The Gibbs–Thomson Effect in DiluteBinary Systems

MA QIAN

The solubility of a precipitate, which is commonly referredto as the composition of the matrix in equilibrium with theprecipitate in the presence of a curved interface, has beenwell documented. For a pure precipitate b, which consistsof component B, in an a matrix that is a dilute solution ofB in A, the Gibbs–Thomson effect is well known as[1,2]

xaB(r) 5 xa

B(`) exp 12sabV bm

RTr 2 [1]

where xaB(r) and xa

B(`) are the equilibrium atomic fractionsof B in a at a curved interface of a spherical precipitate (b )of radius r and a planar interface, respectively. The termsab is the specific interfacial free energy, which is assumedto be isotropic and independent of precipitate size (r), andV b

m is the molar volume of b. When b is not pure B, e.g.,Fig. 3—The temperature dependence of the specific strength for theits composition is given by xb

B, a number of different expres-tested alloyssions have been proposed or employed. These include[3–11]

xaB(r) 5 xa

B(`) exp 12sabV bm

xbBRTr 2 [2]

xaB(r) 5 xa

B(`) exp 12sabV bB

RTr 2 [3]

and

xaB(r) 5 xa

B(`) exp 1 1 2 xaB(`)

xbB(`) 2 xa

B(`)

2sabV bm

RTr 2 [4]

Equation [3], in which V bB is the partial molar volume of

B in b, is given by Gaskell in a well-known textbook.[6]

Unfortunately, this is an incorrect expression as pointed outby Hillert.[12] To illustrate this, one may consider such acase where the partial volume of an interstitial is so small

Fig. 4—Specific strength of (Ir,Rh)75Nb15Ni10 alloys compared with other that it can be neglected. Equation [3] says that xaB(r) should

high-temperature materials. then be independent of precipitate, but from a molar Gibbsenergy diagram, one can see that it is not independent.[12]

The reason Eq. [3] is incorrect may be elaborated as follows.alloys based on combining Ir with Rh, which has higherAs can be seen from Reference 6, until Eq. [77], Gaskell isspecific strengths compared with Ir-, Rh-, and Ta-basedonly considering a single phase and Eq. [77] holds underalloys and is comparable to W-based alloys at 1600 8C.constant composition (i.e., a constant xb

B). However, it shouldbe noted that one cannot define the partial molar volume orany other partial molar quantities of a component in a singleREFERENCES phase with a fixed composition unless the phase under con-sideration is in equilibrium with another phase.[12] Also,1. Y. Yamabe-Mitarai, Y. Koizumi, H. Murakami, Y. Ro, T. Maruko, andwhen considering a two-phase equilibrium, Gaskell’s Eq.H. Harada: Scripta Metall., 1997, vol. 36, pp. 393-98.

2. Y. Yamabe-Mitarai, Y. Ro, T. Maruko, and H. Harada: Metall. Mater. [78], reproduced as Eq. [3] above, is derived under theTrans. A, 1998, vol. 29A, pp. 537-49. incorrect assumption that the two-phase equilibrium is deter-

3. Y.F. Gu, Y. Yamabe-Mitarai, and H. Harada: in Iridium, E.K. Ohriner, mined by equal activity of one component, neglecting theR.D. Lanam, P. Panfilov, and H. Harada, eds., TMS, Warrendale, PA,other one.[12] Thus, it is not by using the common tangent2000, pp. 73-84.

4. Y.F. Gu, Y. Yamabe-Mitarai, Y. Ro, T. Yokokawa, and H. Harada: construction. With a simple molar Gibbs energy diagram,Metall. Mater. Trans. A, 1999, vol. 30A, pp. 2629-39.

5. Y.F. Gu, Y. Yamabe-Mitarai, X.H. Yu, and H. Harada: Mater. Lett.,1999, vol. 41, pp. 45-51.

6. N.S. Stoloff and C.T. Sims: in Superalloy II, C.T. Sims, N.S. Stoloff, MA QIAN, Senior Research Fellow, is with the CRC for Cast MetalsManufacturing (CAST), Department of Mining, Mineral and Materialsand W.C. Hagel, eds., John Wiley & Sons, New York, NY, 1987, p. 519.

