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The effect of third nearest neighbor interactions on the thermodynamic properties of interstitial solid solutions
THE EFFECT OF THIRD NEAREST NEIGHBOR INTERACTIONS ON THE THERMODYNAMIC PROPERTIES OF INTERSTITIAL SOLID SOLUTIONS* K.
ALEX
and R. B. McLELLANt
The effect of third nearest neighbor interactions on the thermodynamic functions of interstitial solid solutions htts been investigated. The free energy of the solution crystal was obtained by the cumulant expansion technique. It has been shown th8t even strong third nearest-neighbor inters&ions have 8 negligible effect on the partial co~g~at~on81 entropy of the solute 8toms. The effect of these interactions on the partial enthalpy is much 18rger, but the general form of the variation of the par&l enth8lpy with temperature 8nd composition round previously in simpler statistical mechanical models is conserved. INFLUENCE DES INTERACTIONS DE TROISIEMES VOISINS SUR LES PROPRIETER THERMODYNAMIQUES DES SOLUTIONS SOLIDES INTERSTITIELLES L’influence des interactions de troisiemes voisins sur les fonctions thermodynsmiques des solutions solides interstitielles a et6 Btudiee. L’energie libre de la solution solide 8 et6 obtenue par la methode d’sddition des develonnements. Les auteurs ont m&&t% que mbme des interactions fortes de troisiemes proches voisins ont un &et nogligeable sur l’entropie partielle de configuration des 8tomes dissous. L’influence de ces interactions sur l’enthalpie p8rtielle eat beaucoup plus importante, mais la forme generale de la variation de l’enthalpie partielle 8vec la temperature et la composition, qui 8v8it et6 trouvee anterieurement dans des modeles plus simpbs de 18 mecanique statistique, peut &re eonservee. DER EINFLUB DER WECHSEL~RKUNG THERMODYNAMISC~N EIGENSC~FTEN
DRITTNACHSTER NACHRARN AUF DIE INTERSTITI~LLER LEGIER~~G~N
Der Einflu3 der We~~lwir~g drittrmchster N8chbar-n 8uf die the~~~amischen Funktionen interstitieller Legierungen wurde untersucht. Die freie Energie des L&umgskristalls wurde mit Hilfe der Methode kumulativer Entwickhmg gewonnen. Es wird gezeigt, da0 selbst starke Wechselwirkung swischen drittnitchsten Nachbern auf die p8rtielle Konfigurationsentropie der Fremdstoffatome einen vernachl&ssigbaren Einflulj hat,. Der Einflu9 dieser Wechselwirkungen auf die partielle Enthalpie ist vie1 groDer, jedoch bleibt die allgemeine Form der Variation der pa&&en Enthalpie mit Temperatur und Zusammensetsung, wie sie schon friiher mit einfaoheren statistischen mechanischen Methoden gewonnen wurde, erhalten.
INTRODUCTION
Recently the method of moment expansions has been used to calculate the therm~~amic properties of interstitial solid solutions.o*a) This technique obviates the deficiencies inherent in simpler statistical models for solid solutions in that no error is made in counting the degeneracy in the largest term in the The previous configurational partition function. calculations for f.c.c. solvent lattices showed that the inclusion of interactions between second nearestneighbor solute atoms had a small effect on the entropy of the crystal for repulsive solute mutual interactions, but quite a large effect on the partial enthalpy of solution. There are several impelling reasons for extending these calculations to include third nearest neighbor interactions. Calculations of the partial enthalpy B,” and excess entropy flUvfor interstitial solute atoms in the limit of infinite dilution using the computer simulation technique,c3v4)involving realistic two-body interatomic * Received Mssch 22, 1972; revised May 30, 1972.
t Department of Mechanical and Aerospace Engineering and Materials Science, William Marsh Rice University, Houston, Texas i7001. ACTA
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potentials compatible with experimental diffusion activ&tion energies and the variation of the lattice parameter with solute content, have indicated that the displacements of the solvent atoms in the third shell surrounding the defect are not negligible and make a significant contribution to &,” , and a larger contribution to “TUv.This effect may of course be due in part to some of the inherent deficiencies involved in using two-body interactions and assuming that the force-law constants are really constant. However, there is a clear indication of long-range interactions. Even if the solute-solvent interaction is truncated at short distances, as in the previous calculations,(3~4) and those of Johnson(5,s) and Johnson et dl(‘) the significant displacements found for solvent atoms in the third or fourth shell, indicate that solute atoms can interact in a long-range manner through solventsolvent interactions. Wagner@) has recently pointed out that the bulk of thermodynamic data for solid solutions has been derived from constant-pressure measurements in which the volume of the solution can vary as a function of solute content. Thus problems arise in correlating the experimental data with the equation 107
ACTA
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METALLURGIC-~,
deduced from constant volume statistical mechanical models. Most of the existing experimental data refers to the isothermal variation of the chemical potential at constant pressure. Furthermore, as Wagner clearly points out, (s) the solute-solute interactions representative of constant volume solutions are much stronger than had previously been supposed. This implies that care must be taken in considering interactions other than those due only to first nearest neighbors. Experimentally observed phenomena in b.c.c. interstitial solid solutions also indicate that often long-range interactions are of great importance. The tetragonal distortions observed at low solute concentrations in single crystals, and the low limiting solute concentration, observed in certain b.c.c.metal-hydrogen systems indicate long-range interactions between the hydrogen atoms.(g) All statistical mechanical solid solution models are complicated by the fact that the non-configurational properties, usually considered to be constant, may vary with com~sition.(l~) Thus an unde~~n~ng of the thermod~amic properties of solid solutions, based on a statistical approach, must involve as rigorous a calculation of the configurational properties as possible. Only then can the measured thermodynamic properties be separated into contributions which have diverse physical origins. Accordingly, the attempt has been made in the present work to calculate the effect of third nearest neighbor interactions of a given strength on the thermodynamic properties of interstitial solid solutions based on an f.c.c. solvent metal IatStice. METHOD
OF
atom.
(1)
where N,, is the number of solute-solvent nearest neighbor pairs and E,, is the u-v nearest neighbor pairwise interaction energy. The first term in equation (1) is constant for a given solute concentration and equal to zN,E,, where z is the number of nearest neighbor solvent atoms surrounding a given solute
1973
N,,
= X
AL,,’ = Y N,,”
so
= Z
(2)
Euu ' =
t1%MA
& uu IF=
t2%4
that, E = zN,E,,
+ e,,(X
+ t,Y + t$).
