Atomic resolution observations of solute-atom segregation effects and phase transitions in stacking faults in dilute cobalt alloys—II. Analyses and discussion

Amr

w~tr//.Vol.

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in Great

33. No. X. ,‘p. I565 1576.IYM Britain. All rights reserved

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0001~6160:85 53.00 +o.oo ,(” IY85 Pergam~n Press Lrd

ATOMiC RESOLUTION OBSERVATIONS OF SOLUTE-ATOM SEGREGATION EFFECTS AND PHASE TRANSITIONS IN STACKING FAULTS IN DILUTE COBALT ALLOYS-II. ANALYSES AND DISCUSSION R. HERSCHtTZf and D. N. SEIDMAN Cornell University, Bard Hall. Department of Materials Science and Engineering and The Materials Center, Ithaca, NY 14853-0121, U.S.A.

Abstract-Solute atom segregation effects to individual stacking faults in Co-O.96at.%Nb and Co-O.98 at.% Fe alloys have been investigated using the atom-probe field-ion microscope (Part I). In this paper a deconvolution procedure is developed. This procedure in combination with the experimental results is used to demonstrate that the solute concentration profile associated with a stacking fault is extremely narrow-it falls off to the solute ~n~ntration of the matrix within fess thun 4A. The the~~ynamics of solute-atom segregation is discussed in light of the experimental observations. It is shown that a simple Langmuir-McLean segregation isotherm-which assumes no interaction between the segregating atoms-can not be used to interpret our experimental results. However, the results can be explained qualitatively by a Bragg-Williams model, which takes into account the first-nearest-neighbor interaction between atoms. This isotherm predicts the occurrence of a phase transition in the plane of the stacking fault. A partial molar heat of segregation of -2OmeV atom-’ and a first-nearest-neighbor interaction energy parameter of -65 mevatom-’ are consistent qualitatively with our experimental observations. The ex~~mentaliy observed transition from an unsa&urat~ (disorders) to a saturated (ordered) stacking fault is broader than the one predicted by the Bragg-Williams model. Possible reasons for this difference are discussed. R&sum&Nous avons itudit les effets de segregation d’atomes de solute sur des dtfauts d’empilement individuels dam. des alliages Co-O,96 at.% Nb et Co-O,98 at.% Fe a I’aide de la sonde atomique d’un microscope & Cmission de champ (Premiere partie). Dam cet article, nous dbveloppons un pro&b de d~nvoiution. Nous utiiisons cc pro&de, combine aux r&hats ex~~menta~, pour d&rnontrer que le profil de concentration en solute associi: au difaut d’empilement est extrsmement ttroit-il tetombe d la concentration en solute de la matrice en moins de 4 A. Nous discutons la thermodynamique de la segregation des atomes de solute a la lumitre des observations experimentales. Nous montrons que l’on ne peut pas utiliser une simple isotherme de segregation de Langmuir et McLean-qui suppose qu’il n’y a pas d’interaction entre les atomes segriges-pour interpreter nos r&mats expirimentaux. Cependant, on peut expliquer qualitativement ccs msultats par un modtle de Bragg et Williams, qui tient compte de ~interaction atomique entre premiers voisins. Ce-tte isotherme pmvoit I’existence dune transition de phases dans le plan du d6faut ~~pilement. Une chaleur molaire partielle de segregation de -20 meV par atome et une parametre d’inergie d’interaction entre premiers voisins de -65meV par atome sont qualitativement cohbrents avec nos observations expirimentales. La transition observ&e exp&imentalement entre un difaut d’empilement non sat& (dbordonnt) et un dbfaut saturl (ordonnt) est plus large que ne le pmvoit le modele de Bragg et Williams. Nous discutons des raisons possibles pour cette difference. Z~m~nf~~-im Teil I wnrden die ~~egation~ffekte en einzelnen Stapelfehlem in den Legierungen Co-O,96 At.-“/, Nb und Co-O,98 At.-“/, Fe mit der Atomsonde des Feldion~mikrosko~ untersucht. In dieser Arbeit wird eine Methode zur Entfaltung entwickelt. Diese Methode wird zusammen mit den experimentellen Ergebnissen zum Nachweis dafiir benutzt, daB das mit dem Stapelfehler zusammenhangende Konzentrationsprofil der segregierten Atome extrem eng ist; es fallt innerhalb von 4 A auf die Konzentration der Matrix ab. Die Thermodynamik der Segregation wird anhand der Beobachtungen diskutiert. Es wird gezeigt, daB die einfache Langmuir-McLean-Segregationsisotherme, die Seine Wechselwirkung zwischen den Segregationsatomen annimmt, zur Interpretation unserer Ergebnisse niclrt herangezogen werden kann. Dagegen konnen sic qualitativ mit einem Bragg-Willies-M~ell erkllrt werden, welches die W~hselwirkung zwischen ngchsten Nachbarn heriicksichtigt. Diem Isotherme sagt voraus, daB ein Phasenfibergang in der Ebene des Stapelfehlers auftritt. Mit unseren Beobachtungen stimmen eine molare partiale SegregationswHrme von -20 meV/Atom und ein Parameter Tur die Wechselwirkungsenergie zwischen nHchsten Nachbam von -65 meV/Atom qualitativ iiberein. Der experimentell beobachtete iibergang von einem ungtittigten (entordneten) zue einem gesiittigten (geordneten) Stapelfehler ist breiter als der vom Bragg-Williams-Model1 vorausgesagte. MBgliche Ursachen f”urdiese Diskrepanz werden diskutiert. tPresent address: R.C.A., Astroelectronics Princeton, NJ 08540, U.S.A.

Division, 1565

1566 HERSCHITZ

and

SEID.MAN:

SOLUTE

AToM

SEGREGATION-II.

