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Simulation of the growth of copper critical nucleus in dilute bcc Fe–Cu alloys
Scripta Materialia 55 (2006) 223–226 www.actamat-journals.com
Simulation of the growth of copper critical nucleus in dilute bcc Fe–Cu alloys Takatoshi Nagano and Masato Enomoto* Department of Materials Science and Engineering, Ibaraki University, 4-12-1, Nakanarusawa, Hitachi 316-8511, Japan Received 12 January 2006; revised 14 April 2006; accepted 17 April 2006 Available online 15 May 2006
The composition of Cu particles formed during isothermal aging in bcc Fe–Cu alloys was simulated using the Cahn–Hilliard nucleation theory and non-linear diffusion equation. Whereas Cu nuclei are almost pure Cu in low Cu content alloys, the Cu concentration in the nucleus is significantly less than unity in higher Cu content alloys. During aging it increases almost to unity at an early stage and the increase in particle radius follows. Experimental evidence in the latter alloys is yet to be obtained. 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Simulation; Nucleation; Growth; Iron alloys; Copper
Copper precipitation in steel has attracted considerable attention for many years because of its marked age hardening behavior in low carbon, low alloy ferritic and martensitic steels [1], as a major cause of radiation embrittlement of pressure vessel steels [2] and more recently, from the concern over environmental issues, that is, the need for efficient use of ferrous scrap. The morphology, crystallography and kinetics of Cu precipitation in iron are well documented [3–5], including the structural transformation that occurs during the growth of Cu particles [6,7]. Concerning the composition of Cu particles at relatively early stages of precipitation, opposing results are available. Some authors reported on the basis of the results of atom probe field ion microscopy (APFIM) analysis that Cu particles contained a significant proportion of iron [2,8]. In contrast, other authors concluded, using other techniques, e.g., small angle neutron scattering (SANS) [9] and positron annihilation [10], that the composition of the particles is almost pure copper. It is well known that Cu precipitates are formed initially as coherent body-centered cubic (bcc) clusters in the aFe (ferrite) matrix, undergo bcc ! 9R ! face-centered cubic (fcc) structural transformation [11], and continue to grow as fcc particles. The time dependence of the composition of Cu particles has not been fully discussed. In other words, the comparison of Cu concentrations between different experimental techniques has * Corresponding author. E-mail: [email protected]
to be made taking the alloy composition and aging time into account. The change in the free energy accompanying the formation of a critical nucleus of Cu in the aFe matrix is written as [12]: Z ð1Þ DF ¼ ½Df ðcÞ þ jðrcÞ2 dm; m
where
of Df ðcÞ ¼ f ðcÞ f ðc Þ ðc c Þ ; oc c0 0
0
ð2Þ
f(c) is the free energy per unit volume of the homogeneous solid solution of composition c. In the presence of coherency strain (isotropic elasticity is assumed) it is given by the equation, 0 þ cf 0Cu þ RT fc ln c þ ð1 cÞ lnð1 cÞg f ¼ ð1 cÞfFe E 2 ðc c0 Þ þ Lcð1 cÞ þ g2 ð3Þ 1m 0 0 where fFe and fCu are, respectively, the free energy of pure bcc Fe and bcc Cu, c0 is the bulk Cu concentration and j is the gradient energy coefficient, L is the regular solution coefficient (=39 258 4.17 T [13]), g = (1/ a0)(da/dc) (=9.5 · 105 nm per at.% Cu [14]), a0 is the lattice parameter at c0, E is the Young’s modulus, m is the Poisson’s ratio and RT has its usual meaning. Under the assumption that j is independent of the concentration, the concentration profile of Cu nucleus is calculated from the Euler equation of variational principle,