7. H. Tanaka, Y. Tan, A. Kasama, and R. Tanaka: Proc. Int. Conf. on Engineering, The University of Queensland, Brisbane QLD 4072, Australia.Contact e-mail: q.ma@minmen.uq.edu.auAdvanced Technology in Experimental Mechanics’ 99, Japan Society

of Mechanical Engineers, Tokyo, Japan, 1999, p. 539. Manuscript submitted September 27, 2001.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 33A, APRIL 2002—1283

Fig. 1—A molar Gibbs free energy diagram constructed to illustrate the equilibrium between an a phase (the matrix) and a spherical b phase (the precipitate)with the radius of b varying from ` to r. AH and DF are two common tangents (B, O, Q, and S are four points of tangency). P is the point of intersectionfor AH and DF corresponding to xa

B on the composition axis. K is the orthogonal projection of point S on the Gibbs free energy curve Gbm(`). The three

dashed lines CE, MG, and NL are all constructed to be parallel to DF. As a result, CD 5 EF, MN 5 GL, and DN 5 FL 5 SK 5 2sabV bm /r.

it is easy to demonstrate that this is wrong because the and the latter equivalent to assuming (1 2 xaB(`))/(xb

B(`) 2xa

B(`)) ' 1,[14] where Eq. [4] reduces to Eq. [1].composition of the precipitate phase must also change. Ifthe precipitate phase cannot change its composition, one Equations [4] and [5], including their variations, have

probably reflected all of the representative expressionsmust still equilibrate both activities, i.e., use a commontangent.[12] A generalized Gibbs–Thomson equation has reported to date for the Gibbs–Thomson effect in a dilute

binary system. However, Eq. [5] is restricted to a constantbeen given by Hillert[9] for this case based on a molar Gibbsenergy diagram, i.e., Eq. [38] of Reference 9. Using Hillert’s xb

B and Eq. [4] is limited to both (xaB(r) 2 xa

B(`))/xaB(`) ¿

1 and (xbB(r) 2 xb

B(`))/xbB(`) ¿ 1. Thus, neither describesEq. [38],[9] and noting that when a is a dilute solution of B

in A, the solute B obeys Henry’s law and the solvent A obeys the general case when xaB and xb

B can freely change with thesize of b. Hillert’s Eqs. [45] and [49],[9] which are given inRaoult’s law, one can obtainterms of d 2Gm/dx2, and therefore valid for either a dilute ora concentrated binary system, are also limited to small

(1 2 xbB(`)) ln 11 2 xa

B(r)

1 2 xaB(`)2 1 xb

B(`) ln 1xaB(r)

xaB(`)2 5

2sabV bm

RTr changes in xaB and xb

B.In this article, using a constructed molar Gibbs energy

diagram, a general description of the Gibbs–Thomson effect[5]in a dilute binary system is introduced for the case whenboth xa

B and xbB can freely change with the size of b. As willEquation [5] is an exact Gibbs–Thomson expression valid

be shown, this description is the same form as Eq. [5] basedfor a dilute binary two-phase system in which the composi-on Hillert’s Eq. [38][9] for a constant xb

B. Then, it is showntion of the precipitate phase, xbB, is constant, i.e., fixed at

that Eq. [4] is a limit form of Eq. [5] when xaB(r) → xa

B(`).xbB(`). When xb

B is a function of r, xbB(`) in Eq. [5] should

Finally, the applicability of Eq. [4] is clarified by comparingbe taken as some value between xbB(`) and xb

B(r).[12] Unfortu-it with Eq. [5].nately, this value is not known explicitly when b is a solu-

