Point defects in B2 alloys
Introduction
I worked on the problem of point defect concentrations in B2 ordered alloys for my Masters thesis. At present, I am working on phase field modelling of microstructural evolution and I am not planning to go back to point defects in ordered intermetallics. However, I still am curious about them. I suppose I am not alone in this respect. Once in a while, I find Prof. Robert Cahn discussing this question - the most recent being his excellent book "The Coming of Materials Science" - which, by the way, I enjoyed very much. Incidentally, this book is probably the nicest starting point in case you are interested in this problem. If you have any comments/suggestions, please feel free to write to me.
Point defects in an ordered alloy such as B2 NiAl are qualitatively different from those in pure metals or disordered alloys. Since point defects control many technologically important properties such as atomic diffusion, creep, hardnes, mechanical properties and sintering, they have been studied extensively.
Experimental and theoretical studies
A host of experimental techniques are used in studying the point defect concentrations in ordered B2 alloys. In NiAl for example, the following techniques have been used:
X-ray diffraction,
density,
magnetic susceptibility,
electrical resistivity,
void formation studies,
perturbed angular correlation of gamma rays, and
in situ neutron diffraction.
Theoretically, electronic structure calculations and empirical interatomic potential calculations have been used for studying the defect energies (in B2-NiAl, for example). Several studies have examined point defect concentrations using alloy models in which atoms interact with each other through pairwise bonds of fixed length and energy. A Bragg-William mean field approximation is then used to extract defect concentrations at finite temperature.
Some mean field studies of point defects (in B2-NiAl, for example) a priori assume certain type of defects to be negligible and then proceed to calculate the concentrations of the remaining defects. However, since vacancies and antisite defects are all essentially substitutional point defects in the same underlying b.c.c lattice, it is possible to formulate a mean field theory which treats all the point defects in a unified way.
In some other mean field studies, pair potentials that are used in the formulation are obtained by fitting experimentally observed defect concentrations. A theoretical formulation is not satisfactory if it relies on parameters which are obtained by fitting the very property which it tried to predict. Given the advances in electronic structure calculations for metals and alloys, it is now possible to obtain interactions energies, which could then be used in Bragg-Williams formulation to yield point defect concentrations.
Our mean field study on NiAl
Keeping the above observations in mind, we have used a Bragg-Williams formulation with pair interaction energies derived from the (continuous) effective pair interaction energies (which were themselves were based on many-body central force potentials of the Finnis-Sinclair type) between Ni-Ni, Al-Al, and Ni-Al. Since, the effective pair potentials become negligible at and beyond the theird nearest neighbour distances, we restrit the interatomic interactions in our model to only the first two mearest neighbour shells.
Our calculations predict B2-NiAl to be a triple defect B2 system. The predominant defect in the stoichiometric alloy is a combination of an an Ni-antisite defect and two vacancies in the Ni sublattice.
The Al-rich alloys exhibit a Bradley-Taylor behaviour; alloys of composition (50+x) at%, contain 2x% vacancies, almost all of which are on the Ni sublattice. Further, these vacancies have no temperature dependence upto nearly 1500 K. Similarly, the Ni-rich alloys of composition (50-x) at% contain x% Ni-antisites, with negligible temperature dependence up to nearly 1500 K.
Further, our vacancy concentrations are in good agreement with the experimental vacancy concentrations reported in the literature, despite the limitations of the formalism.
If you need further information about point defects in B2-NiAl, please refer to
Mean field theory of point defects in $\beta$-NiAl
M. P. Gururajan and T. A. Abinandanan
Intermetallics , 8 (2000), 759-767.
My MSc thesis also contains an extensive literature review as well as discussion of some of our results (and is available for download here).
Our studies on other B2 alloys
From our study of B2-NiAl, to extend the study to other alloys of B2 structure was rather simple. Using our model, and treating the bond energies as parameters, we were able to correlate the bond energies in a nearest neighbour Bragg-Williams mean field model to the point defect behaviour. Using a bond-breaking model, we could also identify the condition on the bond energies for which the alloy would exhibit the so called constitutional vacancies.
The range of vacancy concentration behaviours in B2 intermetallics can be broadly classified into four types. In particular, the Bradely-Taylor behaviour could be obtained for a wide range of bond energies, as long as certain conditions on the bobd energies are satisfied. The difference results reported by various mean field studies on NiAl, FeAl, CoAl, NiGa, CoGa, and AuCd can be rationalised within our scheme of classification. Further, there are indications that the Bradely type behaviour might be an atomic size effect.
If you need further information on our parametric study and our results, please refer to the following papers:
Correlation between bond energies and point defect behaviour in B2 intermetallics
M. P. Gururajan and T. A. Abinandanan
Philosophical Magazine A , 81 (2001), 2785-2795.
Effect of interaction energies on the vacancy behaviour in B2 ordered intermetallics
M. P. Gururajan and T. A. Abinandanan
Materials Science and Engineering A , 329-331 (2002), 388-394.