(3)
The parameters t, and t, measure the strengths of the second and third nearest neighbor interactions. They are both zero when it suffices to consider only first neighbor interactions. As shown previously, t2) the configurational partition function of crystal can be written in the form,
where a = -.s,,lkT and l7 is the total number of configurations of the assembly,
w=
(NJ!
Q-4
(NV- NJ!N,!
The moments Bfl are the a priori averages over all configurations of the i-th power of the quantity (X + t,Y + &Z). Thus, i&=”
The solvent metal f.c.c. lattice containing N, atoms has N, solute atoms distributed in its N, octahedral sites. The tetrahedral sites are assumed to be totally unoccupied. In the present treatment we will consider interactions between first, second, and third nearest neighbor solute atoms. The energy of that set of complexions of the solution crystal containing N,, solute atom first neighbor pairs, N,,’ second neighbor pairs and N,,” third neighbor pairs is,
21,
We will use the abbreviation,
1(X
CALCULATION
E = NUVe,, + Nuue,,, i- Nrrrr’euu’+ Nu,‘s,,”
VOL.
+ QY + $$V 5.Z{(X + t,y + t2Z)i) w
(6)
The configurational Helmholtz free energy F,c can be calculated from equation (4) by expanding in terms of a. Fuc = zN,E,, -
kT In W -
kT
where, il1=% A, =
M2 -
Ml2
(8
n,=M,3M,M, + 2Mi3 etc. Thus knowing the moments Mi enables the configurational free energy to be calculated without introducing error into the counting scheme. It has
ALEX
AND
THERMODYNAMIC
MeLELLAN:
PROPERTIES
OF
IKTERSTITIAL
SOLIDS
tO9
occupation numbers, equal to unity if the site indicated by the subscript is occupied, and zero if it is unoccupied. Thus the total number of third nearest neighbor pairs in the ent,ire solution crystal is given by ~%~“(WQ UP Simple Cubii 1st Neighbor 2nd Neighbor 3rd Neighbor of A of A of A sublottice 0 4 0 8 :
i3
Total
4 0
0 6
4
0
E-
T-
The contribution to the first moment 111,from third nearest neighbor pairs is given by,
(2) = /i
8 0 8 24
FIG. 1. Decomposition of the f.c.c. lattice into four simple cubic lattices.
X,,,,“(ag)\
= i /up.
\
(N,,,“(a/3):
(l@ (11)
From equation (9) we have,
been demonstrated previouslyd*s) that the series in (7) converges sufficiently rapidly for solutions of limited solute concentration where kT > E,,. CALCULATION
OF
THE
MOMENTS
in the f.o.c. lattice, the octahedral interstitial sites form an f.c.c. lattice. The calculation of the moments is fa~ili~ted by the decom~sition of the f.c.c. lattice gas of octahedral sites into four equivalent interpenetrating simple cubic lattices. This is illustrated by the f.c.c. conventional unit cell depicted in Fig. 1. Interstitial sites located on the four independent sublattices are denoted by different symbols. It is important to note that the jlrst nearest neighbors of any interstitial site oocw on different sublattices than that on which the given site is located. However, the second nearest neighbors of a given site are all located on the same sublattice as the given site itself. The third nearest neighbors of the given site are located on the same sublattice as the first nearest neighbors. Now
The Jirst moment The prescriptions for calculating the contributions to the moments from X and Y have been given previously. Let us consider the contributions arising from third nearest neighbors. Consider a given pair of sublattices a and ,9. Let $5 be a third neighbor osculation matrix element such that the number of third nearest neighbor pairs on the u/l pair of sublattices is,
If the sites a and b are third nearest neighbors, y$, = 1, otherwise y$ = 0. The t,” and teB are the
Now the averages of the form (&“[bB} are a priori averages over all complexions in which all states are weighted equally. The a priori distributions on two different lattices are independent and {&,aF,B>can be written as the product of the a priori averages of 5,” and E,,@. This, of course, would not be true of statistical mechanical ensemble averages where there would be correlations between distributions on separate sublattices. Thus
where zs(c$) is the number of third nearest neighbor sites on the p-sublattice when the reference site is on the a-sublattice. [zs(c$l) = 81. Thus {@)
=
2N v
e2
(16)
and (2) = -&S$, a/+
= 12NJ32
(17)
since pairs can be chosen from the four independent sublattices in six distinct ways. It was shown previously@) that (X) = 6N,Ba, (Y) = 3NVe2, so that the total first moment is given by M, = (X) + 4(Y> + &&g)
(16)
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and A1 = 6N,02(1 + at, + 2t,)
(19)
At this point it should be mentioned that averages like (l,“) and more complex averages of the form (5:. . . E,*) have to be evaluated in order to compute the higher cumulants. It is easy to derive a general formula for 8,,, 3 (E,” . . . p), where the bracketed quantity involves n sites on the same sublattice.f2) For a sublattice of NJ4 sites this formula is,
;e(i$H-l) (5,” . . . 5,“) =
:e
VOL.
(25)
. ..eB--n+l)
-1) . . . (2--n+l)
(26)
This equation is used in the subsequent computations involving quantities of the form 8,,,.
a-a’ dfb’
Again the contributions to M,, and thus to A,, due to first and second nearest neighbor pairs have already been calculatecl,(1~2) so that we need only consider third nearest neighbor pairs. By definition M, = ((X + t,Y + t2Z)2), so that M, = (X2) + 2t,(X Y) + t,2( Y2) + t&Z2) + 2b(XZ)
+ 2t,t,(Yq
2t,{Gw
-
+
2t,t,{(YZ)
Gwm1
-
(YW)).
