International Journal of Minerals, Metallurgy and Materials Volume 18, Number 3, June 2011, Page 303 DOI: 10.1007/s12613-011-0438-5
Materials databases for the computational materials scientist Marcel H.F. Sluiter1), Darko Simonovic1,2), and Emre S. Tasci1,3) 1) Materials Science and Engineering 3ME, Delft University of Technology, Delft 2628CD, the Netherlands 2) Materials innovation institute M2i, Delft 2628 CD, the Netherlands 3) Física de la Materia Condensada, Universidad del Pais Vasco, Bilbao 48080, Spain (Received: 10 May 2010; revised: 25 June 2010; accepted: 29 June 2010)
Abstract: Until recently, many computational materials scientists have shown little interest in materials databases. This is now changing because the amount of computational data is rapidly increasing and the potential for data mining provides unique opportunities for discovery and optimization. Here, a few examples of such opportunities are discussed relating to structural analysis and classification, discovery of correlations between materials properties, and discovery of unsuspected compounds. Keywords: database systems; materials; data mining; ab initio prediction; structural analysis
1. Introduction Historically, materials databases have been compilations of experimental data. Until the eighties, some of the main efforts in this area took place in the USA, and later it shifted to other countries [1]. The high cost of constructing and maintaining databases and the difficulty of attracting scientific staff, coupled with the difficulty of generating sustained financial support, generally have led many databases to languish as time goes on. For this reason a new development of generating materials databases using ab initio electronic structure calculations has become of special interest. Computer resources are not only getting cheaper, but also idle frequently. Such unused computer time can be used to compute materials properties for large numbers of materials systems that are not of immediate interest. Because the computed data are all of a similar type, they are easily classified and inserted in a database without human intervention. Special scripts can be used to filter out and set apart suspect data, which is screened later by a human expert. Over time however, as more and more material systems are considered, it becomes possible to data mine the compiled data and search for trends and correlations. This too, can to a large extent be done without human intervention through techCorresponding author: Marcel H.F. Sluiter
niques such as principal component analysis [2]. Aside from reducing the amount of human effort involved in constructing databases, the availability of much more complete and systematic databases obtained without a well-defined (computational) protocol offers new opportunities. There are many systems for which experimentation is hazardous or difficult: when the elements involved are short-lived radioactive elements, highly reactive and refractive (5th row transition metals), or poisonous (Be). When very high pressures or temperatures are to be considered, computation is vastly easier than actual experimentation. Moreover, many compounds or mixtures are unstable and cannot be produced experimentally at all. For these reasons, computational databases potentially can be much more complete than experimental databases. Experimental data is not going out of fashion, however. A computational database needs verification by a critical comparison with experimental data, and only after this critical step is passed can conclusions be drawn with any confidence. As will be shown below, such more complete computational databases allow us to see correlations that have not been apparent from the more scattered and scarce experimental data [3]. In truth, ab initio computations for many properties of practical interest are not yet feasible in an automated fashion
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without human intervention. The properties that currently are amenable for automated computation (for most metallic materials at least) are crystal structure parameters and associated enthalpies of formation, elastic tensors, phonon spectra, energetics of simple point defects, impurity diffusivities, and other derived properties. The first ab initio compilations of compound/structure/formation enthalpy data were assembled on the Internet as early as the mid 1990s [4-6].
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small ordered precipitates from supersaturated solid solutions by heat treatments. Well-known, particularly illustrative examples occur in the Au-Cu phase diagram where the AuCu3 and the AuCu prototype structures are rather simply related to the high temperature fcc Au-Cu solid solution (Fig. 1) and in the Cd-Mg phase diagram where the CdMg3 and the CdMg prototype structures are related to the high temperature hcp Cd-Mg solid solution.
A special case concerns the enthalpy differences between allotropes of the pure elements. These enthalpies are of critical importance for the computation of phase diagrams of alloys, and therefore these numbers have been published in the regular literature also [7-9]. The Ceder group at MIT generated a very extensive database computing the crystal structure parameters and formation enthalpies of more than a hundred prototypical (prototype) structures for about a hundred binary alloys [10]. This extensive database made it possible to predict the likely stability of structures in alloy systems that are not included in the database by using correlations in the data. As an example of such a correlation, of the 87 A-B alloys that have a stable A3B compound with the Fe3C prototype structure, 52 alloys also have a stable AB2 compound with the MgCu2 prototype structure. This work addresses an important challenge in computational materials science: to predict compounds and their structures in mixtures of which only the elemental constituents are known [11-13]. Below, three applications of databases with relevance to ab initio computational materials science will be given: finding superstructures related to bcc, fcc, and hcp solid solution phases using the Pauling file database of experimentally determined structures [14] (see also the contribution by Pierre Villars in these proceedings); finding compounds and their crystal structures by combining ab initio computed structural features of the liquid phase [15-16] with the Pauling file database [14]; and discovery of an empirical rule for impurity diffusion through large scale ab initio computations [3].
