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Antiphase domain growth in Cu3Au: Quantitative comparison between theory and experiment
ANTIPHASE DOMAIN GROWTH IN Cu,Au: QUANTITATIVE COMPARISON BETWEEN THEORY AND EXPERIMENT A. J. ARDELL.
N. MARDESICHt
and C. N. J. WAGSER
Materials Department. University of California. Los Angeles. CA 90024, U.S.A. (Received 4 January 1979)
Abstract-The Hillert theory of normal grain growth has been modified. for antiphase domain (APD) growth in Cu,Au. A semi-empirical correction factor, 8, is introduced into the theory in order to incorporate the effect of the extreme breadth of the experimentally observed APD size distributions on the rate of APD growth. The mobility. .If, of APD boundaries in an ordered alloy. which is proportional to the chemical diffusion coefficient. 3. is shown to be inversely proportional to the long-range order parameter, S. The theory is compared with literature data on APD growth in Cu,Au. as well as our own measurements made at 325,350 and 37X for times up to 10 h on pure Cu,Au and Cu,Au-Co alloys containing 1.5 and 2.5 at.‘?;Co. Using estimated values of 3, measured values of S. values of /3 determined by fitting the theory to existing APD distributions and another parameter (u) which is a measure of the relative values of the average and ‘critical’ APD sizes, the experimentally measured rate constants for APD growth were used to calculate values of the APD boundary interfacial energy, 0, for comparison with recently derived theoretical values. The-quantitative agreement between theory and experiment is satisfactory, considering the uncertainty in 2. The theory also predicts that APD growth should increase in the Co-containing alloys. Such an effect is observed experimentally, but is greater in magnitude than predicted by the theory, for reasons that are not understood. R&urn&-Nous
avons modifie la theorie de Hillert pour la croissance normale des grains afm de l’appliquer a la croissance des domaines antiphase (DAP) dans CusAu. Nous introduisons un facteur semi-empirique /? pour tenir compte de I’effet de la grande dispersion des valeurs experimentales de la taille des domaines sur la vitesse de croissance des DAP. Nous montrons que la mobilite M des parois de DAP, qui est proportionnelle au coefficient de diffusion chimique 3, est inversement proportionnelle au parametre d’ordre a longue distance S. Nous comparons cette theorie avec les donnees de la litterature concernant la croissance des DAP dans Cu,Au, et avec nos propres mesures effectutes a 325, 350 et 37YC, pour des durees pouvant atteindre 10 h, sur le Ct+Au pur et sur des alliages CusAu-Co contenant 1.5 et 2,5 at%Co. En utilisant une valeur estimee de 8. les valeurs experimentales de S. les valeurs de /J determinees en ajustant la theorie aux repartitions existantes des DAP, et un autre parametre (10 qui est une mesure da valeurs relatives des tailles moyennes et “‘critiques” des DAP. ainsi que les vitesses de croissance des DAP mesurees experimentalement, noux avons calcule des valeurs de I’energie u des joints de DAP. afin de les comparer avec des valeurs thioriques recentes. L’accord entre la thtorie et l’expirience est satisfaisant, compte tenu de I’incertitude sur 3. La thtorie privoit igalement une augmentation de la croissance des DAP dans les alliages contenant du cobalt. Cet effet est observe expirimentalement. mais son ordre de grandeur est superieur a celui prlvu par la thtorie. pour des raisons que nous ne comprenons pas. Zusammenfassung-Die
Hillertsche Theorie des normalen Kornwachstums wurde zur Behandlung des Wachstums von Antiphasengrenzen (ADP) in Cu,Au angepal3t. Urn den EinfluB der extremen Breite der experimentell beobachteten APD-GriiDenverteilung auf die APD-Wachstumsgeschwindigkeit zu berhcksichtigen, wurde ein halbempirischer Korrektionsfaktor p in die Theorie eingefnhrt. Es wird gezeigt, daD die Beweglichkeit IMder APD-Grenzen in einer geordneten Legierung, die zum chemischen DilTusionskoefiienten B proportional ist. sich umgekehrt proportional zum Fernordnungsparameter S verhalt. Die Theorie wird mit Literaturangaben iiber APD-Wachstum in CusAu verglichen, ebenso mit unseren Messungen, die bei 325, 350 und 375’C fur eine Dauer bis zu 10 Stunden an reinem Cu,Au u_nd an CusAu-(I,%2.5) At.-?bCo--Legierungen durchgeftihrt wurden. ,Mit abgeschltzten Werten fur 2, gemessenen fur S, mit aus der Anpassung der Theorie an die vorhandenen APD-GrGBenverteilungen ermittelten Werten fur /?, und mit einem welteren Parameter (u)-einem MaB fur die relativen GriiBen der durchschnittlichen und “kritischen” APD-GroBen-wurden aus den gemessenen Ratenkonstanten des APD-Wachstums Werte fur die APD-Grenzfl$ichenenergie D ermittelt und mit friiher theoretisch abgeleiteten Werten verglichen. Die quantitative_Ubereinstimmung zwischen Theorie und Experiment ist befriedigend, wenn man die Unsicherheit in D beriicksichtigt. Die Theorie sagt weiterhin aus, daf3 das APD-Wachstum in den Co-haltigen Legierungen verstarkt sein sollte. Dieser Effekt wird experimentell gefunden ist jedoch aus noch unverstandenen Griinden gr6Ber als theoretisch vorhergesagt.
t Present address: Spectrolab. 12500 Gladstone Avenue, Sylmar, CA 91340. U.S.A. 1261
1262
ARDELL.
MARDESICH
ASO
WAGNER:
ANTIPHASE
1. INTRODUCTION The kinetics of anti-phase domain (APD) growth in ordered Cu,Au have been shown [l-4] to obey an equation of the type
(0)'- (Do)'= Kt,
(1)
(D) isthe average domain ‘diameter’ at time (Do)isits initial value and K is a constant that
where r.
determines the growth rate at some temperature 7: The rate constant K has an Arrhenius-type temperature dependence, with an activation energy, Q, of about 44 kcal mole- I [l], which compares reasonably well with values of Q for tracer diffusion of Au in Cu [S], and for interdiffusion in concentrated Cu-Au alloys [6]. Equation (1) is representative of the kinetics of grain growth in metals and alloys. Physical justification for this type of equation has been offered by Feltham [7] in an early derivation and more recently by Hillert [S], who used a modification of the theory of Ostwald ripening by Lifshitz and Slyozov [9] and Wagner [lo], to derive not only equation (1) but the theoretical distribution of D aswell. Hillert’s theory was adapted later by Sauthoff [3] who attempted to incorporate specifically the known geometry of APDs in Cu,Au, which are approximately rectangular parallelepipeds with interfaces parallel to : 100;. Sauthoff used a clever, but somewhat curious, force balance to estimate the growth rate of a particular domain with respect to APDs of critical size. In this way he was able to circumvent the dilemma concerning the relationship between the surface tension of a boundary and its radius of curvature. SauthofI’s experimental data included measurements of the domain size distributions, which were considerably broader than those predicted by Hillert’s theory. Moreover, Sauthoff concluded that his size distribution data agreed reasonably well with those of Sakai and Mikkola [Z], although their size distributions, determined from X-ray measurements, exhibited pseudo-periodic fluctuations which Sauthoff did not believe were real. Sauthoff attempted to account for the observed excessive breadth of the distributions by invoking the notion that the critical APD size, D*, which is stationary at a given instant of time (see Section 2.1, below), was itself distributed in size throughout a particular sample. In so doing, Sauthoff succeeded in fitting his experimental distributions to a theoretically derived function. In all the work to date on APB growth in Cu,Au, no attempt has been made to compare theory with experiment in a quantitative manner, despite the clearly excellent semiquantitative agreement that has been demonstrated. The main purpose of this paper is to make such a comparison, using extant data as well as our own measurements, which incorporate the effect of Co precipitates on the kinetics of APB growth in Cu,Au. In making such a quantitative com-
DOMAIN
GROWTH
IN Cu,Au
parison it is necessary to take into account the influence of the broad particle size distributions on the kinetics and it is essential to estimate the APB mobility, M, which is one of the most important parameters of the theory. These items are accounted for theoretically in the next section. 2. THEORETICAL
CONSIDERATIONS
2.1 A semi-empirical mociifcotion of Hiflert’s theory We begin with equation (6) of Hillert’s paper, written in terms of D = 2R (where R is the grain ‘radius’), which appears as
where c is the APB energy, M its mobility and z is a constant approximately equal to unity in a threedimensional system (we will use, henceforth, the assumption a = 1, following Hillert). D' represents the critical domain size; APDs with D = D* are instantaneously neither growing nor shrinking. We note that the coefficient of (l/D* - l/D) is a factor of two larger than that derived by Sauthoff. Equation (2) can be rewritten dD’ = 8aM(u - 1). dt where u = DID*. By proceeding from this point using the mathematical approach of Lifshitz and Slyozov (91, Hillert derived the equations governing the time-dependence of D* (hence (D))and the theoretical distribution of particle sizes expected during normal grain growth. However, Sauthoff convincingly demonstrated that the APD size distributions in CusAu are significantly broader than predicted by Hillert’s theory. This must influence the kinetics of APD growth because of the competitive nature of the process (if all the domains have the same size the growth rate is zero, whereas if the distribution is broader than predicted the growth rate should exceed that of Hillert’s theory). To incorporate the influence of the particle size distribution into the theory, it is necessary to introduce a function which, while accomplishing its task, leaves the nature of the kinetic law, equation (1). undisturbed. As we shall demonstrate, this can be achieved by rewriting (3) as dDZ 7 = SaM(u - l)( 1 + &,. where /I is a parameter that varies between zero and unity. While we are unable to justify the multiplication of (3) by the factor 1 + /Iu on physical grounds. we note that a similar procedure was used by Sauthoff and Kahlweit [ll] to modify Lifshitz and Slyozov’s [93 and Wagner’s [lo] theory of Ostwald ripening to fit their data on 7’ particle coarsening in aged Ni$Si alloys. In a later paper, Ardell[12] demonstrated that the factor 1 + fiu could be related to the influence
ARDELL. MARDESICH
AND
WAGNER:
ANTIPHASE
of the 7’ volume fraction on the coarsening kinetics and particle size distributions. In the cases of APD or grain growth 1 + flu can only be regarded as a semi-empirical correction to the theory. Its consequences. however, enable theory and experiment to be brought into remarkably good agreement. The theory proceeds from this point according to the method of Lifshitz and Slyozov, as utilized by Hillert and later by Ardell [E-14]. We first write the equation for du’/dr, which on using (4) becomes
DOMAIN
B 0
0.05 0.10
0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70
(6)
0.75
0.80 0.85 0.90 0.95
and ‘J = 8 aM dr/dD*‘.