oa’b
Obtaining the value of (&n&,P) from the prescription given in equation (20) we obtain:
(21)
Only the terms involving 2 need be calculated. Combining these terms in Z with the terms in Z from Ml gives a total contribution from third nearest neighbor pairs to ii2 of, ( zj21 +
In writing equation (26) we have used the fact that (5,“)2 = &,“. The first term in equation (26) is 12N,e2 [see equation (17)]. NOW,
The second moment
(29 -
1973
are indicated on the equations. Similar restrictions are to be understood, although not specifically mentioned, in subsequent summations when primed and unprimed indexes appear paired. Consider (S$3,. We can write,
(20)
#
21,
(22)
Let us consider this quantity term by term. In the first term, t22{(22) - (2)2}, only (Z2) is unknown. From equation (II),
Since the summation over a and a’ is restricted such that both indexes are distinct, the summation is evaluated by summing over the a-index without the restriction and then subtracting terms occurring when a = a’. The index a* is thus an unrestricted index. The result is, NJ4 ~~b(y~~y:e~=aEa,~EaP) =
Similarly,
In expanding equation (23) to obtain (24), it should be noted that in using the dummy index scheme there are restrictions such that cc # a’ and t3 # 19’. These
x
e,,2$ 2
~3(a8)[~3(~B) -
11 (28)
(30)
ALEX
AND McLELLXN:
THERMODYKAMIC
PROPERTIES
Similar methods were used to evaluate the other terms in equation (24). When all the results were collected and the ~pprop~&te expressions for B’s found using equation (20), we find that, (Z2> -
(Zj2 = $2
zs(~~)~=~~(l -
@N
= 12~,82(1 -
-
OF
INTERSTITIAL
111
SOLIDS
The coefficient of the concentration dependent term (&“tbP&P> in the first summation in equation (37) ca.n be written as,
@a)
ey
(31)
Now consider the third term in equation (22), i.e. { (YZ) - (Y)(Z)]. The factor (Y) is already known.(2) YZ is found as follows:
=
2
z3(a#?)z2(a)
(39)
where the coordination number z2(a) is the number of sites which are second nearest neighbors to a given octahedral site on the same sublattice [z2(a) = 61. Similarly,
Thus,
and
(3%. )No
= ~~(a~)z2(a) * -i_ -
2, 4
.
(39)
Thus, combining equations (37)-(39),
and since
+ (S$S:))
-
(X$>(8$s’)}
(36)
and
Consider the term (~~~~~)}. We have,
and
it, is easy to show, by computing the B’s and rearranging, that (Ls$F?~~‘> - (~~~}(~(2)) Di = 0
so that
and
(41)
so that (YZ) - (Y)(Z) = 0 and the third term of equation (26) makes no contribution. It should be pointed out that in order to show that (YZ) (Y)(Z) is identical to zero, every term in the expansions for e lr,2 and 0E,3has to be considered. This is also true with respect to the evaluation of other similar moments. Similar techniques show, however, that the second term in equation (22) is non-zero, and in fact (XY)
-
(X)(Z)
= -9SiV*e4.
(42)
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1973
Combining equations (31), (41) aud (42), we find the ~ont~butiou to As due to t,hird nearest neighbor interactions is, i&2{12N,P( 1 -
e)s) + 2t,{ -96
N,84)
(43)
6
and combining this quantity with the contributions from first and second nearest neighbor interactions’s) we have the total cumulant, As = 6NW02(l + Izea(l -
0)s -j- 3N~~~~e2(1e)j + t,2{izlv,eyi
3
0)s -
(44)
It should be noted that, since in 2, the cross term in t,t, vanishes, the effect of the third neighbor interactions is additive to F,c and independent of t,. Tke dhird moment
/ Euu=-Zk.cal/mole
0
A preliminary attempt at a rigorous calculation of the contribution of third nearest neighbor interactions to -&Isshowed that the time required would be excessive. However, an approximate calculation of the largest contributions to this term showed that for any reasonable interaction strength the concomitant ~ont~bution to the free energy of the solution crystal would be small compared with the contributions arising from Ml and M,. No attempt was made to calculate the effect of considering third nearest neighbor interactions on the fourth moment. It has however already been demonstrated that the contribution of second nearest neighbors to M4 can be neglected.(s) THERMODYNAMIC
-
0.02
--/
0.04
Wf(0,
a, t,, ts)
(44)
where B,” is the partial enthalpy of the solute atom in the infinitely dilute solution and the function f(0, a, t,, t2) contains all the terms resulting from the
I
0.06
0.10
0.08
OFIG. 2. Variation of the partial con@urational of a solute atom with composition.
entropy
differentiation of the n(0) with respect to N,. The partial configurational entropy and enthalpy are obtained from equation (44) by the standard thermodynamic relations in the form,
fl,C = -k In
a, tr, t2)
(45)
FUNCTIONS
The configurational Helmholtz free energy Fuc is obtained by inserting the values of 1 into equation (7). The contributions to the first four cumulants due to first neighbor interactions, and to the first three cumulants due to second nearest neighbor interactions are given in the previous calculation.(2) The contributions to the first two moments arising from third nearest neighbor interactions are given in this report. Differentiation of F,” with respect to N, at constant T and V yields an expression for the configurational chemical potential ,uUcof a solute atom in the interstitial solutions of the form 8 Puc = aUm + kT In 1--e
!-\
e)y
+ 2t,(-96Nv84)
THE
A
A, _ j&‘= RESULTS
I=:
kT2 AND
we,
(46)
DISCUSSION
The partial entropy The variation of the partial configurational entropy with composition is depicted in Fig. 2. For the sake of comparison, the curve for the ideal solid solution is also included. Values of t, = 0.1 and t, = 0.05 and T = BLOC were chosen. It can be seen that, for repulsive solute mutual interactions, even when E = 3 kcal/mole, the resulting tendency toward a pLtia1 state of ordering has only a small effect on the confi~rational entropy. On the other hand, computing SUOfor an attractive interaction of E,, = -2 kcal/mole shows a considerable reduction in the configurational entropy due to the tendency toward cluster formation. This result is to be expected and can be compared directly to the effect of the formation
ALEX
AND
McLELLAS:
THERMODPKAMIC
PROPERTIES
OF
IXTERSTITIAL
SOLIDS
113
vanish when, &-
u
{t,2[12iVu82(1 -
19)~]+ 2t,( -
96N,04)}l,.u = 0
This occurs when, t, =
32e2 i -
38 + 282
For 8 = 0.05, this critical value of t, is 0.094. This is clearly seen in Fig. 3. The partial enthalpy
-0 ‘5’ -02(
/ 0
0.02
I
I
I
0.04 0.06
0.08
I
I
0.10 0.12
12
FIG. 3. Effect of including third nearest neighbor interactions of varying strength on the partial configurational entropy of a solute atom.
