Dissolving the Periodic Table in Cubic Zirconia: Data Mining to


Dissolving the Periodic Table in Cubic Zirconia: Data Mining to...

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Dissolving the Periodic Table in Cubic Zirconia: Data Mining to Discover Chemical Trends Bryce Meredig† and C. Wolverton*,† †

Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, United States S Supporting Information *

ABSTRACT: Doped zirconias comprise a chemically diverse, technologically important class of materials used in catalysis, energy generation, and other key applications. The thermodynamics of zirconia doping, though extremely important to tuning these materials’ properties, remains poorly understood. We address this issue by performing hundreds of very large-scale density functional theory defect calculations on doped cubic zirconia systems and elucidate the dilute-limit stability of essentially all interesting cations on the cubic zirconia lattice. Although this comprehensive thermodynamics database is useful in its own right, it raises the question: what forces mechanistically drive dopant stability in zirconia? A standard tactic to answering such questions is to identifygenerally by chemical intuitiona simple, easily measured, or predicted descriptor property, such as boiling point, bulk modulus, or density, that strongly correlates with a more complex target quantity (in this case, dopant stability). Thus, descriptors often provide important clues about the underlying chemistry of real-world systems. Here, we create an automated methodology, which we call clustering−ranking−modeling (CRM), for discovering robust chemical descriptors within large property databases and apply CRM to zirconia dopant stability. CRM, which is a general method and operates on both experimental and computational data, identifies electronic structure features of dopant oxides that strongly predict those oxides’ stability when dissolved in zirconia.



INTRODUCTION Doped zirconias are ubiquitous technological materials with a wide variety of applications, from automobile three-way catalysts1 to fuel cells2 to dental implants.3 The best-known doped zirconia is yttria-stabilized zirconia (YSZ),4 in which the addition of aliovalent Y3+ cations creates charge-compensating oxygen vacancies that serve to stabilize the cubic phase of zirconia relative to monoclinic and tetragonal polymorphs.5 Despite these zirconia-based materials’ great practical importance, some of their properties remain poorly understood. In particular, to the best of our knowledge, no complete survey of zirconia-doping thermodynamics exists. Understanding the degree to which various cation dopants incorporate in the zirconia lattice, and why, will shed light on the defect interactions that fundamentally govern doped zirconia behavior in technological applications. In the process of conducting such a study here, whose results will be quite useful in their own right, we further show that existing heuristic models fail to fully describe zirconia-doping thermodynamics, and we employ cutting-edge statistical and data mining methods to identify descriptors that help us understand the underlying physical mechanisms of dopant stability in zirconia. The use of chemical descriptors to predict materials’ behavior dates to the beginnings of modern solid-state chemistry and physics. Perhaps the most familiar descriptor-based materials map is the periodic table itself, wherein atomic number serves © 2014 American Chemical Society

as a descriptor that organizes elemental properties. Other classic examples include Pauling’s rules for ionic crystal structures,6 Hume−Rothery’s metal solubility criteria,7 and Pettifor’s alloy structure maps.8 Today, descriptors continue to be invaluable tools for building predictive models of materials,9,10 and for materials discovery.11,12 Although the practice of folding often-complex underlying chemistry into a small number of predictive higher-level properties is both wellestablished and successful, how do we identify descriptors in the first place? Historically, chemical intuition has been the approach of choice: one develops a list of intuitively “important” physical properties for a given application and manually searches for relationships among these select variables. However, as chemical data sets and databases (both computational and experimental) grow dramatically in size and diversity and as chemistries of interest become more complex, intuition alone is becoming a much less reliable means of identifying descriptors. Indeed, in their recent review, Curtarolo et al. emphasize both the difficulty and importance of identifying effective descriptors in enabling facile computational materials discovery.13 Received: April 3, 2013 Revised: February 23, 2014 Published: March 4, 2014 1985

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CaO, Y2O3, CeO2, etc.), calculate the resulting 70 oxides within DFT, and also insert the corresponding cations into a zirconia supercell. We then determine solution energies by comparing the stability of the dilutely doped supercell to the pure zirconia and dopant oxides, according to Equation S1 in the Supporting Information. We note that this undertaking involved relaxing within DFT several hundred large doped zirconia supercells, in order to account for all possible dopant−vacancy arrangements within those cells. We make additional comments on these calculations, defect geometries, and our reference states in the Supporting Information. The results of our supercell calculations are given in Table 1.

