Mechanism for direct graphite-to-diamond phase transition

Using classical molecular dynamics with a more reliable reactive LCBOPII potential, we have performed a detailed study on the direct graphite-to-diamond phase transition. Our results reveal a new so-called “wave-like buckling and slipping” mechanism, which controls the transformation from hexagonal graphite to cubic diamond. Based on this mechanism, we have explained how polycrystalline cubic diamond is converted from hexagonal graphite, and demonstrated that the initial interlayer distance of compressed hexagonal graphite play a key role to determine the grain size of cubic diamond. These results can broaden our understanding of the high pressure graphite-to-diamond phase transition.

barrier of the graphite-diamond transformation accurately 29 , which gives confidence in the correct description of the kinetics of direct graphite-to-diamond phase transition 30 .
In this paper, we use classical molecular dynamics 31 with LCBOP II potential 24 to investigate the underlying microscopic mechanism for the direct hexagonal graphite-to-diamond phase transition. The crystallographic orientations of the initial hexagonal graphite lattice in the x-, y-and z-axis are taken to be in the directions of [210], [100] and [001], respectively. First, the initial interlayer distance of hexagonal graphite lattice is set to a certain value; then the configuration is equilibrated using molecular dynamics in the isobaric-isothermal (NPT) ensemble to relax the pressure of x and y directions to 0 bar while the interlayer distance of hexagonal graphite lattice hold constant. Finally the configuration is under uniaxial compression along the x direction with a constant speed in the canonical (NVT) ensemble(see Methods).
We find that the compressed hexagonal graphite can directly convert to single or polycrystalline cubic diamond via a new wave-like buckling and slipping mechanism and the initial interlayer distance of compressed hexagonal graphite plays a key role to determine the grain size of polycrystalline cubic diamond. Moreover, our simulations show that hexagonal diamond also can be achieved from hexagonal graphite by a simple slip model, but no wave-like buckling occur during the compression process.

Result
We first report the results of the simulation from compressed hexagonal graphite (with 0.240 nm interlayer distance) to perfect cubic diamond (see Fig. 1). We set pressure along the x[210] direction. At the early stages of structural transformation under compression, the graphite is under elastic compression, in which, the graphite layers hold straight and no slipping occur (Fig. 1a). When the compressive stress along the x[210] direction reaches a critical value, a ''wave-like buckling'' of the graphite layers occur: parts of the graphite rotate clockwise and the remaining parts rotate anticlockwise; at the same time, the slipping of the graphite layers leads to the stacking order of the graphite layers transforming from ''ABAB'' to ''AAAA'' (Fig. 1b). With the increasing of the compressive stress, the inclination angle (h) of the graphite layers becomes larger and larger, meanwhile the bond length between the C-C atoms becomes shorter and shorter [the interlayer distance of the buckled graphite (d) can be calculated as: d 5 d 0 cosh (where d 0 5 0.240 nm is the initial interlayer distance of the hexagonal graphite); and the 'slipping' distance between the adjacent graphite layers (l) is calculated as: l 5 d 0 sinh]. When the inclination angle reaches 30u, the bond length reduces to 0.128 nm; at the same time the interlayer distance of the buckled graphites can be estimated to be 0.208 nm, which is approximately equal to the {111} interplanar distance of cubic diamond (0.206 nm). The 'slipping' distance between the adjacent graphite layers can be calculated as 0.12 nm, this leads to the stacking order of the graphite layers transforming from ''AAA'' to ''ABC'' (Fig. 1c). Then the graphite in ABC stacking begins to transform into a perfect cubic diamond through the same transformation pathways proposed in references 13,16 (Fig. 1d). The graphite layers rotating clockwise and anticlockwise transform into (-11-1) and ( (Fig. 1e). The transformation process in detail is displayed in movie S1 (see supplementary materials). These results suggest a new ''wavelike buckling and slipping'' transformation mechanism, which is distinct from the usual zigzag and armchair buckling conversion mechanism for the cold-compressed graphite phase transformation proposed by Wang et al. 32 .
The stresses are also monitored during the compression process. The stress-strain curves along the three directions are displayed in Fig. 1f. At the initial state, the stresses along the x[210] and y[100] direction are zero, while the stress along the z[001] direction is 53.3 GPa (because the interlayer distance of the graphite has been compressed to 0.240 nm). At the elastic compression stage, the stress along the x direction increases linearly with increasing strain up to  critical wave-like buckling stress, which is larger than the stress along the z direction. Then the stresses of the three directions all increase with the increasing of strain; finally the stress along the x direction reaches its maximum (488.7 GPa) and then drops suddenly; at the same time the graphite begins to transform into cubic diamond.
To understand the transformation mechanism from graphite to polycrystalline diamond, the structural transformation process with interlayer distance of 0.214 nm are also shown in Fig. 2. We can see that the hexagonal graphite layers first buckle at a critical compressive stress, and in the meantime the slipping of the graphite layers leads to the stacking order of the graphite layers transforming from ''ABAB'' to ''AAAA''. With the increasing of the compressive stress, the inclination angle of the graphite layers becomes larger and larger; when the inclination angle reaches a certain value of 23.5u, the stacking order of the graphite layers transforming from ''AAAA'' to ''ABCA'' (Fig. 2b). Then, the rhombohedral graphite begins to transform into polycrystal cubic diamond; the atoms near the buckling place form grain boundary (Fig. 2c) with local displacement. As a result, hexagonal graphite transforms into polycrystal cubic diamond (Fig. 2d) with buckling angle smaller than 30u.
The phase transformation from hexagonal graphite to diamond with different interlayer distances of 0.320 nm-0.208 nm has been studied. Our simulations show that when the interlayer distance of the hexagonal graphite is between 0.240-0.228 nm, the graphite will transform into a perfect crystal of cubic diamond with a critical buckling angle of 30u. However, when the interlayer distance of the hexagonal graphite is larger than 0.240 nm or less than 0.228 nm with the buckling angle of the graphite layers larger or smaller than 30u, the graphite will transform into polycrystalline cubic diamond; Moreover, when the interlayer distance of the hexagonal graphite is less than 0.208 nm, it will be converted into hexagonal diamond rather than cubic diamond. We here illustrate the final configura-tions in Fig. 3 with interlayer distance of 0.315 nm, 0.288 nm, 0.260 nm, 0.228 nm, 0.220 nm and 0.214 nm. The more detailed transformation patterns can be found in movies S2-S7 (see supplementary materials).
To get the best understanding, the conversion process from hexagonal graphite to hexagonal diamond with 0.208 nm interlayer distance is also given in Fig. 4. At the initial stage after 10 ps relaxation 18% atoms of the hexagonal graphite (Fig. 4b) have four nearest neighbors. With the increasing of compressive stress, the hexagonal graphite layers gradually slip into intermediate orthorhombic graphite phase, then the orthorhombic graphite further transforms into hexagonal diamond phase through the same transformation pathways proposed in references 13,16 . It is noted that the wave-like buckling of the graphite layers does not occur during this compression process. The transformation process in detail is shown in movie S8 (see supplementary materials). The stress-strain curves of the three directions during the compression process are displayed in Fig. 4f. At the initial state, the stresses along the x and y directions are zero, and the stress along the z direction is 136.2 GPa. During the compression process, the stress along the x direction increases linearly with increasing strain. When the stress along the x direction reaches 92.1 GPa, the graphite begins to transform into hexagonal diamond. These results show a strong z[001]-axis compression process in contrary to cubic diamond phase conversion shown in Fig. 1f.
The change of per atom potential energy during the transformation process can be calculated following the equation where E c is the current potential energy of the sample, E g is the initial state potential energy of the sample (the hexagonal graphite at 0 bar), N is the total atom number of the sample. Fig. 5 gives the change of per atom potential energies versus the normalized simulation time during the eight compression processes. From the figure it can be found that the maximum energies for interlayer distance from 0.288 nm to 0.220 nm are about 2.11 eV, which is about 0.20 eV lower than that for interlayer distance 0.315 nm and 0.215 nm (2.41 eV); this can be attributed to different distortion of graphite to be made for different interlayer distance. It also can be see that the maximum energy for interlayer distance 0.208 nm (1.18 eV) is much lower than those for other interlayer distances, which indicates that it is much easier to form hexagonal diamond than to form cubic diamond from compressing hexagonal graphite, this is because the high symmetry of the hexagonal graphite to hexagonal diamond transformation pathway.

