Phase-field modelling of spinodal decomposition in TiAlN including the effect of metal vacancies

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ScienceDirect Scripta Materialia 95 (2015) 42–45 www.elsevier.com/locate/scriptamat

Phase-field modelling of spinodal decomposition in TiAlN including the effect of metal vacancies ⇑ ˚ grenb and M. Ode´na K. Gro¨nhagen,a, J. A a

Nanostructured Materials, Department of Physics, Chemistry and Biology (IFM), Linko¨ping University, SE-581 83 Linko¨ping, Sweden b Department of Materials Science and Engineering, Royal Institute of Technology, SE-100 44 Stockholm, Sweden Received 9 August 2014; accepted 24 September 2014 Available online 16 October 2014

Using a CALPHAD approach together with a Cahn–Hilliard model, we describe the microstructure evolution in cubic Ti1xAlxN including vacancies on the metal sublattice. Our results show that vacancy content has a pronounced effect on the decomposition kinetics. Furthermore, vacancies show a strong tendency to segregate to the coherent AlN–TiN interface regions. We illustrate how vacancies anneal to grain boundaries, and finally, we compare our prediction to experimental differential scanning calorimetry data and attribute the second peak in the thermogram to vacancy depletion. Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Phase-field model; Spinodal decomposition; Vacancies; TiAlN

Cubic (c)-Ti1xAlxN is used as a coating material, e.g. on metal cutting tool inserts. At the annealing temperature, usually 900 °C, the c-Ti1xAlxN phase is unstable and the alloy separates into stable c-TiN and metastable c-AlN; the latter subsequently transforms into stable hexagonal (h)-AlN. Theoretically, it has been shown that there exists a miscibility gap in c-Ti1xAlxN [1,2], with a stability range up to around x = 0.7 before h-AlN forms [3]. Experimentally, spinodal decomposition of arc-evaporated c-Ti1xAlxN has been studied extensively by, for example, in situ small angle X-ray scattering [4–6], atom probe tomography [7,8], X-ray diffractometry (XRD) [9,10], scanning transmission electron microscopy [11,12] and differential scanning calorimetry (DSC) [13,14]. The results from these studies show an interconnected structure of Al-rich and Ti-rich regions, typical for materials undergoing spinodal decomposition, and it results in an enhanced performance of the coating when used in cutting applications [3,10,15–17]. In this work, we introduce metal vacancies as a third component when modelling spinodal decomposition of c-Ti1xAlxN. The high-energy of the incident ions (100 eV) during the arc-deposition process is expected to generate a large number of such vacancies in the TiAlN films [18–20]. The experimental findings mentioned above show that spinodal decomposition is taking place throughout the material and progresses faster at grain

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and column boundaries where diffusivities are higher [21]. These high diffusion paths are often considered to be regions with a higher concentration of vacancies and less ordering of atoms. Furthermore, four exothermic reactions are consistently recorded when annealing arc-evaporated c-TiAlN coatings ([Al]/([Ti] + [Al]) P 0.5) from room temperature to 1200 °C [14] (see Fig. 2). The first and the second peaks are postulated to result from stress relaxation, diffusion and annihilation of vacancies, while this has not been confirmed experimentally or theoretically. The third and fourth peaks are well established to originate from the spinodal decomposition [4,6,14] and the transformation of c-AlN to the stable h-AlN phase [6,13], respectively. The aim of his work is to model not only the effect of metal vacancies on spinodal decomposition (peak three in Fig. 2), but also the vacancy-driven diffusion process corresponding to peak two in the DSC diagram (Fig. 2). Phase-field simulations including vacancy diffusion have been performed by others, e.g. Wan et al. [22], who studied the effects of coherency stress and vacancy sources/sinks on interdiffusion across coherent multilayer interfaces. Phase-field modelling including vacancy diffusion has also been studied for sintering [23] and void migration in a temperature gradient [24], but not in the case of spinodal decomposition. In this model, (TiAlVa)N is approximated as a pseudo-ternary system consisting of Al:N, Ti:N and Va:N where the composition of N is supposed to be homogeneous throughout the system, i.e. we present a Cahn–Hilliard model [25] that is expanded to a substitutionally ternary system where Al, Ti and vacancies

http://dx.doi.org/10.1016/j.scriptamat.2014.09.027 1359-6462/Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

