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Ordering potential and chemical short range order in the structure of model liquid binary alloys
132
Journal of Non-Crystalline Solids 117/118 (1990) 132-135 North-Holland
ORDERING P O T E N T I A L A N D C H E M I C A L BINARY ALLOYS
D. J. GONZALEZt,
SHORT RANGE O R D E R IN THE STRUCTURE
OF M O D E L LIQUID
L. E. GONZALEZ t and M. SILBERT #
~Departamento de Fisica Teorica, U n i v e r s i d a d de Valladolid, 47011 Valadolid, #School of Physics, U n i v e r s i t y of East Anglia, N o r w i c h NR4 7TJ, U.K.
SPAIN.
The p a r t i a l structure factors, Sij(q), and thermodynamic properties of a model liquid b i n a r y alloy are c a l c u l a t e d in the mean spherical approximation. It is assumed the ordering potential, v(r) = ~[~1~(r) + ~22(r) - 2~12(r) ], is given by a screened Coulomb potential form, and is dependent on concentration. The functional behaviour of v(r) is chosen so that it describes phase separating tendencies at one c o n c e n t r a t i o n range, and c o m p o u n d forming tendencies at antoher. This type of b e h a v i o u r appears to be p r e s e n t in liquid Na-Cd.
i. I N T R O D U C T I O N Liquid N a - C d has b e e n o b s e r v e d phase s e p a r a t i n g the phase
tendency
at the N a - r i c h
d i a g r a m and a c o m p o u n d
ency at the other end I, have s t u d i e d
to have a
r < 6ij
Aije-zr/r
r > 6ij
(2)
end of
forming tend-
In earlier papers 2, we
the p r o p o s i t i o n
~ ~ij (r) =
that once the
where
the 6ij are the hard core diameters,
that 612 = (611 + 622)/2. also takes on a Y u k a w a
We assume
such
that v(r)
form
ordering p o t e n t i a l v(r) = C e-Zr/r v(r) = ~[~11(r) is fixed, are, fied.
+ ~22 - 2~12 (r)]
then several properties
to all practical
purposes,
If our ideas are correct,
ordering p o t e n t i a l varies;
change
the structure
do not address
speci-
we will see the
and thermodynamic
proper-
In this w o r k we
of a c o m p o s i t i o n
ordering potential.
dependent
This allows us the use of a
simple model
is studied w i t h i n
approximation
uniquely
liquid N a - C d directly but examine
instead the effects
relatively
of the system
sign as the compositon
ties will r e s p o n d accordingly.
which
(I)
liquid b i n a r y alloy, the mean spherical
composition
For the purposes
A22/A~I
= i, z611 = 4.5.
have the c o n c e n t r a t i o n I.
We note
that,
dictates w h e t h e r
Then C is assumed to
dependence
given eq.
first,
negative
shown in Figure
(i), the sign of C
the system has tendencies
hetero or homocoordination;
eharaterized
of the
= 1.2,
positive
for the second.
by eq.
to
for the
The system
(2) has been solved w i t h i n
the M S A for w h i c h Blum and H~ye have obtained an analytic
solution 3.
For this specific
whose
explicit
case
form have been
w r i t t e n down by A r r i e t a et al 4.
2. T H E O R Y AND RESULTS
potentials,
dependent.
present work we assume 622/611
in h a n d it leads to eight non linear algebraic
(MSA).
(i) the ~ij(r)
(3)
where C = (At, + A22 - 2A12)/2 , is assumed
equations,
In eq.
r > Max(6ij)
denote
the interatomic
w h i c h are a s s u m e d to take on Y u k a w a
forms
0022-3093/90/$03.50 Q Elsevier Science Publishers B.V. (North-Holland)
followed
the procedures
which we i n c o r p o r a t e d select our physical
outlined
Pastore's
solution
We have in Ref.
4, to
criteria 5 to
from the several
D. J. Gonzalez et al./ The structure of model liquid binary alloys
1.5
0.6
133
,Sn
0
-_o 0.2
~ 0.5
~
-0,2
0
-0.6 I
-0"5V I
I
0.2
0.4
-1
i
i
(3.6
0.8
0
10
20
30 q~
Concentroti0n
FIGURE 1 Pre exponential factor, C (in units of the coefficient A1,), of the ordering potential v(r) as a function of concentration.
w h i c h are obtained. used to calculate Sij(q)
The physical the partial
from the Fourier
Ornstein-Zernike In figures partial
solution
structure
transforms
FIGURE 2 Partial structure factors Sij(k ) at T* = 1.05, p* z 3.675 and c = 0.5
is
factors
s22 /~
of the
direct c o r r e l a t i o n
1.5
functions 6.
0
2 and 3 we show the A s h c r o f t - L a n g r e t h
structure
factors 6 at a reduced temper-
ature T* = kBT/c11 P* = P6~i/c11
t
= 1.05 and reduced pressure
= 3.675 for concentrations m
c = 0.5 and 0.8 rspeetively. c11 = - A11e-Z~11/~11 ,
and c i
of component
concentration
that c I + c 2 = i.