2. Determination of superstructures Order-disorder phenomena in substitutional alloys are important for understanding phase stability theoretically and as well for age-hardening and other heat treatments of alloys practically. In many alloys ordered superstructures transform to disordered solid solutions as the temperature increases. In alloys such as Al-Cu, Al-Li, and Ni-Al, strengthening can be achieved through the formation of
Fig. 1. AuCu3 (left) and AuCu (right) superstructures. Yellow (blue) colored spheres indicate atomic positions preferably occupied by Au (Cu) atoms. Both ordered structures are based on the fcc lattice, although AuCu3 is simple cubic, and AuCu is tetragonal.
In spite of the technological interest, it is remarkable that there is no standard way to determine whether some ordered structure is related to the structure of a disordered solid solution. Up till now typically a visual inspection of the ordered crystal structure has addressed this question. When there are distortions and when the ordered crystal structure is complex, it will be apparent that it is easy to miss a relationship with an underlying solid solution crystal structure. Here, a simple method is presented, which relies heavily on the database work related to the Pauling file [14]. When a structure orders, a particular site in the crystal takes a higher probability of being occupied by one particular species, whereas in the disordered solid solution all sites have the same probability of being occupied by a particular species. When this ordering takes place, the topology of the sites and their neighbors does not change, certainly not when the preference for a particular atomic species is till weak. It follows then that the local atomic structure or environment around any given atomic position should be the same in the ordered and the disordered structures. The local atomic environment can be classified according to various types, the so-called atomic environment types (AETs) [17] depending on number of the nearest neighbors and the relative arrangement of those neighbors. AETs for all structures observed so far are tabulated in the Pauling file, so that the
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first filter to search for possible superstructures consists of a test that ensures that all inequivalent crystallographic sites in a superstructure have the same AET as that of the parent disordered structure. In practice there are only 3 disordered structures to be considered: fcc, hcp, and bcc. In Fig. 2 the AET relevant for fcc and its superstructures is shown. When the filter described above is applied [18], all the commonly known superstructures are found, and in addition tens of other, not previously recognized superstructures emerge. Examples are Al8Mo3 and Cu10Sb3 (Fig. 3), which is shown to be fcc and hcp superstructures, respectively. In total more than a hundred superstructures are identified.
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Fig. 3. fcc based Al8Mo3 prototype structure (left) with an fcc cube superimposed in green; and hcp based Cu10Sb3 prototype structure (right) with the top and bottom triangles of the hcp AET.
and evolutionary or genetic algorithms coupled with ab initio calculations, such as that proposed by Oganov and Glass [12] and the Zunger group [13].
3. Prediction of compounds and their structures
Here, we will take an entirely different route, however [15-16]. As in many natural processes we start with the liquid state. For certain alloy systems, such as those form Zintl phases, there is a well-documented relationship between local atomic structure in the solid crystalline state and in the liquid state. Therefore, one might suspect that atomic coordination in the liquid phase and in the crystal phase shares certain similarities. Thus, we use the known coordination number distribution in the liquid phase as a criterion for selecting prototype crystal structures as potential compounds in alloy systems with no known compounds. We apply these ideas to Au-Si and Au-Ge alloys, for which no compounds have been reported (Fig. 4).
Accurately predicting the stable compounds and the corresponding crystal structures remains a formidable scientific challenge [11]. Among the many approaches to this problem, we mention the data mining structure predictor combining data-mining with ab initio calculations by Fischer et al. [10],
Using the coordination number distributions computed via ab initio calculations [15-16] shown in Figs. 5 and 6, we set a coordination number range for the constituent elements to act as a filter among the over 2634 reported crystal structure prototypes included in the Pauling file binaries (PFB)
Fig. 2. Atomic environment types (AETs) pertaining to the fcc (left) and the hcp (right) crystal structures. For both fcc and hcp AETs the central atom is surrounded by 12 other atoms. The surrounding atoms form 6 squares and 8 triangles. AETs for fcc and hcp nevertheless clearly differ, in hcp squares share edges, in fcc they do not; also top and bottom triangles in hcp overlap, in fcc they do not.
Fig. 4. Au-Si (a) and Au-Ge (b) phase diagrams.
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database [14]. To further narrow down the list of structures, the requirement that the composition of the structure prototype is close to that of the Au-Si or Au-Ge eutectic is imposed also.
Fig. 5. Coordination number distribution around Au (black) and Si (red) atoms in liquid Au-Si at the eutectic composition at 600 and 700 K as computed by ab initio molecular dynamics. The distribution is in arbitrary units.
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ble compounds in Au-Si and Au-Ge differ also. For Au-Si it has been shown that the Au4Si compound with the Pd4Se prototype structure is stable, whereas for Au-Ge it has been shown that the Au5Ge2 compound with the Pt5B2 prototype structure is stable shown in Fig. 7.
