Prediction of the kinetics of the phase transformations and the ...

J. Phys. IV France 120 (2004) 93-101 Ó EDP Sciences, Les Ulis DOI: 10.1051/jp4:2004120010

Prediction of the kinetics of the phase transformations and the associated microstructure during continuous cooling in the Ti17 J. Da Costa Teixeira1,2, L. Héricher1, B. Appolaire1, E. Aeby-Gautier1, G. Cailletaud3, S. Denis1 and N. Spath2 1

LSG2M – UMR CNRS/INPL/UHP, École des Mines, Parc de Saurupt, 54042 Nancy Cedex, France 2 Snecma Moteurs, YKOG2, 291, avenue d’Argenteuil, BP. 48, 92234 Gennevilliers Cedex, France 3 UMR CNRS 7633, Centre Des Materiaux, École des Mines de Paris, BP. 87, 91003 Evry Cedex, France Abstract. The aim of this paper is to present recent experimental results and related simulation about the b ® aGB + aWGB and b ® aWI transformations which occur in the Ti17 alloy during the thermal treatments following the heating in the b phase field. These phase transformations were experimentally studied under isothermal conditions in samples with negligible thermal gradients. The IT diagram was obtained, on the basis of electrical resistivity measurements and microstructural SEM observations. The kinetics of the phase transformation was further numerically simulated for continuous cooling on the basis of a formerly developed model giving the amount of each morphology (aWGB, aWI). Experimental and calculated results are compared.

1. INTRODUCTION The mechanical properties of Ti17 titanium alloy, as well as other metastable beta titanium alloys are strongly dependent on the microstructure, namely the nature and amount of the phases, their morphology and the way they are distributed [1]. Thus the prediction of the microstructure evolution and their final distribution associated with any thermomechanical treatment would help with the process control as well as with the prediction of the mechanical properties. The relationships between the microstructure evolution and the thermomechanical treatment parameters have been widely studied for several titanium alloys [2-4]. For Ti17 (Al 5%, Mo 4%, Cr 4%, Zr 2% and Sn 2%) only few data are available [5-6]. We have thus performed experiments in order to characterise the experimental transformation kinetics, on the basis of the experimental knowledge gained from previous studies made on a+b titanium alloys. During the cooling from the high temperature b phase (BCC) field, two main precipitation mechanisms of the a (HCP) phase can occur [2]. At high temperatures, the following sequence is observed b ® aGB + aWGB +benriched. First, aGB nucleates only on the most favourable sites which are the grain boundaries and grows by diffusion. Then, colonies constituted of parallel lamellae of aWGB (a Widmanstätten morphology) with only one crystallographic orientation nucleate on aGB. The growth of aWGB is diffusion controlled. In isothermal transformation conditions, the ratio between the a phase amount and the enriched b phase, in the aWGB +benriched colony, is nearly the ratio between the a and b phase at equilibrium for the given transformation temperature. At lower temperatures, b ® aWI + benriched occurs on intragranular nucleation sites. A more displacive character could be associated with the transformation. The nucleation can occur on less favourable sites as the dislocations, because

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of the higher driving force. The orientation relationships between b and both aWGB and aWI are Burgers ones, that is:

