Development of “Macroscopic Composition Gradient Method” and its application to the phase transformation

Progress in Materials Science 57 (2012) 1010–1060

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Development of ‘‘Macroscopic Composition Gradient Method’’ and its application to the phase transformation Toru Miyazaki ⇑ Nagoya Institute of Technology, Nagoya 466-8555, Japan

a r t i c l e

i n f o

Article history: Received 30 June 2009 Received in revised form 3 May 2011 Accepted 10 November 2011 Available online 2 December 2011

a b s t r a c t A new characterization method, ‘‘Macroscopic Composition Gradient (MCG) Method’’ is proposed to investigate the phase transformations near the phase boundaries. The MCG method is a new technique to investigate the phase transformations in various composition alloys by utilizing a single specimen having the macroscopic solute composition gradient. Since the macroscopic composition gradient in the MCG alloy is so prepared as to cross over the phase boundary, the morphological transition of critical phenomena at the phase boundary can continuously be investigated by means of analytical transmission electron microscopy. By utilizing the MCG method, the various kinds of phase transformation, such as the coherent and incoherent precipitation boundaries, the order/disorder phase transition and the morphological change at the spinodal line have successfully been investigated. Furthermore, to an important thing, the critical size of precipitate-nucleus and the nucleation rate near the solubility limit can be experimentally obtained for respective nucleus. The phase decomposition of supersaturated solid solution progresses by a mechanism of spinodal decomposition even in the N-G region of phase diagram. On the basis of experimental results, the application limit of the conventional nucleation theory is investigated, and hence the failure of Boltzmann–Gibbs free energy becomes obvious in the early stage of phase decomposition. It is noteworthy that the present experiment is systematically conducted for the alloy composition range very close to the solubility limit. Such critical phenomena of phase transformation have

⇑ Address: 973-223, Minamiyama, Komenoki-cho, Nisshin 470-0111, Japan. Tel.: +81 561 73 7688; fax: +81 561 73 2242. E-mail address: [email protected] 0079-6425/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.pmatsci.2011.11.002

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been scarcely examined in the past. The MCG method proposed here is considered to open a new way to investigate the critical phenomena in the phase boundary. Ó 2011 Elsevier Ltd. All rights reserved.

Contents 1.

2.

3.

4.

5.

6.

The outline of macroscopic compositional gradient (MCG) method . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. General Idea of MCG method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Preparation of MCG alloy specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Experimental procedures and a typical example of TEM microstructure in MCG Alloys . . . . 1.3.1. Experimental procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2. A typical example of TEM microstructure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Influences of macroscopic composition gradient (MCG) on the phase decomposition . . . . . . . . . . . . 2.1. Profile change of MCG specimen during aging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Influences of existence of MCG on phase transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. The macroscopic composition gradient energy, E grad macro . . . . . . . . . . . . . . . . . . . . . . . . . .......................... 2.2.2. The gradient strain energy of microstructure, E stgrad str 2.3. Computer simulations of microstructure formation based on phase field method. . . . . . . . . 2.3.1. Theoretical basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Computer simulated microstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Experimental verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transition of microstructures at phase boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Determination of coherent precipitation line of Ni–V and Ni–Mo alloy systems . . . . . . . . . . 3.1.1. Ni–V alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. Ni–Mo alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Difference of the solubility Limits between the coherent and incoherent precipitation . . . . 3.3. Microstructure continuity between spinodal and N-G phase decompositions . . . . . . . . . . . . 3.4. A2/B2 order/disorder transition in Fe–Al ordering alloy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Precipitate-nucleation near the edge of miscibility gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Microstructure changes with aging in varies alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Evaluations of nucleus-stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Calculation of the critical stable nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Thermodynamic discussion on the basis of conventional nucleation theory . . . . . . . . . . . . . Kinetic investigations on the nucleation process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Theoretical basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Kinetic investigation on the experimental results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Pre-nucleation phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. High speed growth of big nucleus near the solubility limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6. Problems of nucleus formation in the N-G region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Method of ‘‘composition vs. distance’’ curve in the MCG specimen . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1011 1011 1014 1015 1015 1015 1015 1016 1017 1018 1018 1021 1021 1022 1023 1023 1023 1024 1026 1026 1027 1029 1030 1030 1032 1038 1042 1046 1046 1047 1051 1053 1054 1055 1056 1057 1057 1059

1. The outline of macroscopic compositional gradient (MCG) method 1.1. General Idea of MCG method The comprehensive description of phase transformations should be realized in a form of three dimensional diagram consisting of the temperature (T), time (t) and composition (c) axes [1–4] as shown in Fig. 1. The section parallel to the temperature (T) and the composition (c) axes is well known

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Nomenclature cA cB ca ce ce(r) ce(1) cp

composition of A atom composition of B atom local average composition equilibrium solute concentration at the interface of particle equilibrium solute concentration at interface of particle with radius r equilibrium solute concentration at interface of particle with the infinite size composition of precipitate

Egrad macro

macroscopic composition gradient energy (see Eq. (1))

Estgrad str

gradient elastic strain energy (see Eq. (12))

X eTij hriIðHÞ r1 ij ðXÞ V Vm N Eincl

volume of particle eigenstrain internal stress in precipitate particle internal stress of a particle existing in the infinite matrix volume of alloy molar volume of precipitate number of particle elastic strain energy of a single particle in infinite matrix (see Eq. (4))

Ehom str Gsystem Gc Esurf Estr ci(r) sj(r)

R T Qd

elastic strain energy per unit volume of the composition flat alloy (see Eq. (5)) total free energy of microstructure (see Eq. (13)) chemical free energy interfacial energy elastic strain energy composition parameter of ith component at position r structure parameter of jth component at position r chemical potential surface potential strain potential diffusion potential mobility of atoms inter-diffusion coefficient gradient energy coefficient expansion coefficient with respect to the solute concentration elastic constant radius of precipitates minimum radius of a critical stable precipitate (radius of stable nucleus) interfacial energy density between particle and matrix gas constant temperature activation energy for diffusion of solute atoms

DG(c) U U1 Dc a0 DF m

nucleation energy barrier that must be overcome to form nucleus frequency of nucleation time to nucleation (see Fig. 42 and Table 4) degree of super-saturation lattice parameter energy barrier due to the convex part of free energy–composition curve (see Fig. 46)

lc lsurf lstr v M e D

j g Y r r

cs

as the phase diagram, and the section parallel to the temperature (T) and the time (t) axes is the TTT diagram. A section parallel to the composition (c) and time (t) axes is also important, nevertheless it has not attracted attention yet. In the T–t–c diagram, the experiments whose variables are the T-axis and t-axis are well known as the thermal analysis and the isothermal aging, respectively. Since the

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α α1 + α 2

) am am gr iagr a i d td T T- -T(T 0

Temperature,T

T-c diagram (Phase diagram)

1.0

tim

e, t

c-t diagram

Composition,c Fig. 1. A comprehensive description of phase transformation in a form of three dimensional diagram with temperature (T), time (t) and composition (c) axes.

c-axis is also an important element in the T–t–c diagram, an experiment whose variable is composition c should take more attention. The other view point of phase transformation should be taken into consideration. It has widely been accepted that the researchers pay attention to a typical phenomenon in the realistic complex phenomena and try to linearize it in order to understand it simply. However, since the most of phase transformations may include the non-linear part, the conventional treatment based on the linear theory often gives insufficient results. The non-linearity is very important to understand the phase transformation in the vicinity of phase boundary. Nevertheless, the critical phenomena have hardly been examined on the materials science, particularly in the phase transformation phenomena. A characterization method to investigate the critical phenomena in the vicinity of phase boundary should be developed. On the basis of two considerations above described, we proposed a new characterization method of phase transformation, that is ‘‘Macroscopic Composition Gradient (MCG) Method’’ [4–6]. This is a new characterization method of microstructure by utilizing a macroscopic composition gradient introduced into alloys. In the MCG method the solute atom can be prepared so as to cross over the phase boundary, so the various critical phenomena caused by the phase transformation near the phase boundary are visually investigated by means of analytical transmission electron microscopy. In the conventional research, the alloy characteristic has been investigated for the individual alloy whose composition was fixed. Inevitably there is no continuity in composition. Therefore, the change of characteristics with composition has hardly been investigated systematically. However by using the MCG method, we can continuously observe the change of phase transition with composition. We can study the phase transformation based on a new viewpoint. By utilizing the MCG method, various kinds of phase transformation such as the coherent and incoherent precipitations, the order/disorder phase transition, the spinodal/nucleation-growth phase decomposition and the other many phase transformations can successfully be investigated, as presented in Section 3. Furthermore, to more important thing, the precipitate-nucleation very close to solubility limit is experimentally investigated by utilizing the MCG method (Section 4). On the basis of these important experimental results, the validity of conventional nucleation theories is discussed statistically and kinetically (Sections 4 and 5) and hence the failure of Boltzmann–Gibbs extensive entropy becomes clear for the early stage of phase decomposition. These observations have succeeded for the first time by MCG method and therefore the experimental facts obtained here give us many new knowledge of phase transformation. The MCG method is very useful for investigating various composition-dependent phenomena, especially the non-linear phenomena in the vicinity of a phase boundary.

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1.2. Preparation of MCG alloy specimen There are many preparation methods to create the macroscopic composition gradient in alloys, i.e., a diffusion coupling method, an imperfect arc melting method of sandwiched alloys and an imperfect homogenization method of the coarse discontinuous precipitates and so on [1–6]. In the present investigation, the Ni–Si, Ni–Al, Cu–Co and Cu–Ti MCG binary alloy systems and Ni– Al–Co ternary MCG alloy were investigated. At first, the preparation of Ni–Si MCG alloy was briefly explained in Fig. 2. A Ni–14 at.%Si alloy, prepared by arc melting, was coupled with pure Ni. The surfaces of coupled alloy should be chemically polished to the mirror surface. The coupled alloy was imperfectly homogenized at high temperature for a suitable duration or imperfectly arc-melted for a short time. The heating time or arc-melting time was controlled so as to realize the MCG region over several lm at least. The specimen was cut to thin plate of 0.5–1.0 mm in thickness, along the compositional graduation of solute atoms, as illustrated with a thin solid rectangular in Fig. 2. The specimen must be collected from the region not including the Kirkendall interface. Similarly, the Ni–15 at.%Al and Cu–20 at.%Co alloy were also coupled with pure Ni and Cu, respectively, and then imperfectly arc-melted. The Cu alloy plates were annealed at 1373 K for 30 ks and the Ni alloys were annealed at 1423 K for 40 ks to stabilize the macroscopic composition gradient and then quenched into the iced brine. By this treatment the supersaturated solid solutions having MCG were produced. In the case of arc-melting or couple diffusion method, the crystallographic directions of coupled two mother alloys are usually different, hence the high angle grain boundary may be formed in the MCG portion. This boundary is usually formed at the initial interface (Kirkendall interface), where the lattice defects and impurity atoms are concentrated. The MCG alloy specimen must be a single crystal which does not contain the grain boundary, because the continuity of microstructure is necessary in the MCG method. Thus, the MCG alloy should be extracted from the portion of no-grain boundary in the front side or back side of Kirkendall interface, as illustrated in Fig. 2b. The best way is to use two single crystals whose coupled faces are coherent in crystallography. The macroscopic composition gradient for Cu–Ti alloy was prepared by utilizing the imperfect homogenization of coarse discontinuous precipitates of Cu3Ti-stable phase. A Cu–4 at.%Ti alloy was firstly prepared in a vacuum induction furnace, and then forged and rolled to a thin plate of about 1 mm in thickness, and solution treated at 1173 K for a suitable duration. After homogenization at 1173 K, the specimens were aged at 973 K for a long duration enough to produce the large discontinuous Cu3Ti precipitates whose inter-lamella distance is over 5 lm. These specimens were again heated at 1173 K for a short duration (10–15 min) to make imperfectly homogenized solid solution and then directly quenched into the aging temperatures. By this heat treatment the macroscopic composition gradient of the solute atom (Ti) was formed between the lamellar Cu3Ti precipitates.

Fig. 2. Schematic illustrations of typical preparation method of Ni–Si MCG alloy. (a) Coupled two alloys having different compositions. (b) MCG Ni–Si alloy produced by imperfect arc melting or imperfect annealing at high temperature. The specimen is cut to thin plate of 0.5–1.0 mm in thickness, perpendicular to the equi-concentration direction. As shown by a thin solid rectangular. The specimen must be collected from the region not including the Kirkendall interface.

