Diffusive and massive phase transformations in Ti–Al–Nb alloys – Modelling and experiments

Intermetallics 38 (2013) 126e138

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Diffusive and massive phase transformations in TieAleNb alloys e Modelling and experiments E. Gamsjäger a, Y. Liu a, M. Rester b, c, P. Puschnig d,1, C. Draxl d, 2, H. Clemens e, G. Dehm b, c, 3, F.D. Fischer a, * a

Institute of Mechanics, Montanuniversität Leoben, Franz-Josef-Strasse 18, 8700 Leoben, Austria Department Materials Physics, Montanuniversität Leoben, Franz-Josef-Strasse 18, 8700 Leoben, Austria c Erich Schmid Institute of Materials Science, Austrian Academy of Sciences, Leoben, Austria d Atomistic Modelling and Design of Materials, Montanuniversität Leoben, Franz-Josef-Strasse 18, 8700 Leoben, Austria e Department of Physical Metallurgy and Materials Testing, Montanuniversität Leoben, Franz-Josef-Strasse 18, 8700 Leoben, Austria b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 18 December 2012 Received in revised form 27 February 2013 Accepted 1 March 2013 Available online 29 March 2013

The thermodynamic properties of the TieAleNb system are obtained from recently published thermodynamic assessments. Based on these data the phase boundaries of the (a-Ti þ g-TiAl) two phase region are calculated by utilizing the CALPHAD approach and are compared to those, obtained by ab-initio calculations. It is found that the ab-initio phase boundaries deviate significantly from those based on the CALPHAD fit to experimental data which can be rationalized by the lack of vibrational entropy contributions in the present approach. Consequently a thermodynamic description based on the CALPHAD approach is used to further investigate the kinetics of the massive a / gm phase transformation in the TieAleNb system by means of a recently developed thick-interface model. Simulation of the transformation kinetics results in a massive transformation in the single-phase region only. However, very thin mole fraction spikes are obtained due to comparatively high interface velocities. It is likely that these spikes cannot be fully developed in experiments meaning that diffusion processes are partly suppressed (quasi-diffusionless transformation). A massive transformation in the two-phase region would then be possible. The theoretical predictions are compared to experimental studies performed on a Tie45Ale5Nb alloy (composition in atomic percent). The alloy is heat treated slightly above the a-transus temperature and subsequently oil quenched to room temperature to generate gmea2 interfaces. Energy-dispersive X-ray spectroscopy measurements were performed across gmea2 interfaces in a scanning transmission electron microscope to search for chemical spikes. Ó 2013 Elsevier Ltd. All rights reserved.

Keywords: A. Titanium aluminides, based on TiAl B. Phase transformation B. Thermodynamic and thermochemical properties E. Phase diagram, prediction E. Simulations, atomistic F. Electron microscopy, transmission

1. Introduction Phase transformations occur during processing of industrially relevant materials. The kinetics of these phase transformations define the final microstructure and are thus responsible for the resulting properties. In principle a huge variety of phase transformation may occur in materials and even different classification schemes for phase transformations exist and have their merits [1]. This work is focussed on massive transformations which

* Corresponding author. Tel.: þ43 3842/402 4001; fax: þ43 3842/46048. E-mail address: [email protected] (F.D. Fischer). 1 Present address: Institute of Physics, Karl-Franzens-Universität Graz, Austria. 2 Present address: Physics Department, Humboldt-Universität zu Berlin, 12489 Berlin, Germany. 3 Present address: Max-Planck-Institut für Eisenforschung GmbH, Düsseldorf, Germany. 0966-9795/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.intermet.2013.03.001

constitute a limiting case in the field of diffusive phase transformations. According to Hillert [2], “the massive transformation is defined as the reaction by which a one-phase alloy transforms into a new phase by the growth of blocky or massive grains, such that the whole volume of material may transform”. It is generally accepted that diffusion processes are localized at the interface and its nearest surroundings. Hillert [2] argues that for an “ideal massive transformation local diffusion of atoms inside the migrating interface and in a spike in front of the advancing interface” may occur. On the other hand Massalski [3] states that “due to the lack of experimental evidence of such a compositional spike, and because of predicted spikes that are smaller than 1 interatomic spacing it seems a reasonably safe conclusion to consider composition invariance as an established feature of massive transformations”. Therefore, the motivation of this work is to diminish this apparent discrepancy in defining the characteristics of massive transformations.

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As an example for a massive transformation the features of the transformation from a to gm in Nb-containing TieAl based alloy are investigated in this work by experimental techniques and by modelling. Due to their attractive properties g-TieAl based alloys are considered for high-temperature applications in aerospace and automotive industries. Their advantage is mainly seen in low density (3.9e4.2 g/cm3, depending on composition and constitution), high specific yield strength, high specific stiffness, good oxidation resistance, resistance against “titanium fire”, and good creep properties up to high temperatures [4e6]. From a technological point of view, a massive phase transformation combined with a subsequent heat treatment can be used for the refinement of coarse-grained microstructures and to promote the formation of special types of lamellar microstructures as reported in [7,8]. Detailed information on massive phase transformation mechanisms and nucleation kinetics in TieAl based alloys are reported in [9e13]. In Veeraraghavan et al. [9] it is, for example, found that incoherent nucleation is possible during massive transformations. The thermal treatment necessary to obtain a massive a-to-gm transformation in Tie45Ale7.5Nb alloys is determined based on dilatometric measurements [12]. In addition the orientation relationship between the a and the gm phase is discussed. However, questions related to the details of the growth kinetics during a massive transformation, e.g. the development of the mole fractions of the components during growth, have so far not been addressed. Thus in this work a series of experimental investigations for rapid cooling from the disordered hexagonal a-phase to the tetragonal phase g-TiAl has been performed. Zones of massive transformation from a to gm have been observed. The mole fraction profiles (i.e. the chemical composition profiles) have been analyzed in a small region containing the a2/gm interface (a2-Ti3Al, the ordered form of the a-phase occurs at lower temperatures.) These experiments have been performed by means of energy dispersive X-ray spectroscopy using transmission electron microscopy. In this context numerical simulations based on a recently developed thickinterface model [14] have been carried out to investigate the conditions especially the possible compositional changes during the massive a-to-gm transformation. 2. Material and experimental work 2.1. Material The intermetallic Tie45Ale5Nb alloy (composition in atomic percent) was produced by a powder metallurgical approach. For the powder production, the elemental constituents were alloyed and homogenized by means of a plasma torch in a water cooled copper crucible. The melt was then atomized using the Plasma Inert Gas Atomization (PIGA)-technique. More details about the PIGAtechnique can be found in [15]. The alloy powder has shown the following particle size distribution: d50 w 90 mm, i.e. 50% of the particles are smaller than w90 mm, 85 wt% <180 mm and 19wt% <45 mm. For densification powder of the size fractions 0e180 mm was filled in rectangular Ti-cans, degassed for 16 h at 400  C and finally sealed gas tight. Hot isostatic pressing (HIP) was conducted for 2 h at 1270  C and 200 MPa. Chemical analysis indicated that the composition of both alloy powder and HIPed material match the nominal composition within the experimental error. The mass fractions of nitrogen and oxygen were analyzed in alloy powder as well as HIPed compacts using a conventional LECO melt extraction system. In summary, the impurity levels in nitrogen and oxygen ranged for all sample types around 20e100 mg/g and 370e410 mg/g, respectively. These low levels indicate that the nitrogen and oxygen pick-up during powder handling or one of the subsequent processing steps could largely be avoided [16].

