Energetic balance and kinetics for the decomposition

Acta Materialia 55 (2007) 1441–1446 www.actamat-journals.com

Energetic balance and kinetics for the decomposition of supersaturated Ti1xAlxN P.H. Mayrhofer b

a,b,*

, F.D. Fischer c, H.J. Bo¨hm d, C. Mitterer b, J.M. Schneider

a

a Materials Chemistry, RWTH-Aachen, D-52074 Aachen, Germany Department of Physical Metallurgy and Materials Testing, Montanuniversita¨t Leoben, Franz Josef Strasse 18, A-8700 Leoben, Austria c Institute of Mechanics, Montanuniversita¨t Leoben, A-8700 Leoben, Austria d Institute of Lightweight Design and Structural Biomechanics, Vienna University of Technology, A-1040 Vienna, Austria

Received 25 May 2006; received in revised form 12 September 2006; accepted 12 September 2006 Available online 13 December 2006

Abstract Ti1xAlxN films have been shown to exhibit superior mechanical and thermal properties and are thus widely used for industrial applications. We have recently reported, that metastable NaCl-structure (c) Ti1xAlxN decomposes to form c-TiN and c-AlN domains, and that the chemical requirement for spinodal decomposition is fulfilled over a broad composition and temperature range. Ab initio calculations showed that the maximum metastable solubility limit of AlN in c-TiN is in the range of 0.64–0.74, depending on the configurational entropy. The enthalpy change for decomposition of supersaturated c-Ti1xAlxN into c-TiN and c-AlN has a maximum of 0.146 eV/at (28.18 kJ/mol) at x 0.61. Here, we use continuum mechanical investigations in addition to ab initio calculations to consider also the previously not described contribution of strain and surface energy on the energetic balance for this decomposition process. Based on these results a simple kinetic model for the decomposition process of c-Ti1xAlxN can be developed. 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Metastable phases; Spinodal decomposition; Continuum mechanics; Density functional theory; Ti-Al-N

1. Introduction Hard protective coatings are increasingly required for wear-resistant applications. Most often, e.g., in machining, casting or hot-forming applications, they are subjected to high temperature, causing thermal fatigue [1]. Among many different transition metal nitrides, borides, and carbides, Ti1xAlxN and/or their alloys represent a large market share for several industrial applications. Consequently, a lot of effort is made to investigate and understand the materials properties thereof. Due to the limited atomic assembly kinetics during low temperature plasma assisted vapor deposition (substrate temperature below 0.2–0.3 of the melting point in Kelvin), Ti1xAlxN crystallizes in a supersaturated cubic NaCl (c) modification for AlN mole *

Corresponding author. E-mail address: [email protected] (P.H. Mayrhofer).

fractions (x) below 0.7, where Al substitutes for Ti [2–6]. The metastable c-Ti1xAlxN decomposes during annealing to form c-TiN and c-AlN domains, before phase transforming into the stable constituents c-TiN and ZnS-wurtzite (w) structure AlN. The formation of domains results in a hardness increase by providing additional obstacles for the dislocation movement [5]. Consequently, they allow to counteract the typically occurring hardness loss during annealing of hard coatings with large single phase fields which is based on defect annihilation (defects are generated during growth and act as barriers for dislocation glide) during annealing [7]. We have recently reported on the chemical requirement for the decomposition of c-Ti1xAlxN to form c-TiN and c-AlN domains, which is fulfilled over a broad temperature and composition range [8]. Ab initio calculations showed that the maximum metastable solubility limit of AlN in c-TiN is in the range of 0.64–0.74, depending on

