Abstract
Abstract The effects of an external stress on the formation of ellipsoidal x phase precipitates have been examined for a Ti-20 wt% Mo alloy aged at
Introduction
The effects of a directional stress during aging on the orientation distribution of precipitate structures have been examined widely for Al-Cu system alloys [1] [2] [3] [4] . For example, Eto et al. [2] observed that application of the external stress during aging below 180°C induced the oriented precipitation of disk-shaped Guinier-Preston (GP) zones and plate-shaped h 0 phase in single crystals of an Al-3.71Cu alloy (All compositions in this text are expressed by weight per cent, wt%). Tensile stress along the [001] axis produced the preferential formation of (100) and (010) variants of GP zones and h 0 , and compressive stress (001) variants. Using a two-step aging (aging at 80°C under stress and aging at 170°C under no stress) procedure, they showed that the stress-orienting effect was initiated and determined during the nucleation stage, namely the formation of GP [I] , which will grow to h 0 via GP [II] . On the other hands, Monzen et al. [5] have shown by transmission electron microscopy (TEM) that application of an external stress during aging at 200°C induces the oriented precipitation of disk-shaped GP zones in single crystals of a Cu-0.9Be alloy. The disk-shaped GP zones consist of monolayers of Be atoms on the {100} a planes. A compressive stress along the [001] a direction assists preferential formation of the GP zones perpendicular to the stress axis and tensile stress induces the same parallel to the stress axis. They have also revealed that the effect of applied stress is on the nucleation process of GP zones. In addition, the misfit strains, e 11 = e 22 and e 33 , of the GP zone in directions, respectively, parallel and perpendicular to the plate plane have been estimated from measurements of the length change and lattice parameter during aging. Moreover, Monzen et al. [6] have examined the effects of an external stress on the nucleation and growth of diskshaped c 00 phase for a Cu-1.2Be-0.1Co alloy aged at 320°C. The metastable c 00 phase is composed of alternate Be and Cu matrix layers parallel to the {100} a planes. A compressive stress applied in the [001] direction during aging preferentially accelerates the nucleation and growth of the c 00 variant normal to the [001] axis among crystallographically equivalent three ones. However, a tensile stress does not significantly affect the nucleation and growth of c 00 precipitates. The acceleration of the nucleation and growth of the specific c 00 variants can be well understood through the interaction energy between the stress acting on the c 00 variants and their misfit strains. Prior to this work, Watanabe et al. [7] already estimated the misfit strains, e 11 = e 22 and e 33 , of the disk-shaped c 00 precipitates from measurements of the length change and lattice parameter during aging.
In order to evaluate the effect of an applied tensile stress on the nucleation of x precipitated phase in Ti-20Mo alloy specimens aged at 350°C, Nishizawa et al. [8] have measured the number of x precipitates in constant areas in the stress-free and stress-aged specimens, and concluded that a tensile stress applied during aging promotes the nucleation of x precipitates. The structure of the x phase in an aged Ti-14Mo alloy is hexagonal with a = 0.2820 nm and c = 0.4606 nm [9] . Since the x precipitates have an ellipsoidal shape slightly elongated along h111i b of the b-Ti matrix [10] , in contrast to the diskshaped GP zones and c 00 phase in Cu-Be alloys [5, 6] and the disk-shaped GP [1] zones and h 0 phase in Al-Cu alloys [1] [2] [3] [4] , it is interesting to examine how crystallographically equivalent four x variants are formed under an applied external stress. We have found the stress-oriented nucleation of the x precipitates, which depends on the sense of the applied stress, and furthermore found the promotion of the nucleation of x precipitates under an applied tensile stress but almost no effect of an applied compressive stress on the nucleation, as will be shown later.
Experimental
Sheets of a Ti-20Mo alloy were provided by Daido Steel Co., Ltd. The alloy sheets were cold rolled to a 50 % reduction in thickness and then spark cut into specimen strips. For tensile aging, the specimens had a cross section of 1 mm 9 5 mm and a gage length of 20 mm. For compressive aging, the specimens had a cross section of 2 mm 9 5 mm and a gage length of 7 mm. All the specimens were solution treated at 950°C for 2 h in a vacuum and quenched into water. The solution treatment also caused complete recrystallization of the specimens. The solution-treated specimens were then aged at 300°C for various times either under an applied stress of 450 MPa (stress aging) or under no stress (free aging). The applied stress of 450 MPa is about three-fourths of the yield strength of the solution-treated specimen at 300°C.