K. Gro¨nhagen et al. / Scripta Materialia 95 (2015) 42–45

are allowed to diffuse on the metal sublattice. The total Gibbs energy of the system is based on the functional: Z 1 Gm ðy Al ; y Ti ; y Va ; T Þ  jAl;Ti:N ðry Al ry Ti Þ Vm X  jAl;Va:N ðry Al ry Va Þ  jTi;Va:N ðry Ti ry Va ÞdX

ð1Þ

where Gm is the estimated molar Gibbs energy per formula unit (TiAlVa)N given by: Gm ¼ G0Va:N y Va þ RT ðy Al ln y Al þ y Ti ln y Ti þ y Va ln y Va Þ þ y Al y Ti LAl;Ti:N þ y Al y Va LAl;Va:N þ y Ti y Va LTi;Va:N

ð2Þ

All molar quantities are given per mole of formula unit (TiAlVa)N. V m is the molar volume, calculated from the lattice parameter of the global overall composition and is approximated to be constant. G0Va:N is the molar Gibbs energy of only vacancies on the metal sublattice. The parameters jAl;Ti:N ; jTi;Va:N ; jAl;Va:N are the gradient energy coefficients and y Al ; y Ti ; y Va are the site fractions for each component and their sum is unity. The evolution of the concentrations fields is governed by the normal conservation equation: 1 @y i ¼ r  J i V m @t

ð3Þ

where i ¼ Al; Ti. A third equation can be written for i ¼ Va but it will not be independent of the other two. When the vacancy mechanism is operating, the diffusion flux of a component i in a lattice-fixed frame of reference is described by:   1 X dGm ð4Þ Ji ¼  M ij r Vm j dy i where J Va ¼ ðJ Al þ J Ti Þ and dGm =dy i is the chemical potential of Al:N or Ti:N. M ij denotes the phenomenological coefficients that are related to the individual diffusivities Di of the different elements in the Ti or Al nitride, respectively, as: M Al;Al ¼ y Al y Va DAl =RTy eq Va M Al;Ti ¼ 0 M Ti;Ti ¼

ð5Þ

y Ti y Va DTi =RTy eq Va

y eq Va denotes the equilibrium fraction of vacancies. If the vacancies are treated as a conserved component, the vacancy content could have any value depending on whether there is accumulation or depletion of vacancies. If local equilibrium prevails, the vacancy content is equal to the equilibrium fraction. Thus, the two independent equations to be solved are: @Gm @Gm DAl:N y Al ð1  y Al  y Ti Þ=RTy eq Va r @y Al  @y Va

@y Al ¼r @t 2rjAl;Va:N ry Al  rðjAl;Va:N þ jTi;Va:N  jAl;Ti:N Þry Ti

!

ð6Þ

and @Gm @Gm DTi:N y Ti ð1  y Al  y Ti Þ=RTy eq @y Ti Va r @y Ti  @y Va ¼r @t 2rjTi;Va:N ry Ti  rðjAl;Va:N þ jTi;Va:N  jAl;Ti:N Þry Al

!

ð7Þ

The diffusion coefficient of atomic self-diffusion in a solid is expressed as an Arrhenius equation: D ¼ D0 expðQ=RT Þ, where Q is the activation energy, R

43

is the gas constant, and T is the temperature. D0 is a frequency factor. The activation energy of spinodal decomposition in c-TiAlN was estimated to Q = 303 kJ mol1 and D0 = 1.4  106 m2 s1 by Knutsson et al. [4]. The gradient contribution must always be positive in order to stabilize the homogeneous solution. The gradient coefficient for a binary regular solution was evaluated by Cahn and Hilliard [25] to be j ¼ Lk2 =2, where L is the interaction parameter and k is the interatomic distance calculated from the lattice parameter [1,26] of the overall composition using Vegard’s law. For a system containing a miscibility gap the interaction parameter is positive. However, when a third component is added it may have a negative interaction parameter with one or both of the first two components, and negative gradient coefficients would then be predicted and the total gradient energy may not be positive. From a physical point of view a negative interaction parameter indicates that there is a tendency for ordering rather than miscibility. Such tendency is not included in the present model. In our case it will be shown that the interaction parameters between Al,Va:N and Ti,Va:N are both positive. No ternary interaction parameter was used in the present work. For the pseudo-binary system (Ti,Al)N, we have used the interaction parameter evaluated by Ullbrand [27]. They used a Redlish–Kister polynomial of degree three to fit to ab initio data of the free energy calculated by Alling et al. [28]. Alling et al. used a unified cluster expansion method to allow for some clustering instead of a perfect random alloy. The thermodynamic data used in this work correspond to an equilibrium clustering degree at 2000 K. Furthermore, Ullbrand et al. assumed that the L-parameters are linear functions of temperature. When calculating the interaction parameters for the pseudo-binary system (Ti,Va)N we start with the chemical potential: 2

