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Statistical thermodynamics of long-range order
Scripta
METALLURGICA
V o l . 17, pp. Printed in
53-58, 1983 the U.S.A.
Pergamon
Press
Ltd.
STATISTICAL THERMODYNAMICS OF LONG-RANGE ORDER
N. A. Gokcen Albany Research Center, Bureau of Mines, U.S. Department of the Interior Albany, Oregon 97321 ( R e c e i v e d A p r i l 22, 1 9 8 2 ) (Revised November i0, 1 9 8 2 )
Numerous statistical thermodynamic attempts have been made to formulate the behavior of long-range order (LRO) in alloys after the significant initial work of Gorsky, and Bragg and Williams (GBW) (see Ref. 1-7 at the end). The quasl-chemical method (Q. Ch.), largely due to Bethe (I) and to Guggenheim (2) and based on the permutation of bonds, is often erroneously claimed to be an improvement over the GBW method. The permutation of bonds in the Q. Ch. method is made to become the permutation of atoms for zero exchange energy, or when the temperature is sufficiently high, through a dubious normalization process. It has been shown conclusively by Gokcen and Chang (8, 9) that (i) the normalization process of the Q. Ch. method yields the same equations for the excess Gibbs energy of solution as without normalization, and (ii) the actual enumerations of configurations (8, I0) prove conclusively that the permutable entities are the atoms, not the bonds. The purpose of this communication is twofold: (1) to show that the permutation of atoms yields proper new equations for order-disorder phenomena after reasonable corrections, and (ii) to discuss the errors in the previous regular assembly treatments (i, 2). We consider a binary alloy of equiatomic composition for simplicity in our formulation because the results can easily be extended to nonequiatomic compositions. We limit our discussion to the body-centered cubic structure such as that in the Cu-Zn system; however, the present procedure can be extended to any system. Let N A and N B be the number of atoms of A and B respectively, and N A = N B = L so that 2L is one gram atom of an alloy. Thus the symbol L also represents the number of lattice sites on either a-sublattice (a-sites) for A atoms, or b-sublattice (b--sites) for B atoms, b e a r i n g in mind that in a perfectly ordered alloy all N A are on the as u b l a t t [ c e , and all N B are on the b - s u b l a t t [ c e . Let R be the actual number o~ A atoms on asites, and Q be the remaining A atoms on b-sites, and let the number of nearest neighbors to an atom be Z. The occupancy of B atoms is similar, i.e. R is the number of B atoms on b-sites and Q is the remaining B atoms on a-sites. The long-range order parameter r is defined by R
~(I+r);
Q = ~(l-r)
;
(~
R ~ L; O $ O ~ ~).
(I)
If X is the net number of A atoms on a-sites whose neighbors are A atoms, then R - X is the net number of atoms surronnded by B atoms. In su~nary, for all the atoms, we have (i) (ii) (iii) (iv)
A on A on A on A on Total
a-sites with A neighbors = X, a-sites with B neighbors = R - X, b-sites with A neighbors = X, b-sites with B neighbors = Q - X, number of A atoms = R + Q = L.
(v) B on b-sites (vi) B on b-sites (vii) B on a-sites (viii) B on a-sites Total number of
with B neighbors = X, with A neighbors = R - X, with B neighbors = X, with A neighbors = Q - X, B atoms = R + Q = L.
The net numbers X, R - X, and Q - X represent the numbers of atoms such that when each of these quantities is multiplied by Z/2, which is the number of bonds per atom, we obtain the number of corresponding types of bonds. This concept, based on the previous publications (8, 9) assures that the permutable entities are the net numbers of each type of atoms, conceived to have only one type of neighbors. The correction required to equate the actual permutations to the permutations based on the net numbers of atoms is negligible. This concept, fully justified earlier (8-10) is essential in pursuing the succeeding arguments. The permutation of these atoms yields
53 0036-9748/83/010053-06503.00/0
54
LONG-RANGE
ORDER
Vol.
17,
No.
L~ L{ D = (R_X)!(X!)2(Q_X) ! (R_X) i(XI)2(Q_X)!,
1
(2)
where the first factor is the permutation of L atoms on a-sites consisting of the types given by (i), (ii), (vii), and (viii), and likewise, the second factor is for L atoms on b-sites. For X = O, D(X = O) = (L!/R!Q!).(L!/R!Q!), which
is the distribution
equation
used in the GBW method.
(3)
We use the Stifling
approximation,
(X!) 2 = (2X)!(2) -2X, etc., and
Eq.
(I)
to
convert
(4)
Eq.
(2)
into
D =
L{ L! .(,.2.)._2L ....... (L + Lr - 2X)![(2X)!]2(L - Lr - 2X)!
(5)
When r = i, then R = L; hence, both X and Q must be zero since there are no A-A bonds, and D = i as expected from Eq. (5) as well as from Eq. (3). Conversely, if X = 0 or if there are no A-A bonds, R has to be equal to L by geometrical requirements because when R is less than L it can he easily shown by one- and two-dimenslonal crystal constructions that X cannot remain zero if the permutation given by Eq. (3) were carried out for other values of R than R = L. The GBW method assumes that R and Q may have values unrestricted by the values of X so that Eq. (3) is valid for any value of X, and this assumption is analogous to the random distribution permitted in the zeroth approximation to the long-range disordered (LRD) regular solutions. All A atoms have ZN /2 bonds b e l o n g i n g to A atoms. If for convenience we define Y such that EY/2 are the A-B b3~ds emanating from A atoms, then th~ r ~ bon-ds are the A-A b o n d s ; hence (A-A bonds) + ZY/2 = ZNA/2, but A-A bonds are equal to (2X)Z/2 from (1) and (iii); consequently 2X + Y = N A = L.