(7)
For stationary solutions to exist both du’/dr and d(du’/dr)/du must vanish at a value of u,, corresponding to a constant value of 7. On setting (5) and its hrst derivative w.r.t. u equal to zero at u = ue, we find ug = 2/( 1 - 8,
(8)
y = 4/(1 + /?y.
(9)
and
At this juncture, we note that the kinetic equation describing the increase of D* with t is obtained on integrating (7), which, on substitution of (9), yields
De2- Dz2= 2 aM(1 + /?)2t,
(10)
where Dt is the critical APD size at the onset of the growth process. It is evident that the correction factor 1 + flu has the expected effect of accelerating the kinetics of APB growth by a factor of (1 + b)2 while leaving the parabolic rate law unchanged. Moreover, according to (8) the particle size distributions can be expected to broaden as /I- 1, since u. increases without limit as b increases to unity. The particle size distributions can be derived using the procedures of Lifshitz and Slyozov. Similar derivations are presented in the papers of Ardell[12-141. The probability density function, g(u), is given by
IN Cu,Au
(U, 0.8889 0.8794 0.8701 0.8608 0.8517 0.8427 0.8339 0.8251 0.8165 0.8080 0.7997 0.7915 0.7834 0.7755 0.7677 0.7600 0.7525 0.7451 0.7379 0.7312
which results in the expression G(u)=l
_{2_u;
_fl~t’1+8”“-8’1’
-3u(l + B)’ x exp i (1 - /I)[2 - u(1 - /I)] I . (12)
As the variable measured experimentally is (D), not D*, it is necessary to rewrite equations (10) through (12) in terms of (D) for the purpose of comparing theory with experiment. To this end, we need to calculate 0 (u) = (D)/D*= 44 du, (13) s0 as a function of the parameter 8. The results of numerical calculations for values of /I up to 0.95 are presented in Table 1. Graphical representation of the values of(u) in Table 1 shows that (u) has a limiting value of - 0.725 at p = 1; the value of (u) at b = 0 is 8/9, consistent with Hillert’s theory. It is now convenient to define the variable u’ = D/(D)= u/(u)
The cumulative distribution function. G(u), is readily obtained using the procedure described by Ardell [ 153,
(14)
and the probability functions g’(u’) and G’(u’), which can be readily compared with experimentally determined histograms or cumulative distributions. It is not difficult to show that g’(u’) = (u)g(u) and G’(u3 = G(u). In terms of (D),the kinetic equation for APD growth, (lo), becomes
(D)'- (Do)'= 2 aM(u)‘(l (11)
1263
Table 1. The variation of (u) with the parameter /I
where 5 = In D*z
GROWTH
+ m2t.
(15)
On comparing (15) with (l), we see that the rate constant K isgiven by
K = 2 &f(u)’
(1 + ,3)‘.
(16)
126J
ARDELL.
,MARDESICH
ASD
WAGNER:
1 ’ The APB mobility. hZ -._ In deriving an expression for Jf in terms of measurable physical quantities, we will adapt the definition used by Sun and Bauer [16] to the problem of APB motion in ordered crystals. In general. ,cI can be expressed as (17)
where 2 is the diffusivity of the species responsible for motion of the boundary, n is the number of such species per unit boundary area participating in the jump process and kT has its usual meaning. In a given APB, the boundary advances when the atoms in anti-phase positions in the domain being consumed jump the required distance (the displacement vector of APB) and so become in phase with the other atoms in the growing domain. In a binary ordered alloy, both atomic species will be required to jump in order that the APB advance. Consequently, the appropriate diffusivity will be the interdiffusion coefficient, 3. The value of n in CusAu can be expressed as $‘a’ (where S is the long range-order parameter as defined by Warren [17] and a is the lattice constant) because this is precisely the number of atoms per unit area of [IOO) APB that must be transported across the interface for it to advance. The reason S enters into the expression is that it represents the total probability that the occupancy of Cu and Au on their respective sublattices in the ordered alloy exceeds random 0ccupancy.t Another way of stating this that there is no need for (1 - S)/a2 atoms per unit area to diffuse into new sites in order for the APB to advance. Equation (17) can thus be rewritten as
M=S
SkT’
which on substitution into (16) yields the final expression for K K = 2a2& SkT
3. EXPERIMENTAL
PROCEDURES
3.1 ,411oy preparation Pure CulAu and Cu,Au containing 1.5 and 2.5 at.:/, Co were prepared from elements of 99.999:/, purity. The starting materials were Au splatters and Cu rods obtained from the United Mineral and Chemical Corporation and Co sponge from Elect In Cu,Au S can be expressed as (rc, - 3,‘4)+ (T,” - l/4), where rcU and raU are the fractions of Cu and Au atoms, respectively. occupying Cu and Au sites. 314 is thus the fraction of Cu sites occupied by Cu rC” atoms in excess of random occupancy while rAu- 1/‘4 is the fraction of Au sites occupied by Au atoms in excess of random occupancy.
ANTIPH.\SE
DOMAIN GROWTH
IS Cu,Au
tronic Space Products, Inc. Two master alloys of Au with 10 and 20”, Co were first prepared by thoroughly mixing the constituents and inductionmelting them in a high purity graphite crucible under a Ti-gettered He atmosphere. The alloys were annealed for 48 h and cold-rolled for subsequent use in the final alloy preparation. Stoichiometric CusAu alloys containing various amounts of Co were subsequently prepared from the master alloys and pure Cu and Au by sealing them in quartz tubes at about lo-’ torr and melting them by induction heating. To insure homogeneity, the samples were cold-rolled and then remelted under the same conditions. For final homogenization the alloys were annealed at 925’C in an Hz atmosphere furnace for about 3 days. Filings (- 325 mesh, co.4 pm) were made with a Nicholson No. 4 file. The filings a-ith 1.5 at.“, Co were then encapsulated in an evacuated quartz tube and solution-heated for 15 min at 925:C followed by a water-quench. Even after this relatively brief heattreatment, the filings were partially sintered, but could be separated by filling again. applying a very light pressure. The filings of Cu,Au with 0 and 1.5 at.?; Co were annealed for 1 h at 41O’C and then used to determine the long-range order parameter S between 325 and 385’C [18]. At the conclusion of these measurements, the filings of Cu,Au with 0 and 1.5 at.% Co, as well as the cold-worked filings with 2.5 at.% Co were sealed in evacuated Pyrex tubes, first heat-treated for 40 h at 45O’C. then for 0.5, 2 and 10 h at 325, 350 and 375’C. respectively and water-quenched. 3.2 X-ray analysis The (lOO), (llO), (210) and (211) superstructure reflections of each annealed sample were recorded on a theta-theta diffractometer [19] using filtered Curadiation and a proportional counter, or on a GE diffractometer using Cu-Kz radiation diffracted by a doubly-bent graphite monochromator in the primary beam. The heat-treated filings were mounted in a plexiglass holder using a solution of Duco cement in acetone as a binder. Si and W powder samples were used as standards. The peak protiles of the alloys and of the standard samples were separated into their Kz, and Kz, components using the Rachinger [ZO] method and the Kz, profiles were corrected for instrumental broadening by the Stokes [21] method, using a Fortran IV program. When using the Fourier coefficients corresponding to Ksc, peaks only, it is possible to use standard peaks whose positions in 20 are close to, but not necessarily exactly at, those of the broadened peaks [22]. The profile P’(26) of a powder pattern peak, corrected for instrumental broadening. can be expressed in terms of a Fourier integral [23] P(Z@ x
A(L) exp[ -2rciL(s - se)] dL.
