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Kinetic study of the precipitation in physical vapour depositedMg–12% wt. Ti alloy
Journal of Alloys and Compounds 347 (2002) 188–192
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Kinetic study of the precipitation in physical vapour deposited Mg–12% wt. Ti alloy ´ * ,1 , Paloma Adeva Gerardo Garces Department of Physical Metallurgy, National Centre for Metallurgical Research CENIM, CSIC, Av. De Gregorio del Amo 8, 28040 Madrid, Spain Received 18 March 2002; received in revised form 8 May 2002; accepted 8 May 2002
Abstract Kinetic parameters of titanium precipitation in Mg–12 wt.% Ti obtained by physical vapour deposition (PVD) have been studied using the Johnson–Mehl–Avrami equation extended to a non-isothermal transformation. The titanium precipitation takes place in two stages: the first corresponds to the nucleation and growth of the precipitates by diffusion along the basal plane and the second to their coarsening along the [0001] direction. The Avrami exponent, n, has been calculated from the differental scanning calorimtery curves for the two stages. In both cases values of n around 1 were obtained, which indicate a diffusion process dominant in one dimension. Furthermore, at the last stage of titanium nucleus formation, a decrease in Avrami exponent has been observed associated to a fast and progressive saturation of nucleation sites. 2002 Elsevier Science B.V. All rights reserved. Keywords: Transition metal alloys; Vapour deposition; Precipitation; Kinetics; Thermal analysis
1. Introduction The physical meaning of the n exponent in the Johnson– Mehl–Avrami (JMA) model in terms of microstructural variables, especially when n varies during the transformation is not well understood. Since n depends on the nucleation and growth exponents, at least one of both phenomena has to change during the transformation. The decrease of n during a reaction has been reported previously for several solid-state transformations, described in terms of the JMA kinetics and different interpretations have been given. In precipitation processes [1–3], the decrease of n has been related to the progressive saturation of nucleation sites as well as reduction of the growth dimension during the last stages of the transformation. In recrystalization phenomena, Canh [4] proposed that the continuous decrease of n reported as the reaction proceeds could be attributed to the not random distribution of nucleation sites in the volume assuming an inhomogeneous process. It has been observed frequently that in the initial stages of recrystallisation the new grains appear to *Corresponding author. ´ E-mail address: [email protected] (G. Garces). 1 ¨ Metallforschung, HeisenbergPresent address: Max-Planck-Institut fur strasse 3, D-70569 Stuttgart, Germany.
be clustered in some region while other regions of the specimen still seem to be entirely free of nucleating grains. This fact may be due to an inhomogeneous deformation state of the material as discussed by Rollet et al. [5], which will give rise to a locally stored energy and hence different nucleation and growth conditions. Other authors [6] explain the variation of n as the occurrence of different simultaneous processes as recovery. In crystallisation of amorphous materials, the break in linearity through the decrease in the n exponent has been connected with the saturation of the nucleation sites at the final stage of crystallisation [7,8], analogous to the precipitation case. It can also be caused by a restriction of crystal growth by small size of the particles [9]. The precipitation behaviour during heating at a constant rate of a metastable Mg–12 wt.% Ti alloy obtained by physical vapour deposition (PVD) was studied in a recent paper [10] using differential scanning calorimetry (DSC) and transmission electron microscopy (TEM). Three different exothermal transformations were reported. The first exothermal transformation, at 425 K, was related to stress relaxation. The high density of defects generated during the deposit growth, especially dislocations, results during heating in a reorganisation of these into a stable situation at low temperatures [11,12]. The second transformation, between 550 and 686 K, was related to the precipitation of
0925-8388 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0925-8388( 02 )00793-4
´ , P. Adeva / Journal of Alloys and Compounds 347 (2002) 188–192 G. Garces
titanium in the magnesium matrix. This transformation takes place in two sequences: the formation of coherent titanium precipitates along the basal plane and the consecutive growth of them along the [0001] direction. Finally, the third exothermal transformation characterised by an increase in the flow heat above 703 K was associated with a magnesium oxidation phenomenon in agreement with the experiments of Diplas et al. [13]. This paper is a continuation of the mentioned work and is focused on the study of the precipitation kinetics of the PVD Mg–12% wt. Ti alloy. The main purpose has been to calculate the JMA exponent, n, using a model proposed by Borrego and ´ Gonzalez-Doncel [14]. Furthermore, the variation of n with temperature during the titanium precipitation has been studied and associated with microstructural parameters.
