Effects of Re, W and Co on dislocation nucleation at the crack tip in the γ-phase of Ni-based single-crystal superalloys by atomistic simulation

The effects of Re, W and Co on dislocation nucleation at the crack tip in Ni have been studied by the molecular dynamics method. The results show that the activation energy of dislocation nucleation is lowered by the addition of Re, W and Co; moreover, the activation energy decreases when the alloying element increases from 1 at.% to 2 at.%. The energy landscapes of the atoms are studied to elucidate these effects. Quantification analyses of the bonding strength between Ni and X (X = Re, W or Co) reveal that strong bonding between Ni and X (X = Re, W or Co) in the dislocation nucleation process can suppress the cleavage process and enhance the ability of dislocation nucleation. The surface energy and unstable stacking fault energy are also calculated to understand the alloying effects on the dislocation nucleation process. The results imply that interaction between alloying elements and Ni atoms plays a role in promoting the dislocation nucleation process at the crack tip. The ability of Re, W and Co in improving the ductility of the Ni crack system is in the order W > Re > Co. The results could provide useful information in the design of Ni-based superalloys.


Introduction
Nickel-based single-crystal (SC) superalloys have excellent mechanical properties (such as high-temperature strength, creepresistance behaviour and low crack-growth rates) and are usually used as turbine blade materials of aerospace engines [1][2][3][4]. They

Models and simulation details
The MD simulation within the framework of embedded atom method (EAM) [28,29] is adopted. For the study of dislocation emission at the crack tip in systems containing Re, W or Co, we use the Ni -Al -Re [30], Ni -Al -W [31], Ni -Al-Co [32] ternary EAM potentials in our research. The Ni -Al -X (X ¼ Re, W or Co) EAM potentials fitted the parameters obtained from first-principles calculations or from experimental results, such as lattice constants, cohesive energies, elastic constants of Ni, Al, X (X ¼ Re, W or Co) and their compounds. The Ni -Al-Re, Ni -Al -W, Ni -Al -Co EAM potentials have been applied to predict the physical properties in the respective systems and gave reasonable results [30][31][32]. In particular, the Ni-Al-Re, Ni-Al-W and Ni-Al-Co EAM potentials have been applied to study the systems containing dislocations or/and cracks [6,23,24,30,33] in the Ni-based superalloys.
We adopt the (1 1 1)[1 1 0] crack system in Ni under mode I loading, in which the crack lies in the (1 1 1) plane and the crack front is along the [1 1 0] direction. This crack system is chosen as the crack propagation and dislocations nucleation happens easily for this crack system [6,34 -36]. This crack system had also been studied by Zhu et al. [20] and Liu et al. [6] and the models in current research take a similar form as their models. To study the alloying effect of Re, W or Co on the dislocation nucleation behaviour at the crack tip, 1 at.% or 2 at.% X (X ¼ Re, W or Co) atoms are randomly doped into the Ni matrix of this crack system. The systems in current research thus are Ni matrix, Ni matrices which contain 1 at.%Re, 2 at.%Re, 1 at.%W, 2 at.%W, 1 at.%Co and 2 at.%Co, respectively. The atomistic simulation model is cylindrical with a radius of R ¼ 90 Å and the crack front is along the central (longitudinal) axis as shown in figure 1a. The crack front contains 29 unit cells in [1 1 0] direction (total crack front length is about 72 Å ), which is sufficient to obtain an accurate activation energy of an isolated dislocation loop [6,16,20]. As the atomically sharp crack configuration in (1 1 1)[1 1 0] crack system of Ni is unstable, one layer of atoms in the (1 1 1) plane behind the crack tip is removed to ensure a stable crack tip configuration. For the (1 1 1)[1 1 0] crack system under mode I loading ( pure tensile load applied along the [1 1 1] axis), the activated dislocation nucleation slip plane is (1 1 1) plane, which is inclined at an angle of u ¼ 70.538 with respect to the (1 1 1) crack plane as shown in figure 1b. The model for the Ni system consists of 167 156 atoms. Periodic boundary condition is imposed along the [1 1 0] direction. The atoms within R ¼ 80 Å around the cylindrical central axis (the grey atoms in figure 1a) are allowed to move and the other boundary atoms (the black atoms in figure 1a) are kept fixed during the MD relaxation. The crack systems with alloying element additions adopt the same settings as the model for the Ni system, except that the lattice parameters for the systems after doping with 1 at.% or 2 at.% X (X ¼ Re, W or Co) are used in building the models. The lattice parameters used in the simulations are 3.520, 3.524, 3.527, 3.525, 3.530, 3.519, 3.519 for systems without alloying elements and systems with 1 at.%Re, 2 at.%Re, 1 at.%W, 2 at.%W, 1 at.%Co and 2 at.%Co, respectively. The models for calculating the lattice parameters will be introduced in the following section.
