- Email: [email protected]
Coarsening of precipitates during step-wise and continuous temperature sweeps
Materials Science and Engineering A 496 (2008) 530–532
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Short communication
Coarsening of precipitates during step-wise and continuous temperature sweeps Eckhard Nembach ∗ Institut fuer Materialphysik, Universitaet Muenster, Wilhelm-Klemm-Strasse 10, 48149 Muenster, Germany
a r t i c l e
i n f o
Article history: Received 12 February 2008 Received in revised form 28 April 2008 Accepted 30 April 2008 Keywords: Coarsening Ostwald ripening Phase separation Precipitation Nanostructure
a b s t r a c t Standard theories of isothermal coarsening (Ostwald ripening) of nano-scale precipitates are adapted to describe two types of non-isothermal coarsening processes: (i) several isothermal heat treatments at different temperatures are performed in sequence and (ii) the temperature is varied continuously. Procedures are developed which allow to predict the final average size of the precipitates with an adequate accuracy. The present results are helpful for estimating the lifetime of particle strengthened technical structural components which are exposed to high temperatures. © 2008 Elsevier B.V. All rights reserved.
3
1. Introduction
Actually the original form of Eq. (1) is slightly different: r(t) = 3
Isothermal homogeneous precipitation of a new phase from a supersaturated solid solution involves three distinctly different processes: (i) nucleation of particles of the new phase, (ii) their growth and accompanying depletion of the matrix of solute, and (iii) coarsening of the precipitates. Though all three processes may occur concurrently, there are many systems, e.g. ␥ -strengthened nickel-base superalloys [1–3], in which nucleation and growth are accomplished after very short times, whereas coarsening continues infinitely. During this last stage, the average size of the precipitates increases with time: big precipitates grow at the expense of smaller ones, which dissolve, and the total volume fraction f of the precipitates stays virtually constant. This process reduces the total precipitate–matrix interface energy. Wagner [4] and Lifshitz and Slyozov [5] theoretically analyzed this coarsening process, which is often referred to as Ostwald [6] ripening, and found for the growth of the precipitates: r 3 (t) = r 3 (t = 0) + kt
(1)
where r 3 (t) is the average cube of the precipitate radii after the time t has been spent in coarsening. At its start r 3 (t) equals r 3 (t = 0). The coarsening parameter k varies with temperature T for two entirely different reasons [7,8]: (i) because, e.g. the diffusion constant of the precipitating atoms increases with T and (ii) because k tends to increase with f and f decreases as T is raised.
∗ Tel.: +49 251 83 33570; fax: +49 251 83 38346. E-mail address: [email protected]. 0921-5093/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2008.04.066
r(t = 0) + t, i.e. the cubes of the average radii r¯ are considered. The ratio [r 3 /¯r 3 ] (=average cube of the radii divided by the cube of the average radius) depends on the distribution function of the precipitate radii. If the distribution functions of the precipitate radii are the same at the start and at the end of the coarsening process, both equations are equivalent and k and differ only slightly. In the derivation of Eq. (1) the precipitates were assumed to be of spherical shape. If this is not the case like, e.g. in advanced industrial nickel-base superalloys, which are strengthened by precipitates of the ␥ -phase, r 3 in Eq. (1) is replaced by the average volume of the precipitates. After prolonged coarsening the ␥ -precipitates in superalloys are cuboidal in shape with rounded edges and corners [1]. Subsequent theoretical developments of coarsening have been reviewed in Refs. [7,8]. Even though more advanced models of coarsening are available, because of its simplicity Eq. (1) will be used in the following; it describes all relevant effects with adequate accuracy. 2. Non-isothermal coarsening It is emphasized that so far single-stage isothermal coarsening has been considered. This limits the applicability of Eq. (1) correspondingly. In many practical cases, however, the material is subjected to two or even more successive isothermal ageing treatments performed at different temperatures or to continuous temperature variations. No matter whether the service temperature T2 of a precipitate strengthened technical structural component is above or below that of the original ageing