2. Experimental The PVD alloy Mg–12% wt. Ti of nominal composition, was grown by DERA (Defence and Evaluation Research Agency) using an aluminium collector at 150 8C. The thickness of the deposit was around 2 mm. Details about the synthesis procedure can be found in Ref. [15]. The precipitation kinetics were followed by DSC, using a Perkin-Elmer System 4 Thermal Analyser. The scans were made under argon atmosphere to minimise oxidation at four different heating rates of 10, 20, 30 and 40 K min 21 . High-purity magnesium was used as reference material. To obtain a baseline (which depends, among other factors, on the different heat capacities of the reference and sample pans) two runs were carried out. The first run was conducted using high purity magnesium discs of approximately the same mass in both pans. The second run was carried out after replacing the magnesium disc in the sample pan by a disc of the material under study. Subtraction of the heat flow in the first run from the heat flow in the second run results in a signal, which allows the calculation of the heat flow due to reactions in the sample.
3. Results and discussion The DSC thermograms of the alloy in the as-deposited condition at the four heating rates showing the two exothermal peaks attributed to the precipitation of the titanium phase are presented in Fig. 1. The titanium precipitation is taking place by the nucleation and growth of coherent titanium precipitates in the magnesium basal plane (first peak) and the coarsening of precipitates along the [0001] direction (second peak). The temperature of both peaks at, the four heating rates are shown in Table 1. The temperature of the peaks increases with heating rate as is common in thermally activated transformations. The Kissinger analysis was used to determine the activation
189
Fig. 1. Heat flow vs. temperature from DSC scans for the precipitation transformation at 10, 20, 30 and 40 K min 21 .
energies of both transformations and the values of the apparent activation energies obtained for the fit were 144 and 155 KJ mol 21 for the first and second transformation, respectively [10]. Assuming that the precipitation phenomenon in the PVD Mg–Ti alloy follows a Johnson–Mehl–Avrami theory, the heat flow curves have been fitted following the model ´ deduced by Borrego and Gonzalez-Doncel [14] to calculate the Avrami exponent, n. In this model the heat flow (H~ T ) is given by the expression: T R O A n K S]] FE D 2
H~ T 5
i 51
2
S
n i Ei exp 2 ]] RT i n n i Ei T 2R i n i ]] K 0 i exp 2 ]] F Ei RT
i i
H FS D
exp 2
n i 21
ni 0i
D (1)
D GJ
S
where E is the apparent activation energy for precipitation, T the temperature, n the Avrami exponent, K0 the preexponential factor, R the universal gas constant, F the heating rate and A is a constant. By fitting the experimental heat flow curve with Eq. (1) the n values can be obtained. Fig. 2 shows the fitting for the heating rates of 10 and 40 K min 21 . The value of n calculated for each heating rate is summarised in Table 1. The value of n for the formation of titanium precipitates along the magnesium basal plane (first peak) is lower than for the precipitate coarsening in the [0001] direction (second peak). As is known, the Avrami exponent depends on several factors such as the mechanism of transforma-
Table 1 Values of exothermal peak temperatures and n values obtained from the fitting Heating rate (K min 21 )
T first (K)
10 20 30 40
551 563 568 578
peak
T second (K) 656 665 676 686
peak
n first 0.75 0.90 0.81 0.87
peak
n second 0.93 1.14 1.15 1.15
peak
190
´ , P. Adeva / Journal of Alloys and Compounds 347 (2002) 188–192 G. Garces
´ Gonzalez-Doncel may determine the heat flow for the case in which n decreases during the transformation: dn O A K S]nu 1 ] ln u Du du 2
H~ T 5
ni 0i
i
i 51
i
i
i
i
i
ni i
S
D
Ei n exp 2 ] exps 2 u i id RT (2)
where
S D
S
Ei T 2R ui 5 ]] K0 i exp 2 ] F Ei RT
Fig. 2. Heat flow vs. temperature from the DSC experiments at 10 and 40 K min 21 and fits from the JMA.
