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A simplified derivation of the quasichemical approximation in order-disorder theory
A SIMPLIFIED
DERIVATION
OF THE
QUASICHEMICAL
IN ORDER-DISORDER
APPROXIMATION
THEORY*
J. M. HONIGt A simplilled procedure is presented, whereby the so-called quasichemical approximation in orderdisorder theory is derived; the complexities attendant to other derivations are avoided. Following the procedure of Hijmans and de Boer, the lattice consisting of A and B units is decomposed into representative arrays of “bonds” and “points”; the energy, entropy, and free energy for this system of arrays is then found in terms of probabilities of encountering the possible configurations AA, AB, BB, B, or A. Upon proper minimization of the free energy, the quasichemical result (16) is found.
CALCUL
SIMPLIFIE
DE L’APPROXIMATION
QUASI-CHIMIQUE
DE LA THEORIE
DE
L’ORDRE-DESORDRE Par une methode simplifiee, I’auteur etablit l’approximation quasi-chimique de la theorie de I’ordredesordre, Bvitant ainsi les alea rencontres dans d’autres methodes. Suivant Hijmans et de Boer, le reseau forme d’unites A et B est decomposes en reseau de “liens” (interactions) et “points”; l’energie, l’entropie et I’energie libre de ce systeme sent alors determinees par un calcul de probabilites de frequence des configurations possibles AA, AB, BB, B ou A. En tenant compte du minimum d’energie libre, l’approximation quasi-chimique est alors obtenue (16). EINE
VEREINFACHTE
NAHERUNG
ABLEITUNG
DER THEORIE
DER
DER QUASICHEMISCHEN ORDNUNGSEINSTELLUNG
Ein vereinfachtes Verfahren wird vorgelegt, bei dem die sogenannte quasichemische Niiherung der Theorie der Ordnungseinstellung abgeleitet wird; dabei werden die bei anderen Ableitungen auftretenden Komplikationen vermieden. Nach dem Vorgang von Hijmans und de Boer wird das aus A und B Einheiten bestehende Gitter zerlegt in repriisentative Anordnungen von “Bindungsstrichen” und “Punkten”; die Energie, die Entropie und die freie Energie fiir dieses System van Anordnungen findet man dann ausgedriickt in den Wahrscheinlichkeiten, die moglichen Konfigurationen AA, AB, BB, B, oder A anzutreffen. Sucht man das Minimum der freien Energie in geeigneter Weise auf, so findet man das quasichemische Ergebnis ( 16).
The so-called tensively the
used
quasichemical
application
interpretation
of of
heuristic
is ex-
of spins aligned in parallel and antiparallel
requiring
theory
phenomena.
for The
arguments,
but
provided
The object
of this publication
the
more
the
Each of the species A or B is assumed to be situated
final
at one of the L lattice The approximation
rigorous
for the solid under consideration.
The pro-
quasichemical
cedure here is based on a series of publications by Hijmans and de Boerc3) who established a very powerful
and generalized
quasichemical
approach,
from which the
clear that the number REPRESENTATION
readily
extended
METALLURGICA,
VOL.
1959
the most
the
all inter-
complex
number
repre-
for the lattice,
of bonds
obtainable
lattice is L, it is clear that an overcount
to a collection
7, MAY
one neglects
If it is
by the
decomposition process is (Z/2) L. The above mentioned figure assembly also contains ZL points; since the correct number of sites in the has occurred.
This situation can be rectified bv setting up a figure assembly consisting of (l-2) L members. The fact that the number of constituents in the latter group
* Received July 16, 1958. t Department of Chemistry, Purdue University, Lafayette, Indiana. ACTA
Consequently,
Z is the coordination
As a concrete example, we shall consider a binary alloy consisting of components A and B: the methodcan be very
approximation,
of those
In introducing
lattice sites, the latter often being termed points.
first order results.
ology
assemblies,
sentative array required for the lattice representation is the bond, which represents a pair of adjacent
result follows as a first approximation.
LATTICE
crystal.
the lattice
actions among the species save those between nearest neighbors.
The rather intricate procedure adopted by these writers can be avoided if one is interested only in the
THE
of a perfect
whose statistical properties are representative
is to present a simple
the final expression.
points
consists in decomposing
into a set of simpler arrays, termed$gure
to date(1s213) are quite complex.
for obtaining
directions,
and to other systems of a similar nature.