(20)
ARDELL.
AN3
MARDESICH
WAGNER:
ANTIPHASE
where s = 2 sir@/& so = 2 sin&,/i., 24, being the position of the peak maximum, and L = ndku is the distance normal to the reflecting planes (h&l) with interplanar spacing dktl. In the absence of microstrains, a reasonable assumption for the heat-treated samples, the Fourier coefficients A(L) represent the particle size coefficients A’(L), i.e. A(L) r /lP(L) = [l/D(hkf)]
z (J - ILl)p(J)dJ, I 3 =IL\
(21) where p(J) dJ represents the fraction of sizes between the length J and J + dJ and D(hkl) is the effective particle size normal to the reflecting planes. For small values of L, A’(L) can be written as AP(L) k 1 - L/D(hkl).
(22)
The negative initial slope of the A’(,!,) curve plotted vs L yields D(hkf), i.e. ljD(hk[) = - [d:dA’(L)/d& =,,.
(23)
In the case of a superstructure reflection, the particle size coefficient D(hkl) is a measure of the average APD sizes normal to the reflecting planes (/I&!).As originally shown by Wilson [24] and later generalized by lblikkola and Cohen [ZS], this average APD size is related to the probability of crossing a domain boundary. It is estimated that the error in D(hkl) for values less than 300 A is about + 10%. This is due mainly to errors in estimating the background levels of the relatively weak superstructure peaks, which introduces the so-called ‘hook’ effect [26] into the A(L) vs L curves. For D(M) larger than 300 A, the corrections for instrumental broadening become more important and, consequently, the error in D(M) will increase. 4. RESULTS AND DISCUSSION The measured values of (D) for the three alloys are shown in Table 2, along with the individual values of D(hkf) resulting from the analysis of the four super-
DOMAIN
GROWTH
IN Cu,Au
1265
structure peaks used in the experiments. Following Poquette and Mikkola [l], (D) was calculated as the arithmetic mean of the values of o(hkf). Least-squares analysis of the data in Table 2 according to equation (1) resulted in the values of (Do)’ and K seen in Table 3. The data of Poquette and Mikkola using their D(hkl) values only for the (lOO), (110). (210) and (211) reflections to calculate (D), were similarly analyzed, yielding values of (D,,)’ and K which are also reported in Table 3. The data in Table 3 are represented in Fig. 1, in which only our data points for pure Cu,Au are shown. It is apparent that our data are in good agreement with those of Poquette and Mikkola and that the rate of APD growth increases slightly with increasing Co content, despite the fact that the Co is present in the form of small precipitates. This latter result is consistent with the earlier finding of Ardell and Hovan [27J The results of the transmission electron microscopy investigations of APB growth by Sakai and .Mikkola [Z] and Sauthoff [3] are shown in Fig. 2. In this instance, we used the values of D,,,,, reported in Table 2 of Sakai and Mikkola’s paper, and 2z/3 taken from Fig. 4 of Sauthoff’s paper. Least-squares analysis of these data produced the values of (Do>’ and K reported in Table 4. It is apparent that the values of K in Tables 3 and 4 are in good agreement. Measurements of (D) vs I at 35O’C in pure Cu,Au have also been reported by Morris et al. [4]. For reasons that are not clear to us. their values. determined both by X-ray analysis and transmission electron microscopy, are significantly larger than those of the other investigators and result in much larger values of K. For this reason, they are not reproduced herein. In order to estimate K theoretically, it is necessary to evaluate all the parameters in equation (19), the most troublesome of which are /I. 3 and G. To estimate /I, we compared the experimental cumulative distribution functions of Sauthoff with theoretically
Table 2. Values of D(hkl) determined from the four superstructure reflections analyzed and (0) for the three Cu,Au alloys W4
D fA)
(A)
(hid)
7;co T (33
r x 1o-A (S)
325
0.18 0.36
350
360 375
f1W 0 1.5 2.5
(110) 0 1.5 2.5
0
(210) 1.5 2.5
0
(211) 1.5 2.5
45 45 70 75 -
27 34
27 35 -
29 40
29 40 -
38 44
36 45 -
0.12 3.60 0.18 0.72
82 125 72 132
80 144 80 140
87 148
39 66 34 57
51 72 40 70
42 75
45 83 47 80
52 94 50 90
47 90
52 95 60 105
57 112 60 110
62 111
3.60 0.18 0.12 3.60 0.18 0.72 3.60
235 90 200 375
230 120 190 400 100 210 385
235 100 235 385
106 58 112 200
120 50 75 170 60 105 195
140 65 112 195
175 76 145 330
200 51 100 210 -
200 -
195 100 165 350
210 70 I20 250 -
255 -
0 35 47 55 93 53 94
1.5 2.5 34 49 60 108 58 102
178 190 72 120 250 81 155 31-1 -
60 106
207 -
ARDELL, MARDESICH
1266
AND
WAGNER:
ANTIPHASE
Table 3. The values of (D,)’ and K obtained from leastsquares analysis of the X-ray data. Also included are the data of Poquette and Mikkola [ 11. indicated by the symbol (PM) ?,Co
T (‘c)
O(PM) O(PM)
300 325
0 1.5 O(PM) 0 1.5 2.5 1.5 O(PM) 0
350
360 375
(a,>’
x IO-* (A?
K (AZ/s)
0.784 0.101
0.0392 k 0.0033 0.1925 + 0.0114
0.123 0.116 0.116 0.203 0.260 0.230 0.221 0.221 0.325
0.2064 * 0.0136 0.2811 + 0.0178 0.9317 + 0.0542 0.8272 k 0.0428 0.9356 f 0.0547 1.1308+ 0.0419 1.6739_+0.0050 3.3150 f. 0.0847 2.6572 + 0.0856
calculated functions G’(u’) for several values of fit. A comparison of Sauthoff’s data with G’(u’)for B = 0 and fl = 0.95 is shown in Fig. 3. It is obvious that when B = 0.95, the theoretical function and experimental data are in much closer agreement than they are when /? = 0. We found, in general, that as /3 increased, the fit improved for values of u’ between 0.5 and 1.5, while the fit for ti < 0.4 (where the experimental data are least accurate) became worse. It is t In Ref. [IS] it was stated that G’(u’)could be compared directly with experiment without needing to know the value of (u). This is clearly incorrect.
DOMAIN
GROWTH
IN Cu,Au
likely that some improvement might be obtained for fl > 0.95, but for computational purposes this is of virtually no consequence. Hence, w-e conclude that /I = 0.95 is representative of APD growth in Cu,Au. It would appear that G’(u) with fi = 0.95 does not fit Sauthoff’s data quite as well as his smeared model distribution, but the discrepancy is not significant. The interdiffusion coefficient in pure ordered Cu,Au has not been measured. However, Paulson and Hilliard [6] have measured 5 in samples of average composition 16 at.% Au at temperatures from 200 to 260°C. They have shown that their data are in agreement with values of 3 determined at much higher temperatures (> 700°C). For lack of a better estimate, we have used values of 3 interpolated from the data in Fig. 7 of Paulson and Hilliard’s paper. These values are shown in Table 5. Of the other parameters required. S was obtained from the data of Mardesich et al. [lS] while a was taken from the data compiled by Pearson [27]. Using the values of S and a also shown in Table 5, and 8, u was estimated for pure Cu,Au from the values of K in Tables 3 and 4. These values are compared with theoretical values of Q, recently calculated by Kikuchi and Cahn [29], which are also shown in Table 5. It is apparent that the values of u derived from the data are smaller than those predicted theoretically by about a factor of two at the lower temperatures (300-350°C). However, Riven the uncertainty in 3, the agreement is quite good. It should be
10
I 9
x co -0
--_____
a
I
Poquette 1.5 1Mikkola
A ”
and
111
_/’ /
/
325’C
0
2
1
t x 10 -4
3 (s)
Fig. 1. The results of several X-ray measurements of APD growth in _CusAu, plotted according -. . to equation (1). The data points representing our measurements on pure CusAu are shown. Wther data are represented by the lines determined by least-squares analysis.