Length changes e T on aging were examined by measuring, with a micrometer, the distance between two scribed marks, about 4, 5, or 8 mm apart for the specimens stress-aged at 300°C for 3 h and then free-aged at 350°C for various times, or free-aged at 300°C for 3 h and then free-aged at 350°C for various times. The length change is defined as e T = (l -l 0 )/l 0 , where l 0 and l are the length between the two marks before and after aging, respectively. The measurement accuracy of length change is in the order of 10 -5 in strain. An X-ray analysis was performed to measure the lattice parameters of the solution-treated and aged specimens. The diffractometer with a Cu target was used for the X-ray analysis.
Thin foils, 0.2 mm thick, for TEM observations were prepared by grinding the aged specimens with SiC papers and by electropolishing using a solution of 60 vol% methanol, 35 vol% 2-Butoxyethanol and 5 vol% perchloric acid at -20°C, and 15 V in a twin-jet electropolisher. Microscopy was performed using a JEOL 2010FEF microscope operated at 200 kV.
Results
Formation of x precipitates under external stress SADP. Reflections due to two x variants, I and III, are present. The three dark-field images were taken using three reflections due to the variant I. The variant III exhibited the following orientation relationship to the b-Ti matrix:
. All the four variants had an ellipsoidal shape elongated along h111i b with an aspect ratio of about 2, irrespective of application of external stress. Here, the shape of x precipitates is assumed to be a spheroid described in the x-y-z coordinates by x 2 /r 2 ? y 2 /r 2 ? z 2 /l 2 B 1, where l;2r. It may be noted in Fig. 2a-c that the precipitate sizes r and l for the TSA specimen are larger than those for the FA specimen, and also the sizes for the CSA specimen are slightly larger than those for the FA specimen. Figure 3a -c displays the number density N of the ellipsoidal x precipitates, the lattice parameter of the b-Ti matrix measured as by X-ray analysis, and the average precipitate sizes " r and " l, each as a function of aging time t for the specimens TSA, CSA, and FA at 300°C. The number density of x precipitates is defined as the volume fraction of precipitates divided by the average volume of a single precipitate. The volume fraction was determined by applying the lattice parameters of the TSA, CSA, and FA specimens to the experimental data reported for the dependence of the lattice parameter on the Mo concentration (Da = -0.0002 nm/at.%) in the literature [11] . The concentration of Mo in the x precipitates is assumed to be 10 wt% equal to the equilibrium concentration of Mo at 300°C in the Ti-Mo binary phase diagram [12] . Even if the Mo concentration in the precipitates is assumed to be 20 wt% equal to the Mo concentration in the present alloy, the volume fraction estimated at each aging time is not significantly affected. The average volume of an ellipsoidal precipitate was obtained by measuring r and l of the four crystallographically equivalent variants of x precipitates. About 400 precipitates were used for measurements of l and r at each aging time. Figure 3a demonstrates that the number density of x precipitates is constant over a time span of 1.5 h (5.4 9 10 3 s) in the case of the TSA specimen, and over 3 h for the CSA and FA specimens. Thus the formation of the x precipitates in the TSA, CSA, and FA specimens is completed after either 1.5 or 3 h. More importantly, the number density of x precipitates in the TSA specimen is higher than that in the FA specimen, indicating that the formation of x precipitates is favored by applied tensile stress. There is, however, no significant difference in number density between the CSA and FA specimens. Moreover, it can be seen from Fig. 3c that aging under tension and compression accelerates the growth of the precipitates. The average sizes r and l of x precipitates in the specimen FA at 300°C follow a parabolic growth law (r and l / t 1 2 Þ, whereas aging at 300°C under tension generates an increase in precipitate size that is linear with t. Consequently, while the growth of x precipitates is governed by the diffusion of Mo atoms in x precipitates from the interface between the b-Ti matrix and the x precipitates to the b-Ti matrix [12] , precipitate growth is instead interface controlled under tensile stress [6] . The result of a more detailed investigation of this phenomenon will be reported in a future paper.