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In the ternary alloy system, the MCG should be performed along the tie-line of phase decomposition. The Ni–Cr–Al MCG alloy used was prepared by coupling Ni–24 at.%Cr with Ni–25 at.%Al alloys, so that Al content in the MCG alloy varied along the tie line of c/c0 (see the phase diagram of Fig. 32 in Section 4.1). The macroscopic composition gradient of these solid solutions should be controlled to be less than 10 at.%/l, because 10 at.%/l is the critical value of composition gradient which does not give a wrong influence on the phase transformation. The value of 10 at.%/l is very large compared with the usual macroscopic composition gradient prepared in the usual MCG alloys. The detail is described in Section 2.2. 1.3. Experimental procedures and a typical example of TEM microstructure in MCG Alloys 1.3.1. Experimental procedures The specimens were aged at various temperatures for suitable durations and then quenched into iced brine. The aging should be undertaken at the lower temperatures than the stabilizing temperature at least 300–400 K, for instance, at 973 K for the Ni–Al and Ni–Si alloys and 873 K for the Cu–Co and Cu–Ti alloys. The specimens were quenched into the iced brine after aging. The thin foils for TEM were prepared by electro-polishing after aging in an electrolyte, H2SO4:CH3OH = 1:9 at 240 K for the Ni–Al, Ni–Si and Ni–Al–Co alloys and HNO:CH3OH = 1:3 at 240 K for the Cu–Ti and Cu–Co alloys. Since the solute concentration varies at the portion of MCG specimen, the condition of electro-polishing may often encounter the difficulty. The careful control of polishing-voltage is particularly requested. The platinum mesh with a single hole method should be used for this case. The microscopic observation was performed by the analytical transmission electron microscope (JEOL 2000FX), and the solute concentration analysis by the energy dispersive X-ray spectroscopy (EDS) (the detector: the Tracor Northern Company TN-5500) was concurrently performed at several locations in the same thin foil. The electron microscope was operated at 200 kV. The LaB6 filament was used at an accelerating voltage of 200 kV and the beryllium mesh is used. The K-factor defined by the Cliff–Lorimer method [7] is estimated in the limit of a thin film specimen and determined by the EDS measurement on the standard samples whose chemical compositions are already known. The spurious X-ray was detected for correction of the characteristic X-ray Since the size of the incident electron beam is operated so as to be large enough to cover the area containing several precipitates, the measured values of solute concentration indicates a locally averaged composition ca. The measurement was performed 5–7 times on the same place and then these were averaged The measurement error of each averaged value was within ±0.1 at.%. The identical measurements were performed at the many places in the thin foil. In the MCG method the chemical composition at any place can be estimated from the ‘‘composition vs. distance curve’’ calculated by the least square method of the error function for the many measured values. By utilizing this method, one order higher accuracy than that of as-measured value can be obtained (see Appendix A). 1.3.2. A typical example of TEM microstructure The precipitation behavior and the microstructure formation in the MCG alloys may be unfamiliar for the most researchers. Therefore, the precipitation of Ni3Si particles in the Ni–Si MCG alloy is demonstrated, as an example. Fig. 3 shows a 100 dark field TEM image of microstructure formed by aging at 973 K for 7.2 ks. The white small cuboids in Fig. 3 show Ni3Si precipitates. The gray solid circles in the photograph indicate the EDS measuring points of local average composition, whose values are correspondingly plotted in the small figure of Fig. 3. The solid line in the small figure is determined by using the least square method of the error function. A virtual line connecting two black arrows in the photograph is named as ‘‘precipitation front’’ whose composition is given from the inserted figure. 2. Influences of macroscopic composition gradient (MCG) on the phase decomposition The MCG method is a technique to investigate the phase transformations in various composition alloys by utilizing a single specimen having the macroscopic solute composition gradient. Namely, the macroscopic composition gradient method is assumed to be identical with that of ‘‘the plywood of the thin plates of alloy’’ whose solute composition changes little by little. In order to materialize

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Fig. 3. A 100 dark field TEM image of the microstructure of Ni–Si MCG alloy aged at 973 K for 7.2 ks. The white cuboids are Ni3Si precipitates. The gray solid circles in the photograph indicate the measuring points of locally average solute composition of which values are correspondingly plotted in the small figure inserted. A virtual line connecting two black arrows in the photograph is defined as a ‘‘precipitation front’’ whose composition is given by the inserted figure to be 10.92 at.%Si for this aging condition.

the assumption, the following conditions are indispensable, i.e., the thin plates are thermodynamically independent with each other and are never affected from other surrounding plates during phase transformation. Therefore, it is important to make clear the threshold value of composition gradient for safety use of MCG method. In the present section, we make clear the following two points in order to judge the validity of MCG method: the first point is to make clear that the profile of macroscopic composition gradient does not change during aging at lower temperature. The second point is to make clear the influence of MCG itself on the phase transformation, because the existence of MCG may give some influences on the microstructure formation even if the profile of MCG does not change during aging. Therefore, it is important to know the critical lower limit of MCG which does not give any influence on the microstructure formation. 2.1. Profile change of MCG specimen during aging In order to avoid the profile change of MCG during aging, the aging temperature, TA, must be fairly lower than the stabilizing temperature, TS. The temperature difference between the stabilizing and

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Fig. 4. The composition profile of Mo atoms calculated for a Ni–Mo MCG alloy. A solid curve is for as-stabilized specimen and a dotted line is for the aged specimen after stabilization. The both line are perfectly consistent.

Fig. 5. The macroscopic composition profile measured by XPMA for Fe–Si MCG alloy, where solid circles and squires indicate Siconcentration before and after a long time aging (1023 K for 2.42Ms), respectively. Any significant difference is not recognized between the specimens before and after aging.

aging temperatures should be over 400 K. The diffusion coefficient of solute atoms varies approximately 10 times per each 50 K in the alloy system whose activation energy of diffusion is about 250 kJ/mol. Therefore, the temperature difference, TS  TA, of 400 K results in about 108 times-difference between both diffusion coefficients. Such big difference between the diffusion coefficients is considered to give hardly change in MCG profile even if the MCG specimen is aged for a long duration. Fig. 4 shows theoretically calculated MCG profile of Mo atoms in two Ni alloys; one is the alloy just stabilized at 1373 K for 7.2 ks and the other is that of the alloy aged at 923 K for 864 ks after the stabilization. These lines are theoretically calculated on the basis of the Fick’s 2nd law of diffusion equation. The both lines are perfectly coincident with each other and any difference cannot be recognized between them. The same behavior is also experimentally confirmed. Fig. 5 shows the macroscopic composition profile measured by XPMA for Fe–Si MCG alloy, where solid round circles and squares indicate Si-contents before and after aging for a long time, respectively. Any difference is also not recognized between them, as is theoretically predicted. Thus, it is concluded that the MCG profile does not change during aging, so long as the MCG profile is stabilized at a higher temperature of 350–400 K than the aging temperature. 2.2. Influences of existence of MCG on phase transformation Here, the direct influence of MCG on the phase transformation is discussed. Even if the macroscopic composition profile does not change during aging, the existence itself of MCG may give some influences

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on the microstructure formation. Therefore, it is important to know the critical limit of MCG which does not give any influence on the microstructure formation. In this section, the threshold value is evaluated. The MCG alloy aged has a gradient of microstructure consisting of many precipitate particles, namely the volume fraction of precipitates inclines. Therefore, the gradient energy of the microstructure is considered to consist of the following two gradient energies, i.e., the first is a macroscopic composition gradient energy, Egrad macro , caused by the macroscopic composition gradient itself (hereafter, called as ‘‘macroscopic compositional gradient energy’’ Egrad macro ) and the other is an additional elastic strain energy arising from the gradient of volume fraction of precipitates (hereafter, called as ‘‘microstructure gradient strain energy’’ Estgrad ). str 2.2.1. The macroscopic composition gradient energy, Egrad macro According to Cahn and Hilliard [8,9], the chemical energy change due to the composition fluctuation has been given by the Taylar expansion with respect to composition c. Similarly in the MCG alloy the macroscopic gradient energy Egrad macro is expressed by

Egrad macro ¼

Z



j

2 dc dx dx macro

ð1Þ

where (dc/dx)macro is the macroscopic composition gradient in the alloy and j is the gradient energy coefficient. The typical value of (dc/dx)macro for the macroscopic composition gradient is the order of 105/nm since the composition change is roughly 0.5 at.% per 1 lm distance, as is known from Fig. 3. On the other hand, a microscopic composition gradient energy Egrad micro arises from the interface of precipitate-particle newly produced by aging. The (dc/dx)micro at the precipitate interface is considered to be order of 101/nm because the composition change of several tens of percents occurs at the precipitate interface whose width is about 1 nm. Therefore, the value of ðdc=dxÞ2macro due to the macroscopic composition gradient is extremely small, compared with that of precipitate interface, i.e., approximately, (dc/dx)macro/(dc/dx)micro = 109 [6]. Thus, the effect of (dc/dx)macro is so small that it can be disregarded. 2.2.2. The gradient strain energy of microstructure, Estgrad str Here, we evaluate the elastic strain energy of microstructure Estgrad caused by the gradient of volstr ume fraction of precipitates. In Fig. 6a the particles are distributed uniformly in the matrix, whereas in Fig. 6b the distribution of particles varies with change of the macroscopic composition. The bold rectangulars in Fig. 6a and b show the region where the volume fraction of precipitates are equal. The total elastic strain energies in the rectangular are evaluated here for the two cases. The elastic strain energy in (b) is possibly different from that of (a), because the additional elastic interaction energy among particles may be generated from the gradient of volume fraction of precipitates. Thus, the additional elastic strain gradient energy Estgrad may occur in the case of (b). str The Estgrad is evaluated here. Firstly we consider the case of composition flat alloy in which many str particles distribute uniformly. Each particle volume is X and the eigenstrain is eTij . The particles are assumed to be ellipsoidal revolution in shape and same in size. The volume fraction of particle is f. The mean internal stress of a particle in an infinite matrix is given by [10,11] 1 hrij iIðHÞ ¼ r1 ij ðXÞ  f rij ðXÞ

ð2Þ

Eq. (2) means that hriIðHÞ is sum of the internal stress of precipitate particle existing in an infinite matrix, r1 ij ðXÞ, with the mean internal stress affected by the surrounding particles. Therefore, the elastic strain energy per unit volume of alloy Ehom is expressed by str

Ehom str ¼ 

1 2V

Z

H

rij eTij dx ¼ 

1 NX 1 T 1 T ð1  f Þr1 ij ðXÞeij ¼  f ð1  f Þrij ðXÞeij 2 V 2

ð3Þ

where N is number of precipitate particles, which is related with volume fraction f as f = NX/V. V is a volume of alloy considered. The elastic strain energy of single particle existing in an infinite matrix Eincl is given by [10,11]

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Fig. 6. Schematic illustrations of the microstructure of (a) macroscopically composition flat alloy and (b) composition gradient alloy (MCG alloy). The solid circles represent the coherent precipitates. The bold rectangles in the center indicate the region having the same volume fraction of precipitates in the both alloys. The stability of microstructure is compared for this region.

Eincl ¼ 

1 1 r ðXÞeTij 2 ij

ð4Þ

Hence Ehom of Eq. (3) is simplified as shown in the following equation: str

Ehom str ¼ f ð1  f ÞEincl

ð5Þ

Next, the case of MCG alloy shown in Fig. 6b is discussed. In this case, an additional strain energy term caused by non-uniform distribution of particles should be added to the second term of Eq. (2). The heterogeneity of volume fraction is obtained by utilizing the Taylar expansion of f as a function of distance r.

f inhom ¼ f þ

1 2 2 r r f 2!

ð6Þ

By using Eq. (6), the elastic strain energy per unit volume in the rectangular region of Fig. 6b Einhom is given by

Einhom ¼ str

  1NX 1 1 2 2 T 1  f  r 2 r2 f r1 ij ðXÞeij ¼ f ð1  f ÞEincl  r fEincl  r f 2 2 2V

ð7Þ

The Taylar expansion of Einhom with respect to the volume fraction f and its derivative terms (df/dx), str (d2f/dx2), gives 2 2 Einhom ðf ; rf ; r2 f ; . . .Þ ¼ Ehom str str ðf Þ þ e1 r f þ e2 ðrf Þ þ   

ð8Þ

The terms over third power are omitted here, and the term of 5f is disregard because of symmetry of energy. Here, the integral sign is omitted for simplifying the expression of formula. By applying Gaussian divergent theorem to the second term in the right side of Eq. (8), Eq. (8) is transferred to 2 Einhom ðf ; rf ; r2 f ; . . .Þ ¼ Ehom str str ðf Þ þ eðrf Þ

where e ¼ de1 =df þ e2 . Comparing Eq. (7) with Eq. (8), the following equations are known; Therefore, e is given by

1 2

e ¼ r2 Eincl

ð9Þ

e1 ¼  12 r2 Eincl f and e2 = 0. ð10Þ

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Defined to be Estgrad ¼ Einhom  Ehom and compared Eq. (9) with Eq. (10), the gradient elastic strain str str str stgrad energy Estr is consequently given by

Estgrad ¼ str

 2 1 2 1 df 1 r ðrf Þ2 Eincl ¼ r 2 Eincl ¼ ðdf Þ2 Eincl 2 2 r 2

ð11Þ

where df is a variation of the volume fraction with respect to distance r. The gradient elastic strain energy is numerically evaluated on the basis of Eq. (11). The macroscopic composition gradient in the actual MCG specimen is approximately 0.5 at.%/lm [1,6], and the composition difference between precipitate and matrix is about 10 at.% [4]. Using these values and the fol@f @f @c lowing equation: @x ¼ @c  @f ¼ 0:5 at:%=lm and @c ¼ 10 1at:%, we obtain @x ¼ 0:5  103 =lm. @f @x Therefore, taking the interparticle distance to be 5 nm, we get df ¼ 2:5  104 . . hom E Thus, the ratio of Estgrad is evaluated for f = 0.5 as follows: str str