To explain zirconia-doping thermodynamics, and to generally aid in the discovery of materials descriptors, we introduce the CRM framework. The CRM method consists of one prerequisite and three process steps, which we illustrate in the context of doped zirconia: 0. Construction of a training database, which includes all candidate descriptors and the property to be predicted (here, solution energy). 1. Clustering materials based on all candidate descriptors. 2. Ranking the performance of all candidate descriptors on each cluster using a simple regression (in our case, quadratic). 3. Modeling the behavior of materials within each cluster using the best-performing descriptors identified in the previous step.

Table 1. First-Principles Dilute-Limit Solution Energies on the Cubic Zirconia Cation Sublattice for All Dopants Considered in This Work, Determined According to Equation S1 (Supporting Information)a



PREREQUISITE: CONSTRUCTING A DATABASE OF TARGET PROPERTY AND DESCRIPTORS CRM requires a training database consisting of (a) the value of the target property to be predicted by CRM-identified descriptors and (b) the value of candidate descriptor properties for the same collection of materials. For our zirconia study, we obtain both of these requirements from density functional theory (DFT) calculations; see the Supporting Information for calculation details. In this case, the target property is a particular dopant’s solution energy in cubic zirconia, and the candidate descriptors are properties of the dopant binary oxides that we derive from DFT calculations. Examples of these possible descriptors include cation d-band center, oxygen atomic volume, oxygen Bader charge, band gap, and so forth. A full list is provided in the Supporting Information. An effective descriptor maps a feature of the pure dopant binary oxides to those oxides’ stability on the cubic zirconia lattice; in other words, it should indicate which property of the dopant oxides most strongly governs those oxides’ thermodynamic interactions with cubic zirconia. To construct the training database for our zirconia case study, we perform high-throughput first-principles density functional theory (DFT) calculations on dopant binary oxides and large cells of dilutely doped cubic zirconia. We scan the entire periodic table (Figure 1) for metals with stable, wellcharacterized divalent, trivalent, or tetravalent oxides (e.g.,

tetravalent (+4)

dilute solution energy (eV/ dopant)

Si* Ti* V Cr Mn Ge* Zr Nb Mo Tc Ru Sn Hf Ta W Re Ir Pb Ce* Pr Tb Th Pa U Np Pu

5.13 1.22 1.77 1.49 1.96 2.59 0.00 0.78 1.11 1.77 1.50 0.93 0.29 0.96 1.48 2.21 2.85 0.74 0.38 0.17 −0.38 1.03 0.55 0.86 0.44 0.67

trivalent (+3)

dilute solution energy (eV/ dopant)

Al Sc* Ti V Cr Mn Fe Ga Y* In La* Ce* Pr* Nd* Pm Sm* Eu* Gd* Tb Dy Ho* Er* Tm* Yb* Bi

0.78 0.55 1.18 0.68 1.16 0.95 0.62 0.49 0.36 0.88 0.71 0.35 0.30 0.30 0.34 0.25 −0.04 0.30 0.31 0.32 0.34 0.36 0.37 −0.12 0.56

divalent (+2)

dilute solution energy (eV/ dopant)

Be Mg* Ca* Ti V Cr Mn Fe* Co Ni Cu Zn Sr* Pd Cd Sn Ba* Pt Hg Pb*

1.85 1.13 0.81 1.12 0.84 1.23 1.04 0.39 0.34 1.37 1.77 1.40 0.83 2.14 1.35 1.12 1.26 2.00 1.88 1.46

a

Cations indicated by asterisks form at least one known compound with zirconia and generally (though not always; the observed compound may have a structure very different from fluorite) exhibit more favorable solution energies.