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2.2. Experimental work In order to adjust a microstructure which contains gmea2 interfaces cooling experiments from the single a phase field regions were conducted. To this end samples of 1  1  1 cm3 were cut by a universal precision cut-off machine. The samples were then annealed for 6 min at 1330  C, which corresponds to a temperature slightly above the a-transus temperature (1295  C [17],), and subsequently cooled employing different quenching conditions (water quenching, oil quenching and air cooling). The microstructures of the heat treated samples were observed in a field emission scanning electron microscope (LEO 1525) utilizing back scattered electrons (BSE). Depending on the applied heat treatment different microstructures appear as shown in Fig. 1. The water quenched sample is characterized by a microstructure which consists of supersaturated a2 grains only (Fig. 1a), while oil quenching of the sample leads to a2 grains with a small amount of irregular shaped gm-domains at grain boundaries and triple junctions (Fig. 1b). However, air cooling of the sample leads to a fine lamellar microstructure, consisting of a2 and g laths with embedded gm-domains (see Fig. 1c for the gm-domains). In the following only the oil quenched samples were investigated, since they exhibit the largest amount of gmea2 interfaces, and most probably no diffusion process has triggered the formation of g-lamellae within the supersaturated a2 grains. The local chemical analyses of the Al, Ti, and Nb content in the a2-phase, gm-phase and across the a2/gm interface was performed by energy dispersive X-ray spectroscopy (EDS) using transmission electron microscopy (TEM, Jeol 2100F image-side aberration corrected). The high resolution field emission TEM was operated for the EDS measurements in scanning TEM (STEM) mode using a minimum spot size of 1 nm and a minimal step size of 0.5 nm at 200 kV accelerating voltage. To discriminate between chemical spikes and local chemical inhomogeneities, the Al, Ti, and Nb contents were measured by EDS linescans in the a2- and gm-grains and across the a2/gm-interface. The TEM samples were prepared using two different routes. One set of samples was prepared conventionally by cutting 3 mm disks by spark erosion, grinding to a remaining thickness of w100 mm, dimpling and Ar ion milling at 6 and 3 kV accelerating voltage. The second set of samples was made by focussed ion beam lift-out technique, which allows obtaining specimens site-specific and with a rather constant foil thickness in the electron transparent region. Initially, STEM/EDS linescans were recorded with a step size of 10 nm to detect how homogenous the Al, Ti, and Nb content is within the a2- and gm- grains. Fig. 2 shows a high angle annular dark field STEM image with the positions of 4 linescans, denominated there as sites 0e3, each with a length of 200 nm. As can be seen from the mole fraction profiles presented in Fig. 3aed, the chemical composition is subjected to quite strong variations even within one grain when measurements are performed at different positions. While at site 1 the Al content fluctuates between 45 and 48 at% it is with 42e44 at% lower at site 3. Similar variations are encountered at sites labeled 0 and 2 and as well for the Ti content. The Nb content is in general found to lie between 4 and 7 at% and varies in the linescans at distances over 200 nm by more than 0.5 at %. The chemical inhomogeneities observed in Fig. 2 for a conventionally prepared TEM foil remain similar for FIB prepared TEM samples, see Fig. 4. The corresponding linescans, see Fig. 5aed, reveal chemical fluctuations of 1e2% for Ti and Al within individual linescans. The fluctuations for Nb are slightly smaller than 1%, which means that a possible spike must significantly exceed this value to be detectable. It must be concluded that the measured inhomogeneities in chemical composition most likely stem from the materials synthesis and heat treatment route. The unexpected

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Fig. 2. STEM high angle annular dark field (HAADF) image of a a2/gm-interface. The positions of the EDS linescans are indicated as sites 0e3.

fraction in the a2- and in the gm-phase are within the range of the error bars. The Nb mole fraction is almost constant at 6.5% within the a2- andgm-phase and at the a2/gm-interface. None of the EDS measurements revealed systematic changes at the interface. Thus, it must be concluded that the various analytical STEM studies are devoid of a mole fraction spike at the a2/gm interface. This fact is also supported by the FIB-prepared TEM specimen where no spike at the transformation interface was detected. Similarly as for the conventionally prepared TEM specimen mole fraction changes were found across the grains, although to a somewhat smaller extent. 3. Theory 3.1. Motivation The experimental results outlined in Section 2 demonstrate in detail, specifically with the linescans, unexpected strong chemical fluctuations so that a spike, indicating massive transformation, cannot be localized. The situation is complicated by the fact that the width of the spike may be less than a few nm, which may make it difficult to obtain a high signal to noise ratio for EDS measurements at high spatial resolution. Therefore, a simulation concept is engaged helping to identify possible mole fraction spikes in front of the migrating interface. The necessary steps to provide such a tool are outlined in the next subsections. Fig. 1. SEM images taken in BSE mode of Tie45Ale5Nb after (a) water quenching, (b) oil quenching and (c) air cooling.

strong compositional fluctuations in the a2-grains and gm-phase makes the detection of a possible spike at the a2/gm-interface very difficult, see the discussion in the later sections. In order to average over the chemical fluctuations in the a2- and gm-grains EDS measurements were recorded by scanning areas of 44 nm  4 nm parallel to and at the interface (Fig. 6). The gm phase reveals with 48.9% a slightly higher Al mole fraction compared to the a2-phase with 44.4%. No indication for a mole fraction spike at the interface has been found, as the differences between the mole