of interstitial solute atom clusters surrounding substitutional solute atoms in ternary solid solutions. It has been shown that this kind of clustering decreases the partial configurational entropy of the interstitial solute species much more than the decrease caused by the ordering tendency due to repulsive interactions between the interstitial and substitutional solute atoms.dm2’ The flue-values calculated for 1000°C using a series of positive and negative e,,-values and including only first and second neighbor interactions were close to those values found when the third nearest neighbor interactions were included. The effect is seen in Fig. 3 which depicts plots of the difference, Sfl,, between fiUc including, and not including, third nearest neighbor interactions, varying with the strength of the third neighbor interaction. The variation of Ss, with t, is symmetrical about Sfl, = 0 for repulsive and attractive interactions of the same magnitude. However, the most pertinent result is that the effect on the configurational entropy of considering third nearest neighbor interactions is small, even when t, is larger than t,. The zero value of 6#, for a given value oft, is a consequence of the absence of the cross term involving t,t, in I,. This can be seen in equation (43). Since the contribution to Fuc due to third nearest neighbors is linear in A,, the contribution to fl, is proportional to (WWJ,vV~ Th us, for any value of a and 8, Sfl, will
The effect of including third nearest neighbor solute atom interactions on the partial configurational enthalpy is however much larger than the effect on the entropy. This is illustrated in Figs. 4 and 5. Figure 4 shows the variation of B,, with composition for a repulsive interaction of E,,, = 1.0 kcal/mole. For the sake of comparison, the corresponding linear variation of B, with 0 pertaining to the simple zerothorder solution model is included.os) It is clear that, when t, = $t, (although this may represent an upper limit), the third nearest neighbor interactions have a large effect on BUGand should not be ignored. Similar behavior is apparent in Fig. 5 (for E,, = 2 kcal/mole). Figure 5 also contains HUGcalculated from the firstorder solution model.04) Although the variation with 8 of Due in the more refined calculations is not linear, the curvature is quite small and accuracy limits of
II
‘=0.0 0.6
04 0.2 0
0.01
a.02 0.03
e-
0.04
0.05 0.06
0.07
FIG. 4. Variation of the partial configurational enthalpy of a solute atom with composition.
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I I I I I 1000°C First n.n. only ,.*_ ----- Inclu&ng second n.n. with t, =O.I ~~~=ilk col/mole -_-+-* Including third n.n. 1.6with t, =O.l, tz =0.05 -I-I- Zeroth approximation
2.0
I
VOL.
21,
1953
although gWC is an explicit function of temperature, the functional form is such that for reasonable values of E,u, the variation of B,c in the temperature range 700-1100°C is too small to be depicted graphically. The tem~rature variation of BEeCin the momentexpansion calculations is shown in Fig. 6. Again the variation with temperature is quite small and probably undetectable experimentally except for data covering wide t,emperature and composition ranges. It should perhaps be mentioned that the closed expressions for c%, depend only on a and not on E,,,, and T independently. The plots in Fig. 6 have been given however because the values of E,, and T chosen 0.5-
I
I
0.02 I
0.04 I
I
I I T = 1000°C
0.4-
0
0.01
0.02 0.03
0.04
0.05
0.06 0.07
eFIG.5. Variation of the partial configurational ent,halpy of a solute atom with composition.
most thermodynamic data are probably such that this nonlinearity could not be detected experimentally. This linear (or virtually linear) variation of B,,c with 0 has been found in the recent data for carbon austenite.(15,16) The zeroth-order model predicts that E,c is independent of temperature. In the first-order model, -0.4-
-“‘50 1.6
0.06I
0.08 f
0.10 I
C 2
t2 -
FIG. 7. Effsct of including third nearest neighbor interactions of varying strength on the psrtial oonfigurationai enthalpy of a solute at,om.
al I.4 z E 1 1.2 z J 1.0 .c 8!0.6
I ‘==0.6 Partial Entholpie for t,xO.f ond t, Units for k. col/mole
0.02
0.04
0.06
0.08
0.10
EUU a
0.12
e-
FIQ. 6. Temperature variation of the partial configure. tional enthalpy of E solute atom.
have been shown to be appropriate for some interstitial solid solutions.~z*l*,l*) Figure 7 shows the virtually linear effect of increasing the third nearest neighbor interaction strength on the difference, c%,, between i7,c oalculated by including and omitting third nearest neighbor interactions. Note that in general @, is two orders of magnitude larger than SS,. The general technique of Kirkwood expansion is useful and, as shown, capable of enabling the effects of in~raGtions of a given strength between solute atoms in various neighbor shells on the thermodynamic properties of the solution to be calculated accurately. There are two immediately obvious extensions of this
ALEX
THERMODTBAMIC
AND McLELLAS:
work which are currently
receiving
attention.
Com-
puter simulation
studies are being made in the hope of
throwing
light
more
potentials
appropriate
atoms
various
in
on the for
configurations
At the same time the Kirkwood being
applied
where
the
complexity
to
forms
small
interstitial
of non-cubic
(and interest)
of solute
in f.c.c.
lattices.
expansion technique is
b.c.c.-based
presence
of interatomic
groups
solutions
distortions
adds
to the problems.
ACKNOWLEDGEMENT
The
authors
provided
by
gratefully the
Robert
acknowledge A.
Welch
the
support
Foundation
(Grant C-281). REFERENCES 1. K. ALEX and R. B. MCLELLARActa Met. 19,439 (1971). 2. K. ALEX and R. B. MCLELLAN, Acla Met. 20,ll (1972).
PROPERTIES
OF
INTERSTITIAL
SOLIDS
115
3. R. H. SILLER and R. B. MCLELLA~, Acta illet. 19,193 (1971). 4. R. H. SILLER and R. B. MCLELLAS. Acta iVet. 19,671 (19il). 5. R. A. Joaasox, Phys. Rev. A134, 1329 (1964). 6. R. A. JOHNSON,Aeto Met. 15,513 (1967). 7. R. A. JOHNSON,G. J. DIExEs and A. C. DAMASK,Acta Met. 12, 1215 (1964). 8. C. WAGNER, Acta Alet. 19, 843 (1971). 9. H. BUCK and G. ALEFELD. Whys. Status Solidi B47, 193 (1971). 10. R. B. MCLELLA~, Mat. Sci. Engr 9, 121 (1952). 11. K. ALEX and R. B. MCLELLAS, J. Phys. Chem. Solids 32, 449 (19il). 12. K. ALEX and R. B. MCLELLAS, Mat. Sci. E?lgr 7, 77 119711. 13. lk A&X and R. B. MCLELLAN. J. Phys. Chem. Solids 31, 2751 (1970). 14. R. B. MCLELLAN and W. u’. Du~ix, J. Phys. Chem. sO1id830. 2631 (1969). 15. S. BAN-YA, J. F: ELLIOTTand J. CHIPMAN,!haw. metall. Sot. A.I.M.E. 245, 1199 (1969). 16. S. BAN-YA, J. F. ELLIOTT and J. Cmmfm, lllet !hatt.s. 1, 1313 (1970). 17. R. B. MCLELLAN and W. W. DUNN, Scripta Met. 4, 321 (1970). 18. K. ALEX and R. B. MCLELLAN,Scripta Alet. 4,964
(1970).