. Ehom Estgrad str str ¼

 104 Þ2 Eincl ¼ 1:25  107 ð0:5  ð1  0:5ÞÞEincl 1 ð2:5 2

ð12Þ

In the case of f = 0.5, Estgrad is very small compared with the elastic strain energy Ehom for composistr str tion flat alloy. However, in the alloy whose composition is close to the edge of miscibility gap (f is small), Eq. (12) has a fairly large value, for example 4.0  105 for f = 0.01%. Therefore, Estgrad is not str all ways negligibly small but may be effective for the alloy having a small f. Using Eq. (12), we estimate roughly the critical limit of the macroscopic composition gradient which does not give any influence on the microstructure formation. We assume that, when the stgrad Estgrad is less than 1/1000 of the elastic interaction energy Ehom does not affect the microstructure str str , Estr formation. The elastic interaction energy is about 5% of Ehom , the necessarily condition for safety use of str MCG alloys is approximately given by

Estrgrad ¼ str

1 Ehom 20000 str

ð120 Þ

After repeating a trial and error, we determined the critical limit of macroscopic composition gradient approximately to be 10 at.%/lm, detail of which is described in the next section. We call this value the allowable maximum composition gradient MCGmax. It is noteworthy that, since the Fe–Mo alloy is a very strong elastically constrained system, the MCGmax value evaluated for Fe–Mo is available for the most alloy systems. As described above, the heterogeneity of microstructure is influenced by the gradient elastic strain energy Estgrad , which is str determined with the elastic strain energy Eincl (see Eq. (11)). Eincl is approximately determined by  2  2 the square of expansion coefficient with respect to the solute concentration g2 ¼ 1a @a (a is lattice @c constant) of alloy system. Thus, the alloys having larger g tend to form a strong heterogeneous microstructure. Table 1 lists up g-values for several alloy systems [12–16] where the phase decomposition takes place. An Au–Ni alloy system has so extraordinarily large g-value which leads to breakdown the lattice coherence at the particle interface on the mid-way of phase decomposition [34]. The Fe–Mo alloy has a very large g except Au–Ni alloy. Therefore, the MCGmax estimated for Fe–Mo alloy is valid enough to be effective for the most alloy systems. Furthermore, it should be noted that the macroscopic composition gradient of 10 at.%/lm is fairly large than the gradient of the typical MCG alloys used in the present work (about 0.5 at.%/lm (see Fig. 3)). On the basis of these simulations it is satisfactory to consider 10 at.%/lm as the MCGmax. Table 1 g-Value of various alloy systems. Alloy

Cu–Co [12]

Al–Zn [12]

Ni–Al [13]

Ni–Si [13]

Fe–Mo [16]

Au–Ni [14]

g

0.019

0.026

0.043

0.057

0.083

0.15

g¼

1@a a

@c

. Brackets [] show the reference number.

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2.3. Computer simulations of microstructure formation based on phase field method On the basis of consideration of Section 2.2, the time-development of microstructure in the area of bold rectangle of Fig. 6 is simulated on the basis of phase field method. 2.3.1. Theoretical basis The theoretical basis of the phase field method has already been described in our papers [17,18]. In the phase field method, the total free energy of microstructure Gsystem is expressed in terms of various order parameters prescribing the microstructure, i.e., composition, degree of order, crystal structure and so on

Gsystem ¼

Z

½Gc fci ðrÞ; sj ðrÞ; Tg þ Esurf fci ðrÞ; sj ðrÞ; Tg þ Estr fci ðrÞ; sj ðrÞ; Tgdr

ð13Þ

r

where ci and sj are a composition parameter and a structure parameter, respectively. Gc is the chemical free energy, Esurf is the interfacial energy and Estr is the elastic strain energy. The gradient strain energy Estgrad given by Eq. (11) is included in the elastic strain energy Estr. str The time-dependencies of c and s are evaluated by using following two kinetic Eqs. (14a) and (14b) in the phase field method. Eq. (14a) is available for the conservative parameter such as a solute composition, while Eq. (14b) is available for the non-conservative order parameter such as a longrange order parameter. Eq. (14a) is usually called as Cahn–Hilliard equation [8] and Eq. (14b) is called as Allen–Cahn equation [19].

  

 @sj ðr; tÞ @ci ðr; tÞ dGsystem ¼ r M ci fci ðr; tÞ; T g r þ nci ðr; T; tÞ þ K c ci ðr; tÞ; sj ðr; tÞ; T @t @t dci ðr; tÞ

ð14aÞ

 

 dGsystem

 @ci ðr; tÞ @sj ðr; tÞ ¼ Lsj sj ðr; tÞ; T þ nsj ðr; T; tÞ þ K s ci ðr; tÞ; sj ðr; tÞ; T @t @t dsj ðr; tÞ

ð14bÞ

The ci(r, t) and sj(r, t) are functions of position r and time t in three dimensional. The interaction between two order parameters ci(r, t) and sj(r, t) proceeds through the total free energy of microstructure Gsystem shown in Eq. (13). Mci{ci(r, t), T} and LSj{sj(r, t), T} are the mobility of order parameters, ci(r, t) and sj(r, t), respectively, and are assumed to be a function of temperature T. v(r, t) is the diffusion potential and n(r, T, t) is a so-called Gaussian noise term for the order parameters ci and sj. The final terms in Eqs. (14a) and (14b) are a dynamic coupling terms in the phase field, i.e., a dynamic feed back term. The coupling coefficients Kc and Ks in Eqs. (14a) and (14b) are usually zero for most phase transformations not so much deviating from the equilibrium state. The phase transformation where the dynamic coupling term cannot be ignored is only for a few phenomena highly deviated such as dendrite growth in solidification and fractal pattern formation. The precise evaluation of diffusion potential v(r, t) is most essential and important for the present simulation. Since Gsystem is the total free energy of microstructure consisting of the chemical free energy Gc, the interfacial energy Esurf and the elastic strain energy Estr, the each potential is given by the following equations, respectively.

lc fcðr; tÞg 

  dGc ; dc r¼r0

lsurf ðr; tÞ 



dEsurf dc

 ; r¼r0

lstr ðr; tÞ 

  dEstr dc r¼r0

ð15Þ

Thus, the diffusion potentials v(r, t) are given by

dGsystem c cp c ðr; tÞ þ lstrp ðr; tÞ ¼ lcp ðr; tÞ þ lsurf dcp ðr; tÞ dG s q vsq ðr; tÞ  system ¼ lscq ðr; tÞ þ lssurf ðr; tÞ þ lstrq ðr; tÞ dsq ðr; tÞ

vcp ðr; tÞ 

ð16a; 16bÞ

Consequently, the diffusion potential v(r, t) at position r’ in microstructure is obtained from Eqs. (16a) and (16b), and then @c/@t and @s/@t are evaluated from Eqs. (14a) and (14b). Thus, we can calculate changes in c and s with progress of phase transformation by repeating Eq. (17a,17b).

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T. Miyazaki / Progress in Materials Science 57 (2012) 1010–1060 Table 2 The numerical values used for the calculation of microstructure formation in a MCG alloy. The values are corresponding to the Fe–Mo binary alloy system. Temperature, T/K Alloy composition, c0 Composition gradient, Dc/lm Elastic stiffness, Cij/104 MN m2

773 0.3 0, 0.1, 0.5

Mo C Fe 11 C 11

23.3, 46.3

Mo C Fe 12 C 12

13.5, 16.1

Mo C Fe 44 C 44 Expansion coefficient with respect to the solute concentration, g Calculation area, L/109 m Interaction distance, d1/1010 m Number of Fourier wave, N

11.8, 10.9

cp ðr; t þ DtÞ ¼ cp ðr; tÞ þ f@cp ðr; tÞ=@tgDt sq ðr; t þ DtÞ ¼ sq ðr; tÞ þ f@sq ðr; tÞ=@tgDt

0.083 120 2.86 256  256

ð17a; 17bÞ

2.3.2. Computer simulated microstructures On the basis of Section 2.2.1, the time-development of microstructures for MCG alloy are demonstrated. The numerical values used for calculation are listed in Table 2, each of which corresponds to the thermodynamic data of Fe–Mo alloy system [16,18]. Fig. 7 shows time-developments of microstructures calculated for the flat composition alloy A(a–c) and two MCG alloys B(d–f) and C(g–i) of which macroscopic composition gradient are 10 at.%/lm. and

Fig. 7. The time development of microstructure formation of Fe–30 at.%Mo aged at 773 K, calculated on the basis of the phase field method. The microstructures A (a–c) are composition flat alloy. The B (d–f) and C (g–i) are for Fe–30 at.%Mo MCG alloy whose composition gradient are 10 at.%Mo/lm and 50 at.%Mo/lm, respectively. The small arrows in (d) and (g) indicate the solute concentration at the both ends of microstructure of the MCG alloys.

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50 at.%/lm, respectively. The calculating area corresponds to the bold rectangular area of Fig. 6a and b. The macroscopic composition gradient decreases linearly from the right hand site to the left side, and the Mo contents are just same at the centre of rectangular box of A, B and C alloys in Fig. 6. The initial composition fluctuations were given by the same Fourier waves generated by the same seeds of the random function. These are added to the macro composition profile. Therefore, the difference of initial composition profiles between the composition flat alloy and MCG alloy is only for the macroscopic composition profile. It is obvious from Fig. 7 that the microstructures do not show any difference between the alloy A and alloy B through the all aging time. Hence, it is clearly known that the macro composition gradient of 10 at.%/lm does not give any influence. Thus, It approved that the macroscopic composition gradient less than 10 at.%/lm does not give any influence on microstructure formation. When the MCG becomes larger, the influence appears clearly. The alloy C in Fig. 7 demonstrates the simulated microstructure of the alloy having a steeper composition slope, i.e. 50 at.%/lm. In early stages of aging, the remarkable difference is not recognized between the both alloys, but in the later stage the microstructure of MCG alloy C becomes different from that of composition flat alloy, as is clearly known by comparing the long aging time (1600s’). The precipitate particles in Fig. 7i are extended to the [0 1 0] direction. This means that the atom diffusion along [1 0 0] direction becomes predominant than [0 1 0] direction by existence of macroscopic composition gradient. 2.4. Experimental verification Here, we perform experimental verification on the basis of the theoretical evaluations. The Fe–15– 20 at.%Mo alloys are well known to decompose spinodally so as to make h1 0 0i modulated structure [18]. Fig. 8 shows changes in wavelength of modulated structure with progress of aging, experimentally obtained for the compositionally flat Fe–19.2 at.%Mo alloy and the Fe–Mo MCG alloy whose macro composition gradient is 8.9 at.%/lm which is less than the MCGmax 10 at.%/lm. The wavelength in the MCG alloy was detected from the composition area of 19.2 at.%Mo. The wavelengths of compositionally gradient alloy coincide well with that of flat alloy for all aging time. Thus, it is experimentally proved that the macro composition gradient does not affect the microstructure formation so long as the gradient is less than 10 at.%/lm. The wavelengths of MCG alloy are well consistent with that of composition flat alloy for all aging time. Thus, it is experimentally proved that the macro composition gradient does not give any influence on the microstructure formation so long as the gradient is less than 10 at.%/lm. Consequently, on the basis of theoretical and experimental investigations it is confirmed that the MCG method does not give any influences on the phase transformation and microstructure formation unless the MCG is over the critical composition gradient 10 at.%/lm. 3. Transition of microstructures at phase boundary The MCG method is very useful to investigate the composition-dependence of phase transformation, particularly phase transformations in the vicinity of phase boundary, because of the drastic change of chemical free energy at the edge of miscibility gap, compared with other energies such as elastic strain energy and interfacial energy. In the present section the several representative microstructure transformations which demonstrate superiority of MCG method are demonstrated as follows; the precipitation limit of coherent and incoherent precipitates, the order/disorder phase transition, the morphological transition of the microstructure from the spinodal to the N-G type phase decomposition and so on. 3.1. Determination of coherent precipitation line of Ni–V and Ni–Mo alloy systems It is examined for Ni–V and Ni–Mo MCG alloys whether the coherent solubility limit experimentally obtained by the MCG method is precisely consistent with the conventional phase diagram or

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Fig. 8. Change in wavelength of the modulated structure with aging time for the Fe–19.2 at.%Mo composition flat alloy (usual composition-fixed alloy) and Fe–Mo MCG alloy. The average composition (rectangular region of Fig. 5b) in MCG alloy is 19.2 at.%Mo and the average composition gradient of (oc/ox)MCG is 8.9 at.%Mo/lm. The wavelength of modulated structure is examined by means of the satellite of 200 electron reflection spot.

Fig. 9. A TEM microstructure formed by aging at 873 K for 1.38 Ms. in the Ni–V MCG alloys, indicating that the solubility limit for coherent Ni3V precipitates is 15.5 at.%V.

not. A Ni–34 at.%V and a Ni–20 at.%Mo alloys were coupled with pure Ni-plate, respectively and then the coupled specimens were imperfectly arc-melted for a short time. After that, the alloy was heated at 1373 K for 3.6 ks for stabilization of the macroscopic composition gradient and then quenched into the iced brine. By this treatment the macroscopic composition gradient were formed in the Ni–V and Ni–Mo supersaturated solid solutions. The alloys were isothermally aged at several temperatures of 873–1173 K for very long duration so as to reach the equilibrium solubility limit. 3.1.1. Ni–V alloy Fig. 9 shows a TEM bright field image of Ni–V MCG alloy aged at 873 K for a very long time, 1.38 Ms. It is clear that the strain contrasts due to the Ni3V coherent precipitates are uniformly observed in the high V-concentration area, while in the lower concentration area the precipitates cannot be seen except the dislocations. The microstructure is obviously separated into two parts by the virtual line connecting with two white arrows in the micrograph. The precipitation front is determined from the inserted figure to be 15.5 at.%Mo for 1.38 Ms aging at 873 K. The similar investigations are performed

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Fig. 10. A coherent solubility limit for the Ni3V precipitation, which is firstly proposed by utilizing the MCG method. Our experimental results are consistent precisely with Moreen’s coherent precipitation line.