1359-6462/$ - see front matter 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2006.04.015
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T. Nagano, M. Enomoto / Scripta Materialia 55 (2006) 223–226
o2 c 4j oc oDf ¼ ð4Þ þ or2 r or oc with the boundary conditions, oc ¼ 0 at r ¼ 0 and r ¼ 1 or ð5Þ c ¼ c0 at r ¼ 1 In a bcc solid solution the first and second neighbor interactions have to be taken into account. The gradient energy coefficient can be related to the regular solution interaction coefficient as j ¼ 0:14 La20 =V m for a bcc solid solution [15] assuming constant interaction energies, where Vm is the molar volume of the solution. The Cahn–Hilliard diffusion equation is written as [16], oc o2 v 2 oc dM ov ¼ r fMðcÞrvg ¼ M 2 þ M þ ð6Þ ot or r or dc or 2j
where M is the mobility given by, MðcÞ ¼ fM Cu ð1 cÞ þ M Fe cgcð1 cÞ
ð7Þ
profile from that of the critical nucleus was less than 10% of kT. The incubation time is defined as the time at which the nucleus evolves through the region in which the free energy of the nucleus is within kT from that of the critical nucleus. Thus, the assumed initial profile is considered to be reasonably close to the profile of the critical nucleus. Figure 2 shows the variation with time of the Cu concentration at the center of the nucleus c(r = 0), denoted cN, for the case of contraction. It stays at the initial value for some time and decreases to the bulk concentration c0 fairly rapidly once it starts to decrease. Figure 3(a) and (b) show the results for the case of growth. Whilst in the 1% Cu alloy cN is close to unity and the radius started to increase from the beginning, in the 3% Cu alloy cN increased first without a significant increase in particle radius and then the increase in radius occurred. It is seen in Figure 4 that the time for cN to reach almost unity (denoted t1) is 200 s in the 3Cu alloy. It can be shown that the variation of j that arises from the change in the assumed interaction energies, e.g., the ratio of the first and second interaction energies
and MFe and MCu are the atomic mobility of Fe and Cu, respectively, which can be related to the intrinsic diffusion coefficient [17]. v is the diffusion potential defined by, oDf 2jr2 c ð8Þ oc Eq. (6) was solved by the finite difference method with the grid size Dr = 0.01 nm. The size of the system was taken to be so large that it had a very small influence on the results, e.g., 0.1% for the radius of a Cu nucleus at the longest simulation time. Figure 1 illustrates the Cu profiles of critical nuclei calculated at 550 C in Fe-1.0 and 3.0 wt.% Cu alloys. To begin with, calculation was conducted ignoring coherency strain. Whilst in the 1% Cu alloy the Cu concentration is almost unity in a sizable range, in the latter alloy it is significantly less than pure Cu even at the center of the nucleus. This tendency increases as the bulk concentration of Cu is increased at a constant temperature, and as the temperature is lowered at a fixed bulk Cu content [12]. Neither growth nor dissolution takes place from the concentration profile of a critical nucleus. Thus, somewhat arbitrarily, simulation was started with the profile of critical nucleus moved to the left or right by one grid. The difference of the free energy associated with this
v¼
Figure 1. Concentration profile of coherent Cu nuclei in bcc iron matrix calculated from Cahn–Hilliard non-classical nucleation theory.
Figure 2. Variation of Cu concentration at the center of a nucleus cN with time for the case of dissolution.
Figure 3. Time evolution of Cu concentration profile for the case of growth, calculated for (a) a Fe–1Cu alloy, and (b) a Fe–3Cu alloy.
T. Nagano, M. Enomoto / Scripta Materialia 55 (2006) 223–226
Figure 4. Variation of cN with time calculated at 550 C in Fe–1Cu and 3Cu alloys.
