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Trace element diffusion in bulk NiAl

S. Mukherjee and B. R. Cooper
Department of Physics, West Virginia University, Morgantown, West Virginia
E-mail:
sonali@cooper.phys.wvu.edu

The use of NiAl as a high temperature structural material is hindered due to its limited ductility. Experiments have shown that the activation energy for the motion of dislocations in NiAl is the same as that of the self-diffusional activation energy of Ni in NiAl, thus pointing to the dependence of dislocation motion on self-diffusion of Ni in NiAl. One of the ways to increase the mechanical strength of NiAl would be to mechanically block dislocation movement by adding trace elements which have higher diffusional activation energy in NiAl as compared to the self-diffusional activation energy of Ni in NiAl. The aim of this work is to understand the factors which determine diffusion of typical trace elements in NiAl. The dynamics of the diffusion process is determined, in part, by the diffusion activation energy of the trace elements in the host alloy of NiAl. The activation energy in turn depends on the size and other consequences (e.g., stiffness), of the electronic structure of the trace elements. We have calculated the diffusional activation energy of typical trace elements like Cr, Mo and Nb in NiAl using full potential, LDA based total energy calculations. Furthermore, we have also calculated the self-diffusional activation energy of Ni in NiAl.

 

Fig (1) Figure of a NiAl lattice showing vacancy position

NiAl has a BCC (CsCl) structure with the Ni and the Al atoms occupying the sites of two inter-penetrating simple cubic lattices respectively. The nearest neighbors of a Ni atom are 8 Al atoms and vice versa (see Fig(1)). As diffusion in alloys is through atom-vacancy exchanges, the diffusional activation energy is calculated using 16 atom supercell with a Ni vacancy as shown in Fig(1). We have chosen Ni vacancy over an Al vacancy on the basis of earlier calculations [1], which have shown that Al vacancy formation energy is much higher (2.14 eV) as compared to Ni vacancy formation energy (0.93 eV) in NiAl. The difference in the Ni and Al vacancy formation energies translate to a Ni vacancy concentration of 10-3 to 10-4 at 1000oC as compared to 10-7 to 10-9 Al vacancy concentration at the same temperature. Thus even at 1000oC an Al vacancy is highly improbable.

 

For trace element diffusion, the trace element atom substitutes one of the Ni atoms. We have chosen a Ni site as the preferred site for occupation by the trace element, as compared to the Al site, because of the presence of much larger number of Ni vacancies as compared to Al vacancies. For both self-diffusion and trace element diffusion we have assumed that the diffusing atom follows a direct pathway during a typical diffusional jump, wherein the diffusing atom attempts to hop to the second nearest neighbor Ni vacancy without going to the other Al sites as shown by the arrow in Fig(1). Though, in principle the diffusional pathway can involve many intermediate steps wherein the diffusing atom first jumps to nearest neighbor Al vacancies forming anti-site defects before reaching the Ni vacancy, the low probability of finding an Al vacancy makes the direct pathway the most likely pathway. Other diffusional pathways involving concerted jumps of more than one atom which might avoid creation of Al vacancies is also highly improbable because of the high cost of anti-site defect formation at the Ni sites.

 

Fig (2) Figure of a NiAl lattice showing saddle position

The diffusional activation energy associated with the direct pathway is given by: Esaddle - Eoriginal. Eoriginal is the total energy of the system (supercell in this case), when the diffusing atom is at the original position as shown in Fig(1). Esaddle is the system energy when the diffusing atom is at the saddle point position. The saddle point configuration is obtained when the diffusing atom is at the face of the cube formed by the Al atoms as shown in Fig(2). (Only part of the supercell is shown in Fig(2) for clarity). The system has the highest total energy at the saddle point because of the close proximity between the Al atoms and the diffusing atom. (The diffusing atom in both Fig(1) and Fig(2) is Ni. The trace elements are assumed to follow the same pathway and hence the Eoriginal and Esaddle for trace atom diffusion can be obtained by replacing the diffusing Ni atom by the trace atom). Static relaxations of the atoms are included in the calculations of Eoriginal and Esaddle. The relaxation of the Al atoms away from the diffusing atom, at the saddle point position is shown by the arrows in Fig(2).

 

We find that along the direct pathway, the size of the diffusing atom plays the dominant role in determining the diffusional activation energy. Because of its small size, Cr atom feels the least proximity to the Al atoms at the saddle point, resulting in a lower saddle point energy and consequently lower activation energy as compared to the other trace elements. A Mo atom has a higher activation energy because of its larger size and consequently because of the larger saddle point energy. With even further increase in the trace element size, for instance in Nb, the activation energy reduces and drops below the value of the self-diffusion of Ni in NiAl. Because of the large size of the Nb atom, even at the original system configuration the proximity of the 8 nearest neighbor Al atoms makes Eoriginal not much less compared to Esaddle. This reduces the difference between the two energies and consequently the activation energy. Note that at the original position the diffusing atom has 8 nearest neighbor Al atoms as compared to 4 in the saddle point. Though at the saddle point the diffusing atom is closer to the Al atoms than at the original point, if the size of the diffusing atom is increased beyond certain limit, the energy required to be at the original position can become comparable to the saddle point configuration thus reducing the activation energy. Thus we find that the activation energy increases with increase in size of the trace element; but beyond certain trace element size, the activation energy reduces upon further size increase.

References

[1] C. L. Fu, M. H. Yoo and K. M. Ho, Phys. Rev. B. 48, 6712 (1993).


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