ALEX
and R. B. McLELLANt
The effect of third nearest neighbor interactions on the thermodynamic functions of interstitial solid solutions htts been investigated. The free energy of the solution crystal was obtained by the cumulant expansion technique. It has been shown th8t even strong third nearest-neighbor inters&ions have 8 negligible effect on the partial co~g~at~on81 entropy of the solute 8toms. The effect of these interactions on the partial enthalpy is much 18rger, but the general form of the variation of the par&l enth8lpy with temperature 8nd composition round previously in simpler statistical mechanical models is conserved. INFLUENCE DES INTERACTIONS DE TROISIEMES VOISINS SUR LES PROPRIETER THERMODYNAMIQUES DES SOLUTIONS SOLIDES INTERSTITIELLES L’influence des interactions de troisiemes voisins sur les fonctions thermodynsmiques des solutions solides interstitielles a et6 Btudiee. L’energie libre de la solution solide 8 et6 obtenue par la methode d’sddition des develonnements. Les auteurs ont m&&t% que mbme des interactions fortes de troisiemes proches voisins ont un &et nogligeable sur l’entropie partielle de configuration des 8tomes dissous. L’influence de ces interactions sur l’enthalpie p8rtielle eat beaucoup plus importante, mais la forme generale de la variation de l’enthalpie partielle 8vec la temperature et la composition, qui 8v8it et6 trouvee anterieurement dans des modeles plus simpbs de 18 mecanique statistique, peut &re eonservee. DER EINFLUB DER WECHSEL~RKUNG THERMODYNAMISC~N EIGENSC~FTEN
DRITTNACHSTER NACHRARN AUF DIE INTERSTITI~LLER LEGIER~~G~N
Der Einflu3 der We~~lwir~g drittrmchster N8chbar-n 8uf die the~~~amischen Funktionen interstitieller Legierungen wurde untersucht. Die freie Energie des L&umgskristalls wurde mit Hilfe der Methode kumulativer Entwickhmg gewonnen. Es wird gezeigt, da0 selbst starke Wechselwirkung swischen drittnitchsten Nachbern auf die p8rtielle Konfigurationsentropie der Fremdstoffatome einen vernachl&ssigbaren Einflulj hat,. Der Einflu9 dieser Wechselwirkungen auf die partielle Enthalpie ist vie1 groDer, jedoch bleibt die allgemeine Form der Variation der pa&&en Enthalpie mit Temperatur und Zusammensetsung, wie sie schon friiher mit einfaoheren statistischen mechanischen Methoden gewonnen wurde, erhalten.
INTRODUCTION
Recently the method of moment expansions has been used to calculate the therm~~amic properties of interstitial solid solutions.o*a) This technique obviates the deficiencies inherent in simpler statistical models for solid solutions in that no error is made in counting the degeneracy in the largest term in the The previous configurational partition function. calculations for f.c.c. solvent lattices showed that the inclusion of interactions between second nearestneighbor solute atoms had a small effect on the entropy of the crystal for repulsive solute mutual interactions, but quite a large effect on the partial enthalpy of solution. There are several impelling reasons for extending these calculations to include third nearest neighbor interactions. Calculations of the partial enthalpy B,” and excess entropy flUvfor interstitial solute atoms in the limit of infinite dilution using the computer simulation technique,c3v4)involving realistic two-body interatomic * Received Mssch 22, 1972; revised May 30, 1972.
t Department of Mechanical and Aerospace Engineering and Materials Science, William Marsh Rice University, Houston, Texas i7001. ACTA
~TALLURGI~A,
VOL.
21, FERRUARY
1973
potentials compatible with experimental diffusion activ&tion energies and the variation of the lattice parameter with solute content, have indicated that the displacements of the solvent atoms in the third shell surrounding the defect are not negligible and make a significant contribution to &,” , and a larger contribution to “TUv.This effect may of course be due in part to some of the inherent deficiencies involved in using two-body interactions and assuming that the force-law constants are really constant. However, there is a clear indication of long-range interactions. Even if the solute-solvent interaction is truncated at short distances, as in the previous calculations,(3~4) and those of Johnson(5,s) and Johnson et dl(‘) the significant displacements found for solvent atoms in the third or fourth shell, indicate that solute atoms can interact in a long-range manner through solventsolvent interactions. Wagner@) has recently pointed out that the bulk of thermodynamic data for solid solutions has been derived from constant-pressure measurements in which the volume of the solution can vary as a function of solute content. Thus problems arise in correlating the experimental data with the equation 107
ACTA
10s
METALLURGIC-~,
deduced from constant volume statistical mechanical models. Most of the existing experimental data refers to the isothermal variation of the chemical potential at constant pressure. Furthermore, as Wagner clearly points out, (s) the solute-solute interactions representative of constant volume solutions are much stronger than had previously been supposed. This implies that care must be taken in considering interactions other than those due only to first nearest neighbors. Experimentally observed phenomena in b.c.c. interstitial solid solutions also indicate that often long-range interactions are of great importance. The tetragonal distortions observed at low solute concentrations in single crystals, and the low limiting solute concentration, observed in certain b.c.c.metal-hydrogen systems indicate long-range interactions between the hydrogen atoms.(g) All statistical mechanical solid solution models are complicated by the fact that the non-configurational properties, usually considered to be constant, may vary with com~sition.(l~) Thus an unde~~n~ng of the thermod~amic properties of solid solutions, based on a statistical approach, must involve as rigorous a calculation of the configurational properties as possible. Only then can the measured thermodynamic properties be separated into contributions which have diverse physical origins. Accordingly, the attempt has been made in the present work to calculate the effect of third nearest neighbor interactions of a given strength on the thermodynamic properties of interstitial solid solutions based on an f.c.c. solvent metal IatStice. METHOD
OF
atom.
(1)
where N,, is the number of solute-solvent nearest neighbor pairs and E,, is the u-v nearest neighbor pairwise interaction energy. The first term in equation (1) is constant for a given solute concentration and equal to zN,E,, where z is the number of nearest neighbor solvent atoms surrounding a given solute
1973
N,,
= X
AL,,’ = Y N,,”
so
= Z
(2)
Euu ' =
t1%MA
& uu IF=
t2%4
that, E = zN,E,,
+ e,,(X
+ t,Y + t$).
(3)
The parameters t, and t, measure the strengths of the second and third nearest neighbor interactions. They are both zero when it suffices to consider only first neighbor interactions. As shown previously, t2) the configurational partition function of crystal can be written in the form,
where a = -.s,,lkT and l7 is the total number of configurations of the assembly,
w=
(NJ!