E. Nembach / Materials Science and Engineering A 496 (2008) 530–532
temperature T1 , if T2 is sufficiently high, Ostwald ripening will continue during the component’s service life. Mrotzek and Nembach [2] dealt with coarsening during double-ageing for the times t1 and t2 at the two temperatures T1 and T2 , respectively. The volume fraction f did not exceed 0.1. The temperature dependence of f complicates the calculation of the final average precipitate radius: f decreases as T is raised. This change of f affects the precipitates’ size and number per unit volume. On the basis of Eq. (1) and k derived from data taken for single-stage isothermally aged specimens, the latter cited authors calculated the final average cube r 3 (t1 , T1 , t2 , T2 ) of the precipitate radii expected after coarsening during the two successive isothermal ageing treatments. First r 3 (t1 , T1 ) was calculated for the isothermal treatment for the time t1 at T1 . It suggests itself to calculate the final value r 3 (t1 , T1 , t2 , T2 ) with Eq. (1) by inserting r 3 (t1 , T1 ) for r 3 (t2 = 0, T2 ) and using k of T2 . However, the change of T from T1 to T2 causes one of f1 to f2 and consequently one of the average cube of the precipitate radii. The cited authors analyzed this problem theoretically and compared the results of three alternative models with experimental data taken after double-ageing the industrial superalloy NIMONIC PE16 containing spherical ␥ -precipitates and found that it is quite accurate to entirely disregard any effects of the quoted change in f on r 3 (t2 = 0, T2 ). This was the result of their Model 1. Since their two alternative models Models 2 and 3 yielded nearly the same results for r 3 (t1 , T1 , t2 , T2 ) as Model 1 did and Model 1 is the most simple one, only this latter model is considered in the present communication: at the start of the second coarsening treatment the number of the precipitates and the size of each of them are assumed to be exactly the same as at the end of the first treatment. Hence r 3 (t1 , T1 , t2 , T2 ) can be accurately calculated by inserting (i) r 3 (t1 , T1 ) for r 3 (t2 = 0, T2 ) and (ii) k(T2 ) for k into Eq. (1). “accurately” means that the average radius r(t1 , T1 , t2 , T2 ) obtained in this way agrees within about 6% with that measured by standard transmission electron microscopy. The accuracy of the latter method is around 5% [9]. Since in Model 3, Mrotzek and Nembach [2] also considered the case that the change from T1 to T2 alters the number of precipitates and disturbs the distribution function of their radii, the authors preferred the present form of Eq. (1) to the 3
3
original version r(t) = r(t = 0) + t. Even though in the presently exclusively considered Model 1, the temperature change from T1 to T2 does not affect the distribution function of the precipitate radii, for the sake of conformity with Ref. [2] the form of Eq. (1) is presently kept. As stated above, Eq. (1) has the advantage, that – in the case of non-spherical precipitates – r 3 can directly be replaced by their average volume. In the present communication extensions from Ostwald ripening during two successive isothermal ageing treatments to multi-stage ageing and to ageing during continuous temperature variations are described. The latter ones occur, e.g. during slowly heating or cooling or during temperature cycles. The technical relevance of these considerations derives from the fact that many structural components which are strengthened by nano-scale precipitates of a second phase, are used at so high temperatures that Ostwald ripening continues during their use. This has to be taken into consideration if the lifetime of such components is estimated. All subsequent derivations will be based on Eq. (1), which serves the present purpose well. If the temperature Tn+1 of the (n + 1)th heat treatment is lower than Tn , at Tn+1 a second population of very fine precipitates may nucleate in between those which grew already at Tn [2,3]. Since (i) in general the probability of nucleation is difficult to predict and (ii) such secondary precipitates may dissolve again at a subsequent treatment performed at a temperature above Tn+1 , in all following analyses it will be assumed that no secondary precipitates form.