The model is incrementally loaded from K I ¼ 0.60K Ic with an increment of 0.01K Ic until the energy barrier of dislocation nucleation is overcome and dislocation emission occurs at K emit (K emit ¼ 0.86K Ic for Ni), where K Ic denotes the critical stress intensity factor (theoretical Griffith stress intensity factor) and K emit denotes the stress intensity factor when the energy barrier of dislocation nucleation is overcome. The critical stress intensity factor characterizes the onset of the crack extension. At each loading step, the atom configurations of the model are initially determined according to linear elastic solutions in an anisotropic continuum [37,38], then the conjugate gradient method is used to relax the model while keeping the outer boundary atoms fixed.
We use the climbing image nudged elastic band (CI-NEB) method [39 -42] to determine the minimum energy path (MEP) of dislocation loop emission at the crack tip. The CI-NEB method we adopted is implemented in the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [43]. The initial states for CI-NEB calculation at each K I are given by the above loading scheme. The final states at each K I for the NEB calculation are determined by unloading from K emit to K I ¼ 0.60K Ic with a decrement of 0.01K Ic . The boundary conditions are the same for the initial and final configurations at each specific level of stress intensity factor (SIF) K I . In the CI-NEB method, system configurations (images) are inserted between the initial and final states. This method inserts system images which are interconnected by linear 'springs' in 3N-dimensional (N is the number of atoms) configurational space and this mimics an elastic band. Each configuration can be treated as an image that the spring connects. For current research 24 images (or replicas) including the initial state and the final state are used to perform the NEB calculation. The activation energy for dislocation nucleation DE act can be obtained by subtracting the energy at the initial state from the energy at the royalsocietypublishing.org/journal/rsos R. Soc. open sci. 6: 190441 saddle-point state (saddle state) of the MEP, where the saddle-point state is the state with the highest energy in the MEP. The convergence criterion for CI-NEB calculation in current research is that the potential force on each replica vertical to the path is smaller than 0.005 eV Å 21 .
We adopt OVITO software [44] and the common neighbour analysis (CNA) [45] algorithm implemented in OVITO to visualize atomic structures and dislocations.