ARDELL. MARDESICH
AND
WAGNER:
ANTIPHASE
10
DOMAIN
A Sauthoff
GROWTH
IN Cu,Au
1267
[3]
CIA Sakai and Mikkola [2]
Fig. 2. The results of measurements of APD growth in pure CusAu, plotted according to equation (1). as determined by transmission electron microscopy. The lines shown were determined by leastsquares analysis of the data.
mentioned that a similar analysis using SauthotTs theory would result in values of cr greater by a factor of two than those in Table 5. As a result of these calculations, we conclude that Hillert’s theory of APD growth, as modified herein for consistency with the broad APD size distributions observed experimentally, is in very good quantitative agreement with experimentally measured rates of APD growth. As is often the case, many of the parameters needed to test the theory rigorously are poorly known, and it is perhaps for this reason that the derived values of u in Table 5 are not as strongly temperature dependent as those predicted by Kikuchi and Cahn’s theory (the values of g derived from the data of Poquette and Mikkola are not temperature dependent at all). It should also be mentioned that over the time interval of our measurements, S in-
Table 4. The values of (Do>z and K obtained from leastsquares analysis of the electron microscope data on APD growth in pure Cu,Au. (SM) refers to the data of Sakai and Mikkola [Z], while (S) refers to those of Sauthoff [3] T (C) 350 375
(o,# (S) (SM) (SM)
x IO-~ (AZ) 0.273 0.356 3.264
K (A*:s) 0.9594 + 0.0497 1.2652 + 0.029 3.1848 f 0.3628
creases by as much as 15% at all three temperatures [30], although we used equilibrium values of S in the calculations. Given these qualifications the agreement between theory and experiment is as good as can be expected. The accelerating effect of Co precipitates on the kinetics of APD growth can also be rationalized qualitatively by the theory. The simplest effect of the Co atoms that we can envision is that they are equivalent, in whatever form they happen to have. to an additional amount of disorder encountered by a moving APB. Thus, instead of requiring the transport of S/a2 atoms per unit area for boundary motion, this number will be reduced by the factor 1 - Xc,, where Xc,, is the atom fraction of Co, since Xc,, atoms do not need to participate in the transport process. The mobility of a Co-containing alloy should therefore be greater than that of pure Cu,Au by a factor of (1 - Xc,,)-‘. While this qualitatively accounts for the acceleration observed in Fig. 1, it fails quantitatively; the increase in K is much greater than predicted. One possible explanation for this is that u increases in the Co-containing alloys. This is not unreasonable, although any increase in cr must be large enough to overcome the potential effects of interactions between Co particles and APBs, which would be expected to retard APD growth as do Al,O, particles in Cu,Au [31]. It is fair to state that the magnitude of
1268
ARDELL. MARDESICH
AND
WAGNER:
ANTIPHASE
DOMAIN GROWTH
IN Cu,Au
1.0
0 0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
u'
Fig. 3. Illustration of the comparison between Sauthoff’s [3] experimentally measured cumulative distributions, G’(u) and those predicted by Hillert’s theory (/I = 0) and the present theory with ,!I= 0.95.
Table 5. Values of the APB interfacial free energy in pure Cu,Au obtained from equation (19) using the experimentally determined values of K in Tables 2 and 3. in all the calculations, the parameters fi and (u) were taken as 0.95 and 0.731. respectively T(‘K)
S
a {A)
B (cm’s_‘)
573 598
0.94 0.92
3.757 3.759
3.24 x IO-i* 1.55 x 10-I’
623
0.88
3.761
6.56 x lo-“
d (erg cm - ‘) (PM) (PM) (PM) (S) (SM)
648
0.83
3.763
2.48 x lo-” (PM) (SM)
the effect of Co, while relatively small, is much larger than expected and not well understood. In closing, we consider the theory herein in light of a recent paper by Cahn and Allen [32]. They point out that since u-+0 as the critical temperature, T,, is approached, in ordered alloys in which the orderdisorder transformation is second-order or higher, domain growth should theoretically tend to zero. Since such is not the case experimentally, they conclude that any theory that predicts that the rate of APD growth approaches zero as T, is approached must be incorrect. While we do not dispute this, we point out that in such alloys S also approaches zero as T--+ T, so that the ratio a/S* to which K is proportional [equation (19)], should remain finite. Therefore, Hillert’s modified theory of APD growth should apply equally well to ordered alloys with higher-order tra~formatio~. Acknowledgements-We are very grateful to Dr. G. Sauthoff for valuable correspondence and to Dr. R. Kikuchi for bringing the results of Ref. (291 to our attention prior to publication. Financial support for this research was provided by the National Science Foundation under grant number GH-31818.
REFERENCES 1. G. E. Poquette and 13. E. Mikkola, Trans. met& Sot. A.I.M.E. 245, 743 (1969).
15.7 + 2.7 17.6 i: 1.2 16.4 f 1.0 16.6 f 0.9 18.7 f: 1.1 19.2 f 1.0 25.4 & 0.6 13.8 + 0.4 17.2 f 0.4 16.6 + 1.9
atk (erg cm-‘) 39.5 33.7 26.5
18.1
2. M. Sakai and D. E. Mikkola ~~feroll. Trans. 2. 1635 (1971). 3. G. Sauthoff, Acta metall. 21, 273 (1973). 4. D. H. Morris, F. M. C. Besag and R. E. Smallman. Phfl. Mac?. 29, 43 (1974). 5. A. B. Martin. R. D. Johnson and F. Asaro, 3. appl. Phys. 25, 364 (1954). 6. W. IM. Paulson and J. E. HilIiard, J. uppl. Phys. 48, 2117 (1977). 7. P. Feitham, Acta metall. 5, 97 (1957). 8. M. HiBert, Acta merall. 13, 227 (1965). 9. I. M. Lifshitt and V. V. Slyozov. J. pkys. Ckem. Sotids 19, 35 (1961). 10. C. Wagner, 2. Elektrochem. 65, 581 (1961). 11. G. Sauthoff and M. Kahlweit. rlcta metall. 17, 1501 (1969). 12. A. J. Ardell, Acta merali. 20, 61 (1972). 13. A. J. Ardell, Acta met& 20, 601 (1972). 14. A. J. Ardell, Metal/. Trans. 3, 1395 (1972). 15. A. J. Ardell, Met&. 5, 285 (1972). 16. R. C. Sun and C. L. Bauer, Acta metall. 18, 639 (1970). 17. B. E. Warren, X-ray ~~uct~ff~ p_ 208. Addison-Wesley, Reading, Mass. (1969). 18. N. Mardesich, C. N. J. Wagner and A. J. Ardell, .r. appl. Cryrtallogr. 10, 468 (1977). 19. C. N. J. Wagner. Adc. X-ray Anal. 12, 50 (1969). 20. W. A. Rachinger, J. Sci. Inst. 25, 254 (1948). 21. A. R. Stokes. Proc. Phys. Sot. London 61, 382 (1948). 22. C. N. J. Wagner and E. N. Aqua, Adu. X-ray Anal. 7, 46 (1964). 23. C. N. J. Wagner, Local Atomic Arrangements Studied by X-ray Diffracrion (edited by J. B. Cohen and J. E. HiIIiard), p. 219. Gordon and Breach. New York (1966).
ARDELL. MARDESICH
AND
WAGNER:
24. A. J. C. Wilson Proc. R. Sot. A 181, 360 (1943). 3. D. E. Mikkola and J. 9. Cohen. Local Atomic &rangemenfs Studied by X-ray Difiacrion (edited by J. 9. Cohen and J. E. Hilliard), p. 289. Gordon and Breach, New York (1966). 26. 9. E. Warren. Prog. Metal Phps. 8, 147 (1959). 27. A. J. Ardell and M. J. Hovan, ;Morer. Sci. Eng. 9, 163 (1972).
ANTIPHASE
DOMAIN
GROWTH
IN Cu,Au
1269
28. W. 9. Pearson. Handbook of Lattice Spacings and Srructures of .%letals, p. 414. Pergamon, Oxford (1958). 29. R. Kikuchi and J. W. Cahn, to be published. 30. N. cMardesich, MS. Thesis, University of California at Los Angeles (1975). 31. S. M. L. Sastry and B. Ramaswami Mater. Sci. Eng. 14, 93 (1974). 32. J. W. Cahn and S. M. Allen. J. Physique 38, C7 (1977).