ANALYSES

AND

DISCUSSION

means an urrcorrected value. The value of . is a her limit to the actual concentration of solute This paper is the second in a two part series on a atoms in a stacking fault ((c!>*). The reason for this study of soIute-atom segregation effects to stacking is that a stacking fault may have a solute concenfaults in dilute cobalt alloys. In this paper we present tration profile [c,(x)] perpendicular to the plane of the analyses used and give a discussion of the thermothe fault, and this concentration profile is lost in the dynamics of solute-atom segregation in light of the manner in which the data was recorded [see Fig. (2) experimental observations. in particular we describe of Part I]. Therefore, the data has to be analyzed to a method which allows us to determine the actual obtain (c$* values from the experimental values of solute composition of a stacking fault and to show ”quantitatively that the solute concentration profile We now consider a model which allows us to falls off very quickly with distan~w~th~n Iesr &cm calculate Cc<>* from the.ex~~men~l data. For e&r 4 A-from the plane of the fauit, It is shown that the atomic plane chemically analyzed one measures the local solute fluctuations present in the integral integral value of the solute concentration profile in a profiles for the specimens annealed at T, > 450°C are circular area (,$)-this area is determined by the due to solute-rich regions of two-dimensional (2[D]) projection of the probe hole on the FIM tip. The ordered-phases in the plane of the stacking fault, A quantity (c!), is, therefore, the mean value of the simple statistical model which justifies this inter- integral solute concentration for a num&r of cryspretation is presented. talIographic planes which are perpendicular to the Various isotherms for solute-atom segregation to stacking fault. The calculated mean value of the stacking faults are discussed. It is shown that a simple solute con~entratjon profile is taken to be equal to the ~angmuir-McLean isotherm--it assumes no inter- ex~~menta1 value of (c$, in our analysis. Hence, action between the segregating atoms-cannot ‘be the quantity (cjs), is given by the expression used to interpret our experimental results. However, the results can be explained q~~lit~tivel~by a Bragg-Williams model of segregation which includes the effect of coverage on the amount of solute-atom segregation. This isotherm predicts the occurrence of a phase transition as it takes into account the fit-ne~~t-neigh~r interactions between the soiute and sofvent atoms. The experimentally observed To evaluate equation (1) we employed the eoncentransition from an unsaturated (disordered) to a tration profile exhibited in Fig, l(a). This profilet is saturated (ordered) stacking fault is broader than the described by the equations one predicted by the Bragg-Williams model. Possible -mx in O$;xsv; and reasons for, this difference are discussed. (2) G(X) = <0 2. ANALYSES

2. I, Determinationofthe actual s&.&ecompsition of a stackingfuuit In Part I we presented integral profiles for solute atoms (Nb or Fe) from a cylinder of alloy that contained a sing/e stacking fault. An integral profile consists of a plot of the cumulative number of solute atoms (ordinate) versus the cumulative number of solute plus solvent atoms (abscissa). The average slope of an integral profile corresponds to the aueruge solute composition ((ok),,) of the cylinder of alloy analyzed, where the superscript and subscript f and s stand for stacking fault and solute species (Nb or Fe), respectively. and the subscript u on the bracket ~_.._ .___ -.-_ II tThe linear profile is an approximation IOthe

actual profile. which must be stepped. as the solute atoms sit substilutionally in atomic planes which are parallel to the plane of the f&k. If one chooses an exponeilti~t function for c’,{s) then equation (1) does nc~f have a closed-form solution. hcnLy we emplo>- ;t linear Function for c.(s). The initial slope of nu exponential function is linear. and since 6.,(.x ) falls oll’vcry quickly with dist;wcc the choice of a lincx profile should tt,~r strongly ;tfkct the onlculatcd v:rtue of’ (c:)*,

G(X) = (c)

in

rl S x I; D,f2.

(3)

The quantity (c,) is the mean solute concentration of the alloy and m is the slope of the linear region. The c,*(x) vs x profile is actually perpendicular to the x-y pfane, but it has been folded into this plane for calculational purposes. The parameter 9 is, the half width of the solute concentration profile at its base. The final equation for (c{}* (see Appendix A) is

where the expressions for $ and [ are given in Appendix A. The easiest case to consider is a step function, i.e. all the solute atoms reside in the plane of the stacking fault and M = 0. The exact solution (see Appendix A) for this case is

and when (%l/n,,)c< 1 this reduces to

HERSCHIT

and

SEIDMAN:

SOLUTE

ATOM

S~~;~~~~ATl~N~-~I.

ANALj’SES

AND DISCUSSION

1567

Fig. I. A schematic diagram showing the relationship between the solute concentration profile associated with a stacking fauit and the probe hole. See text for the definitions of the physical quantities indicated in this figure.

Equation (5b) is identical to equation (2) given in our first report on this work, which was derived on the basis of conservation of mass. The difference between the value of (c{)* calculated employing equations (5a) and (Sb) is only approximately 0.04% and, therefore, one can very safely use equation (5b) to calculate (4)’ for a step-function distribution. The second case of interest is M does not equal to

Co - 0.96 at.%

Nb

(SF2.

r, = 45O.C)

I

I

I

I

I

1

2

3

4

5

?

in units of do,,,

Fig. 2. A plot of the actual solute composition of a stacking fault (c<>* vs the half width {q) of the solute concentration profile. The upper bound to (&* corresponds to the step-function case, i.e. when all solute atoms reside in the plane of the stacking fault. TThis corresponds to the pure 2-[D] case. When solute atoms reside in adjacent planes then we refer to this as quasi 2-[D].

zero. In Appendix A \ve show that for (2q/D,)<< t equation (4) reduces to

cc!>*Z

(co.($+(c.,(i-2).

(6)

The on/y unmeasured parameter on the right-hand side of equations (41, (5) and (6) is the quantity g. Thus it is possible to calculate (c$* for a range of reasonable tl values. Figure 2 exhibits (c{)* vs q for the Co(Nbf and CofFe) alloys. The upper bound to (c{)* corrresponds to the step-function case, i.e. when all the solute atoms reside in the stacking fault plane.? There is strong experimental and theoretical evidence which indicates that the solute concentration profile at a surface falls to (c,) within a few interplanar distances [2-61 at most. Thus, we have only calculated (cc>* out to 5~&,~; that is, a maximum of the first four planes to either side of the stacking fault are enriched in sofute atoms. in Part I we performed experiments which demonstrate that the solute concentration profile associated with a stacking fault is extremely narrow; see Fig. 2(c) and (d) of Part I for the principle of this experiment and Figs 16, 18,20 and 2 1 (Part I) for the corresponding experimental results. Figure I(b) illustrates the relationship between the position of the probe hole and the c,(x) profiie associated with a stacking fault for our ex~rimen~l conditions. We modeled this physical situation (see Appendix A) to obtain a relationship between (c{), and q. The objective is to examine the effect of the solute concentration profile on (c<).. The final expression is (C!)” = (c,V)(I - Z) + (c:>*E

(7)

where the value of H is given by equation (36 j-it is only a function of q, Figure 3 shows a plot of (c{>, vs q. These curves were calculated using equation (7) emptoying the experimental values of (c,) and D,. This figure demonstrates that the c.,(x) profile must be very narrow; otherwise we would have measured appreciably larger values of (c{)” in the experiments illustrated in Fig. Z(C) and (d) of Part I.