Similar to the molar Gibbs energy diagrams used bytion phase.Hillert,[9,13] Christian[10] and Martin et al.[11] for the formula-Equation [4] is derived for the limiting case when bothtion of Eq. [4], Figure 1 shows a molar Gibbs energy diagram(xa

B(r) 2 xaB(`))/xa

B(`) ¿ 1 for the composition of a andconstructed for the equilibrium between an a phase and a(xb

B(r) 2 xbB(`))/xb

B(`) ¿ 1 (including xbB 5 constant) for

spherical b phase. This analysis assumes that (a) the secondthat of b are satisfied.[7–11] Equation [2] can be regarded asphase is incompressible and of isotropic surface propertiesa variation of Eq. [4] when xa

B(`) ¿ xbB(`). Two other varia-

and (b) there are no stresses within the solid matrix phase.tions of Eq. [4] have also been given for the case when aThe latter assumption could happen either if all the molarand b are both dilute solutions of B in A[13] and for the casevolumes are equal and there will be no local volume changeswhen a is a dilute solution of B in A but b is a dilute solutionor if the matrix phase is soft enough (e.g., by the temperatureof A in B.[14] The former is equivalent to assuming (1 2

xaB(`))/(xb

B(`) 2 xaB(`)) ' 1/(xb

B(`) 2 xaB(`)) in Eq. [4][13] being high) to relieve all stresses. Note that, in Figure 1,

1284—VOLUME 33A, APRIL 2002 METALLURGICAL AND MATERIALS TRANSACTIONS A

both xaB and xb

B can freely change with the radius of b. Apparently, if xaB ¿ xa

A holds, i.e., if a is a dilute solutionof B in A,[15,16] then Eq. [13] always holds and therefore MNComparing triangles ABC and EBH yields¿ AN.

Equation [9] can thus be written with sufficient accuracy,xaB(`)

1 2 xaB(`)

5ACEH

[6] when a is a dilute solution of B in A, in the form

xaB(a )

xbB(a )

'ma

A(`) 2 maA(r) 2 CD

AN[16]Substituting AC 5 ma

A(`) 2 maA(r) 2 CD and EH 5

maB(r) 2 ma

B(`) 1 CD leads to

Substituting Eqs. [7] and [10] with some rearrangementCD 5 xa

A(`)(maA(r) 2 ma

A(`)) 2 xaB(`)(ma

B(r) 2 maB(`)) gives

[7]xb

A(`)(maA(r) 2 ma

A(`)) 1 xbB(`)(ma

B(r) 2 maB(`))

[17]Then, comparing triangles ABC and AOM gives

52sabV b

m

rxa

B(`)

xbB(`)

5ACAM

[8] which is identical to Hillert’s Eq. [38],[9] on which Eq. [5]is based, except that in Eq. [17], V b

m is the molar volume ofb corresponding to xb

B(r) (Eq. [18] gives a description ofThat is,xa

b(r) when b is also a dilute solution). Equation [17] demon-strates that the expression of Eq. [5] derived for a constantxb

B based on Hillert’s Eq. [38][9] can be extended to thexaB(`)

xbB(`)

5ma

A(`) 2 maA(r) 2 CD

AN 2 MN[9]

general case where both xaB and xb

B can freely change withthe size of b. In other words, the expression of Eq. [5] canbe used as a general description of the Gibbs–Thomsonin whicheffect in a dilute binary system.

It should be noted that Eq. [5] is not an exact expressionAN 5 AD 1 SK 5 maA(`) 2 ma

A(r) 1 2sabV bm /r [10]

unless xbB is constant. However, it is a sufficiently accurate

expression so long as xaB ¿ xa

A or xaB ¿ 1 holds for theIn the following, we shall prove that if a is a dilute

system under consideration as rigorously derived earlier (i.e.,solution of B in A, i.e., when xaB ¿ 1 or xa