in the bcc solid solution, does not have a significant influence on the j value and simulation results. Figure 5 illustrates the concentration profile of Cu nuclei in a Fe–1.5 wt.%Cu alloy calculated at 400 C with and without coherency strain. The critical temperature of the miscibility gap is lowered from 1587 C to 1503 C and, as a result, the size of the critical nucleus was increased in the presence of coherency strain. The evolution of the Cu profile of these nuclei was simulated at the same temperature and the variation of cN with time is shown in Figure 6. It is seen that t1 became shorter when coherency strain was included, whereas the difference in cN did not become larger than 0.05 for most of the aging time. Goodman et al. [8] and Worrall et al. [2] conducted APFIM analyses and reported that the Cu concentration of precipitates was considerably smaller than unity in dilute Fe–Cu alloys aged at 500 C for 3 h. cN is calculated to be 0.94 in a 1.4 Cu alloy studied by Worrall
225
et al. [2]. From the curves in Figure 6, t1 is expected to be much less than a few hours since the diffusivity of Cu is nearly two orders of magnitude greater at 500 C. Thus, the Cu concentrations of the particles in these reports are likely almost pure Cu. The precipitation of Cu particles has been studied in a Fe–1.5Cu alloy by positron annihilation [18]. The Wparameter, the ratio of high momentum region to the total region in the coincidence Doppler broadening (CDB) spectrum, varies with the species of element by which positrons are captured and annihilated. Figure 7 shows the variation with aging time of the W-parameter obtained from the CDB spectra after aging for 1.2 ks at various temperatures. At 500 C and 600 C it is seen to be close to the value of pure Cu and this may indicate that the Cu particles did not contain significant numbers of Fe atoms. At lower temperatures the W-parameter decreased toward pure iron. However, this does not necessarily imply that Cu particles contain Fe atoms. Even if the particles do not contain Fe atoms, it can deviate from the value of pure Cu when the number of precipitates is not enough so that positrons are annihilated in the Fe matrix. It can also be smaller than the value of pure Cu when the precipitates are not large enough so that positrons are not totally confined within the precipitates. The critical diameter for this is 1 nm [10]. Under these circumstances the composition and size of Cu particles were calculated at each temperature and results are shown in Figure 8(a) and (b). Here, the particle radius was defined by the half width of the concentration profile [12]. It is seen that after aging for 1.2 ks the concentration and radius of Cu particles were not much different from those of critical nuclei except at higher temperatures. It is also noted that the radius of a non-classical nucleus first decreased and then, increased with increasing undercooling [12] (dashed curves in Figure 8(a)). The slight decrease in the particle radius at t = 1.2 ks from the critical radius at 300 C and 400 C (Fig. 8(a)) was because the Cu concentration near the center of the nucleus increased by diffusion of Cu atoms from the outer region of the nucleus. In an fcc Cu–Co alloy it was shown that the concentration profiles of nucleus calculated using the Cahn– Hilliard continuum and discrete lattice point approaches were similar at temperatures as low as T/Tc 0.25 [19]. Assuming that it is also the case with a bcc alloy, the present approach may at least give a qualitative account
Figure 5. Concentration profile of bcc Cu nuclei with and without incorporating coherency strain.
Figure 6. Influence of coherency strain on the time evolution of cN.
Figure 7. W-parameter of coincidence Dopler broadening spectrum of positron annihilation measured in a Fe–1.5Cu solution treated at 800 C and aged for 20 min at each temperature [18].
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T. Nagano, M. Enomoto / Scripta Materialia 55 (2006) 223–226
content alloys the Cu concentration increases to unity at first in the interior of the nucleus without a significant increase in the particle radius and then the particle radius starts to increase. The time for the nucleus to reach almost pure Cu is a few tens of seconds at a higher temperature in high Cu alloys, whereas it becomes very long at lower temperatures. The present treatment may be useful in simulating growth behavior. However, it may not yield an exact quantitative account of the evolution of the concentration in Cu particles when the particle size (diameter) is significantly less than a nanometer.
Figure 8. (a) Radius and (b) cN of critical nuclei (open circles) and those after holding for 20 min (solid circles) calculated for a Fe–1.5Cu alloy.