(16) can readily be justified
derivations method
analyses
order-disorder
physical
result, given by equation by
approximation
in mathematical
354
HONIG:
THE
QUASICHEMICAL
APPROXIMATION
is negative need cause no undue concern; it should be noted that the point figure assembly was set up for a corrective purpose and that the total number of sites, obtained from the two assemblies described above, is the positive quantity L. Since each lattice site may be occupied by species A or B, it is clear that the bonds may be encountered in the configurations AA, AB or BA, and BB. Let us represent by &,, /?r, pZ the fraction of the (Z/2) L bonds in the above configurations. Similarly, let us designate by u0 and a, the fraction of the (1-Z) L points in the configurations A and B. The following interrelations are then immediately obtained : a0 + cc1= 1
(1.)
B0 + 2/% + 82 = 1
(2)
al = & + Bs
(3)
Equations (1) and (2) represent normalization conditions; these simply state that the fractional parts of the assembly, when added together, are equivalent to the entire assembly. The factor 2 in equation (2) results from the fact that the configurations AB and BA are equivalent. Equation (3) is known as consistency condition; it expresses the fact that the ~o~guration B is enco~te~ed in the bond states BA or BB. THE
FREE
ENERGY
OF THE FIGURE
ASSEMBLIES
The next step consists in obtaining an expression for the free energy of the bond and point figure assemblies. We define by Ed and .eB the energy of a point in configuration A and B respectively; likewise, cAA, &AUand .sBBwill represent the energies of the bond in the various indicated ~on~gurations. The total energy for the (Z/2) L bonds and (l-2) L points is thus: E = Li@/2)
(Boeaa + 2#4&_4, + l%%B) +
wherein each energy term is multiplied by the number of units in the corresponding configuration. The various contributions are then summed. Next, the entropy of the system of figure assemblies will be considered. We introduce here the well known relation S = k In W, where W is the number of complexions consistent with the distribution of (2~2)L~~ (j = 0, 1, 2) bond types among (2/2)L bonds, and of (1 - Z)Lcc, (j = 0,l) point types among
IM ORDER-DISORDER
THEORY
356
(1 - 2) L points. Thus, X=kIn
(LZ/2)! 1(/90Lz/2)!(p,Lz/2)!s(/92Lz/2)! (I-Z)L!
x
x (1-Z)~~L!(l-Z)a~L!
(5)
i
Since L is large, we introduce Stirling’s approximation to find S = ~~(ZL~2) hi (ZL/S) -(&gL)
m (B,zL/2) -
(~sZL~2) hl(~~Z~~2) -
(F2zL/2) ln (P,zLI2) +
+(l-Z)Lln(l-Z)L-(1-Z)aOLln(l-Z)fx,L-(1
-
Z)or,L ln (I -
Z)a,L)
(6)
If terms such as ln (fi,L/2) are now written out as In L/2 + In pi, further cancellations occur and one is left with the simple expression x = -~~(Lzl2)(~~
m Bo i
+L(l
-
28, h B, + A? m Ba) +
z)(a, m x0 + a1 ln ar)}
(7)
The free energy expression may now readily be found by combining (4) and (7). It should be noted that the resulting equation is indicative of the properties of the figure assemblies. It is still necessary to determine the statistical properties for the lattice itself. To keep matters as simple as possible, a new assumption will now be ~troduc~, to the effect that the free energy expression for the figure assemblies also applies to the lattice proper. This assumption shows very clearly the drastic simplification that is required to obtain the desired end result. MINIMIZATION
OF
THE
FREE
ENERGY
To find the equilibrium value of the free energy it is necessary to minimize the appropriate expression. Before carrying out the differentiation, one must take account of the fact that the various ai and bj are not independent, since interrelations are provided by equations (l), (2) and (3). We shall arbitrarily select x1 and /la as the independent variables and express the rema~~g quantities in terms of these. This leads to the relations cc0= 1 - a1 (8) a, = PO=1
x1-al+
B2
(9) B2
WV
Thus, the free energy, F = E - TX will first be expressed in terms of $‘(a,, fi,); subsequent minimization leads to the condition
ACTA
356
METALLURGICA,
Since (11) is to apply for all conceivable independent variation in a1 and &, each coefficient must vanish identically. In view of (@-(lo) this leads to the two conditions. t3(E -
TX)
aaI
+
a(E -
a(E -
TS) aa,
ag,
a,+1
aa, +
TX) + a(E -
aif4
TS) a/t?,
+ 0
(12)
aal
a(E -
ag+2
w,
&
atE- Ts)a@l ai4
a(E -
TS) aj3,
TS) ag, = o
aa
a~, (13)
Differentials such as apo/aal may be computed from relations @-(lo); terms such as a(E - TS)/a& are found from (4) and (7). Straightforward algebraic manipulations then lead to the following set of equilibrium conditions : Equation (12) simplifies to (1 -
Z)[(Eg -
+ m,I3
eA) + kT(ln al -
h
a&l + - &axd + kT@ A - h /%,I = 0 (14)
and equation (13) results in (Z/2)[(&AA + kT h B,) -
2(&~ + kT h 81) +
+ (EBB + kT h
/&)I = 0 (15)
The last relation may be rewritten as 808d81” = K
K = exl? {-km
+
&AA -
2EAR)/kT) (16)
which will immediately be recognized as one formulation of the quasichemical approximation.