Q-4
(NV- NJ!N,!
The moments Bfl are the a priori averages over all configurations of the i-th power of the quantity (X + t,Y + &Z). Thus, i&=”
The solvent metal f.c.c. lattice containing N, atoms has N, solute atoms distributed in its N, octahedral sites. The tetrahedral sites are assumed to be totally unoccupied. In the present treatment we will consider interactions between first, second, and third nearest neighbor solute atoms. The energy of that set of complexions of the solution crystal containing N,, solute atom first neighbor pairs, N,,’ second neighbor pairs and N,,” third neighbor pairs is,
21,
We will use the abbreviation,
1(X
CALCULATION
E = NUVe,, + Nuue,,, i- Nrrrr’euu’+ Nu,‘s,,”
VOL.
+ QY + $$V 5.Z{(X + t,y + t2Z)i) w
(6)
The configurational Helmholtz free energy F,c can be calculated from equation (4) by expanding in terms of a. Fuc = zN,E,, -
kT In W -
kT
where, il1=% A, =
M2 -
Ml2
(8
n,=M,3M,M, + 2Mi3 etc. Thus knowing the moments Mi enables the configurational free energy to be calculated without introducing error into the counting scheme. It has
ALEX
AND
THERMODYNAMIC
MeLELLAN:
PROPERTIES
OF
IKTERSTITIAL
SOLIDS
tO9
occupation numbers, equal to unity if the site indicated by the subscript is occupied, and zero if it is unoccupied. Thus the total number of third nearest neighbor pairs in the ent,ire solution crystal is given by ~%~“(WQ UP Simple Cubii 1st Neighbor 2nd Neighbor 3rd Neighbor of A of A of A sublottice 0 4 0 8 :
i3
Total
4 0
0 6
4
0
E-
T-
The contribution to the first moment 111,from third nearest neighbor pairs is given by,
(2) = /i
8 0 8 24
FIG. 1. Decomposition of the f.c.c. lattice into four simple cubic lattices.
X,,,,“(ag)\
= i /up.
\
(N,,,“(a/3):
(l@ (11)
From equation (9) we have,
been demonstrated previouslyd*s) that the series in (7) converges sufficiently rapidly for solutions of limited solute concentration where kT > E,,. CALCULATION
OF
THE
MOMENTS
in the f.o.c. lattice, the octahedral interstitial sites form an f.c.c. lattice. The calculation of the moments is fa~ili~ted by the decom~sition of the f.c.c. lattice gas of octahedral sites into four equivalent interpenetrating simple cubic lattices. This is illustrated by the f.c.c. conventional unit cell depicted in Fig. 1. Interstitial sites located on the four independent sublattices are denoted by different symbols. It is important to note that the jlrst nearest neighbors of any interstitial site oocw on different sublattices than that on which the given site is located. However, the second nearest neighbors of a given site are all located on the same sublattice as the given site itself. The third nearest neighbors of the given site are located on the same sublattice as the first nearest neighbors. Now
The Jirst moment The prescriptions for calculating the contributions to the moments from X and Y have been given previously. Let us consider the contributions arising from third nearest neighbors. Consider a given pair of sublattices a and ,9. Let $5 be a third neighbor osculation matrix element such that the number of third nearest neighbor pairs on the u/l pair of sublattices is,
If the sites a and b are third nearest neighbors, y$, = 1, otherwise y$ = 0. The t,” and teB are the
Now the averages of the form (&“[bB} are a priori averages over all complexions in which all states are weighted equally. The a priori distributions on two different lattices are independent and {&,aF,B>can be written as the product of the a priori averages of 5,” and E,,@. This, of course, would not be true of statistical mechanical ensemble averages where there would be correlations between distributions on separate sublattices. Thus
where zs(c$) is the number of third nearest neighbor sites on the p-sublattice when the reference site is on the a-sublattice. [zs(c$l) = 81. Thus {@)
=
2N v
e2
(16)
and (2) = -&S$, a/+
= 12NJ32
(17)
since pairs can be chosen from the four independent sublattices in six distinct ways. It was shown previously@) that (X) = 6N,Ba, (Y) = 3NVe2, so that the total first moment is given by M, = (X) + 4(Y> + &&g)
(16)
ACTA
110
MET_4LLURGICA.
and A1 = 6N,02(1 + at, + 2t,)
(19)
At this point it should be mentioned that averages like (l,“) and more complex averages of the form (5:. . . E,*) have to be evaluated in order to compute the higher cumulants. It is easy to derive a general formula for 8,,, 3 (E,” . . . p), where the bracketed quantity involves n sites on the same sublattice.f2) For a sublattice of NJ4 sites this formula is,
;e(i$H-l) (5,” . . . 5,“) =
:e
VOL.
(25)
. ..eB--n+l)
-1) . . . (2--n+l)
(26)
This equation is used in the subsequent computations involving quantities of the form 8,,,.
a-a’ dfb’
Again the contributions to M,, and thus to A,, due to first and second nearest neighbor pairs have already been calculatecl,(1~2) so that we need only consider third nearest neighbor pairs. By definition M, = ((X + t,Y + t2Z)2), so that M, = (X2) + 2t,(X Y) + t,2( Y2) + t&Z2) + 2b(XZ)
+ 2t,t,(Yq
2t,{Gw
-
+
2t,t,{(YZ)
Gwm1
-
(YW)).