eq lVa:N  loVa:N ¼ G0Va:N þ RT ln y eq Va þ ð1  y Va Þ LTi;Va:N

ð8Þ

where we can write lVa:N ¼ lVa þ lN : Furthermore, we assume that the following holds: lVa ¼ 0; lN ¼ 12 lN 2 ; eq 1 1bar lo;ref Va:N  2 lN 2 and y Va << 1. We can then rewrite the lefthand side of Eq. (8) in terms of temperature and partial pressure: 1 1 1 P N2 l  l1bar ¼ RT ln 2 N2 2 N2 2 1bar ¼ G0Va:N þ RT ln y eq Va þ LTi;Va:N

ð9Þ

Hence, the final expression of the interaction parameter can be expressed as: y eq Va LTi;Va:N ¼ RT ln pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  G0Va:N P N 2 =1bar

ð10Þ

The interaction parameter is thus dependent on the equilibrium concentration of vacancies, the temperature, the partial pressure of nitrogen and G0Va:N . If we consider thermal vacancies in a crystal of a pure element the equilibrium vacancy concentration may be expressed as y eq Va ¼ expðG0Va =RT Þ [29], where G0Va is the Gibbs energy of formation of a vacancy expressed per mole of vacancies. In our case, the quantity G0Va:N þ LTi;Va:N is identified as the Gibbs energy of vacancy formation. We have arbitrarily chosen an equilibrium value of vacancies in the nitride to be 106. This value might be a bit high, but due to numerical problems with very small numbers this value was

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K. Gro¨nhagen et al. / Scripta Materialia 95 (2015) 42–45

chosen to give a low enough concentration. The partial pressure of nitrogen was chosen to be atmospheric pressure and G0Va:N = 30 kJ mol1. The interaction parameter for (Al,Va)N is calculated in the same way. Two cases have been studied for a closed system at a constant temperature of 900 °C: the first case has an equilibrium concentration of vacancies on the metal sublattice, and in the second case, the vacancy concentration is 10 times the equilibrium concentration. As expected, the rate for spinodal decomposition is highly affected by the vacancy concentration: a higher vacancy concentration increases the phase separation rate. With an excess number of vacancies the phase separation starts after 5 s and with equilibrium concentration of vacancies it starts after 50 s. The phase separation is also completed faster for the high vacancy concentration. Furthermore, we note that Al:N reaches equilibrium concentration before Ti:N, meaning that the driving force to form Al:N is larger than to form Ti:N. Since we have the same diffusivities for Al and Ti in this system, we conclude that this is due to the asymmetric shape of the Gibbs energy curve, where the curvature is larger on the Al-rich side, as also discussed in Ref. [27]. Figure 1 shows the concentration as a function of x (y is constant) for Al:N and Va:N in the same plot for the case in which we have an excess number of vacancies annealed at 900 °C for 9 s. It can be seen from Figure 1 that vacancies segregate to the phase boundaries. As mentioned above, pure Al:N forms faster than Ti:N. In these pure Al-rich areas the vacancy concentration is very low, but we have a higher concentration of both aluminum and vacancies in the Ti-rich domains. Gradients of aluminum extending further into the Ti-rich domains have been experimentally observed in the work by Johnson et al. [8]. As also discussed in Johnson et al.’s paper, a Kirkendall effect, based on the assumption that Al has a higher diffusivity than Ti, implies a net flow of metal vacancies from the Al-rich side of the composition gradient to the Ti-rich side. In the present model, the diffusivity is equal for Al and Ti but due to the asymmetric Gibbs energy curve Al diffuses faster, leading to the same result. In Figure 2 we compare the heat flow in the model with that of DSC measurements [14]. To imitate the DSC measurement as much as possible we ramp in temperature at 20 °C min1 from room temperature up to 850 °C. Arc-deposited TiAlN coatings often display a columnar growth with column widths in the range

Figure 1. Concentration profile of Al:N and Va:N (normalized) for 10y eq Va and y Al ¼ 0:67 after 9 s of annealing at 900 °C.