[110 ]b //[0001]a

and 111

-

b

// 11 2 0 a

The calculation of the phase transformation kinetics during continuous cooling is based on the modelling of the transformations in isothermal conditions and the hypothesis of linear cumulation. Such models have been extensively developed for the calculation of transformation kinetics in steels [7]. These models require the experimental determination of the transformation kinetics and the microstructural analysis for isothermal treatments. In our study, the isothermal precipitation kinetics were studied experimentally by in-situ electrical resistivity measurements because dilatometry is not accurate enough to determine the transformation kinetics. The obtained results are modelled using the Johnson – Mehl – Avrami (JMA) law. Additional experiments were also performed under continuous cooling and compared with numerical simulations. 2. THE TRANSFORMATION KINETICS MODEL Our main goal is to develop a numerical tool able to predict the microstructural evolutions associated with the b ® a transformations during continuous cooling following the forging in the b field. In the present case, we consider an unstrained alloy of a given grain size with uniform temperature. The model used is based on a description of the transformation kinetics using JMA laws, developed and tested previously on the b-cez alloy [4]. During the cooling from the single b phase temperature range, transformation only occurs at temperatures lower than the equilibrium temperature (Tb). For a quench to a constant temperature lower than Tb, the transformation starts after an incubation time, which depends on the temperature. This incubation time is necessary to form the a phase nuclei. Further nucleation and growth of the new phase is described by the JMA law. For continuous cooling, we have to describe the two sequences of incubation and further nucleation and growth. Moreover two distinct transformation kinetics have to be considered. Indeed, as two main mechanisms of transformation are observed, one has to consider how they interact. The assumptions of this model and the way the transformation progression is calculated are recalled. Assumptions The mechanisms of transformation of aWGB and aWI are independent. Indeed the distance of diffusion ahead of the reaction front of the lamellae of aWGB, has been measured in the b-cez alloy at different transformation temperatures. It was shorter than 20 µm at a temperature of 830°C and shorter than 3µm at 750°C [4] (the grain size is about 200 µm). Thus, during a continuous cooling treatment, the aWI precipitates in b phase with the initial chemical composition of b phase in the present case, the nominal chemical composition of the alloy. For each transformation sequence, the incubation time can be calculated using a linear cumulation rule (i.e. additivity rule). The variable characterising incubation, g, is defined through its rate as : ·

g=

1 t d (T )

(1)

With td the incubation time at temperature T. Its initial value is zero. The end of the incubation period corresponds to g = 1, so that the time needed ti is given by : 1

ò

0

ti ·

ti

0

0

dg = ò g dt = ò

dt =1 t d (T )

(2)

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This formulation, which is convenient for performing numerical integration, is the continuous form of the well known sum of Scheil rule [8]. For each transformation sequence, the linear cumulation rule works also during nucleation and growth, i.e., at any time, the transformation rate is a state variable depending only on the temperature and the a phase amount (i.e. the untransformed b phase amount). It is independent on the thermal path [9]. ·

z = f (T , z )

(3)

With z the amount of a phase and T the transformation temperature. The kinetics of the isothermal transformation is described by the JMA law Several studies on titanium alloys have shown that the JMA law is able to describe the experimental isothermal transformation kinetics. [4-3,11-14]. The particular case of simultaneous transformations has already been treated [4,15,16]. In order to calculate the transformation progress we have to consider that, at the beginning of the transformation, the transformed volume, containing new α phase and enriched β phase, can be expressed as [8]:

Vi e = Vk i (t - t di ) i n

(4)

Vie

the extended volume of the a phase with morphology i (either aWGB or aWI), V the With volume to be transformed, ki and ni the coefficients of the JMA law and tdi the incubation time. This volume, called the extended volume, describes the nucleation and growth of the new mixture (a+b enriched) without taking into account the impingement. The true increase of volume needs to consider an impingement factor taken as the fraction of untransformed volume.

æ å Vi ç dVi = ç1 - i V ç è

ö ÷ e ÷dVi ÷ ø

(5)

With Vi the transformed volume of the morphology i, including enriched β phase. Let us define Vi* such as :

æ V* ö dVi * = çç1 - i ÷÷dVi e V ø è

(6)

Considering equation (6), we can see that Vi* represents the transformed volume of the morphology i if there was a single transformation mechanism (i.e. Vj=0 for j¹i). Then, from (5) and (6) :

1 dVi * æ Vi * ö ÷÷ni k i ni = çç1 V V ø è

é æ ê ç 1 êlnç * ê ç Vi ê çè 1 - V ë

öù ÷ú ÷ú ÷ú ÷ú øû

ni -1 ni

dt

(7)

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and :

æ å Vi ö ç ÷ i ç1 ÷ V ÷ ç dVi è dVi* ø = V æ Vi * ö V ç1 ÷ ç ÷ V è ø

(8)

The evolution of the transformed volume is obtained by integrating equations (7) and (8). Under isothermal conditions, the a phase amount inside the transformed volume is the equilibrium one. Thus:

dz i =

dVi z eq (T ) V

(9)

With zi the amount of a phase of morphology i and zeq(T) the equilibrium a phase amount at the isothermal transformation temperature considered. If we consider anisothermal treatment conditions, we will have to take into account the strong dependence of zeq on temperature in the Ti17 (see § 3.2). Indeed, the volume Vi transformed at temperatures higher than the current temperature T will have an amount (z) of a phase lower than the equilibrium one at the current temperature. The a phase amount inside the transformed volume will tend towards the equilibrium one mainly by diffusional processes. Thus there will be two origins for the increase of a phase amount: i) the growth of the transformed volume Vi (in the untransformed one), whose associated increase in a amount is given by equation (9), and ii) the diffusion inside the transformed volume Vi. This result can be obtained as followed:

