Alloying-related trends from first principles: An application to the Ti--Al--X--N system
David Holec, Liangcai Zhou, Richard Rachbauer, Paul H. Mayrhofer
aa r X i v : . [ c ond - m a t . m t r l - s c i ] F e b Alloying-related trends from first principles: An application to the Ti–Al–X–N system
David Holec, ∗ Liangcai Zhou, Richard Rachbauer, † and Paul H. Mayrhofer
1, 2, 3, 4 Department of Physical Metallurgy and Materials Testing,Montanuniversit¨at Leoben, A-8700 Leoben, Austria Institute of Materials Science and Technology, Vienna University of Technology, A-1040 Vienna, Austria Christian Doppler Laboratory for Application Oriented Coating Developmentat the Department of Physical Metallurgy and Materials Testing,Montanuniversitt Leoben, A-8700 Leoben, Austria Christian Doppler Laboratory for Application Oriented CoatingDevelopment at the Institute of Materials Science and Technology,Vienna University of Technology, A-1040 Vienna, Austria (Dated: August 1, 2018)Tailoring and improving material properties by alloying is a long-known and used concept. Recentresearch has demonstrated the potential of ab initio calculations in understanding the materialproperties at the nanoscale. Here we present a systematic overview of alloying trends when early-transition metals (Y, Zr, Nb, Hf, Ta) are added in the Ti − x Al x N system, routinely used as aprotective hard coating. The alloy lattice parameters tend to be larger than the correspondinglinearised Vegard’s estimation, with the largest deviation more than 2 .
5% obtained for Y . Al . N.The chemical strengthening is most pronounced for Ta and Nb, although also causing smallest elasticdistortions of the lattice due to their atomic radii being comparable with Ti and Al. This is furthersupported by the analysis of the electronic density of states. Finally, mixing enthalpy as a measureof the driving force for decomposition into the stable constituents, is enhanced by adding Y, Zr andNb, suggesting that the onset of spinodal decomposition will appear in these cases for lower thermalloads than for Hf and Ta alloyed Ti − x Al x N. PACS numbers: 61.66.Dk 68.60.Dv 71.15.Mb 71.20.Be 81.05.JeKeywords: Density Functional Theory; alloys; TiAlN; phase stability
I. INTRODUCTION Ti − x Al x N is nowadays a well-established materialused as a protective hard coating due to its high hardness,relatively low coefficient of friction, and good oxidationand corrosion resistance [1, 2]. As such it has attractedsignificant attention both in basic as well as in appliedresearch areas, resulting also in several quantum mechan-ical studies on phase stability, mechanical and electronicproperties of the ternary Ti − x Al x N system [3–11].One of the possible ways how to further tune the ma-terial properties is the concept of multicomponent alloy-ing. For example, Y, Zr, and Hf have been shown toimprove the oxidation resistance of Ti − x Al x N [12–14]and Cr − x Al x N coatings [15], Ta to increase the hightemperature durability [16], Cr, Nb, and Ta to retardthe decomposition process to higher thermal loads [17–19]. The most recent successful approach is to combinetheoretical studies with experimental work in order togain deeper understanding of the experimental observa-tions (e.g., Refs. [11, 13–15, 17–22]). Good examplesdemonstrating the ability of ab initio calculations forguiding the experiment by predicting chemistry-relatedtrends are e.g. Refs. [6, 9, 22] for various TM–Al–N ∗ [email protected] † currently employed at OC Oerlikon Balzers AG, Iramali 18, LI-9496 Balzers, Liechtenstein ternary systems. However, similar systematic theoreti-cal study showing the alloying-related trends for quater-nary Ti − x − y Al x X y N (or, more precisely, pseudo-ternaryTiN–AlN–XN) systems, is missing.In this paper we demonstrate the potential of the firstprinciple calculations for providing such systematic andexhaustive information for Ti − x Al x N alloyed with Y,Zr, Hf, Nb, and Ta throughout the whole quaternaryphase field. It is clearly shown that with some care, thecalculated results can serve as reliable “trend-givers” forthe materials design. Lastly, it should be noted thatin the paper we discuss only some related topics like theground state properties, chemical strengthening, or onsetof the spinodal decomposition; others, such as the finaldecomposition phase (precipitation of AlN in its stablewurtzite (B4, ZnS prototype) structure or influence onoxidation resistance, remain for the future studies.