B ¿ xaA,[15,16] then

from Eqs. [10] to [17]). Note that to metallurgists, a diluteMN ¿ AN always holds. Consequently, solving Eq. [9] cansolution is commonly considered to mean that in a smallresult in a general description of the Gibbs–Thomson effect.concentration range the activity coefficient of the solute isThe length of MN can be determined by comparing trian-approximately constant.[17] It is therefore possible that thegles AOM and GOHconstraint xa

B ¿ 1 or xaB ¿ xa

A may not always be satisfiedin such a practically dilute system. Accordingly, the use ofEq. [5] may become less accurate under these circumstances,

xbB(`)

1 2 xbB(`)

5AMGH

[11]depending upon to what extent the approximation MN ¿AN is satisfied according to Figure 1.

where AM 5 maA(`) 2 ma

A(r) 1 SK 2 MN and GH 5 A similar expression can be obtained for the dependencyma

B(r) 2 maB(`) 2 SK 1 MN (SK 5 2sabV b

m /r). Therefore, of xbB on r from Eq. [17] when b is also a dilute solution

(either B in A or A in B), by noting that mai (r) 2

mai (`) 5 mb

i (r) 2 mbi (`) at equilibrium, as follows:MN 5 2xb

A(`)(maA(r) 2 ma

A(`)) 2 xbB(`)(ma

B(r)[12]

2 maB(`)) 1

2sabV bm

r (1 2 xbB(`)) ln 11 2 xb

B(r)

1 2 xbB(`)2 1 xb

B(`) ln 1xbB(r)

xbB(`)2

[18]Then, comparing Eqs. [12] and [10], the assumption MN ¿

52sabV b

m

RTrAN leads to

For easy comparison with Eq. [4], Eq. [5] can be written asmaA(r) 2 ma

A(`) ¿ maB(r) 2 ma

B(`) [13]

On the other hand, comparing triangles APD and FPH gives xaB(r) 5 xa

B(`) exp 1 2sabV bm

(1 2 « )xbB(`)RTr2 [19]

in whichxa

B

1 2 xaB

5ADFH

[14]

Namely,« 5 2

(1 2 xbB(`)) ln 11 2 xa

B(r)

1 2 xaB(`)2

xbB(`) ln 1xa

B(r)

xaB(`)2

[20]xa

B

xaA

5ma

A(`) 2 maA(r)

maB(r) 2 ma

B(`)[15]

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 33A, APRIL 2002—1285

Table I. Comparison between the Solubilities Calculated Based on Equations [4] and [5] (xaB(`) 5 0.05)

(xaB(r)8 2 xa

B(r))/xaB(r) (Pct)

2sabV bm /RT 5 1028 m 2sabV b

m /RT 5 1029 mPrecipitate Radii(r) (nm) xb

B(`) 5 0.2 xbB(`) 5 0.5 xb

B(`) 5 0.9 xbB(`) 5 0.2 xb

B(`) 5 0.5 xbB(`) 5 0.9

15 * 13.37 0.23 3.45 0.062 1.77 3 1023

20 * 5.67 0.12 1.77 0.034 9.87 3 1024

50 * 0.63 0.017 0.25 0.0053 1.55 3 1024

100 9.72 0.14 0.004 0.059 0.0013 3.88 3 1025

200 1.77 0.034 9.87 3 1024 0.014 3.27 3 1024 9.69 3 1026

500 0.25 0.0053 1.55 3 1024 0.0022 5.21 3 1025 1.55 3 1026

1000 0.059 0.0013 3.88 3 1025 5.65 3 1024 1.31 3 1025 3.88 3 1027

5000 0.0022 5.21 3 1025 1.55 3 1026 2.25 3 1025 5.21 3 1027 1.55 3 1028

*No solution to Eq. [5] for xaB(r) due to the selected large capillary length.

xaB(r)8: solubility calculated based on Eq. [4].

xaB(r): solubility calculated based on Eq. [5].