Discussion
In summary, a new ''wave-like buckling and slipping'' mechanism is proposed to probe into the atomistic mechanism for the direct graphite-to-cubic diamond phase transition. Based on the ''wave-like buckling and slipping'' mechanism, we have explained how polycrystalline cubic diamond can be converted from hexagonal graphite. We found that the initial interlayer distance of the compressed hexagonal graphite determines the grain size of cubic diamond. Our simulation   work provides a possible pathway for the high pressure graphite-todiamond phase transition.

Methods
The MD simulations are performed using the LAMMPS code 31 with LCBOP II potential 24 . Simulations are carried out by integrating Newton's equations of motion for all atoms using a time step of 0.5 fs. The crystallographic orientations of the initial hexagonal graphite lattice in the x-, y-and z-axis are taken to be in the directions of [210], [100] and [001], respectively (the initial length of x direction is 13.2 nm, while the y and z directions' length are 10.2 nm and 13.6 nm, respectively, containing 192,000 atoms). The initial interlayer distance of hexagonal graphite lattice is set to a certain value; then the configuration is equilibrated using molecular dynamics in the isobaric-isothermal (NPT) ensemble at certain temperature (2000 K) for 10 ps; during the thermally equilibrium, the interlayer distance of hexagonal graphite lattice hold constant and relaxed the pressure of x and y directions to 0 bar (LAMMPS code has the ability to hold specific dimension not changed in the isobaric-isothermal (NPT) ensemble). Finally the configuration was compressed along the x direction with a constant speed (100 m/s) in the canonical (NVT) ensemble (fix indent command of the LAMMPS code is used to realize the compression process). During the compression process, the periodic boundary conditions were used in y and z directions; and the temperature of the system was 2,000 K. In this paper, engineering strain is defined as e 5 (l 2l 0 )/l 0 , where l is the current length and l 0 is the length along the X direction after the thermally equilibrium process. The stresses reported in this work were calculated using the virial theorem, which takes the form: Where V is the current volume of the sample, N is the total number of atoms, _ x a i is the ith component of velocity of atom a, m a is the mass of atom a, r ab is the distance between two atoms a and b, Dx ab i~x a i {x b i , U is the potential energy function.