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One of the many singularities of FrankKasper phases is their ability to accommodate extremely large composition ranges by atom mixing on the different sites of the crystal structures. This phenomenon will be reviewed in the present paper with special emphasis on the experimental demonstration of this phenomenon, the theoretical calculation of disordered structures and the modeling of these phases.
FrankKasper (FK) phases are fascinating compounds. They not only represent one of the largest group of intermetallics, but they are also characterized by the richness of their structural and physical properties. Their application as high temperature structural materials (see e.g., Laves, σ, χ and
FK phases are generally characterized by the association of (at least) two elements of different size and different electronic properties (for a review see [
Single elements may also crystallize in one of the FK structures. For example, this is the case for βU (structure of the σ phase), αMn (structure of the χ phase) and βMn. This is also the case for different metastable structures of Ta (
While the size ratio between the two elements may be quite flexible in certain phases (Laves phases), it is restricted to a very small range for other phases (σ). It seems to play a key role for the site preference, with the general rule being that the larger element prefers the sites with higher coordination numbers. Very often the more electropositive atom corresponds also to the atom with larger atomic radius, so it may be difficult to deconvoluate the electronic effect from the size effect. In the following, the larger radius element will be designated
Homogeneity ranges of various FrankKasper (FK) phases in the combined systems in which they crystallize represented as a function of the composition of the larger (
From what is known experimentally, the major structural defect responsible for the stoichiometry deviation from the ideal, ordered crystal structure is substitution. In the case where the deviation is very limited, this can be called an antisite defect since the major element at each given site is clearly identified (e.g., Laves phases). For other systems for which the nonstoichiometry is much larger, it may not be easy to identify the stoichiometric ordered composition, and some sites may completely switch as a function of composition from the occupation by one atom to the occupation by the other atom (see e.g., the case of the µ phase in
A common idea concerning the site occupancies of
Ideal site occupancies following Kasper rules for the σ phase. As a function of
Among the different experimental techniques that can be used to study the nonstoichiometry, we can cite the Mössbauer spectroscopy. It has proved to be useful in order to study the magnetic properties of the σ phase [
However, most interesting systems (like σ phase for example) are too complex for this technique to be used by itself to determine with certainty quantitative values for the site occupancies. However, it may be used in combination with crystallographic studies with success [
In the case of FK phases, the most suited technique is the Rietveld refinement of powder diffraction data. This method is in general very useful to determine site occupancies when atoms mix on different sites [
In the case of ternary compounds, in which three elements may mix on some sites, more sophisticated techniques have to be used since there are two site occupancies per site to be determined and single diffraction data can only yield one average scattering per site. Therefore, combined analysis of different diffraction data sets in which the contrast is different has to be used. Two techniques are in this case available: resonant diffraction [
A systematic study of the site occupancies for different FK phases has been conducted as a function of composition. The existing data has been reviewed in the following papers: µ [
The µ phase presents two sites of equal coordination numbers (3
On the other hand, the σ phase presents sites of distinct coordination (CN14 and CN15) that behave rather similarly as shown in
Experimental site occupancies for the σ phase in all the studied systems (redrawn from [
Finally, the χ phase, contrary to the other phases, obeys in a rather strict manner the Kasper rule with sequential
First principle calculations are generally limited to the calculation of perfectly ordered periodic structures which seems, at first, not to be very useful to study the nonstoichiometry and disorder related to atomic mixing. However, several solutions to this problem have been found which are summarized in the following.
One of the first attempts to compute thermodynamically properties of FK phases from first principle techniques has been made by Turchi
The first significant work done on the σ phase has been performed on the two systems CrFe [
In [
W site fraction in σ ReW, obtained by CWMCVM at 500K (
(
Prototype σ CrFe has been by far the most investigated system by many different firstprinciples calculations. 10 years after the first CWMCVM study [
Fe site fraction in σ CrFe, obtained by CWMCVM at 500K (
Increased computer performance has allowed researchers to investigate more complex problems. For the first time, the total energies of every ordered configuration of a ternary σ phase,
The aim of the Calphad modeling is to describe the Gibbs energies of each phase of a given system. This yields the possibility of combining these phases and systems in a database allowing to make predictions of the phase equilibria in a multicomponent system. In the Calphad approach, generally, phases are described not only in their stable ranges of stability, but also outside in metastable states. The most convenient way of describing the Gibbs energy of a nonstoichiometric ordered phase is to use the socalled compound energy formalism, also called sublattice model [
Let’s take the example of a phase of stoichiometric composition
One should be able, with this method, to describe both the composition range of the phase (in the preceding example, one may note that the phase is defined from pure
However, this approach requires the knowledge of the mechanism for nonstoichiometry (number of sites on which disorder is present and type of disorder). Let’s imagine that the nonstoichiometry on the
On the other hand, for crystal structures containing more than two crystallographic sites on which disorder occurs, the number of endmembers to be evaluated varies exponentially with the number of sublattices. This yields the necessity to simplify the description and reduce the number of sublattices by (
Note that a review on this subject is given in [
The case of Laves phase
On the other hand, phases like the δ phase are so complex (14 sites) that they have to be simplified. A common model for this phase is
Between these extreme examples, phases with a moderately high number of crystal sites (four to five) may be discussed.
The µ phase has been commonly described by grouping the two sites of CN12 in a single sublattice [
Over the years, many different models have been used for the σ phase (for a review see [
For the χ phase, since the observed site occupancies are well accounted for by the Kasper rules, the issued proposal was (
A summary of these recommendations can be found in
Proposed Calphad sublattice models for selected FK phases issued from crystallographic analysis. Bold characters indicate the major element, if any.
Phase  Site coordination and multiplicity per cell  Ideal composition(s)  Sublattice model 

(CN14)_{6}(CN12)_{2} 

( 

(CN16)_{8}(CN12)_{16} 

( 

(CN16)_{4}(CN12)_{2}(CN12)_{6} 

( 

( 

µ  (CN16)_{6}(CN_{15})_{6}(CN14)_{6}(CN12)_{3}(CN12)_{18}  ( 

σ  (CN15)_{4}(CN14)_{8}(CN14)_{8}(CN12)_{8}(CN12)_{2} 

( 
χ  (CN16)_{2}(CN16)_{8}(CN13)_{24}(CN12)_{24}  ( 
Due to the development of both thermodynamic calculation softwares and DFT calculations, models can now be defined using one sublattice per site. Of course this necessitates the use of the DFT calculated enthalpies of formation in order to be able to assign values for the stability of each endmember. Examples of such calculations for the σ and the µ phase, each with 5 sublattices have been presented in
We have shown that nonstoichiometry in FK phases could be described and analyzed from experimental measurements. We have shown that the simple application of CN rules for the prediction of site occupancy is misleading and experimental measurement should support any proposed scheme. Nowadays, with the increase of computation facilities, these complex phases can not only be calculated by DFT but also their nonstoichiometry predicted. Finally, Calphad models help in describing these phases in comparison with the other phases in the same systems, allowing the description of the complete phase diagram. Multicomponent databases may then be built that succeed in predicting the precipitation of these phases in complex alloys. Modern approaches such as those described in the present paper may be used in the future in new thermodynamic databases for increased accuracy in their predictions.
The contribution of Pierre Joubert to the proof reading of the manuscript is greatly appreciated. Financial support from the Agence Nationale de la Recherche (Project Armide 2010 BLAN 912 01) is also acknowledged.
The authors declare no conflict of interest.