1568 HERSCHITZ

and SEIDMAN:

SOLUTE ATOM SEGREGATION--II.

ANALYSES AND DiSCUSSiON

The lower limit to (cf),t is equnl to :: = ((&.,,

nn; - CC\>) -@ ( )

I - 5 . fW t > ( This equation is derived in the same way as equation + (co*

(5). 2.3. Probability of the occurrence of a specific solute ~uctuation in a random solid solution

Fig. 3. A plot of the expecfed value of the measured solute concentration (c/s), vs the half width (q) of the solute concentration profile. This figure demonstrates that the solute concentration profile must be very narrow. Otherwise, we would have measured appreciably larger values of (I$ in the experiments illustrated in Figs 2{c) and (d) of

Part I. 2.2. Determination of the actual composition of a solute-rich juctuation In order to obtain the actual composition of a solute-rich fluctuation {c$z it is necessary to know its exact shape. The manner in which the data is recorded, however, allows us to determine only one dimension (I) of the fluctuation. Thus we have calculated upper and lower limits to (c{)z . The lower limit is obtained by assuming the fluctuation is a rectangle of dimensions I x D,. And the upper limit is calculated by assuming the fluctuation is a square of dimensions 1 x 1. The values of 1 and (4): for each fluctuation in both alloys are given in Tables 3 and 5 of Part I. The upper limit to (c$)z is equal to (see Appendix C)

(z )-

e-sl.ir

(0

x (c_‘>? f LI =

(s-g+w(s-$)

(*)

d/2

where the expression for S is given in Appendix When (Zq/D,,)<< 1 equation (8) reduces to

C.

The nature of the integral profiles presented in Part I changed in two fundamental ways as the annealing temperature (T,) was increased. First, the. average slope [and correspondingly both (4). and (c{)*] decreased as T, was increased. Second, as T, was increased from 450 to 575”C, local fluctuations ag pear in the integral profiles and the average concentration of these fluctuations ((c<>$) is significantly different from the value of (c$*. In the case of the Co-096 at.%Nb alloy the value of {r$)* is -31 at.% Nb at 450”C-the lowest T,--and the values of the lower and upper limits to (c<)z of the fluctuations at 525 and 575°C are approximately equal to 34 and 75 at.% Nb, and 23 and 50 at.% Nb respective1y.t In the case of the Co-O.98 at.% Fe alloy the extreme values of (c<)z of the fluctuations at 500 and 575°C are approximately equal to 23 and 74at.x Fe, and 13 and 33 at.% Fe respectively. Thus these experimental results demonstrate fairly conclusively that solid state composition fluctuations exist in stacking faults. These fluctuations are temperature dependent and their magnitude is large enough such that they cannot simply be random statistical fluctuations. The experimental evidence indicates that they are due to solute-rich regions of a 2-[D] (or quasi 2-[D]) phase in the stacking fault. High resolution scanning transmission electron microscopy in combination with electron energy loss spectroscopy may be a way to determine precisely the exact shape(s) and stoichiometries of these solute-rich regions. We now describe a simple statistical model which is used to calculate the probability of occurrence of a specific solute fluctuation in an integral profile. In a random binary solid solution the probability [fN..(F,j(n)] of finding n solute atoms in a cluster containing a total of N atoms (solute plus solvent) is given by the binomial distribution fv.&n)

=

0

;

= ,7,(;; Now consider

tin the case of SF4 rhe upper limit to (I*!): is greater than 100% (see Tubks ? ;Ind S in Part I). This implies that the actilill dimcnsitms of the fluctuations are greater than I x /.

((c,J)“(l

- Cc,>)“-*

n)! (
(11)

an integral profile which contains a total cumulative number (N,) of solvent and solute atoms. We arbitrarily divide the integral profile up into a number of bins (0 each oi’ which contains N atoms. The probability OPMU finding a cluster (r) of size N. with ;I specitic v;llue of U, amotig the N,

HEKSCHITZ solvent

and SEIDMAN:

SOl_li.TT: ATO\!

plus solute atoms sampled

is ci\cn

St:(;KI‘:
ANALYSES

cncrgy 01‘ a solute

I~!

plant. The equation I-=

lj[I I-

-r;

-./,,;,,\(n,]‘=[I

(II)]..

(!2)

AND

DISCUSSION

1569

atom in the plane i = 0 or i # 0 for the chemical potential

of the

\~~lute atom in the bulk (11”“‘“) is

I

,lh”lh = ,ly + l;,T In (c,,) + i;:‘,htl’k.

The probability of finding (x) such a cluster is sinlplj I - r. Thus the final expression Ihr the probability 01‘

Al

detecting

I(!

a cluster

of N atoms containing

atoms. among N, atoms sampled. mean solute concentration

(c$)

II solute

in an alloy wth

(G;,%;,

(ci ,>* = (c,) rxp

I- ,,,(NN; ,,)! (((~,))“(I - C(,$>)‘ ” .. (13) I [ . The specification of 1; presents a problem. as ideally one would like to have L: values which are chosen independently of the experimental data. ‘This is a difficult problem to solve and therefore. in lieu of an exact solution, we took [ to be equal to N,/N. where tie used the specific value of N for each fluctuation in question. We have calculated x for the fluctuations shown in Figs 15-18 and 21-23 (see Part I) and find that the values are in the range of O.OI-l”/O and, therefore, they cannot simply be random statistical fluctuations. On the other hand, the values of x of the fluctuations present in the integral profiles of the matrix are typically greater than lo%, which indicates that they are random statistical fluctuations. 3. DISCUSSION

segregation isotherm

An isotherm equation for solute atom segregation to stacking faults has been derived by a number of authors [7-l I]. This equation is of the same form as the Langmuir or McLean isotherm equation for surface or grain boundary segregation [12, 131. We now reconsider briefly the isotherm equation for solute atom segregation to a stacking fault under the assumption of a substitutional model, that is, the segregating solute atoms substitute for the solvent atoms in the plane of the fault and, perhaps, also in several planes which are parallel to it. This model, however, excludes vacancies. For the stacking fault or any plane i which is parallel to the fault? the chemical potential of a solute atom (&) is given by (14)

where pe is the standard state chemical potential of a solute atom, k,T has is usual significance, (c$)* is the corrected concentration of solute atoms in the fault plane (i = 0) or a plane parallel to the fault (i # 0), and G$ is the excess partial molar free tFor a stacking fault i has both positive and negative values as the concentration profile is symmetric with respect to the fault plane. fTVs implies a pure 2-[Dlsituation. The substitutional model [equations (16) and (17)] diverges as Tapproaches 0 K. However, in the temperature range of interest both the substitutional and interstitial [equation (l9)] models yield the same values of As<,,, and AR<,,,.