oa’b
Obtaining the value of (&n&,P) from the prescription given in equation (20) we obtain:
(21)
Only the terms involving 2 need be calculated. Combining these terms in Z with the terms in Z from Ml gives a total contribution from third nearest neighbor pairs to ii2 of, ( zj21 +
In writing equation (26) we have used the fact that (5,“)2 = &,“. The first term in equation (26) is 12N,e2 [see equation (17)]. NOW,
The second moment
(29 -
1973
are indicated on the equations. Similar restrictions are to be understood, although not specifically mentioned, in subsequent summations when primed and unprimed indexes appear paired. Consider (S$3,. We can write,
(20)
#
21,
(22)
Let us consider this quantity term by term. In the first term, t22{(22) - (2)2}, only (Z2) is unknown. From equation (II),
Since the summation over a and a’ is restricted such that both indexes are distinct, the summation is evaluated by summing over the a-index without the restriction and then subtracting terms occurring when a = a’. The index a* is thus an unrestricted index. The result is, NJ4 ~~b(y~~y:e~=aEa,~EaP) =
Similarly,
In expanding equation (23) to obtain (24), it should be noted that in using the dummy index scheme there are restrictions such that cc # a’ and t3 # 19’. These
x
e,,2$ 2
~3(a8)[~3(~B) -
11 (28)
(30)
ALEX
AND McLELLXN:
THERMODYKAMIC
PROPERTIES
Similar methods were used to evaluate the other terms in equation (24). When all the results were collected and the ~pprop~&te expressions for B’s found using equation (20), we find that, (Z2> -
(Zj2 = $2
zs(~~)~=~~(l -
@N
= 12~,82(1 -
-
OF
INTERSTITIAL
111
SOLIDS
The coefficient of the concentration dependent term (&“tbP&P> in the first summation in equation (37) ca.n be written as,
@a)
ey
(31)
Now consider the third term in equation (22), i.e. { (YZ) - (Y)(Z)]. The factor (Y) is already known.(2) YZ is found as follows:
=
2
z3(a#?)z2(a)
(39)
where the coordination number z2(a) is the number of sites which are second nearest neighbors to a given octahedral site on the same sublattice [z2(a) = 61. Similarly,
Thus,
and
(3%. )No
= ~~(a~)z2(a) * -i_ -
2, 4
.
(39)
Thus, combining equations (37)-(39),
and since
+ (S$S:))
-
(X$>(8$s’)}
(36)
and
Consider the term (~~~~~)}. We have,
and
it, is easy to show, by computing the B’s and rearranging, that (Ls$F?~~‘> - (~~~}(~(2)) Di = 0
so that
and
(41)
so that (YZ) - (Y)(Z) = 0 and the third term of equation (26) makes no contribution. It should be pointed out that in order to show that (YZ) (Y)(Z) is identical to zero, every term in the expansions for e lr,2 and 0E,3has to be considered. This is also true with respect to the evaluation of other similar moments. Similar techniques show, however, that the second term in equation (22) is non-zero, and in fact (XY)
-
(X)(Z)
= -9SiV*e4.
(42)
ACT_%
113
METALLURGICA.
VOL.
21.
1973
Combining equations (31), (41) aud (42), we find the ~ont~butiou to As due to t,hird nearest neighbor interactions is, i&2{12N,P( 1 -
e)s) + 2t,{ -96
N,84)
(43)
6
and combining this quantity with the contributions from first and second nearest neighbor interactions’s) we have the total cumulant, As = 6NW02(l + Izea(l -
0)s -j- 3N~~~~e2(1e)j + t,2{izlv,eyi
3
0)s -
(44)
It should be noted that, since in 2, the cross term in t,t, vanishes, the effect of the third neighbor interactions is additive to F,c and independent of t,. Tke dhird moment
/ Euu=-Zk.cal/mole
0
A preliminary attempt at a rigorous calculation of the contribution of third nearest neighbor interactions to -&Isshowed that the time required would be excessive. However, an approximate calculation of the largest contributions to this term showed that for any reasonable interaction strength the concomitant ~ont~bution to the free energy of the solution crystal would be small compared with the contributions arising from Ml and M,. No attempt was made to calculate the effect of considering third nearest neighbor interactions on the fourth moment. It has however already been demonstrated that the contribution of second nearest neighbors to M4 can be neglected.(s) THERMODYNAMIC
-
0.02
--/
0.04
Wf(0,
a, t,, ts)
(44)
where B,” is the partial enthalpy of the solute atom in the infinitely dilute solution and the function f(0, a, t,, t2) contains all the terms resulting from the
I
0.06
0.10
0.08
OFIG. 2. Variation of the partial con@urational of a solute atom with composition.
entropy
differentiation of the n(0) with respect to N,. The partial configurational entropy and enthalpy are obtained from equation (44) by the standard thermodynamic relations in the form,
fl,C = -k In
a, tr, t2)
(45)
FUNCTIONS
The configurational Helmholtz free energy Fuc is obtained by inserting the values of 1 into equation (7). The contributions to the first four cumulants due to first neighbor interactions, and to the first three cumulants due to second nearest neighbor interactions are given in the previous calculation.(2) The contributions to the first two moments arising from third nearest neighbor interactions are given in this report. Differentiation of F,” with respect to N, at constant T and V yields an expression for the configurational chemical potential ,uUcof a solute atom in the interstitial solutions of the form 8 Puc = aUm + kT In 1--e
!-\
e)y
+ 2t,(-96Nv84)
THE
A
A, _ j&‘= RESULTS
I=:
kT2 AND
we,
(46)
DISCUSSION
The partial entropy The variation of the partial configurational entropy with composition is depicted in Fig. 2. For the sake of comparison, the curve for the ideal solid solution is also included. Values of t, = 0.1 and t, = 0.05 and T = BLOC were chosen. It can be seen that, for repulsive solute mutual interactions, even when E = 3 kcal/mole, the resulting tendency toward a pLtia1 state of ordering has only a small effect on the confi~rational entropy. On the other hand, computing SUOfor an attractive interaction of E,, = -2 kcal/mole shows a considerable reduction in the configurational entropy due to the tendency toward cluster formation. This result is to be expected and can be compared directly to the effect of the formation
ALEX
AND
McLELLAS:
THERMODPKAMIC
PROPERTIES
OF
IXTERSTITIAL
SOLIDS
113
vanish when, &-
u
{t,2[12iVu82(1 -
19)~]+ 2t,( -
96N,04)}l,.u = 0
This occurs when, t, =
32e2 i -
38 + 282
For 8 = 0.05, this critical value of t, is 0.094. This is clearly seen in Fig. 3. The partial enthalpy
-0 ‘5’ -02(
/ 0
0.02
I
I
I
0.04 0.06
0.08
I
I
0.10 0.12
12
FIG. 3. Effect of including third nearest neighbor interactions of varying strength on the partial configurational entropy of a solute atom.