tion, the nucleation condition, the geometry and growth of the precipitates, etc. It is interesting to note that for the two transformations studied, the values of JMA exponent, n, obtained are close to 1. These values could characterise a precipitation process where growth is controlled by diffusion and one-dimensional needles and plates are formed (first peak) followed by growth of these ones along the [0001] direction after the complete end of impingement (second peak) [16]. In a previous paper [10] related to the Mg–12% wt. Ti precipitation, a value of n51 was assumed to calculate the activation energy for the diffusion of titanium in the basal plane and along the [0001] direction in magnesium. Therefore, from the present results obtained, n around 1, it can be concluded that the assumption made in Ref. [10] has been a correct one. On the other hand, a slight misfit between both experimental and theoretical DSC curves of Fig. 2 is observed at the end of the first peak and the start of the second one, i.e. when nucleation of the precipitates in the basal plane is taking place. According to Borrego and ´ Gonzalez-Doncel, the differences between the experimental and theoretical curves in the DSC experiments could be attributed to a change of the Avrami exponent n because the transformation progress during the first peak results in a spread of this peak when the reaction comes to an end [3]. The decrease in n during this first stage of transformation can be explained as a progressive saturation of nucleation site. The expression proposed by Borrego and
D
(3)
In the case of titanium precipitation in the magnesium matrix, the variation of n will only take place during the formation of titanium precipitates along the basal plane. No changes in the n value, however, can be expected in the second precipitation stage since this stage is characterised by the coarsening of the initial clusters along the [0001] direction. This argument agrees with the good fitting observed between the experimental and theoretical curves in the case of the second peak. To calculate the n variation during the first stage of Ti precipitation and assuming that H~ T is given by the experimental DSC data, the following differential equation can be obtained (Appendix A): H~ Tu 1 dn 1 ] 5 ]] ]]]]]]]]] 2n dT Y(T ) E AK nu n21 exp 2 ] exp(2u n) RT
5
S
D
6
(4)
where, the boundary condition is chosen for the value of n (obtained from Eq. (1)) for which the theoretical curve starts to deviate from the experimental data. Solving Eq. (4) numerically a decrease in n is observed at each heating rate as the temperature increases (Fig. 3). These figures show a rapid, when the theoretical curve starts to deviate from the experimental data, followed by a stationary stage. This result could be explained on the basis of a very fast nucleation process, assuming Enuc ¯0, and agrees with the hypothesis made in a previous study on Mg–Ti precipitation [10]. Nuclei of the precipitated Ti phase form because of localised compositional fluctuations that occur statistically within the supersaturated matrix. Provided that the effect of elastic coherency strains around the nucleated precipitates can be ignored during nucleation, the energy barrier for heterogeneous nucleation, DG *het , is given by [17]: A3 * 5 ]]]]]]2 DG het Cm ] sRTd 2 ln C0
FS DG
(5)
where Cm is the mean solute content in the matrix, C0 is the equilibrium solute content at the particle / matrix interface (given by the phase diagram), R is the universal gas constant and A is a constant. Since the equilibrium solute content of titanium in magnesium is neglected, the energy
´ , P. Adeva / Journal of Alloys and Compounds 347 (2002) 188–192 G. Garces
191
dq dn ] 5 2 2] dT dT
(9)
The decrease of n at the end of the first stage of the titanium precipitation would result in an increase of q and ~ an decrease of the nucleation rate N(t). Thus, the decrease in the Avrami exponent is clearly due to the progressive site saturation during nucleation. This observation is in agreement with the work of other authors [1–3]. Bratland et al. [1] showed in the case of Mg 2 Si precipitation in an aluminium matrix that for different annealing temperatures the coefficient of the Avrami is constant (¯0.7), decreasing for long annealing times. 4. Conclusions
Fig. 3. Variation of the JMA exponent n with the temperature at 10 and 40 K min 21 .