VOL.
7, 1959 CONCLUDING
REMARKS
Equation (14) represents an equilibrium condition not normally encountered in elementary treatments of order-disorder phenomena. It arises from the fact that, throughout, a, was regarded as a variable. It is clear that this viewpoint is applicable, so long as one deals with a system such as a collection of spins, where the number of spins aligned with or against a predetermined axis is not known in advance. On the other hand, in the case of binary alloys, the composition of the material is controlled experimentally, and a1 is therefore predetermined. In this particular case, only &, can be regarded as a variable. Equations (12) and (14) drop out and only equation (16) remains. With the aid of equations (l), (2), (3) and (16) it is now possible to solve for &, the result being B = -1 1
& (1 + 4(K - l)a,a,,)l/2 2(K - 1)
(17)
where the minus sign should be discarded in order that /$ might remain within the limits (0, 1) as K assumes over all permissible values in the range (0, CO). The quantity oZ may be found by using (17) in (3), and POmay be obtained from the normalization requirement (2). The above indicates that the method of Hijmans and de Boert3)is verywell suited for obtaining the wellknown quasichemical results in an elementary fashion. REFERENCES 1. R. H. FOWLER and E. A. GUGGENHEIM, Statistic& Thennodynamics. Cambridge University Press (1939). 2. E. A. GUGGENHEIM, Mixtures. Cambridge University Press (1948). 3. J. HIJMANS and J. DE BOER, Physica 21, 471, 455, 499 (1955).
DERIVATION
OF THE
QUASICHEMICAL
IN ORDER-DISORDER
APPROXIMATION
THEORY*
J. M. HONIGt A simplilled procedure is presented, whereby the so-called quasichemical approximation in orderdisorder theory is derived; the complexities attendant to other derivations are avoided. Following the procedure of Hijmans and de Boer, the lattice consisting of A and B units is decomposed into representative arrays of “bonds” and “points”; the energy, entropy, and free energy for this system of arrays is then found in terms of probabilities of encountering the possible configurations AA, AB, BB, B, or A. Upon proper minimization of the free energy, the quasichemical result (16) is found.
CALCUL
SIMPLIFIE
DE L’APPROXIMATION
QUASI-CHIMIQUE
DE LA THEORIE
DE
L’ORDRE-DESORDRE Par une methode simplifiee, I’auteur etablit l’approximation quasi-chimique de la theorie de I’ordredesordre, Bvitant ainsi les alea rencontres dans d’autres methodes. Suivant Hijmans et de Boer, le reseau forme d’unites A et B est decomposes en reseau de “liens” (interactions) et “points”; l’energie, l’entropie et I’energie libre de ce systeme sent alors determinees par un calcul de probabilites de frequence des configurations possibles AA, AB, BB, B ou A. En tenant compte du minimum d’energie libre, l’approximation quasi-chimique est alors obtenue (16). EINE
VEREINFACHTE
NAHERUNG
ABLEITUNG
DER THEORIE
DER
DER QUASICHEMISCHEN ORDNUNGSEINSTELLUNG
Ein vereinfachtes Verfahren wird vorgelegt, bei dem die sogenannte quasichemische Niiherung der Theorie der Ordnungseinstellung abgeleitet wird; dabei werden die bei anderen Ableitungen auftretenden Komplikationen vermieden. Nach dem Vorgang von Hijmans und de Boer wird das aus A und B Einheiten bestehende Gitter zerlegt in repriisentative Anordnungen von “Bindungsstrichen” und “Punkten”; die Energie, die Entropie und die freie Energie fiir dieses System van Anordnungen findet man dann ausgedriickt in den Wahrscheinlichkeiten, die moglichen Konfigurationen AA, AB, BB, B, oder A anzutreffen. Sucht man das Minimum der freien Energie in geeigneter Weise auf, so findet man das quasichemische Ergebnis ( 16).