531
2.1. Multi-stage coarsening First Ostwald ripening during multiple-ageing treatments is considered. N successive isothermal treatments are performed for the times tj at Tj , 1 ≤ j ≤ N. Generalizing the procedures developed for double-ageing [2] to multiple-ageing, one obtains with Eq. (1) for the average cube rj3 of the precipitate radii after treatment no. j: 3 + k(T )t rj3 = rj−1 j j
(2)
and for the final average cube rN3 of the precipitate radii: rN3 = r03 +
N
k(Tj )tj
(3)
j=1
At the start of coarsening at T1 , the average cube of the radii equals r03 . Evidently rN3 is independent of the sequence in which the ageing treatments are performed. 2.2. Coarsening during continuous temperature variations For the description of coarsening during the continuous temperature variation from Ts (start) to Te (end), the sum in Eq. (3) is replaced by an integral and the sweep rate v(T ) = dT/dt of the temperature is introduced. rs3 is the average cube of the radii at the start of the temperature sweep and re3 that at its end:
Te
1 k(T ) dT v(T )
re3 = rs3 + Ts
(4)
If the continuous temperature sweep is preceded by one or more isothermal ageing treatments, rs3 equals rN3 given by Eq. (3). Analogously if the sweep is followed by isothermal treatments, a term of N the form { j=1 k(Tj )tj } is added on the right hand side of Eq. (4). A sequence of sweeps is represented by a sum of integrals. 2.3. Temperature dependence of the coarsening parameter k The coarsening parameter k(T) is derived from data measured after single-stage isothermal treatments. Mrotzek and Nembach [2] published data k(T) for the industrial ␥ -strengthened nickel-base superalloy NIMONIC PE16 for the temperature range 949–1119 K. A simple approximate analytical representation of their results for k(T) is
A 2
k(T ) = A1 exp −
T
(5)
with A1 = 0.165 × 1019 nm3 /h and A2 = 0.391 × 105 K. The form of Eq. (5) has been chosen on the basis of theories of isothermal coarsening [7,8]. Since Eq. (5) has been fitted to actual experimental data, the indirect T-dependence of k(f(T)) due to the variation of the volume fraction f with T (see Section 1) is allowed for. 3. Discussion As stated above the accuracy of average radii r(t1 , T1 , t2 , T2 ) calculated for coarsening during two successive isothermal ageing treatments, i.e. N = 2 in Eq. (3), is around 6% [2]. This is nearly the same accuracy as that of experimental data obtained by standard transmission electron microscopy, which is about 5% [9]. Since the calculations of rN3 , N > 2 (multiple-ageing, Eq. (3)) and re3 (continuous temperature variation, Eq. (4)) involve coarsening parameters k of more than two different temperatures, the accuracy of the
532
E. Nembach / Materials Science and Engineering A 496 (2008) 530–532
respective average radii rN and re is estimated to be somewhat lower, around 10%. Before applying Eqs. (3) and (4) to coarsening of particles in a new material, a cursory check of their accuracy must be performed; it must, e.g. be proved that no secondary particle dispersions precipitate, see Section 2. Eqs. (1)–(4) yield the average cube r 3 of the precipitate radii, whereas the average radius r¯ is what one really wants. The ratio (r 3 /¯r 3 ) (=average cube of the radii divided by the cube of the average radius) depends on the distribution function of the precipitate radii r. It has been found experimentally that after two successive isothermal Ostwald ripening treatments performed at different temperatures, the distribution function can be reasonably well represented by the function g(x) [3] which Wagner [4] and Lifshitz and Slyozov [5] derived for single-stage isothermal Ostwald ripening: g(x) =
4 2 3 x 9 3+x
for x < 1.5
and
7/3 1.5 11/3 1.5 − x g(x) = 0,
exp
for x ≥ 1.5
x x − 1.5
, (6)
where x is the normalized radius r/¯r . For g(x) the ratio (r 3 /¯r 3 ) equals 1.13 [2,3]. Hence it is assumed that also after coarsening during multi-stage ageing treatments and during continuous temperature variations (r 3 /¯r 3 ) equals 1.13. Even if multi-stage ageing disturbs the distribution function of the radii somewhat, the effect on r¯ will √ 3 be small; | 1.13 − 1.00|, e.g. amounts to only 4.2%. In general the actual distribution function tends to broaden as the volume fraction of the precipitates increases [7,8]. Marsh and Glicksman [10], however, have shown that this broadening affects the ratio (r 3 /¯r 3 ) only slightly. In advanced studies of coarsening [7,8] it was stressed that Eq. (1) is actually an asymptotic approximation holding for long ageing times. In the case of Eq. (3) this holds for each temperature Tj . Mrotzek and Nembach [2] reported that before subjecting a NIMONIC PE16 specimen to a coarsening treatment at or above 1029 K, each of them was given a ␥ -nucleation treatment: 2 h at