N. MARDESICHt
and C. N. J. WAGSER
Materials Department. University of California. Los Angeles. CA 90024, U.S.A. (Received 4 January 1979)
Abstract-The Hillert theory of normal grain growth has been modified. for antiphase domain (APD) growth in Cu,Au. A semi-empirical correction factor, 8, is introduced into the theory in order to incorporate the effect of the extreme breadth of the experimentally observed APD size distributions on the rate of APD growth. The mobility. .If, of APD boundaries in an ordered alloy. which is proportional to the chemical diffusion coefficient. 3. is shown to be inversely proportional to the long-range order parameter, S. The theory is compared with literature data on APD growth in Cu,Au. as well as our own measurements made at 325,350 and 37X for times up to 10 h on pure Cu,Au and Cu,Au-Co alloys containing 1.5 and 2.5 at.‘?;Co. Using estimated values of 3, measured values of S. values of /3 determined by fitting the theory to existing APD distributions and another parameter (u) which is a measure of the relative values of the average and ‘critical’ APD sizes, the experimentally measured rate constants for APD growth were used to calculate values of the APD boundary interfacial energy, 0, for comparison with recently derived theoretical values. The-quantitative agreement between theory and experiment is satisfactory, considering the uncertainty in 2. The theory also predicts that APD growth should increase in the Co-containing alloys. Such an effect is observed experimentally, but is greater in magnitude than predicted by the theory, for reasons that are not understood. R&urn&-Nous
avons modifie la theorie de Hillert pour la croissance normale des grains afm de l’appliquer a la croissance des domaines antiphase (DAP) dans CusAu. Nous introduisons un facteur semi-empirique /? pour tenir compte de I’effet de la grande dispersion des valeurs experimentales de la taille des domaines sur la vitesse de croissance des DAP. Nous montrons que la mobilite M des parois de DAP, qui est proportionnelle au coefficient de diffusion chimique 3, est inversement proportionnelle au parametre d’ordre a longue distance S. Nous comparons cette theorie avec les donnees de la litterature concernant la croissance des DAP dans Cu,Au, et avec nos propres mesures effectutes a 325, 350 et 37YC, pour des durees pouvant atteindre 10 h, sur le Ct+Au pur et sur des alliages CusAu-Co contenant 1.5 et 2,5 at%Co. En utilisant une valeur estimee de 8. les valeurs experimentales de S. les valeurs de /J determinees en ajustant la theorie aux repartitions existantes des DAP, et un autre parametre (10 qui est une mesure da valeurs relatives des tailles moyennes et “‘critiques” des DAP. ainsi que les vitesses de croissance des DAP mesurees experimentalement, noux avons calcule des valeurs de I’energie u des joints de DAP. afin de les comparer avec des valeurs thioriques recentes. L’accord entre la thtorie et l’expirience est satisfaisant, compte tenu de I’incertitude sur 3. La thtorie privoit igalement une augmentation de la croissance des DAP dans les alliages contenant du cobalt. Cet effet est observe expirimentalement. mais son ordre de grandeur est superieur a celui prlvu par la thtorie. pour des raisons que nous ne comprenons pas. Zusammenfassung-Die
Hillertsche Theorie des normalen Kornwachstums wurde zur Behandlung des Wachstums von Antiphasengrenzen (ADP) in Cu,Au angepal3t. Urn den EinfluB der extremen Breite der experimentell beobachteten APD-GriiDenverteilung auf die APD-Wachstumsgeschwindigkeit zu berhcksichtigen, wurde ein halbempirischer Korrektionsfaktor p in die Theorie eingefnhrt. Es wird gezeigt, daD die Beweglichkeit IMder APD-Grenzen in einer geordneten Legierung, die zum chemischen DilTusionskoefiienten B proportional ist. sich umgekehrt proportional zum Fernordnungsparameter S verhalt. Die Theorie wird mit Literaturangaben iiber APD-Wachstum in CusAu verglichen, ebenso mit unseren Messungen, die bei 325, 350 und 375’C fur eine Dauer bis zu 10 Stunden an reinem Cu,Au u_nd an CusAu-(I,%2.5) At.-?bCo--Legierungen durchgeftihrt wurden. ,Mit abgeschltzten Werten fur 2, gemessenen fur S, mit aus der Anpassung der Theorie an die vorhandenen APD-GrGBenverteilungen ermittelten Werten fur /?, und mit einem welteren Parameter (u)-einem MaB fur die relativen GriiBen der durchschnittlichen und “kritischen” APD-GroBen-wurden aus den gemessenen Ratenkonstanten des APD-Wachstums Werte fur die APD-Grenzfl$ichenenergie D ermittelt und mit friiher theoretisch abgeleiteten Werten verglichen. Die quantitative_Ubereinstimmung zwischen Theorie und Experiment ist befriedigend, wenn man die Unsicherheit in D beriicksichtigt. Die Theorie sagt weiterhin aus, daf3 das APD-Wachstum in den Co-haltigen Legierungen verstarkt sein sollte. Dieser Effekt wird experimentell gefunden ist jedoch aus noch unverstandenen Griinden gr6Ber als theoretisch vorhergesagt.
t Present address: Spectrolab. 12500 Gladstone Avenue, Sylmar, CA 91340. U.S.A. 1261
1262
ARDELL.
MARDESICH
ASO
WAGNER:
ANTIPHASE
1. INTRODUCTION The kinetics of anti-phase domain (APD) growth in ordered Cu,Au have been shown [l-4] to obey an equation of the type
(0)'- (Do)'= Kt,
(1)
(D) isthe average domain ‘diameter’ at time (Do)isits initial value and K is a constant that
where r.
determines the growth rate at some temperature 7: The rate constant K has an Arrhenius-type temperature dependence, with an activation energy, Q, of about 44 kcal mole- I [l], which compares reasonably well with values of Q for tracer diffusion of Au in Cu [S], and for interdiffusion in concentrated Cu-Au alloys [6]. Equation (1) is representative of the kinetics of grain growth in metals and alloys. Physical justification for this type of equation has been offered by Feltham [7] in an early derivation and more recently by Hillert [S], who used a modification of the theory of Ostwald ripening by Lifshitz and Slyozov [9] and Wagner [lo], to derive not only equation (1) but the theoretical distribution of D aswell. Hillert’s theory was adapted later by Sauthoff [3] who attempted to incorporate specifically the known geometry of APDs in Cu,Au, which are approximately rectangular parallelepipeds with interfaces parallel to : 100;. Sauthoff used a clever, but somewhat curious, force balance to estimate the growth rate of a particular domain with respect to APDs of critical size. In this way he was able to circumvent the dilemma concerning the relationship between the surface tension of a boundary and its radius of curvature. SauthofI’s experimental data included measurements of the domain size distributions, which were considerably broader than those predicted by Hillert’s theory. Moreover, Sauthoff concluded that his size distribution data agreed reasonably well with those of Sakai and Mikkola [Z], although their size distributions, determined from X-ray measurements, exhibited pseudo-periodic fluctuations which Sauthoff did not believe were real. Sauthoff attempted to account for the observed excessive breadth of the distributions by invoking the notion that the critical APD size, D*, which is stationary at a given instant of time (see Section 2.1, below), was itself distributed in size throughout a particular sample. In so doing, Sauthoff succeeded in fitting his experimental distributions to a theoretically derived function. In all the work to date on APB growth in Cu,Au, no attempt has been made to compare theory with experiment in a quantitative manner, despite the clearly excellent semiquantitative agreement that has been demonstrated. The main purpose of this paper is to make such a comparison, using extant data as well as our own measurements, which incorporate the effect of Co precipitates on the kinetics of APB growth in Cu,Au. In making such a quantitative com-
DOMAIN
GROWTH
IN Cu,Au
parison it is necessary to take into account the influence of the broad particle size distributions on the kinetics and it is essential to estimate the APB mobility, M, which is one of the most important parameters of the theory. These items are accounted for theoretically in the next section. 2. THEORETICAL
CONSIDERATIONS
2.1 A semi-empirical mociifcotion of Hiflert’s theory We begin with equation (6) of Hillert’s paper, written in terms of D = 2R (where R is the grain ‘radius’), which appears as
where c is the APB energy, M its mobility and z is a constant approximately equal to unity in a threedimensional system (we will use, henceforth, the assumption a = 1, following Hillert). D' represents the critical domain size; APDs with D = D* are instantaneously neither growing nor shrinking. We note that the coefficient of (l/D* - l/D) is a factor of two larger than that derived by Sauthoff. Equation (2) can be rewritten dD’ = 8aM(u - 1). dt where u = DID*. By proceeding from this point using the mathematical approach of Lifshitz and Slyozov (91, Hillert derived the equations governing the time-dependence of D* (hence (D))and the theoretical distribution of particle sizes expected during normal grain growth. However, Sauthoff convincingly demonstrated that the APD size distributions in CusAu are significantly broader than predicted by Hillert’s theory. This must influence the kinetics of APD growth because of the competitive nature of the process (if all the domains have the same size the growth rate is zero, whereas if the distribution is broader than predicted the growth rate should exceed that of Hillert’s theory). To incorporate the influence of the particle size distribution into the theory, it is necessary to introduce a function which, while accomplishing its task, leaves the nature of the kinetic law, equation (1). undisturbed. As we shall demonstrate, this can be achieved by rewriting (3) as dDZ 7 = SaM(u - l)( 1 + &,. where /I is a parameter that varies between zero and unity. While we are unable to justify the multiplication of (3) by the factor 1 + /Iu on physical grounds. we note that a similar procedure was used by Sauthoff and Kahlweit [ll] to modify Lifshitz and Slyozov’s [93 and Wagner’s [lo] theory of Ostwald ripening to fit their data on 7’ particle coarsening in aged Ni$Si alloys. In a later paper, Ardell[12] demonstrated that the factor 1 + fiu could be related to the influence
ARDELL. MARDESICH
AND
WAGNER:
ANTIPHASE
of the 7’ volume fraction on the coarsening kinetics and particle size distributions. In the cases of APD or grain growth 1 + flu can only be regarded as a semi-empirical correction to the theory. Its consequences. however, enable theory and experiment to be brought into remarkably good agreement. The theory proceeds from this point according to the method of Lifshitz and Slyozov, as utilized by Hillert and later by Ardell [E-14]. We first write the equation for du’/dr, which on using (4) becomes
DOMAIN
B 0
0.05 0.10
0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70
(6)
0.75
0.80 0.85 0.90 0.95
and ‘J = 8 aM dr/dD*‘.