1359-6454/$30.00 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.09.045

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P.H. Mayrhofer et al. / Acta Materialia 55 (2007) 1441–1446

the configuration entropy [6]. The enthalpy change for decomposition of supersaturated c-Ti1xAlxN into c-TiN and c-AlN has a maximum of 0.146 eV/at (28.18 kJ/ mol) at x0.61. Here, continuum mechanical approaches, in addition to ab initio calculations, are used to consider the previously not described influence of elastic strain energy and surface energy on this decomposition process. Based on these data a simple description of the decomposition kinetics of c-Ti1xAlxN into c-TiN and c-AlN was developed. 2. Theoretical methods Ab initio calculations of Ti1xAlxN are based on the density-functional theory, using the VASP code [9,10], in conjunction with the generalized-gradient approximations projector augmented wave potentials [11]. Relaxation convergence of 1.0 meV for ions and 0.1 meV for electrons, reciprocal-space integration with a Monkhorst–Pack scheme [12], energy cutoff of 500 eV, and tetrahedron method with Blo¨chl corrections [13] for the energy were used in the calculations. Relaxations for both the lattice parameter as well as the ion positions were performed. A k-points grid of 5 · 5 · 5 was used for all calculations of the supercells built of 32 and 64 atoms. The equilibrium total energies were obtained by a least-square fit of the calculated total energy over volume curves employing the Birch–Murnaghan’s equation of state [14] for Ti1xAlxN composites in their NaCl modification with AlN mole fractions of 0, 1/16, 2/16, 4/16, 6/16, 8/16, 10/16, 11/16, 12/16, 14/16, 15/16, and 1. The energy of formation Ef is calculated from the equilibrium total energy difference between the probed phases and the elements. More details on the performed ab initio calculations can be found in Refs. [6,8]. 3. Results and discussion 3.1. Energetic balance The decomposition of c-Ti1xAlxN into c-TiN and cAlN, before recrystallization occurs and w-AlN forms has been reported in Ref. [5]. Here we consider three major contributions to the energetic balance during the decomposition process of c-Ti1xAlxN to form c-TiN and c-AlN precipitates. These specific contributions (by volume) are the difference in chemical energy before and after decomposition, DGchem, as discussed in Refs. [6,8] as well as the previously not considered contributions of the elastic strain energy caused by the formation of particles with different specific volumes and elastic constants, W, and the energy associated with the formation of new surfaces (interfaces), O. Thus, the change of the total specific Gibbs free energy, energetic balance for the decomposition process, can be written depending on the fraction of transformation, n, as DG ¼ ðDGchem W OÞn:

ð1Þ

The evolution law for the two families of particles, c-TiN and c-AlN precipitates, in a c-Ti1xAlxN matrix, m, (or parent phase) is derived based on a concept which was developed by Svoboda et al. [15] by exploiting the thermodynamic extremum principle going back to Onsager in 1931 for heat conduction [16] and in 1945 for diffusion [17]. A general framework for the thermodynamic extremum principle can be taken from Refs. [18,19], and applications for a diffusional process can be found in Refs. [15,20]. Here, we assume that spherical c-TiN and c-AlN particles with isotropic materials properties precipitate, which are referred to as TiN and AlN throughout this manuscript. During the decomposition, the amount of n growing particles in a unit volume consists of (1 x)nTiN and xnAlN particles, according to the chemical composition of c-Ti1xAlxN. Hence, the extent of transformation n to form spherical precipitates with radius r is calculated as n¼n

4p 3 r: 3

ð2Þ

3.1.1. Chemical contribution The chemical contribution to DG during precipitation (transformation) is expressed by the difference in specific chemical energy of the untransformed system and the fully transformed system. Since the chemical energy of the (partially) transformed system is smaller than that of the parent phase, DGchem is positive and can be considered as the chemical driving force. The quantity DGchem depends strongly on x and temperature T and is found in Ref. [8] as TiN DGchem ¼ Em þ xEAlN f ½ð1 xÞEf f

þ TR½x lnðxÞ þ ð1 xÞ lnð1 xÞ ¼ DEf þ TR½x lnðxÞ þ ð1 xÞ lnð1 xÞ:

ð3Þ

TiN , and EAlN are the energy of formation for Here, Em f , Ef f the matrix c-Ti1xAlxN, c-TiN, and c-AlN, respectively, which were determined in Ref. [6] and are calculated per Ti1xAlxN formula unit, R is the gas constant. Here we assume that only contributions from the metal sublattice affect the configurational entropy. Fig. 1 shows that DGchem reaches a maximum for x 0.63 which is 21.73 kJ/mol for T = 900 C.