II. COMPUTATIONAL APPROACH
We used Vienna Ab initio Simulation Package [23, 24]employing the Density Functional Theory [25, 26] to per-form quantum mechanical calculations. The exchangeand correlation effects were described with GeneralisedGradient Approximation as parametrised by Wang andPerdew [27], and implemented in projector augmentedplane-wave pseudo-potentials [28]. We used the supercellapproach to model random alloys, in particular 3 × × × × ≈ k -point · atoms guarantee the total energy accu-racy in the order of meV per atom. The stability ofsystems can be described using energy of formation, E f ,calculated as E f = E (Ti − x − y Al x X y N) − h (1 − x − y ) E (Ti hcp )+ xE (Al fcc ) + yE (X ξ ) + 12 E (N ) i . (1)Here, E (Ti − x − y Al x X y N) is the total energy per atom ofc-Ti − x − y Al x X y N, E (Ti hcp ), E (Al fcc ) and E (X ξ ) are thetotal energies of Ti in hexagonal close-packed (hcp, A3),Al in face-centered cubic (fcc, A1), X=Y, Zr, and Hf inhcp, and X=Nb, and Ta in body-centered cubic (bcc, A2)structures, respectively. E (N ) denotes the total energyof a nitrogen molecule. III. RESULTS AND DISCUSSIONA. Bulk properties and phase stability ofquaternary Ti–Al–X–N
Fitting the energy–volume curve with the Birch-Murnaghan equation of state (EOS) [30] yields theground state properties: lattice parameter, a , total en-ergy, E , and bulk modulus, B . The cubic lattice pa-rameters of five quaternary systems investigated here asa function of AlN mole fraction, x , and XN mole frac-tion, y , exhibit almost linear behaviour. Straight equallyspaced contours demonstrate this in Fig. 1a on the exam-ple of Ti − x − y Al x Hf y N. The linearly interpolated latticeparameter according to the Vegard’s empirical rule [31]reads a V ( x, y ) = xa AlN + ya XN + (1 − x − y ) a TiN . (2)The difference ∆ a = a − a V is shown in Fig. 1b againfor the Ti − x − y Al x Hf y N system. It follows that thelinear Vegard’s estimate is erroneous by as much as1 .
2% in the middle of the HfN–AlN tie-line. Similarly,differences of ≈ . ≈ . ≈ .
0% are ob-tained also for Ti − x − y Al x Zr y N, Ti − x − y Al x Nb y N, andTi − x − y Al x Ta y N systems, respectively, with the maxi-mum appearing always in the middle of the AlN–XN tie-line (X=Zr, Nb, Ta). Ti − x − y Al x Y y N exhibits an errorof > .
5% for Y . Al . N, reflecting the fact that Y isa significantly larger atom than Al. These deviations should be considered when using the Vegard’s linearisa-tion as an estimate of a . The reason for the non-linearbehaviour of a can be traced back to the gradual changesin the bonding (e.g., from ionic to metallic character), asdemonstrated for Ti − x Al x N in Ref. [8].In all five cases of Ti − x − y Al x X y N, alloying an elementX to Ti − x Al x N increases the lattice constant when theAl content on the metallic sublattice, x , is kept constantas well as for the constant Al-to-Ti ratio, x/ (1 − x − y ).These trends are in a good agreement with previousexperimental observations and theoretical studies [12–14, 18–21]. The calculated lattice parameters are sum-marised in Table I.Bulk moduli, being the measure of volume compress-ibility (the elastic behaviour) of the system, are plot-ted in Fig. 2. In all cases, the variation is non-linearbut smooth. The results can be classified into threedifferent compositional dependencies, based on the va-lency of the alloying element X. For the group IVB el-ements, Zr and Hf, it seems that the main parame-ter controlling B is the TiN mole fraction, z = 1 − x − y (there is only a minor variation along the tie-lines with a fixed TiN mole fraction). For exam-ple, bulk modulus of Ti − x − y Al x Hf y N varies between ≈
240 GPa obtained for Al-excess Hf − x Al x N, and ≈
290 GPa for TiN [32] (see Fig. 2b). For the isova-lent alloy Ti − x − y Al x Zr y N, a similar behaviour is ob-tained (Fig. 2d). Situation is somewhat different forthe Ti − x − y Al x Nb y N and Ti − x − y Al x Ta y N alloys (groupVB), as NbN ( B = 307 GPa) and TaN ( B = 330 GPa)[32], respectively, yield the highest bulk moduli of thequaternary systems investigated (Figs. 2c and d). Forthese systems, the main parameter controlling B is theAlN mole fraction, x . Finally, yet a completely differ-ent elastic response is obtained for Ti − x − y Al x Y y N alloy( B YN = 130 GPa), where the bulk modulus reaches val-ues well below 100 GPa for compositions in the centreof the compositional triangle (Fig. 2a). This is likely tobe related to a strong instability of such material due tohuge differences in the atomic size of individual species.A comparison of Figs. 1a and 2d together with the datain Table I offers the possibility to design lattice matchedsuperlattice materials (i.e. multilayer arrangements withlayer thicknesses within the order of the lattice parame-ter) with variations in bulk modulus. For example, TiNand Hf . Al . N have the same equilibrium lattice param-eters of ≈ .