Fig. 11. A microstructure of TEM dark field image of Ni–Mo MCG alloy aged at 923 K for 864 ks shows that the precipitation front is 13.0 at.%Mo.

at other temperatures. The solubility limits experimentally determined are plotted in Fig. 10 for several aging temperatures. Our experimental results are well consistent with the reliable coherent precipitation line which was obtained by the work of Moreen et al. [20]. The Pearson’s result [21] is contradictory to the coherent precipitation line, possibly because of inaccurate analysis. It is noteworthy that the precipitation front shifts gradually to the lower concentration region with longer time aging and finally stops at the equilibrium phase boundary. (The experimental data of this phenomenon is described in detail and discussed in Section 4.) Therefore, in order to investigate the equilibrium state, we should adopt a long aging time enough to realize the equilibrium solubility limit.

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Fig. 12. A part of phase diagram of Ni–Mo system, showing the coherent solubility limit of Ni4Mo precipitates obtained by several researchers, and our experimental results of Ni–Mo MCG alloy.

3.1.2. Ni–Mo alloy Fig. 11 shows a dark field TEM image of Ni–Mo MCG alloy aged at 923 K for a long time aging, 864 ks. The Ni4Mo particles are observed uniformly in the Mo-high composition area, but the number of precipitates decreases gradually in the lower composition area and finally no precipitates can be seen. The composition of precipitation front is determined to be 13.0 at.%Mo. The similar experiments were performed at the other aging temperatures (923–1123 K). The solubility limits experimentally determined are shown in Fig. 12. The incoherent solubility lines proposed by Heiwegen and Rieck [22] and by Gust et al. [23] and Casselton and Hume-Rothery [24] are also plotted, which were investigated by means of the diffusion-coupling [22] and by the X-ray measurement of lattice change with composition [23,24]. These experimental results are adopted by Massalski [25] into the Ni–Mo equilibrium phase diagram. Our coherent solubility line is slightly shifted to low temperature compared with the incoherent solubility lines of Heiwegen’ work and Gust’ work, because the elastic strain energy due to the coherency forces down the miscibility gap to the lower temperature. 3.2. Difference of the solubility Limits between the coherent and incoherent precipitation Fig. 13 shows an incoherent precipitate heterogeneously nucleated on the dislocation in the Cu–Ti alloy, which is indicated by a large arrow. This photograph clearly shows the difference in solubility limit between the coherent and incoherent precipitates. The solubility limit of incoherent precipitation is given by the virtual line connecting with two paired arrows which are numbered as No. 2, while the coherent precipitate is indicated by the two arrows numbered as No. 1. From the virtual lines of 1 and 2, the coherent and incoherent precipitation limits are determined to be 1.75 at.%Ti and 2.15 at.%Ti, respectively, at 873 K. Such solubility-difference is also observed in other MCG alloy system. Fig. 14 shows the homogeneous coherent Ni3Al precipitates and the heterogeneous incoherent Ni3Al formed on a dislocation. The composition difference between both is 0.4 at.%Al from the inserted figure. It has been generally conceptualized that the elastic strain energy forces down the miscibility gap to the lower temperature [26–28]. Figs. 13 and 14 show the clear evidence to verify the concept, and may give the quantitative evaluation of the elastic strain energy.

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Fig. 13. Coherent Cu4Ti precipitates and incoherent precipitates heterogeneously nucleated on a dislocation. The each precipitation front is indicated by the arrow numbered 1 (2.15 at.%Ti) and 2 (1.75 at.%Ti), respectively. The incoherent precipitate is isolated apart from the coherent precipitation front.

Fig. 14. Homogeneously dispersed coherent Ni3Al precipitates and predominant precipitates on the dislocation line.

3.3. Microstructure continuity between spinodal and N-G phase decompositions On the basis of Boltzmann–Gibbs’s free energy [29], the process of phase decomposition has been considered to be divided into the two types of phase decomposition, i.e., spinodal decomposition and the Nucleation-Growth type decomposition, The theoretical research on the spinodal decomposition started from the linear theory of Cahn–Hilliard [8,9], and has resulted in the dynamical investigations based on the nonlinear diffusion equation. In the linear theory the spinodal and the NucleationGrowth are separate type phase decomposition. However, by recent theoretical considerations, it is predicted that the both are continuously shifted gradually [30]. There have been a lot of experimental investigation of spinodal decomposition for many alloys, i.e., Fe–W [31], Fe–Cr [32], Fe–Mo [16], Nb–Zr [33], Au–Pt [34], Cu–Ni–Fe [35], Cu–Ni–Si [36] and so on. Nevertheless the empirical proof of the continuity of microstructure has not be seen in a TEM photograph, so long as author know. The MCG alloy is suitable to verify the continuity of microstructure. By utilizing the Fe–Mo MCG alloy the morphological transition from the spinodal region to N-G region is investigated. The Fe–Mo alloy system is well known to form the typical h1 0 0i modulated structure by the spinodal decomposition [16]. Fig. 15 is the phase diagram of Fe–Mo alloy system, showing

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Fig. 15. A Equilibrium phase diagram of the Fe–Mo binary system. The dotted and chain lines are meta-stable coherent bimodal and spinodal lines of bcc A2 phase, respectively.

Fig. 16. A TEM microstructure and the electron reflection satellites around 200 spot for Fe–Mo MCG alloy aged at 823 K for 10.8 ks, showing the continuous transition of microstructure from the h1 0 0i modulated structure to the mottled structure which is considered to arise from a so-called Nucleation-Growth phase decomposition. The two arrows indicate the area of spinodal line (19.0 at.%Mo) given by the Fe–Mo phase diagram [16].

the equilibrium phase diagram and a meta-stable miscibility gap of Fe–Mo supersaturated solid solution [16]. The spinodal and binodal lines are asymmetric with composition c = 0.5, because of asymmetric change of elastic stiffness with Mo content [37]. The spinodal point of meta-stable miscibility gap at 823 K is 19.0 at.%Mo. Fig. 16 shows a TEM microstructure in the vicinity of spinodal line of Fe–Mo MCG alloy aged at 823 K.. Many contrasts caused by the h1 0 0i lattice modulation can be observed in the high Mo concentration area, but gradually decrease in contrast with decrease of Mo concentration. The microstructure of spinodal point is marked by two arrows, i.e. 19.0 at.%Mo. The microstructural morphology seems to change gradually from the typical h1 0 0i modulated structure in the right side to the mottled structure. The 100 satellites around the 200 electron reflection spot are also obtained in the modulated structure area, whereas the satellites becomes diffuse and finally cannot be recognized in the left side of photograph. The extinction of satellite signifies disappearance of the periodic distribution of solute atom, that takes place usually in a so-called N-G type phase decomposition. Thus, the microstructure changes continuously from the h1 0 0i modulated structure to the non-periodic mottled structure without a distinct boundary. The experimental fact shown here is a clear proof of continuity of microstructure from spinodal to N-G phase decompositions [30,38,39] signify.

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Fig. 17. A part of phase diagram of Fe–Al alloy system, showing a heat treatment temperature for Fe–Al MCG alloy.

Fig. 18. A TEM 100 dark field image of Fe–Al MCG alloy aged at 1023 K for 86.4 ks in the vicinity of the B2/A2 2nd order transition lines, showing the microstructure change of ordered phase with composition. The bright region in the figure corresponds to the B2 ordered domain and the black smoothly curved lines show the anti-phase boundary. The A2/B2 transition line is determined to be 24.7 at.%Al at 1023 K, which is good agreement with the phase diagram of Fe–Al binary alloy system.

3.4. A2/B2 order/disorder transition in Fe–Al ordering alloy In the present section, the microstructural change accompanying with the A2/B2 2nd order transition is demonstrated. Fig. 17 shows a part of Fe–Al equilibrium phase diagram.[40,41] The 2nd order transition A2/B2 can be seen at the high temperature, although the coexistence with two phase A2 + B2 and A2 + D03 exist at the lower temperature. The change of microstructure with 2nd order transition was investigated at 1023 K.

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Fig. 18, showing a dark field 100 TEM image of Fe–Al MCG alloy aged at 1023 K for 86.4 ks [19], demonstrates the morphological change of domain of anti-phase boundary (APB) near the A2/B2 transition point. In the region of high Al concentration the ordered domain has a large size with bright contrast. However, with decreasing Al content, the size of ordered domain becomes gradually smaller, while the volume fraction of APB domain (black part) increases and finally reaches to 100%, as is shown in the left edge of Fig. 18. As a matter of course the degree of long range order gradually reduces. It was experimentally confirmed that the domain size of microstructure does not vary even if the aging time and quenching rate are changed. Thus, the microstructure in Fig. 18 is considered to have been in equilibrium state. The A2/B2 transition in Fe–Al alloy system has been well known as the 2nd order transition. Therefore, the contrast of B2 ordered domain ought to decrease gradually without change of ordered domain size, because the brightness is related to the degree of order. However, Fig. 18 shows clearly that the ordered domain keeps the contrast still bright even at the region close to A2 boundary. The morphological behavior shown in Fig. 18 does not imply the feature of the 2nd order transition but the 1st order transition [42]. This experimental fact may suggest the necessity of reconsidering over the 2nd order transformation of Fe–Al alloy system. 4. Precipitate-nucleation near the edge of miscibility gap A huge number of researches have theoretically and experimentally investigated the precipitatenucleation of alloys in the past 100 years. The first development of nucleation theory was performed by Volmer and Weber [43]. They introduced a concept that the local composition fluctuation brought to the embryo and critical stable nucleus. On the basis of Volmer’s theory, Becker and Döring [44–46] evolved largely the kinetic theory of nucleation. They, assuming that the solute composition and structure of embryo are same as nucleus, proposed that the free energy change due to nucleation is only caused by the embryo size On the other hand, Borelius [47,48], assuming the embryo to keep at constant size, proposed a new kinetic theory that the free energy change due to nucleation was estimated only by solute composition. Furthermore, Höbstetter [49,50] proposed the change of free energy for a case where size and composition of embryo are changeable. A big theoretical development on the spinodal decomposition was performed by Cahn and Hilliard [8,51]. They introduced a new concept of diffuse interface changeable continuously from the precipitate to matrix, and described the interfacial energy as a function of solute composition gradient @c=@x. The kinetic theories of nucleation and coarsening were largely developed by Wagner [52] and Lifshiz–Slyozov [53]. A comprehensive research has been performed on the homogeneous nucleation by Binder and Stauffer [54]. A large number of detailed experimental investigations have been performed [55–58]. Furthermore, excellent comprehensive reviews and books on the nucleation have been also proposed [59–61]. However, such previous investigations have been examined on the alloy specimens whose solute atom distribution is macroscopically flat. Hence, the alloy composition of each specimen was discontinuous. As described in Section 1.1, the phase transformation should be expressed by the three dimensional space consisting of three axes of composition, time, and temperature. Thus the continuity on the composition axis should be desired. However, it is very difficult to prepare many alloy specimens each of which have a little composition intervals such as less than 0.01 at.%. The discontinuity of composition may give a large error to the composition-dependence of phase transformation, particularly phase transformation in the vicinity of phase boundary where the thermodynamic variables change drastically. In the present chapter, by utilizing the MCG method, it is possible to obtain the following experimental data; composition-dependence of the size of precipitate-nucleus, the time to nucleate and the equilibrium solute concentration at the particle interface for each nucleus. These are very important to understand the stability of precipitate-nucleus and mechanism of nucleation. The MCG method will contribute greatly to our knowledge of the phase transformation of alloys. 4.1. Microstructure changes with aging in varies alloys Firstly, the precipitation of Ni3Al particles in the Ni–Al MCG alloy is examined. Fig. 19 shows a 100 dark field TEM image of Ni3Al particles formed near the solubility limit by aging at 973 K for 10.8 ks.

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Fig. 19. A 100 dark field image of TEM microstructure of Ni–Al MCG alloy aged at 973 K for 10.8 ks. The small white cuboids show Ni3Al precipitates. The white circles in the photograph indicate the measuring points of the locally averaged solute composition, whose values are correspondingly plotted in the small figure inserted.

The white small cuboids show Ni3Al precipitates. The gray solid circles in the photograph indicate the EDS measuring points of the solute composition. The solid line in the inserted figure is evaluated by using the least square method for the error function. It is obvious that the Al concentration decreases gradually from the right side of photograph to the left over several lm. Many coherent Ni3Al particles can be seen in the high concentration region, but the number of particles decreases in the lower concentration area and finally the particles are extinguished. Thus, a line connecting the two white arrows of photograph is defined as the ‘‘precipitation front’’ of coherent particles for aging time of 10.8 ks at

Fig. 20. A 100 dark field TEM image of the Ni–Al MCG alloy aged at the same temperature with Fig. 19 but for a longer time, 86.4 ks. The precipitation front for 86.4 ks, indicated by the gray big arrows, is 11.7 at.%Al. The precipitation front is clearly moves to the lower Al side by the longer time aging.