of the Cu concentration in the nuclei if the temperature is not too low, say, above 300 C (Tc = 1587C in the absence of strain energy), where the particles may contain more than 10 Cu atoms if it is pure Cu. Whereas the variation of cN with aging temperature (Fig. 8(b)) qualitatively agrees with the variation of the experimental W-parameter, the particles are likely to be too small to totally confine the positrons within them. Thus, one can not unequivocally conclude the composition of these particles. It is tentatively concluded that Cu particles formed at 200–400 C may have contained a significant proportion of Fe atoms because the critical nucleus composition is very low and diffusion of atoms to achieve local equilibrium is extremely sluggish at these temperatures. In the above discussion the incubation time for nucleation was ignored. According to the classical nucleation theory, the incubation time is calculated from the equation [20], tinc ¼
8kT ra4 v2a ðDGv Þ2 Dc
ð9Þ
where va is the volume of one atom, DGv is the driving force for nucleation, and k is the Boltzmann constant. Assuming that the particle/matrix interfacial energy is r = 0.28 J/m2, which was calculated from the nearest neighbor broken bond model [21], it amounts to 103 s and 105 s at 550 C and 400 C, respectively. This is comparable with the time for dissolution of a critical nucleus in Figure 2. Actually, however, Cu particles are observed at a shorter aging time probably because a considerable amount of defects are introduced during quenching from solution temperature and reheating. Thus, the incubation time is likely considerably less than theoretical prediction, although it may not be totally neglected. In summary, in low Cu content alloys the composition of Cu nuclei is close to pure Cu and the particle radius increases slowly from the beginning. In higher Cu
This investigation was conducted as a part of the research activities of the Nanometal project under the auspices of the Japan Research and Development Center for Metals (JRCM), Tokyo. The authors are thankful to Dr. Nagai, Institute of Materials Research, Tohoku University for valuable discussions. [1] N. Maruyama, M. Sugiyama, T. Hara, H. Tamehiro, Mater. Trans. JIM 40 (1999) 268. [2] G.M. Worral, J.T. Buswell, C.A. English, M.G. Hetherington, G.D.W. Smith, J. Nucl. Mater. 148 (1987) 107. [3] E. Hornbogen, R.C. Glenn, Trans. TMS-AIME 218 (1960) 1064. [4] U. Dahmen, P. Ferguson, K.H. Westmacott, Acta Metall. 32 (1984) 803. [5] G.R. Speich, R.A. Oriani, Trans. TMS-AIME 233 (1965) 623. [6] P.J. Othen, M.L. Jenkins, G.D.W. Smith, Philos. Mag. 70 (1994) 1. [7] R. Monzen, M.L. Jenkins, A.P. Sutton, Philos. Mag. A 80 (2000) 137. [8] S.R. Goodman, S.S. Brenner, J.R. Low, Metall. Trans. 4 (1973) 2371. [9] K. Osamura, H. Okuda, M. Takashima, K. Asano, M. Furusaka, Mater. Trans. JIM 34 (1993) 305. [10] Y. Nagai, M. Hasegawa, Z. Tang, A. Hempel, K. Yubuta, T. Shimamura, Y. Kawazoe, A. Kawai, F. Kano, Phys. Rev. B 61 (2000) 6574. [11] Y. Le Bauer, Acta Mater. 49 (2001) 2661. [12] J.W. Cahn, J.E. Hilliard, J. Chem. Phys. 31 (1959) 688. [13] B. Sundman (Ed.), SGTE solution database, Royal Institute of Technology, Stockholm, 1992. [14] L.J. Swartzendruber, Phase Diagrams of Binary Iron Alloys, in: H. Okamoto (Ed.), ASM, Metals Park, OH, 1993, p. 131. [15] C. Zhang, M. Enomoto, T. Yamashita, N. Sano, Metall. Trans. A 35A (2004) 1263. [16] J.E. Hilliard, Phase Transformations, in: H.I. Aaronson (Ed.), ASM, Metals Park, OH, 1970, p. 497. [17] E.A. Brandes, G.B. Brook (Eds.), Smithells Metals Reference Book, seventh ed., Butterworth–Heinemann, Oxford, 1992, pp. 1–13. [18] H. Yasuhara, S. Sato, Y. Hosoda, Y. Nagai, M. Hasegawa, Unpublished research at JFE Steel Corporation and the Oarai Branch, Institute for Materials Research, Tohoku University, 2004. [19] F.K. LeGoues, Y.W. Lee, H.I. Aaronson, Acta Metall. 32 (1984) 1837. [20] H.I. Aaronson, J.K. Lee, Lectures on the Theory of Phase Transformations, in: H.I. Aaronson (Ed.), second ed., TMS, Warrendale, PA, 1999, p. 165. [21] Z.G. Yang, M. Enomoto, Mater. Sci. Eng. A 332 (2002) 184.
