J Mol Model (2006) 12: 611–619DOI 10.1007/s00894-005-0068-9

ORIGINAL PAPER

Simone Sieg . Bernhard Stutz . Timm Schmidt .Fred Hamprecht . Wilhelm F. Maier

A QCAR-approach to materials modeling

Received: 25 April 2005 / Accepted: 8 September 2005 / Published online: 25 April 2006# Springer-Verlag 2006

Abstract Little is known about the relationship between thefunction and structure of materials. Materials (solids with afunction) are complex entities and a better knowledge of theparameters that contribute to function is desirable. Here, wepresent modeling approaches that correlate chemical compo-sition with function of heterogeneous catalysts. The completecomposition space of the mixed oxides of Ni–Cr–Mn and ofNi–Co–Mo–Mn (10% spacing) have been measured for theoxidation of propene to acroleine. The data have beencollected, visualized and modeled. Different mathematicalapproaches such as Support Vector Machines, multilevel B-splines approximation and Kriging have been applied tomodel this relationship. High-throughput screening data ofternary and quaternary composition spreads are approxi-mated to locate catalysts of high activity within the searchspace. For quaternary systems, slice plots offer a good toolfor visualization of the results. Using these approximationtechniques, the composition of the most active catalysts canbe predicted. The study documents that distinct relation-ships between chemical composition and catalytic functionexist and can be described by mathematical models.

Keywords Selective oxidation . Acroleine materialsmodeling . Kriging . Heterogeneous catalysts .Composition–activity relationship . QCAR

Introduction

In the late seventies, quantum mechanical and force-fieldcalculations had a low reputation and experimentalists did

not accept these calculations as serious and reliable data.During the last three decades through the work of PaulSchleyer and his colleagues around the globe this haschanged completely and calculations and molecular mod-eling now complement experiments as well as predict newmolecules. Today, nobody is surprised any longer if ex-periments confirm predictions from calculations or if cal-culations prove experiments to be wrong. Calculationshave become an accepted tool in molecular science. Oneof the important acronyms here is QSAR (quantitativestructure–activity relationships), where correlations be-tween molecular structure and activity are established andused for prediction and understanding. Molecular struc-tures are a well defined parameter space in organic andinorganic chemistry. In organometallic chemistry theparameter space becomes even larger, due to the manycentral atoms and their larger variation in coordinationand electronic states.

However, drugs and molecular structures representonly a small portion of the needs of our society for betteror new substances. Clearly, the greatest needs are inmaterials. Materials, solids with a function, are stronglyaffected by chemistry, but can mostly not be described bya structure. Instead, materials are often poorly definedcomposites containing amorphous, polymeric, and crystal-line components. Materials development in the past hasbeen a time-consuming process of sequential materialssynthesis, property measurements and analysis in the “oneat a time” approach. During the last 10 years high-throughput technologies have spread from pure drug-related applications in the pharmaceutical industry tomaterials laboratories in industry and academia. A largenumber of technological applications have been developedand materials can now be optimized rapidly and newmaterials are discovered [1–6].

There has been significant resistance to the use of HighThroughput Experimentation (HTE) in many materials ap-plications. Especially complex materials, such as catalysts,polymers or materials based on formulations have beenpredicted not to lend themselves to HTE. In the meantimeHTE has been applied successfully to an ever-increasing

S. Sieg . B. Stutz . T. Schmidt . W. F. Maier (*)Lehrstuhl für Technische Chemie,Universität des Saarlandes,66123 Saarbrücken, Germanye-mail: w.f.maier@mx.uni-saarland.de

F. HamprechtInterdisciplinary Center for Scientific Computing (IWR),University of Heidelberg,D-69120 Heidelberg, Germany

number of complex materials-development problems in-cluding polymers, electronic materials and catalysts.