ö dz i 1 d æç ÷ = y dv i ÷ dt V dt çè Dòi ø

(10)

With yi the local a phase amount inside the transformed volume and Di the domain including the transformed volume. Thus, using the Leibniz rule :

dz i 1 = dt V

dy i 1 dv + dt V Di

ò

ò y uds * i

(11)

Si

With Si the moving surface of the domain Di, yi* the α phase amount at the interface between the transformed volume and the non transformed β matrix and u the velocity of the interface. Considering an average value for the α phase amount inside the transformed volume, , and the equilibrium α phase amount everywhere at the interface,

dz i Vi dy i dV 1 = + z eq (T ) i dt V dt V dt

(12)

If the diffusion is quick enough, the α phase amount of the transformed volume reaches immediately the equilibrium one. For the further calculations, we have made this assumption. Thus :

zi V = i z eq (T ) V

(13)

Then the amount of each morphology is obtained by integrating (7), (8) and (13). There are 7 material parameters whose value has to be determined between 400 and 800°C : tdi, ki, ni and zeq. The following section deals with the determination of these data.

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3. EXPERIMENTAL MEASUREMENTS 3.1 Experimental procedure The initial microstructure of the Ti17 samples contains equiaxed a grains in a b matrix. The samples were cylinders 30 mm long and 3 mm in diameter. Oxidation during thermal treatments was controlled by working within a secondary vacuum (5. 10-5 mbar). Contamination by oxygen was limited to 50 µm from the surface. The first sequence of the thermal treatment consisted in a solution treatment in the b field at 920°C for 30 min. The heating rate from room temperature to 920°C was 10°C.s-1. This treatment was followed by either quenching to isothermal transformation temperature or continuous cooling. For isothermal treatments, the cooling was obtained by blowing gas helium to the isothermal treatment temperature. The time required to this cooling was less than 5 seconds. The device used to achieve the thermal treatments made possible the limitation of thermal gradient to 2°C [4]. The b grain size after homogenizing treatment is 200 µm. 3.2 Determination of the equilibrium data The transus temperature (Tb) was characterised by dilatometric and electrical resistivity changes associated to the a + b ® b transformation during the heating from the initial microstructure. These measurements have then been compared to results of microstructural observations made by the supplier of the alloy. After these comparisons, Tb of the Ti17 was taken as equal to 880°C. The variations of equilibrium a and b phase amounts and chemical compositions were determined at various temperatures after long isothermal treatments. As shown in Figure 1, the values of zaeq increase with decreasing temperature. The observed variations are similar to the ones obtained for the b-cez alloy [2-4]. zaeq was also calculated with the software Thermocalc, with the Saunder’s database, and correcting the chemical composition of the a phase which is almost constant [17]. The results are in good agreement with the measured ones in the range of temperature investigated. 3.3 Isothermal kinetics of transformation The studied temperature range is 450-800°C with a step of 50°C between each isothermal treatment. The evolution of the a phase amount during isothermal treatments has been followed by in-situ electrical resistivity measurements. The electrical resistivity is sensitive to the phase transformation [2-4,11-14]. Assuming that the isothermal electrical resistivity variation is proportional to the a phase amount, we determined the incubation time and the evolution of the global a phase amount, normalised to its value at the end of transformation as a function of temperature. The TTT diagram of the Ti17 alloy was established, giving, as a function of the temperature of isothermal transformation, the incubation time and the times for 10 and 90% of transformation. For each temperature we verified that the kinetics can be described by the JMA law, and determined the k and n parameters. The n values are between 1 and 2. These kinetics measurements have been completed with observations of the microstructure of the alloy at the end of the transformation by optical and scanning electron microscopy, after helium quenching to room temperature. The different morphologies were observed in the following temperature ranges : between 800 and 750°C, b ® aGB + aWGB. Under 700°C, we obtained b ® aWI. Between 750 and 700°C, the two transformation mechanisms are competitive. These results confirm previous observations made on the b-cez [3-4]. The determination of these temperature domains made possible the extrapolation of the IT diagram proper to each morphology and thus an estimation of the values of ki, ni and tdi.

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aGB + aWGB aWI

Figure 1. Evolution of the a phase amount at the end of isothermal treatment and at equilibrium.

