strength of polycrystalline monolithic diamond. The calculated stress-strain curve of the whole diamond is shown in Figure 4. The yield strength of monolithic diamond is equal to 165 GPa, the yield strength of nt diamonds with twin thicknesses of 5.5 and 1.2 nm is 154 and 161 GPa, and 140 is GPa of ng-diamond, respectively. Although the strain rate (5 × 108 s−1) in the MD simulation is higher than that in the experiment, these results qualitatively confirm that the yield strength of monolithic diamond is greater than that of nt diamond, and further confirm the simulation results of our Sachs model.
The red, yellow, blue and cyan lines are the stress of int-diamond with twin thickness λ1 = 5.5 nm and λ2 = 1.2 nm, and nt-diamond and ng-diamond with twin thickness λ of 5.5 and 1.2 nm Strain curves, respectively.
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 ng-diamond. TEM observations show that, in nt-diamond, very large A part of the grains contains intersecting nano twins, forming the tweed-like pattern characteristics of int-diamond (see Figure 2 in Reference 7 and Supplementary Figure 2 in References). 8). These observations indicate that monolithic diamonds are easy to manufacture. The challenge is how to consistently produce solid diamonds in bulk samples. According to previous experimental results, two potentially important parameters are the grain size of the onion carbon precursor (the highly deformed graphene layer in the precursor increases the chance of TB formation in diamond formation) and the pressure and temperature conditions (diamond formation The high nucleation rate also increases the chance of tuberculosis). Therefore, pre-deformation of the onion carbon precursor through uniaxial compression, large shear deformation and increased synthesis pressure is a feasible method to form a monolithic diamond in the experiment.
We have studied the mechanical properties of diamond with a new microstructure by introducing cross twin boundaries in ng-diamond. A systematic analysis of 39 slip systems in the designer's int-diamond's four slip modes. Based on the critically resolved shear stress of the four modes, we used the Sachs model to calculate the hardness of the bulk diamond, and showed that the diamond is much harder than the nt diamond. The hardening mechanism of int-diamond is attributed to the crossed TBs, which prevents the movement of dislocations, leading to an increase in CRSS. These results are further confirmed by direct polycrystalline MD simulations. This work provides a new strategy for designing new superhard materials in experiments.
The dislocation reaction with TB is a process of kink formation and migration, which is schematically drawn in Supplementary Figure 1. In order to simulate this process, a cuboid diamond twin structure model was first established. In this model, their x, y, and z axes are along the .\([11\bar2]\), \([\bar 110]\) and [111] directions of the diamond matrix, the x, y dimensions, The z-axis is 20.9, 2.5, and 13.8 nm, and contains approximately 120,000 carbon atoms. Next, by using the dislocation displacement field method (in Supplementary Figure 1), a series of kink shuffle settings with different kink pair widths are introduced in the twin plane of the cuboid diamond twin structure model. 0° and 60° perfect Dislocation 39,40.
Then use the LAMMPS program for MD simulation, and the CC bonding interaction is described by the LCBOP potential. Periodic boundary conditions are applied only in the y direction, and the free surface is applied in the x and z directions. The structure of all these structures relaxes by minimizing energy under different shear stress conditions. After relaxation, the system energy related to the kink width is obtained, and the maximum excess energy can be regarded as the kink formation energy (2Ef) under a given shear stress. At the same time, the kink migration energy (Em) was calculated using the NEB method. Finally, according to Q = 2Ef Em27, the activation energy Q for the reaction with TB dislocation is obtained. The shear stress-dependent activation energy of the dislocation reaction with TB is plotted in Figure 1d. As shown in Figure 1d, when the activation energy of the reaction between dislocations and TB reaches zero, the corresponding shear stress can be considered as the barrier strength of the reaction between dislocations and TB.