Note thatm ln 11 2 xa

B(r)

1 2 xaB(`)2 1 n ln 1xa

B(r)

xaB(`)2 5

2sabV bm

RTr[22]

and that when (xaB(r) 2 xa

B(`))/xaB(`) ¿ 1,

« (xaB(`)) 5 lim

xaB(r)→xa

B(`)2

(1 2 xbB(`)) ln 11 2 xa

B(r)

1 2 xaB(`)2

xbB(`) ln 1xa

B(r)

xaB(`)2 xa

B(r) 5 xaB(`) exp 1 1 2 xa

B(`)

n/(m 1 n) 2 xaB(`)

2sabV bm

(m 1 n)RTr2[23]5

xaB(`)(1 2 xb

B(`))

xbB(`)(1 2 xa

B(`))

[21]

in which V bm is the volume of one mole of formula units

AmBn . Equation [22] was first given by Bhattacharyya andwhere L’Hopital’s rule[18] was used in the course of solvingEq. [21]. It is interesting to note that on substituting Eq. Russell[19] (i.e., Eq. [8] of Reference 19). It may also be

derived by means of the Gibbs–Duhem relation.[20,21] Equa-[21] into Eq. [19], one at once obtains Eq. [4], demonstratingthat Eq. [4] is actually a limit form of Eq. [5] when tion [23] was first derived by Trivedi[8] (Appendix III of

Reference 8) and has been used by the present author toxaB(r) → xa

B(`). When « ¿ 1, Eq. [19], i.e., Eq. [5], willreduce to Eq. [2]. The condition « ¿ 1 thus serves as an describe the coarsening of intermetallic or compound precip-

itates in binary systems.[22] Similarly, when 2sabV bm /(m 1appropriate constraint on Eq. [2]. This has been neglected

before. n)RT # 1029 m, Eq. [23] can be used as a good approxima-tion of Eq. [22] according to Table I. However, whenAlthough Eq. [4] is a limit form of Eq. [5] when

xaB(r) → xa

B(`), owing to its simpler form, it would be useful 2sabV bm /(m 1 n)RT . 1029 m and r , 200 nm, Eq. [22]

should be employed rather than Eq. [23]. A detailed compari-to identify the circumstances where Eq. [4] can be used toreplace Eq. [5] in practice. Table I shows the results of son between Eq. [22] and the reported experimental measure-

ments of r and xaTi(r) for different sizes of Cu4Ti particlescomparison between Eqs. [4] and [5] for different dilute

binary systems (the capillary length of a metallic system, in a Cu-4Ti alloy has been reported previously.[20] It wasshown that Eq. [22] is in good agreement with the reported2sabV b

m /RT, is usually less than 1028 m when typical valuesare substituted[11]). The value of xa

B(`) is fixed at 0.05 in experimental results.Like most previous analyses, this analysis is also aimedthese calculations because changing xa

B(`) only slightlyinfluences the results. As is evident from Table I, both equa- at solid-solid systems. However, it should be noted that the

thermodynamic equilibrium between a solid matrix and ations lead to almost the same results in most cases of practicalinterest, even when the constraint, (xa

B(r) 2 xaB(`))/xa

B(`) ¿ solid second-phase particle is very complicated.[23] The com-mon tangent approach used in this analysis is strictly applica-1, on the composition of the matrix (a ) is not met for

Eq. [4]. The notable exceptions are when the system under ble to fluid-fluid systems, where surface stress and surfaceenergy are identical for a fluid-fluid interface.[8,23,24] A recentconsideration is characterized by 2sabV b

m /RT . 1029 m andxb

B(`) # 0.5. Under these circumstances, applying Eq. [4] detailed examination of the thermodynamic basis for thecommon tangent approach has shown that this approach isto precipitates smaller than 200 nm in radius tends to result

in significant errors in the calculated values of xaB(r). Other- also applicable to fluid-solid systems if the solid second-

phase particle under consideration is incompressible and ofwise, Eq. [4] can always be used as a good approximationof Eq. [5]. isotropic surface properties.[25] With the assumptions made

earlier, i.e., (a) the second phase is incompressible and ofFor a compound precipitate of stoichiometric compositionsuch as AmBn , following the same methodology used for isotropic surface properties and (b) there are no stresses

within the solid matrix phase, this analysis has approximatedderiving Eq. [17] or using Hillert’s Eq. [38],[9] it can bederived that the solid-solid systems under consideration as if the matrix