barrier for nucleation is practically zero. Then the nucleation process occurs rapidly leading to saturation of the nucleation sites. Since a high concentration of vacancies is generated during the PVD process, they act as preferential nucleation sites for the formation of titanium clusters. These vacancies are homogeneously distributed, which results in a homogeneous precipitation. On the other hand, if the decrease of n at the end of the first peak is related to the saturation of nucleation site, it will be interesting to connect n with the nucleation process. The nucleation rate is commonly expressed as [2,18]:
~ 5 C0t 2q N(t)
(6)
where C0 is a time constant with an Arrhenian temperature dependence:
S
Qn C0 5 C9exp 2 ] RT
D
(7)
It had been proved the reliability of the model proposed ´ by Borrego and Gonzalez-Doncel [3,14] based on the Johnson–Mehl–Avrami approach, to study the precipitation kinetics of PVD Mg–Ti. The JMA exponent, n, as well as the nucleation rate decrease at the end of the first stage of precipitation gradually towards a stationary stage in which the Avrami exponent becomes constant again and the nucleation rate is zero. These results ratify the assumption of Enuc ¯0 due to a fast and progressive saturation of nucleation sites which are vacancies, mostly. Acknowledgements The authors would like to thank Dr. S.B. Dodd, S. Morris and R.C. Piller of DERA (Defence and Evaluation Research Agency), Farnborough, UK, for providing the vapour deposited alloys and useful information on process´ and Mr. J.M. ing conditions as well as J.M. Badıa ´ Antonanz, of E.T.S.I. Aeronauticos, Madrid, Spain, for their assistance in the DSC experiments. We gratefully acknowledge the support of the CICYT MAT 981620-CE.
Appendix A The heat flow where n depends on temperature for the precipitation transformation is given by the expression: H~ T 5H~ Tu 1 2H~ Tu 2
The constant q could be related with the Avrami exponent, n, by means of the following expression deduced ´ by Borrego and Gonzalez-Doncel [14]: 2n 5 5 2 q
(8)
Thus the variation of the nucleation exponent with the temperature will be given by:
dn O A K S]nu 1 ] ln u Du du 2
5
i
i 51
ni 0i
i
i
i
i
ni i
i
S
D
Ei n exp 2 ] exps 2 u i id RT (A.1)
where 1 and 2 represent the first and second peak, respectively, and:
S D
S
Ei T 2R ui (T ) 5 ]] K0 i exp 2 ] F Ei RT
D
(A.2)
´ , P. Adeva / Journal of Alloys and Compounds 347 (2002) 188–192 G. Garces
192
Assuming that the nucleation takes place during cluster formation in the magnesium basal plane, n only will change at this stage. Therefore:
S S D
D
n 1 dn 1 n n H~ T 2H~ Tu 2 5H~ Tu 1 5 A 1 K 0 11 ] 1 ] ln u u 1 u du E1 n 3 exp 2 ] exps 2 u 1d RT
(A.3)
Thus, the following differential equation is obtained:
H
H~ Tu 1 dn 1 ] 5 ]] ]]]]]] 2n n n21 du u ln u AK u exp(2u n)
J
(A.4)
From Eq. (A.4): dn dn dT dn 1 ] 5 ] ] 5 ] ]]]]] du dT du dT 2 E u ] 1 ]]2 T RT
F
(A.5)
G
and substituting Eq. (A.5) into Eq. (A.4): H~ Tu 1 dn 1 ] 5 ]] ]]]]]]]]] 2n dT Y(T ) E1 n n21 n AK u exp 2 ] exp(2u ) RT
5
S
D
6 (A.6)
where: ln u (T ) Y(T ) 5 ]]]] 2 E ] 1 ]]2 T RT
F
G
(A.7)
References [1] D.H. Bratland, Ø. Grong, H. Shercliff, O.R. Myhr, S. Tjøtta, Acta Mater. 45 (1997) 1. [2] I. Dutta, D.L. Bourell, Acta Metall. Mater. 38 (1990) 204. ´ [3] A. Borrego, G. Gonzalez-Doncel, Mater. Sci. Eng. A252 (1998) 149. [4] J.W. Cahn, Acta Metall. 4 (1956) 449. [5] A.D. Rollet, D.J. Srolovitz, R.D. Doherty, M.P. Anderson, Acta Metall. 37 (1989) 62. [6] C.W. Price, Acta Metall. Mater. 38 (1990) 727. [7] J. Colmenero, J.M. Barandiran, J. Non-Cryst. Solids 30 (1979) 263. [8] P. Duhaj, D. Baranock, A. Ondrejka, J. Non-Cryst. Solids 21 (1976) 641. [9] R.F. Speyer, S.H. Risbud, Phys. Chem. Glass. 24 (1983) 26. ´ P. Adeva, Phil. Mag. 82 (2002) 699. [10] G. Garces, [11] S. Diplas, P. Tsakiropoulos, R.M.D. Brydson, Mater. Sci. Technol. 15 (1999) 1349. [12] J.P. Chu, C.H. Chung, P.Y. Lee, J.M. Rigsbee, J.Y. Wang, Metall. Mater. Trans. A 29A (1998) 647. [13] S. Diplas, P. Tsakiropoulos, R.M.D. Brydson, Mater. Sci. Technol. 14 (1998) 689. ´ [14] A. Borrego, G. Gonzalez-Doncel, Mater. Sci. Eng. A245 (1998) 10. [15] S.B. Dodd, R.W. Gardiner, in: G.W. Lorimer (Ed.), Proceedings of the Third International Magnesium Conference, Institute of Metals, Manchester, 1996, p. 271. [16] J.W. Christian, in: The Theory of Transformation in Metals and Alloys, Pergamon Press, Oxford, 1965, p. 489. [17] Ø. Grong, O.R. Myhr, Acta Mater. 48 (2000) 445. [18] J.W. Christian, in: The Theory of Transformation in Metals and Alloys, Pergamon Press, Oxford, 1965, p. 477.