For the shuffle-set 0° perfect dislocation slip in diamond, it can also be considered as the process of kink formation and migration. The schematic diagram of this process is drawn in the insert in Supplementary Figure 2. In order to simulate this process, a diamond structure model was first established. In this mode, its x, y, and z axes are along the \([11\bar 2]\), \([\bar 110]\) and [111] directions, and the dimensions are 20.9, 2.5, and 13.8 nanometers ,respectively. Then, a series of kink shuffle-set 0° perfect dislocations with different kink pair widths were introduced in the slip plane at the center of the diamond structure model. On the basis of these models, the shear stress is added to the diamond structure model by using the above method (the method part of the barrier strength) to obtain the shear stress-dependent kink formation and migration energy. Finally, the shear stress-dependent activation energy of the 0° perfect dislocation slip randomly set in diamond is plotted in Supplementary Figure 2. When the activation energy of dislocation slip reaches zero, the corresponding stress is lattice friction stress.
Sachs model is an effective method to study the yield strength of polycrystalline materials with anisotropic slip system. In the Sachs model, the yield strength of each grain can be expressed as:
Where \(\sigma _n^m\) represents the yield strength of the m-th slip system in the n-th grain, which can be expressed as:
Where \(\tau _m^{{\mathrm{CRSS}}}\) is the CRSS of the m-th sliding system; \(\mu _n^m\) is the m-th sliding system in the n-th grain Schmid factor.
In this work, a polycrystalline model with 6000 randomly oriented grains was considered. The yield strength of each grain can be obtained by using an equation. 8 and 9. Based on these critical yield strengths, we determine whether the grains yield under a given uniaxial stress condition. As shown in Figure 3a, the proportion of yield grains increases with the increase of uniaxial stress. When the proportion of yield gain reaches 90%, the corresponding stress can be considered as the yield strength of the polycrystalline material. In addition, its hardness can be obtained by three times its yield strength 33, 34, 35.
In this work, the int-diamond, nt-diamond and ng-diamond atomic models were constructed using the Voronoi polyhedron method. As shown in Supplementary Figure 3, each model contains 20 grains with an average grain size of 16.23 nm. For the int-diamond model, there are two types of twin boundary TB1: Σ3(111) and Σ27(115), and the twin boundary TB2 is Σ3(111). In this structural model, the ratio of twin boundary Σ3(111) is about 75%, and the ratio of twin boundary Σ3(111) can be further improved to approach 145 asymptotically. The twin thickness of TB1 is 5.5 nm, and that of TB2 is 1.2 nm. In the nt-diamond model, the twin thickness is 5.5 and 1.2 nm.
Then use the popular LAMMPS code to perform MD simulations on these atomic models, and use the OVITO package to visualize and analyze the atomic configuration. In this MD simulation, the CC bonding interaction is described by the Tersoff potential 47 and the isothermal-equal pressure (NPT) scheme 48 is used. The time step is set to 0.001 ps, and the relaxation time is 200 ps. After structural optimization at 300 K and ambient pressure, compressive deformation is applied along the x direction at a constant strain rate of 5 × 108 s-1, the total true strain is 0.3, and the corresponding stress-strain curve is recorded. The maximum stress in the recorded stress-strain curve can be regarded as the corresponding yield strength.
The author declares that the data supporting the results of this study can be found in the paper and its supplementary information files.
All atomic simulations are performed using the open source LAMMPS code.
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This work was supported by the National Natural Science Foundation of China (NSFC, Grant Numbers 51925105, 51771165 and 51525205).
High Pressure Science Center, State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao 066004
Xiao Jianwei, Wen Bin, Xu Bo, Zhang Xiangyi and Tian Yongjun
Advanced Radiation Source Center, University of Chicago, Chicago, Illinois, 60439, USA
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BW conceived this project. JX performed all calculations. JX and BW analyzed the calculation results. JX, BW, BX, XZ, YW and YT co-authored this paper. All authors discussed the results and commented on the paper.
The author declares no competing interests.
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Xiao, J., Wen, B., Xu, B. etc. Crossed nano twin diamond-the hardest polycrystalline diamond in the design. npj Comput Mater 6, 119 (2020). https://doi.org/10.1038/s41524-020-00387-3
DOI: https://doi.org/10.1038/s41524-020-00387-3
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