Results and discussions
3.1. Influence of alloying elements Re, W and Co on activation energy of dislocation nucleation DE act The minimum energy path, which is the energy variation DE with respect to the initial state energy, of dislocation loop emission at the loads K I ¼ 0.64K Ic and K I ¼ 0.80K Ic for Ni is shown in figure 2. The normalized reaction coordinate is used to represent the reaction coordinate. It is represented as the ratio between the hyperspace arc length along the MEP from the initial state to current state and the total hyperspace arc length along the MEP [20]. As observed from figure 2, there exists an energy barrier for dislocation nucleation in the MEP. This energy barrier characterizes the activation energy of dislocation nucleation DE act and is a critical parameter that labels the onset of plastic deformation. As the loading increases from K I ¼ 0.64K Ic to K I ¼ 0.80K Ic , the DE act decreases from 6.32 eV to 0.53 eV. The activation energy of dislocation nucleation (DE act ) for all the systems is listed in table 1 with the loads K I /K Ic ¼ 0.64, 0.68, 0.72, 0.76 and 0.80, respectively. The simulations for each concentration of randomly doped alloying element are repeated four times. The DE act are calculated from the average value of the simulated results. Figure 3 shows the interpolated curves of the DE act for each system, in which the cubic-spline interpolation is used. The results show that with the increase of the load K I , the activation energy for dislocation nucleation DE act decreases. All the systems with alloying element additions show a decrease of DE act with respect to the Ni matrix in the loading range studied. The DE act for 2 at.% X (X ¼ Re, W or Co) addition is lower than that for 1 at.% X (X ¼ Re, W or Co) addition (We also observe the phenomena that the DE act of the 2 at.% Co system is larger than that of the 1 at.% Co system as seen in the electronic supplementary material, table S4. (A similar phenomena is also seen in the work of Liu et al. [6].) It is due to the random fluctuation of atoms that the number of Co atoms at the dislocation loop in the 2 at.% Co system is lower than that in the 1 at.% Co system. The overall comparative results of the averaged DE act are not influenced by this fluctuation and the averaged DE act is reasonable to reduce the influence of the random fluctuation of atoms.). Among the systems studied, W has the strongest ability in reducing DE act than systems containing Re or Co when concentration changes from 1 at.% to 2 at.%, which indicates that W may be a better element in improving the ductility of the crack system in Ni compared with Re or Co.

Dislocation loop nucleation at the crack tip
In our study, we observed the dislocation nucleation at the crack tip, and the (1 1 1) slip plane for dislocation nucleation under mode I loading is inclined at u ¼ 70.538 with respect to the (1 1 1) crack plane. The saddle state dislocation loop for the Ni matrix system under K I ¼ 0.64K Ic is shown in figure 4 and the face-centred cubic (FCC) structure atoms are removed for a clearer view of the atomic structure. The saddle state shows an incipient dislocation loop bowing out at the crack tip. This dislocation is a a 0 /6k1 1 2l Shockley partial dislocation, where a 0 is the lattice constant. We observed that the Ni matrices with X (X ¼ Re, W or Co) addition also show similar a 0 /6k1 1 2l type dislocation loops. In order to examine the saddle state configuration for dislocation nucleation, the relative shear and normal displacement of the adjacent upper and lower atomic layers across the inclined (1 1 1) slip       represent the Ni atoms, and the pink triangles and inverted pink triangles represent the X (X ¼ Re, W or Co) atoms. As shown in figure 5, the contour line with the displacement approximately around b/2 is the locus of the dislocation loop, which is a a 0 /6k1 1 2l Shockley partial dislocation. It is evident that in forming the dislocation loop, the relative shear displacement has a non-uniform distribution. This non-uniform distribution of shear displacement across the slip plane, instead of forming a uniform shear displacement distribution, can make the dislocation emission process easier. It can also be seen that the dislocation loop near the crack tip is localized in the saddle state. Thus, the fixed boundary effect on our transition state is expected to be weak and the in-plain radius R in our model is reasonable from this respect. In the area enclosed by the dislocation loop and