one

(15)

musl

have

]-

(16)

/ = pi”“” and therelhre

is

pf.r., = p”.1+ k,T In (c/ s.()* + G.“d I.,

equilibrium

it

x=1-

3.1. Langmuir-McLean

thermodynamic

-

_

~;‘.l”“h)

k,T

Let us now restrict the solute atom segregation to the is analogous to the monolayer surface segregation model--then (&) = (c.) for all planes except the fault plane and equation (16) becomes

plane of the Paul+-this

(CO* =
exp~-AC&B~l

(17)

where AC{,,, is the partial molar Gibbs free energy of segregation for a stacking fault. The quantity is equal to As:.,, - TAs.{.,,; where AR& AK, and AS!.,, are the partial molar enthalpy and entropy of segregation, respectively, of solute atoms to a stacking fault. Equation (17) can be finally rewritten as * = (c,> exp(AS~.,,/k,)exp(-A~!,,,lk,T)

(18)

to obtain the isotherm equation for substitutional solute atom segregation to an atomic plane of a stacking fault. It one uses an interstitial model for solute atom segregation then the isotherm equation is

cc’,>* =- (C.l> exp(A%&d cd>* 1-cc,> x exp(--A~~,&J). (1%

1-

Physically the quantities Aff& and As<=, are the enthalpy and entropy changes which occur when a solute atom in the bulk exchanges its place with a solvent atom in the stacking fault. If AC<.,, is independent of concentration then equation (19) is of the form of the classical McLean equation for grain boundary segregation or the Langmuir equation for adsorption on a free surface. Both McLean and Langmuir assumed no interactions between the segregating or adsorbing species. And, therefore, the values of AR<,,, and As:,,, are independent of coverage. We now demonstrate for our dilute alloys that AK.*, must depend on the degree of coverage of the stacking fault. Figure 4 is a plot of (c<)* vs 103/T for the two cobalt alloys. The values of AR’,,,, and AK, deduced from this plot are are equal to -0.6 and -0.3eVatom-’ (13.8 and 6.9 kcalmol-‘), and -6.0 k, and -2.7 k,, respectively, for the Co-O.96 at.% Nb and Co-O.98 at.% Fe alloys. The values of AR[,lcl and As{.,, are almost identical if one uses the interstitial segregation isotherm [equation (19)]. These values of Afl’,,,, are - 120 to 60 and -60 to 30 times greater than the stacking fault energy of cobalt (5.2 to 9.4 meV atom-‘) for the

1570 HERSCHITZ

and SEIDMAN: Temperature

00 I

600 550 I,

‘o-096at

500 ,

SOLUTE ATOM SEGREGATION--II.

energy is - 0.5 eV atom _‘, while for the Co(Fe) alloy

("C 1

450 I

ANALYSES AND DISCUSSION

400 I

350 I

1

% Nb

it is _ 0.01 eV atom-’ (Appendix D). The observation of an enormous segregation effect in the Co(Fe) alloy demonstrates that the elastic strain energy term can not make a significant contribution to AR&. The fact that the elastic strain energy for the Co(Nb) alloy is similar to the measured value of AJ%&Cr is purely fortuitous in our opinion. From the above discussion we conclude that a simple Langmuir-McLean segregation isotherm can not be used to interpret our measurements of solute atom segregation effects to stacking faults in these dilute Co alloys, and that one needs a model which allows for solute-solute and solute-solvent atom interactions in the plane of the stacking fault. 3.2. Fowler-Guggenheim

001 10

I

I

I

1.1

1.2

13

I 1.4

I 1.5

I 1.6

J

1 .7

1000/T~K-'l

Fig. 4. A plot of the corrected solute concentration in a stacking fault (e!)* vs lO’/T for two cobalt alloys. The values of AR{=r and AS{.,, are equal to -0.6 and -0.3 eV atom-’ and -6.0 kB and -2.7 k,, respectively, for the Co-O.96 at.% Nb and Co-O.98 at.% Fe alloys. Each data point corresponds to an individual measurement of (4)‘.

Co(Nb) and Co(Fe) alloys, respectively. These AR{.,, values are also very large for solute atom segregation to stacking faults when one considers the models of Defay et al. [14], McLean [12], or the unified model of Wynblatt and Ku [IS]?. The first nearest-neighbor regular solution model of Defay et al. [14] predicts that A&, is equal to zero-since the common stacking faults (Part I) in cobalt have no first or second-nearest-neighbor violations. The WynblattKu unified model includes the relaxation of the elastic strain energy associated with a solute atom when it segregates-following McLean [12J-as an additional contribution to the partial molar heat of segregation. The Wynblatt-Ku model assumes that all of a solute atom’s elastic strain energy is relaxed when a solvent atom in the surface is exchanged for a solute atom in the bulk. It is ~rorclear that a significant fraction of a solute atom’s strain energy can be relaxed in the case of a stacking fault. Therefore, it does not seem physically reasonable to include the elastic strain energy as a major contribution to A/?{,,,. Specifically in the case of the Co(Nb) alloy the elastic strain tThese models are only applicable when the surface or interface is in the state of a dilute solution. as they only consider the interaction of ow solute atom with a phmar defect. No allownnce is mndc for the possibility 01 solute-solute or sohnc-solvent atom interactions. $lf ‘f is equal to ?&,: ( - 4 fi) tlwn (t,‘,,)* corrcspontls lo Co,Nb.

segregation isotherm

The experimental data indicate that a high temperatures 2-[D] (or quasi 2-[D]) regions of ordered phases form in the plane of the stacking fault and that these regions grow and cover the entire plane of the fault as the annealing temperature (T,) is decreased from 575 to 450°C. This is particularly clear in the case of the Co-O.96at.%Nb alloy. The value of m 31 at.% at 450°C; this corresponds to the
[{ERSCHITZ

and SEIDMAN:

SOLllT‘E

ATOM

SEGREGATION--II.