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973 K. The alloy composition at the precipitation front is determined to be 12.0 at.%Al from the inserted figure. It is remarkable that the particle size gradually increases large and then becomes a maximum size at the precipitation front. The further aging gives a forward movement of the precipitation front to the lower composition side. In Fig. 20 the precipitation front is given to be 11.7 at.%Al for 86.4 ks aging. It is obviously known by comparing Fig. 20 with Fig. 19 that the precipitation front shifts toward the lower Al side and the particle size becomes larger. The similar behavior of precipitation is also recognized in other alloy systems. Fig. 21, showing 100 dark field TEM images of Ni–Si MCG alloys aged at 973 K for various aging times, the precipitation front shifts gradually toward the lower Si composition side with progress of aging. The characteristic of microstructures described above is not only in Ni–Al and Ni–Si alloys but also general in the alloy systems of Cu–Co and Cu–Ti. Fig. 22 shows a TEM bright field image in the vicinity of coherent phase boundary of the Cu–Ti MCG alloy aged at 873 k for 30s. Many coherent Cu4Ti precipitates disperse in the high Ti region, but the particle gradually decreases in number toward the left side. Since the photograph was taken under the condition of g = 111, a so-called butterfly-contrast is observed around the coherent precipitates. Such contrast may lead to a fallacious particle size. Therefore, the square contrast taken under the multi beam condition was also used to estimate the accurate particle size. Fig. 22B shows a microstructure taken under a multi-beam condition of B = 100 for same specimen with Fig. 22A. The square contrast in Fig. 22B shows a real particle shape and size. Thus, the solute concentration of precipitation front of Fig. 22 is given to be 2.30 at.%Ti and the particle size is r = 15 nm. The dark field TEM image of Ni–Cr–Al ternary MCG alloy aged at 973 K for 173 Ks is represented in Fig. 23. In a case of ternary system, the tie-line of phase decomposition should be taken into consideration. The direction of tie-line and phase compositions will be discussed later (see Fig. 32). The reason why the shape of precipitate of Ni–Cr–Al alloy system is spherical comes from very small elastic strain energy, i.e., g = 0.008[15]. Thus, the identical behavior of microstructure is recognized not only in the binary MCG alloy system but also in the ternary MCG alloy system. Here, it is noteworthy that the movement of precipitation front results from the composition dependence of incubation time for nucleation, not caused by the change of macroscopic gradient profile during aging (see Section 2). The composition dependence of incubation time is schematically represented in Fig. 24. According to Darken [62], the mobility of atoms MðcA ; cB Þ in the interdiffusion e for A–B binary system is given by coefficient D

MðcA ; cB Þ  ðM B cA þ M A cB ÞcA cB

ð18Þ

By assuming that MA = MB  M0, the mobility M(cA, cB) is expressed as

MðcA ; cB Þ ¼ M 0 cð1  cÞ

ð19Þ

Therefore, the diffusion of atoms is faster in the center part of miscibility gap, whereas becomes slower in the lower solute composition area and remarkably slow in the edge of miscibility gap. Thus, the time-lag of precipitation arises as a result. Therefore, when the MCG alloy is aged, the precipitation front moves gradually to the lower composition side and stops eventually at the equilibrium solubility limit, although the attainment has need of a long time aging. 4.2. Evaluations of nucleus-stability By utilizing the changes of microstructure represented in Figs. 19–21, we are able to obtain relationships between the equilibrium solute concentration ce at the particle interface and the particle radius r. Fig. 25 shows a schematic illustration to explain how to get them. In Fig. 25 the local average compositions ca, the schematic microstructures formed for different aging times t1, t2, t3 (t1 < t2 < t3) and the solute composition profiles around the particle locating at positions r and s are described in the top, middle and bottom parts, respectively. The volume fraction of precipitate f with respect to the matrix is given by

f ¼

ca  ce cp  ce

ð20Þ

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Fig. 21. Dark field TEM images of Ni–Si MCG alloys aged at 973 K for (a) 0.3, (b) 1.8 and (c) 7.2 ks, respectively. The precipitation fronts are 11.30, 11.11 and 10.92 at.%Si, respectively.

Here, a case of aging time t3 is taken into consideration. The composition profile of solute atom in the high composition area must be as that of s in Fig. 25. In this case the local average composition ca is not equal to the equilibrium composition ce. This profile is usual for the microstructure consisting of two phases. However, at the precipitation front the volume fraction is nearly equal to zero, i.e., f ; 0, so

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Fig. 22. TEM microstructures formed in Cu–Ti MCG alloy aged at 873 K for 30sec, showing many coherent Cu4Ti precipitates. (A) The dark circles show the measuring points of average solute composition, whose values are correspondingly plotted in the inserted figure. The photograph (A) was taken under a condition of g = 111, showing the butterfly contrast. The photograph (B) was taken under g = 100 and multi-beam condition, showing cubic contrasts.

Fig. 23. A 100 dark field TEM micrograph of Ni–Cr–Al MCG ternary alloy aged at 973 K for 173 ks. The white particles are c0 -precipitate of which composition is Ni–11.32 at.%Cr–13.70 at.%Al (see Fig. 32). The shape of precipitate particle is spherical and the distribution of particles is in random, because of small lattice mismatch, g = 0.008%. The composition gradient is along the tie-line shown by a thick solid line.

the average composition ca should be approximately equal to ce, as is clearly known from Eq. (20). The composition profile is given by r. Thus, the relationship between the equilibrium composition ce (ca at the precipitation front) and the diameter of a precipitate particle 2r are experimentally determined by means of the analytical transmission electron microscope.

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Fig. 24. A schematic representation of change of incubation time for the precipitate-nucleation with alloy composition, indicating that the nucleation time depends on the alloy composition.

Fig. 25. A schematic illustration how to get the equilibrium composition at the particle interface ce whose diameter is 2r, where the local average composition ca, the microstructures with different aging time t1, t2, t3 (t1 < t2 < t3) and the solute composition profiles around the particle locating at positions r and s, are schematically represented in the top, middle and bottom parts, respectively. The particle r at the precipitation front is called as a critical stable precipitate hereafter.

Such relationship between the precipitate radius r and the equilibrium concentration ce holds always at the precipitation front, because the thermodynamic equilibrium is always maintained at

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the precipitation front. Therefore, by conducting aging of various times, we can obtain the systematic values of r and ce for various particle size. It is noteworthy that this relation is only available for the precipitation front. In the higher composition area than the precipitation front, the competitive growth among particles has already elapsed, so that the distribution of particle size is different from that of just nucleated particles. The experimental facts above described bring up a following rule that the precipitate particle whose diameter is 2r is only able to nucleate at the precipitation front. The particles smaller than the critical diameter 2r are unstable because ce > ca, thereby dissolve into the matrix. On the other hand, the larger particles than 2r are produced as a result of particle coarsening starting from the size of 2r. The competitive growth among the particles is very slow in the precipitation front. Therefore, the particles in the precipitation front show the size as just nucleated. Thus, the particle observed at the precipitate front corresponds to the thermodynamically equilibrium size. The precipitation front shifts gradually to the lower concentration side with aging time and a new nucleus is generated in a new front. Fig. 26 shows the experimentally determined relationship between the radius of precipitate r at the precipitation front and the equilibrium composition ce (ca) for the Ni–Si MCG alloy. A thick solid curve, described along the lower limit of measuring points, shows the boundary of stability of Ni3Si particle. The upper side of the curved line shows the thermodynamically stable region of Ni3Si precipitates, whereas the lower side shows the unstable region. The ‘‘precipitate-nucleus’’ is defined to be the ‘‘minimum stable precipitate-particle’’. Therefore, the solid curve in Fig. 26 represents the change of nucleus size with alloy composition. The vertical broken line in Fig. 26 indicates the equilibrium solubility limit for the particle of infinite radius, i.e., the equilibrium phase boundary. It is clearly known from the figure that the size of stable particle steeply increases up to several ten nm with approaching to the phase boundary and such a rapid increase takes place in an extremely narrow composition range less than 0.3 at.% from the equilibrium phase boundary. The similar relationships are also obtained for the Ni–Si binary alloys aged at different temperatures, 923 K and 873 K, as represented in Fig. 27. Furthermore, the identical behavior is found in Ni–Al, Cu–Co, Cu–Ti binary MCG alloys and Ni–Cr–Al ternary MCG alloy. Figs. 28 and 29 show the nucleus size changes with composition Ce for Ni–Al and Cu–Co MCG alloys, respectively. Both MCG alloys rapidly increase their particle size in the area of very vicinity of solubility limit, although the equilibrium solubility limit is separate each other.

Fig. 26. A relationship between the critical radius of precipitate r and the equilibrium solute composition ce for the Ni–Si alloy aged at 973 K. The solid circles in the figure indicate the radii of the particles observed in the vicinity of precipitation front for various aging times and a thick solid curve (stability boundary) is described along the lower limit of these measured values.

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Fig. 27. The relationships between the critical radius of precipitate r and the equilibrium solute composition ce for the Ni–Si MCG alloys aged at 923 and 823 K. The solid circles in the figure indicate the radii of the particles observed in the vicinity of precipitation front for various aging times and the stability boundary (thick solid line) is described along the lower limit of these measured values.

Fig. 30a shows a composition-dependence of nucleus size in Cu–Ti MCG alloy aged at 873 K for various times, clearly indicating rapid increase of size near the solubility limit. The experimental results at 873 K and other aging temperatures are plotted on Cu–Ti phase diagram as shown in Fig. 30b. The spinodal point of the alloy has been reported to be 2.77%Ti for 873 K [16], as indicated in the figure. Any discontinuity in nucleus size is not recognized at the spinodal point, as seen in Fig. 30a. Fig. 31 shows composition-dependence of r on the tie-line in the Ni–Cr–Al ternary MCG alloy aged at 973 K. The rapid increase of particle nucleus is also recognized in ternary alloy system. This alloy was prepared by coupling Ni–24 at.%Cr and Ni–25 at.%Al binary alloys, so that the phase decomposition proceeds along the tie line of c/c0 described with a thick solid line in the phase diagram of Fig. 32. The composition of c-matrix is Ni–21.13Cr–3.42Al(at.%) and the precipitate is Ni–11.32Cr– 13.70Al(at.%). Here, we again argue the advantage of the MCG method. As described above, the MCG method can systematically detect the drastic increase of nucleus size in the very narrow composition range near the solubility limit. If we try to get the same result by using the usual alloy specimen whose composition is macroscopically flat, we have to prepare many specimens whose compositional interval changes precisely less than 0.05 at.% at least. Such precise control of alloy composition is actually impractical. By contrast, in the MCG method the microstructure change is detectable even for extremely small composition interval. Thus, it becomes possible for us to observe in detail the nucleation behavior at the very vicinity of phase boundary.

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Fig. 28. A relationship between the critical radius of stable particle r and equilibrium solute composition Ce for Ni–Al MCG alloy aged at 973 K.

Fig. 29. A relationship between the critical radius of stable particle r and equilibrium solute composition Ce for Cu–Co MCG alloys aged at 823 K.

4.3. Calculation of the critical stable nucleus The critical nuclei experimentally obtained for several MCG alloys are theoretically evaluated in the present section. The Ni–Al MCG alloy is adopted for calculation. The solute composition profile of a globular shaped precipitate in the matrix is expressed by using several parameters which are shown by Eq. (21) and Fig. 33.

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Fig. 30. (a) A relationship between the critical radius of stable particle r and equilibrium solute composition Ce for Cu–Ti MCG alloy aged at 873 K. (b) A part of phase diagram showing meta-stable coherent binodal and spinodal lines.

cðrÞ ¼ cp ð0 r 6 r 2 Þ c  c 2ðr  r Þ n p e 2 cðrÞ ¼  ¼ cp ðr 2 6 r 6 r3 Þ r1  r2 2 n c  c 2ðr  r Þ p e 2  2 ¼ ce ðr 2 6 r 6 r 3 Þ cðrÞ ¼  r1  r2 2 cðrÞ ¼ ce

ð21Þ

ðr 1 6 r 6 r0 Þ

Fig. 31. Change of the critical radius of precipitate nucleus r with composition ce for Ni–Cr–Al ternary MCG alloy system.

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Fig. 32. A partial equilibrium phase diagram of Ni–Al–Cr ternary alloy system, indicating the tie line (a thick solid line) of phase equilibrium of c/c0 . The MCG alloy is prepared by the imperfect arc melting on the interface of stacked Ni–25 at.%Al and Ni– 24 at.%Cr binary alloys, so that the phase decomposition proceeds on the tie line connecting with both mother binary alloys.

Fig. 33. A Solute composition profile of three-dimensional spherical precipitate particle in the matrix, which is used for the theoretical estimation of particle stability.

The term n is a parameter determining the slope of composition gradient at the particle surface (odd number). The total free energy change due to the nucleation DG is given by Eq. (22), proposed by Cahn and co-workers [8,9,51], where the first, second and third terms are a chemical, gradient and coherent elastic strain energies, respectively.