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Kinetic study of the precipitation in physical vapour deposited Mg–12% wt. Ti alloy ´ * ,1 , Paloma Adeva Gerardo Garces Department of Physical Metallurgy, National Centre for Metallurgical Research CENIM, CSIC, Av. De Gregorio del Amo 8, 28040 Madrid, Spain Received 18 March 2002; received in revised form 8 May 2002; accepted 8 May 2002
Abstract Kinetic parameters of titanium precipitation in Mg–12 wt.% Ti obtained by physical vapour deposition (PVD) have been studied using the Johnson–Mehl–Avrami equation extended to a non-isothermal transformation. The titanium precipitation takes place in two stages: the first corresponds to the nucleation and growth of the precipitates by diffusion along the basal plane and the second to their coarsening along the [0001] direction. The Avrami exponent, n, has been calculated from the differental scanning calorimtery curves for the two stages. In both cases values of n around 1 were obtained, which indicate a diffusion process dominant in one dimension. Furthermore, at the last stage of titanium nucleus formation, a decrease in Avrami exponent has been observed associated to a fast and progressive saturation of nucleation sites. 2002 Elsevier Science B.V. All rights reserved. Keywords: Transition metal alloys; Vapour deposition; Precipitation; Kinetics; Thermal analysis
1. Introduction The physical meaning of the n exponent in the Johnson– Mehl–Avrami (JMA) model in terms of microstructural variables, especially when n varies during the transformation is not well understood. Since n depends on the nucleation and growth exponents, at least one of both phenomena has to change during the transformation. The decrease of n during a reaction has been reported previously for several solid-state transformations, described in terms of the JMA kinetics and different interpretations have been given. In precipitation processes [1–3], the decrease of n has been related to the progressive saturation of nucleation sites as well as reduction of the growth dimension during the last stages of the transformation. In recrystalization phenomena, Canh [4] proposed that the continuous decrease of n reported as the reaction proceeds could be attributed to the not random distribution of nucleation sites in the volume assuming an inhomogeneous process. It has been observed frequently that in the initial stages of recrystallisation the new grains appear to *Corresponding author. ´ E-mail address: [email protected] (G. Garces). 1 ¨ Metallforschung, HeisenbergPresent address: Max-Planck-Institut fur strasse 3, D-70569 Stuttgart, Germany.