25 ˚A, but about 40 GPa difference in B . An-other example is Ti . Al . Zr . N and TaN, both hav-ing lattice parameter a ≈ .
42 ˚A, but the respective bulkmoduli being 230 GPa and 330 GPa. Systems consistingof varying softer and harder components are known to beable to e.g., hinder dislocation motion, or stop or deflectpropagation of cracks [33, 34]. Here, an additional ben-efit stems from expected perfect interfaces (same crystalstructures, lattice-matched).It has been shown that for cubic structured Ti − x Al x Nthin films, the hardness reaches a maximum for the high-est Al contents for which the cubic structure is still main-
TTiN AlN
HfN a [Å]a) iN AlN HfN Δ a b) FIG. 1. a) Lattice parameter, a , and b) its deviation, ∆ a , from the Vegard’s estimate for the quasi-ternary TiN–AlN–HfNsystem.
60 150 285 330 B [GPa] HfN d) xy z TiN AlN YN a) xy z TiN AlN
NbN c) xy z TiN AlN
ZrN b) xy z TiN AlN e) TaN y z x
TiN AlN
FIG. 2. Bulk modulus, B , as a function of the composition of pseudo-ternary alloys: (a) Ti z Al x Y y N, (b) Ti z Al x Zr y N, (c)Ti z Al x Nb y N, (d) Ti z Al x Nb y N, and (e) Ti z Al x Ta y N. Countours are shown every 25 GPa for Ti z Al x Y y N (a), and every 5 GPafor the other four systems (b)–(e). tained [35]. The chemical compositions with hardnessmaximum are close to Ti . Al . N for physical vapor-deposited coatings. These compositions provide also thehighest driving force for decomposing towards their sta-ble constituents, cubic TiN and wurtzite AlN, via a spin-odal decomposition route. The calculated energies of for-mation, E f , of the cubic and wurtzite phases were inter-polated with cubic polynomials in x and y , as imple-mented in the software package Mathematica. Equating E cub f = E wur f yields the cross-over, an estimation of thecubic and wurtzite single phase fields. Considering only small amounts of X (i.e., Y, Zr, Nb, Hf, or Ta) up to ≈ . x ≈ . x ≈ .
56 [12, 20].The lowering of the solubility limit in the case of Zr andHf is a result of widening of the dual-phase region in theZr − x Al x N and Hf − x Al x N phase diagrams, as discussedin Ref. [9]. The lowered solubility limit of Y is likely to beconnected with the large differences in the atomic sizes
Ti-excessAl-excess Ti-excessAl-excess Ti-excessAl-excessTi-excessAl-excess Ti-excessAl-excess
VEC [e - /f.u.] -1.42-1.40-1.38-1.36-1.34-1.32-1.30 ene r g y o f f o r m a t i on [ e V / a t.] Ti Al N+Y+Zr+Nb+Hf+Ta
FIG. 3. Energy of formation, E f as a function of the va-lence electron concentration (VEC) for Ti . Al . N, Ti-excess(Ti . Al . X . N), and Al-excess (Ti . Al . X . N) al-loys. ( r Al = 1 .
432 ˚A, r Ti = 1 .
448 ˚A, and r Y = 1 .
80 ˚A) destabil-ising the close packed rock-salt structure while favouringthe more open wurtzite structure.