3.2. Chemical potentials Two approaches are chosen to obtain the chemical potentials of the components Ti, Al, Nb. The first one follows well-established CALPHAD-method [18] being a fitting procedure based on experimental investigations. The second one is based on ab-initio thermodynamics. 3.2.1. CALPHAD method Analytical expressions for the chemical potentials mai ; i ¼ Ti; Al; Nb for the a-phase (and the liquid phase) can be g taken from [19,20]. The chemical potentials mi ; i ¼ Ti; Al; Nb, can

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Fig. 3. EDS linescans for the sites 0e3 (aed) in the gm and a2 grains and across the a2/gminterface as indicated in Fig. 2. Note the mole fraction fluctuations in Ti, Al and Nb which provide no indication of a clear mole fraction spike occurring only at the interface.

be described by polynomials of second order in the mole fractions xAl and xNb as

mgi ¼ a0 þ a1 xNb þ a2 x2Nb þ b1 xAl þ b2 x2Al þ cxAl xNb

(1)

Fig. 4. STEM bright field (BF) image of a a2/gm-interface prepared by FIB. The positions of the linescans (site 0 and site 1) are located in the bigger red circle, the linescans (site 2 and site 3) in the smaller one. The linescans have been performed perpendicular to the supposed a2/gm-interface. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

A grid of distinct values of the chemical potentials has been extracted from Thermo-Calc [21] using [19,20]. The functions (1) are polynomial fits of these composition dependent chemical potentials, see Appendix A, assessed at a certain constant temperature T. Here, T ¼ 1500 K has been selected and thus all the simulations are performed for this temperature. The coefficients of Eq. (1) and g the according chemical potentials mi are given in Table 1. The chemical potentials obtained from the two dimensional regression analysis approximate the Thermo-Calc values quite well as demonstrated in Fig. 7aec. These fit functions for the chemical potentials in the g-phase, Eq. (1) and the chemical potentials of the components in the other phases, a and liquid (the interface is assumed to be an undercooled liquid) are then implemented into Fortran-subroutines. Thus, it is possible to access the chemical potentials of the components of each phase by a fairly small computational effort. g Since the minimization of the Gibbs energy yields mai ¼ mi , these relations allow to calculate the phase boundaries separating the phases in the according phase diagram. The according procedure is described in Appendix A. The main program, a routine to calculate the transformation kinetics, has to access the values of the chemical potentials of the phases in each time step, so that the actual driving force can be calculated. 3.2.2. Ab-initio methods Density-functional theory (DFT) has evolved into an extremely powerful methodology for calculating materials properties by starting from a first-principles description of the quantum mechanical many-electron problem. Formally, it is based on a theorem by Hohenberg and Kohn from 1964 stating that the ground-state total energy of a system of interacting electrons can be obtained from a non-interacting counterpart in an effective potential [22]. In

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Fig. 5. EDS linescans across the a2/gm-interface of an FIB prepared TEM sample with the linescan positions indicated in the corresponding image shown in Fig. 4. Note the mole fraction fluctuations of 1e2% for Al and Ti, and 0.5e1% for Nb within individual linescans.

the formulation of Kohn and Sham [23], a set of single-particle Schrödinger equations needs to be solved permitting a practical application of DFT. As demonstrated by a vast number of applications of DFT in the past decades, the advent of powerful computers over the past decades has enabled the calculation of materials properties by starting from the knowledge of the atomic numbers of the constituents. However, DFT treats a system in its ground state, i.e., at T ¼ 0 K, while undoubtedly many practical questions in materials science, such as mechanical properties at elevated

temperatures or phase stability of alloys, require a finite temperature treatment. By combining the quantum-mechanical total energy calculations in the framework of DFT and the statisticalmechanical modelling, thermodynamic quantities of alloys, e.g., the Gibbs free energy, can be obtained. Compared to traditional techniques to calculate thermodynamic quantities such as the CALPHAD approach, DFT-based calculations get along without any empirical models and fitting parameters but merely start from knowledge of the atomic structure and chemical composition of the

Fig. 6. (a) STEM bright field image of a a2/gm-interface. The measurement areas are indicated by red areas (“windows”) which are not to scale. (b) The composition obtained by averaging over a rectangular window of 44 nm  4 nm is plotted for windows 1, 2 and 3 (see Fig. 6a for the location of the windows). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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! define the formation energy DEð s Þ of a configuration with respect to the energies of the pure components as

Table 1 Coefficients a0, a1, a2, b1, b2, c of Eq. (1) for T ¼ 1500 K. a0/kJ mol1 a1/kJ mol1 a2/kJ mol1 b1/kJ mol1 b2/kJ mol1 c/kJ mol1

mAl 190.1 mTi 76.45 mNb 185.8

93.05 98.65 562.5

1.136 181.8 79.78 29.89 3391.0 219.2

30.39 174.0 400.4

250.4 208.6 358.9

material in question. In this context, a particularly useful technique in order to obtain phase diagrams of alloys is the so-called cluster expansion (CE) formalism [24]. Details to this formalism can be found in Appendix B. Within this work, we have computed a cluster expansion for the ternary TieAleNb system by making use of the Alloy Theoretic Automated Toolkit (ATAT) [25]. For a ternary TieAleNb system, we

a

-92 x = 0.40

-94

x = 0.41 x = 0.42

µ Ti / kJ mol

-1

-96

x = 0.43

-98

x = 0.44

-100

x = 0.46 x = 0.48

-102

x = 0.49 x = 0.50 x = 0.51

0.00

0.02

0.04

0.06

0.08

0.10

xNb

-1

(2)

where the energy terms are given per atom site. Cluster expansions were computed separately for L10 and hcp lattices. For the structures based on hcp-lattice, the procedure to obtain the CE calculations included the following steps. First, one has to enumerate all possible structures which have maximum first atoms per unit. The obtained results show that totally 57 structures are necessary for the TieAleNb system. Second, one has to calculate the total energies of all the relaxed and unrelaxed structures. In the third step, the overrelaxed structures are removed from the expansion and not considered for the calculation of effective cluster interactions (ECIs). Finally, a multi-component CE is performed only on the structures within a composition range of xAl  0.5 to calculate the ECIs. The results show that totally one zero-point cluster, two one-point clusters, six pair clusters, and six triplet clusters are necessary to reach a satisfying convergence. The cross validation score (CV) value is 0.027 eV.