(7)
For stationary solutions to exist both du’/dr and d(du’/dr)/du must vanish at a value of u,, corresponding to a constant value of 7. On setting (5) and its hrst derivative w.r.t. u equal to zero at u = ue, we find ug = 2/( 1 - 8,
(8)
y = 4/(1 + /?y.
(9)
and
At this juncture, we note that the kinetic equation describing the increase of D* with t is obtained on integrating (7), which, on substitution of (9), yields
De2- Dz2= 2 aM(1 + /?)2t,
(10)
where Dt is the critical APD size at the onset of the growth process. It is evident that the correction factor 1 + flu has the expected effect of accelerating the kinetics of APB growth by a factor of (1 + b)2 while leaving the parabolic rate law unchanged. Moreover, according to (8) the particle size distributions can be expected to broaden as /I- 1, since u. increases without limit as b increases to unity. The particle size distributions can be derived using the procedures of Lifshitz and Slyozov. Similar derivations are presented in the papers of Ardell[12-141. The probability density function, g(u), is given by
IN Cu,Au
(U, 0.8889 0.8794 0.8701 0.8608 0.8517 0.8427 0.8339 0.8251 0.8165 0.8080 0.7997 0.7915 0.7834 0.7755 0.7677 0.7600 0.7525 0.7451 0.7379 0.7312
which results in the expression G(u)=l
_{2_u;
_fl~t’1+8”“-8’1’
-3u(l + B)’ x exp i (1 - /I)[2 - u(1 - /I)] I . (12)
As the variable measured experimentally is (D), not D*, it is necessary to rewrite equations (10) through (12) in terms of (D) for the purpose of comparing theory with experiment. To this end, we need to calculate 0 (u) = (D)/D*= 44 du, (13) s0 as a function of the parameter 8. The results of numerical calculations for values of /I up to 0.95 are presented in Table 1. Graphical representation of the values of(u) in Table 1 shows that (u) has a limiting value of - 0.725 at p = 1; the value of (u) at b = 0 is 8/9, consistent with Hillert’s theory. It is now convenient to define the variable u’ = D/(D)= u/(u)
The cumulative distribution function. G(u), is readily obtained using the procedure described by Ardell [ 153,
(14)
and the probability functions g’(u’) and G’(u’), which can be readily compared with experimentally determined histograms or cumulative distributions. It is not difficult to show that g’(u’) = (u)g(u) and G’(u3 = G(u). In terms of (D),the kinetic equation for APD growth, (lo), becomes
(D)'- (Do)'= 2 aM(u)‘(l (11)
1263
Table 1. The variation of (u) with the parameter /I
where 5 = In D*z
GROWTH
+ m2t.
(15)
On comparing (15) with (l), we see that the rate constant K isgiven by
K = 2 &f(u)’
(1 + ,3)‘.
(16)
126J
ARDELL.
,MARDESICH
ASD
WAGNER:
1 ’ The APB mobility. hZ -._ In deriving an expression for Jf in terms of measurable physical quantities, we will adapt the definition used by Sun and Bauer [16] to the problem of APB motion in ordered crystals. In general. ,cI can be expressed as (17)
where 2 is the diffusivity of the species responsible for motion of the boundary, n is the number of such species per unit boundary area participating in the jump process and kT has its usual meaning. In a given APB, the boundary advances when the atoms in anti-phase positions in the domain being consumed jump the required distance (the displacement vector of APB) and so become in phase with the other atoms in the growing domain. In a binary ordered alloy, both atomic species will be required to jump in order that the APB advance. Consequently, the appropriate diffusivity will be the interdiffusion coefficient, 3. The value of n in CusAu can be expressed as $‘a’ (where S is the long range-order parameter as defined by Warren [17] and a is the lattice constant) because this is precisely the number of atoms per unit area of [IOO) APB that must be transported across the interface for it to advance. The reason S enters into the expression is that it represents the total probability that the occupancy of Cu and Au on their respective sublattices in the ordered alloy exceeds random 0ccupancy.t Another way of stating this that there is no need for (1 - S)/a2 atoms per unit area to diffuse into new sites in order for the APB to advance. Equation (17) can thus be rewritten as
M=S
SkT’
which on substitution into (16) yields the final expression for K K = 2a2& SkT
3. EXPERIMENTAL
PROCEDURES
3.1 ,411oy preparation Pure CulAu and Cu,Au containing 1.5 and 2.5 at.:/, Co were prepared from elements of 99.999:/, purity. The starting materials were Au splatters and Cu rods obtained from the United Mineral and Chemical Corporation and Co sponge from Elect In Cu,Au S can be expressed as (rc, - 3,‘4)+ (T,” - l/4), where rcU and raU are the fractions of Cu and Au atoms, respectively. occupying Cu and Au sites. 314 is thus the fraction of Cu sites occupied by Cu rC” atoms in excess of random occupancy while rAu- 1/‘4 is the fraction of Au sites occupied by Au atoms in excess of random occupancy.
ANTIPH.\SE
DOMAIN GROWTH
IS Cu,Au
tronic Space Products, Inc. Two master alloys of Au with 10 and 20”, Co were first prepared by thoroughly mixing the constituents and inductionmelting them in a high purity graphite crucible under a Ti-gettered He atmosphere. The alloys were annealed for 48 h and cold-rolled for subsequent use in the final alloy preparation. Stoichiometric CusAu alloys containing various amounts of Co were subsequently prepared from the master alloys and pure Cu and Au by sealing them in quartz tubes at about lo-’ torr and melting them by induction heating. To insure homogeneity, the samples were cold-rolled and then remelted under the same conditions. For final homogenization the alloys were annealed at 925’C in an Hz atmosphere furnace for about 3 days. Filings (- 325 mesh, co.4 pm) were made with a Nicholson No. 4 file. The filings a-ith 1.5 at.“, Co were then encapsulated in an evacuated quartz tube and solution-heated for 15 min at 925:C followed by a water-quench. Even after this relatively brief heattreatment, the filings were partially sintered, but could be separated by filling again. applying a very light pressure. The filings of Cu,Au with 0 and 1.5 at.?; Co were annealed for 1 h at 41O’C and then used to determine the long-range order parameter S between 325 and 385’C [18]. At the conclusion of these measurements, the filings of Cu,Au with 0 and 1.5 at.% Co, as well as the cold-worked filings with 2.5 at.% Co were sealed in evacuated Pyrex tubes, first heat-treated for 40 h at 45O’C. then for 0.5, 2 and 10 h at 325, 350 and 375’C. respectively and water-quenched. 3.2 X-ray analysis The (lOO), (llO), (210) and (211) superstructure reflections of each annealed sample were recorded on a theta-theta diffractometer [19] using filtered Curadiation and a proportional counter, or on a GE diffractometer using Cu-Kz radiation diffracted by a doubly-bent graphite monochromator in the primary beam. The heat-treated filings were mounted in a plexiglass holder using a solution of Duco cement in acetone as a binder. Si and W powder samples were used as standards. The peak protiles of the alloys and of the standard samples were separated into their Kz, and Kz, components using the Rachinger [ZO] method and the Kz, profiles were corrected for instrumental broadening by the Stokes [21] method, using a Fortran IV program. When using the Fourier coefficients corresponding to Ksc, peaks only, it is possible to use standard peaks whose positions in 20 are close to, but not necessarily exactly at, those of the broadened peaks [22]. The profile P’(26) of a powder pattern peak, corrected for instrumental broadening. can be expressed in terms of a Fourier integral [23] P(Z@ x
A(L) exp[ -2rciL(s - se)] dL.