B. Alloying effects on Ti . Al . N In order to elucidate the role of individual ele-ments, we use c-Ti . Al . N as a test composition.By replacing 1 Al or 1 Ti (out of 18 on the metal-lic sublattice) with 1 X atom, the alloy composi-tion is changed to Ti . Al . X . N (Ti-excess) orTi . Al . X . N (Al-excess).
1. Chemical strength and elastic distortions
The relative phase stability change by alloying maybe quantified by energy of formation expressing the en-ergy gain of forming an alloy of a given composition withrespect to the individual elements. The results are plot-ted in Fig. 3 against the valence electron concentration(VEC) which is calculated as an average value of va-lence electrons per formula unit (f.u., 1 metal and 1 ni-trogen atom). Ti ([Ar]3 d s ), Al ([Ne]3 s p ), and N([He]2 s p ) have 4, 3, and 5 valence electrons, respec-tively, hence VEC of Ti . Al . N is ( (4 + 3) + 5) = 8 . E c = E (Ti − x − y Al x X y N) − h (1 − x − y ) E (Ti)+ xE (Al) + yE (X) + E (N) i (3)where E ( X ) is the total energy of an isolated atom X .Based on this quantity, two energy terms are discussed:relative chemical strength, calculated as the cohesive en-ergy difference between relaxed Ti . Al . N and (unre-laxed) Ti . Al . N+1X, where either one Ti or Al atom(out of 18 in the supercell) is replaced with one X atom.In the second step we allow for a full structural relax-ation. The total energy (or cohesive energy) decreasedue to the relaxation (with fixed chemical compositionof the alloy) corresponds to the elastic energy related tothe local distortions caused by the foreign atom. Thesetwo contributions are shown in Figs. 4a and b.The relative chemical strengthening (Fig. 4a) demon-strates the detrimental influence of Y. For all theother elements the relative chemical strength of(Ti . Al . ) − y X y N with respect to Ti . Al . N becomesmore negative. The cohesive energy, as defined in Eq. 3,expresses the energy (per atom) needed to de-assemblethe crystalline material into individual atoms. Since theelastic relaxation of the atom positions can only lowerthe total or cohesive energy, a negative/positive valueof the relative chemical strengthening is an indicator forstronger/weaker bonded atoms in the crystal. This isa purely electronic effect (a local change of the elec-tronic structure) and as such we interpreted it as elec-tronic strengthening. Consequently, the quaternary al-loys with Zr, Hf, Nb, and Ta are expected to be moreresistant against plastic deformation, leading e.g., to achange in hardness of the alloys (although there are manyother aspects influencing material’s hardness). Indeed,a comparison with the experimental values taken fromthe literature, H TiAlN ≈
33 GPa [13], H TiAlYN ≈
23 GPa[20], H TiAlZrN ≈
38 GPa [13], H TiAlNbN ≈
37 GPa [20], H TiAlHfN ≈
35 GPa [14], and H TiAlTaN ≈
37 GPa [19],supports this qualitative trend. It should be mentionedthat the experimental value of H TiAlYN corresponds toa wurtzite single phase structure, a phase change causedby the addition of ≈
10 mol% YN to Ti . Al . N. Sincethe wurtzite phase is in general softer than the cubic one,this significant decrease in hardness is likely to combineboth effect, the phase change as well as the weakening ofthe chemical bonding due to Y. Finally, the linear guide-for-the eye suggest that the chemical strength increaseswith increasing VEC, at least in the investigated region.Figure 4b shows how much elastic energy is intro-duced by local distortions into the material by alloy-ing. The elastic energy is plotted against the volume (a)
Ti-excessAl-excess Ti-excessAl-excess Ti-excessAl-excessTi-excessAl-excessTi-excessAl-excess
VEC [e - /f.u.] -4-20246 c he m i c a l s t r eng t hen i ng [ e V / . % X N m o l . f r a c .] Ti Al N+Y+Zr+Nb+Hf+Ta (b)
Ti-excessAl-excess Ti-excessAl-excessTi-excessAl-excess Ti-excessAl-excess Ti-excessAl-excess volume [Å /at.] d i s t o r t i on e l a s t i c ene r g y [ e V / . % X N m o l . f r a c .] Ti Al N+Y+Zr+Nb+Hf+Ta
FIG. 4. (a) Chemical strengthening and (b) alloying-related elastic energy caused by the local distortions ofTi . Al . N, Ti-excess (Ti . Al . X . N), and Al-excess(Ti . Al . X . N) alloys. per atom. The volume expansion in the first approxima-tion equals to 3 ε , where ε is the homogeneous isotropicstrain. Assuming the elastic constants are not signifi-cantly influenced by alloying, the elastic energy is pro-portional to ε . The quadratic fit gives qualitativelya good description as can be seen in Fig. 4b. Takinginto account the metallic radii of individual metal atoms, r Al = 1 .