Simulation of the growth of copper critical nucleus in dilute bcc Fe–Cu alloys Takatoshi Nagano and Masato Enomoto* Department of Materials Science and Engineering, Ibaraki University, 4-12-1, Nakanarusawa, Hitachi 316-8511, Japan Received 12 January 2006; revised 14 April 2006; accepted 17 April 2006 Available online 15 May 2006
The composition of Cu particles formed during isothermal aging in bcc Fe–Cu alloys was simulated using the Cahn–Hilliard nucleation theory and non-linear diffusion equation. Whereas Cu nuclei are almost pure Cu in low Cu content alloys, the Cu concentration in the nucleus is significantly less than unity in higher Cu content alloys. During aging it increases almost to unity at an early stage and the increase in particle radius follows. Experimental evidence in the latter alloys is yet to be obtained. 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Simulation; Nucleation; Growth; Iron alloys; Copper
Copper precipitation in steel has attracted considerable attention for many years because of its marked age hardening behavior in low carbon, low alloy ferritic and martensitic steels [1], as a major cause of radiation embrittlement of pressure vessel steels [2] and more recently, from the concern over environmental issues, that is, the need for efficient use of ferrous scrap. The morphology, crystallography and kinetics of Cu precipitation in iron are well documented [3–5], including the structural transformation that occurs during the growth of Cu particles [6,7]. Concerning the composition of Cu particles at relatively early stages of precipitation, opposing results are available. Some authors reported on the basis of the results of atom probe field ion microscopy (APFIM) analysis that Cu particles contained a significant proportion of iron [2,8]. In contrast, other authors concluded, using other techniques, e.g., small angle neutron scattering (SANS) [9] and positron annihilation [10], that the composition of the particles is almost pure copper. It is well known that Cu precipitates are formed initially as coherent body-centered cubic (bcc) clusters in the aFe (ferrite) matrix, undergo bcc ! 9R ! face-centered cubic (fcc) structural transformation [11], and continue to grow as fcc particles. The time dependence of the composition of Cu particles has not been fully discussed. In other words, the comparison of Cu concentrations between different experimental techniques has * Corresponding author. E-mail: [email protected]
to be made taking the alloy composition and aging time into account. The change in the free energy accompanying the formation of a critical nucleus of Cu in the aFe matrix is written as [12]: Z ð1Þ DF ¼ ½Df ðcÞ þ jðrcÞ2 dm; m
where
of Df ðcÞ ¼ f ðcÞ f ðc Þ ðc c Þ ; oc c0 0
0
ð2Þ
f(c) is the free energy per unit volume of the homogeneous solid solution of composition c. In the presence of coherency strain (isotropic elasticity is assumed) it is given by the equation, 0 þ cf 0Cu þ RT fc ln c þ ð1 cÞ lnð1 cÞg f ¼ ð1 cÞfFe E 2 ðc c0 Þ þ Lcð1 cÞ þ g2 ð3Þ 1m 0 0 where fFe and fCu are, respectively, the free energy of pure bcc Fe and bcc Cu, c0 is the bulk Cu concentration and j is the gradient energy coefficient, L is the regular solution coefficient (=39 258 4.17 T [13]), g = (1/ a0)(da/dc) (=9.5 · 105 nm per at.% Cu [14]), a0 is the lattice parameter at c0, E is the Young’s modulus, m is the Poisson’s ratio and RT has its usual meaning. Under the assumption that j is independent of the concentration, the concentration profile of Cu nucleus is calculated from the Euler equation of variational principle,
1359-6462/$ - see front matter 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2006.04.015
224
T. Nagano, M. Enomoto / Scripta Materialia 55 (2006) 223–226
o2 c 4j oc oDf ¼ ð4Þ þ or2 r or oc with the boundary conditions, oc ¼ 0 at r ¼ 0 and r ¼ 1 or ð5Þ c ¼ c0 at r ¼ 1 In a bcc solid solution the first and second neighbor interactions have to be taken into account. The gradient energy coefficient can be related to the regular solution interaction coefficient as j ¼ 0:14 La20 =V m for a bcc solid solution [15] assuming constant interaction energies, where Vm is the molar volume of the solution. The Cahn–Hilliard diffusion equation is written as [16], oc o2 v 2 oc dM ov ¼ r fMðcÞrvg ¼ M 2 þ M þ ð6Þ ot or r or dc or 2j
where M is the mobility given by, MðcÞ ¼ fM Cu ð1 cÞ þ M Fe cgcð1 cÞ
ð7Þ