Some of the promises not delivered yet by HTE arediscovery of knowledge and correlation of the function ofa solid with its structure. Here things become difficult.While most drugs are clearly identified by their molecularstructure, materials are complex entities that do not lendthemselves to the idea of structure–activity relationships.Heterogeneous catalysts are typical complex materials andwere chosen as models in this study. Today’s state oftheory does not allow us to predict the catalytic activity ofnew compositions. Catalyst optimization is therefore atedious process of materials preparation followed byindividual testing. Correlating activity with useful de-scriptors, such as composition, would open the complexfield of materials to theoretical modeling and predictions.In an interesting study, the groups of Schüth andMirodatos investigated 467 catalysts and their catalyticperformance for the oxidation of propene with oxygen.The catalysts were of different origin and prepared byvarious methods [7]. The authors searched for generaldescriptors that correlate with the catalytic activitydescribed by the dataset [8]. Because of the difficulty inassigning appropriate structure parameters to materials, acomplex strategy was developed. In the dataset, param-eters such as mean electron affinity, mean electroneg-ativity or mean molar mass of all elements in the catalystswere identified as descriptors, while chemical composi-tion does not correlate with the function of the materials[7]. This was to be expected, since catalytic activity is verysensitive to the preparation method of the catalyst. Thefunction of a catalyst is not defined by composition, but bythe exact synthesis protocol and the synthesis procedure,such as co-precipitation, impregnation, sol–gel-synthesisor powder synthesis and the precursors used. Changingthe sequence of additions and allowing different dryingtimes and calcination temperatures will result in a varietyof different materials of identical composition, butdifferent catalytic activity and selectivity. Such a datasetcannot be used in the search for the optimal compositionof a catalyst. When optimization of chemical compositionbecomes the issue of HTE, great care has to be takenthat the synthesis of the materials of varying compositionis carried out under identical synthetic conditions. Thisimplies synthesis recipes that tolerate a broad variation inchemical composition. We have found that this can beachieved very generally with specially optimized sol–gel-procedures. With such a set of materials, catalytic activity,measured under identical reaction conditions, shouldcorrelate with chemical composition and can be calledQCAR (quantitative composition–activity relationship)[9]. If such correlations can be found, modeling catalyticactivity becomes possible, as would be the prediction ofoptimal compositions. One essential aspect for such anapproach is that the dependence of the material’s function(catalytic activity or other desired properties) on chemicalcomposition is continuous and not erratic.

QCAR modeling applied to heterogeneous catalysts

The main goal of this work is the modeling of the functionof materials dependent on their composition. As modelmaterials a discrete set of catalysts has been prepared. Thecomplete composition space of the mixed oxides of Ni, Cr,Mo, Co and Mn has been synthesized (spacing steps of10%, 1,001 catalyst samples) based on a compositionallytolerant sol–gel-recipe developed for this purpose. Thematerials have been prepared with the help of a pipettingrobot in arrays of HPLC-vials through the control of thesoftware Plattenbau [10]. The thus prepared sols wereallowed to gel, then calcined at 300°C and manuallytransferred into the wells of a 207-well slate plate. Prior tothe measurement, the reactor was heated to 400°C insynthetic air for 3 h. Catalytic activity for the oxidation ofpropene to acroleine was monitored at 350°C (28.6 vol.%propene in synthetic air at ambient pressure) on a high-throughput reactor by sequential rapid analysis with amicro GC [11]. This dataset not only contains the activityand selectivity of the mixed oxides composed of fivemetals, but also all lower mixed oxides (pure oxides, binary,ternary and quaternary mixed oxides).

Parameter space

Mixed metal oxides are potential heterogeneous catalysts,composed of n different metals in various contents andoxidation states. Since the oxidation states, and thereforethe oxygen content of the materials, may vary with prep-aration conditions, pretreatment or during catalysis, it isnot specified. The total metal content (defined as 100%) isuniquely determined if the contents of n−1 metals arespecified. From a mathematical point of view, a catalystof n metals can be regarded as a point from an n-di-mensional space where each coordinate axis corresponds toone element. Figure 1 illustrates this for ternary catalysts.Ternary mixtures can be regarded as points lying on thediagonal of the unit cube in the Cartesian coordinate systemif the composition is interpreted as Cartesian coordinateswithin the range from 0 to 1 (0–100%).

Because of the redundancy of the nth dimension, thepoints can be transformed by an adequate coordinate trans-form into a (n−1)th dimensional representation. Figure 2shows the three-dimensional case.

Thus, a ternary composition spread can be illustrated intwo dimensions. Analogously, catalysts consisting of fourdifferent metals can be illustrated in three dimensions: theyform a tetrahedron as illustrated in Fig. 3. Clearly visibleis the rather steady catalytic activity. A clear region ofhigher activity is in the tetrahedron towards higher Co andCr contents. This also verifies that the catalytic activitychanges smoothly with composition and may provide agood basis for modeling.