Figure 2. TTT diagram of the Ti17 alloy, obtained by electrical resistivity measurements.

a)

b)

c) Figure 3. Microstructures at the end of the isothermal treatments at a) 800°C, b) 750°C and c) 700°C.

3.4 Alpha phase amount at the end of transformation Once the electrical resistivity variations in isothermal conditions were negligible, the specimens were quenched to room temperature. The a phase amount was measured using X-ray diffraction and quantitative image analysis. The obtained values were compared to electrical resistivity variations. The X-ray diffraction patterns were obtained in the following conditions: cobalt anode (l =1.78897 Å), 35 kV, 20 mA. The diffraction peaks of the a and b phases were analysed in a 2q range of [35°,110°]. For each analysis, 9 q-2q scatterings were made with c varying between 0 and 45°. The amount of each phase was supposed to be proportional to the area under the associated diffraction

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peaks. The experimental error associated to this method is high because the b phase is highly textured and the grains have a large size. Analysis of the electrical resistivity curves: It was experimentally shown for the b-cez alloy that, when the transformation is completed, the amount of a phase is proportional to the relative variation of the electrical resistivity [3-4].

zmax = K

DR(T) Rtd

(14)

Moreover, the K constant does not depend on the temperature. Similar results were obtained for Ti17. The value of K was determined from the resistivity curve at 750°C and the corresponding a phase amount was measured by quantitative image analysis. The retained value is 10. Image analysis The a phase amounts were measured on BSE-SEM micrographs, using a Philips SEM-FEG. The microstructure corresponds to the one obtained after quenching at the end of the transformation. Assuming that the microstructure is isotropic, the fraction surface is equal to the volumic fraction of a phase. The absolute uncertainty associated to this method was evaluated to 1%. It was applied to the isothermal treatment temperatures of 700, 750 and 800°C. The comparison of the amount of a phase measured at the end of the isothermal transformation, at temperatures above 700°C shows that for the different methods used, we obtain the same value as the ones determined in §3.2. For temperatures lower than 700°C, for which aWI forms, the electrical resistivity values lead to larger values than the one estimated from the equilibrium diagram. These differences have to be further investigated because electrical resitivity variations are sensitive to chemical composition variations as well as with structural changes and defects. For the model, we consider that the amount of a phase at the end of transformation corresponds to equilibrium. 3.5 Experimental verification of the cumulative rule The model assumes a linear cumulative rule for the nucleation and growth. An experimental method has been proposed in order to verify this assumption [10] and we applied it to the Ti17 alloy. A sample has been transformed in isothermal conditions at two successive temperatures. The sample was first isothermally transformed at 815°C (holding for 4315 s) and then quickly quenched to 750°C, for 6700 s. The amount of a phase was followed by tracking the relative variations of the electrical resistivity. Figure 4 shows the relative variations of the electrical resistivity for isothermal treatments at 800 and 750°C in a single step, and the ones for the two steps 815-750°C treatment. The time scale was shifted to allow a better comparison. We observe that the transformation rate changes very quickly when the transformation temperature decreases. Moreover, the transformation rate after the thermal step corresponds to the rate at 700°C for the already formed a phase amount at the time considered.

Figure 4. Experimental verification of the additivity of the phase transformation for the high temperature a morphology.

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4. NUMERICAL SIMULATIONS The model presented above was implemented in the FEM code Zebulon through Zebfront interface [18]. A first group of calculations was done considering the data obtained from the above TTT diagram. Continuous cooling treatments ranging from 0.02 to 1°C.s-1 were simulated and then compared to experiments for which the kinetics of phase transformation was also determined by electrical resistivity measurements using the method proposed in [2]. Two domains of transformation are observable (Figure 5). At high temperature, the precipitation of a phase begins with the aGB + aWGB morphology whose kinetics of transformation is slow, compared to that of the aWI morphology whose precipitation begins at about 680°C.

Figure 5. Simulations of continuous cooling treatments at 0.1 and 0.05 °C.s-1 and comparison to experience.

Figure 6. Calculated and experimental alpha phase amount at the end of transformation. The total experimental a phase amounts have been determined by X-Ray Diffraction. The amount of aWGB morphology has been determined by image analysis.