The so-called tensively the
used
quasichemical
application
interpretation
of of
heuristic
is ex-
of spins aligned in parallel and antiparallel
requiring
theory
phenomena.
for The
arguments,
but
provided
The object
of this publication
the
more
the
Each of the species A or B is assumed to be situated
final
at one of the L lattice The approximation
rigorous
for the solid under consideration.
The pro-
quasichemical
cedure here is based on a series of publications by Hijmans and de Boerc3) who established a very powerful
and generalized
quasichemical
approach,
from which the
clear that the number REPRESENTATION
readily
extended
METALLURGICA,
VOL.
1959
the most
the
all inter-
complex
number
repre-
for the lattice,
of bonds
obtainable
lattice is L, it is clear that an overcount
to a collection
7, MAY
one neglects
If it is
by the
decomposition process is (Z/2) L. The above mentioned figure assembly also contains ZL points; since the correct number of sites in the has occurred.
This situation can be rectified bv setting up a figure assembly consisting of (l-2) L members. The fact that the number of constituents in the latter group
* Received July 16, 1958. t Department of Chemistry, Purdue University, Lafayette, Indiana. ACTA
Consequently,
Z is the coordination
As a concrete example, we shall consider a binary alloy consisting of components A and B: the methodcan be very
approximation,
of those
In introducing
lattice sites, the latter often being termed points.
first order results.
ology
assemblies,
sentative array required for the lattice representation is the bond, which represents a pair of adjacent
result follows as a first approximation.
LATTICE
crystal.
the lattice
actions among the species save those between nearest neighbors.
The rather intricate procedure adopted by these writers can be avoided if one is interested only in the
THE
of a perfect
whose statistical properties are representative
is to present a simple
the final expression.
points
consists in decomposing
into a set of simpler arrays, termed$gure
to date(1s213) are quite complex.
for obtaining
directions,
and to other systems of a similar nature.
(16) can readily be justified
derivations method
analyses
order-disorder
physical
result, given by equation by
approximation
in mathematical
354
HONIG:
THE
QUASICHEMICAL
APPROXIMATION
is negative need cause no undue concern; it should be noted that the point figure assembly was set up for a corrective purpose and that the total number of sites, obtained from the two assemblies described above, is the positive quantity L. Since each lattice site may be occupied by species A or B, it is clear that the bonds may be encountered in the configurations AA, AB or BA, and BB. Let us represent by &,, /?r, pZ the fraction of the (Z/2) L bonds in the above configurations. Similarly, let us designate by u0 and a, the fraction of the (1-Z) L points in the configurations A and B. The following interrelations are then immediately obtained : a0 + cc1= 1
(1.)
B0 + 2/% + 82 = 1
(2)
al = & + Bs
(3)
Equations (1) and (2) represent normalization conditions; these simply state that the fractional parts of the assembly, when added together, are equivalent to the entire assembly. The factor 2 in equation (2) results from the fact that the configurations AB and BA are equivalent. Equation (3) is known as consistency condition; it expresses the fact that the ~o~guration B is enco~te~ed in the bond states BA or BB. THE
FREE
ENERGY
OF THE FIGURE
ASSEMBLIES
The next step consists in obtaining an expression for the free energy of the bond and point figure assemblies. We define by Ed and .eB the energy of a point in configuration A and B respectively; likewise, cAA, &AUand .sBBwill represent the energies of the bond in the various indicated ~on~gurations. The total energy for the (Z/2) L bonds and (l-2) L points is thus: E = Li@/2)
(Boeaa + 2#4&_4, + l%%B) +
wherein each energy term is multiplied by the number of units in the corresponding configuration. The various contributions are then summed. Next, the entropy of the system of figure assemblies will be considered. We introduce here the well known relation S = k In W, where W is the number of complexions consistent with the distribution of (2~2)L~~ (j = 0, 1, 2) bond types among (2/2)L bonds, and of (1 - Z)Lcc, (j = 0,l) point types among
IM ORDER-DISORDER
THEORY
356
(1 - 2) L points. Thus, X=kIn
(LZ/2)! 1(/90Lz/2)!(p,Lz/2)!s(/92Lz/2)! (I-Z)L!
x
x (1-Z)~~L!(l-Z)a~L!