979 K. This applied to isothermally as well as to double-aged specimens. However, if in the case of double-ageing, T1 was 949 K, this ␥ -nucleation treatment was not performed. Even if the actual coarsening time was below 1 h (in the case of double-coarsening even as low as 0.38 h), Eqs. (1) and (3) with N = 2 described single and double isothermal coarsening well. This proves that the coarsening process adjusted itself rather fast to the new temperature. Analogously, in the case of Eq. (4) the ratio (|Ts − Te |/) must be sufficiently high. The above word of caution is reiterated: before applying Eqs. (3) and (4) to coarsening of particles in a new material, a cursory check of the applicability of these equations must be performed. 4. Conclusion On the basis of experimental data taken for single-stage isothermal Ostwald ripening (coarsening), the average radius of precipitates expected after ripening treatments performed at more than two successive different temperatures or after ripening during continuous temperature variations can be predicted with satisfactory accuracy. The relevance of this result for estimates of the lifetime of particle strengthened technical components which are exposed to high temperatures, is emphasized. References [1] E.W. Ross, C.T. Sims, in: C.T. Sims, N.S. Stoloff, W.C. Hagel (Eds.), Superalloys II, Wiley, New York, 1987, pp. 97–134. [2] M. Mrotzek, E. Nembach, Acta Mater. 56 (2008) 150–154. [3] E. Nembach, C.K. Chow, Mater. Sci. Eng. 36 (1978) 271–279. [4] C. Wagner, Z. Elektrochem. 65 (1961) 581–591. [5] I.M. Lifshitz, V.V. Slyozov, J. Phys. Chem. Solids 19 (1961) 35–50. [6] W. Ostwald, Z. Phys. Chem. 34 (1900) 495–503. [7] R. Wagner, R. Kampmann, P.W. Voorhees, in: G. Kostorz (Ed.), Phase Transformations in Materials, Wiley–VCH Verlag GmbH, Weinheim, 2001, pp. 309–410. [8] L. Ratke, P.W. Voorhees, Growth and Coarsening, Springer-Verlag, Berlin, 2002. [9] C. Schlesier, E. Nembach, Mater. Sci. Eng. A 119 (1989) 199–210. [10] S.P. Marsh, M.E. Glicksman, Acta Mater. 44 (1996) 3761–3771.
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Short communication
Coarsening of precipitates during step-wise and continuous temperature sweeps Eckhard Nembach ∗ Institut fuer Materialphysik, Universitaet Muenster, Wilhelm-Klemm-Strasse 10, 48149 Muenster, Germany
a r t i c l e
i n f o
Article history: Received 12 February 2008 Received in revised form 28 April 2008 Accepted 30 April 2008 Keywords: Coarsening Ostwald ripening Phase separation Precipitation Nanostructure
a b s t r a c t Standard theories of isothermal coarsening (Ostwald ripening) of nano-scale precipitates are adapted to describe two types of non-isothermal coarsening processes: (i) several isothermal heat treatments at different temperatures are performed in sequence and (ii) the temperature is varied continuously. Procedures are developed which allow to predict the final average size of the precipitates with an adequate accuracy. The present results are helpful for estimating the lifetime of particle strengthened technical structural components which are exposed to high temperatures. © 2008 Elsevier B.V. All rights reserved.
3
1. Introduction
Actually the original form of Eq. (1) is slightly different: r(t) = 3
Isothermal homogeneous precipitation of a new phase from a supersaturated solid solution involves three distinctly different processes: (i) nucleation of particles of the new phase, (ii) their growth and accompanying depletion of the matrix of solute, and (iii) coarsening of the precipitates. Though all three processes may occur concurrently, there are many systems, e.g. ␥ -strengthened nickel-base superalloys [1–3], in which nucleation and growth are accomplished after very short times, whereas coarsening continues infinitely. During this last stage, the average size of the precipitates increases with time: big precipitates grow at the expense of smaller ones, which dissolve, and the total volume fraction f of the precipitates stays virtually constant. This process reduces the total precipitate–matrix interface energy. Wagner [4] and Lifshitz and Slyozov [5] theoretically analyzed this coarsening process, which is often referred to as Ostwald [6] ripening, and found for the growth of the precipitates: r 3 (t) = r 3 (t = 0) + kt
(1)
where r 3 (t) is the average cube of the precipitate radii after the time t has been spent in coarsening. At its start r 3 (t) equals r 3 (t = 0). The coarsening parameter k varies with temperature T for two entirely different reasons [7,8]: (i) because, e.g. the diffusion constant of the precipitating atoms increases with T and (ii) because k tends to increase with f and f decreases as T is raised.