Table 1 represents by far the largest survey of doped zirconia thermodynamics to date. In general, our data agree well with earlier computational and experimental investigations of doped zirconia. Comparing our results on specific dopants with previous work,14−17 we find, for example, that the stabilizer Ca is the most stable alkaline earth dopant on the ZrO2 lattice (i.e, it has the lowest dilute solution energy), that the common stabilizers Sc and (as expected) Y are quite stable compared to other dopants, that nearly all rare earths are among the most stable dopants in our data set, and that fluorite-based oxides (Y,

Figure 1. Dopant elements whose binary oxides were included in our study. For each of these elements, we used density functional theory to calculate a variety of properties of their oxides and also their dilutelimit solution energy in cubic zirconia. Several of the highlighted elements have more than one available charge state (e.g., +2, +3, and +4 Ti), leading to a total of 70 distinct dopants. 1986

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Figure 2. Application of two well-known descriptors to our comprehensive doped zirconia data. (A) Relationship between dopant Shannon ionic radius20 and solution energy. (B) Relationship between dopant oxide electronic d-band center and dopant solution energy. Although both descriptors do provide discernible trends with R2 > 0.4, neither descriptor alone gives a completely satisfactory picture.

Hf, Ce, and several other rare earths) tend to have relatively high stability in zirconia. With the dopant solution energies in Table 1 (our target property), we proceed to identifying which dopant oxide characteristics (i.e., which descriptors) best capture the chemistry of zirconia doping. Some earlier studies have also attempted to discover dopant stability descriptors for zirconia; we will now show that existing descriptors perform poorly on our large dopant database. Bogicevic et al.15 and Khan et al.14 explained dopant stability in zirconia with a simple ionic size argument. In particular, Bogicevic et al. showed for a particular set of closed-shell cations that dopants whose ionic radii were slightly larger than that of the host zirconium were most stable. Their result matches the empirical argument that oversized dopants more readily establish a preferred local oxygen-deficient 7-fold coordination for zirconium.18 With this intuitively reasonable view of zirconia in mind, we tested the ionic size hypothesis with our much more extensive data set of 70 dopants. Further, we applied another well-known descriptor, d-band theory best known for its descriptions of catalyst performance19to the zirconia dopant stability question. This model associates the properties of materials with the energy of those materials’ d electron band centers of mass, on the basis of the rationale that d electrons are the most likely to participate in chemistry in transition-metal compounds.9 The performance of these descriptors for our data is shown in Figure 2. Figure 2 shows that, though the ionic size and d-band descriptors might have limited utility in our case, they are far from the complete picture. The trend in Figure 2a matches the findings of Bogicevic et al.:15 the most stable dopantsthose with lower solution energiesare those whose ionic radii are slightly greater than that of Zr4+ (i.e., in the range 0.9−1.0 Å). In Figure 2b, we find that, generally, oxides with more positive d-band centers (more d states above the Fermi level) are more stable in cubic zirconia; in other words, less-occupied d-bands are conducive to stability. Nonetheless, we see that, with a large data set of 70 dopants, neither descriptor alone offers a satisfying quantitative explanation for dopant stability in zirconia. To address this shortcoming, and thereby illustrate our method, we now apply CRM to identify a much-improved set of descriptors.