profile from that of the critical nucleus was less than 10% of kT. The incubation time is defined as the time at which the nucleus evolves through the region in which the free energy of the nucleus is within kT from that of the critical nucleus. Thus, the assumed initial profile is considered to be reasonably close to the profile of the critical nucleus. Figure 2 shows the variation with time of the Cu concentration at the center of the nucleus c(r = 0), denoted cN, for the case of contraction. It stays at the initial value for some time and decreases to the bulk concentration c0 fairly rapidly once it starts to decrease. Figure 3(a) and (b) show the results for the case of growth. Whilst in the 1% Cu alloy cN is close to unity and the radius started to increase from the beginning, in the 3% Cu alloy cN increased first without a significant increase in particle radius and then the increase in radius occurred. It is seen in Figure 4 that the time for cN to reach almost unity (denoted t1) is 200 s in the 3Cu alloy. It can be shown that the variation of j that arises from the change in the assumed interaction energies, e.g., the ratio of the first and second interaction energies
and MFe and MCu are the atomic mobility of Fe and Cu, respectively, which can be related to the intrinsic diffusion coefficient [17]. v is the diffusion potential defined by, oDf 2jr2 c ð8Þ oc Eq. (6) was solved by the finite difference method with the grid size Dr = 0.01 nm. The size of the system was taken to be so large that it had a very small influence on the results, e.g., 0.1% for the radius of a Cu nucleus at the longest simulation time. Figure 1 illustrates the Cu profiles of critical nuclei calculated at 550 C in Fe-1.0 and 3.0 wt.% Cu alloys. To begin with, calculation was conducted ignoring coherency strain. Whilst in the 1% Cu alloy the Cu concentration is almost unity in a sizable range, in the latter alloy it is significantly less than pure Cu even at the center of the nucleus. This tendency increases as the bulk concentration of Cu is increased at a constant temperature, and as the temperature is lowered at a fixed bulk Cu content [12]. Neither growth nor dissolution takes place from the concentration profile of a critical nucleus. Thus, somewhat arbitrarily, simulation was started with the profile of critical nucleus moved to the left or right by one grid. The difference of the free energy associated with this
v¼
Figure 1. Concentration profile of coherent Cu nuclei in bcc iron matrix calculated from Cahn–Hilliard non-classical nucleation theory.
Figure 2. Variation of Cu concentration at the center of a nucleus cN with time for the case of dissolution.
Figure 3. Time evolution of Cu concentration profile for the case of growth, calculated for (a) a Fe–1Cu alloy, and (b) a Fe–3Cu alloy.
T. Nagano, M. Enomoto / Scripta Materialia 55 (2006) 223–226
Figure 4. Variation of cN with time calculated at 550 C in Fe–1Cu and 3Cu alloys.
in the bcc solid solution, does not have a significant influence on the j value and simulation results. Figure 5 illustrates the concentration profile of Cu nuclei in a Fe–1.5 wt.%Cu alloy calculated at 400 C with and without coherency strain. The critical temperature of the miscibility gap is lowered from 1587 C to 1503 C and, as a result, the size of the critical nucleus was increased in the presence of coherency strain. The evolution of the Cu profile of these nuclei was simulated at the same temperature and the variation of cN with time is shown in Figure 6. It is seen that t1 became shorter when coherency strain was included, whereas the difference in cN did not become larger than 0.05 for most of the aging time. Goodman et al. [8] and Worrall et al. [2] conducted APFIM analyses and reported that the Cu concentration of precipitates was considerably smaller than unity in dilute Fe–Cu alloys aged at 500 C for 3 h. cN is calculated to be 0.94 in a 1.4 Cu alloy studied by Worrall
225
et al. [2]. From the curves in Figure 6, t1 is expected to be much less than a few hours since the diffusivity of Cu is nearly two orders of magnitude greater at 500 C. Thus, the Cu concentrations of the particles in these reports are likely almost pure Cu. The precipitation of Cu particles has been studied in a Fe–1.5Cu alloy by positron annihilation [18]. The Wparameter, the ratio of high momentum region to the total region in the coincidence Doppler broadening (CDB) spectrum, varies with the species of element by which positrons are captured and annihilated. Figure 7 shows the variation with aging time of the W-parameter obtained from the CDB spectra after aging for 1.2 ks at various temperatures. At 500 C and 600 C it is seen to be close to the value of pure Cu and this may indicate that the Cu particles did not contain significant numbers of Fe atoms. At lower temperatures the W-parameter decreased toward pure iron. However, this does not necessarily imply that Cu particles contain Fe