1286—VOLUME 33A, APRIL 2002 METALLURGICAL AND MATERIALS TRANSACTIONS A

17. C.H.P. Lupis: Chemical Thermodynamics of Materials, North-Holland,were liquid. In other words, this analysis has assumed thatNew York, NY, 1983, p. 161.the common tangent approach is applicable to the solid-

18. C.E. Pearson: Handbook of Applied Mathematics, 2nd ed., Van Nos-solid systems analyzed. trand Reinhold Company Inc., New York, NY, 1983, pp. 96-97.

In summary, the following points can be concluded from 19. S.K. Bhattacharyya and K.C. Russell: Metall. Trans., 1972, vol. 3,pp. 2195-99.this work.

20. M. Qian and L.C. Lim: Scripta Mater., 1998, vol. 39, pp. 1451-55.21. M. Qian and L.C. Lim: Metall. Mater. Trans. A, 2000, vol. 31A, pp.1. The expression of Eq. [5] can be used as a general descrip-

2659-61.tion of the Gibbs–Thomson effect in a dilute binary sys-22. M. Qian: Metall. Mater. Trans. A, 2000, vol. 31A, pp. 3195-97.tem, wherein the compositions of both the matrix (a ) 23. J.W. Cahn and F. Larche: Acta Metall., 1982, vol. 30, pp. 51-56.

and the precipitate (b ) can freely change with the size 24. J.W. Cahn: Acta Metall., 1980, vol. 28, pp. 1333-38.25. M. Hillert and J. Agren: Acta Mater., in press.of b, as long as the matrix (a ) remains to be a dilute

solution when xaB varies.

2. Although Eq. [4] is a limit form of Eq. [5] whenxa

B(r) → xaB(`), it can be used as a good approximation

of Eq. [5] in most cases of practical interest. The notable Infrared Imaging during Low-Cycleexceptions are when the system under consideration is

Fatigue of HR-120 Alloycharacterized by 2sabV bm /RT . 1029 m and xb

B(`) # 0.5,especially, when the precipitate particles are smaller than200 nm in radius. H. WANG, L. JIANG, Y.H. HE, L.J. CHEN, P.K. LIAW,

3. For a binary compound precipitate of stoichiometric com- R.R. SEELEY, and D.L. KLARSTROMposition such as AmBn , Eq. [23] can serve as a good

Low-cycle fatigue behavior is an important engineeringapproximation of Eq. [22] when 2sabV bm /(m 1 n)RT ,

parameter for superalloys under cyclic loading conditions.1029. However, when 2sabV bm /(m 1 n)RT . 1029 m

While many researchers have conducted extensive studiesand r , 200 nm, Eq. [22] should be used rather thanregarding low-cycle fatigue behavior of superalloys,[1–6]

Eq. [23].very limited work was performed on HR-120 alloy. A typicallow-cycle fatigue study of high-temperature alloys includes(1) fatigue tests at different strain levels as a function of

The author is most grateful for many helpful discussions temperature, to generate a fatigue life vs total strain rangewith Professor Mats Hillert during the preparation of this plot; (2) peak stress amplitude vs cycles to study cyclicwork and for his many helpful comments on the first and hardening or softening; and (3) microstructural studies ofsecond versions of this article. The key reader’s suggestion specimens with different degrees of fatigue damage. Nonde-of mentioning the difference in thermodynamic equilibrium structive evaluation tools, such as optical imaging and acous-conditions between fluid-fluid and solid-solid systems is tic emission, are often used to monitor the fatigue process.acknowledged. This suggestion has led to the addition of During strain-controlled fatigue tests, a significant increasethe entire paragraph immediately before summary. in specimen temperature occurred, especially at high strain

levels. This rise in specimen temperature is a well-knownfact, because almost all the energy of cyclic loading is expec-