3.3. Phase diagrams

x = 0.47

-106

µ Al / kJ mol

DEð! s Þ ¼ Etot ð! s Þ  ð1  x  yÞETi  yEAl  xENb

x = 0.45

-104

b

131

-106

x = 0.40

-108

x = 0.41

-110

x = 0.42

The phase boundaries separating the a-hcp phase and the g-TiAl phase in the TieAleNb ternary system at T ¼ 1500 K are calculated on the basis outlined in Section 3.2.1, and are depicted in Fig. 8, top panel, denoted as CALPHAD results. It has been necessary to use the thermodynamic description of [19] for this comparison with the abinitio results, since only the a- and the g-phase have been computed by ab-intitio methods. However, in [20] it is shown that for not too low mole fractions of Nb the beta phase becomes stable.

x = 0.43

-112

x = 0.44

-114

x = 0.45 x = 0.46

-116

x = 0.47

-118

x = 0.48

-120

x = 0.49 x = 0.50

-122

x = 0.51

-124 0.00

0.02

0.04

0.06

0.08

0.10

xNb

c

-120 x = 0.40 x = 0.41

-130

x = 0.42

µ Nb / kJ mol

-1

x = 0.43

-140

x = 0.44 x = 0.45 x = 0.46

-150

x = 0.47 x = 0.48

-160

x = 0.49 x = 0.50 x = 0.51

-170 0.00

0.02

0.04

0.06

0.08

0.10

xNb g

Fig. 7. (a) Values for the chemical potential mTi of Ti in the g-phase, calculated with Thermo-Calc [21] in comparison with the curves obtained by the polynomial fit, Eq. (1). g (b) Values for the chemical potential mAl of Al in the g-phase calculated with ThermoCalc [21] in comparison with the curves obtained by the polynomial fit, Eq. (1). (c) g Values for the chemical potential mNb of Nb in the g-phase calculated with ThermoCalc [21] in comparison with the curves obtained by the polynomial fit, Eq. (1).

Fig. 8. Top panel: calculated phase boundaries as a function of Al and Nb mole fractions between the a-hcp phase and the g-TiAl phase at T ¼ 1500 K obtained from the CALPHAD approach. Bottom panel: phase boundaries between the a-hcp phase and the g-TiAl phase at T ¼ 1500 K (solid lines) and 1300 K (dashed lines) obtained from ab-initio calculations.

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These phase boundaries are also calculated from ab-initio simulations and presented in Fig. 8, bottom panel. For the Monte-Carlo simulations the size of the simulation cell is set to be at least as big as 20 times of the distance of the nearest-neighbor of the interatomic spacing. 1000 steps per site for each pair of (T, mNb) values and a temperature step of 10 K are used to obtain the target precision. The ab-initio simulations are started from low temperature expansion within chemical potential ranges that are close to their values on the stable ground state structures. Compared to the CALPHAD results at T ¼ 1500 K, the ab-initio phase boundaries for both a and g are shifted to lower Al mole fractions, and the two-phase region a þ g appears to be much broader. Since the typical range of accuracy that can be reached with state-of-the-art ab-initio calculations is in the order of a range of few meV e which corresponds to a range of few tens up to hundred Kelvin e we also show the phase boundaries at a somewhat lower temperature, i.e., 1300 K (dashed lines). We observe that both the a- and g-phase boundaries are shifted towards larger Al mole fractions. Nevertheless, the width of the two-phase region remains too large compared to the CALPHAD results, which are thermodynamically reasonable fits and extrapolations based on a huge amount of experimental data, for example see [19]. We attribute the differences between ab-initio and CALPHAD results to neglecting vibrational entropy contributions in our current ab-initio approach. This is corroborated by the fact that the inclusion of vibrational entropy contributions has been shown to improve the agreement with experiment in the binary TieAl phase diagram [26]. Since at the current state the CALPHAD results appear to be more reliable with respect to the experimental situation, we have decided to use the chemical potentials of TieAleNb system from the CALPHAD technique as an input for the transformation kinetics reported below. 4. Modelling the kinetics

be assessed by a main programme, in this work a routine to simulate the aegm transformation kinetics by means of a thickinterface model introduced in [14]. A one-dimensional geometrical setting is selected, see Fig. 9. The spatial coordinate is denoted as z, the left boundary of the system is located at zL, the right boundary at zR. The system consists of three components Ti, Al and Nb and of two phases a-Ti and g-TiAl separated by an interface of finite thickness h. The mole fraction profiles of the components i in the interface are approximated by parabolic functions parameterized by the mole fractions xi at the left side of the interface, the mole fractions xi in the centre of the interface and xi at the right side of the interface. The chemical potentials in the interface change from those valid in the product g-phase at the left side of the interface (z ¼ h/2) to those valid in the parent a-phase at the right side of the interface (z ¼ h/2). The chemical potentials in the centre of the interface I (z ¼ 0) are approximated by the chemical potentials in the liquid phase undercooled to the actual temperature of the system so that the centre of the interface behaves like an amorphous microstructure. A smooth transition of the chemical potentials from the left side of the interface to the right side of the interface via the value in its centre is accomplished by using Hermite polynomials as weight functions (see also [28]). In case of a positive driving force Df, i.e. a transforms to g, the material moves with a velocity v, as depicted in Fig. 9, towards the left side for a spatially fixed interface. Diffusive fluxes jqi of the components i occur in the system with q representing the a-phase, or the g-phase or the interfacial region. A closed thermodynamic system is investigated, thus the diffusive fluxes become zero at the system boundaries zL and zR. As described in [14] the transformation kinetics is determined by the evolution equations of the independent kinetic variables; these are the independent fluxes in the a-phase, the independent fluxes in the g-phase, the rates of the mole fractions x_ iL , x_ iI , x_ iR and the interface velocity v.