In the zirconia case, the dopant oxides include three cation charge states, both metals and insulators, many different crystal structures and coordination environments, magnetic and nonmagnetic elements, and so forth. It seems reasonable to suspect that no single descriptor will predict the behavior of all these dopant oxides. In data mining, the practice of classifying diverse data points into like groups is known as clustering, and this is the first step in CRM. To simplify the problem of finding descriptors, we apply the X-means clustering method21 as implemented in the Weka data mining package22 to our list of dopant oxides. This approach subdivides a data set into an algorithmically determined number of smaller clusters (in our case, four) based on minimizing the distance between the vectors representing members of each cluster and the centroid of that cluster. The vector for a given dopant oxide is simply the values of all candidate descriptors for that oxide; in our case, as noted before, we use numerous properties derived from DFT calculations on the oxides (band centers, Bader analysis outputs, magnetic moments, etc.) as well as an empirical quantity of interest, the dopant cation’s Shannon radius;23 we give a full list and description of these descriptor candidates in the Supporting Information. Dopants in a particular cluster have chemically similar parent binary oxides, in the same sense that elements in a column on the periodic table are related. We show the results of our clustering procedure on a periodic table in Figure 3. Figure 3 depicts the four dopant oxide clusters identified automatically by the clustering stage of CRM. These clusters match our intuition in satisfying ways: dopants that we might expect to be chemically similar appear in the same clusters. Cluster 1 consists of mostly closed-shell dopants from the s and p blocks of the table. Cluster 2 includes the largely chemically inert lanthanide series and closed-shell, large-ionic radius oxides from columns II−IV; it encompasses many of the most commonly used zirconia dopants (Ca, Sc, Y, and Ce). Cluster 3 consists mostly of early transition metals, although one moredistant element (Pa) also falls into Cluster 3. Finally, Cluster 4 contains all of the divalent transition metals, plus Fe3+. That these clusters are physically sensible gives us confidence in relying on the clusters for further analysis.

CRM STEP 1: CLUSTERING MATERIALS We first turn our attention to the issue of wide chemical diversity in the training database and to the possibility that no single descriptor property will capture trends across all the data.

CRM STEP 2: RANKING DESCRIPTORS Though we have successfully grouped the dopants into chemically similar clusters that we expect to share common descriptors, we must now identify the appropriate descriptor for





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automatically identified descriptor in Figure 5, which represents the output of CRM. Figure 5 demonstrates that our CRM descriptor search approach has, indeed, meaningfully subdivided and explained our dopant stability data set. Figure 5a, c, and d depicts cases in which we are able to explain a very large fraction of the variance in a dopant cluster’s stability in zirconia with a single-descriptor quadratic relationship, a remarkable outcome for such a chemically diverse set of dopants spanning the entire periodic table. Figure 5b gives a lower R2 score, butas noted above the heavy elements cluster is uniquely homogeneous compared to the other three clusters. We highlight the sharp contrast between Figure 2, in which two intuitively appealing descriptors failed to successfully model our data, and Figure 5, in which CRM-derived descriptors provide a quantitative picture of zirconia-doping thermodynamics.



CRM PREDICTIVE PERFORMANCE Although the descriptive power of CRMits ability to identify trends in a given set of data pointsis of course important, we are also very interested in its predictive strength. In other words, how well can we predict characteristics of a new dopant using the CRM framework? We address this matter with two tests. First, we study the “persistence” of our three unique descriptors (oxygen p center, oxygen charge, and oxygen volume, which was separately selected for two clusters) as we randomly remove dopants from our data set and rerun the descriptor regressions. If the optimal descriptors were very unstable based on which dopants happen to exist in the fitting set, we would question the robustness of CRM. Second, we calculate CRM’s leave-one-out cross-validation score to obtain a quantitative estimate of CRM’s predictive accuracy for new dopants. We turn first to the question of descriptor robustness. We perform a series of trials in which we withhold a random set of dopants from our data set, cluster the remaining dopants, rank all descriptors for the remaining dopants as in Figure 4, and finally determine the best-observed ranking for each of our three originally selected descriptors (we re-emphasize the fact that we examine only three descriptors, not four, because two of our clusters share oxygen volume as their optimal descriptor, as shown in Figure 5). We perform 1000 such random trials at each level of data removal and present the averaged results in Figure 6. Figure 6 indicates that, indeed, the descriptors that CRM selects when provided with the entire data set are very similar to the descriptors it selects when given fewer data. Only oxygen p center shows any notable decline in ranking as we remove a greater fraction of the fitting data, but as discussed previously, this descriptor is associated with the least-distinctive (and, hence, least in need of modeling) cluster among our original four clusters. The minimal RMSE variation in Figure 4b illustrates this feature of the heavy elements cluster. Our second predictive test, leave-one-out cross-validation (LOOCV), involves removing each dopant from the data set in turn and then using CRM constructed on the remaining dopants to quantitatively predict the withheld dopant’s solution energy. The LOOCV score is given by