From Figs. 1 and 2 it becomes clear what the mathe-matical model should achieve. Modeling the three dimen-

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sional case, the result is a triangular surface providingcatalytic activity for any composition within the parameterspace. In more than two dimensions, the result will be ahyperplane describing the data. The goal of interpolation orapproximation is to reconstruct an underlying function (e.g.surface) that may be evaluated at any desired set of po-sitions. This serves to propagate the information associatedwith the discrete data smoothly onto all positions in thedomain. The field of mathematical approximation tech-niques is very large and the approximation in itself can bedone in a variety of ways such that no unique solutionexists. Using different approaches, the approximating sur-faces differ in shape and smoothness according to themodel chosen. As many real-world applications, thisproblem ranks among the so-called ill-posed problems.The following figure stresses the spectrum of possibleinterpolations and gives an idea what ill-posed means inthis context.

Figure 4 illustrates in a simple way what the difficultiesof interpolation and approximation problems are. All fourcurves interpolate or approximate the data points butpossess different shapes. As with catalyst data the under-lying composition–activity relationship is an unknownfunction, the goodness of fit cannot be calculated directly(e.g. least-squares distance, mean squared error etc.) butone has to think about a criterion that handles the problemstructure.

For the interpolation of catalyst data, we assume thatsamples of the same composition should show nearly thesame catalytic behavior within the range of synthesis andmeasurement errors. Samples of similar compositions areassumed to cause a similar catalytic activity such that the

approximating curve should not exhibit jumps or sharpbends but is assumed to be of rather smooth shape.

Due to synthesis and measurement errors, the catalystdata cannot be considered as absolute values but as valuesplus a certain amount of noise and also outliers. Therefore,the use of approximation techniques instead of interpola-tion approaches to model the catalytic surfaces appears tobe more suitable.

Data

For our modeling purpose we chose a data set consisting of66 ternary heterogenous catalysts containing Cr, Mo and Niin 10%-wise variation without replication. The activityvalues for the oxidation of propene can be seen in Fig. 2.Among these catalysts the main activity occurred for mix-tures containing Mo and Ni in nearly equal contents andlittle Cr. The data point containing only Cr and Mo as wellas the point having the same Cr content but containingmore Ni show higher activity indicative of local maxima.Both points were suspected to be outliers due to mea-surement errors since there are no more active sampleswithin their direct neighborhoods. To prove this, 50 rep-licates including 2–3 repetitions were prepared and ana-lyzed together with more local samples (5 and 10%spacing) in the direct neighborhood. The data show nosignificant differences between the data points and theirdirect neighborhood, which identifies the two originalmeasurement points in the left middle and the middle of thebottom of Fig. 2b as outliers. In the following sections theresults of three different models are discussed and theirbehavior in coping with possible outliers is stressed.

Fig. 1 Ternary compositionspread interpreted as Cartesiancoordinates in three dimensions

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For our approach, we have chosen variations of threeparameters (ternary composition spread) for demonstra-tion, because these can be visualized easily by projectiononto a triangular plane (see above). Parameter sets of higherorder are clearly of interest and a tetrahedral presentation(quaternary composition spread) has also been shownabove. A more attractive visualization approach is pre-

sented below based on the modeling of the parameter spaceof quaternary mixed oxide catalysts.

Support vector machines (SVM)

The use of support vector machines was first introducedby V. Vapnik and his coworkers in 1992 [12]. Similar

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Fig. 2 a Ternary compositionspread illustrated in three di-mensions. The color corre-sponds to the measured activityat 350°C. b Ternary composi-tion spread projected onto twodimensions. The color corre-sponds to the measured activity

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approaches have been known in machine learning theorysince 1960, but it was not until 1992 that all of the relevantfeatures were put together to define the so-called “maximalmargin classifier”, the basic Support Vector Machine. SVMcan be used to solve classification problems but also re-gression problems and are often employed for feature ex-traction. For a detailed introduction to the theory of SVMand the mathematical background we refer to [13] and [14].

SVM have been applied to our catalytic data to performregression, i.e. by means of SVM regression. The activitiesof non-synthesized catalyst samples have been estimatedand checked whether calculated activities correlated with

measured activities (cf. [15]). The local fitting procedure ofSVM seems to be an advantage, since there is no reason toassume that catalytic activity across the complete param-eter space can be described by a simple function.

Figure 5 illustrates the application of support vector re-gression to the dataset introduced in Fig. 2. Among the 66heterogenous catalysts selected, Cr, Mo and Ni composi-tion varies from 0–100% with an increment of 10%.

Compared to Fig. 2 it can be seen that the most activeregions are well identified by the model. Variation of theresolution of the method can be used to smooth the activitysurface. In Fig. 5 the isolated points of activity have beenconfirmed as outliers (s.a.).