∗ Tel.: +49 251 83 33570; fax: +49 251 83 38346. E-mail address: [email protected]. 0921-5093/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2008.04.066
r(t = 0) + t, i.e. the cubes of the average radii r¯ are considered. The ratio [r 3 /¯r 3 ] (=average cube of the radii divided by the cube of the average radius) depends on the distribution function of the precipitate radii. If the distribution functions of the precipitate radii are the same at the start and at the end of the coarsening process, both equations are equivalent and k and differ only slightly. In the derivation of Eq. (1) the precipitates were assumed to be of spherical shape. If this is not the case like, e.g. in advanced industrial nickel-base superalloys, which are strengthened by precipitates of the ␥ -phase, r 3 in Eq. (1) is replaced by the average volume of the precipitates. After prolonged coarsening the ␥ -precipitates in superalloys are cuboidal in shape with rounded edges and corners [1]. Subsequent theoretical developments of coarsening have been reviewed in Refs. [7,8]. Even though more advanced models of coarsening are available, because of its simplicity Eq. (1) will be used in the following; it describes all relevant effects with adequate accuracy. 2. Non-isothermal coarsening It is emphasized that so far single-stage isothermal coarsening has been considered. This limits the applicability of Eq. (1) correspondingly. In many practical cases, however, the material is subjected to two or even more successive isothermal ageing treatments performed at different temperatures or to continuous temperature variations. No matter whether the service temperature T2 of a precipitate strengthened technical structural component is above or below that of the original ageing
E. Nembach / Materials Science and Engineering A 496 (2008) 530–532
temperature T1 , if T2 is sufficiently high, Ostwald ripening will continue during the component’s service life. Mrotzek and Nembach [2] dealt with coarsening during double-ageing for the times t1 and t2 at the two temperatures T1 and T2 , respectively. The volume fraction f did not exceed 0.1. The temperature dependence of f complicates the calculation of the final average precipitate radius: f decreases as T is raised. This change of f affects the precipitates’ size and number per unit volume. On the basis of Eq. (1) and k derived from data taken for single-stage isothermally aged specimens, the latter cited authors calculated the final average cube r 3 (t1 , T1 , t2 , T2 ) of the precipitate radii expected after coarsening during the two successive isothermal ageing treatments. First r 3 (t1 , T1 ) was calculated for the isothermal treatment for the time t1 at T1 . It suggests itself to calculate the final value r 3 (t1 , T1 , t2 , T2 ) with Eq. (1) by inserting r 3 (t1 , T1 ) for r 3 (t2 = 0, T2 ) and using k of T2 . However, the change of T from T1 to T2 causes one of f1 to f2 and consequently one of the average cube of the precipitate radii. The cited authors analyzed this problem theoretically and compared the results of three alternative models with experimental data taken after double-ageing the industrial superalloy NIMONIC PE16 containing spherical ␥ -precipitates and found that it is quite accurate to entirely disregard any effects of the quoted change in f on r 3 (t2 = 0, T2 ). This was the result of their Model 1. Since their two alternative models Models 2 and 3 yielded nearly the same results for r 3 (t1 , T1 , t2 , T2 ) as Model 1 did and Model 1 is the most simple one, only this latter model is considered in the present communication: at the start of the second coarsening treatment the number of the precipitates and the size of each of them are assumed to be exactly the same as at the end of the first treatment. Hence r 3 (t1 , T1 , t2 , T2 ) can be accurately calculated by inserting (i) r 3 (t1 , T1 ) for r 3 (t2 = 0, T2 ) and (ii) k(T2 ) for k into Eq. (1). “accurately” means that the average radius r(t1 , T1 , t2 , T2 ) obtained in this way agrees within about 6% with that measured by standard transmission electron microscopy. The accuracy of the latter method is around 5% [9]. Since in Model 3, Mrotzek and Nembach [2] also considered the case that the change from T1 to T2 alters the number of precipitates and disturbs the distribution function of their radii, the authors preferred the present form of Eq. (1) to the 3