be clustered in some region while other regions of the specimen still seem to be entirely free of nucleating grains. This fact may be due to an inhomogeneous deformation state of the material as discussed by Rollet et al. [5], which will give rise to a locally stored energy and hence different nucleation and growth conditions. Other authors [6] explain the variation of n as the occurrence of different simultaneous processes as recovery. In crystallisation of amorphous materials, the break in linearity through the decrease in the n exponent has been connected with the saturation of the nucleation sites at the final stage of crystallisation [7,8], analogous to the precipitation case. It can also be caused by a restriction of crystal growth by small size of the particles [9]. The precipitation behaviour during heating at a constant rate of a metastable Mg–12 wt.% Ti alloy obtained by physical vapour deposition (PVD) was studied in a recent paper [10] using differential scanning calorimetry (DSC) and transmission electron microscopy (TEM). Three different exothermal transformations were reported. The first exothermal transformation, at 425 K, was related to stress relaxation. The high density of defects generated during the deposit growth, especially dislocations, results during heating in a reorganisation of these into a stable situation at low temperatures [11,12]. The second transformation, between 550 and 686 K, was related to the precipitation of
0925-8388 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0925-8388( 02 )00793-4
´ , P. Adeva / Journal of Alloys and Compounds 347 (2002) 188–192 G. Garces
titanium in the magnesium matrix. This transformation takes place in two sequences: the formation of coherent titanium precipitates along the basal plane and the consecutive growth of them along the [0001] direction. Finally, the third exothermal transformation characterised by an increase in the flow heat above 703 K was associated with a magnesium oxidation phenomenon in agreement with the experiments of Diplas et al. [13]. This paper is a continuation of the mentioned work and is focused on the study of the precipitation kinetics of the PVD Mg–12% wt. Ti alloy. The main purpose has been to calculate the JMA exponent, n, using a model proposed by Borrego and ´ Gonzalez-Doncel [14]. Furthermore, the variation of n with temperature during the titanium precipitation has been studied and associated with microstructural parameters.
2. Experimental The PVD alloy Mg–12% wt. Ti of nominal composition, was grown by DERA (Defence and Evaluation Research Agency) using an aluminium collector at 150 8C. The thickness of the deposit was around 2 mm. Details about the synthesis procedure can be found in Ref. [15]. The precipitation kinetics were followed by DSC, using a Perkin-Elmer System 4 Thermal Analyser. The scans were made under argon atmosphere to minimise oxidation at four different heating rates of 10, 20, 30 and 40 K min 21 . High-purity magnesium was used as reference material. To obtain a baseline (which depends, among other factors, on the different heat capacities of the reference and sample pans) two runs were carried out. The first run was conducted using high purity magnesium discs of approximately the same mass in both pans. The second run was carried out after replacing the magnesium disc in the sample pan by a disc of the material under study. Subtraction of the heat flow in the first run from the heat flow in the second run results in a signal, which allows the calculation of the heat flow due to reactions in the sample.
3. Results and discussion The DSC thermograms of the alloy in the as-deposited condition at the four heating rates showing the two exothermal peaks attributed to the precipitation of the titanium phase are presented in Fig. 1. The titanium precipitation is taking place by the nucleation and growth of coherent titanium precipitates in the magnesium basal plane (first peak) and the coarsening of precipitates along the [0001] direction (second peak). The temperature of both peaks at, the four heating rates are shown in Table 1. The temperature of the peaks increases with heating rate as is common in thermally activated transformations. The Kissinger analysis was used to determine the activation
189
Fig. 1. Heat flow vs. temperature from DSC scans for the precipitation transformation at 10, 20, 30 and 40 K min 21 .
energies of both transformations and the values of the apparent activation energies obtained for the fit were 144 and 155 KJ mol 21 for the first and second transformation, respectively [10]. Assuming that the precipitation phenomenon in the PVD Mg–Ti alloy follows a Johnson–Mehl–Avrami theory, the heat flow curves have been fitted following the model ´ deduced by Borrego and Gonzalez-Doncel [14] to calculate the Avrami exponent, n. In this model the heat flow (H~ T ) is given by the expression: T R O A n K S]] FE D 2
H~ T 5
i 51
2
S
n i Ei exp 2 ]] RT i n n i Ei T 2R i n i ]] K 0 i exp 2 ]] F Ei RT
i i
H FS D
exp 2
n i 21
ni 0i
D (1)
D GJ
S
where E is the apparent activation energy for precipitation, T the temperature, n the Avrami exponent, K0 the preexponential factor, R the universal gas constant, F the heating rate and A is a constant. By fitting the experimental heat flow curve with Eq. (1) the n values can be obtained. Fig. 2 shows the fitting for the heating rates of 10 and 40 K min 21 . The value of n calculated for each heating rate is summarised in Table 1. The value of n for the formation of titanium precipitates along the magnesium basal plane (first peak) is lower than for the precipitate coarsening in the [0001] direction (second peak). As is known, the Avrami exponent depends on several factors such as the mechanism of transforma-
Table 1 Values of exothermal peak temperatures and n values obtained from the fitting Heating rate (K min 21 )
T first (K)
10 20 30 40
551 563 568 578
peak
T second (K) 656 665 676 686
peak
n first 0.75 0.90 0.81 0.87
peak
n second 0.93 1.14 1.15 1.15
peak
190
´ , P. Adeva / Journal of Alloys and Compounds 347 (2002) 188–192 G. Garces
´ Gonzalez-Doncel may determine the heat flow for the case in which n decreases during the transformation: dn O A K S]nu 1 ] ln u Du du 2
H~ T 5
ni 0i
i
i 51
i
i
i
i
i
ni i
S
D
Ei n exp 2 ] exps 2 u i id RT (2)
where
S D
S
Ei T 2R ui 5 ]] K0 i exp 2 ] F Ei RT
Fig. 2. Heat flow vs. temperature from the DSC experiments at 10 and 40 K min 21 and fits from the JMA.