3.1.2. Elastic strain energy As the specific volumes V are different for the matrix, TiN, and AlN, both particle families are associated with a change dp of their specific volumes Vp (p = TiN, AlN) with respect to that of the matrix (V m) dp ¼

VmVp : Vm

ð4Þ

This volume change is calculated from ab initio results of V as a function of x and presented in Fig. 2. As the absolute value of (1 x )dTiN is always smaller than that of


Fig. 1. Difference in free energy DGchem of c-Ti1xAlxN and the decomposed quantity of (1 x) c-TiN + x c-AlN as a function of x for T = 700, 800, and 900 C.

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tions) or vacancy migration (to and from the areas with tensile and compressive stresses, respectively) [21] is considered during our calculations. The specific elastic strain energy, W, due to the accommodation of the transformation volume change ep ij (Eq. (5)) by the matrix and the particles themselves can be calculated in a mean field setting according to Ref. [22]. For a fully constrained condition the elastic strain energy has a maximum of 2.99 kJ/mol at x = 0.50, see Fig. 3. Further decreasing or increasing in x results in decreasing elastic strain energy, which reaches zero at the extreme cases of x = 0 and 1, respectively, where the Ti1xAlxN system degenerates to the binary TiN and AlN systems, respectively. For the c-Ti1xAlxN coatings with x being 1/4, 1/2, and 2/3, which are significant for industrial application [3,4], the specific elastic strain energy W is 2.12, 2.99, and 2.72 kJ/mol, respectively. It is important to note that W is an average value. If the local thermodynamic behaviour in a point of the interface is investigated, the local elastic strain energy, together with the strain and stress states on both sides of the interface, is the relevant quantity, which differs from the average value W, see e.g. Ref. [23]. 3.1.3. Surface energy The total specific surface energy, Ot, at the interfaces between the particles and the matrix, represented by a surface energy term, c, can be calculated by Ot ¼ nð4r2 pÞc ¼ n

4p 3 3c 3c r ¼ n ¼ nO: 3 r r

ð6Þ

The interfacial energy c can be estimated by using the nearest-neighbour-broken-bond hypothesis [24,25] nS z S c¼ DEi : ð7Þ NzL Here, N is the number of atoms per unit volume, nS is the number of atoms per unit area at the interface, zS is the

Fig. 2. Volume change dTiN and dAlN for the decomposition of c-Ti1x AlxN to form c-TiN and c-AlN, respectively.

xdAlN the total volume change associated with the formation of (1 x)nTiN and xnAlN particles is negative. This is, because the lattice parameters of c-Ti1xAlxN have a positive deviation from Vegard’s law [6]. The corresponding misfit strain tensor ep ij writes with dij being the unity tensor as p ep ij ¼ ðd =3Þdij :

ð5Þ

This eigenstrain must be accommodated by the elastic deformation of the neighbouring matrix and the precipitates themselves. The elastic moduli E of c-Ti1xAlxN as a function of x(0 6 x 6 1) can be found in Ref. [6]. It should be mentioned that no relaxation of the generated stresses either due to plastification (generation of disloca-

Fig. 3. Specific elastic strain energy W for c-Ti1xAlxN as a function of x.

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number of bonds per interface atom across the interface, zL is the coordination number denoting the numbers of nearest neighbors for each atom, and DEi is the difference between the inner energy of the system before and after decomposition. For the NaCl structure c-Ti1xAlxN matrix and c-TiN and c-AlN particles the following values are used: N 1.106 · 1029 at/m3, nS 2.347 · 1019 at/m2, zS 4, zL = 12, and DEi = jDEfj in J/m3 (for DEf see Eq. (3) or Ref. [8]). Thus, the interfacial energy c can be estimated to be 0.182, 0.170, and 0.090 J/m2 for x = 2/3, 1/2, and 1/4, respectively. For comparison, the energy of symmetrical twin boundaries in intermetallic TiAl alloys is reported to be 0.076 J/m2 [26]. Finally, the change of the total Gibbs free energy (Eq. (1)) of the system can be written with Eqs. (2) and (6) as DG ¼ nðDGchem W OÞ 4p 3 3c ¼ n r DGchem W : 3 r