(6)
[Cf. Eq. (17) of Ref. 8]. For the long-range disordered alloy (LRD), r is zero but the solution is not necessarily random, and substitution of r = 0 in Eq. (5) yields D = ----L-L!L! ~2-1~-L[(L-y) ! ] 2(y!) 2"
(7)
The factor (2) 2L, or 4 L, should be unity when r = 0 as required by the correct first approximation to the LRD regular solutions [cf. Eq. (15) of Ref. 8] based on actual enumerations of configurations. Further, all the possible permutations of all atoms are equal to (2L)!/(L!L!), and we note, as a simple example with r = 0.2 and 2X = 0.2L, that Eq. (5) gives = D
L!L{22L
[( 2~)--'t { >> = 22L L!(O.2L{)2(0.6L! ) [ L!L[
(8)
and this result is simply impossible. The factor 22L=4 L cannot be constant for a given value of L; it should therefore be a function of r, i.e. iF(r)] L where F(r) = 4 for r = i, and F(r) = i, for r = O. This requirement now unifies the equations for LRO and LRD to one equation: L!Li[F(r)]) L D =
iF(r)=4 (L + Lr
- 2X)![(2X)!]2(L
for
r=l;
F(r)=l
for
r=0]
(9)
- Lr - 2X)!'
The energy E of the alloy is Z e / 2 times each term in (i) - (viii) with the subscripts and q representing A and B, and epq b e i ~ the bond energy; thus, E
where W, the exchange energy,
is defined by
=
ZLeAB- 2XW
p
(i0)
Vol.
17,
No.
i
LONG-RANGE
ORDER
55
(it)
W = (Z/2)(2eAB-eAA-eBB). The partltfon
functfon
(P.F.)
can be written P.F.
by using Eq.
(9) and (i0)
in the usual
form:
=x!0De -E/kr,
(12)
where k is the Boltzmann constant and T is the temperature in K. The maximum term in this equation is obtained by setting b~n(P.F.)/DX to zero where ~n denotes the natural logarithm; the result is (R - X)(Q - X) = X2e -W/kT= x2n; [n=e-W/kT]. (13) The value of X from this equation yields the maximum term in Eq. (12), the term that replaces the summation in Eq. (12). The solution for X from Eq. (13) and substitution from Eq. (i) for R and Q, and then simplification yield 2X I - r2 -~" = l--+--[iV-$1--r2i-(-~_l~,5 where b is used
for brevity
I - r2 = l--#--b--'
(14)
to denote [I + (l-r2)(n-l)] 0.5 = b.
(15)
We now digress from the usual derivation for the Gibbs energy G(r,T) at constant composition and constant r by following the same procedure as that in Ref. 8. We assume that the enthalpy H is nearly identical with E of Eq. (i0) for condensed phases and write T-I fT_l=0 m i n i
T-I = fT_l=0 H d(r-l),
(16)
where the integration is carried out at constant composition and constant r and T +~ is written as T -I = O. The lower integration limit for the left side is [G--(-r-~l~]r_1=O
= [~)r_l=o
for
- S(r,r-l=o).
simplicity
(17)
Since H = E is finite according to Eq. (i0), H/T is zero for T -I = 0, and the next term in Eq. (17) can be obtained from Eq. (9) by writing S(r,T -I = 0) = k%nD. The value of X for T -I = 0 can be obtained from Eq. (13) by setting n = exp (-W/kT) = I for T -I= 0. The result is x(r -I = O) = RQ/L = (l-r2)L/4. Substitution
of Eq.
S(r,T-I=0)
(18) in Eq.
(9) and in S(r,T-I=o)
= k%nD = Lk%nF(r)
= k~D
- 2Lk(l+r) in(l+r)
(18)
yields
- 2Lk(l-r)%n(l-r)
+ 2Lk%n2.
(19)
For the right side of Eq. (16), the integration procedure is similar to that used for the LRD regular solutions (2,8,9) having r = 0. For integration of the right side of Eq. (16) we express d(T -I) by using Eq. (15): bdb - ~_--~ _
d(- W T ) Substitution
of Eq.
(I0),
G(r,T) T Substitution
of Eq.
+
value
+ (l+r)L
(21) and rearrangement
- ~nF(r) + (l+r)~n(l+r)(b+r) of r is obtained i
(20)
and (20) in (16) and integration = Lk
(19) in Eq.
= ~kT The equilibrium
(14),
•
by setting
by parts yield
+
of the result
2
.
(21)
give
+ (l-r)~n(l-r)(b-r)
- 2~n(l+b).
(22)
~G/~r = 0:
~G i .dF(r) rr l+r ~r b+r ~] + %n[Ll-~J~-m-fJJ = O, ~F = -F-(-r)-- r ~
(23)
56
LONG-RANGE
ORDER
Vol.
17, No.
I
where all the remaining terms including those containing Db/Dr cancel out. For r = 0, this equation requires that dF(r)/dr be zero since F(r=0)=l according to Eq. (9). Differentiation of Eq. (23) and thereafter substitution of r = 0 at the critical temperature gives --[--- ~-2-G-= - d-2-%n-F-[r3-+ 2 + 2 = 0; (r = 0, T = rc, b c = b at Tc). LkT c Dr 2 dr 2 bc For
r = O, Eq.
(15) gives b c = n 0"5 = exp (-W/2kT),
and the rearrangement
%n[0.5 d-2-~n-F- - I] = W _ _ ; dr 2 2kTc We now d[gress disordered solutions
briefly to show when r = O, i.e.
that Eq.