Figure 2. Heat flow developed in the simulations for different vacancy concentrations compared to the DSC measurement.

of 20–200 nm and a length extending much longer in the growth direction [30,31]. Therefore we have chosen a simulation box of 25  25 nm to imitate one-fourth of a grain in the x–y plane if the z-direction is considered to be along the growth direction of the film. In the following simulations, we set two boundaries (x = 25 nm, y = 0) to the equilibrium value of vacancy concentration. These boundaries will act as grain boundaries where we assume we always have an equilibrium vacancy concentration. We have the same composition of aluminum (0.67) as in the experiments and an arbitrarily chosen value of 0.003 initial vacancies. That means that excess vacancies will diffuse out through the grain boundaries to reach equilibrium. Figure 2 clearly shows that we can capture two different phenomena with our model: energy recovery in terms of metal vacancy diffusion, and spinodal decomposition of metastable c-TiAlN into stable c-TiN and metastable c-AlN. At low temperature, no diffusion occurs, but at 550 °C the vacancies start to move, which corresponds to the first peak in the simulations (second peak in the DSC measurement) (see Fig. 2). Later, phase separation occurs where Tiand Al-rich domains form, which corresponds to the second peak in the simulations (third peak in the DSC measurement). There are several parameters that affect the size and the height of the first peak. A higher initial vacancy concentration gives rise to higher peaks, but will also decrease the set-off temperature of that peak. A lower equilibrium vacancy concentration will have the same effect. Keeping the ratio y Va =y eq Va constant will keep the position of the first peak, cf. Eq. (5). Furthermore, a lower equilibrium vacancy concentration gives a higher interaction parameter, which in turn results in a higher peak. In the simulations, we have an equilibrium vacancy concentration of 1  106 due to numerical issues. When comparing the experimental and simulated data in Figure 2, we note that such a high value of the equilibrium vacancy concentration results in a too small first peak. Decreasing the equilibrium vacancy concentration will increase the height of the first peak through the expression of the interaction parameter L. Moreover, we see that with higher initial vacancy concentration the set-off temperature of the spinodal decomposition may decrease. This is because for higher initial vacancy concentrations there will remain some excess vacancies far from the grain boundary where kinetics is locally enhanced. In these areas spinodal decomposition starts earlier (see also Fig. 3). That can be compared with the experimental DSC peak for spinodal decomposition

K. Gro¨nhagen et al. / Scripta Materialia 95 (2015) 42–45

Figure 3. Composition field of Al:N at 790 °C in an open system. Initial y Va  0:004: The concentration of vacancies is higher in the upper left corner.

which is broad and not sharp. That would indicate that there is some variation in vacancy concentration in the real sample and spinodal decomposition starts earlier in areas in which the vacancy concentration is higher. One further observation is that we could not capture the first peak in the DSC diagram (Fig. 2). The first peak is assumed to come from lattice relaxation [14]. For arc-evaporated films, the compressive stresses usually arise from vacancy interstitials, which have lower activation energy than metal vacancies. That implies that interstitials would diffuse at lower temperatures compared to metal vacancies. Furthermore, interstitials only need to jump to a neighboring metal vacant site to relax. In this work we are not considering interstitials but these will be the subject of coming reports. It is also worth mentioning that we are only considering single vacancies in the present model. Divacancies, vacancy–vacancy interactions and vacancy clustering are not considered and how these appear is today unknown. In conclusion, we have shown that by including metal vacancies as a third component, dynamically important processes can be captured. With our model we can simulate two out of four peaks in the DSC diagram of TiAlN. Moreover, the model predicts the dynamic behavior of the spinodal decomposition process in TiAlN. The authors gratefully acknowledge the Swedish Foundation for Strategic Research (SSF) Project: Designed multicomponent coatings – Multifilms, the Swedish Governmental Agency for Innovation Systems and (Vinnova) M-ERA.NET Project: Multi-scale Computational-driven design of novel hard nanostructured Coatings – MC2, and the Swedish Research Council are gratefully acknowledged for financial support.

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