ð8Þ

3c Consequently, DG becomes positive for r > DGchem , W which is the requirement to initiate decomposition. This minimum radius rmin increases with decreasing Al content and is 0.31, 0.34, and 0.45 nm for x = 2/3, 1/2, and 1/4, respectively, at T = 900 C. Slightly decreasing values for all compositions are obtained with decreasing T, which are 0.30 nm for 800 and 700 C with x = 2/3. Fig. 4 shows the total Gibbs free energy change for T = 900 C and n = 1 as a function of the chemical composition and for particle radii of 0.5, 1, 5, 10, and 20 nm. For r = 1 nm the specific interfacial energy O is around 5.88, 5.61, and 3.06 kJ/mol for x = 2/3, 1/2, and 1/4, respectively. Due to the reciprocal dependence of O on r, this term is below 0.59 kJ/mol for r P 10 nm and x = 2/3, hence there is almost no difference in DG for r > 10 nm

(see Fig. 4). For x P 0.21 with r = 0.5 nm (this value is slightly larger than the calculated lattice parameters a for c-Ti1xAlxN with a = 0.4255–0.4069 nm for x = 0–1, respectively) [6] the chemical driving force is larger than the retarding force (sum of strain and surface energy). Therefore, the decomposition of c-Ti1xAlxN into c-TiN and c-AlN will occur if sufficient activation energy for diffusion of the involved elements is provided. As the chemical driving force for decomposition continuously decreases with increasing temperature [8], also the total change in Gibbs free energy DG decreases with increasing T. It has to be mentioned that the decomposition of c-Ti1xAlxN into their stable constituents c-TiN and w-AlN is not considered here. Instead, the formation of c-TiN and metastable c-AlN is the focus of this work as it effectively compensates for the hardness loss due to recovery effects [5], which is observed for coatings with large single phase fields where no decomposition takes place (e.g., TiN) [5]. 3.2. Kinetics 3.2.1. Definition For the kinetic model, both particle families are assumed to have approximately the same evolution of their radius r(t) with time t, being the time derivative r_ ¼ dr=dt. The concentrations of Ti and Al in the particles, cpTi and cpAl , respectively, with p = TiN, AlN, and their concentrations m in the matrix, cm Ti and cAl , are defined by the corresponding molar volumes Xp and Xm, respectively, as ð1 xÞ x ; cm ; m Al ¼ X Xm 1 cTiN ; cTiN Ti ¼ Al ¼ 0; XTiN 1 cAlN cAlN : Ti ¼ 0; Al ¼ XAlN cm Ti ¼

ð9Þ

3.2.2. Dissipation For simplicity of the mathematical treatment we consider that no diffusion takes place inside the precipitates and that the N atoms are essentially fixed at their positions. The latter is valid as the matrix and the formed domains are NaCl structured with chemical differences only in the metal sublattice. Consequently, the main dissipative process is the diffusion of Ti and Al as two substitutional components in the Ti1xAlxN matrix. The mobilities of Ti and Al in the matrix is described by their diffusion m coefficients Dm Ti and DAl , respectively. As outlined in Ref. [15] Section 5, the according total energy dissipation, Q, denoted in Ref. [15] as Q3, follows with the notation of this paper as ( Q ¼ n4pRTr Fig. 4. Change of the total Gibbs free energy DG as a function of x for different particle radii r and T = 900 C.