4.1. Geometry and thermodynamic situation

4.2. Interface mobility and diffusion coefficients

Ziegler’s principle of maximum dissipation is used to calculate the kinetics of the aegm phase transformation in the TieAleNb system. Details about this thermodynamic extremal principle can, e.g. be found in [27]. As already mentioned the chemical potentials are taken from Section 3.2.2. A Fortran code has been written, so that the chemical potentials of both phases, a-hcp and g-TiAl, can

As processes associated with the migration of an interface are rate-controlling during massive phase transformations, the interface mobility M is an essential input quantity of the model. For processes not too far from equilibrium a linear relationship between the interface velocity v and the driving force Df for interface migration is reasonable,

Fig. 9. Schematic sketch of the geometrical setting.

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v ¼

M $Df Vm

(3)

with Vm being the molar volume of the material estimated as Vm ¼ (0.5MTi þ 0.5MAl)/rTiAl. The quantities MTi and MAl are the molar masses of component Ti and Al, respectively; the density of g-TiAl is denoted by rTiAl. The density of a substance usually depends on temperature. According to the experimental results reported in [4], the change of the lattice constants of the g-TiAl alloys from room temperature to w1500 K is about 1% and rTiAl taken as 3.8  103 kg m3 [29] yielding Vm ¼ 1  105 m3 mol1 (for Nb the same molar volume as for Ti is taken). Veeraraghavan et al. [30] performed electrical resistivity measurements during continuous cooling in a binary TieAl alloy with a mole fraction x ¼ 0.475 and (see also the thermodynamic analysis provided by Perepezko and Massalski [31]) estimated the temperature dependence of the interface velocity during massive a/gmphase transformation as


 Df a/gm Q v ¼ v0 exp  $ RT RT

(4)

with v0 ¼ 2.302  104 m s1, the activation energy for interface transport, Q ¼ 155.25 kJ mol1, and the temperature-dependent driving force for a/gm transformation,

Df a/gm ¼




3712 

2:312T K



J mol1

(5)

The interface mobility for the a/gm-phase transformation follows from Eq. (3) as M ¼ 7.24 1011 m2 s kg-1. The diffusion coefficients in the TieAleNb system are listed in Table 2. Unfortunately, the values of the diffusion coefficients in the a-phase are not well-known. Experimental data for these tracer diffusion coefficients in a-Ti are obtained up to 1373 K as this is the stability limit of this phase. However, the calculations are performed at a temperature of T ¼ 1500 K, i.e. the experimental values have to be extrapolated over a temperature range of more than 100 K. The extrapolated values have been used in the calculations. There are no data available for the diffusion coefficients in the interfacial region. However it is likely that the diffusion coefficients in the interface are high compared to those in the resulting product phase (g), when the lattice is already rearranged. The diffusion D coefficient is given as D ¼ D0exp(QA/RT) with RT ¼ 12.471 kJ mol1 for 1500 K.

133

This approach is also motivated by the fact that the composition in the polycrystalline samples is not uniform, so that locally different processes might occur during the transformation. Therefore, the kinetics of the a / g phase transformation is at T ¼ 1500 K studied for different initial compositions denoted as:  Case A: xTi ¼ 0.5, xNb ¼ 0.05 (which corresponds to the composition of the specimen, at 1500 K this composition lies in the g/a-two phase region; g-TiAl becomes more stable at lower temperatures).  Case B: xTi ¼ 0.475, xNb ¼ 0.05 (composition in the a/g-two phase region).  Case C: xTi ¼ 0.465, xNb ¼ 0.05 (composition in the g-one phase region). These compositions are marked in the isopleth depicted in Fig. 10. The following initial data concerning the geometry and the phase arrangement are assumed. The initial volume fraction x, defined as the ratio of the volume of the g-region to the total volume, is 0.3. The length of the system is set to 100 nm. The thickness of the interface is set to h ¼ 0.5 nm. The values of the thermally activated diffusion coefficients are specified in Table 2. The interface mobility is also provided in Section 4.2. 4.4. Modelling the transformation kinetics The transformation kinetics has been studied for the three different initial compositions case A, case B and case C mentioned above. Mole fraction profiles are plotted for various transformation times using, for sake of easier reading, a coordinate system ~z with the origin fixed to the left side of the interface. The evolution of the interface velocity during transformation has also been analyzed. 4.4.1. Case A For initial mole fractions xTi ¼ 0.5, xNb ¼ 0.05 in the bulk phases, an initial mole fraction xTi,I ¼ 0.51, xNb,I ¼ 0.045 in the center of the interfacial region and an initial volume fraction x ¼ 0.3 a transformation from g to a occurs at T ¼ 1500 K until the system is equilibrated, i.e. the interface velocity is negative (Fig. 11c). The mole fraction profiles of Ti and Nb are plotted for the initial situation (t ¼ 0 s) and at several transformation times in Fig. 11a and b, respectively. Neither depletion nor segregation of Ti in the

4.3. Initial configuration This section is devoted to study the type of transformation for systematically changing the mole fraction of the initial composition from the two-phase region to the one-phase region.

Table 2 Diffusion coefficients in the TieAleNb system and in the interface. D0 (m2 s1) Ti in g phase Al in g phase Nb in g phase Ti in a phase Al in a phase Nb in a phase Ti in interface Al in interface Nb in interface a

6

1.5  10 2.1  102 1.5  104 1.35  103 6.6  103 1.1  104 1.5  103 2.1  101 1.5  101

No experimental data are available.

QA (kJ mol1)

Ref.

250 360 324 303 329 401 303 329 401

[32] [32] [33] [34] [34] [35] a a a

Fig. 10. Isopleth of the TieAleNb ternary system at xNb ¼ 0.05 based on database given in [20]. The initial mole fractions xTi are the same in both bulk phases. Full triangle (case A) and open square (case B) represent the two compositions in the two-phase region, full circle that in the single-phase region of g-TiAl.