(5)
i
Since L is large, we introduce Stirling’s approximation to find S = ~~(ZL~2) hi (ZL/S) -(&gL)
m (B,zL/2) -
(~sZL~2) hl(~~Z~~2) -
(F2zL/2) ln (P,zLI2) +
+(l-Z)Lln(l-Z)L-(1-Z)aOLln(l-Z)fx,L-(1
-
Z)or,L ln (I -
Z)a,L)
(6)
If terms such as ln (fi,L/2) are now written out as In L/2 + In pi, further cancellations occur and one is left with the simple expression x = -~~(Lzl2)(~~
m Bo i
+L(l
-
28, h B, + A? m Ba) +
z)(a, m x0 + a1 ln ar)}
(7)
The free energy expression may now readily be found by combining (4) and (7). It should be noted that the resulting equation is indicative of the properties of the figure assemblies. It is still necessary to determine the statistical properties for the lattice itself. To keep matters as simple as possible, a new assumption will now be ~troduc~, to the effect that the free energy expression for the figure assemblies also applies to the lattice proper. This assumption shows very clearly the drastic simplification that is required to obtain the desired end result. MINIMIZATION
OF
THE
FREE
ENERGY
To find the equilibrium value of the free energy it is necessary to minimize the appropriate expression. Before carrying out the differentiation, one must take account of the fact that the various ai and bj are not independent, since interrelations are provided by equations (l), (2) and (3). We shall arbitrarily select x1 and /la as the independent variables and express the rema~~g quantities in terms of these. This leads to the relations cc0= 1 - a1 (8) a, = PO=1
x1-al+
B2
(9) B2
WV
Thus, the free energy, F = E - TX will first be expressed in terms of $‘(a,, fi,); subsequent minimization leads to the condition
ACTA
356
METALLURGICA,
Since (11) is to apply for all conceivable independent variation in a1 and &, each coefficient must vanish identically. In view of (@-(lo) this leads to the two conditions. t3(E -
TX)
aaI
+
a(E -
a(E -
TS) aa,
ag,
a,+1
aa, +
TX) + a(E -
aif4
TS) a/t?,
+ 0
(12)
aal
a(E -
ag+2
w,
&
atE- Ts)a@l ai4
a(E -
TS) aj3,
TS) ag, = o
aa
a~, (13)
Differentials such as apo/aal may be computed from relations @-(lo); terms such as a(E - TS)/a& are found from (4) and (7). Straightforward algebraic manipulations then lead to the following set of equilibrium conditions : Equation (12) simplifies to (1 -
Z)[(Eg -
+ m,I3
eA) + kT(ln al -
h
a&l + - &axd + kT@ A - h /%,I = 0 (14)
and equation (13) results in (Z/2)[(&AA + kT h B,) -
2(&~ + kT h 81) +
+ (EBB + kT h
/&)I = 0 (15)
The last relation may be rewritten as 808d81” = K
K = exl? {-km
+
&AA -
2EAR)/kT) (16)
which will immediately be recognized as one formulation of the quasichemical approximation.
VOL.
7, 1959 CONCLUDING
REMARKS
Equation (14) represents an equilibrium condition not normally encountered in elementary treatments of order-disorder phenomena. It arises from the fact that, throughout, a, was regarded as a variable. It is clear that this viewpoint is applicable, so long as one deals with a system such as a collection of spins, where the number of spins aligned with or against a predetermined axis is not known in advance. On the other hand, in the case of binary alloys, the composition of the material is controlled experimentally, and a1 is therefore predetermined. In this particular case, only &, can be regarded as a variable. Equations (12) and (14) drop out and only equation (16) remains. With the aid of equations (l), (2), (3) and (16) it is now possible to solve for &, the result being B = -1 1
& (1 + 4(K - l)a,a,,)l/2 2(K - 1)
(17)
where the minus sign should be discarded in order that /$ might remain within the limits (0, 1) as K assumes over all permissible values in the range (0, CO). The quantity oZ may be found by using (17) in (3), and POmay be obtained from the normalization requirement (2). The above indicates that the method of Hijmans and de Boert3)is verywell suited for obtaining the wellknown quasichemical results in an elementary fashion. REFERENCES 1. R. H. FOWLER and E. A. GUGGENHEIM, Statistic& Thennodynamics. Cambridge University Press (1939). 2. E. A. GUGGENHEIM, Mixtures. Cambridge University Press (1948). 3. J. HIJMANS and J. DE BOER, Physica 21, 471, 455, 499 (1955).