(22)
of Eq. (24) yields
(r = 0, r = Tc).
is also
applicable
(24)
(25)
to the ordinary
G~r_--O_,T) = Z e__AB_ %!_- 2%n(l+b). LkT kT kT
regular
(26)
For an ideal equiatomic solution the enthalpy H is equal to the enthalpy of pure components, H =(eAA + e B B ) Z L / 2 , and the c o n f l g u r a t i o n a l entropy is the ideal entropy of the alloy so that S(ideal) = 2Lk%n2; hence the ideal Gibbs energy of solution is G(id__eal)_LkT = (eAA+ e B B ) ~ k T Subtraction of this equation from Eq. (26) and substitution lowing excess Gibbs energy of solution (II), Ge: Ge _ . . . . 2~n2 - 2%n(l+b) LkT This equation,
where 2Lk is the gas constant;
2~n2. for W from Eq.
(27) (ii) yields
e 2~-~= 22nI----~-- I.
~+e~~
the fol-
(28)
is identical with Eq. (45) of Ref. 8.
We return to Eq. (23) and (25) and their application to the Cu-Zn system for which some data are available. The value of W/k is the only parameter permitted to be determined from experimental data in any approximation to the regular LRD solutions. The compilation of Hultgren et al. (12) e x t r a p o l a t e d a short distance to the equiatomic composition yields Ge/2LkT = -1.202 at 773 K, which substituted in Eq. (28), gives W/k = -2678. The regular solution model assumes that W/k is independent of temperature. The critical temperature T c is 742 K. The explicit functional form of F(r) is not known and its determination would require extensive and laborious enumeration of configurations for one- and two-dimensional crystals as in Ref. (i0). Such a task is expected to be very difficult and exceedingly time-consuming. Meanwhile, we suggest tentatively the following equation with three parameters, after having attempted the use of numerous other equations: %nF(r) = g(l+r-frm)%n(l+r-fr m) + g(l-r+frm)£n(l-r+frm).
(29)
The parameters f, g, and m in this equation must satisfy three conditions required by F(r=0)=l, F(r=l)=4, and Eq. (25) at T c. Substitution of %nF(r) in Eq. (25) and then setting r=0 yields g = 1.16454, and then using this result in %nF(r=l)=%n4 gives f=O.040. The values of f and g are independent of m, and since f is small, fr m is relatively small for very large values of m when r < I. Therefore, we neglect the term fr m for arbitrarily and adequately large values of m for r < I, and rewrite £nF(r) = g(l+r)%n(l+r)
+ g(l-r)%n(l-r).
(3o)
Substitution in Eq. (23) yields the values of r at various temperatures as shown in Fig. i. There are limited sets of data available for wide ranges of values for r versus T/T c but not for Cu-Zn, and such data for other systems appear to favor (7, 13) the curve resulting from Eq. (23). The functional form of F(r), obtained without recourse to Eq. (23) and (30), is exceedingly sensitive to all the results obtained by using F(r) in the related equations. For example, if we
Vol.
17,
No.
i
LONG-RANGE
....
Author Zeroth . approximahon
ORDER
57
1.0 .8
~. '~\
.6 .4 .2
0
I
1
I
0.2
0.4
0.6
KEY
\
~'~
I
4
--'--
._~
.....
0.1 3 ~.
...........
\
1
0.8
T/T c FIG. I Variation of long-range order parameter with temperature. T c is 742 K. Curve for quasichemical approximation is not shown because it follows zeroth approximation curve very closely.
I0
~_)
Experpmentol Autho r Zerot~ opprox
~g*
or,
I /
1,~as - c n e m , : ~ aporox,mot or
/
0.4
05
06
0.7 0 8 T/T c
0.9
10
I
FIG. 2 Variation of heat capacity with temperature for equiatomic Cu-Zn alloy. T c is 742 K, and measured heat capacities of pure components were subtracted from that of alloy to obtain experimental curve.
impose g = I on F(r) in Eq. (30), then Eq. (25) would yield b = % i.e. W/k = - % and if we set g = i.I, then b ffi I0, i.e. W/k ffi -3417. A similar sensitivity is observed when F(r) is expressed as a power series in r. The heat capacity C divided by 2Lk, C/2Lk, involves the derivatives of [~(G/T)]/[~(I/T)] with respect to T and r, and therefore it is even more sensitive to the functional form of F(r) than b, G, and H. The results for Cu-Zn are shown in Fig. 2, and compared with the closely concordant experimental values of Moser (14), and Sykes and Wilkinson (15). It is possible to duplicate the experimental results for the heat capacity by using a power series for F(r) with the coefficients satisfying Eq. (9), (23) and (25) but such an equation would have only an interim utility until a firm statistical significance can be attached to F(r) by enumeration of configurations as mentioned earlier (8, I0). We now justify F(r) in Eq. (30) by way of summary of the foregoing procedure. Eq. (2), or (5) is exact only when rffil, which makes XffiO, and Eq. (3) is applicable only when XffiO, or RffiL, i.e. when perfect order exists. As a result F(rffil) must be 4. When r=0, previous work in Ref. 8 and I0 by actual enumerations of configurations show that F(rffi0) must be I, and between i and 4 for this function it is possible to start with any value of X dependent on Y in Eq. (6), where Y is determined by W and T as in Eq. (27) of Ref. 8. Since for any starting value of X at r=O to rffil, F(r) still varies between i and 4, F(r) should be independent of X. The functional form of F(r) should be such that, in addition to F(rffil)ffi4 and F(rffiO)ffil, it should satisfy Eq. (23) so that dF/dr10 when rffiO, and further, d2~nF/dr 2 must satisfy Eq. (24) when r=0. These stringent requirements are likely to make determination of F(r) by statistical means an extremely difficult task. At present, consistent results can be obtained only with any function of r satisfying the foregoing requirements, and Eq. (30) may therefore be regarded as a useful preliminary equation. Comments on Existin~ Approximations: The Q. Ch. method assumes that the bonds associated with (i)-(vlii) are permutable, i.e. each factor such as X! in Eq. (2) should be written as (ZX/2)!. The resulting D is then normalized so that for X = X 0 = RQ/L, D becomes identical with our Eq. (3) for the GBW method. However, Eq. (3) is valid only for X = 0 and not for X 0 = RQ/L as discussed earlier; therefore, the normalization process contains a significant contradiction. In addition, it was shown by Gokcen and Chang (8-10) that the permutation of bonds is incorrect since it creates impossible spliced molecules and contradicts the numbers of configurations obtained by actual enumerations. Further, as a trival matter and contrary to simple geometrical facts, the Q. Ch. method predicts that there can be no order when Z = 2. Therefore, the Q. Ch. method, despite the claims by numerous investigators, is not an improvement over the GBW method.