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Fig. 11. (a) Ti-mole fraction profiles at increasing transformation times for case A: initial composition: xTi ¼ 0.5, xNb ¼ 0.05. The vertical lines separate the interfacial region I from the bulk phases a and g. (b) Nb-mole fraction profiles at increasing transformation times for case A: initial composition: xTi ¼ 0.5, xNb ¼ 0.05. The vertical lines separate the interfacial region I from the bulk phases a and g. (c) Interface velocity v as function of time t for case A: initial composition: xTi ¼ 0.5, xNb ¼ 0.05.

interfacial region is observed; the Ti profile in the interface is almost linear (Fig. 11a) and the depletion of component Nb in the interface is stabilized during the transformation (Fig. 11b). 4.4.2. Case B Similar as in case A, a bulk diffusion controlled transformation process, at least with respect to the components Ti and Al, occurs also in case B: xTi ¼ 0.475, xNb ¼ 0.05, xTi,I ¼ 0.485, xNb,I ¼ 0.045, x ¼ 0.3 and T ¼ 1500 K. However, the interface velocity becomes positive at 1500 K (Fig. 12c), i.e., a transformation from the a-phase to the g-phase is observed. The mole fraction profiles of Ti and Nb are plotted for the initial situation (t ¼ 0 s) and at several transformation times in Fig. 12a and b, respectively. The Ti-profiles in

Fig. 12. (a) Ti-mole fraction profiles at increasing transformation times for case B: initial composition: xTi ¼ 0.475, xNb ¼ 0.05. The vertical lines separate the interfacial region I from the bulk phases a and g. (b) Nb-mole fraction profiles at increasing transformation times for case B: initial composition: xTi ¼ 0.475, xNb ¼ 0.05. The vertical lines separate the interfacial region I from the bulk phases a and g. (c) Interface velocity v as function of time t for case B: initial composition: xTi ¼ 0.475, xNb ¼ 0.05.

Fig. 12a show that component Ti partitions during the transformation process so that after a sufficient transformation time the profiles in the bulk phases are flat. The mole fractions will finally reach their equilibrium value in both phases. No partitioning with respect to the component Nb is necessary, see Fig. 12b. The interface velocity decreases until the transformation process stops when equilibration is reached (Fig. 12c). 4.4.3. Case C The a-to-gtransformation behaviour changes, if the initial composition is in the g-one phase region with an initial mole fraction; xTi ¼ 0.465, xNb ¼ 0.05, xTi,I ¼ 0.475, xNb,I ¼ 0.045, and an initial volume fraction x ¼ 0.3. After a certain transformation time a steady state is obtained. Thus the mole fraction profiles for all

E. Gamsjäger et al. / Intermetallics 38 (2013) 126e138

components do not change after this transformation time is reached. Ti-mole fraction profiles for t ¼ 0 s and at various transformation times are plotted in Fig. 13a, where a large part of the system is shown and Fig. 13b, which presents the details in the interfacial region and its vicinity. Nb-mole fraction profiles for t ¼ 0 s and at various transformation times are plotted in Fig. 13c. It is worth mentioning that Ti required to segregate in the interface and build a spike in front of the migrating interface is taken from the g-TieAl phase according to mass balance. After an initial transition period both the Ti- and the Nb-mole fraction profiles become stationary. The interface velocity converges to a constant value when the system runs into steady state (Fig. 13d). 4.5. Discussion of modelling results Three different cases have been investigated. For an initial composition in the two-phase region (case A and case B) the system evolves until it is equilibrated. The driving force is negative for case A, g-TiAl transforms to a. With a positive driving force, as available in case B, the a-phase transforms to g-TiAl. In case that the initial composition lies in the a-one phase field as in case C, a massive transformation becomes possible. By a smaller amount of xTi the driving force for the a/g-TiAl transformation is increased. Similarly the driving force for this transformation can also be increased by lowering the transformation temperature T. A rather small spike in the Ti-mole (Fig. 13b) occurs in the a-bulk material (on the right side of the interface, which is marked by two dotted lines). Its width at half

135

maximum is smaller than 0.5 nm, and at the top of the spike the mole fraction is 0.05 higher than at the bottom. For an increasing driving force and thus an increasing interface velocity v, the width di of the spikes of the components become even smaller during a massive phase transformation as di being proportional to Div1 (see e.g. [45].) assuming constant diffusion coefficients Di at a certain temperature. Changing the diffusion coefficients in the interface does not seem to have a significant influence on the kinetics as long as these diffusion coefficients are higher than the bulk values, since the mole fractions at the interface and the driving force remain almost unchanged. By increasing the thickness of the interface, however, dissipation due to the diffusion processes in the interface can be significantly retarding the transformation kinetics. The latter point has been demonstrated for the kinetics of the g/a transformation in the FeeNi system (see [46]). 5. Comparison between experimental and modelling results The massive transformations from the a- to the gm-phase observed in the experiments (see Section 2.2) occurred during relatively high cooling rates at high temperatures. The samples, however, have been investigated at room temperature with the consequence that the mole fraction profiles might deviate from the in-situ high temperature situation. However, due to high cooling rates and a slow substitutional diffusion mechanism, one can expect that the high temperature mole fraction profiles are preserved. We also face limitations with respect to modelling the massive transformation. High cooling rates down to room

Fig. 13. (a) Ti-mole fraction profiles at increasing transformation times for case C: initial composition: xTi ¼ 0.465, xNb ¼ 0.05. Representation of a large part of the system. (b) Timole fraction profiles at increasing transformation times for case C: initial composition: xTi ¼ 0.465, xNb ¼ 0.05. Detailed view of the interface and its surroundings. (c) Nb-mole fraction profiles at increasing transformation times for case C: initial composition: xTi ¼ 0.475, xNb ¼ 0.05. (d) Interface velocity v as a function of time t for case C: initial composition: xTi ¼ 0.475, xNb ¼ 0.05.