tion, the nucleation condition, the geometry and growth of the precipitates, etc. It is interesting to note that for the two transformations studied, the values of JMA exponent, n, obtained are close to 1. These values could characterise a precipitation process where growth is controlled by diffusion and one-dimensional needles and plates are formed (first peak) followed by growth of these ones along the [0001] direction after the complete end of impingement (second peak) [16]. In a previous paper [10] related to the Mg–12% wt. Ti precipitation, a value of n51 was assumed to calculate the activation energy for the diffusion of titanium in the basal plane and along the [0001] direction in magnesium. Therefore, from the present results obtained, n around 1, it can be concluded that the assumption made in Ref. [10] has been a correct one. On the other hand, a slight misfit between both experimental and theoretical DSC curves of Fig. 2 is observed at the end of the first peak and the start of the second one, i.e. when nucleation of the precipitates in the basal plane is taking place. According to Borrego and ´ Gonzalez-Doncel, the differences between the experimental and theoretical curves in the DSC experiments could be attributed to a change of the Avrami exponent n because the transformation progress during the first peak results in a spread of this peak when the reaction comes to an end [3]. The decrease in n during this first stage of transformation can be explained as a progressive saturation of nucleation site. The expression proposed by Borrego and
D
(3)
In the case of titanium precipitation in the magnesium matrix, the variation of n will only take place during the formation of titanium precipitates along the basal plane. No changes in the n value, however, can be expected in the second precipitation stage since this stage is characterised by the coarsening of the initial clusters along the [0001] direction. This argument agrees with the good fitting observed between the experimental and theoretical curves in the case of the second peak. To calculate the n variation during the first stage of Ti precipitation and assuming that H~ T is given by the experimental DSC data, the following differential equation can be obtained (Appendix A): H~ Tu 1 dn 1 ] 5 ]] ]]]]]]]]] 2n dT Y(T ) E AK nu n21 exp 2 ] exp(2u n) RT
5
S
D
6
(4)
where, the boundary condition is chosen for the value of n (obtained from Eq. (1)) for which the theoretical curve starts to deviate from the experimental data. Solving Eq. (4) numerically a decrease in n is observed at each heating rate as the temperature increases (Fig. 3). These figures show a rapid, when the theoretical curve starts to deviate from the experimental data, followed by a stationary stage. This result could be explained on the basis of a very fast nucleation process, assuming Enuc ¯0, and agrees with the hypothesis made in a previous study on Mg–Ti precipitation [10]. Nuclei of the precipitated Ti phase form because of localised compositional fluctuations that occur statistically within the supersaturated matrix. Provided that the effect of elastic coherency strains around the nucleated precipitates can be ignored during nucleation, the energy barrier for heterogeneous nucleation, DG *het , is given by [17]: A3 * 5 ]]]]]]2 DG het Cm ] sRTd 2 ln C0
FS DG
(5)
where Cm is the mean solute content in the matrix, C0 is the equilibrium solute content at the particle / matrix interface (given by the phase diagram), R is the universal gas constant and A is a constant. Since the equilibrium solute content of titanium in magnesium is neglected, the energy
´ , P. Adeva / Journal of Alloys and Compounds 347 (2002) 188–192 G. Garces
191
dq dn ] 5 2 2] dT dT
(9)
The decrease of n at the end of the first stage of the titanium precipitation would result in an increase of q and ~ an decrease of the nucleation rate N(t). Thus, the decrease in the Avrami exponent is clearly due to the progressive site saturation during nucleation. This observation is in agreement with the work of other authors [1–3]. Bratland et al. [1] showed in the case of Mg 2 Si precipitation in an aluminium matrix that for different annealing temperatures the coefficient of the Avrami is constant (¯0.7), decreasing for long annealing times. 4. Conclusions
Fig. 3. Variation of the JMA exponent n with the temperature at 10 and 40 K min 21 .