:\NALYSES

AND

DISCUSSION

1571

0=022 No-0 co- l

Saiurallon

I@ -

coverage

1)

corresponds

To Co3Fe \

\ (b) L

I

____

I

I

I

I

400

500

600

700

Temperature

PC)

Fig. 5. The coverage of a stacking fault (0) vs temperature (T) for two cobalt alloys. The saturation coverage (0 = I) corresponds to Co,Nb (a) and Co,Fe (b). The occurrence of a 2-[D] phase transition in the stacking fault plane is evident from this figure.

fault may not have been in local equilibrium with its solute reservoir [l]. If it had been in local equilibrium then 0 would have been greater. The Bragg-Williams model assumes the segregating atoms are distributed randomly on the surface at localized sites and they can interact with one another over a first-nearest-neighbor distance [ 16-191. We can apply this model to a stacking fault by assuming a first-nearest-neighbor interaction energy parameter (co), with o defined by stacking

o = EAB-

WAA+ EBB) 2

(20)

Fig. 7. A schematic illustration of what we believe occurs in the (1I I) plane of a stacking fault for a Co(Nb) alloy. The large open and small solid circles indicate Nb and Co atoms, respectively, and the cross-hatched regions correspond to islands of the 2-[D] phase Co,Nb. At high temperatures these islands form in the plane of the fault and they grow and cover the entire plane of the stacking fault as the temperature is decreased from 575 to 450°C. where the solvent-solute, solvent-solvent and solute-solute interaction energies are given by EAB, EAA and EBB, respectively. A negative value of o implies a tendency toward compound formation. The chemical potential of a solute atom in a stacking fault in this interacting case differs from the noninteracting case [Langmuir-McLean] by the term (zo[(c_l)* - (c,)]), where z is the total number of first nearest-neighbors. Thus, we can now write p!= PQ+ kBT In (c{>* + (ZMc!>*

-

(CA) + GY. (21)

Equation (15) for p p” is unchanged. Equating these chemical potentials we obtain the following segregation isotherm equation for a substitutional model, which excludes vacancies, in the presence of first nearest-neighbor interactions cc<>* = (0

exp(A%&,)

x exp[ - (AR:.,, + zNcQ*

_-Fig. 6. A schematic diagram of 2-[D] ordered phases ot compositions (a) Co,Nb and (b) Co,Nb in the (1 I I) plane of an f.c.c. lattice.

-

(c,>lYk9~1. (22)

This equation is of the same form as the Fowler-Guggenheim adsorption isotherm as it includes the effect of coverage-through the (zw[(c{)* - (cJ]) term in the Boltzmann factor--on the amount of solute atom segregation [19]. Bernard and Lupis [20] and Guttmann [21] have derived adsorption (segregation) isotherm equations for surfaces and interfaces, respectively, based on

1572 HERSCHITZ psL~r~c~(,-interstitial

and SEIDMAN: models.

These

SOLUTE ATOM SEGREGATION--II. models

aswne

a

particular ordered stoichiometry for the surFAce or interface with one of the two species absent. The solute-atom segregation occurs as a result of the continuous filling of the empty sites. The resulting segregation isotherms include the same physical efEectsas equation (22). That is, the Boltzmann factor contains an interaction energy term and a coverage parameter, aithough the detailed expressions are somewhat different. All of them predict a phase transition with a transition temperature that is a function of the stoichiometry. The substitutional segregation isotherm equation (22) diverges as T approaches zero. Therefore, we use an intersitial segregation isotherm for analyzing our experimentat results. This isotherm is

x exp[ - (A&,,

+ ZW[(~>* - (~,N)lW’l.

ANALYSES AND DISCUSSION Temperature

(“Cl

200

401

600

1

I

0

1 0,

b

800

100

1200

\ \ 09

06

_

-.

’

I I

’ I

i

;

I

-.

07-pQJ

@ %

I

?;l: 3

131

06I

’

e zi

s II 3

l

I 05-

I

u”

04

-

/’

:!’ I

I

I

03-

1)

(23)

Figure 8 is an approximate plot of 0 vs T for different values of AH{,_ and o, with As<_ set equal to zero. The segregation isotherm with AH{_ = -20 meV atom-’ and o = 0 corresponds to the Lancer-McL~n adsorption isotherm. This value of A&<, is approximately two to four times the stacking fault energy of cobalt (9.4 to 5.2 meV atom-‘). This segregation isotherm does got predict a phase transition. The other adsorption isotherms are for AR!,,, = -20 meV atom-’ and different values of w. The combination and w = -65 meV A&.,, = -2OmeV atom-’ atom-’ predicts a critical temperature (T’,) of 600°C. Appendix E describes two methods of estimating the values of EAA, EBB and EAB. Table A2 lists the interaction energies for Co(Nb) and Co(Fe) alloys. Tine agreement between the values calculated by two different methods is not good. However, an important point to note is that the values of o calculated by the two different methods are essentially the same. This indicates that even a small value of o will lead to a phase transition. The value of AA{,,, = - 20 meV atom- ’ is approximately two to four times the stacking fault energy (- 10 to 5 meV atom-‘) of cobaft. This may seem to be a large factor but there is at least one other case where a similar result has also been found. McCarty and Wise [24,25] have studied sulfur chemisorption on nickel and have deduced, from a thermodynamic cycle, a heat segregation of-2.0eVatom-’ in the temperature range 1173-1473 K. The solid surface energy of nickel at 1523 K is -0. t I eV atom-’ 1261. This implies that the heat of segregation of sulphur is approxiln~lteiy twenty times the solid surface energy of pure nickel. The calculated segregation isotherms exhibited in Fig. 8 all have a sharp transition from the saturated (0 2 1) to an unsaturated state (0 ~0). This is the result of employing a small value of ANI!,,,,

200

400

600

Temperature

100

1000

Qoo

1400

1

(K)

Fig. 8. A plot of the coverage vs temperature for the Fowier~uggenheim adsorption isotherm. The isotherms were caiculated for A#& = -20meV atom-’ and values of w which range from zero to -75 meV atom-‘. The segregation isotherm for w =O corresponds to the Langmuir-McLean isotherm. The value of A?& is equal to zero for these isotherms. (- 20 meV atom-‘) and values of o which are comparable to or greater than AR<.,. Alternatively, a large value of A/?& in combination with values of o which are significantly less than AR{._ yields a broad transition from the saturated to ~saturated state. However, the values of w we employed are reasonable ones (see Table A2), and we do not have a fundamental justification for using an extremely large value of AH/;., (see Section 3.1).