58
LONG-RANGE
ORDER
Vol.
17,
No.
1
The value of W/k for an ordered solution is calculated from the critical temperature T c in the GBW and Q. Ch. approximations. For the same solution in the disordered state, W/k is determined from experimental results, and we now proceed to show that each of these methods leads to inconsistent results within itself. For Cu-Zn, the GBW method requires 2kT c = -W(GBW) from which W(GBW)/k = -1484, and the zeroth approximation to the regular LRD solutions, the analog of GBW approximation, yields W(zeroth)/k = -3717 from Ge/2LkT = -1.202 with T = 773 and G e = 2LXA%B W = 0.5 LW, (YA = mol fraction of A = 0.5 = ~B). The Q. Ch. approximation for ordered Cu-Zn gives W/k = -1708 from W/k = TcZ%n[(Z-2)/Z], and the same approximation for the disordered solution gives W/k = -3285 from Ge/2LkT = -1.202 = (Z/2)%n{2 exp(W/ZkT)/[l+exp(W/ZkT)]}, with T = 773 and Z = 8. Similar calculations for other binary systems show similar discrepancies. The results for r versus T/T c and for C/2Lk versus T/Tc, obtained by using Tc=742 K, are presented in Fig. i and Fig. 2 for comparison. A significant improvement over the Q. Ch. method is the cluster variation method developed (16) and perfected (17-19) by Kikuchi. A simplified version of the earlier development by Kikuchi was presented by Guggenheim and McGlashan (20). The simplified version for the square as the cluster would require writing Q. Ch. equilibria and their pseudo-equillbrium constants for pseudoreactions such as
and accounting for all the possible lower hierachy of configurations down to single bonds and points. While with increasingly complex clusters the method would minimize the error originating from the permutation of single bonds and formation of unrealistic spliced atoms, clearly a great number of different clusters must be used as shown in Ref. I0. Further, much larger and varied clusters than those used by Kikuchi pose vastly increasing mathematical complexities (21). As a concluding remark, we wish to emphasize that the permutation promises the optimal degree of realistic success. The main reason for the resulting function for D of Eq. (9) would be obtained by deductive enumeration of D for one- and two-dimensional crystals as shown in Ref.
of "net numbers of atoms" this possibility is that reasoning based on actual I0 for L ~I00.
Statistical thermodynamics of long-range order is part of the Bureau provide a scientific base for use in developing new alloys and alloy phases. References i. 2. 3. 4. 5. 6. 7. 8. 9. I0.
ii. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
of Mines
program
to
H. A. Bethe, Froc. Roy. Soc. ALSO, 552 (1935). E. A. Guggenheim, "'Mixtures," Oxford Univ. Press, Chapter IV and VII (1952). R. Fowler and E. A. Guggenheim, "Statistical Thermodynamics," Cambridge Univ. Press, Chapter XII (1956). M. A. Krivoglaz and A. A. Smirnov, "The Theory of Order-Disorder in Alloys," Macdonald, London (1965). L. Guttman, Solid St. Phys. 3, 146 (1956). I. Prigogine, "The Molecular Theory of Solutions," North Holland, Amsterdam (1957). D. de Fontalne, Solid St. Phys., 34, 73 (1979). N. A. Gokcen and E. T. Chang, J. Chem. Phys., 55, 2279 (1971). N. A. Gokcen and E. T. Chang, Scripta Metall. 4, 941 (1970). N. A. Gokcen and E. T. Chang, "A New Method for Enumerating Molecular Configurations in Propellant Mixtures," Aerospace Report No: TR-0172 (2210-10)-1, Aerospace Corp., E1 Segundo, Calif. (1971). N. A. Gokcen, "Thermodynamics," Techscience, Inc., Chapter XI, Hawthorne, Calif. (1975). R. Hultgren, P. D. Desal, D. T. Hawkins, M. Glelser, and K. K. Kelley, "Selected Values of the Thermodynamic Properties of Binary Alloys," Am. Soc. Met., Metals Park, Ohio (1973). L. M. Falicov and R. C. Kittler, "Theory of Alloy Phase Formation," L. H. Bennett, Ed. Conference Proc., AIME; p. 303 (1980). H. Moser, Physikal. Zeit. 37, 737 (1936). C. Sykes and H. Wilkinson, J. Inst. Met. 61, 223 (1937). R. Kikuchi, Phys. Rev. 81, 988 (1951). R. Kikuchi and S. G. Brush, J. Chem. Phys., 4_7_7, 195 (1967). R. Kikuchi and C. M. van Vaal, Scripta Metall. 8, 425 (1974). R. Kikuchi and H. Sato, Acta Metall., 22, 1099 ~1974). E. A. Guggenhelm and M. L. McGlashan, Molec. Phys. 5, 433 (1962). R. Kikuchi, J. Chem. Phys. 60, 1071 (1974).