The change of kinetics of phase transformation as a function of time is well reproduced. Moreover, the calculated amounts of aWGB morphology at the end of continuous cooling treatments for 0.05 and 0.1 °C.s-1 are in good agreement with the experimental results (Figure 6). The results of continuous cooling treatment simulations show that a full metastable b phase is obtained at the end of the treatment for cooling rates larger than 10 °C.s-1. At very low cooling rates, the aWGB morphology is favoured. But there is always an amount of intragranular morphology. This result has been verified qualitatively by micrographic observation for a sample cooled at 0.02 °C.s-1.

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5. CONCLUSION The global TTT diagram of the Ti17 alloy was established, by in-situ electrical resistivity measurements. Between 800 and 450°C, the phase transformation kinetics follow the JMA law. The IT diagrams for each morphology of a phase were extracted from metallurgical observations. The transformations kinetics model was implemented in the FEM Zebulon 3D software. With this metallurgical model we are able to progress in the prediction of the microstructural evolutions during any thermal treatment after solution treatment in the b field even in large parts if the different couplings (thermal, mechanical) are included. References [1] O. Gourbesville, Caractérisation par DRX de la microstructure d'alliages à base de nickel (718) et à base de titane (Ti-17) forgés et traités. Prévision des propriétés mécaniques. PhD thesis, ENSAM, Paris, 2000 [2] C. Angelier, S. Bein and J. Bechet, Metall. Trans. A, 1997, 28A, pp. 2467-75 [3] S. Bein, Transformation de phases dans les alliages de titane a+ b quasi b. Approches comparatives des évolutions morphologiques et des mécanismes de précipitation observés dans l'alliage b-cez, PhD thesis, CNAM, Paris, 1996 [4] E. Laude, E. Gautier, S. Denis, “Calculation of transformation kinetics of titanium alloys during continuous cooling. Application to the b-cez alloy.” P.A. Blenkinsop, W. J. Evans, H. M. Flowers (Eds.), Titanium 1995 : Science and Technology, Proceedings of the Eighth World Conference on Titanium, The Institute of Materials, London, 1996, pp. 2330-2337 [5] J. Bechet, B. Hocheid, “Decomposition of the b titanium alloy Ti17”. G. Lütjering, U. Zwicker, W. Bunk (Eds.), Titanium 1984 Science and technology, Proceedings of the Fifth International Conference on Titanium, Deutsche Gesellschaft für Metalkunde, 1985, pp. 1613-1619 [6] H. Patiès, Etude expérimentale des cinétiques globales de transformation au refroidissement d'alliages de titane, Stage de DEA, INPL, Nancy, 1998 [7] F. Fernandes, S. Denis, A. Simon “Prévision de l’évolution thermique et structurale des aciers au cours de leur refroidissement continu”. Mém. et Etudes Scient. Rev. Mét., juillet-août (1986) p. 355-366. [8] J. W. Christian, The theory of transformations in metals and alloys, Part I, Second edition, Pergamon Press [9] J. W. Cahn, Acta Metall, 1956, 4, p. 572 [10] R. G. Kamat, E. B. Hawbolt, L. C. Brown, J. K. Brimacombe, Metall. Trans. A, 1992, 23A, pp. 2469-80 [11] S. Malinov, Z. Guo , W. Sha and A. Wilson, Metall. Trans. A, 2001, 32A, pp. 879-887 [12] S. Malinov, P. Markovsky, W. Sha, Z. Guo, Journal of Alloys and Compounds 314 (2001) pp. 181-192 [13] S. Malinov, P. Markovsky et W. Sha, Journal of Alloys and Compounds 333 (2002) pp.122-132 [14] S. Malinov, W. Sha et P. Markovsky, Journal of Alloys and Compounds 348 (2003) pp. 110-118 [15] S. J. Jones and H. K. D. H. Bhadeshia, Acta mater. 45, No. 7, pp. 2911-2920, 1997 [16] T. Reti, Z. Fried and I. Felde, Computational Materials Science, 22, Issues 3-4 (2001) pp. 261278 [17] L. Hericher, B. Appolaire, E. Gautier, “Modelling of growth of a phase in a metastable Ti-alloy”. Journées d’automne 2001 SF2M, La revue de métallurgie, 2001, p. 171. [18] J. Besson, R. Le Riche, R. Foerch, and G. Cailletaud, Object-Oriented Programming Applied to the Finite Element Method: Application to Material Behaviors, Revue européenne des éléments finis, 1998, 7, number 5, pp.567-588