432 ˚A and r Ti = 1 .
448 ˚A, allows to explain whythe smallest elastic deformations are caused by alloyingwith Nb ( r Nb = 1 .
46 ˚A) and Ta ( r Ta = 1 .
46 ˚A), fol- lowed by Hf ( r Hf = 1 .
564 ˚A) and Zr ( r Zr = 1 .
59 ˚A), andthe largest distortions being caused by alloying with Y( r Y = 1 .
80 ˚A), as in this sequence the atomic radii in-crease.In conclusion, Ta and Nb are predicted to be the mostdominant strengthening elements, which at the sametime introduce also the smallest lattice distortions.
2. Density of states
The total density of states (DOS) for Ti . Al . N andcorresponding Al-excess alloys is shown in Fig. 5a. Thetop of the valence band, as shown, can be divided intothree distinct regions: (i) −
10 to ≈ − sp d covalent bonding (as demonstratedby the overlap of s -, p − , and d -projected DOS in Fig. 5a),(ii) − ≈ − p –TM- d bonding, and (iii) ≈ − d electrons.The term “covalent” in the following analysis refers to theregions (i) and (ii) together (i.e., energy between −
10 and ≈ − R E max E min ρ ( E ) E d E R E max E min ρ ( E ) d E (4)where ρ ( E ) is DOS in the band between E min and E max .In assessing the alloying effects, the relative decreaseof COM in particular energy range contributes towardslower total energy, and consequently leads to chemicalstrengthening as discussed in Section III B 1. Addition-ally, each energy range (or band) can be characterised bya “band width” (BW) defined as:BW = 2 vuut R E max E min ρ ( E ) E d E R E max E min ρ ( E ) d E . (5)Such quantity reflects the width of the band weighted bythe density of states distribution. Smaller value of BWcorresponds to better overlap of the hybridised states,and hence to increased degree of hybridisation.COM of the metallic range of Ti . Al . N is − .
684 eV.This value changes to − . − . − . − . − .
762 eV for Y, Zr, Nb, Hf, and Ta, respectively,as evaluated for the Al-excess compositions (see Fig. 5b).Here, a significant strengthening is predicted by alloyingof Nb and Ta, related to an increased concentration of d electrons forming the d – d bonds ( t g symmetry) along h i directions [19, 20]. The same analysis of COM forthe covalent region yields − .
148 eV for Ti . Al . N, and − . − . − . − . − .
256 for Y, Zr, (a) -10 -8 -6 -4 -2 0 E − E F [eV] den s i t y o f s t a t e s [ a . u .] Ti Al N+Y+Zr+Nb+Hf+Ta covalenthybridised metallics d p (b) -0.78-0.75-0.72-0.69-0.66 C O M m e t a lli c [ e V ] -5.25-5.20-5.15-5.10-5.05 C O M c o v a l en t [ e V ] T i . A l . N + Y + Z r + N b + H f + T a B W h y b r i d i s ed [ e V ] FIG. 5. (a) Density of states for pure, Y-, Zr-, Hf-, Nb-,and Ta-containing Ti . Al . N for the Al-excess compositions( y = 0 . s -, p -, and d -projected density of statesof Ti . Al . N is shown on the very top. E F denotes theFermi energy. (b) Calculated corresponding centres of massfor metallic and convalent regions, and the band width of the sp d hybridised region ( −
10 and ≈ −
Nb, Hf, and Ta, respectively, in the Al-excess configura-tions (Fig. 5b). These numbers suggest that Y consid-erably weakens the covalent interaction, while the otherfour elements (Zr, Nb, Hf, and Ta) all strengthen the co-valent bonding. Finally, a consistent picture is obtainedalso from BW of the hybridised region: Y significantlybroadens the electronic states distribution while Ta con-fines the states resulting in an increased degree of the sp d hybridisation.In summary, the most significant strengthening is ob-tained by alloying with Ta, followed by Nb and Hf. Stilla small strengthening is obtained by alloying with Zr,while Y softens the material. These trends are in linewith the chemical strength considerations based on thetotal energy changes, as discussed in Section III B 1.The density of states at the Fermi level is an indica-tor of the (relative) alloy stability. It reaches a value of0 .