atoms. Even if the particles do not contain Fe atoms, it can deviate from the value of pure Cu when the number of precipitates is not enough so that positrons are annihilated in the Fe matrix. It can also be smaller than the value of pure Cu when the precipitates are not large enough so that positrons are not totally confined within the precipitates. The critical diameter for this is 1 nm [10]. Under these circumstances the composition and size of Cu particles were calculated at each temperature and results are shown in Figure 8(a) and (b). Here, the particle radius was defined by the half width of the concentration profile [12]. It is seen that after aging for 1.2 ks the concentration and radius of Cu particles were not much different from those of critical nuclei except at higher temperatures. It is also noted that the radius of a non-classical nucleus first decreased and then, increased with increasing undercooling [12] (dashed curves in Figure 8(a)). The slight decrease in the particle radius at t = 1.2 ks from the critical radius at 300 C and 400 C (Fig. 8(a)) was because the Cu concentration near the center of the nucleus increased by diffusion of Cu atoms from the outer region of the nucleus. In an fcc Cu–Co alloy it was shown that the concentration profiles of nucleus calculated using the Cahn– Hilliard continuum and discrete lattice point approaches were similar at temperatures as low as T/Tc 0.25 [19]. Assuming that it is also the case with a bcc alloy, the present approach may at least give a qualitative account
Figure 5. Concentration profile of bcc Cu nuclei with and without incorporating coherency strain.
Figure 6. Influence of coherency strain on the time evolution of cN.
Figure 7. W-parameter of coincidence Dopler broadening spectrum of positron annihilation measured in a Fe–1.5Cu solution treated at 800 C and aged for 20 min at each temperature [18].
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T. Nagano, M. Enomoto / Scripta Materialia 55 (2006) 223–226
content alloys the Cu concentration increases to unity at first in the interior of the nucleus without a significant increase in the particle radius and then the particle radius starts to increase. The time for the nucleus to reach almost pure Cu is a few tens of seconds at a higher temperature in high Cu alloys, whereas it becomes very long at lower temperatures. The present treatment may be useful in simulating growth behavior. However, it may not yield an exact quantitative account of the evolution of the concentration in Cu particles when the particle size (diameter) is significantly less than a nanometer.
Figure 8. (a) Radius and (b) cN of critical nuclei (open circles) and those after holding for 20 min (solid circles) calculated for a Fe–1.5Cu alloy.
of the Cu concentration in the nuclei if the temperature is not too low, say, above 300 C (Tc = 1587C in the absence of strain energy), where the particles may contain more than 10 Cu atoms if it is pure Cu. Whereas the variation of cN with aging temperature (Fig. 8(b)) qualitatively agrees with the variation of the experimental W-parameter, the particles are likely to be too small to totally confine the positrons within them. Thus, one can not unequivocally conclude the composition of these particles. It is tentatively concluded that Cu particles formed at 200–400 C may have contained a significant proportion of Fe atoms because the critical nucleus composition is very low and diffusion of atoms to achieve local equilibrium is extremely sluggish at these temperatures. In the above discussion the incubation time for nucleation was ignored. According to the classical nucleation theory, the incubation time is calculated from the equation [20], tinc ¼
8kT ra4 v2a ðDGv Þ2 Dc
ð9Þ
where va is the volume of one atom, DGv is the driving force for nucleation, and k is the Boltzmann constant. Assuming that the particle/matrix interfacial energy is r = 0.28 J/m2, which was calculated from the nearest neighbor broken bond model [21], it amounts to 103 s and 105 s at 550 C and 400 C, respectively. This is comparable with the time for dissolution of a critical nucleus in Figure 2. Actually, however, Cu particles are observed at a shorter aging time probably because a considerable amount of defects are introduced during quenching from solution temperature and reheating. Thus, the incubation time is likely considerably less than theoretical prediction, although it may not be totally neglected. In summary, in low Cu content alloys the composition of Cu nuclei is close to pure Cu and the particle radius increases slowly from the beginning. In higher Cu
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