barrier for nucleation is practically zero. Then the nucleation process occurs rapidly leading to saturation of the nucleation sites. Since a high concentration of vacancies is generated during the PVD process, they act as preferential nucleation sites for the formation of titanium clusters. These vacancies are homogeneously distributed, which results in a homogeneous precipitation. On the other hand, if the decrease of n at the end of the first peak is related to the saturation of nucleation site, it will be interesting to connect n with the nucleation process. The nucleation rate is commonly expressed as [2,18]:
~ 5 C0t 2q N(t)
(6)
where C0 is a time constant with an Arrhenian temperature dependence:
S
Qn C0 5 C9exp 2 ] RT
D
(7)
It had been proved the reliability of the model proposed ´ by Borrego and Gonzalez-Doncel [3,14] based on the Johnson–Mehl–Avrami approach, to study the precipitation kinetics of PVD Mg–Ti. The JMA exponent, n, as well as the nucleation rate decrease at the end of the first stage of precipitation gradually towards a stationary stage in which the Avrami exponent becomes constant again and the nucleation rate is zero. These results ratify the assumption of Enuc ¯0 due to a fast and progressive saturation of nucleation sites which are vacancies, mostly. Acknowledgements The authors would like to thank Dr. S.B. Dodd, S. Morris and R.C. Piller of DERA (Defence and Evaluation Research Agency), Farnborough, UK, for providing the vapour deposited alloys and useful information on process´ and Mr. J.M. ing conditions as well as J.M. Badıa ´ Antonanz, of E.T.S.I. Aeronauticos, Madrid, Spain, for their assistance in the DSC experiments. We gratefully acknowledge the support of the CICYT MAT 981620-CE.
Appendix A The heat flow where n depends on temperature for the precipitation transformation is given by the expression: H~ T 5H~ Tu 1 2H~ Tu 2
The constant q could be related with the Avrami exponent, n, by means of the following expression deduced ´ by Borrego and Gonzalez-Doncel [14]: 2n 5 5 2 q
(8)
Thus the variation of the nucleation exponent with the temperature will be given by:
dn O A K S]nu 1 ] ln u Du du 2
5
i
i 51
ni 0i
i
i
i
i
ni i
i
S
D
Ei n exp 2 ] exps 2 u i id RT (A.1)
where 1 and 2 represent the first and second peak, respectively, and:
S D
S
Ei T 2R ui (T ) 5 ]] K0 i exp 2 ] F Ei RT
D
(A.2)
´ , P. Adeva / Journal of Alloys and Compounds 347 (2002) 188–192 G. Garces
192
Assuming that the nucleation takes place during cluster formation in the magnesium basal plane, n only will change at this stage. Therefore:
S S D
D
n 1 dn 1 n n H~ T 2H~ Tu 2 5H~ Tu 1 5 A 1 K 0 11 ] 1 ] ln u u 1 u du E1 n 3 exp 2 ] exps 2 u 1d RT
(A.3)
Thus, the following differential equation is obtained:
H
H~ Tu 1 dn 1 ] 5 ]] ]]]]]] 2n n n21 du u ln u AK u exp(2u n)
J
(A.4)
From Eq. (A.4): dn dn dT dn 1 ] 5 ] ] 5 ] ]]]]] du dT du dT 2 E u ] 1 ]]2 T RT
F
(A.5)
G
and substituting Eq. (A.5) into Eq. (A.4): H~ Tu 1 dn 1 ] 5 ]] ]]]]]]]]] 2n dT Y(T ) E1 n n21 n AK u exp 2 ] exp(2u ) RT
5
S
D
6 (A.6)
where: ln u (T ) Y(T ) 5 ]]]] 2 E ] 1 ]]2 T RT
F
G
(A.7)
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