Specimen length change Figure 4 presents the length change e T along the loading direction (LD) and transverse direction (TD), plotted as a function of aging time t for the specimen at 300°C for 3 h and then FA at 350°C (TSA-FA), the specimen CSA at 300°C for 3 h and then FA at 350°C (CSA-FA), and the specimen FA at 300°C for 3 h and then again FA at 350°C (FA-FA). The specimens TSA, CSA, and FA at 300°C for 3 h each exhibit a slight decrease in length change. Length change along the LD and TD for the TSA-FA, CSA-FA, and FA-FA specimens rapidly decreases and then eventually plateaus after about 53 h (1.908 9 10 5 s). The plateau value of e T along the LD for the TSA-FA specimen is larger than that along the TD for the same specimen, whereas, in the case of the CSA-FA specimen, the plateau value along the LD is smaller. In addition, the values of e T measured along several directions of the specimen FA at 300°C for 3 h and then FA at 350°C for 53 h were identical within experimental error, indicating that there was no anisotropy in length change of the FA-FA specimen. It should also be noted in Fig. 4 that length change along the LD for the specimens TSA and CSA at 300°C for 3 h exhibits a slight decrease. This indicates that the creep strain due to dislocation motion does not contribute to the length change. Figure 5 presents the dependence of the lattice parameter a of the b-Ti matrix on aging time t for the TSA-FA, CSA-FA, and FA-FA specimens. The lattice parameter decreases with increasing aging time for the three specimens. Diffusion of Mo atoms in x precipitates from the x/ b interface to the b-Ti matrix occurs as the x precipitates grow [12] . Since the size of Mo atoms is smaller than that of Ti atoms, the increase in Mo atoms within the b-Ti matrix causes the decrease in the lattice parameter. The lattice parameter of the TSA-FA specimen decreases more rapidly than that of the FA-FA specimen, although eventually the lattice parameters of both specimens converge on the same value of 0.3232 nm. This value is close to the lattice parameter, 0.3230 nm [11] , of a b-Ti solid solution containing 38 wt% Mo that is the equilibrium solubility of Mo in b-Ti at 350°C in the Ti-Mo binary phase diagram [12] . The more rapid decrease in the lattice parameter of the TSA-FA specimen than the FA-FA specimen is attributed to the higher number density and the larger size of precipitates in the TSA-FA specimen (Fig. 3a, c) . However, the lattice parameter of the CSA-FA specimen decreases slightly rapidly compared to that of the FA-FA specimen. This is attributable to the fact that the CSA-FA specimen has almost the same number density as the FA-FA specimen, and the slightly larger size of x precipitates than the FA-FA specimen (Fig. 3a, c) .
The specimen length change e T along any direction can be written as functions of the average misfit strain e a along the same direction caused by the misfit strains of precipitates, the volume fraction f of the precipitates, and the dimensional change e l due to the increase in dissolved Mo atoms in the solid solution, as follows [7, 13] :
where e l = (a -a 0 )/a 0 . Here a 0 and a are the lattice parameters of the specimens before and after aging. We are able to estimate the average misfit strains e aL and e aT along the LD and TD for the TSA-FA and CSA-FA specimens, as well as the average misfit strain e aF for the FA-FA specimen. A value for f of 0.14 was obtained by applying the values a 0 = 0.3253 nm and a = 0.3232 nm before and after aging at 350°C for 53 h (Fig. 5) to the experimental data regarding the dependence of the lattice parameter on Mo concentration [11] . The Mo concentration in the x precipitates was assumed to be 10 wt%, as noted above. . These values were estimated from the value of e T in Fig. 5 , as well as from f and e l . The value of e aF = 0.010 is in good agreement with the misfit strain of e c = 0.0096, calculated from the lattice parameters a = 0.3232 nm for the b-Ti matrix and a = 0.2835 nm and c = 0.4676 nm for the x precipitates, as determined by X-ray analysis after aging at 350°C for 53 h, using the following equation:
Here V x and V b are the molar volumes of the x precipitates and b matrix, respectively. The measured values of a and c for the x precipitates arnucleation and growthe close to the values a = 0.2820 nm and c = 0.4606 nm for the x precipitates in a Ti-14Mo alloy [9] . The e c value of 0.010 also is nearly identical to the value of e c = 0.0088 obtained by Hickman [14] from the lattice parameters of the x precipitates and b matrix using Eq. 2. For the TSA-FA 0 specimen, e aT = 0.008 \ e aF = 0.010 \ e aL = 0.028, specimen, e aL = 0.004 \ e aF = 0.010 \ e aT = 0.016. These data indicate that tensile stress preferentially aids the formation of specific x variants out of the crystallographically equivalent four ones, having positive misfit strains along the LD greater than e aF = 0.010, and that compressive stress induces the preferential formation of specific x variants with smaller positive and/or negative misfit strains along the LD. Below, the dependence of the number density of the x variants on the misfit strain along the LD is examined for the specimens TSA, CSA, and FA at 300°C for 12, 48, and 48 h, respectively.