3

2

2

m m ð1 xÞ½_rðcTiN x½_rðcAlN Ti cTi Þ Al cAl Þ þ m m m m cTi DTi cAl DAl

) : ð10Þ


Inserting Eq. (9) into Eq. (10) yields 8h i2 h m i2 9 > > Xm X = < ð1 xÞ x 4p 3 2 3RT XAlN XTiN : þ Q ¼ n r r_ m m m > 3 X > DTi DAl ; :

1445

ð11Þ

3.2.3. System evolution Since no further dissipation mechanism is active, the total dissipation Q has to be equal to the rate of DG (Eq. (8)). Note that DG_ is the negative rate of the Gibbs free energy G of the system and n is held as constant with time. With Eqs. (8) and (11) the equality Q ¼ DG_ gives the relationship m

r_r ¼

X RT

8h i2 h m i2 91 > > Xm X < = ð1 xÞ x AlN TiN 2c X X DGchem W þ : m m > > r : DTi DAl ;

ð12Þ

Standard nonequilibrium thermodynamics would yield an evolution equation of the type oG 2c r_ ¼ L ¼ L DGchem W ; ð13Þ on r with L being the interface mobility, see Ref. [27]. The advantage of the procedure outlined here is that the mobility term is directly related to the dissipation in the system. Furthermore, if the molar volumes can be assumed to be identical (XTiN = 11.60 cm3/mol, XAlN = 10.15 cm3/mol, and Xm is between these extremes) a simple evolution equation follows as # " 2 2 1 Xm 2c x ð1 xÞ B DGchem W þ ¼A : r_r ¼ m r r RT Dm D Ti Al ð14Þ A and B are constants to be found by comparison between the second term and third term of Eq. (14). Solving this differential equation for r yields 2 A2 2 A A A3 r þ ln r 1 r þ 2 ¼ 2 t: ð15Þ B B B2 B2 According to Eq. (8) any decomposition is just possible 3c for r > DGchem ¼ 3B . This means that the application of 2A W . Hence, by using Eq. (14) makes only sense for r > 3B 2A A 2 A 3 r > in Eq. (15) the term ln r 1 is below 27% of B 2 B 2 A 2 A r þ 2 B r and rapidly decreases below 10% for B2 A r > 1:75, where the radius is 17% larger B than2the minimum value. Thus, by disregarding ln AB r 1 in Eq. (15), as its contribution rapidly decreases with increasing A r, a simplified evolution equation for r of the form B sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B2 B rffi þ 2At ; ð16Þ 2 A A can be developed. Fig. 5 shows that the radii for c-TiN and c-AlN precipitates obtained by small angle neutron scattering (SANS)

Fig. 5. Evolution of the particle radius r with time and temperature of annealed c-Ti0.34Al0.66 N coatings obtained by SANS measurements (open symbols) compared to the results for c-Ti1xAlxN with x = 2/3 calculated after Eqs. (15) and (16).

of Ti0.34Al0.66N coatings are 0.5, 1, and 4 nm after annealing at 700, 800, and 900 C for 90 min [28]. Dynamical annealing to 890 C, with a heating and cooling rate of 50 C min1, yields a precipitate radius of 0.7 nm [28], which is added to Fig. 5 for T = 900 C and t = 2 min. The evolution of r with time and temperature can be described by Eq. (15) using diffusion coefficients for Ti 18 m and Al in c-Ti1xAlxN of Dm , Ti ¼ DAl ¼ 5 10 19 19 2 4 · 10 , and 1 · 10 cm /s for 900, 800, and 700 C, respectively. Unfortunately, the diffusion coefficients Dm Ti and Dm Al are not available in literature, but the used ones fit the observations made for metal interdiffusion in TiN/ NbN superlattice films [29,30]. The results obtained by the simplified evolution equation for r (Eq. (16)) deviate from those obtained by Eq. (15) by 0.45, 0.34, and 0.31 nm for x = 1/4, 1/2, and 2/3, respectively, at 900 C. Based on these results a simple kinetic model for the temporal dependence of coherent precipitates with temperature is developed as the main dissipative process in the cTi1xAlxN matrix is the diffusion of their substitutional components Ti and Al. The derived kinetic model allows to describe the evolution of the c-TiN and c-AlN radii in c-Ti0.34Al0.66N obtained by small angle neutron scattering. For this compound with x 2/3 the time before decomposition is initiated increases from 0.8 to 8.5 to 28.5 min with temperature decreasing from 900 to 800 to 700 C, respectively. At 900 C the precipitate radii increase with time from the initial value of 0.31 nm after 0.8 min to 4.1 nm after 90 min. As all other chemical compositions have smaller positive DG values, they also have smaller r values with time and temperature as well as longer incubation times. The obtained results show that the energetic balance and the kinetics of the decomposition of supersaturated multinary phases can be described, and allow their prediction, by a combination of ab initio calculations and continuum mechanical approaches. Acknowledgements The Erwin Schro¨dinger Program (project J2469-N02) of the Austrian Science Fund (FWF) is acknowledged by