DGðca ; ce ; cp ; r 0 ; r 1 ; r2 ; nÞ ¼

1 V

Z " r

#  2 @cðrÞ DGc þ j þ g2 YðcðrÞ  ca Þ2 dr @r

ð22Þ

where V is the volume of precipitate, j is the gradient energy coefficient, g is the expansion coefficient with respect to the solute concentration c, and Y is the elastic constant. The chemical free energy of Ni–Ni3Al alloy at 973 K is proposed by Fig. 34 according to Hultgren et al.’s research [63]. The equilibrium compositions of matrix and precipitate phases given by the free energy curves are Ni– 11.68 at.%Al solid solution (c) and Ni–24.54 at.%Al (c0 ), which precisely correspond to the actual phase boundaries [63]. The chemical free energy DGc, the gradient energy j(oc(r)/or)2 and the elastic strain energy g2Y(c  c0)2 are functions of the parameters ca, ce, cp, r0, r1, r2 and n. Therefore, the parameters are determined so as to give the minimum DG for a given particle size r1. If the particle of size r1 is thermodynamically stable, the calculated composition profile has a suitable finite value. On the other hand, when the particle is unstable, the composition profile should be flat. The critical size of stable precipitate r1 is obtained by calculating the composition profile for various particle sizes r1, The composition profiles calculated for various sizes of c0 -precipitate are represented in Fig. 35a for the Ni–12.35 at.%Al and (b) for Ni–11.75 at.%Al. The numerical values used for calculation are listed up

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Fig. 34. The chemical free energy of c and c0 phases in Ni–Al alloy system (Hultgren et al. [63]).

Fig. 35. The solute composition profiles of particle with various size, calculated for a Ni3Al particle in (a) Ni–12.35 at.%Al and (b) Ni–11.75 at.%Al alloys aged at 973 K, showing that the critical radii of stable particle r1 are 3.0 nm and 22.0 nm, respectively. The r1 values show the outer radius defined by Fig. 33.

in Table 3, It is clearly known from Fig. 35a that the particles larger than 3.0 nm in radius r1 P 3.0 nm are energetically stable, whereas the particles smaller than r1 = 2.9 nm are unstable. Therefore, the critical radius of stable precipitate r 1 is determined to be 3.0 nm for the Ni–12.35 at.%Al alloy at 973 K. On the other hand, the composition profiles of Ni–11.75 at.%Al alloy described in Fig. 35b, which is very close to the solubility limit, shows that the particles larger than r1 = 22.0 nm are stable, while the particle of r1 = 21.0 nm is unstable. The critical radius r 1 ¼ 22:0 nm is very large compared with that of Ni–12.35 at.%Al alloy, because the chemical driving force DGc is extremely small in the

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T. Miyazaki / Progress in Materials Science 57 (2012) 1010–1060 Table 3 Numerical values used for the calculation of stable precipitate-nucleus in Ni–Al system. Molar volume, VM/105 m3 mol1 Elastic stiffness, Cij/104 MN m2

2.791 [18,19]

Ni Ni C Ni 11 ; C 12 ; C 44 Lattice mismatch, g Interfacial energy density, cS/J m2

20.21, 14.79, 9.74 [21] 0.0434 [20] 0.014 [16]

Fig. 36. Theoretical and experimental critical size of stable Ni3Al precipitate in Ni–Al MCG alloy.

vicinity of edge of miscibility gap. Thus, the critical radii of precipitate nucleus are theoretically evaluated for various alloy compositions. The calculated values are described by open circles in Fig. 36. The calculated curve almost coincide with the solid line experimentally determined (see Fig. 28). Consequently, the critical sizes of precipitate nucleus experimentally obtained are theoretically proven to be proper on the basis of energetic evaluation. Thus, it becomes experimentally and theoretically clear that the solid curves of Figs. 26–28 and 30 represent the boundary of thermodynamic stability of precipitate-nucleus. Here, it should be noted that such discussion is restricted only for the stability of formed nucleus, not for embryo on the way of nucleation. The energy barrier for the nucleation is not taking into consideration at all in this calculation. 4.4. Thermodynamic discussion on the basis of conventional nucleation theory The thermodynamic stability of big nucleus formed near the phase boundary is discussed in this section. According to the conventional nucleation theories [43], the critical nucleus radius, i.e. the maximum embryo radius r em is given by Eq. (23).

r em ¼ 2cs =DGvc ol

ð23Þ v ol

The chemical free energy change due to the nucleation DGc solution.

DGev ol ¼ ð1=V m RT lnðce ðrÞ=C e ð1ÞÞ

is given by Eq. (24) in the dilute solid

ð24Þ

where cs is the interfacial energy density between the particle and matrix, Vm is the molar volume of precipitate, R is the gas constant and T is the temperature. The ce(r) and ce(1) are the equilibrium solute compositions at the interface of particle of radius r and infinite size, respectively.

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Substituting Eq. (23) to Eq. (24), Eqs. (25) and (26) are introduced

rem RT=2cs V m ¼ 1= lnðce ðrÞ=ce ð1ÞÞ

ð25Þ

ce ðrÞ ¼ ce ð1Þ expð2cs V m =r em RTÞ

ð26Þ

Eq. (26) is linearized for a large radius to the well known Eq. (27) [64].

 2c V m ce ðrÞ ¼ ce ð1Þ 1 þ s r em RT

ð27Þ

Since the particle size obtained experimentally in the present work is that of the critical stable nucleus r, Eq. (27) should be replaced by Eq. (28) where the critical size remb is replaced by r.

 3c V m ce ðrÞ ¼ ce ð1Þ 1 þ s r RT

ð28Þ

where r = 1.5rem Eqs. (26)–(28) are called as the Gibbs–Thomson equation. Eq. (28) shows a linear relationship between the inverse particle size 1/r and equilibrium solute concentration ce(r). The Gibbs–Thomson relations are demonstrated for Ni–Si alloys whose particle size are fairly large and Cu–Ti alloy systems whose particle size are small. Fig. 37 represents the relationships between ce(r) and r1 for the Ni–Si alloys aged at three temperatures. The experimental values of ce(r) are obviously straight for the three lines. The equilibrium compositions for the infinite particle size ce(1) are given to be 10.77 for 973 K, 10.48 for 923 K and 9.91 for 823 K, which are precisely consistent with the solubility limits shown by vertical dotted lines in Figs. 26 and 27. By giving Vm = 2.596  105 m3/mol and R = 8.314 J/K mol, the interfacial energy density cs is evaluated to be 0.012 J/m2 for 973 K, 0.010 J/m2 for 923 K and 0.011 J/m2 for 823 K, respectively. The interfacial energy density has been reported to be the order of 0.01 J/m2 for the coherent L12 type precipitate particle [64,65], so that the interfacial energies cs evaluated here are considered to be proper. Next, we show the case of Cu–Ti alloy whose nucleus radius is fairly smaller than that of Ni–Si alloy. Fig. 38 shows the G–T relations between ce(r) and r1 for the Cu–Ti alloy aged at 873 K and 823 K. The experimental values of ce(r) are not straight but bend in the small particle area, as predicted by Eq. (26). For the large particle region the straight lines are recognized, and so the equilibrium concentration ce(1) is given by the intercept of ordinate of Fig. 38 to be 2.12 at.%Ti for 873 K, which is perfectly consistent with the solubility limit 2.12 at.%Ti at 873 K (see Fig. 30a). By giving Vm = 7.12  106 m3/mol and R = 8.314 J/K mol, the interfacial energy cs is estimated from the slope of the straight line to be 0.11 J/m2 for 873 K and 0.10 J/m2 for 823 K. Since the interfacial energy density of the coherent precipitate particles cs has been known to be the order of 0.1 J/m2 [65], the evaluated values are considered to be proper.

Fig. 37. The Gibbs–Thomson’s relations for Ni3Si precipitates of Ni–Si MCG alloy aged at the three temperatures. The three numerical values of the vertical axis indicate the equilibrium solute concentration at the interface of infinite particle, i.e. the equilibrium solubility limit.

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Fig. 38. The Gibbs–Thomson’s relationships between the particle radius r and equilibrium composition at the particle interface ce for Cu4Ti particles in Cu–Ti MCG alloys. The deviation from the straight line for the small particles is rationalized by Eq. (27) in the text. The interfacial energies evaluated for 873 K and 823 K are 0.11 and 0.10 J/m2, respectively.

Fig. 39. A relationship between the normalized nucleus size and the equilibrium solute composition at the particle interface. These are scaled for different aging temperatures of Ni–Ni3Si MCG alloys, based on the Gibbs–Thomson’s equation given by Eq. (27).

On the basis of Eq. (26), the critical nucleus size r can be scaled with respect to the supersaturation of solute atom. Fig. 39 demonstrates that the Ni3Si precipitate particles nucleated are well scaled by the Gibbs–Thomson’s equation even though the aging temperature is different. Similarly, the nucleation for different alloy system is well scaled as represented in Fig. 40. The consideration above explained is also verified by Fig. 41 which shows the relationship between critical nucleus size r and the solute supersaturation Dc for the Cu–Co alloys. The nucleus size of Cu– Co alloy was obtained by Wagner [66] and Wentd and Haasen [67]. They determined the smallest radii in several Cu–Co alloys aged at 783–833 K. Fig. 41 shows the experimental results (open marks) deter-

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Fig. 40. The critical particle sizes scaled for all alloy systems with respect to the super-saturation of solute atom. This figure clearly shows that the big precipitate-nuclei in the vicinity of the miscibility gap are well explained by the Gibbs–Thomson’s equation, i.e. the conventional nucleation theory.

Fig. 41. The relationship between the critical nucleus size r and the solute supersaturation Dc in Cu–Co alloys. The solid circles are experimentally obtained by present work for Cu–Co MCG specimens and the open marks are obtained by P. Haarsen and R. Wagner [58] and arranged by H.I. Arronson and F.K. LeGoues [59] for the composition flat alloys. The solid straight line is theoretically given by Wagner (see [52]) on the basis of Eq. (24).

mined for the usual composition flat Cu–Co alloys which were obtained by Haarsen and Wagner and arranged by Aaronson and LeGoues [59]. The solid straight line shows the changes of critical nucleus radius with composition change, estimated on the basis of Eq. (23) by Wentd and Haasen [67]. The solid circles in Fig. 41 show our experimental results for the Cu–Co MCG alloy. The our results of Cu–Co MCG alloy are clearly on the extension of the theoretical line of nucleation. In the usual composition flat specimen, the detectable lowest limit of alloy composition remains at about 0.1 at.%, but by utilizing the MCG method the detectable limit is improved up to the order of 0.01 at.%. Consequently, the experimental results of the nucleus size formed in the region of solubility limit are summarized as follows; the nucleus size increases rapidly up to several hundreds of nanometers in a very narrow composition range less than 0.3 at.% from the equilibrium solubility limit. The diameter of nucleus size can reach over 500 nm in the edge composition very close to the precipitation limit. Such the nuclei are energetically predicted by the conventional nucleation theory and the Gibbs– Thomson’s equation.

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However, it is noteworthy that, though the stability of nucleus is thermodynamically rationalized, it is impossible that the conventional nucleation theory explains kinetically why the big nucleus is formed for so short duration of aging. The kinetic investigation is dealt with in the next Section 5.

5. Kinetic investigations on the nucleation process As described in Section 4, the thermodynamic stability of the big nucleus is rationalized by the classical nucleation theory. However, it seems difficult for the conventional nucleation theory to rationalize that the big nucleus appears within a fairly short aging time, for instance, t = 86.4 ks for the nucleus radius r = 86 nm (see Fig. 20). The nuclei smaller than the stability-boundary (see Figs. 27–30) are thermodynamically unstable, as described in Section 4. Thus, the nucleus, starting from the supersaturated solid solution, must pass through the unstable region and then reach to the stable region. We discuss the kinetics of nucleation in the present section on the basis of experimental results. 5.1. Experimental results The kinetic data on the nucleation are summarized firstly. Fig. 42 represents the radius of critical stable nucleus r, the equilibrium composition ce and the time to nucleate Ni3Si nucleus in Ni–Si MCG alloy aged at 973 K. The upper and lower regions of the solid line show the stable and unstable regions, respectively. Hence, the line means the critical minimum size of stable nucleus. ‘‘Aging time’’ in Fig. 42 is identical with ‘‘time for nucleation’’, i.e., the incubation time for each nucleation. The representative experimental results obtained for three aging temperatures are listed up in Table 4. It is clear that the nucleation time needs more time with lower composition for all temperatures. Fig. 43 and Table 5 show the radius of critical stable nuclei r, the equilibrium composition Ce and the time to nucleate Ni3Al nucleus in Ni–Al MCG alloy aged at 973 K for various times. The similar results are demonstrated for Cu–Ti MCG alloy in Fig. 44 and Table 6. In this manner, the nucleus size and the formation time were measured to each nucleus. Such the results have obtained first by utilizing the MCG method. These experimental results are very impor-

Fig. 42. A composition dependence of the critical radius of stable particle r, the equilibrium solute concentration ce and the aging time required to nucleate Ni3Si-nucleus in the Ni–Si alloy system aged at 973 K. A solid curved line, described along the lower limit of experimental particle radius, shows the boundary for nucleus stability. The upper and lower regions of the solid line show the stable and unstable regions, respectively. Hence, the line means the critical minimum size of stable nucleus. ‘‘Aging time’’ is identical with ‘‘time to precipitate’’, i.e., incubation time for nucleation.