Misfit strain and number density of x variants We often observed grains in which the (113) b plane was nearly parallel to the surfaces of the TSA, CSA, and FA specimens. The number densities of the x variants I and II were determined by taking dark-field images using reflections due to these variants I and II in the [113] b SADPs (see Fig. 1 ), and measuring the thicknesses of TEM foils using thickness fringes. In addition, the misfit strains of the variants I and II along the LD on (113) b were obtained by measuring the angle / between the LD and the elongated Fig. 6 . However, the actual misfit strains may be small, since a large misfit strain can be relieved by an array of misfit dislocations at the interface between different phases [6, 7, [15] [16] [17] . The values of N for the TSA specimen are larger than those for the FA specimen when e M [ 0, but are smaller when e M \ 0. On the other hand, the values of N for the CSA specimen are larger than those for the FA specimen when e M \ 0, but are smaller when e M [ 0. It is also noted that the average of the N values obtained for the TSA specimen is larger than that for the CSA specimen. This fact agrees with the observation that the number density of x precipitates in the TSA specimen is higher than that in the CSA specimen (Fig. 3a) .
Discussion
The origin of the dependences of the formation of x precipitates (Fig. 3a) , the relationship between the number density of x variants and the misfit strain along the LD (Fig. 6 ) and the average misfit strains along the LD and TD (Table 1 ) on the sense of the applied stress can be understood to arise through the interaction energy between an external stress r ij and a misfit strain e ij (stress-free transformation strain). That is, the cause of the present results is understood to be due to the work done by the external stress during the nucleation of x precipitates. The interaction energy F I is expressed as [18] 
where V is the volume of the x precipitate. According to a classical nucleation theory [19] , the interaction energy F I affects the nucleation rate R as written by
where G* is the activation energy for nucleation in the absence of a stress, F I * is the interaction energy associated with nucleation of the x precipitate of a critical nucleus volume, k is Boltzmann's constant, and T is the absolute temperature.
Equations 3 and 4 predict that applied tensile stress promotes the nucleation of x precipitates because the average misfit strain e aF = 0.010 for the FA specimen, but compressive stress does not affect it. This prediction is in agreement with the result shown in Fig. 3a . Similarly, it may be expected from Eqs. 3 and 4 that tensile and compressive stresses promote the nucleation of a x variant when the misfit strain of the variant along the LD is positive and negative, respectively. This is coincident with the dependence of the number density of x variants on the misfit strain in Fig. 6 . Moreover, Eqs. 3 and 4 predict that, for the TSA specimen, the average misfit strain e aL along Fig. 6 Number density N of x variants in Ti-Mo specimens, TSA, CSA, and FA at 300°C for 12, 48, and 48 h respectively as a function of the misfit strain e M of the variants along the LD the LD is larger than e aF = 0.010 for the FA specimen, and, for the CSA specimen, e aL \ 0.010. This is the case, as shown in Table 1 .
Conclusions

Application of tensile stress during aging at 300°C
promotes the nucleation of ellipsoidal x precipitates in a Ti-20 wt% Mo alloy, but compressive stress does not essentially affect the same. 2. From measurements of the length change and the lattice parameter during aging, the average misfit strains along the loading and the transverse directions, caused by the misfit strains of x precipitates, have been estimated for the TSA and CSA specimens. The estimates reveal that specific x variants among crystallographically equivalent ones are preferentially formed, depending on the sense of the applied stress. This is supported by the result of the dependence of the number density of x variants formed under the applied stress on the misfit strain of the variants along the loading direction. 3. The dependences of the nucleation of x precipitates and the formation of specific x variants on the sense of the applied stress can be understood through the interaction energy due to the presence of misfit strains between the applied stress and x precipitate.

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