P.H.M for financial support. F.D.F. very much appreciates the discussions with Dr. J. Svoboda, Brno. J.M.S. gratefully acknowledges funding from DFG (Schn 735/9-1). References [1] Komanduri R, Hou ZB. Trib Int 2001;34:653. [2] Ho¨rling A, Hultman L, Odeń M, Sjo¨leń J, Karlsson L. J Vac Sci Technol A 2002;20:1815. [3] Ho¨rling A, Hultman L, Odeń M, Sjo¨leń J, Karlsson L. Surf Coat Technol 2005;191:384. [4] Kutschej K, Mayrhofer PH, Kathrein M, Polcik P, Tessadri R, Mitterer C. Surf Coat Technol 2005;200:2358. [5] Mayrhofer PH, Ho¨rling A, Kalrsson L, Sjo¨leń J, Larsson T, Mitterer C, et al. Appl Phys Lett 2003;83:2049. [6] Mayrhofer PH, Music D, Schneider JM. J Appl Phys 2006;100:094906. [7] Mayrhofer PH, Mitterer C, Clemens H, Hultman L. Prog Mater Sci 2006;51:1032. [8] Mayrhofer PH, Music D, Schneider JM. Appl Phys Lett 2006;88:71922. [9] Kresse G, Hafner J. Phys Rev B 1993;48:13115. [10] Kresse G, Hafner J. Phys Rev B 1994;49:14251. [11] Kresse G, Joubert J. Phys Rev B 1999;59:1758. [12] Monkhorst HJ, Pack JD. Phys Rev B 1976;13:5188. [13] Blo¨chl PE. Phys Rev B 1994;50:17953. [14] Birch F. J Geophys Res 1978;83:1257. [15] Svoboda J, Fischer FD, Fratzl P, Kozeschnik E. Mater Sci Eng A 2004;385:166. [16] Onsager L. Phys Rev 1931;37:405. [17] Onsager L. Ann New York Acad Sci 1945;46:241. [18] Svoboda J, Turek I. Phil Mag B 1991;64:749. [19] Svoboda J, Turek I, Fischer FD. Phil Mag 2005;85:3699. [20] Svoboda J, Fischer FD, Fratzl P, Kroupa A. Acta Mater 2002;50:1369. [21] Svoboda J, Gamsja¨ger E, Fischer FD. Phil Mag Lett 2005;85: 473. [22] Bo¨hm HJ. Stress and strain fields due to multiple transforming inhomogeneities, ILSB Report 190, Institute of Light Weight Design and Structural Biomechanics, Vienna University of Technology, Vienna, Austria, 2005. [23] Fischer FD, Simha NK, Svoboda J. ASME J Eng Mater Technol 2003;125:266. [24] Bragg WL, Williams EJ. Proc Roy Soc London 1934;145:699. [25] Becker R. Ann Phys 1938;32:128. [26] Murr LE. Interfacial phenomena in metals and alloys. Reading, MA: Addison-Wesley; 1975. [27] Suo Z. In: Hutchinson JW, Wu ThY, editors. Advances in applied mechanics, 33. San Diego et al.: Academic Press; 1997. p. 193. [28] Mayrhofer PH, Staron P, Eckerlebe H, Clemens H, Mitterer C, Hultman L. unpublished results (2005). [29] Engstro¨m C, Birch J, Hultman L, Lavoie C, Cabral C, Jordan-Sweet JL, et al. J Vac Sci Technol A 1999;17:2920. [30] Hultman L, Engstro¨m C, Odeń M. Surf Coat Technol 2000;133: 227.