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Table 4 The equilibrium solute concentration at nucleus interface ce, the nucleus size r and the incubation time to nucleation t, experimentally obtained for Ni–Si MCG alloy aged at 823, 923 and 973 K. Ni–Si MCG alloy T = 973 K

T = 923 K

T = 823 K

Dce (at.%Si)

r⁄ (nm)

t (ks)

Dce (at.%Si)

r⁄ (nm)

t (ks)

Dce (at.%Si)

r⁄ (nm)

t (ks)

1.13 0.74 0.53 0.29 0.23 0.135 0.068 0.041

2.1 2.6 8.0 19.5 25.1 40.2 61.5 85.5

0.06 0.13 0.30 1.15 1.80 7.20 20.81 61.89

0.932 0.466 0.325 0.205 0.150 0.096 0.060 0.049

5.1 11.1 17.6 22.2 35.2 43.1 45.0 78.7

1.02 3.83 7.08 22.4 35.48 89.12 237.14 442.38

1.02 0.851 0.53 0.30 0.183 0.100 0.067

5.1 7.0 10.1 20.0 34.3 61.5 82.0

18.62 21.54 48.64 146.78 301.30 681.24 1467.57

Fig. 43. A composition dependence of the critical radius of stable particle r, the equilibrium solute concentration ce and the aging time required to nucleate Ni3Al-nucleus in the Ni–Al alloy system aged at 973 K. The upper and lower regions of the solid line show the thermodynamically stable and unstable regions, respectively. Aging time is identical with the incubation time for nucleation.

tant to analyze the process of nucleation. We investigate the kinetics of nucleation on the basis of these experimental results. 5.2. Theoretical basis The composition-dependence of the incubation time has been qualitatively well known. However, the quantitative evaluation, particularly in the vicinity of solubility limit, has not been investigated, although the feature of nucleation comes out most clear in the region of phase boundary, because of the drastic change of energy barrier associated with composition- change. According to the kinetic equation for nucleation, the frequency of nucleation U is well known to be expressed by Eq. (29) [42].



U ¼ Ad MðcÞ exp 

   Qd DG ðcÞ  Af exp  kT kT

ð29Þ

where Qd is the activation energy for the diffusion of solute atoms and DG(c) is the activation barrier for nucleation, i.e., the nucleation energy barrier that must be overcome to form a nucleus. Ad and Af

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T. Miyazaki / Progress in Materials Science 57 (2012) 1010–1060 Table 5 The equilibrium solute concentration at nucleus interface ce, the nucleus size r and the incubation time to nucleation t, experimentally obtained for Ni–Al MCG alloy aged at 973 K (ce = 11.65 at.%Al).

Dce (at.%Al)

r⁄ (nm)

t (ks)

0.71 0.56 0.38 0.18 0.081 0.040 0.040 0.025

2.5 3.2 5.1 12.0 28.0 40.0 49.1 62.0

0.22 0.64 1.57 4.53 10.22 36.00 54.11 100.0

Fig. 44. A composition dependence of the critical radius of stable particle r, the equilibrium solute concentration ce and the aging time required to nucleate Cu4Ti-nucleus in the Cu–Ti alloy system aged at 873 K. A solid curved line shows the boundary for nucleus stability. The upper and lower regions of the solid line show the thermodynamically stable and unstable regions, respectively. Aging time is identical with the incubation time for nucleation.

are constant parameters. The mobility for atom-diffusion M(c) has been proposed by Darken [62] as M(c) = M0c(1  c) (see Eq. (19)). The first term of Eq. (29) is rewritten here as Eq. (30).



Ud ¼ Ad MðcÞ exp 

Qd kT

 ð30Þ

Eq. (30) is available for the phase decomposition phenomenon of which rate-controlling process is the atomic diffusion, i.e., the spinodal decomposition, the microstructure coarsening such as Ostwald ripening of precipitate-particles. On the other hand, Eq. (31) which is the second term of Eq. (29), is caused by the energy barrier for nucleation, and is available for the phase decompositions of ‘‘nucleation-growth’’ type.



Uf ¼ Af exp 

 DG ðcÞ kT

ð31Þ

Firstly, Eq. (30) is taken into discussion. At a constant temperature, Ud is only a function of composition c and expressed by a quadratic function of composition, because M(c) = M0c(1  c). Therefore, the time to nucleation U1 d is expressed by a hyperbolic curve which increases with decrease of super-

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T. Miyazaki / Progress in Materials Science 57 (2012) 1010–1060 Table 6 The equilibrium solute concentration at nucleus interface ce, the nucleus size r and the incubation time for nucleation t, experimentally obtained for Cu–Ti MCG alloy aged at 873 K.

Dce (at.%) (ce = 2.12 at.%Ti)

r⁄ (nm)

t (s)

0.755 0.615 0.501 0.205 0.125 0.065 0.050

3.00 4.55 4.70 5.01 7.50 15.3 26.0

3 5 10 15 20 30 60

Fig. 45. Typical two types of nucleation process; (a) size growth of small particle (Becker type) and (b) increase of solute atom concentration (Borelius type).

saturation Dc and diverges at Dc = 0. Thus, the relationship between log U1 d and log. Dc is given by a straight line in the region of low solute concentration. Even if the aging is carried out at different temperature, the straight line moves only in parallel. Next, we consider Eq. (31) which arises from the energy barrier for nucleation. Two typical energy barriers have been commonly known in a mechanism of nucleation, as illustrated by Fig. 45. One is a case where a small embryo with a high solute concentration and a sharp interface is initially formed and then widens its size to the stable size. Another is a case where an embryo increases its solute concentration with progress of nucleation. The former is well known as Becker type nucleation [44,45] and the latter is called as Borelius type nucleation [47,48]. In either way, the nucleation frequency Uf is given by Eq. (31), because the nucleation progresses by a mechanism of thermal activation process. The energy barrier DG(c) is given by Eqs. (32) and (33) for the Becker type and the Borelius type, respectively.

4pr 2 cs 3  3 r DG ðcÞ ¼ 4p DF v a0

DG ðcÞ ¼

ð32Þ ð33Þ

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Free Enregy

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Composition Fig. 46. A schematic illustration of the concave part of free energy–composition curve, which is essentially caused by the Boltzmann–Gibbs extensive entropy.

Table 7 Increments of the energy barrier DG(c) with nucleus size for Ni–Si alloys. Nucleus radius (nm) 1 5 10 20 30 50

Becker type DG(c) (kJ/mol) 2

1.0  10 2.5  103 1.0  104 4.0  104 9.0  104 2.5  105

Borelius type DG(c) (kJ/mol) 0.8  102 1.0  104 8.0  104 6.4  105 2.2  106 1.0  107

Fig. 47. A schematic illustration of the nucleation times, which are theoretically given by Eqs. (29)–(31) for the alloy whose composition is very close to the solubility limit. It is noteworthy that the line of U1 must be bend near the solubility limit so long as based on the Gibbs–Boltzmann free energy.

where r is the radius of nucleus, a0 is the lattice constant and DF m is the energy barrier arising from a concave part of the free energy–composition curve (see Fig. 46). Since formulas are the functions of nucleus size r, DG(c) depends greatly on the size. Table 7 shows the energy barriers estimated for various nucleus size of Ni3Si nucleus. When the nucleus radius is small (1 nm), the energy barrier is about 80 kJ/mol, that is consistent with the conventional activation energy for nucleation, e.g., 86 kJ/mol [68]. However, DG(c), increasing exponentially with size increase, reaches to an incredibly huge value, for instance, 107 kJ/mol for r = 50 nm. In the two mechanisms, the Becker’s type barrier is artificially generated from the assumption that the interface of nucleus is sharp at the starting point of nucleation. Since the interfacial energy is a function of composition gradient (oc/or)2 [8,9], it is not necessary that the nucleus has a sharp interface of high energy from the beginning of nucleation. This means that the premise of Becker’s type energy barrier collapses. Therefore, it is not necessary to take the Becker type barrier into consideration. On the other hand, the Borelius type energy barrier is caused by the concave part of the free energy–composition curve. Since the concave part originates in the Boltzmann–Gibbs free energy essentially, the

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Borelius type barrier is inevitable. The non-classical nucleation theory proposed by Cahn–Hilliard [8,9] is consistent with the classical nucleation theory in the vicinity of the edge of miscibility gap. Consequently, any nucleation theory cannot avoid the Borelius type energy barrier so far as the Boltzmann– Gibbs free energy is taken into the consideration. As shown in Table 7 the nucleation energy barrier DG(c) is enormously high for the big nucleus in either nucleation mechanism. Hence, the composition-dependence of Eq. (31) must be enormously large than that of Eq. (30) for the big nucleus. Accordingly, the line of log U1 must be strongly bent in the vicinity of edge of miscibility gap, as is schematically drawn in Fig. 47. 5.3. Kinetic investigation on the experimental results Nevertheless, the experimental results never show such bending. Fig. 48 shows composition1 dependences of the nucleation time U1 and U1, which are calculated for Ni3Si nucleus in Ni– d , Uf Si MCG alloy aged at 823 K. In Fig. 48 the two solid lines are theoretically given by Eqs. (30) and (31), respectively, and the experimental data are plotted by solid circles whose numerical values are given in Table 3. The experimental data are perfectly compatible with the straight line theoretically given by Eq. (30), whereas never show any consistence with Eq. (31). The identical experimental facts are also obtained in other alloys examined in the present work, as represented in Fig. 49a–c. Therefore, the experimental facts described above clearly prove that the first term of Eq. (28) is only effective, whereas the 2nd term does not contribute at all. These straight lines are parallel each other. The activation energies estimated by means of Ahrenius plot of them are 221 K J/mol for Ni–Si alloy system, 234 K J/mol for Ni–Al system and 190 K J/mol for Cu–Ti alloy system. These experimental activation energies approximately consist with the activation energies for solute atom diffusion [69]; 250 K J/mol for Ni alloys and 204 K J/mol for Cu–Ti alloys. Thus, it is proved that the nucleation processes shown in Fig. 49a–c are resulting from the solute atom diffusion without energy barrier for nucleation, i.e. spinodal-like phase decomposition. Thus, it is necessarily approved that the phase decomposition of supersaturated solid solution progresses by a mechanism of spinodal even in the N-G region. Fig. 50 illustrates a schematic phase diagram showing that the phase decomposition proceeds spinodally in whole area of miscibility gap. The

Fig. 48. A composition dependence of the time to nucleation U1, experimentally obtained for Ni3Si precipitates. A straight solid line and a curved solid line are theoretically given by the 1st and 2nd terms of Eq. (29), respectively, and the experimental results are represented by solid circles. The theoretical line is fit with the experimental data at the point of star mark.

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Fig. 49. The relationships between the time to nucleation U1 and the super-saturation of solute atoms Dc for (a) Ni–Si, (b) Cu– Ti and (c) Ni–Al alloys, respectively.

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experimental fact denies inevitably the existence of concave part in the free energy–composition curve described in Fig. 46. The concave arises originally from the Boltzmann–Gibbs’s extensive entropy. Therefore, the denial of concave part is synonymous with denial of the Boltzmann–Gibbs free energy consisting of the extensive entropy. Thus, the behavior of phase decomposition demonstrated in Fig. 50 is never rationalized by the Gibbs–Boltzmann free energy. A comprehensive consideration on the nucleation is opened in Section 5.5. 5.4. Pre-nucleation phenomena As described above, the big nucleus appears at the precipitation front, and never be observed in the lower composition region than the precipitation front. Namely, the precipitation front is the front line of nucleus precipitation. When the further aging is progressed, the bigger nucleus appears newly in the lower region. Therefore, it is inferred that a pre-nucleation phenomenon is progressing in the lower composition region. The pre-nucleation phenomenon is experimentally investigated. Fig. 51 is schematic illustrations of predictable microstructures when the MCG alloy is heat-treated by the two step aging. When the MCG specimen is aged at high temperature T2, the precipitate particles are formed till the precipitation front, as illustrated in Fig. 51a. In the usual sense, when the specimen is rapidly quenched from the high temperature T2 to the lower temperature T1 and then hold at T1 for a short duration, very fine precipitates ought to appear newly in the lower composition area for the cause of difference of solubility limit between the two temperatures, as illustrated in Fig. 51b. However, in the actual microstructure the several big precipitates are coexistent with fine precipitates, as schematically illustrated in Fig. 51c. Fig. 52 shows a 100 dark field TEM image of Ni–Si MCG alloy aged at 823 K for 6 ks after aging at 973 K for 54 ks. The precipitation front of 973 K aging is indicated by the two arrows in Fig. 52. It is obviously recognized in Fig. 52 that the big precipitates mingle with the fine precipitates in the lower composition side. The similar microstructure is also recognized in the other alloy system. Fig. 53 shows a dark field image of Ni–Al MCG alloy formed by two step aging, namely aged at 823 K for 180 s after aged at 973 K for 54 ks. Those micrographs certainly show the mingled structure of very big particles with the fine particles. The side planes of large square particles seem to dent in the side planes, which are possibly the beginning of particle-splitting [70,71]. These experimental facts imply that big clusters inhere in the lower composition area, although the cluster is not recognized by TEM observation. Such the solute rich clusters have already been reported.