3.3. Effecrs segregation

of ferromagnetism

on

solute

atom

Recently Szklarz and Wayman [27] have studied the effect of the ferroma~etic-to-paramagnetic transition on the segregation of solute atoms to an interface in dilute iron-base alloys. From earlier research [28,29] it is known that solute atoms which decrease the Curie temperature have a very high chemical potential; both Nb and Fe decrease the Curie temperature of cobalt. This fact results in a strong dimunition of the solubility of the solute atom in the primary solid solution, as an alloy’s temperature is dropped below the Curie temperature. The general result for those solute atoms which depress the bulk Curie temperature, is that the segregation isotherm is shifted to higher temperatures (see Fig. 7 in Ref. (271) and, they also develop a sharp step hclow the bulk Curie temperature. The transition from the saturated to the unsaturated interl’acc is also

HERSCHITZ

and SEIDMAN:

SOLUTE ATOM SEGREGATION---II.

broader than the prediction of the Bragg- W~lli;uns model. Thus. perhaps another contribution to the

broad tcmpcruturc transition we have observed is due to ;I tcmperaturc dependent magnetic contribution to the total Gibbs plus intcrfacc].

free energy

of the system

[resewoil

4. SUMMARY (I) We present a method which allows us to deconvolute the experimental data in order to obtain the actual solute concentration in a stacking fault as a function of the width of the solute concentration protile. A deconvolution procedure is necessary because of a matrix contribution to the measured solute concentration of a stacking fault. (2) The deconvolution procedure in combination with the experimental results show that the solute concentration profile associated with a stacking fault is extremely narrow-it falls off to the solute concentration of the matrix within fess than 4A. (3) The experimental results demonstrate fairly conclusively that solid state composition fluctuations exist in the stacking faults for the specimens annealed at 7’, > 450°C. These fluctuations are temperature dependent and their magnitude is large enough such that they cannot be random statistical fluctuations, but are rather due to solute-rich regions of a 2-[D] (or quasi 2-[D]) phase in the stacking fault plane. A simple statistical model which quantifies this interpretation is presented. (4) The commonly used Langmuir-McLean segregation isotherm equation, which assumes no interactions among the segregating atoms, cannot be used to interpret our experimental results. (5) The fact that we have observed substantial solute atom segregation to a Co-O.98 at.% Fe alloy, even though the value of the elastic strain energy is negligibly small-the atomic volumes of Co and Fe are almost identical-indicates that elastic strain energy does not play a major role in the driving force for solute atom segregation to stacking faults. (6) We can qualitatively explain our experimental results using a Bragg-Williams type of segregation isotherm which includes the effect of coverage on the amount of solute atom segregation. This isotherm predicts the occurrence of a phase transition, as it takes into account the first-near-neighbor interactions between atoms. A partial molar heat of segregation of - 20 meV atom-’ and an w value of -65 meV atom-’ are qualitatively consistent with our experimental observations. These values predict a sharp transition from the saturated (0 z 1) to an unsaturated state (0 G O)-see Fig. 8. Whereas the experimental coverage curves (Fig. 5) are consid-

erably broader. This is the result of employing a small value of the partial molar heat of segregation (A.H,&,), in conjunction with a large value of the parameter w. A broader transition from the saturated to the unsaturated state can be obtained by employ-

ANALYSES AND DISCUSSION

1573

ing ;I large \‘aluc of AR:,,,

in conjunction with a the values of w we ClllplOyCd are reasonable OllCS (see Table ,42), and we cl0 not Il8v~ ;I I‘lllld;llllC:nt~l justification for using an cxtrcmcly 1;1rgc wlw ol‘ Afi&. (7) All 01‘ the csisting classical thermodynamic descriptions [7 I I] of solute atom segregation to stacking 111~11 ts ;lre im&c,~u/c for explaining our observations. Furthermore, all of the simple “one particle” models of surface or grain boundary segregation elrects [ 12. 14, 151 are unable to quantiSIXIII

WIUC

01’ to.

However.

tatively explain the magnitude of the effects we have observed. (8) The results obtained in this work are for a specific pair of dilute Co alloys. However, we believe the implications of this work are quite general for all internal interfaces in alloys. That is, it is thermodynamically possible for any interface to exhibit extensive solute enrichment and a phase transition. Internal interfaces in alloys can have a phase diagram just as external surfaces of alloys have a phase diagram [13]. In the case of grain boundaries one also has the possibility of segregation effects which are related to the structure of the grain boundary [30]. Note added in proqf-A.

Brokman and D. N. Seidman (to be published) have used a statistical mechanical model to derive a Fowler-Guggenheim type segregation isotherm for a stacking fault in a binary alloy, which employs a regular solution model for the primary solid solution and a “perturbed” regular solution model for the stacking fault. The source of the “perturbation” is due to violations in the number of third nearest-neighbors and higher violations our of the plane of the fault plane. The standard first nearestneighbor regular solution model can’t be used for a stacking fault, since it doesn’t exhibit the necessary driving force for segregation-which has its origins in the third nearestneighbor and higher violations. This model yields directly that the chemical potential of a solute atom in the stacking fault contains the additional term-zo[(&* - (c,)]; see equation (21). Acknowledgemenrs-This work has been supported by the National Science Foundation through the Materials Science Center at Cornell University. The support of the U.S. Department of Energy is acknowledged for certain technical facilities. We wish to thank Professor J. M. Blakely for many illuminating discussions on the analogous subject of surface segregation, Professor M. E. Fisher and Dr D. Huse for a discussion on the subject of solid-state fluctuations, Dr A. Brokman for useful discussions concerning twodimensional phase transformations and Professor H. H. Johnson for enthusiastic encouragement. REFERENCES

I. R. Herschitz and D. N. Seidman, Acfa mefall. 33, 1547 (1985). 2. S. Ono, Memoirs of the Faculty of Engineering, Kyushu Uniuersilv Vol. 12. D. I (1950). 3. F. L. I%lliams anh D: Naion, SurJ Sci. 45, 377 (1974).

4. Y. S. Ng and T. T. Tsong, Surj. Sci. 78, 419 (1978). 5. T. T. Tsong, Y. S. Ng and S. B. McLane, J. Chem. Phys. 73, 1464 (1980). 6. Y. S. Ng, T. T. Tsong and S. B. McLane, Phys. Rev. &VI. 42, 588 (1979).

1574 HERSCHITZ

and SEIDMAN:

SOLUTE ATOM SEGREGATION-II.