In Reference [15], another ternary data set has beentested and modeled using SVM. After the calculations,several new samples were synthesized and tested by IR-thermography, gas chromatography or mass spectroscopyfor certain reactions. Good agreement between predictedand experimental values was found, confirming the validityof the approach.

Data approximation with multilevel b-splines

As mentioned above, data approximation refers to theproblem of fitting a smooth surface through a scattered oruniform distribution of data. This subject is of practicalimportance in many areas of science and engineering suchas physics, geology, chemistry, meteorology, mining etc.All these fields require data approximation or interpolationtechniques to determine values at arbitrary positions. Thisis exactly what will be done with catalyst data and thereforesome techniques from this mathematical field are alsoapplied. Publications dedicated to the theory of scattereddata interpolation are for example [16, 17], which givegood surveys on this topic. For modeling the catalyst data,a multilevel B-spline approach is followed according to[18].

The basic idea of this approximation technique can beseen as follows.

For a three-dimensional application, consider a datasetP={(xc, yc, zc)} in 3D-space where (xc, yc) is a point in thexy-plane and zc is considered as the corresponding activityvalue. Let � ¼ x; yð Þj0 � x < m; 0 � y < nf g be a rect-angular domain in the xy-plane. To approximate the datapoints, an approximation function f is formulated as auniform bicubic B-spline function, which is defined by acontrol lattice Φ overlaid on domain Ω. Without loss ofgenerality, it is assumed that Φ is an mþ 3ð Þ � nþ 3ð Þlattice which spans the integer grid in Ω, cf. Fig. 6.

Let ϕij be the value of the ij-th control point on lattice Φ,located at (i,j) for i=−1, 0,…, m+1 and j=−1, 0,…, n+1.The approximation function is defined in terms of thesecontrol points by

f x; yð Þ ¼X3

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Fig. 3 Quaternary composition spread in three dimensions. Thecolor corresponds to the measured activity

Fig. 4 Possibilities to interpolate and approximate given data points

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where i ¼ xb c � 1 , j ¼ yb c � 1 , s ¼ x� xb c and t ¼ y�yb c . Bk and Bl are uniform cubic B-spline basis functions

defined as:

B0 tð Þ ¼ 1� tð Þ36

B1 tð Þ ¼ 3t3 � 6t2 þ 4ð Þ6

B2 tð Þ ¼ �3t3 þ 3t2 þ 3t � 1ð Þ6

B3 tð Þ ¼ t3

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where 0 � t < 1 . They serve to weight the contribution ofeach control point to f(x, y) based on its distance to (x, y).With this formulation, the problem of deriving function f isreduced to solving for the control points in Φ that bestapproximate the data points in the least square sense.According to Eq. 1 each point in P influences 4×4 neigh-boring control points in Φ and analogously, 4×4 controlpoints of the lattice Φ are needed to determine theapproximated value at any location in Ω.

It is clear that the resulting surface depends on thefineness of lattice Φ. As Φ becomes finer, the influence of a

data point is limited to smaller neighborhoods. This enablesP to be more closely approximated, although f will tend tocontain local peaks near the data points (danger of localbiases). Figure 7 illustrates this behavior.

Using this approach, there is a trade-off between theshape smoothness and accuracy of the approximationfunction that has to be handled. This is solved by applyinga multilevel B-spline approach with a hierarchy of controllattices to generate a sequence of functions fk whose sumapproaches the desired approximation function. In thesequence, a function from a coarse lattice provides a roughapproximation, which is further refined in accuracy byfunctions derived from fine lattices. When Φ is sufficientlyfine compared to the data distribution, f interpolates thedata.

The application of this approach to catalyst data isexemplified in Fig. 8, where the same ternary compositionspread of catalysts used for SVM above is approximated.Comparing the result to Fig. 2 (discrete data points) revealsthat the structure of the data set is well represented. Incontrast to Fig. 5 (SVM regression) this approach leads tomaxima that tend to be more local than those of SVM.

Simulated Gauss functions have been used to comparethe approximation accuracy of SVM to B-splines [15].The SVM regression yielded a smaller mean squarederror (MSE) in this test, although we cannot concludethat in general SVM regression is more suitable toapproximate catalyst data since the real function f is notknown as it was in [15]. The B-spline approximation ispresently limited by its application to a maximum ofthree dimensions. Challenges lie in the extension of thismethod to datasets that range in spaces with more thanthree dimensions The formalism for applications tospaces of higher order has not yet been developed.Another possibility could be the use of a polynomial-response surface with interaction terms. For our purposethis is not reasonable due to a lack of data. Furthermore,these techniques cause responses outside the sampledspace yielding +/− infinity instead of zero. Therefore, wehave chosen to try Kriging as another approach.