REFERENCES ted to convert into heat. The specimen temperature can bemeasured by attaching a thermocouple. However, contact

1. D.V. Ragone: Thermodynamics of Materials, John Wiley & Sons. Inc., temperature measurements at a single point are not alwaysNew York, NY, 1995, vol. II, pp. 106-08.

practical, and they provide limited useful information to2. D.A. Porter and K.E. Easterling: Phase Transformations in Metalsfatigue studies. Therefore, the temperature evolution duringand Alloys, 2nd ed., Chapman & Hall, London, 1997, pp. 44-47.

3. H.B. Aaron and G.R. Kotler: Metall. Trans., 1971, vol. 2, pp. 393-408. the fatigue process has not been very well documented, and4. C.M. Sargent and G.R. Purdy: Scripta Metall., 1974, vol. 8, pp. 569-72. it has not been using to interpret low-cycle fatigue data.5. G. Sauthoff : J. Phys. IV (Coll. CI), 1996, vol. 6, pp. 87-97. Infrared (IR) imaging, also called IR thermography, can6. D.R. Gaskell: Physical Metallurgy, 4th ed., R.W. Cahn and P. Haasen,

provide temperature maps of a specimen subjected to cycliceds., North-Holland, Amsterdam, 1996, Part I, pp. 458-60.7. G.R. Purdy: Met. Sci. J., 1971, vol. 5, pp. 81-85. loading. The IR images show the energy flow and heat8. R.K. Trivedi: in Lectures on the Theory of Phase Transformations, conduction in real time during the entire fatigue process.

H.I. Aaronson, ed., TMS-AIME, New York, NY, 1975, pp. 51-81. Much more information can be derived from the temperature9. M. Hillert: in Lectures on the Theory of Phase Transformations, H.I.

maps as compared to thermocouple measurements. OnlyAaronson, ed., TMS-AIME, New York, NY, 1975, pp. 1-50.limited studies using IR thermography during fatigue10. J.W. Christian: The Theory of Transformations in Metals and Alloys,

2nd ed., Pergamon Press, Oxford, United Kingdom, 1975, pp. 179-82. tests,[7,8] with temperature resolution of 0.1 8C can be found11. J.W. Martin, R.D. Doherty, and B. Cantor: Stability of Microstructures

in Metallic Systems, 2nd ed., Cambridge University Press, Cambridge,United Kingdom, 1997, pp. 242-47.

12. M. Hillert: private communications, 2001. H. WANG, Research Staff Member, is with the Oak Ridge NationalLaboratory, Oak Ridge, TN 78931-6064. L. JIANG, Metallurgist, is with13. M. Hillert: Phase Equilibria, Phase Diagrams and Phase Transforma-

tions, Cambridge University Press, Cambridge, United Kingdom, GE CR&D, Schenectady, NY. Y.H. HE, Professor, is with National KeyLaboratory of Powder Metallurgy, South-Central China University. L.J.1998, pp. 143-50.

14. R.T. Dehoff : Thermodynamics in Materials Science, McGraw-Hill, CHEN, Visiting Scientist, and P.K. LIAW, Professor, are with the Depart-ment of Materials Science and Engineering, University of Tennessee atInc., New York, NY, 1993, pp. 377-82.

15. R.G. Mortimer: Physical Chemistry, 2nd ed., Academic Press, San Knoxville, Knoxville, TN 37996. R.R. SEELEY, manager, and D.L.KLARSTROM, Director of Technology, are with Haynes International,Diego, CA, 2000, pp. 209-11.

16. R.A. Alberty: Physical Chemistry, 7th ed., John Wiley & Sons, New Inc., Kokomo, IN 46904.Manuscript submitted January 17, 2001.York, NY, 1995, pp. 217-18.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 33A, APRIL 2002—1287