ANALYSES AND DISCUSSION

7. H. Suzuki, J. P/I~s. Sac,. Jtr/‘un 17, 322 ( 1962). 8. J. E. Dorn, Acru mesa//. II, 2 IX (1963). 9. R. Dewitt and R. E. Howard, Ac/cr ntc,ru//. 13, 655 (1965). IO. T. Ericson, Acra rnertdl. 14, IO73 (I 966). I I. J. P. Hirth, Me/o//. Truns. I, 2367 (I 970). 12. D. McLean, Grain Bounduries in Metals, pp. Il&l49. Oxford Univ. Press (1957). 13. J. M. Blakely and M. Eizenberg, in Tile Cllemical Physics

of

Solid &r/aces

(editedby D. A.

and Heterogeneous

King and D. P. Woodruff),

Carulysis

Vol. I, pp.

l-80. Elsevier, Amsterdam (1981). 14. R. Defay, 1. Prigogine, A. Bellmans and D. H. Everett, Surface Tension and Adsorption. Longmans, London (1966). 15. P. Wynblatt and R. C. Ku, in Interfacial Segregation (edited by W. C. Johnson and J. M. Blakely), pp. 115-136. Am. See. Metals, Metals Park, OH (1979). 16. T. L. Hill, An Introduction to Statistical Thermodynamics, Chap. 14, pp. 235-260. Addison-Wesley, Reading, MA (1960). 17. A. Clark, The Theory of Adsorption and Catalysis. Academic Press, New York (1970). 18. J. C. Shelton, H. R. Patil and J. M. Blakely, Surf. Sci. 43, 493 (1974). 19. R. H. Fowler and E. A. Guggenheim, Statistical Thermodynamics. Oxford Univ. Press (1939). 20. G. Bernard and C. H. P. Lupis, Surf. Sci. 42,61 (1974). 21. M. Guttmann, Metal/. Trans. 8A, 1383 (1977). 22. R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser and K. K. Kelley, Selected Values of The Thermodynamic Properties of Binary Alloys. Am. Sot. Metals, Metals Park, OH (1973). 23. A. R. Miedema, R. Boom and F. R. DeBoer, J. less-common Metals 24. J. G. McCarty and (1980). 25. J. G. McCarty and (1981). 26. E. D. Hondros, in

41, 283 (1975).

H. Wise, J. them. Phys. 74, 5877

29. T. Takayama, M. Y. Wey and T. Nishizawa, Trans. Jaoan Inst. Metall. 22. 315 (1981). 30. R.‘ W. Balluffi, in Inierfacial Shgregation (edited by

34. 35.

l‘lc(x)dxdy=I::~~~,~~~(~e~~*-~)dyd~ bJ4

W. C. Johnson and J. M. Blakely), pp. 193-237. Am. Sot. Metals, Metals Park, OH (1979). J. W. Christian, The Theory of Transformations in Metals and Alloys, Chap. 6, pp. 198-206. Pergamon Press, Oxford (1975). H. W. King, J. mater. Sri. 1, 79 (1966). K. A. Gschneidner. So/id Sfafe Ph.vsics (edited by F. Seitz and D. Turnbull). Academic Press, New York (1964). J. W. Brooks, M. H. Loretto and R. E. Smallman, Acra merall. 27, I839 (I 979). R. Hultgren, P. D. Desai. D. T. Hawkins, M. Gleiser and K. K. Kelly. Sekcrecl Values of The Thenno&namic Properties qf’Bincrr_vAllo.vs. pp. 656-663, 672-673. Am. Sot. Metals. Metals Park. OH (1973). APPENDIX

A

Determinurion of /he trcwcrl sohr~c composirion ofa srtrcking fault

In this appendix WCderive equation (3) Ihr the situation described in Fig. l(a). The basic prohlcm is to cvalua~e the double integral in the numerator of equation (I). The symmetry of the c,(.v) proflc. \\ith rcspcc‘t IO the .x - vaxes,

.x-D.,/2

,-,,@/ij=ii

+

s x-,

dy

dx.

(Al)

I J-0

In two of the above three integrals it is necessary to use the substitution x = arcsinf-and to appropriately change the limits on the integrals-to evaluate the integrals. After evaluating the integrals and performing some algebra the exact solution for (c$* is

<4>*=

(4). +
where

Precipitation Processes in Solids

817 (1979).

32. 33.

for this situation allows us to perform the integration over only one-quarter of the circular probe hole. Therefore, the denominator in equation (I) is simply xDa/16. And the double integral in the numerator is given by

H. Wise, J. them. Phys. 72, 6332

(edited by K. C. Russell and H. I. Aaronson), pp. l-30. Metal]. Sot. A.I.M.E., Warrendale, PA (1978). 27. K. E. Szklarz and M. L. Wayman, Acra metall. 29,341 (1981). 28. T. Nishizawa, M. Hasebe and M. Ko, Acta metall. 27,

31.

Fig. Al. A schematic diagram showing the relationship between a solute-rich fluctuation and the probe hole.

+;~)[(I-~)?],

(A3)

$ =z[arcsin($)+$/q].

(A4)

and

When (2q ID,)<< 1 the values of < and $ are approximately equal to (4~ /no.) and (8~ /nD,,), respectively, to,first order. Therefore equation (A2) reduces to

When all the solute atoms reside in the plane of the stacking fault c,(x) is simply a step function, this implies m = 0 in equation (A I) and, therefore, for this case we have


(C!,“G> +(c,)( I -;>.

(A6)

When (2~ /D,)<< 1 equation (A6) reduces to

The approximate equations (A5) and (A7) yield values of (d>* which are within a few-hundredths of a percent of the values given by the exact equations (A2) and (A6) for the values of (2q/D,) relevant to our experiments. APPENDIX

B

Determination o/‘ the expecved o&e qf /he measun~tl solute concenlralian ./iv Ihe case when lhe periphery of /he prohc hale i.s pasitioned nert IO II srocking ,/iru/f We now consider the case

where the stacking faulr resides

HERSCHITZ

at

the

and SEIDMAN:

cdgc of the prohc

equation

SOLIJTE

ATOk

Fig.

;1nc1dcrt\c

hole--WC

(7). In this siluation

In

SEGKEGATION-I[.

ANALYSES

AND

DISCUSSION

I j7j

the prolilc is dcscrib
cqualions

+ ((, )

n,,-

in

2

1, 5 .\ s

I),, 2

(IS’)

The

symmetry is lower [or this second cxpcr~mcnt;~l \iluation and therefore the double integrals in equation (I ) :II-c evaluated over one-half of the circular prohc hole. Thcrcli>rc equation (I) becomes

(c!),, = ;;

JJ

c,(s) d.v dy.