3
original version r(t) = r(t = 0) + t. Even though in the presently exclusively considered Model 1, the temperature change from T1 to T2 does not affect the distribution function of the precipitate radii, for the sake of conformity with Ref. [2] the form of Eq. (1) is presently kept. As stated above, Eq. (1) has the advantage, that – in the case of non-spherical precipitates – r 3 can directly be replaced by their average volume. In the present communication extensions from Ostwald ripening during two successive isothermal ageing treatments to multi-stage ageing and to ageing during continuous temperature variations are described. The latter ones occur, e.g. during slowly heating or cooling or during temperature cycles. The technical relevance of these considerations derives from the fact that many structural components which are strengthened by nano-scale precipitates of a second phase, are used at so high temperatures that Ostwald ripening continues during their use. This has to be taken into consideration if the lifetime of such components is estimated. All subsequent derivations will be based on Eq. (1), which serves the present purpose well. If the temperature Tn+1 of the (n + 1)th heat treatment is lower than Tn , at Tn+1 a second population of very fine precipitates may nucleate in between those which grew already at Tn [2,3]. Since (i) in general the probability of nucleation is difficult to predict and (ii) such secondary precipitates may dissolve again at a subsequent treatment performed at a temperature above Tn+1 , in all following analyses it will be assumed that no secondary precipitates form.
531
2.1. Multi-stage coarsening First Ostwald ripening during multiple-ageing treatments is considered. N successive isothermal treatments are performed for the times tj at Tj , 1 ≤ j ≤ N. Generalizing the procedures developed for double-ageing [2] to multiple-ageing, one obtains with Eq. (1) for the average cube rj3 of the precipitate radii after treatment no. j: 3 + k(T )t rj3 = rj−1 j j
(2)
and for the final average cube rN3 of the precipitate radii: rN3 = r03 +
N
k(Tj )tj
(3)
j=1
At the start of coarsening at T1 , the average cube of the radii equals r03 . Evidently rN3 is independent of the sequence in which the ageing treatments are performed. 2.2. Coarsening during continuous temperature variations For the description of coarsening during the continuous temperature variation from Ts (start) to Te (end), the sum in Eq. (3) is replaced by an integral and the sweep rate v(T ) = dT/dt of the temperature is introduced. rs3 is the average cube of the radii at the start of the temperature sweep and re3 that at its end:
Te
1 k(T ) dT v(T )
re3 = rs3 + Ts
(4)
If the continuous temperature sweep is preceded by one or more isothermal ageing treatments, rs3 equals rN3 given by Eq. (3). Analogously if the sweep is followed by isothermal treatments, a term of N the form { j=1 k(Tj )tj } is added on the right hand side of Eq. (4). A sequence of sweeps is represented by a sum of integrals. 2.3. Temperature dependence of the coarsening parameter k The coarsening parameter k(T) is derived from data measured after single-stage isothermal treatments. Mrotzek and Nembach [2] published data k(T) for the industrial ␥ -strengthened nickel-base superalloy NIMONIC PE16 for the temperature range 949–1119 K. A simple approximate analytical representation of their results for k(T) is
A 2
k(T ) = A1 exp −
T
(5)
with A1 = 0.165 × 1019 nm3 /h and A2 = 0.391 × 105 K. The form of Eq. (5) has been chosen on the basis of theories of isothermal coarsening [7,8]. Since Eq. (5) has been fitted to actual experimental data, the indirect T-dependence of k(f(T)) due to the variation of the volume fraction f with T (see Section 1) is allowed for. 3. Discussion As stated above the accuracy of average radii r(t1 , T1 , t2 , T2 ) calculated for coarsening during two successive isothermal ageing treatments, i.e. N = 2 in Eq. (3), is around 6% [2]. This is nearly the same accuracy as that of experimental data obtained by standard transmission electron microscopy, which is about 5% [9]. Since the calculations of rN3 , N > 2 (multiple-ageing, Eq. (3)) and re3 (continuous temperature variation, Eq. (4)) involve coarsening parameters k of more than two different temperatures, the accuracy of the