the crack front, the maximum relative shear displacement is around 1b. This slipped region is identified to be swept by a a 0 /6k1 1 2l Shockley partial dislocation. Across the inclined (1 1 1) slip plane, the area with the relative shear displacement smaller than b/2 is not swept by the dislocation. The projected atoms labelled as triangles and inverted triangles on the inclined (1 1 1) slip plane also show the arrangement of atoms across the slip plane in the dislocation core has a different structure from the atoms in the region swept by the dislocation and the region not swept by the dislocation. The relative shear displacement is slightly affected by alloying element additions of X (X ¼ Re, W or Co) atoms. Before proceeding to the following discussions, there is the need to visualize the local atomic structure of atoms across the inclined (1 1 1) slip plane. This will be helpful for understanding the local atomic structure across the inclined (1 1 1) slip plane and will be beneficial for the following discussions. The atomic structure of atoms across the inclined (1 1 1) slip plane is shown in figure 6. The alloying element X (X ¼ Re, W or Co) in the lower atomic layer and the upper atomic layer across the inclined (1 1 1) slip plane are shown as magenta coloured atoms. The 12 nearest-neighbour atoms to the alloying element X (X ¼ Re, W or Co) in the lower atomic layer across the inclined (1 1 1) slip plane is denoted as L1 2 L12. Correspondingly, the 12 nearest-neighbour atoms to the alloying element X (X ¼ Re, W or Co) in the upper atomic layer across the inclined (1 1 1) slip plane is denoted as U1 2 U12. Furthermore, the Ni atoms in figure 6a,b labelled with black dots are denoted as L3 0 and U6 0 , respectively. When the system configuration evolves from the initial state to the saddle state, the distance between L3 0 Ni atom and the alloying atom in figure 6a decreases. Correspondingly, the distance between U6 0 Ni atom and the alloying atom in figure 6b also decreases as the system configuration evolves from the initial to the saddle state. As shown in figure 5, for the atoms in the region already swept by the dislocation the L3 0 atom becomes the first nearest neighbour of the x 1 /b  (1 1 1) slip plane at the load of K I ¼ 0.64K Ic . The relative shear displacement is given with respect to the initial state. (a) Ni matrix system; (b) 2 at.%Co system; (c) 2 at.%Re system and (d ) 2 at.%W system. The coordinates and the relative shear displacement d are normalized by the Burgers vector b. The triangles and inverted triangles represent the atoms in the upper atom layer and the lower atomic layer across the inclined (1 1 1)  alloying element instead of the L3 atom. The same situation also applies to the U6 0 and U6 atom in figure 6b. When the atoms are swept by the dislocation, the U6 0 atom becomes the first nearest-neighbour atom of the alloying element instead of the U6 atom. In the following discussions, if the atom is termed with L1 2 L12, U1 2 U12, L3 0 or U6 0 , it refers to the neighbouring Ni atom of the alloying atom as indicated by figure 6.
The relative normal displacement of the adjacent upper and lower atomic layers across the inclined (1 1 1) slip plane is shown in figure 7 and the displacement is normalized with respect to the (1 1 1) interplanar spacing h. It is observed that in the saddle state the atoms at or near the dislocation core across the inclined (1 1 1) slip plane have the maximum normal displacement, which indicates a slightly expanded local structure near the dislocation core. The alloying elements studied in current paper can affect the local normal displacement of their neighbouring Ni atoms across the inclined (1 1 1)

3.3.
Reason for the effects of alloying element X (X ¼ Re, W or Co) on decreasing activation energy of dislocation nucleation at the crack tip The reason that alloying elements can decrease the activation energy of dislocation nucleation can be explained by the energy difference of atom j between the saddle state and the initial state (DE j ). The sum of energy difference of atom j between the saddle state and the initial state gives the energy difference between the saddle state and the initial state of the system. The relations are given as follows [6]: and where equation (3.1) gives the energy difference of atom j between the saddle state and the initial state, E sad j and E ini j denote the energy of the atom j in the saddle state and the initial state, respectively. N is the number of atoms in the system.