21 st. · eV − · at. − for Ti . Al . N. Zr keeps it unalteredfor the Al-excess composition, while Hf, Nb, Ta, and Yincrease it to 0 .
24, 0 .
25, 0 .
25, and 0 .
29 st. · eV − · at. − ,suggesting a gradual destabilisation of Ti − x Al x N by al-loying. This is in qualitative agreement with the trendsin energy of formation as shown in Fig. 3.
3. Decomposition of the unstable alloys
The mixing enthalpy, H mix , is calculated as H mix = E (Ti − x − y Al x X y N) − h (1 − x − y ) E (TiN)+ xE (AlN) + yE (XN) i , (6)where E ( XN ) is the total energy (or energy of for-mation, or cohesive energy) of binary cubic XN, and E (Ti − x − y Al x X y N) is the corresponding energy of thequaternary alloy. The mixing enthalpy expresses the en-ergy gain (when negative) of forming the alloy with re-spect to the binary nitrides, hence it quantifies whetherthe alloy is stable ( H mix <
0) or unstable ( H mix > ≤ x ≤
1, 0 ≤ y ≤
1, and0 ≤ z ≤ x + y + z = 1, for all five quasi-ternary systems.The driving force for decomposition is the largest forTi − x − y Al x Y y N near to the YN–AlN tie-line (Fig. 6a).The lattice difference between YN and AlN is about 19%,a value larger than what is “allowed” for solid solutionsby Hume-Rothery rules [36]. H mix is positive in the wholecompositional range except for alloys close to the quasi-binary TiN–NbN and TiN–TaN tie-line (Figs. 6c and e).This suggests that the Ti − y Nb y N and Ti − y Ta y N al-loys are stable, a result previously shown in literature[21, 37, 38]. The calculated mixing enthalpies are sum-marised in Table I.Adding a small amount (up to y = 0 .
1) of Y, Zr, or Hfto Ti . Al . N while either keeping the Al-to-Ti ratio, or -50 50 150 250 H mix [meV/at.] YN a) .8 0.6 0.4 0.2. xy z TiN AlN
ZrN b) xy TiN z N HfN d) .8 0.6 0.4 0.2. xy z TiN AlN
TaN e) .8 0.6 0.4 0.2. xy z AlN TiN AlN
NbN c) .8 0.6 0.4 0.2. xy z TiN AlN
FIG. 6. Mixing enthalpy, H mix , as a function of the composition of pseudo-quaternary alloys: (a) Ti − x − y Al x Y y N, (b)Ti − x − y Al x Zr y N, (c) Ti − x − y Al x Nb y N, (d) Ti − x − y Al x Nb y N, and (e) Ti − x − y Al x Ta y N. Contours are every 50 meV/at.. keeping the Al amount x = 0 . H mix increases. Ti − x Al x N has beenshown to decompose spinodally before the stable wurtziteAlN precipitates appear [39], and also the quaternary al-loys are expected to do so. Therefore, the increased forcefor decomposition is interpreted as increased force forthe isostructural decomposition, thus shifting the onsetof spinodal decomposition to lower temperatures (lowerthermal loads). Consequently, age-hardening of these al-loys is predicted to occur at lower temperatures thanfor Ti . Al . N. The peak broadening in X-ray diffrac-tograms suggesting the spinodal decomposition is indeedobserved for lower annealing temperatures when Zr [13]is added . On the contrary, addition of Nb and Ta lowersthe mixing enthalpy thus the onset of the spinodal de-composition is predicted to be shifted to slightly higherthermal loads as compared with Ti . Al . N, as in excel-lent agreement with experiments, see e.g. Ref. [19].
IV. CONCLUSIONS
We employed first principle calculations to study al-loying effects of early transition metals on the groundstate properties and stability of Ti − x Al x N alloys usedas a protective hard coating material. The calculatedlattice parameters of the quaternary alloys exhibit a de- viation from Vegard’s-like linear relationship, being thelargest with ≈ .