Fig. 7. Au4Si compound with the Pd4Se prototype structure (left) and Au5Ge2 compound with the Pt5B2 prototype structure (right) as found using the “local liquid structure data mining method” as described in the text. Yellow and blue spheres indicate Au and Ge atoms, respectively.
Notably, both compounds of Si and Ge have completely surrounded by Au atoms. This suggests a strong chemical attraction between Si-Au and Ge-Au. This was already surmised by Turnbull and Chen [20] in 1967 as an explanation for the deep eutectics found in these alloy systems. It should be emphasized that this ab initio compound prediction method relies on the availability of a structural database [14].
4. An ab initio correlation between diffusivity and partial molar volume
Fig. 6. Coordination number probability distribution around Au (black) and Ge (red) atoms in liquid Au-Ge at the eutectic composition at 700 K as computed by ab initio molecular dynamics.
These selection criteria produces a set of 27 (Au-Si) or a set of 11 (Au-Ge) prototype structures, which then are structurally refined by standard ab initio methods. After the prototype crystal structures are refined according to Au-Si or Au-Ge compositions, the formation enthalpies relative to the pure elements in their standard states (Au: fcc; Si and Ge: diamond cubic) can be evaluated. For both Au-Si and Au-Ge a single compound is found to be stable only. Surprisingly, although the Au-Si and Au-Ge phase diagrams look very similar (Fig. 4), the local atomic structure in the liquid is quite different (Figs. 5 and 6), so that the sta-
As a final example of roles that databases can play in materials science is the completely computed database: here we examine the diffusion parameters of impurities in fcc Al. By examining a good fraction of all the elements in the periodic table we get a very complete picture of which atomic characteristics determine diffusivity in Al. The ab initio calculations require significant computer resources, but are otherwise relatively straightforward, seeing Ref. [3] for details. For about 50 elements the activation energy Q for diffusion in fcc Al was calculated. The results are displayed in Fig. 8. It is evident that there is a reasonable agreement with many scattered results in the literature, but more importantly a strong trend emerges: transition metals, especially those in the middle of the transition metal series having much higher activation energies than other elements. Non-transitionmetals all remain within a rather narrow band around the ac-
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tivation energy for Al self diffusion. In fact, as Fig. 9 shown, non-transition-metals have activation energies that are rather nicely correlated with their partial molar volume in dilute Al alloys. The reason for this behavior could also be established [3], generally the Al-vacancy interchanges (jumps) are diffusion rate limiting. It is then no surprise that all non-transition element impurities have about the same diffusion frequency factor Do that is rather close to that of Al self diffusion (Do=0.035-1.7 cm2/s). The activation energy Q is lower when elements are larger relative to Al, the reason is that large impurities attract vacancies to reduce elastic strain, and another reason is that their migration energy is reduced because large impurities relax towards the vacancy so that the jump into the vacancy requires less structural distortion in the neighborhood. As Fig. 9 shown, the activation energy Q(X) for non-transition element X impurities in Al follows roughly a linear relation [3]
that typically much larger values are found than for Al self-diffusion.
Q(X)/Q(Al)=1+0.3[1–V(X)/V(Al)] ,
Fig. 9. Activation energy Q for impurity diffusion in fcc Al as a function of partial molar volume relative to molar volume of Al. Arrows pointing to light blue circles indicate effect of spin-polarization, light blue spheres represent spin-polarized V, Cr, and Mn. Fe, Co, and Ni (and other elements) are not spin-polarized as isolated impurities in fcc Al.
where Q(Al) is the activation energy for Al self-diffusion, V(X) the partial molar volume of element X in dilute Al-X alloys, and V(Al) the molar volume of Al.
It is evident that the large amount of data, even data for elements that are by themselves of little interest, allows us to see tendencies and rules that are not apparent from isolated data.
5. Conclusion
Fig. 8. Activation energy Q for impurity diffusion in fcc Al. Elements are ordered according to their Mendeleev number [21]. Dark and light blue squares indicate current ab initio results: light blue squares represent spin-polarized V, Cr, and Mn. Other elements, such as Fe, Co and Ni are found to be not spin-polarized as isolated impurities in fcc Al. Green triangles represent other ab initio calculations and red circles represent experimental measurements (see [3] for details).
Transition metals in Al do not follow these rules because their strong bonding with Al leads to repulsion of vacancies irrespective of their partial molar volume, and moreover makes the migration energy high because in the transition state several favorable Al-TM bonds are lost. The high bond strength does also increase the diffusion frequency factor, so
Some applications of materials databases for the computational materials scientist have been described such as the determination of superstructures of the common fcc, hcp, and bcc lattices of many solid solutions; the determination of candidate compounds and structures for alloy systems where such information is lacking or controversial. Moreover, it is predicted that databases with ab initio computed data will soon emerge. Such databases will allow data mining for trends and tendencies that now still remain hidden due to gaps in experimental data and scatter associated with variability in circumstances and methodology.
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