Figure 3. Results of our X-means clustering analysis, which we applied to data on the 70 dopant binary oxides in this study. The color of each element on the periodic table represents the assigned cluster for one or more of that element’s binary oxides; that is, Sc refers to Sc2O3, whereas Fe includes both FeO and Fe2O3. The clustering algorithm identifies chemically meaningful relationships that appear on the periodic table, without requiring any human input.

each cluster. To do so, we regress dopant solution energy against all candidate descriptors within each cluster and select the descriptor that gives the lowest root mean square error (RMSE) for that cluster. We elect to use a quadratic fit, as we expect that some descriptor-solution energy relationships could be nonlinear. The results of our descriptor-ranking procedure are illustrated in Figure 4. We use the rankings shown in Figure 4 to select a bestperforming (lowest-RMSE) descriptor for each cluster. Our selections, from the top of each cluster’s list, are (a) s/p block, oxygen Bader volume, (b) heavy elements, oxygen p center, (c) early transition metals, oxygen Bader charge, and (d) divalent transition metals, oxygen Bader volume. Interestingly, we find that oxygen ion properties emerge as top descriptors across all four clusters. We also note that, generally, properties associated with ionic charge, ionic size, and d electrons rank at or near the top for all clusters, suggesting that electrostatics, strain, and covalent bonding all play key roles in zirconia-doping chemistry. Finally, we observe that, although three of the four clusters exhibit robust differentiation among descriptors (i.e., large variations in RMSE across the descriptors) this differentiation is weaker for the heavy elements cluster (Figure 4c). That cluster is, however, by far the most homogeneous of the four, with a solution energy standard deviation of just 0.35 eV compared to 0.57, 0.59, and 1.29 eV for the other three clusters. Thus, that cluster exhibits much less variation than the others and is less in need of a descriptor in the first place.



CRM STEP 3: MODELING WITH OPTIMAL DESCRIPTORS The final step in our automated descriptor search is to use the predictive properties we identified through descriptor ranking (Figure 4) to analyze the dopant stability trends within the four clusters from Figure 3. We plot dopant dilute-limit solution energies for all four clusters as a function of each cluster’s

LOOCV =

1 n

dopant tot



(ΔEpred, n − ΔEactual, n)

n=1

where n is the total number of dopants in our data set, ΔEpred,n is the predicted solution energy for the nth dopant based on a 1988

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Figure 4. Results of our descriptor-ranking procedure within each cluster. The 19 candidate descriptors are arranged in order of increasing RMSE for quadratic fits to all possible solution energy vs descriptor plots. We select the lowest RMSE descriptor for each cluster: (a) s/p block, oxygen Bader volume, RMSE = 0.41 eV; (b) heavy elements, oxygen p center, RMSE = 0.26 eV; (c) early transition metals, oxygen Bader charge, RMSE = 0.33 eV; (d) divalent transition metals, oxygen Bader volume, RMSE = 0.27 eV.

CRM model trained on the other n − 1 dopants, and ΔEactual,n is the true solution energy for the withheld dopant (Supporting Information eq S1). The LOOCV score is then simply the average error of these n models computed across the entire data set. This LOOCV approach simultaneously challenges the robustness of the clustering procedure itself and the regressions constructed within each cluster. If CRM either assigns an incorrect cluster to a new dopant, or overfits its regressions to spurious trends, its resulting predictions will be badly in error. We find that the LOOCV score for CRM in this study is 0.41 eV; such accuracy is certainly sufficient for determining whether

a new dopant is relatively stable or unstable on the zirconia lattice and speaks to semiquantitative predictive capability for our CRM model for zirconia.