-0.5
e2, 2, such
It can be seen in Fig.
for C < 0, the phase
separating
2
w i t h the positions
0
the other hand,
for C > 0, the compound forming
which brings
On
as well as the interplay
and c o n c e n t r a t i o n
fluctuations
down the value of the principal
peak of S11(q), component,
size difference.
are r e f l e c t e d both by the p r e p e a k
b e t w e e n number
while
30
of
the atoms
tendencies
I
2O
of the principal
peaks m i r r o r i n g
exhibited b y S11(q)
I
10
qz,
tendenc-
ies are r e f l e c t e d b y the large low q values Sij(q),
0
Here we define
the number
that,
0.5
the partial
for the smaller
increases
the value of the
Same as in Fig.
FIGURE 3 2 but for c = 0.8
D. J. Gonzalez et a l . / T h e structure of model liquid binary alloys
134
principal peak of $22(q) , without affecting
0.60
their relative positions. We know turn to the thermodynamic properti
G _
ies, which have been evaluated via the internal
o,~s
energy route, and using the expressions developed by H~ye and Stell 7.
The two properties
030
presented in this work are the volume of mixing,
0.15 0.075 ¢
|
i
/
0.2
0.4
0.6
O.8
0.050 C oncentrotion .x_
0.025 FIGURE 5 Scc(0 ) as a function of concentration for the thermodynamic states as in Fig. 4. Ideal mixture result Scc(0 ) = c(l-c) is also shown.
-0.025
0
0.2
0,4 0.6 Concentration
0.8 out effecting its qualitative behaviour.
Fig.
5 shows Scc(0) for the same thermodynamic states as in Fig. 4, where we have also inFIGURE 4 Volume of mixing, in units of ~ i , at T* = 1.05 solid line: P* 3.675; broken line: P* = 5.775
cluded the ideal mixture behaviour for comparison.
Scc(0) also follows closely the
behaviour of v(r), with phase separating tendency at c = 0.5 and weak compound forming tendVmi x - V - cIV I - c2V2, and the long-wavelength
encies at c = 0.2 and 0.8, independently of the
limit of the Bhatia-Thornton concentration-
size difference.
concentration partial structure factor,
surprisingly,
Scc(0) - NkBT/(82G/aC2)T,p, N.
in units of ~ I ,
at two different pressures.
is
The results are
shown in Figures 4 and 5 respectively. shows Vmix,
The affect of pressure,
to enhance these features.
Fig. 4
at T* = 1.05 and We note that the
3. CONCLUSIONS The calculation presented above support the proposition that the ordering potential deter-
behaviour of Vmi x follows that of the ordering
mines the qualitative behavfour of both the
potential, with a maximum at c = 0.5 and minima
structure and thermodynamics of liquid binary
at 0.2 and 0.8 irrespective of the size
alloys.
difference.
actual liquid binary alloys depends on the
It is not yet clear why Vmi x
However,
quantitative agreement with
changes sign again at both concentration ends
development of reliable effective interionic
but,
potentials.
from preliminary calculations,
we tenta-
tively attribute this to size difference. effect due to pressure
The
is to depress Vmi x with-
In this respect,
the work by
Hasegawa et al 8 reported in these Proceedings are very promising.
D. J. Gonzalez et al./ The structure of model liquid binary alloys
135
REFERENCES
ACKNOWLEDGEMENTS We thank S Tamaki and W H Young for helpful
i. H. Hoshino and H. Endo, Phys. Chem. Liq. ii (1982) 327.
discussions and valuable advice concerning this work.
This work has been supported by the
DGICYT, Spain (Grant PB-86-0654-C02),
the Junta
de Castilla y Leon, and the SERC, U K (Grant
2. D. J. Gonzalez and M. Silbert, Z. Phys. Chemie 156 (1988) 657; J. Phys. F 18 (1988) 2353. 3. L.Blum and J. S. H~ye, J. Stat. Phys. 19 (1978) 317.
GR/E84174). 4. E. Arrieta, C. Jedrejek and K. N. Marsh, J. Chem. Phys. 86 (1987) 3607. 5. G. Pastore, Mol. Phys. 63 (1988) 731. 6. W. H. Young, Can. J. Phys. 65 (1987) 241. 7. J. H~ye and G. Stell, J. Chem. Phys. 67 (1977) 439. 8. M. Hasegawa, K. Hoshino, M. Watabe and W. H. Young. These Proceedings.
Journal of Non-Crystalline Solids 117/118 (1990) 132-135 North-Holland
ORDERING P O T E N T I A L A N D C H E M I C A L BINARY ALLOYS
D. J. GONZALEZt,
SHORT RANGE O R D E R IN THE STRUCTURE
OF M O D E L LIQUID
L. E. GONZALEZ t and M. SILBERT #
~Departamento de Fisica Teorica, U n i v e r s i d a d de Valladolid, 47011 Valadolid, #School of Physics, U n i v e r s i t y of East Anglia, N o r w i c h NR4 7TJ, U.K.