The systems for studying the DE j contour in current research are the Ni matrix and the Ni matrices with 2 at.%X (X ¼ Re, W or Co) addition at the load K I ¼ 0.64K Ic . The contours of DE j for the lower and upper atomic layers across the inclined (1 1 1) slip plane are shown in figures 8 and 9, respectively. From the results, we can identify the dislocation core region from the arrangement of the projected triangle and inverted triangle atoms on the inclined (1 1 1) slip plane. The relatively higher value of DE j distribution in the contours can also be used to identify the dislocation core region. In the region near the dislocation loop, the upper atomic layer atoms across the inclined (1 1 1) slip plane have relatively higher values of DE j than that of the lower atomic layer atoms. This indicates that near the dislocation loop the lower atomic layer atoms across the inclined (1 1 1) slip plane are generally more stable than the upper atomic layer atoms when the system evolves from the initial to the saddle state. It can be seen from figures 8 and 9 that the atoms with relatively lower DE j are mainly located at or near the dislocation loop region. We observe that at some special atomic site the DE j is increased, this may result from two alloying elements being too close. For example, there is a W atom at the lower atomic plane of figure 8 that shows a relatively higher DE j . This relatively higher DE j does not change the general trend of most W atoms decreasing the DE j and the increase in the value of DE j is not so large because the atoms at the dislocation core have a relatively higher background DE j . This relatively higher 'background' DE j in the dislocation core and the relatively low possibility of two alloying elements being too close means the contribution of the increase in DE j by special W atoms has little effect on the overall energy decrease of DE act . The effects of Re, W and Co on decreasing the DE j of atoms mainly comes from two aspects. First, for the alloying elements in the lower and upper atomic layers across the inclined (1 1 1) slip plane, the DE j for the neighbouring Ni (L3, L3 0 , U6 or U6 0 ) atoms of the alloying element across the inclined (1 1 1) slip plane are decreased. The ability of Re, W and Co to lower the DE j of their neighbouring Ni (L3, L3 0 , U6 or U6 0 ) atoms is in the order W . Re . Co. The result here shows that the interaction between alloying elements and their neighbouring Ni atoms has an influence on the DE j . The Co atoms show relatively limited ability to affect the DE j of their neighbouring Ni (L3, L3 0 , U6, U6 0 ) atoms across the inclined (1 1 1) slip plane. This may be related to the relatively weak interaction of Co with its neighbouring Ni atoms with respect to Re or W. Second, the Re, W and Co atoms have relatively lower DE j at their own atomic sites. The X (X ¼ Re or W) atoms in the lower atomic layer across the inclined (1 1 1) slip plane have relatively lower DE j especially at or near the dislocation loop region and the X (X ¼ Re or W) atoms in the upper atomic layer across the inclined (1 1 1) slip plane does not show such an effect. The Co atoms in the upper atomic layer across the inclined (1 1 1) slip plane have relatively lower DE j especially at or near the dislocation loop region, and the Co atoms in the lower atomic layer across the inclined (1 1 1) slip plane do not show this effect. This different behaviour of Co compared with Re or W needs to be further explored. The ability of Re, W and Co to lower the DE j at their own atomic sites also follows the order W . Re . Co. Other randomly doped systems with the concentration of 2 at.% X (X ¼ Re, W or Co) at the load of K I ¼ 0.64K Ic are shown in figure 10. It is shown that the above results and discussions also apply to these randomly doped systems. We can see from the results that the interaction between alloying elements and their neighbouring Ni atoms plays a role in decreasing the DE act . The static equilibrium crack can be treated as a reversible thermodynamic system according to Griffith's theory [46], in which the equilibrium between crack advance and new surface creation is established. The above equilibrium is given by the relation G ¼ 2g s , where G is the mechanical energy release upon crack advance and 2g s is the energy required for creating two new