CHEMICAL INSIGHTS The ultimate goal of applying CRM to zirconia-doping thermodynamics was to gain insight into a chemically complex problem that to some extent defies straightforward intuition. Interestingly, Figure 5 shows that dopant size and dopant charge, long known to play a large role in thermodynamics and kinetics of stabilized zirconias,14,15,17,18,24 are indeed strong predictors of dopant stability. However, our CRM analysis 1989

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reasonable given the remarkably covalent nature of the Zr O bond.25



CONCLUSION



ASSOCIATED CONTENT

CRM has revealed for a chemically diverse data set a robust set of descriptors that achieve two important goals: the descriptors (a) predictively map dopant binary oxide properties (i.e., outputs of small, fast calculations) to dopant solution energetics (outputs of much larger, slower calculations that we generally prefer to avoid) and (b) offer nonobvious physical insight into the system we chose to study, which we could not have obtained without CRM. In particular, we suggest that the covalent nature of bonding in zirconia leads to changes in the material’s electronic structure upon doping, which is captured by Bader values for dopant size and ionic charge much better than by empirical hard-sphere values for these properties. Although we selected zirconia doping as an example application for CRM, each step in the approach is entirely general, and the framework enables fully automatic descriptor searching for chemical and materials design tasks in cases where human intuition is insufficient to detect exploitable patterns in properties data. We do note, however, that some user input is required to generate a list of candidate descriptors through which CRM can search. We further note that our CRM approach requires a set of training data in order to identify descriptors and, therefore, cannot make a priori predictions for systems for which no chemical data is supplied (i.e., the results here do not allow us to make generalizations about descriptors for solution energetics in other oxides besides ZrO2; one would need to run CRM in each system). We surmise that, as researchers across a variety of fields discover additional materials descriptors,26 a standard library of such descriptors will emerge that can serve as typical input for CRM.

Figure 5. Results of our automated descriptor search, based on data mining and statistical analysis, for doped zirconia thermodynamics. Our selected descriptor for each cluster’s solution energy trend is given as the x axis for each plot.

S Supporting Information *

Additional details on the calculation methodology. This material is available free of charge via the Internet at http:// pubs.acs.org.



Figure 6. Robustness of the three unique descriptors from Figure 5 against random variations in CRM fitting data. As we remove more dopants from our data set, oxygen charge and oxygen atomic volume remain extremely highly ranked, whereas oxygen p centerassociated with the more homogeneous heavy elements clusterdecays in importance more rapidly.

AUTHOR INFORMATION

Corresponding Author

*C. Wolverton. E-mail: [email protected] Notes

The authors declare no competing financial interest.



ACKNOWLEDGMENTS B.W.M. was partly supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program. The authors also acknowledge financial support from the U. S. Department of Energy under grant DE-FG02-07ER46433 and a Laboratory Directed Research and Development program at Sandia National Laboratories, in the form of a Grand Challenge project entitled Reimagining Liquid Transportation Fuels: Sunshine to Petrol. The calculations in this work were performed on Northwestern University’s Quest high-performance computing system.

provides a crucial twist: the electronic structure-derived Bader ionic sizes and charges were, overall, much better descriptors than the empirical Shannon radius and empirical formal charge (i.e., +4, +3, or +2) properties, as illustrated in Figure 4. Indeed, seven of the eight top-two descriptors from our four clusters are quantum mechanical, not empirical, in nature. This result indicates that zirconia dopant stability is more nuanced than has been assumed in the past and cannot be cast as a simple competition between hard-sphere strain effects and electrostatic preferences.15 Instead, the full electronic structures of the dopant oxides are in fact crucial to explaining how those oxides behave when dissolved on the zirconia lattice, which is 1990

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