of interstitial solute atom clusters surrounding substitutional solute atoms in ternary solid solutions. It has been shown that this kind of clustering decreases the partial configurational entropy of the interstitial solute species much more than the decrease caused by the ordering tendency due to repulsive interactions between the interstitial and substitutional solute atoms.dm2’ The flue-values calculated for 1000°C using a series of positive and negative e,,-values and including only first and second neighbor interactions were close to those values found when the third nearest neighbor interactions were included. The effect is seen in Fig. 3 which depicts plots of the difference, Sfl,, between fiUc including, and not including, third nearest neighbor interactions, varying with the strength of the third neighbor interaction. The variation of Ss, with t, is symmetrical about Sfl, = 0 for repulsive and attractive interactions of the same magnitude. However, the most pertinent result is that the effect on the configurational entropy of considering third nearest neighbor interactions is small, even when t, is larger than t,. The zero value of 6#, for a given value oft, is a consequence of the absence of the cross term involving t,t, in I,. This can be seen in equation (43). Since the contribution to Fuc due to third nearest neighbors is linear in A,, the contribution to fl, is proportional to (WWJ,vV~ Th us, for any value of a and 8, Sfl, will
The effect of including third nearest neighbor solute atom interactions on the partial configurational enthalpy is however much larger than the effect on the entropy. This is illustrated in Figs. 4 and 5. Figure 4 shows the variation of B,, with composition for a repulsive interaction of E,,, = 1.0 kcal/mole. For the sake of comparison, the corresponding linear variation of B, with 0 pertaining to the simple zerothorder solution model is included.os) It is clear that, when t, = $t, (although this may represent an upper limit), the third nearest neighbor interactions have a large effect on BUGand should not be ignored. Similar behavior is apparent in Fig. 5 (for E,, = 2 kcal/mole). Figure 5 also contains HUGcalculated from the firstorder solution model.04) Although the variation with 8 of Due in the more refined calculations is not linear, the curvature is quite small and accuracy limits of
II
‘=0.0 0.6
04 0.2 0
0.01
a.02 0.03
e-
0.04
0.05 0.06
0.07
FIG. 4. Variation of the partial configurational enthalpy of a solute atom with composition.
ACTA
114
METALLURGICA,
I I I I I 1000°C First n.n. only ,.*_ ----- Inclu&ng second n.n. with t, =O.I ~~~=ilk col/mole -_-+-* Including third n.n. 1.6with t, =O.l, tz =0.05 -I-I- Zeroth approximation
2.0
I
VOL.
21,
1953
although gWC is an explicit function of temperature, the functional form is such that for reasonable values of E,u, the variation of B,c in the temperature range 700-1100°C is too small to be depicted graphically. The tem~rature variation of BEeCin the momentexpansion calculations is shown in Fig. 6. Again the variation with temperature is quite small and probably undetectable experimentally except for data covering wide t,emperature and composition ranges. It should perhaps be mentioned that the closed expressions for c%, depend only on a and not on E,,,, and T independently. The plots in Fig. 6 have been given however because the values of E,, and T chosen 0.5-
I
I
0.02 I
0.04 I
I
I I T = 1000°C
0.4-
0
0.01
0.02 0.03
0.04
0.05
0.06 0.07
eFIG.5. Variation of the partial configurational ent,halpy of a solute atom with composition.
most thermodynamic data are probably such that this nonlinearity could not be detected experimentally. This linear (or virtually linear) variation of B,,c with 0 has been found in the recent data for carbon austenite.(15,16) The zeroth-order model predicts that E,c is independent of temperature. In the first-order model, -0.4-
-“‘50 1.6
0.06I
0.08 f
0.10 I
C 2
t2 -
FIG. 7. Effsct of including third nearest neighbor interactions of varying strength on the psrtial oonfigurationai enthalpy of a solute at,om.
al I.4 z E 1 1.2 z J 1.0 .c 8!0.6
I ‘==0.6 Partial Entholpie for t,xO.f ond t, Units for k. col/mole
0.02
0.04
0.06
0.08
0.10
EUU a
0.12
e-
FIQ. 6. Temperature variation of the partial configure. tional enthalpy of E solute atom.
have been shown to be appropriate for some interstitial solid solutions.~z*l*,l*) Figure 7 shows the virtually linear effect of increasing the third nearest neighbor interaction strength on the difference, c%,, between i7,c oalculated by including and omitting third nearest neighbor interactions. Note that in general @, is two orders of magnitude larger than SS,. The general technique of Kirkwood expansion is useful and, as shown, capable of enabling the effects of in~raGtions of a given strength between solute atoms in various neighbor shells on the thermodynamic properties of the solution to be calculated accurately. There are two immediately obvious extensions of this
ALEX
THERMODTBAMIC
AND McLELLAS:
work which are currently
receiving
attention.
Com-
puter simulation
studies are being made in the hope of
throwing
light
more
potentials
appropriate
atoms
various
in
on the for
configurations
At the same time the Kirkwood being
applied
where
the
complexity
to
forms
small
interstitial
of non-cubic
(and interest)
of solute
in f.c.c.
lattices.
expansion technique is
b.c.c.-based
presence
of interatomic
groups
solutions
distortions
adds
to the problems.
ACKNOWLEDGEMENT
The
authors
provided
by
gratefully the
Robert
acknowledge A.
Welch
the
support
Foundation
(Grant C-281). REFERENCES 1. K. ALEX and R. B. MCLELLARActa Met. 19,439 (1971). 2. K. ALEX and R. B. MCLELLAN, Acla Met. 20,ll (1972).
PROPERTIES
OF
INTERSTITIAL
SOLIDS
115
3. R. H. SILLER and R. B. MCLELLA~, Acta illet. 19,193 (1971). 4. R. H. SILLER and R. B. MCLELLAS. Acta iVet. 19,671 (19il). 5. R. A. Joaasox, Phys. Rev. A134, 1329 (1964). 6. R. A. JOHNSON,Aeto Met. 15,513 (1967). 7. R. A. JOHNSON,G. J. DIExEs and A. C. DAMASK,Acta Met. 12, 1215 (1964). 8. C. WAGNER, Acta Alet. 19, 843 (1971). 9. H. BUCK and G. ALEFELD. Whys. Status Solidi B47, 193 (1971). 10. R. B. MCLELLA~, Mat. Sci. Engr 9, 121 (1952). 11. K. ALEX and R. B. MCLELLAS, J. Phys. Chem. Solids 32, 449 (19il). 12. K. ALEX and R. B. MCLELLAS, Mat. Sci. E?lgr 7, 77 119711. 13. lk A&X and R. B. MCLELLAN. J. Phys. Chem. Solids 31, 2751 (1970). 14. R. B. MCLELLAN and W. u’. Du~ix, J. Phys. Chem. sO1id830. 2631 (1969). 15. S. BAN-YA, J. F: ELLIOTTand J. CHIPMAN,!haw. metall. Sot. A.I.M.E. 245, 1199 (1969). 16. S. BAN-YA, J. F. ELLIOTT and J. Cmmfm, lllet !hatt.s. 1, 1313 (1970). 17. R. B. MCLELLAN and W. W. DUNN, Scripta Met. 4, 321 (1970). 18. K. ALEX and R. B. MCLELLAN,Scripta Alet. 4,964
(1970).