surfaces. The above relation is the necessary condition for fracture and it relates the easy cleavage plane with a relatively lower surface energy (g s ) [47]. The unstable stacking fault energy (g us ) was first proposed by Rice [11], and this quantity can be used to approximately estimate the energy barrier of partial dislocation nucleation. The ratio of g s and g us can be used to evaluate the competition between brittle and ductile behaviour [11,24,33]. A larger value of g s /g us may result in easy dislocation emission relative to cleavage. Molecular dynamics simulation is used for the calculation of surface energy of the (1 1 1) plane and the calculation of unstable stacking fault energy. The model for calculating the g s has the dimension 27:5a 0 [1 1 2] Â 39a 0 [1 1 1] Â 48a 0 [1 1 0], where a 0 is the lattice constant for each system. The model contains 1 235 520 atoms. The surface energy is defined as g s ¼ (E surf 2 E 0 )/2S, where E 0 is the energy of the system without free surface and E surf is the energy of the system after a new surface is created. S is the area of the surface. E 0 is obtained by applying periodic boundary conditions in [1 1 2], [1 1 1] and [1 1 0] directions. The lattice parameters for all the systems are also calculated based on the models for calculating E 0 and the calculated lattice parameters are consistent with the previous result [24]. The calculated results of g s , g us and g s /g us are listed in table 2. The Re or W addition increases the g s of the Ni matrix. The higher the concentration of Re or W, the higher the g s . W addition has a stronger effect on increasing the value of g s than Re addition at the same concentration level. The addition of Co can slightly decrease the g s of the Ni matrix. When the concentration of Co changes from 1 at.% to 2 at.%, the g s is lowered. With the addition of Re, W and Co, the g us are lowered and the value of g us decreases when the alloying element increases from 1 at.% to 2 at.%. The addition of Re, W and Co all increase the value of g s /g us . 2 at.% addition of Re, W or Co has a better effect on increasing the value of g s /g us than 1 at.% addition. At the same concentration level, the ability of alloying elements in increasing the value of g s /g us is in the order W . Re . Co. This indicates that the ability of alloying elements in promoting the dislocation nucleation at the crack tip in the Ni matrix is in the order W . Re . Co. This is consistent with the previous results.

Bonding between atoms calculated by discrete variational method and the implications on dislocation nucleation at the crack tip
Generally, the alloying atoms at or near the dislocation loop have a low value of DE act as shown in the previous sections. We now examine the bonding between alloying atom and its neighbouring Ni atoms when the dislocation passes them using the ab initio calculation. The discrete variational method (DVM) [48,49] is used to calculate the interatomic energy (IE) [50][51][52] between atoms. The IE is a quantity that can be used to evaluate the bonding and interaction between neighbouring atoms, and it had been used successfully in studying the electronic structure and properties in metals and alloys [23,24,53,54]. The IE is defined as where N n is the occupation number for the molecular orbital c n , a nal given by a nal ¼ kf al (r)jc n (r)l, and H a 0 l 0 al is the interaction Hamiltonian matrix elements between atoms. A relatively larger absolute value of IE indicates a stronger interaction and bonding between neighbouring atoms [24,52,55]. More details of IE can be found in the literature [52]. We are interested in the IE of the Ni-X (X ¼ Re, W or Co) atom pair across the inclined (1 1 1) slip plane when the dislocation passes them in the saddle state and the IE of the same Ni-X (X ¼ Re, W or Co) atom pair across the inclined (1 1 1) slip plane in the initial state without dislocation passing by. Two kinds of Ni-X (Re, W or Co) atom pairs across the inclined (1 1 1) slip plane are considered. One kind is the Ni-X (X ¼ Re, W or Co) atom pair where the X (X ¼ Re, W or Co) atom resides in the lower atomic layer across the inclined (1 1 1) slip plane; the other kind is the Ni-X (X ¼ Re, W or Co) atom pair where the X (X ¼ Re, W or Co) atom resides in the upper atomic layer across the inclined (1 1 1) slip plane.