Fig. 50. A schematic phase diagram showing that the phase decomposition proceeds spinodally in whole area of miscibility gap. A solid and dotted lines are the binodal and spinodal lines which are described on the basis of Boltzmann–Gibbs free energy. The solid circles in the figure indicates the qualitative change of nucleus size with c and T.

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Fig. 51. A schematic illustration of expected microstructures when the Ni–Si MCG alloy is heat treated by the two-step aging.

Precipitation front of aging at 973K for 54ks. Fig. 52. A TEM dark field image of Ni–Si MCG alloy aged at 823 K for 6 ks after aging at 973 K for 54 ks, showing the nonuniform large particles produced in the lower composition area from the precipitation front, probably arising from the prenucleation phenomenon.

On the basis of investigation by the small angle scattering, Wagner [72–74] argued that the Co-rich cluster exists in the matrix before formation of nucleation. Furthermore, Rzdilsky et al. [75] and Blavet et al. [76] found the Co-concentrated region existing in the Cu matrix by means of AP-FIM observation. Consequently, it is considered that the solute-rich diffuse droplet has been formed in the matrix as a pre-stage of nucleus, in which the solute atoms may be chained, not dispersed randomly. However, it should be noted that such large droplets are also not thermodynamically unstable, so far as based on Boltzmann–Gibbs free energy. The comprehensive discussion is in Section 5.6. 5.5. High speed growth of big nucleus near the solubility limit We discuss whether the formation-rate of big precipitate particle is rationalized or not by the diffusion equation. We concentrate a big Ni3Al particle observed in the Ni–Al MCG alloy. Fig. 20 shows that a particle with radius of r⁄ ; 50nm was formed in the Ni–Al MCG alloy aged at 973 K for 86.4 ks. According to Zener [77] and Zener and Wert [78], the growth of precipitate particle due to the atom diffusion is given by

ðcp  ce Þ

  @r @c ¼D @r r¼r @t

ð34Þ

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Fig. 53. A TEM micrograph of Ni–Al MCG alloy aged at 823 K for 1 ks after aging at 973 K for 54 ks, showing that the nonuniform large particles mingle with very fine particles formed during 823 K aging.

The curvature effect of particle surface is taken into consideration, so this formula will turn into

r  r0 ¼

  D c0  ce ðt  t 0 Þ 2r D cp  ce

ð35Þ

The numerical values used for calculation are listed up in Table 8, which are corresponding to the microstructure in Ni–Al MCG alloys shown in Figs. 19 and 20. we get the particle radius r = 6.47 nm as by calculation. This value is very small, compared with actual radius r = 50 nm. Namely, even if the fastest condition of particle growth is assumed, Eq. (35) cannot rationalize the formation of big particle for a short aging time. The diffusion process is supposed to be accelerated for some mechanism. 5.6. Problems of nucleus formation in the N-G region In the present section, the problems of nucleation process are comprehensively reconsidered on the basis of the experimental facts obtained by MCG method. As described in Section 4, the thermodynamic behavior of ‘‘nucleus created’’ is explained well by the conventional thermodynamic theories based upon the Boltzmann–Gibbs free energy, such as the equilibrium phase diagram, the Gibbs– Thomson relationship, and the composition dependence of nucleus size. However, it is impossible for the conventional nucleation theory to rationalize the kinetics of formation-process, particularly for the big nucleus in the vicinity of solubility limit. Table 8 Numerical values used for calculation of Eq. (44). Temp (K) Composition (at.%Al) Diffusion coeff. D (m2/s) Time t, t0

T = 973 K ca = 11.75 ce = 11.70 1.87  104 exp(268,000/RT) t = 86.4 ks t0 = 10.8 ks

cp = 23.00

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The problem of the nucleate process is to find out how the nucleus passes through the unstable region and reach the stability region, as demonstrated in Figs. 42–44. In the bottom area than the stability boundary the nucleus is thermodynamically unstable. The critical nucleus size depends on the alloy composition ce, that is, the nucleus size is small at the deep area of miscibility gap, whereas in the vicinity of the solubility line the big nucleus of hundreds nm is only stable. Thus, the embryo cannot grow up to the stable big nucleus so far as based upon the conventional free energy of Boltzmann–Gibbs. The Boltzmann–Gibbs free energy accompanies necessarily with the concave portion, as explained in Section 4. Even if any modification is introduced into the Boltzmann–Gibbs extensive entropy, the concave part of free energy must appear in the Boltzmann–Gibbs free energy, because the differential coefficient of extensive entropy becomes infinite in the very vicinity of pure metals. Hence, the energy barrier resulting from the concave portion appear always and should be enormously high for a big nucleus, so that the thermal fluctuation is not able to overcome the extremely high energy barrier, as demonstrated in Sections 5.2 and 5.3. There has been a general concept that the nucleation in the N-G region is progressed through the compositional fluctuation resulting from the thermal energy. The experimental nuclei are continuous from a small size up to a huge size, as demonstrated in Figs. 42–44. Therefore, it is necessary that the nucleation mechanism must be unified for all nucleus size including the big nucleus. Therefore, the formation of the big nucleus must be also explained by the compositional fluctuation. However, it is impossible that the compositional fluctuation creates such big nucleus in the extremely low concentration alloy, because the nucleus size is so large that the solute atoms in the very wide range must gather up to one place by the fluctuation. Consequently, it is impossible to cross over the unstable domain by the usual nucleation mechanism and also impossible by the thermal fluctuation. However, the big nucleus is actually formed within a comparatively short aging time. Since all nuclei including a big nucleus are produced by mechanism of spinodal decomposition, a free energy curve having no concave portion can only rationalize the process of phase decomposition. It is noteworthy that the behavior of nucleus in the equilibrium state is rationalized by the conventional Boltzmann–Gibbs free energy. Hence it is supposed that the separate free energies should be applied for the equilibrium and non-equilibrium states respectively. A free energy having such multiplicity has not been established currently. It is considered that the conventional Boltzmann–Gibbs free energy has a problem in estimation of the entropy of atom-configuration. The extensive entropy of atom configuration has been introduced into the Boltzmann–Gibbs free energy. The extensive entropy defines that a small atom-ensemble consisting of several atoms represents the solute atom arrangement of whole system. However, in a case of very low concentration alloy, since the solute atoms disperse very sparsely in the matrix, the size of the small domain may be changed with the solute concentration [79]. Such free energy consisting of the non-extensive entropy has been proposed by Tsallis [79,80]. However, the evaluation of Tsallis’s theory has not been decided yet for the phase transformations of alloy. The nucleation mechanism based on the Tsallis theory will be reported in near future.

6. Summary A new characterization method, ‘‘Macroscopic Composition Gradient (MCG) Method’’ was proposed to investigate the critical phenomena of phase transformation. The distinctive feature of MCG method is to investigate systematically the phase transformations in the various composition alloys by utilizing a single specimen which has a macroscopic composition gradient. Since the macroscopic composition gradient in the alloy is prepared so as to step over the phase boundary, the morphological observation of critical phenomena at the phase boundary can be realized by means of analytical transmission electron microscopy. The Macroscopic Composition Gradient (MCG) Method was theoretically and experimentally confirmed that it never affects on the phase transformation and microstructure formation.

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By using this method, various kinds of phase transition such as the coherent and incoherent precipitation lines, the order/disorder phase transition and morphological change at the spinodal line, have successfully been evaluated. Furthermore, the critical size of precipitate nucleus and the nucleation rate near the solubility limit are experimentally investigated for several binary alloy systems. The nucleus size shows a steep increase up to several tens of nm in a very narrow composition range less than 0.3 at.% from the phase boundary. The diameter of nucleus size can reach over 500 nm in the composition very close to the precipitation line. The Gibbs–Thomson relation and the conventional nucleation theory thermodynamically rationalize such composition-dependence of nucleus size change. However, the kinetics of nucleation is never explained by the conventional nucleation theories. The kinetic experimental results show distinctly that the nucleation time is only controlled by the atom diffusion and the phase decomposition of supersaturated solid solution is progresses spinodally without energy barrier for the nucleation, even in the so-called Nucleation-Growth region. On the basis of experimental results the application limit of conventional nucleation theory is discussed, and hence the failure of Boltzmann–Gibbs’s extensive entropy becomes clear for the early stage of phase decomposition. The N-G region is brought from the artificial effect arising from the Boltzmann–Gibbs’s extensive entropy and appears artificially only in the equilibrium phase diagram. It is noteworthy that the experiments presented here have not been performed in the past. The MCG method proposed here is considered to open a new way to study the microstructure evaluation, particularly for the critical phenomena near the phase boundary. Acknowledgements We have been constructing a new characterization method of the phase transformation, namely, ‘‘Macroscopic Composition Gradient (MCG) Method’’ with many colleagues for over 15 years. We have developed this technique uniquely and believed that the MCG method will open a new way to investigate the phase transformation, particularly the critical phenomena in materials science. The author is very grateful to many colleague for their cooperation and assistance, particularly Dr. Sengo Kobayashi in the Ehime University, Prof. Toshiyuki Koyama in the National Institute of Materials Science (now Nagoya Institute of Technology), Prof. Takao Kozakai in Nagoya Institute of Technology and many postgraduate students over 60 members of my laboratory. I am also grateful to Dr. Claudio G. Schön in the University of Sao Paolo in Brazil for his partly cooperation in discussion. The author is grateful to Prof. Tetsuo Mohri in Hokkaido University and Dr. John W Cahn in NIST of USA for their encouragement to promote the investigation.

Appendix A. Method of ‘‘composition vs. distance’’ curve in the MCG specimen The MCG specimen is prepared by utilizing the diffusion of solute atoms. Therefore, ‘‘the composition-distance’’ curve (see the inserted figure in Fig. 3, for instance) ought to be given by the Gaussian error function

cðx; tÞ ¼

" #    Z x=2pffiffiffiffi Dx ce 2 ce x 2 ex dx ¼ 1  pffiffiffiffi 1  erf pffiffiffiffiffiffi 2 2 p 0 2 Dx

ðA1Þ

where the initial condition is given as follows; c = ce for x 5 0 and c = 0 for x > 0, when Eq. (A1) is introduced from the Fick’s diffusion equation. However, since the MCG specimen is usually collected from the portion apart from the Kirkendall interface (see Fig. 2) it may be difficult to fit the measured data to Eq. (A1), because Eq. (A1) expresses the solute concentration profile starting from the Kirkendall interface. Hence, by introducing the parameters p and q, Eq. (A1) is generalized to be applicable for the actual measured data, as described in the following equation:

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Fig. A1. Changes of the standard deviations of composition error for the independently measured values cm and the evaluated composition value from the least square curve cfit with increase of composition data.

" !# q q ce p c ðx; t; p; qÞ ¼ 1  erf pffiffiffiffiffiffi 2 2 Dx

2 X fit 2 m ci  ci di ¼ fit

ðA2Þ ðA3Þ

i

Eq. (A3) shows the square of difference between the experimental solute concentration cm i ðxi Þ and the estimated value from Eq. (A2) cfit i ðxi Þ. The parameters p and q in Eq. (A2) are so determined as to minimize d2i . In the experiment, a computer program for the calculation has been arranged, and the composition profile is simultaneously calculated on the occasion of composition measurement. The solute concentration at any position can be estimated from the composition vs. distance curve. By utilizing this method, we obtain the more accurate composition than that of as-measured value. Next, we evaluate a concrete improvement of accuracy of cfit(x) which is given by Eqs. (A2) and (A3). We assume a true value of composition ctrut(x)e at position x, given by Eq. (A4). The parameters are fixed to p = 2 and q = 1.

c

true

  x 1 ce 2 p ffiffiffiffiffiffi ðxÞ ¼ 1  erf 2 2 Dx

ðA4Þ

Thus, we obtain experimentally cm(x) and evaluate ctrue(x) from Eq. (A4). On the basis of these values, we evaluate the accuracy of cfit(x) by utilizing the following two equations.

  true 2 ðdm Þ2 ¼ cm x  cx  2  cfit ðdfit Þ2 ¼ ctrue x x

ðA5Þ ðA6Þ fit

The accuracy of composition profile c (x) is expressed in the form of standard deviation predicted composition value.

pffiffiffiffiffiffi

rx

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # uP 2 " u d 1 ðx  xi Þ i ¼t 1þ þP 2 n2 ðxi  xi Þ2

pffiffiffiffi

r of the

ðA7Þ

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pffiffiffiffi It is clearly understood from Eq. (A7) that the standard deviation r decreases with increase of fit number of data n, that is, the accuracy of c (x) ispimproved with increase of number of data. pffiffiffiffiffiffiffi ffiffiffiffiffiffiffi Fig. A1 shows changes of the two standard deviations rm and rfit with number of measuring point, which are obtained by computer calculation for the model based on Eq. (A4). In the calculation, theffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffi fit and m, open and solid circles represents average of rm is assumed to be 0.5. In Fig. A1 the p r r ffiffiffiffiffiffiffi pffiffiffiffi m is scattered around respectively. It is clear from the solid circles that prffiffiffiffiffiffiffi r ¼ 0:5 even though the data n increases in number. On the other hand, rfit clearly decreases with increase of number of composition data n. Consequently, it is obvious that the cfit(x) given by the least square formula of error function has a higher accuracy than independently isolated data cm.

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