Fig. 7 B-spline approximation using different control latticeresolutions

Fig. 5 Estimated activities by support vector machines

Fig. 6 The configuration of control lattice Φ over Ω (cf. [18])

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Kriging

In this section the technique of Kriging is discussed tosolve the approximation problem introduced above. Theterm Kriging is known in the field of geostatistics orspatial statistics and describes modeling by random pro-cesses. Kriging was named after a South African miningengineer, D. Krige, who first popularized stochasticmethods for spatial predictions, especially for mining.Although originally developed for mining problems,Kriging can also be applied to many other fields andapplications with similar problem structure. A briefoverview of Kriging is given in this section. For acomprehensive review the reader is referred to theliterature [19–21].

A central problem in geostatistics is the reconstructionof a phenomenon over a domain on the basis of valuesobserved at a limited number of points. The main differ-ence to the other techniques described above is thatKriging starts from a statistical model rather than a modelof the approximation function. Formally, Kriging appearsto be similar to the prediction problem in time serieswhere the signal is to be predicted out of given values ofthe past, usually at regular time intervals, for some time inthe future. With time series, the signal is analyzed first bycomputing and modeling the spectrum, and then a pre-dictor is designed. Kriging follows a similar approach butin a spatial setting, where there is no general concept ofpast and future. Within this section the use of Kriging isapplied to model catalyst data to be able to predict theactivity of untested samples. In contrast to Reference [9],the results of the Kriging model are discussed in com-parison to the other approximation techniques describedabove.

The basic idea of Kriging can be described as follows:Using the Kriging approach, catalyst data are modeled as

a random process

Z xð Þ ¼ Z0 xð Þ þ e xð Þ

with x∈D⊂IRn and D is the experiment’s design region. Z0,some smoothly varying term, is assumed to depend only on

the position x and e is a realization of a random processwith zero mean and covariance C:

C eð Þ ¼ C xi; xj� �

:

Let S={x1,…,xN} be the set of all experimental samples,i.e. S contains all measured catalysts. Usually, the covari-ance is assumed to depend only on distance h of twosamples, i.e.,

C xi; xj� � ¼ C hij

� �

For modeling catalyst data, mean and covariance areunknown and have to be estimated out of the data. Theprediction of unmeasured samples is done by calculating abest linear unbiased estimator (BLUE) for each location.This denotation can be explained as follows:

– Best: estimator of smallest variance among all otherestimators.

– Linear: according to its construction, linear combina-tion of all observations.

– Unbiased: no bias (i.e. the expected value of therandom error is zero).

Although Kriging is generally considered “unbiased”,this holds only as long as a stochastic process is studied andthe correct covariance is used.

Another assumption is made concerning the measuredvalues: data points that are quite close together are assumedto be somehow “linked” to each other and this spatial in-fluence decreases with increasing distance of points until itebbs away. To analyze this spatial behavior, the covariancebetween two samples is to be considered. Instead of thecovariance, the sample variogram can also be considered tomodel the spatial behavior of the dataset. The samplevariogram is calculated by:

� hð Þ ¼ 1

2N hð ÞXN hð Þ

i¼1

Z xið Þ � Z xi þ hð Þð Þ2

where N(h) is the number of data points with distance h.Figure 9 illustrates a typical sample variogram and explainsits common descriptors range, sill and nugget effect.

The range can be considered as the distance where co-variance disappears and randomness dominates. This meansthat points that lie more than the range apart from eachother are considered to be uncorrelated. The nugget effectdescribes a discontinuity at the origin. For example, in golddeposits, gold commonly occurs as nuggets of pure metalthat might be much smaller than the size of a sample. Thisresults in a strong grade variability in the sample, evenwhen physically very close. In that case, the variogramshows a discontinuity at the origin. In practice, the nuggeteffect might be due to some kind of microstructure, namelya component of the phenomenon with a range shorter thanthe sampling distance or due to a structure with a rangeshorter than the smallest inter-point distance. However,

Fig. 8 Estimated activities by a multilevel B-spline approximation

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measurement and synthesis errors also cause a nuggeteffect.