METALLURGICA
V o l . 17, pp. Printed in
53-58, 1983 the U.S.A.
Pergamon
Press
Ltd.
STATISTICAL THERMODYNAMICS OF LONG-RANGE ORDER
N. A. Gokcen Albany Research Center, Bureau of Mines, U.S. Department of the Interior Albany, Oregon 97321 ( R e c e i v e d A p r i l 22, 1 9 8 2 ) (Revised November i0, 1 9 8 2 )
Numerous statistical thermodynamic attempts have been made to formulate the behavior of long-range order (LRO) in alloys after the significant initial work of Gorsky, and Bragg and Williams (GBW) (see Ref. 1-7 at the end). The quasl-chemical method (Q. Ch.), largely due to Bethe (I) and to Guggenheim (2) and based on the permutation of bonds, is often erroneously claimed to be an improvement over the GBW method. The permutation of bonds in the Q. Ch. method is made to become the permutation of atoms for zero exchange energy, or when the temperature is sufficiently high, through a dubious normalization process. It has been shown conclusively by Gokcen and Chang (8, 9) that (i) the normalization process of the Q. Ch. method yields the same equations for the excess Gibbs energy of solution as without normalization, and (ii) the actual enumerations of configurations (8, I0) prove conclusively that the permutable entities are the atoms, not the bonds. The purpose of this communication is twofold: (1) to show that the permutation of atoms yields proper new equations for order-disorder phenomena after reasonable corrections, and (ii) to discuss the errors in the previous regular assembly treatments (i, 2). We consider a binary alloy of equiatomic composition for simplicity in our formulation because the results can easily be extended to nonequiatomic compositions. We limit our discussion to the body-centered cubic structure such as that in the Cu-Zn system; however, the present procedure can be extended to any system. Let N A and N B be the number of atoms of A and B respectively, and N A = N B = L so that 2L is one gram atom of an alloy. Thus the symbol L also represents the number of lattice sites on either a-sublattice (a-sites) for A atoms, or b-sublattice (b--sites) for B atoms, b e a r i n g in mind that in a perfectly ordered alloy all N A are on the as u b l a t t [ c e , and all N B are on the b - s u b l a t t [ c e . Let R be the actual number o~ A atoms on asites, and Q be the remaining A atoms on b-sites, and let the number of nearest neighbors to an atom be Z. The occupancy of B atoms is similar, i.e. R is the number of B atoms on b-sites and Q is the remaining B atoms on a-sites. The long-range order parameter r is defined by R
~(I+r);
Q = ~(l-r)
;
(~
R ~ L; O $ O ~ ~).
(I)
If X is the net number of A atoms on a-sites whose neighbors are A atoms, then R - X is the net number of atoms surronnded by B atoms. In su~nary, for all the atoms, we have (i) (ii) (iii) (iv)
A on A on A on A on Total
a-sites with A neighbors = X, a-sites with B neighbors = R - X, b-sites with A neighbors = X, b-sites with B neighbors = Q - X, number of A atoms = R + Q = L.
(v) B on b-sites (vi) B on b-sites (vii) B on a-sites (viii) B on a-sites Total number of
with B neighbors = X, with A neighbors = R - X, with B neighbors = X, with A neighbors = Q - X, B atoms = R + Q = L.
The net numbers X, R - X, and Q - X represent the numbers of atoms such that when each of these quantities is multiplied by Z/2, which is the number of bonds per atom, we obtain the number of corresponding types of bonds. This concept, based on the previous publications (8, 9) assures that the permutable entities are the net numbers of each type of atoms, conceived to have only one type of neighbors. The correction required to equate the actual permutations to the permutations based on the net numbers of atoms is negligible. This concept, fully justified earlier (8-10) is essential in pursuing the succeeding arguments. The permutation of these atoms yields
53 0036-9748/83/010053-06503.00/0
54
LONG-RANGE
ORDER
Vol.
17,
No.
L~ L{ D = (R_X)!(X!)2(Q_X) ! (R_X) i(XI)2(Q_X)!,
1
(2)
where the first factor is the permutation of L atoms on a-sites consisting of the types given by (i), (ii), (vii), and (viii), and likewise, the second factor is for L atoms on b-sites. For X = O, D(X = O) = (L!/R!Q!).(L!/R!Q!), which
is the distribution
equation
used in the GBW method.
(3)
We use the Stifling
approximation,
(X!) 2 = (2X)!(2) -2X, etc., and
Eq.
(I)
to
convert
(4)
Eq.
(2)
into
D =
L{ L! .(,.2.)._2L ....... (L + Lr - 2X)![(2X)!]2(L - Lr - 2X)!
(5)
When r = i, then R = L; hence, both X and Q must be zero since there are no A-A bonds, and D = i as expected from Eq. (5) as well as from Eq. (3). Conversely, if X = 0 or if there are no A-A bonds, R has to be equal to L by geometrical requirements because when R is less than L it can he easily shown by one- and two-dimenslonal crystal constructions that X cannot remain zero if the permutation given by Eq. (3) were carried out for other values of R than R = L. The GBW method assumes that R and Q may have values unrestricted by the values of X so that Eq. (3) is valid for any value of X, and this assumption is analogous to the random distribution permitted in the zeroth approximation to the long-range disordered (LRD) regular solutions. All A atoms have ZN /2 bonds b e l o n g i n g to A atoms. If for convenience we define Y such that EY/2 are the A-B b3~ds emanating from A atoms, then th~ r ~ bon-ds are the A-A b o n d s ; hence (A-A bonds) + ZY/2 = ZNA/2, but A-A bonds are equal to (2X)Z/2 from (1) and (iii); consequently 2X + Y = N A = L.