(20)
ARDELL.
AN3
MARDESICH
WAGNER:
ANTIPHASE
where s = 2 sir@/& so = 2 sin&,/i., 24, being the position of the peak maximum, and L = ndku is the distance normal to the reflecting planes (h&l) with interplanar spacing dktl. In the absence of microstrains, a reasonable assumption for the heat-treated samples, the Fourier coefficients A(L) represent the particle size coefficients A’(L), i.e. A(L) r /lP(L) = [l/D(hkf)]
z (J - ILl)p(J)dJ, I 3 =IL\
(21) where p(J) dJ represents the fraction of sizes between the length J and J + dJ and D(hkl) is the effective particle size normal to the reflecting planes. For small values of L, A’(L) can be written as AP(L) k 1 - L/D(hkl).
(22)
The negative initial slope of the A’(,!,) curve plotted vs L yields D(hkf), i.e. ljD(hk[) = - [d:dA’(L)/d& =,,.
(23)
In the case of a superstructure reflection, the particle size coefficient D(hkl) is a measure of the average APD sizes normal to the reflecting planes (/I&!).As originally shown by Wilson [24] and later generalized by lblikkola and Cohen [ZS], this average APD size is related to the probability of crossing a domain boundary. It is estimated that the error in D(hkl) for values less than 300 A is about + 10%. This is due mainly to errors in estimating the background levels of the relatively weak superstructure peaks, which introduces the so-called ‘hook’ effect [26] into the A(L) vs L curves. For D(M) larger than 300 A, the corrections for instrumental broadening become more important and, consequently, the error in D(M) will increase. 4. RESULTS AND DISCUSSION The measured values of (D) for the three alloys are shown in Table 2, along with the individual values of D(hkf) resulting from the analysis of the four super-
DOMAIN
GROWTH
IN Cu,Au
1265
structure peaks used in the experiments. Following Poquette and Mikkola [l], (D) was calculated as the arithmetic mean of the values of o(hkf). Least-squares analysis of the data in Table 2 according to equation (1) resulted in the values of (Do)’ and K seen in Table 3. The data of Poquette and Mikkola using their D(hkl) values only for the (lOO), (110). (210) and (211) reflections to calculate (D), were similarly analyzed, yielding values of (D,,)’ and K which are also reported in Table 3. The data in Table 3 are represented in Fig. 1, in which only our data points for pure Cu,Au are shown. It is apparent that our data are in good agreement with those of Poquette and Mikkola and that the rate of APD growth increases slightly with increasing Co content, despite the fact that the Co is present in the form of small precipitates. This latter result is consistent with the earlier finding of Ardell and Hovan [27J The results of the transmission electron microscopy investigations of APB growth by Sakai and .Mikkola [Z] and Sauthoff [3] are shown in Fig. 2. In this instance, we used the values of D,,,,, reported in Table 2 of Sakai and Mikkola’s paper, and 2z/3 taken from Fig. 4 of Sauthoff’s paper. Least-squares analysis of these data produced the values of (Do>’ and K reported in Table 4. It is apparent that the values of K in Tables 3 and 4 are in good agreement. Measurements of (D) vs I at 35O’C in pure Cu,Au have also been reported by Morris et al. [4]. For reasons that are not clear to us. their values. determined both by X-ray analysis and transmission electron microscopy, are significantly larger than those of the other investigators and result in much larger values of K. For this reason, they are not reproduced herein. In order to estimate K theoretically, it is necessary to evaluate all the parameters in equation (19), the most troublesome of which are /I. 3 and G. To estimate /I, we compared the experimental cumulative distribution functions of Sauthoff with theoretically
Table 2. Values of D(hkl) determined from the four superstructure reflections analyzed and (0) for the three Cu,Au alloys W4
D fA)
(A)
(hid)
7;co T (33
r x 1o-A (S)
325
0.18 0.36
350
360 375
f1W 0 1.5 2.5
(110) 0 1.5 2.5
0
(210) 1.5 2.5
0
(211) 1.5 2.5
45 45 70 75 -
27 34
27 35 -
29 40
29 40 -
38 44
36 45 -
0.12 3.60 0.18 0.72
82 125 72 132
80 144 80 140
87 148
39 66 34 57
51 72 40 70
42 75
45 83 47 80
52 94 50 90
47 90
52 95 60 105
57 112 60 110
62 111
3.60 0.18 0.12 3.60 0.18 0.72 3.60
235 90 200 375
230 120 190 400 100 210 385
235 100 235 385
106 58 112 200
120 50 75 170 60 105 195
140 65 112 195
175 76 145 330
200 51 100 210 -
200 -
195 100 165 350
210 70 I20 250 -
255 -
0 35 47 55 93 53 94
1.5 2.5 34 49 60 108 58 102
178 190 72 120 250 81 155 31-1 -
60 106
207 -
ARDELL, MARDESICH
1266
AND
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ANTIPHASE
Table 3. The values of (D,)’ and K obtained from leastsquares analysis of the X-ray data. Also included are the data of Poquette and Mikkola [ 11. indicated by the symbol (PM) ?,Co
T (‘c)
O(PM) O(PM)
300 325
0 1.5 O(PM) 0 1.5 2.5 1.5 O(PM) 0
350
360 375
(a,>’
x IO-* (A?
K (AZ/s)
0.784 0.101
0.0392 k 0.0033 0.1925 + 0.0114
0.123 0.116 0.116 0.203 0.260 0.230 0.221 0.221 0.325
0.2064 * 0.0136 0.2811 + 0.0178 0.9317 + 0.0542 0.8272 k 0.0428 0.9356 f 0.0547 1.1308+ 0.0419 1.6739_+0.0050 3.3150 f. 0.0847 2.6572 + 0.0856
calculated functions G’(u’) for several values of fit. A comparison of Sauthoff’s data with G’(u’)for B = 0 and fl = 0.95 is shown in Fig. 3. It is obvious that when B = 0.95, the theoretical function and experimental data are in much closer agreement than they are when /? = 0. We found, in general, that as /3 increased, the fit improved for values of u’ between 0.5 and 1.5, while the fit for ti < 0.4 (where the experimental data are least accurate) became worse. It is t In Ref. [IS] it was stated that G’(u’)could be compared directly with experiment without needing to know the value of (u). This is clearly incorrect.
DOMAIN
GROWTH
IN Cu,Au
likely that some improvement might be obtained for fl > 0.95, but for computational purposes this is of virtually no consequence. Hence, w-e conclude that /I = 0.95 is representative of APD growth in Cu,Au. It would appear that G’(u) with fi = 0.95 does not fit Sauthoff’s data quite as well as his smeared model distribution, but the discrepancy is not significant. The interdiffusion coefficient in pure ordered Cu,Au has not been measured. However, Paulson and Hilliard [6] have measured 5 in samples of average composition 16 at.% Au at temperatures from 200 to 260°C. They have shown that their data are in agreement with values of 3 determined at much higher temperatures (> 700°C). For lack of a better estimate, we have used values of 3 interpolated from the data in Fig. 7 of Paulson and Hilliard’s paper. These values are shown in Table 5. Of the other parameters required. S was obtained from the data of Mardesich et al. [lS] while a was taken from the data compiled by Pearson [27]. Using the values of S and a also shown in Table 5, and 8, u was estimated for pure Cu,Au from the values of K in Tables 3 and 4. These values are compared with theoretical values of Q, recently calculated by Kikuchi and Cahn [29], which are also shown in Table 5. It is apparent that the values of u derived from the data are smaller than those predicted theoretically by about a factor of two at the lower temperatures (300-350°C). However, Riven the uncertainty in 3, the agreement is quite good. It should be
10
I 9
x co -0
--_____
a
I
Poquette 1.5 1Mikkola
A ”
and
111
_/’ /
/
325’C
0
2
1
t x 10 -4
3 (s)
Fig. 1. The results of several X-ray measurements of APD growth in _CusAu, plotted according -. . to equation (1). The data points representing our measurements on pure CusAu are shown. Wther data are represented by the lines determined by least-squares analysis.