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temperature drive the system to a state far from equilibrium. As long as linear non-equilibrium thermodynamics is valid the evolution of the system can be modelled in principle, but the computational demands prevent a solution. However, with increasing deviation from equilibrium conditions at the interface it becomes more and more difficult to model the kinetics of diffusive phase transformations in such systems due to the fact that the interfacial reactions and the sluggish bulk diffusion occur on completely different time scales. It is, however, possible to simulate the a-to-g transformation at a high temperature (T ¼ 1500 K) and investigate the system for different initial compositions of the material. This concept has been followed in the current paper. The relatively small changes in the initial composition (Cases A, B and C see Figs. 10e13) lead to different mole fraction profiles at the a/g interface and even to a change in the transformation direction. These small changes in composition are in the range of the chemical inhomogeneities observed by STEM-EDS measurements for the investigated Ti45Al5Nb alloy. In accordance with the modeling results the experimentally observed mole fraction profiles during cooling might result from locally different transformation mechanisms with the transformation rate being controlled by the interfacial reaction or by diffusion in the bulk phases. Pronounced mole fraction spikes that might occur at high temperature, should also be detectable at room temperature. The simulated Ti-mole fraction profiles, however, are rather small and the half width of these spikes will further decrease during cooling, as the diffusion coefficient decreases and the interface velocity increases; di being proportional to. According to the simulation results it is thus unlikely that mole fraction spikes formed during massive a/gm transformation in Ti45Al5Nb will be observed. In case that narrow mole fraction spikes (in the sub-nm-range) are obtained in front of the migrating interface as suggested by modelling results it is reasonable that such spikes do not appear in the real material during a massive transformation. It is likely that such a sharp mole fraction spike does not occur but is partly suppressed which has already been concluded by Hillert [36], and describes the situation of the Ti45Al5Nb alloy investigated here by STEM-EDS. However, segregation to the interface is realistic and yields a solute drag effect. 6. Conclusions and outlook Ab-initio methods have been used to model the phase boundaries of the a-hcp phase and the g-TiAl phase in the ternary TieAleNb system. The results have been evaluated by CALPHAD-based calculations from recently assessed thermodynamic data. This comparison has demonstrated that a quantitative prediction of the phase boundaries in such a complex system based on ab-initio methods requires vibrational entropy contributions to be incorporated. Unfortunately, the full ab-initio treatment of configurational as well as vibrational free energy contributions is computationally still quite challenging and was beyond the scope of the present work. Therefore, the CALPHAD-method was used to calculate the chemical potentials as functions of composition and temperature. For the ordered TiAl phase it was necessary to use a generalized polynomial regression analysis to fit Thermo-Calc data and obtain comparably simple polynomials for the chemical potentials of the components in this phase. The transformation kinetics of the massive a / gm phase transformation has been calculated by means of a thick-interface model [14]. The simulations show that Ti-mole fraction spikes occur during the massive transformation, which are very small, i.e. their width is smaller than 0.5 nm in front of the migrating interface. Segregation in the migrating interface and solute drag phenomena are likely to occur during the a-hcp to gm-TiAl phase

transformation. These thin mole fraction spikes in front of the massively transforming material have not been observed experimentally. On the one hand the theoretically found spikes are at or even below the limit of experimental resolution due to the unexpected large fluctuations in chemical composition within the individual grains. On the other hand the sluggish bulk diffusion process of the substitutional components could be partly suppressed as being frequently the case during the austenite-to-ferrite transformation in low alloyed steels, see e.g. Hillert’s textbook [36] and the references therein. From a theoretical point of view a transformation with the composition in the two-phase region has to equilibrate after a sufficiently long transformation time [37] and therefore near steady state conditions cannot be preserved ad infinitum. Consequently, a massive transformation in the two-phase region e as it has been observed experimentally (an example for the a-to-gm transformation in TieAl based alloys can be found in Ref. [30]) e is only possible when the slow process of substitutional bulk diffusion is at least partly suppressed. This means that for high enough driving forces the interface will migrate, even if the solubility limit of a certain component in the product phase is exceeded. In this sense the diffusional phase transformation partly becomes a displacive transformation (e.g. carbon diffusion is suppressed during a martensitic phase transformation in the FeeC system). The massive transformation could first occur at sites which facilitate nucleation and with a composition located in the thermodynamic one-phase region of g-TiAl. The migrating front, when sufficiently fast, will enter a micro-region with a composition in the two phase region below the T0-line in a massive way, and the interface can continue to migrate without building up mole fraction spikes. Acknowledgements The authors appreciate the funding by Austrian FWF under the project “Massive Transformation e Experiments and Simulations”, Project number P20709-N20. Appendix A. Classical thermodynamics The equilibrium conditions of a thermodynamic system can be found by calculating the extremum value of an appropriate thermodynamic potential. In case of a closed system at constant pressure and at constant temperature the molar Gibbs energy g of the system has to be minimized. The molar Gibbs energy gF of phase F usually consists of a Gibbs energy term gF,id, according the ideal solution model and a term gF,eaccounting for the excess Gibbs energy,

gF ¼ g F;id þ gF;e

(A.1)

The Gibbs energy terms gF,id and gF,e are temperaturedependent functions of the mole fractions xF ðk ¼ 1; .; nÞ of the k

components k. The parameters of these special functions are fitted to experimental data by the CALPHAD-method [18]. For a given Gibbs energy gF the chemical potentials mF i of phase F are calculated as follows. The chemical potential mF 1 of component 1 (frequently called solvent) follows as

mF1 ¼ gF 

n X k¼2

xF k

vgF vxF k

(A.2a)

and the chemical potentials of the other components (solutes) are obtained by

E. Gamsjäger et al. / Intermetallics 38 (2013) 126e138

mFi ¼ gF þ

n X vg F  xF k F vxi vxF k¼2 k

vg F

Appendix B. Cluster expansion (CE) formalism

(A.2b)

if the site fraction of lattice vacancies yVa is small and can be neglected. A treatment, where lattice vacancies are considered, can be found in [38]. According to the thermodynamic assessment of Witusiewicz et al. [19] the ternary TieAleNb system consists of 18 phases. These phases can be classified as disordered solution phases, ordered intermetallic phases and stoichiometric phases according to Kattner et al. [39]. The Gibbs energy ga of the a-Ti phase, a disordered solution phase, and the Gibbs energy gg of the g-TiAl phase, an ordered intermetallic compound with a certain range of homogeneity, have to be determined in this work. Compared to the thermodynamic description of the disordered solution phases accurate modelling of the ordered intermetallic compounds requires additional efforts. Kattner et al. [39] employ two sublattices where the components Ti, Al and in the ternary case also Nb can occur in both sublattices. The equilibrium mole fractions of the components on each sublattice have to be calculated by minimizing the Gibbs energy of the g-phase, gg, for a given composition. The Gibbs energy of the g-phase as a function of composition at a certain temperature is obtained by minimizing the Gibbs energy with respect to the independent compositional variable between steps where the composition is changed incrementally. For the ternary TieAleNb-system this procedure becomes more complicated also from a numerical point of view, and we were not successful with MAPLE [40]. However, it is possible to calculate the chemical potentials for the g-phase for a certain composition by means of Thermo-Calc [21] utilizing the thermodynamic assessment provided in [20], which is based on the prior assessment [19]. The chemical potentials of Al, Ti and Nb in the g-phase are calculated in the compositional range of interest, i.e. (0.4  xAl  0.5) and (0.01  xNb0.1) point by point changing the mole fraction xAl or xNb by 0.01 between each point. The obtained chemical potentials are fitted by the following nonlinear function with the mole fractions xAl and xNb as independent variables (see also Eq. (1)):

mi ¼ a0 þ a1 xNb þ a2 x2Nb þ b1 xAl þ b2 x2Al þ cxAl xNb ; with i replacing Ti; Al or Nb