To calculate the Kriging estimators, the discrete vario-gram points must be modeled by a variogram function.There exist many approaches and it depends on the prob-lem structure to choose the right variogram function. Werefer to [19–21] for the description of different variogrammodels. By means of the sample variogram, the Krigingsystem can be set up and solved for the Kriging weights wi

that assign a weight to each sample value xi.For the sake of simplicity, the derivation of the Kriging

system is not discussed in detail here but we refer to [19–21] for extensive studies.

The Kriging estimator at location x*, Z(x*) can bederived as a linear combination of all data points togetherwith their calculated weights, i.e.,

Z x�ð Þ ¼XN

i¼1

wixi

where N denotes the number of given data points.Figure 10 illustrates the result of a Kriging approximationapplied to the catalyst data described above. Compared toFigs. 2, 5 and 6 it can be concluded, that all three methodsprovide comparable models of the activity–compositionrelationship.

Determining the nugget effect of a Kriging model mustbe done thoroughly because this parameter controls theextent of approximation. Models with zero nugget effectyield an interpolation of data and the larger the nuggeteffect the smoother the fitted surface becomes. This enablesresearchers to adapt the smoothness of the desired surfacein a very comfortable way. Another advantage of Kriginglies in its excellent ability also to approximate data in morethan three dimensions. The computational effort of Krigingdirectly corresponds to the number of dimensions. Forexample, datasets containing four elements lead to Krigingsystems with matrix sizes of 287×287 entries, which can beeasily solved on a PC. This means Kriging can be used

excellently in more than four dimensions provided thatthere are datasets available. For more than five elements thesynthesizing and screening time of such a catalytic libraryis still too long to establish valuable datasets.

Visualization of a quaternary composition spread

According to Fig. 3, a quaternary composition spread canbe illustrated as a tetrahedron. The Kriging approachdescribed above is applied to a dataset in four dimensionscontaining the elements molybdenum, manganese, cobaltand chromium. Fig. 11 illustrates how the structure of theapproximated values within the tetrahedron can be vi-sualized by using slice plots. The catalysts were tested forthe oxidation of propene at 300°C. We are mainly inter-ested in catalysts that show high selectivity to acroleineamong other products such as acetone, propionaldehyde or1,5-hexadiene.

From Fig. 11 it can be seen, for example, how the nickelcontent influences the selectivity of the catalysts. The firstplot of Fig. 11 illustrates the selectivity of catalysts withoutany nickel. In the second plot of Fig. 11 all catalysts con-taining 10% nickel are shown and it can be seen that thereis the region of highest selectivity (colored in red). In thefollowing slices of higher Ni-content (Fig. 11) the selec-tivity to acroleine decreases. Clearly all four elementscontribute to the area of highest selectivity. Very important

Fig. 10 Approximated surface using the Kriging approach

Fig. 11 Visualization of a ternary catalyst system and its approx-imation using slice plots

Fig. 9 Sample variogram corresponding to the data set illustrated inFig. 2 with fitted variogram function

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here is that the areas of higher activities are supported bythe activity of the neighboring points. Points of isolatedhigh activity are most likely outliers, since in all our com-position spread studies of the last few years, the surfaceshave all been very smooth and continuous. This way ofvisualization helps to look for trends and redundant ele-ments among quaternary catalysts and enables the re-searcher to plan the next experiments for a more efficientsearch of active catalysts. In practice, this visualization canbe done more effectively directly on the monitor using thefourth (here Ni) and higher parameters as sliders.

Conclusions and outlook

Through a systematic preparation of mixed oxides coveringthe whole composition space by a sol–gel preparationprocedure tolerant of composition, the catalytic activity hasbeen determined. It has been demonstrated that catalyticactivity for propene oxidation to acroleine with air is afunction of chemical composition if the materials studiedare prepared by the same method. Three different mathe-matical methods, support vector machines, B-splines andKriging have been applied to model the datasets. All meth-ods provide comparable results and can therefore be ap-plied for QCAR. While B-splines can, up to now, only beapplied to datasets with functional relationships depen-dent on up to three parameters, SVM require a largercomputational effort. Kriging is favored because it can beapplied readily to multiple parameter spaces while itscomputation is still fast enough. Parts of our future workwill deal with the evaluation of the predictive power ofeach method. Furthermore, replications are now includedin all experiments.

This QCAR-modeling provides a new entry into pre-diction of materials properties. It can be used for materialsoptimization as well as for the design of experiments inextended high-throughput searches for new materials.

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