The alloying atoms in the centre of the red circles in figures 8 and 10 and the atoms surround them are selected as the DVM models and the models are shown in figure 11. The alloying atoms within the red circles in figure 8 are located in the lower atomic layer across the inclined (1 1 1) slip plane and the alloying atoms within the red circles in figure 10 are located in the upper atomic layer across the inclined (1 1 1) slip plane. Because the saddle state dislocation configuration may not be the same for different system models, the dislocation did not pass the Co atom within the white circle in figure 8. So, we choose the Co atom in the lower atomic layer across the inclined (1 1 1) slip plane within the red circle in figure 8 and its neighbouring atoms as the model for DVM calculation. For the Co atom in the red circle as shown in figure 8, we do not expect the general trends of the IE compared with Re and W to be vastly different. The alloying atoms all reside in the dislocation region. Furthermore, in the models shown in figure 11 the alloying atoms are not close to each other and the Co atom is in the 3d row of the periodic table, while Re and W are in the 5d row of the periodic table. The interaction of Ni-Co may not closely resemble that of Ni-X (X ¼ Re or W) and previous calculation also showed that there is an apparent difference between the IE of Ni-Co and that of Ni-X (Re,W) [24]. The models containing the Ni-Ni atom pair Table 2. Values of surface energy (g s ), unstable stacking fault energy (g us ) and g s /g us .  Figure 12. Absolute value of the interatomic energy (IE) between X (X ¼ Re, W or Co) and its neighbouring Ni (labelled with L1, L3, U4 or U6 in figure 11) atoms across the inclined (1 1 1) slip plane in the initial state and the saddle state. The absolute values of IE are shown in figure 12. We can see that the absolute values of IE for Ni (L1, L3, U4 or U6)-X (X ¼ Re, W or Co) atom pairs across the (1 1 1) slip plane are generally larger than that for the corresponding Ni -Ni atom pairs. This indicates a stronger bonding between Ni (L1, L3, U4 or U6)-X (X ¼ Re, W or Co) atom pairs compared with the corresponding Ni -Ni atom pairs when the system evolves from the initial state to the saddle state. The absolute value of IE between the alloying element and the neighbouring Ni (L1, U4) atom across the inclined (1 1 1) slip plane becomes larger when the system evolves from the initial state to the saddle state. This can be beneficial for the dislocation emission process because the stronger interaction between the alloying element and the Ni (L1, U4) atom may make the cleavage process more difficult in terms of competition between cleavage and dislocation nucleation. The absolute value of IE between the alloying element and the neighbouring Ni (L3, U6) atom across the inclined (1 1 1) slip plane becomes smaller because the distance between these two atoms becomes larger as the model evolves from the initial state to the saddle state. We can also observe that generally W exhibits the largest absolute IE and Co the smallest. This indicates that the bonding of Ni -W is the strongest and Ni -Co the weakest among the three alloying elements studied. The stronger interaction and bonding between X (X ¼ Re, W or Co) and Ni atoms across the inclined (1 1 1) slip plane can contribute to the stronger ability to resist the cleavage process and promote dislocation nucleation. This interatomic energy calculation validates the fact that the ability of Re, W and Co to promote the dislocation nucleation in the crack tip of Ni follows the order W . Re . Co and it is consistent with the results of previous sections. 3.6. Influence of alloying element X (X ¼ Re, W or Co) and temperature on nucleation frequency of dislocation at the crack tip The nucleation frequency, which defines the rate of dislocation nucleation, can be estimated using the activation energy. Frequency of nucleation events per unit distance along the crack front can be estimated from [15] n ¼ n 0 exp À DE act where n 0 is an appropriate attempt frequency and is defined as n 0 ¼ n(c shear /b). c shear ¼ 3 km s 21 is the shear wave speed. n ¼ 1/45b is the number of nucleation sites per unit length of crack front. The 45b is adopted in terms of the lateral spread length of the dislocation loop along the crack front. The room temperature is taken as 300K, k B is Boltzmann's constant and b is the Burgers vector. When the dislocation emission occurs without a nucleation barrier, the dislocation emission becomes spontaneous. Taking n spon % 10 6 (s . mm) 21 to be the thermally activated spontaneous nucleation threshold of metal in laboratory measurements [15], if the value of n exceeds the spontaneous nucleation threshold n spon , then it can be identified as spontaneous nucleation. The n T relations of different systems are shown in figure 13. To better visualize and compare the magnitude of nucleation