ðA:3Þ

It should be pointed out that the unknown parameters a0, a1, a2, b1, b2, and c of Eq. (A.3) have no physical meaning but are utilized to describe the chemical potentials as functions of composition in the mole fraction range mentioned above. The unknown coefficients a0, a1, a2, b1, b2, and c are found by a generalized polynomial regression analysis using the software tool GNUPLOT [41]. As soon as the Gibbs energies ga and gg are known, the phase boundaries can be calculated by minimizing the Gibbs energy g of the system

g ¼ xg a þ ð1  xÞgg

(A.4)

with x denoting the extent of reaction constrained by mass balance equations. As a result of this Gibbs energy minimization it follows that the chemical potentials are the same for each component i,

mai ¼ mgi

137

(A.5)

Thus, a system of non-linear equations with respect to the mole fractions has to be solved to obtain the (a/a þ g/g)-phase boundaries. This problem has been solved for the binary TieAl system by symbolic computation aided by MAPLE [40].

! We define the formation energy DEð s Þ of a binary alloy char! acterized by the configuration vector, s , in the following way as

DEð! s Þ ¼ Eð! s Þ  ½xA EA ð! s Þ þ xB EB ð! s Þ

(B.1)

! ! here, Eð s Þ is the total energy in configuration s , xA and xB represent the mole fractions of elements A and B, respectively, and ! ! EA ð s Þ and EB ð s Þ are the total energies of a system with component pure A and pure B, respectively. Sanchez et al. [24] have shown that the formation energy of any alloy configuration may ! be expanded in terms of so-called cluster functions Fa ð s Þ, and the effective cluster interactions (ECI), Ea. It should be pointed out that ! Ea are scalar variables and are independent of the configuration s yielding

DEð! sÞ ¼

X

! Ea Fa ð s Þ

(B.2)

a

where a ¼ 1, 2, 3. denote the clusters, as pair interactions such as 1st, 2nd, etc. nearest neighbor pairs, or triplet interactions including three sites and so on. Within the CE formalism the ele! ments si of the configuration vector s are called occupation variables. For a binary system, si takes the values þ1 (atom A) or 1 (atom B), depending on the type of atom occupying site i. Note that a multi-component cluster expansion can be treated as a direct extension of the binary CE. The main difference between the binary CE and the multi-component CE is that in the multi-component CE the different sites of the lattice should be allowed to host arbitrary number of elements instead of only two kinds of atoms as being defined in binary CE. Therefore, the elements si could take any ! value in Fa ð s Þ from 0 to Mi  1 instead of 1, where Mi is the components number that are hosted on site i. ! While the cluster functions Fa ð s Þ appearing in Eq. (B.2) are simply given by products of the occupation values si over a certain cluster, the effective cluster interactions, Ea, are calculated with an ab-initio DFT framework. In the so-called structure inverse method, the ECIs are obtained by computing the formation energies for a set of ordered alloy configurations and mole fractions XA and XB. By employing sophisticated algorithms [25], a set of clusters and corresponding ECIs is determined which minimizes the sum of least squares differences between the ab-initio formation energies and the cluster expansion representation of Eq. (B.2). With a converged cluster expansion at hand, one can then compute total energies of arbitrary alloy configurations in an extremely efficient manner. In order to compute the alloy configuration for given external thermodynamic parameters, such as temperature and mole fractions xi (or chemical potentials), one utilizes the lattice gas MonteCarlo (MC) method. Here, the formation energy of an arbitrary alloy configuration, defined in a simulation cell with given chemical composition (or chemical potentials), is calculated from the cluster expansion as described in the previous section. Other parameters, for instance, the simulation cell size or the target precision, are treated as convergence parameters. The grand-canonical ensemble is employed to perform the MC simulations as implemented in the ATAT package [25]. It should be stressed that in this ensemble, the energy and mole fraction of an alloy with a fixed total number of atoms (N) are allowed to fluctuate while temperature and chemical potentials are externally imposed. The grand canonical potential 4 as a function of the reciprocal temperature b ¼ 1/kBT and the chemical potential m is given by

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E. Gamsjäger et al. / Intermetallics 38 (2013) 126e138

1

fðb; mÞ ¼  ln bN

X

! expð  bNðEi  mxi ÞÞ

(B.3)

i

Here, xi is the mole fraction of the component A at state i. First one needs to calculate the total differential of the product between reciprocal temperature and the grand canonical potential

dðbfÞ ¼ ðE  mxÞdb  bx dm

(B.4)

In this expression, E and x are the system’s average internal energy (per atom) and mole fraction of component A. From Eq. (B.4), we can obtain the thermodynamic function 4(b, m) by integration as

fðbnþ1 ; mnþ1 Þ ¼ fðbn ; mn Þ þ

1

ðbnþ1 Z ;mnþ1 Þ

ðE  mx; bxÞ$dðb; mÞ

b ðbn ;mn Þ

(B.5) where the integral is performed along a continuous path joining points (bn, mn) and (bnþ1, mnþ1) that does not encounter a phase transition. The quantities in the integral can be readily obtained from a Monte-Carlo simulation at a given temperature and chemical potential. In order to define a starting point for the grand canonical potential exact expressions valid either at high or low temperature are used. Once the grand canonical potential 4 has been obtained from the integration of Eq. (B.4) the Helmholtz free energy F is defined as

F ¼ fþ

X

mi xi

(B.6)

i

DFT calculations are performed by using the VASP code [42,43]. The projector-augmented wave method [44] has been used to treat coreeelectron interactions, and the generalized gradient approximation (GGA) [45] has been employed to take into account exchange-correlation effects. Formation energies are numerically converged to approximately 1 meV/atom (w1 J/mol) using a plane wave cutoff of 350 eV and dense k-meshes corresponding to at least (14  14  14)/Natoms k-points in the first Brillouin zone, where Natoms is the number of atoms in the supercell. Gaussian smearing of 0.1 eV was used for relaxation runs, followed by the highly accurate tetrahedron method with Blöchl corrections for the final relaxed geometries. Since the external pressure is zero, the calculated formation energies are the same as the formation enthalpies.

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