532
E. Nembach / Materials Science and Engineering A 496 (2008) 530–532
respective average radii rN and re is estimated to be somewhat lower, around 10%. Before applying Eqs. (3) and (4) to coarsening of particles in a new material, a cursory check of their accuracy must be performed; it must, e.g. be proved that no secondary particle dispersions precipitate, see Section 2. Eqs. (1)–(4) yield the average cube r 3 of the precipitate radii, whereas the average radius r¯ is what one really wants. The ratio (r 3 /¯r 3 ) (=average cube of the radii divided by the cube of the average radius) depends on the distribution function of the precipitate radii r. It has been found experimentally that after two successive isothermal Ostwald ripening treatments performed at different temperatures, the distribution function can be reasonably well represented by the function g(x) [3] which Wagner [4] and Lifshitz and Slyozov [5] derived for single-stage isothermal Ostwald ripening: g(x) =
4 2 3 x 9 3+x
for x < 1.5
and
7/3 1.5 11/3 1.5 − x g(x) = 0,
exp
for x ≥ 1.5
x x − 1.5
, (6)
where x is the normalized radius r/¯r . For g(x) the ratio (r 3 /¯r 3 ) equals 1.13 [2,3]. Hence it is assumed that also after coarsening during multi-stage ageing treatments and during continuous temperature variations (r 3 /¯r 3 ) equals 1.13. Even if multi-stage ageing disturbs the distribution function of the radii somewhat, the effect on r¯ will √ 3 be small; | 1.13 − 1.00|, e.g. amounts to only 4.2%. In general the actual distribution function tends to broaden as the volume fraction of the precipitates increases [7,8]. Marsh and Glicksman [10], however, have shown that this broadening affects the ratio (r 3 /¯r 3 ) only slightly. In advanced studies of coarsening [7,8] it was stressed that Eq. (1) is actually an asymptotic approximation holding for long ageing times. In the case of Eq. (3) this holds for each temperature Tj . Mrotzek and Nembach [2] reported that before subjecting a NIMONIC PE16 specimen to a coarsening treatment at or above 1029 K, each of them was given a ␥ -nucleation treatment: 2 h at
979 K. This applied to isothermally as well as to double-aged specimens. However, if in the case of double-ageing, T1 was 949 K, this ␥ -nucleation treatment was not performed. Even if the actual coarsening time was below 1 h (in the case of double-coarsening even as low as 0.38 h), Eqs. (1) and (3) with N = 2 described single and double isothermal coarsening well. This proves that the coarsening process adjusted itself rather fast to the new temperature. Analogously, in the case of Eq. (4) the ratio (|Ts − Te |/) must be sufficiently high. The above word of caution is reiterated: before applying Eqs. (3) and (4) to coarsening of particles in a new material, a cursory check of the applicability of these equations must be performed. 4. Conclusion On the basis of experimental data taken for single-stage isothermal Ostwald ripening (coarsening), the average radius of precipitates expected after ripening treatments performed at more than two successive different temperatures or after ripening during continuous temperature variations can be predicted with satisfactory accuracy. The relevance of this result for estimates of the lifetime of particle strengthened technical components which are exposed to high temperatures, is emphasized. References [1] E.W. Ross, C.T. Sims, in: C.T. Sims, N.S. Stoloff, W.C. Hagel (Eds.), Superalloys II, Wiley, New York, 1987, pp. 97–134. [2] M. Mrotzek, E. Nembach, Acta Mater. 56 (2008) 150–154. [3] E. Nembach, C.K. Chow, Mater. Sci. Eng. 36 (1978) 271–279. [4] C. Wagner, Z. Elektrochem. 65 (1961) 581–591. [5] I.M. Lifshitz, V.V. Slyozov, J. Phys. Chem. Solids 19 (1961) 35–50. [6] W. Ostwald, Z. Phys. Chem. 34 (1900) 495–503. [7] R. Wagner, R. Kampmann, P.W. Voorhees, in: G. Kostorz (Ed.), Phase Transformations in Materials, Wiley–VCH Verlag GmbH, Weinheim, 2001, pp. 309–410. [8] L. Ratke, P.W. Voorhees, Growth and Coarsening, Springer-Verlag, Berlin, 2002. [9] C. Schlesier, E. Nembach, Mater. Sci. Eng. A 119 (1989) 199–210. [10] S.P. Marsh, M.E. Glicksman, Acta Mater. 44 (1996) 3761–3771.