(6)
[Cf. Eq. (17) of Ref. 8]. For the long-range disordered alloy (LRD), r is zero but the solution is not necessarily random, and substitution of r = 0 in Eq. (5) yields D = ----L-L!L! ~2-1~-L[(L-y) ! ] 2(y!) 2"
(7)
The factor (2) 2L, or 4 L, should be unity when r = 0 as required by the correct first approximation to the LRD regular solutions [cf. Eq. (15) of Ref. 8] based on actual enumerations of configurations. Further, all the possible permutations of all atoms are equal to (2L)!/(L!L!), and we note, as a simple example with r = 0.2 and 2X = 0.2L, that Eq. (5) gives = D
L!L{22L
[( 2~)--'t { >> = 22L L!(O.2L{)2(0.6L! ) [ L!L[
(8)
and this result is simply impossible. The factor 22L=4 L cannot be constant for a given value of L; it should therefore be a function of r, i.e. iF(r)] L where F(r) = 4 for r = i, and F(r) = i, for r = O. This requirement now unifies the equations for LRO and LRD to one equation: L!Li[F(r)]) L D =
iF(r)=4 (L + Lr
- 2X)![(2X)!]2(L
for
r=l;
F(r)=l
for
r=0]
(9)
- Lr - 2X)!'
The energy E of the alloy is Z e / 2 times each term in (i) - (viii) with the subscripts and q representing A and B, and epq b e i ~ the bond energy; thus, E
where W, the exchange energy,
is defined by
=
ZLeAB- 2XW
p
(i0)
Vol.
17,
No.
i
LONG-RANGE
ORDER
55
(it)
W = (Z/2)(2eAB-eAA-eBB). The partltfon
functfon
(P.F.)
can be written P.F.
by using Eq.
(9) and (i0)
in the usual
form:
=x!0De -E/kr,
(12)
where k is the Boltzmann constant and T is the temperature in K. The maximum term in this equation is obtained by setting b~n(P.F.)/DX to zero where ~n denotes the natural logarithm; the result is (R - X)(Q - X) = X2e -W/kT= x2n; [n=e-W/kT]. (13) The value of X from this equation yields the maximum term in Eq. (12), the term that replaces the summation in Eq. (12). The solution for X from Eq. (13) and substitution from Eq. (i) for R and Q, and then simplification yield 2X I - r2 -~" = l--+--[iV-$1--r2i-(-~_l~,5 where b is used
for brevity
I - r2 = l--#--b--'
(14)
to denote [I + (l-r2)(n-l)] 0.5 = b.
(15)
We now digress from the usual derivation for the Gibbs energy G(r,T) at constant composition and constant r by following the same procedure as that in Ref. 8. We assume that the enthalpy H is nearly identical with E of Eq. (i0) for condensed phases and write T-I fT_l=0 m i n i
T-I = fT_l=0 H d(r-l),
(16)
where the integration is carried out at constant composition and constant r and T +~ is written as T -I = O. The lower integration limit for the left side is [G--(-r-~l~]r_1=O
= [~)r_l=o
for
- S(r,r-l=o).
simplicity
(17)
Since H = E is finite according to Eq. (i0), H/T is zero for T -I = 0, and the next term in Eq. (17) can be obtained from Eq. (9) by writing S(r,T -I = 0) = k%nD. The value of X for T -I = 0 can be obtained from Eq. (13) by setting n = exp (-W/kT) = I for T -I= 0. The result is x(r -I = O) = RQ/L = (l-r2)L/4. Substitution
of Eq.
S(r,T-I=0)
(18) in Eq.
(9) and in S(r,T-I=o)
= k%nD = Lk%nF(r)
= k~D
- 2Lk(l+r) in(l+r)
(18)
yields
- 2Lk(l-r)%n(l-r)
+ 2Lk%n2.
(19)
For the right side of Eq. (16), the integration procedure is similar to that used for the LRD regular solutions (2,8,9) having r = 0. For integration of the right side of Eq. (16) we express d(T -I) by using Eq. (15): bdb - ~_--~ _
d(- W T ) Substitution
of Eq.
(I0),
G(r,T) T Substitution
of Eq.
+
value
+ (l+r)L
(21) and rearrangement
- ~nF(r) + (l+r)~n(l+r)(b+r) of r is obtained i
(20)
and (20) in (16) and integration = Lk
(19) in Eq.
= ~kT The equilibrium
(14),
•
by setting
by parts yield
+
of the result
2
.
(21)
give
+ (l-r)~n(l-r)(b-r)
- 2~n(l+b).
(22)
~G/~r = 0:
~G i .dF(r) rr l+r ~r b+r ~] + %n[Ll-~J~-m-fJJ = O, ~F = -F-(-r)-- r ~
(23)
56
LONG-RANGE
ORDER
Vol.
17, No.
I
where all the remaining terms including those containing Db/Dr cancel out. For r = 0, this equation requires that dF(r)/dr be zero since F(r=0)=l according to Eq. (9). Differentiation of Eq. (23) and thereafter substitution of r = 0 at the critical temperature gives --[--- ~-2-G-= - d-2-%n-F-[r3-+ 2 + 2 = 0; (r = 0, T = rc, b c = b at Tc). LkT c Dr 2 dr 2 bc For
r = O, Eq.
(15) gives b c = n 0"5 = exp (-W/2kT),
and the rearrangement
%n[0.5 d-2-~n-F- - I] = W _ _ ; dr 2 2kTc We now d[gress disordered solutions
briefly to show when r = O, i.e.
that Eq.