ARDELL. MARDESICH
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WAGNER:
ANTIPHASE
10
DOMAIN
A Sauthoff
GROWTH
IN Cu,Au
1267
[3]
CIA Sakai and Mikkola [2]
Fig. 2. The results of measurements of APD growth in pure CusAu, plotted according to equation (1). as determined by transmission electron microscopy. The lines shown were determined by leastsquares analysis of the data.
mentioned that a similar analysis using SauthotTs theory would result in values of cr greater by a factor of two than those in Table 5. As a result of these calculations, we conclude that Hillert’s theory of APD growth, as modified herein for consistency with the broad APD size distributions observed experimentally, is in very good quantitative agreement with experimentally measured rates of APD growth. As is often the case, many of the parameters needed to test the theory rigorously are poorly known, and it is perhaps for this reason that the derived values of u in Table 5 are not as strongly temperature dependent as those predicted by Kikuchi and Cahn’s theory (the values of g derived from the data of Poquette and Mikkola are not temperature dependent at all). It should also be mentioned that over the time interval of our measurements, S in-
Table 4. The values of (Do>z and K obtained from leastsquares analysis of the electron microscope data on APD growth in pure Cu,Au. (SM) refers to the data of Sakai and Mikkola [Z], while (S) refers to those of Sauthoff [3] T (C) 350 375
(o,# (S) (SM) (SM)
x IO-~ (AZ) 0.273 0.356 3.264
K (A*:s) 0.9594 + 0.0497 1.2652 + 0.029 3.1848 f 0.3628
creases by as much as 15% at all three temperatures [30], although we used equilibrium values of S in the calculations. Given these qualifications the agreement between theory and experiment is as good as can be expected. The accelerating effect of Co precipitates on the kinetics of APD growth can also be rationalized qualitatively by the theory. The simplest effect of the Co atoms that we can envision is that they are equivalent, in whatever form they happen to have. to an additional amount of disorder encountered by a moving APB. Thus, instead of requiring the transport of S/a2 atoms per unit area for boundary motion, this number will be reduced by the factor 1 - Xc,, where Xc,, is the atom fraction of Co, since Xc,, atoms do not need to participate in the transport process. The mobility of a Co-containing alloy should therefore be greater than that of pure Cu,Au by a factor of (1 - Xc,,)-‘. While this qualitatively accounts for the acceleration observed in Fig. 1, it fails quantitatively; the increase in K is much greater than predicted. One possible explanation for this is that u increases in the Co-containing alloys. This is not unreasonable, although any increase in cr must be large enough to overcome the potential effects of interactions between Co particles and APBs, which would be expected to retard APD growth as do Al,O, particles in Cu,Au [31]. It is fair to state that the magnitude of
1268
ARDELL. MARDESICH
AND
WAGNER:
ANTIPHASE
DOMAIN GROWTH
IN Cu,Au
1.0
0 0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
u'
Fig. 3. Illustration of the comparison between Sauthoff’s [3] experimentally measured cumulative distributions, G’(u) and those predicted by Hillert’s theory (/I = 0) and the present theory with ,!I= 0.95.
Table 5. Values of the APB interfacial free energy in pure Cu,Au obtained from equation (19) using the experimentally determined values of K in Tables 2 and 3. in all the calculations, the parameters fi and (u) were taken as 0.95 and 0.731. respectively T(‘K)
S
a {A)
B (cm’s_‘)
573 598
0.94 0.92
3.757 3.759
3.24 x IO-i* 1.55 x 10-I’
623
0.88
3.761
6.56 x lo-“
d (erg cm - ‘) (PM) (PM) (PM) (S) (SM)
648
0.83
3.763
2.48 x lo-” (PM) (SM)
the effect of Co, while relatively small, is much larger than expected and not well understood. In closing, we consider the theory herein in light of a recent paper by Cahn and Allen [32]. They point out that since u-+0 as the critical temperature, T,, is approached, in ordered alloys in which the orderdisorder transformation is second-order or higher, domain growth should theoretically tend to zero. Since such is not the case experimentally, they conclude that any theory that predicts that the rate of APD growth approaches zero as T, is approached must be incorrect. While we do not dispute this, we point out that in such alloys S also approaches zero as T--+ T, so that the ratio a/S* to which K is proportional [equation (19)], should remain finite. Therefore, Hillert’s modified theory of APD growth should apply equally well to ordered alloys with higher-order tra~formatio~. Acknowledgements-We are very grateful to Dr. G. Sauthoff for valuable correspondence and to Dr. R. Kikuchi for bringing the results of Ref. (291 to our attention prior to publication. Financial support for this research was provided by the National Science Foundation under grant number GH-31818.
REFERENCES 1. G. E. Poquette and 13. E. Mikkola, Trans. met& Sot. A.I.M.E. 245, 743 (1969).
15.7 + 2.7 17.6 i: 1.2 16.4 f 1.0 16.6 f 0.9 18.7 f: 1.1 19.2 f 1.0 25.4 & 0.6 13.8 + 0.4 17.2 f 0.4 16.6 + 1.9
atk (erg cm-‘) 39.5 33.7 26.5
18.1
2. M. Sakai and D. E. Mikkola ~~feroll. Trans. 2. 1635 (1971). 3. G. Sauthoff, Acta metall. 21, 273 (1973). 4. D. H. Morris, F. M. C. Besag and R. E. Smallman. Phfl. Mac?. 29, 43 (1974). 5. A. B. Martin. R. D. Johnson and F. Asaro, 3. appl. Phys. 25, 364 (1954). 6. W. IM. Paulson and J. E. HilIiard, J. uppl. Phys. 48, 2117 (1977). 7. P. Feitham, Acta metall. 5, 97 (1957). 8. M. HiBert, Acta merall. 13, 227 (1965). 9. I. M. Lifshitt and V. V. Slyozov. J. pkys. Ckem. Sotids 19, 35 (1961). 10. C. Wagner, 2. Elektrochem. 65, 581 (1961). 11. G. Sauthoff and M. Kahlweit. rlcta metall. 17, 1501 (1969). 12. A. J. Ardell, Acta merali. 20, 61 (1972). 13. A. J. Ardell, Acta met& 20, 601 (1972). 14. A. J. Ardell, Metal/. Trans. 3, 1395 (1972). 15. A. J. Ardell, Met&. 5, 285 (1972). 16. R. C. Sun and C. L. Bauer, Acta metall. 18, 639 (1970). 17. B. E. Warren, X-ray ~~uct~ff~ p_ 208. Addison-Wesley, Reading, Mass. (1969). 18. N. Mardesich, C. N. J. Wagner and A. J. Ardell, .r. appl. Cryrtallogr. 10, 468 (1977). 19. C. N. J. Wagner. Adc. X-ray Anal. 12, 50 (1969). 20. W. A. Rachinger, J. Sci. Inst. 25, 254 (1948). 21. A. R. Stokes. Proc. Phys. Sot. London 61, 382 (1948). 22. C. N. J. Wagner and E. N. Aqua, Adu. X-ray Anal. 7, 46 (1964). 23. C. N. J. Wagner, Local Atomic Arrangements Studied by X-ray Diffracrion (edited by J. B. Cohen and J. E. HiIIiard), p. 219. Gordon and Breach. New York (1966).
ARDELL. MARDESICH
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WAGNER:
24. A. J. C. Wilson Proc. R. Sot. A 181, 360 (1943). 3. D. E. Mikkola and J. 9. Cohen. Local Atomic &rangemenfs Studied by X-ray Difiacrion (edited by J. 9. Cohen and J. E. Hilliard), p. 289. Gordon and Breach, New York (1966). 26. 9. E. Warren. Prog. Metal Phps. 8, 147 (1959). 27. A. J. Ardell and M. J. Hovan, ;Morer. Sci. Eng. 9, 163 (1972).
ANTIPHASE
DOMAIN
GROWTH
IN Cu,Au
1269
28. W. 9. Pearson. Handbook of Lattice Spacings and Srructures of .%letals, p. 414. Pergamon, Oxford (1958). 29. R. Kikuchi and J. W. Cahn, to be published. 30. N. cMardesich, MS. Thesis, University of California at Los Angeles (1975). 31. S. M. L. Sastry and B. Ramaswami Mater. Sci. Eng. 14, 93 (1974). 32. J. W. Cahn and S. M. Allen. J. Physique 38, C7 (1977).