II

(83)

*/ 11 xI/ (‘,(I. I),: ,: 4Idl. dI +(r,!)’ I-0 i-i? J‘ I 4II(11, / -,I., 2-\,I,; +cc,,> Jli,JJ/-0 d.v. (C3) By evaluating the above integrals and solving li)r (c!): we obtain the following C.YWI rchttionship

A$ The

double integral in equation (B3)

is given

by

~J~~~)d~d~=J~~~,~J~~~~~~ll.‘(C.)dldr A,/2 ,_I&/2

pJ@i4-_;-~

where

[m(x-A)+

+ 5 r-l

(c,>]dy

dr

(B4)

I r-0

where A = (Da/2 -q). The above three integrals were evaluated in the same way as the one in Appendix D. The exact solution for (ci), is (c’)SU = (c I )(I -a)

+ (c(>‘Z

(W

where Z is a constant given by

This is an upper limit to (c()z because we have assumed that the shape of the region is a square (I x I), where I is determined from the integral profile. When (2q/D,)<< I the value of S is approximately equal to (D,n/2). Therefore, equation (C4) reduces to

E2(3)[*_(?;yy~) x[i-sin-‘(:)-g/z].

+(c,,,(+‘)( I - $).

(B6)

A Taylor series expansion cunnof be used in the case of equation (B6) because (DJ29) and (A/n) are greater than unity. APPENDIX

C

Determination of an upper limit to the actual composition of a solute-rich fluctuation

In this appendix we derive an expression which allows us to calculate an upper limit to the solute composition of the local fluctuations present in the integral profiles. We assume that all the solute atoms reside in the plane of the stacking fault and the shape of the island is a square (see Fig. AS). By analogy with equation (1) we can write

APPENDIX

AF

The symmetry of the c,(x, y) profile with respect to the x -y axis, for this situation allows us to perform the integration over only one quarter of the circular probe hole. Therefore, the denominator in equation (Cl) is simply nD3/16. The profile is now described by the following equations c,(x) = (c!)~

in

05 x 5 n

c,(x) = (c{)*

in

0 s x s rj and

and

0 sy $ //2

l/2 5 y 5 Jm

(C2a)

(C2b)

D

Elastic strain energy conlribution IO solure atom segregation to stacking faults

In this appendix we estimate the value of the elastic strain energy (E,,,,,,,) associated with the difference in atomic radii between the solute and the solvent cobalt atoms. The expression for I$,,,,,, is 24rrKGr,r,(r, - re)2 E C!alUC = 3 Kr, + 4Gr,

(D1)

where K and G are the bulk and shear modulii of the solute and solvent atoms, respectively. And rA and rs are the atomic radii of the solute and solvent atoms, respectively. Equation (Gl) has been derived using the elastic continuum model of a crystal [31]. It has been used to interpret solute atom segregation effects to both grain boundaries and free surfaces. The assumption is made that the Ec,,S,iccan be fully relaxed when the solute atom migrates from the bulk to either of these two planar defects. This assumption is unverified and it is not at all clear that: (a) all of E,,,,,,, can be relaxed; and (b) the fraction of Elh,tk relaxed is independent of the detailed atomic structure of the free surface or grain boundary. The calculated value of ,Qnie is equal to 0.5 and 0.01 eV atom-’ for Co(Nb) and Co(Fe) alloys, res ectively. The r values are equal to 1.385, 1.625 and 1.41I 8: for Co, Nb and Fe, respectively [32]. The values of K and G are equal to Kr_,b= 1.07x 106kgcm-I. &=2.14x 106kg cm-‘, and G, = 0.78 x IO”kgcm-’ [33]. Because of the large difference in the atomic radii of Co and Nb atoms the value of E,,,,,,, in the case of the Co(Nb)

c.,(A VI dx dy JJ 61)
W

1576 HERSCHTZ

and SEIDMAN:

SOLUTE ATOM SEGREGATION-II.

alloy is quite large. By contrast, because the values of rco and rrc are almost identical, the value of E,,,,,, is small in the case of the Co(Fe) alloy. The fact that we have observed substantial solute atom segregation effects in the Co-O.Y8at.%Fe alloy, even though the value of E,,,,,, is negligibly smalt, indicates that the strain energy does nor play a major role in the driving force for solute segregation to stacking faults. Recently Brooks et al. [34] have shown by transmission electron microscopy that a small contraction (-0.25”/,) in interplanar spacing is associated with a stacking fault in a Co-6.5 wt% Fe alloy., (The exact thermal history of the specimen used for this study was not specified.) If this is true for all stacking faults then it does not seem physically plausible to assume that the strain energy associated with a solute atom can be relaxed in a stacking fault. APPENDIX E The esf~afion gies

of the~rsf-neuresf-nejgh~r

~nferacfjon ener-

The first-nearest-neighbor

interaction energy between the solvent-solvent or solute-solute atoms is frequently related to the sublimation energies of pure elements by the following relationship E AA

244 =

where A, is the sublimation energy of component A. The first-nearest-neighbor interaction energy between the solvent-~Iute atoms is simply obtained by rearranging equation (20) E

_,~E;A+~BB)

AB

2

Table Al. The values of the heat of sublimation (A), surface energy (7’) and surface area per atom (a) for cobalt, niobium and iron Heat of sublimation Surface energyt A ‘i’ Element (eV atom-‘) (ergcm-‘) _ - ---- . ..- -__-..I _^.._...._. ._ -. CO 4.42 I970 Nb 7.58 2100 Fe 4.33 2090

w

.

The value of w is deduced from a relationship based on

Equation (E4)

(IOw~cm’) 6.02 8.29 6.25

a regular solution approximation AHmi, eJz-----053) Z(X,%) where AH,, is the heat of mixing and x, and xg are the mole fractions of the solvent and solute atoms, respectively. It is assumed that o is independent of temperature. The values of A for Co, Nb and Fe are listed in Table Al. The values of w for Co-O.96 at.% Nb and Co-O.98 at.% Fe alloys are equal to -0.07 and -0.01 eV atom-‘, respectively 1351.The calculated interaction energies are given in Table A2. The second method gives a relationship between the first-nearest-neighbor interaction energy and the surface energy of a pure component based on the following equation E AA” --

2y’a 2,.

y’ is the surface energy of component

A -either solute or solvent, cr is the surface area per atom and r, is the number of vertical bonds in the lattice. The values of y’ and ci are given in Table Al, and the calculated values of the interaction energies are listed in Table A2. It is clear from Table A2 that the agreement between the interaction energies calculated by the two different methods is not good.

Table A2. The calculatedvaluesof the interaction energiesin units of eV atom-i Equation (El)

Surface area per atom

tmJm-f = ergcm-2.

(El)

J-

ANALYSES AND DISCUSSION

ECO-CO

ENb.Nb

Ef&3

b-NI

EC0.R

-0.74 - 0.26

-1.26 -0.31

-0.72 -0.22

-1.07 -0.36

-0.74 -0.25