(22)
of Eq. (24) yields
(r = 0, r = Tc).
is also
applicable
(24)
(25)
to the ordinary
G~r_--O_,T) = Z e__AB_ %!_- 2%n(l+b). LkT kT kT
regular
(26)
For an ideal equiatomic solution the enthalpy H is equal to the enthalpy of pure components, H =(eAA + e B B ) Z L / 2 , and the c o n f l g u r a t i o n a l entropy is the ideal entropy of the alloy so that S(ideal) = 2Lk%n2; hence the ideal Gibbs energy of solution is G(id__eal)_LkT = (eAA+ e B B ) ~ k T Subtraction of this equation from Eq. (26) and substitution lowing excess Gibbs energy of solution (II), Ge: Ge _ . . . . 2~n2 - 2%n(l+b) LkT This equation,
where 2Lk is the gas constant;
2~n2. for W from Eq.
(27) (ii) yields
e 2~-~= 22nI----~-- I.
~+e~~
the fol-
(28)
is identical with Eq. (45) of Ref. 8.
We return to Eq. (23) and (25) and their application to the Cu-Zn system for which some data are available. The value of W/k is the only parameter permitted to be determined from experimental data in any approximation to the regular LRD solutions. The compilation of Hultgren et al. (12) e x t r a p o l a t e d a short distance to the equiatomic composition yields Ge/2LkT = -1.202 at 773 K, which substituted in Eq. (28), gives W/k = -2678. The regular solution model assumes that W/k is independent of temperature. The critical temperature T c is 742 K. The explicit functional form of F(r) is not known and its determination would require extensive and laborious enumeration of configurations for one- and two-dimensional crystals as in Ref. (i0). Such a task is expected to be very difficult and exceedingly time-consuming. Meanwhile, we suggest tentatively the following equation with three parameters, after having attempted the use of numerous other equations: %nF(r) = g(l+r-frm)%n(l+r-fr m) + g(l-r+frm)£n(l-r+frm).
(29)
The parameters f, g, and m in this equation must satisfy three conditions required by F(r=0)=l, F(r=l)=4, and Eq. (25) at T c. Substitution of %nF(r) in Eq. (25) and then setting r=0 yields g = 1.16454, and then using this result in %nF(r=l)=%n4 gives f=O.040. The values of f and g are independent of m, and since f is small, fr m is relatively small for very large values of m when r < I. Therefore, we neglect the term fr m for arbitrarily and adequately large values of m for r < I, and rewrite £nF(r) = g(l+r)%n(l+r)
+ g(l-r)%n(l-r).
(3o)
Substitution in Eq. (23) yields the values of r at various temperatures as shown in Fig. i. There are limited sets of data available for wide ranges of values for r versus T/T c but not for Cu-Zn, and such data for other systems appear to favor (7, 13) the curve resulting from Eq. (23). The functional form of F(r), obtained without recourse to Eq. (23) and (30), is exceedingly sensitive to all the results obtained by using F(r) in the related equations. For example, if we
Vol.
17,
No.
i
LONG-RANGE
....
Author Zeroth . approximahon
ORDER
57
1.0 .8
~. '~\
.6 .4 .2
0
I
1
I
0.2
0.4
0.6
KEY
\
~'~
I
4
--'--
._~
.....
0.1 3 ~.
...........
\
1
0.8
T/T c FIG. I Variation of long-range order parameter with temperature. T c is 742 K. Curve for quasichemical approximation is not shown because it follows zeroth approximation curve very closely.
I0
~_)
Experpmentol Autho r Zerot~ opprox
~g*
or,
I /
1,~as - c n e m , : ~ aporox,mot or
/
0.4
05
06
0.7 0 8 T/T c
0.9
10
I
FIG. 2 Variation of heat capacity with temperature for equiatomic Cu-Zn alloy. T c is 742 K, and measured heat capacities of pure components were subtracted from that of alloy to obtain experimental curve.
impose g = I on F(r) in Eq. (30), then Eq. (25) would yield b = % i.e. W/k = - % and if we set g = i.I, then b ffi I0, i.e. W/k ffi -3417. A similar sensitivity is observed when F(r) is expressed as a power series in r. The heat capacity C divided by 2Lk, C/2Lk, involves the derivatives of [~(G/T)]/[~(I/T)] with respect to T and r, and therefore it is even more sensitive to the functional form of F(r) than b, G, and H. The results for Cu-Zn are shown in Fig. 2, and compared with the closely concordant experimental values of Moser (14), and Sykes and Wilkinson (15). It is possible to duplicate the experimental results for the heat capacity by using a power series for F(r) with the coefficients satisfying Eq. (9), (23) and (25) but such an equation would have only an interim utility until a firm statistical significance can be attached to F(r) by enumeration of configurations as mentioned earlier (8, I0). We now justify F(r) in Eq. (30) by way of summary of the foregoing procedure. Eq. (2), or (5) is exact only when rffil, which makes XffiO, and Eq. (3) is applicable only when XffiO, or RffiL, i.e. when perfect order exists. As a result F(rffil) must be 4. When r=0, previous work in Ref. 8 and I0 by actual enumerations of configurations show that F(rffi0) must be I, and between i and 4 for this function it is possible to start with any value of X dependent on Y in Eq. (6), where Y is determined by W and T as in Eq. (27) of Ref. 8. Since for any starting value of X at r=O to rffil, F(r) still varies between i and 4, F(r) should be independent of X. The functional form of F(r) should be such that, in addition to F(rffil)ffi4 and F(rffiO)ffil, it should satisfy Eq. (23) so that dF/dr10 when rffiO, and further, d2~nF/dr 2 must satisfy Eq. (24) when r=0. These stringent requirements are likely to make determination of F(r) by statistical means an extremely difficult task. At present, consistent results can be obtained only with any function of r satisfying the foregoing requirements, and Eq. (30) may therefore be regarded as a useful preliminary equation. Comments on Existin~ Approximations: The Q. Ch. method assumes that the bonds associated with (i)-(vlii) are permutable, i.e. each factor such as X! in Eq. (2) should be written as (ZX/2)!. The resulting D is then normalized so that for X = X 0 = RQ/L, D becomes identical with our Eq. (3) for the GBW method. However, Eq. (3) is valid only for X = 0 and not for X 0 = RQ/L as discussed earlier; therefore, the normalization process contains a significant contradiction. In addition, it was shown by Gokcen and Chang (8-10) that the permutation of bonds is incorrect since it creates impossible spliced molecules and contradicts the numbers of configurations obtained by actual enumerations. Further, as a trival matter and contrary to simple geometrical facts, the Q. Ch. method predicts that there can be no order when Z = 2. Therefore, the Q. Ch. method, despite the claims by numerous investigators, is not an improvement over the GBW method.