5% for Y . Al . N. The additional com-positional degree of freedom of the quaternary alloys (ascompared with Ti − x Al x N) is suggested to allow for spe-cially designed lattice-matched multilayers with alternat-ing soft and hard components.Analysis of the chemical strength and local elastic dis-tortions showed that Ta and Nb are the most promisingstrengtheners, closely followed by Zr and Hf. These con-clusions agree well with the published experimental re-sults of the increased hardness of Ti − x Al x N alloyed withZr, Nb, Hf, or Ta. Finally, the analysis of the mixingenthalpy as a measure of the driving force for decompo-sition suggests that addition of Y, Zr and Hf leads to itsincrease, thus earlier onset of the spinodal decompositionand the related age-hardening process.
ACKNOWLEDGEMENTS
Financial support by the START Program (Y371) ofthe Austrian Science Fund (FWF) is greatly acknowl-edged.
Appendix A: Properties of the quasi-ternary alloys [1] W.-D. M¨unz, J. Vac. Sci. Technol. A , 2717 (1986). [2] S. PalDey and S. C. Deevi,Mat. Sci. Eng. A , 58 (2003). Ti z Al x Y y N Ti z Al x Zr y N Ti z Al x Nb y N Ti z Al x Hf y N Ti z Al x Ta y N z x y a B H mix a B H mix a B H mix a B H mix a B H mix [˚A] [GPa] [meV/at.] [˚A] [GPa] [meV/at.] [˚A] [GPa] [meV/at.] [˚A] [GPa] [meV/at.] [˚A] [GPa] [meV/at.]0.0 0.0 1.0 4.918 159 0 4.621 245 0 4.458 305 0 4.538 269 0 4.421 336 00.0 0.2 0.8 4.828 135 176 4.545 236 160 4.398 268 74 4.473 252 136 4.378 279 540.0 0.4 0.6 4.670 158 264 4.453 233 240 4.339 247 110 4.394 241 203 4.312 276 820.0 0.6 0.4 4.527 159 264 4.350 233 240 4.265 253 112 4.308 240 203 4.251 259 820.0 0.8 0.2 4.297 216 176 4.228 237 160 4.185 248 75 4.205 241 136 4.177 240 540.0 1.0 0.0 4.070 253 0 4.070 253 0 4.070 253 0 4.070 253 0 4.070 253 00.2 0.0 0.8 4.806 140 94 4.560 242 35 4.425 296 -12 4.495 258 2 4.399 315 -330.2 0.2 0.6 4.683 60 236 4.470 236 151 4.357 279 49 4.419 253 104 4.336 300 200.2 0.4 0.4 4.564 60 278 4.373 233 197 4.290 263 83 4.336 245 150 4.275 277 550.2 0.6 0.2 4.363 94 229 4.263 232 173 4.217 254 92 4.242 243 141 4.208 262 730.2 0.8 0.0 4.124 252 80 4.124 252 78 4.124 252 75 4.124 252 77 4.124 252 740.4 0.0 0.6 4.673 155 141 4.493 245 52 4.388 279 -17 4.445 260 3 4.371 269 -490.4 0.2 0.4 4.547 60 241 4.398 244 135 4.318 276 40 4.365 254 83 4.303 291 110.4 0.4 0.2 4.348 95 233 4.293 245 156 4.248 264 83 4.274 253 121 4.240 272 640.4 0.6 0.0 4.162 260 121 4.162 260 118 4.162 260 113 4.162 260 116 4.162 260 1110.6 0.0 0.4 4.546 160 141 4.421 256 52 4.346 276 -17 4.390 264 3 4.335 269 -490.6 0.2 0.2 4.378 95 188 4.319 259 110 4.275 280 49 4.301 265 74 4.268 286 290.6 0.4 0.0 4.196 269 121 4.196 269 118 4.196 269 113 4.196 269 116 4.196 269 1110.8 0.0 0.2 4.401 219 94 4.342 271 35 4.302 291 -12 4.327 274 2 4.296 295 -330.8 0.2 0.0 4.227 281 80 4.227 281 78 4.227 281 75 4.227 281 77 4.227 281 741.0 0.0 0.0 4.256 292 0 4.256 292 0 4.256 292 0 4.256 292 0 4.256 292 0TABLE I. 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