58
LONG-RANGE
ORDER
Vol.
17,
No.
1
The value of W/k for an ordered solution is calculated from the critical temperature T c in the GBW and Q. Ch. approximations. For the same solution in the disordered state, W/k is determined from experimental results, and we now proceed to show that each of these methods leads to inconsistent results within itself. For Cu-Zn, the GBW method requires 2kT c = -W(GBW) from which W(GBW)/k = -1484, and the zeroth approximation to the regular LRD solutions, the analog of GBW approximation, yields W(zeroth)/k = -3717 from Ge/2LkT = -1.202 with T = 773 and G e = 2LXA%B W = 0.5 LW, (YA = mol fraction of A = 0.5 = ~B). The Q. Ch. approximation for ordered Cu-Zn gives W/k = -1708 from W/k = TcZ%n[(Z-2)/Z], and the same approximation for the disordered solution gives W/k = -3285 from Ge/2LkT = -1.202 = (Z/2)%n{2 exp(W/ZkT)/[l+exp(W/ZkT)]}, with T = 773 and Z = 8. Similar calculations for other binary systems show similar discrepancies. The results for r versus T/T c and for C/2Lk versus T/Tc, obtained by using Tc=742 K, are presented in Fig. i and Fig. 2 for comparison. A significant improvement over the Q. Ch. method is the cluster variation method developed (16) and perfected (17-19) by Kikuchi. A simplified version of the earlier development by Kikuchi was presented by Guggenheim and McGlashan (20). The simplified version for the square as the cluster would require writing Q. Ch. equilibria and their pseudo-equillbrium constants for pseudoreactions such as
and accounting for all the possible lower hierachy of configurations down to single bonds and points. While with increasingly complex clusters the method would minimize the error originating from the permutation of single bonds and formation of unrealistic spliced atoms, clearly a great number of different clusters must be used as shown in Ref. I0. Further, much larger and varied clusters than those used by Kikuchi pose vastly increasing mathematical complexities (21). As a concluding remark, we wish to emphasize that the permutation promises the optimal degree of realistic success. The main reason for the resulting function for D of Eq. (9) would be obtained by deductive enumeration of D for one- and two-dimensional crystals as shown in Ref.
of "net numbers of atoms" this possibility is that reasoning based on actual I0 for L ~I00.
Statistical thermodynamics of long-range order is part of the Bureau provide a scientific base for use in developing new alloys and alloy phases. References i. 2. 3. 4. 5. 6. 7. 8. 9. I0.
ii. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
of Mines
program
to
H. A. Bethe, Froc. Roy. Soc. ALSO, 552 (1935). E. A. Guggenheim, "'Mixtures," Oxford Univ. Press, Chapter IV and VII (1952). R. Fowler and E. A. Guggenheim, "Statistical Thermodynamics," Cambridge Univ. Press, Chapter XII (1956). M. A. Krivoglaz and A. A. Smirnov, "The Theory of Order-Disorder in Alloys," Macdonald, London (1965). L. Guttman, Solid St. Phys. 3, 146 (1956). I. Prigogine, "The Molecular Theory of Solutions," North Holland, Amsterdam (1957). D. de Fontalne, Solid St. Phys., 34, 73 (1979). N. A. Gokcen and E. T. Chang, J. Chem. Phys., 55, 2279 (1971). N. A. Gokcen and E. T. Chang, Scripta Metall. 4, 941 (1970). N. A. Gokcen and E. T. Chang, "A New Method for Enumerating Molecular Configurations in Propellant Mixtures," Aerospace Report No: TR-0172 (2210-10)-1, Aerospace Corp., E1 Segundo, Calif. (1971). N. A. Gokcen, "Thermodynamics," Techscience, Inc., Chapter XI, Hawthorne, Calif. (1975). R. Hultgren, P. D. Desal, D. T. Hawkins, M. Glelser, and K. K. Kelley, "Selected Values of the Thermodynamic Properties of Binary Alloys," Am. Soc. Met., Metals Park, Ohio (1973). L. M. Falicov and R. C. Kittler, "Theory of Alloy Phase Formation," L. H. Bennett, Ed. Conference Proc., AIME; p. 303 (1980). H. Moser, Physikal. Zeit. 37, 737 (1936). C. Sykes and H. Wilkinson, J. Inst. Met. 61, 223 (1937). R. Kikuchi, Phys. Rev. 81, 988 (1951). R. Kikuchi and S. G. Brush, J. Chem. Phys., 4_7_7, 195 (1967). R. Kikuchi and C. M. van Vaal, Scripta Metall. 8, 425 (1974). R. Kikuchi and H. Sato, Acta Metall., 22, 1099 ~1974). E. A. Guggenhelm and M. L. McGlashan, Molec. Phys. 5, 433 (1962). R. Kikuchi, J. Chem. Phys. 60, 1071 (1974).