SPAIN.
The p a r t i a l structure factors, Sij(q), and thermodynamic properties of a model liquid b i n a r y alloy are c a l c u l a t e d in the mean spherical approximation. It is assumed the ordering potential, v(r) = ~[~1~(r) + ~22(r) - 2~12(r) ], is given by a screened Coulomb potential form, and is dependent on concentration. The functional behaviour of v(r) is chosen so that it describes phase separating tendencies at one c o n c e n t r a t i o n range, and c o m p o u n d forming tendencies at antoher. This type of b e h a v i o u r appears to be p r e s e n t in liquid Na-Cd.
i. I N T R O D U C T I O N Liquid N a - C d has b e e n o b s e r v e d phase s e p a r a t i n g the phase
tendency
at the N a - r i c h
d i a g r a m and a c o m p o u n d
ency at the other end I, have s t u d i e d
to have a
r < 6ij
Aije-zr/r
r > 6ij
(2)
end of
forming tend-
In earlier papers 2, we
the p r o p o s i t i o n
~ ~ij (r) =
that once the
where
the 6ij are the hard core diameters,
that 612 = (611 + 622)/2. also takes on a Y u k a w a
We assume
such
that v(r)
form
ordering p o t e n t i a l v(r) = C e-Zr/r v(r) = ~[~11(r) is fixed, are, fied.
+ ~22 - 2~12 (r)]
then several properties
to all practical
purposes,
If our ideas are correct,
ordering p o t e n t i a l varies;
change
the structure
do not address
speci-
we will see the
and thermodynamic
proper-
In this w o r k we
of a c o m p o s i t i o n
ordering potential.
dependent
This allows us the use of a
simple model
is studied w i t h i n
approximation
uniquely
liquid N a - C d directly but examine
instead the effects
relatively
of the system
sign as the compositon
ties will r e s p o n d accordingly.
which
(I)
liquid b i n a r y alloy, the mean spherical
composition
For the purposes
A22/A~I
= i, z611 = 4.5.
have the c o n c e n t r a t i o n I.
We note
that,
dictates w h e t h e r
Then C is assumed to
dependence
given eq.
first,
negative
shown in Figure
(i), the sign of C
the system has tendencies
hetero or homocoordination;
eharaterized
of the
= 1.2,
positive
for the second.
by eq.
to
for the
The system
(2) has been solved w i t h i n
the M S A for w h i c h Blum and H~ye have obtained an analytic
solution 3.
For this specific
whose
explicit
case
form have been
w r i t t e n down by A r r i e t a et al 4.
2. T H E O R Y AND RESULTS
potentials,
dependent.
present work we assume 622/611
in h a n d it leads to eight non linear algebraic
(MSA).
(i) the ~ij(r)
(3)
where C = (At, + A22 - 2A12)/2 , is assumed
equations,
In eq.
r > Max(6ij)
denote
the interatomic
w h i c h are a s s u m e d to take on Y u k a w a
forms
0022-3093/90/$03.50 Q Elsevier Science Publishers B.V. (North-Holland)
followed
the procedures
which we i n c o r p o r a t e d select our physical
outlined
Pastore's
solution
We have in Ref.
4, to
criteria 5 to
from the several
D. J. Gonzalez et al./ The structure of model liquid binary alloys
1.5
0.6
133
,Sn
0
-_o 0.2
~ 0.5
~
-0,2
0
-0.6 I
-0"5V I
I
0.2
0.4
-1
i
i
(3.6
0.8
0
10
20
30 q~
Concentroti0n
FIGURE 1 Pre exponential factor, C (in units of the coefficient A1,), of the ordering potential v(r) as a function of concentration.
w h i c h are obtained. used to calculate Sij(q)
The physical the partial
from the Fourier
Ornstein-Zernike In figures partial
solution
structure
transforms
FIGURE 2 Partial structure factors Sij(k ) at T* = 1.05, p* z 3.675 and c = 0.5
is
factors
s22 /~
of the
direct c o r r e l a t i o n
1.5
functions 6.
0
2 and 3 we show the A s h c r o f t - L a n g r e t h
structure
factors 6 at a reduced temper-
ature T* = kBT/c11 P* = P6~i/c11
t
= 1.05 and reduced pressure
= 3.675 for concentrations m
c = 0.5 and 0.8 rspeetively. c11 = - A11e-Z~11/~11 ,
and c i
of component
concentration
that c I + c 2 = i.
-0.5
e2, 2, such
It can be seen in Fig.
for C < 0, the phase
separating
2
w i t h the positions
0
the other hand,
for C > 0, the compound forming
which brings
On
as well as the interplay
and c o n c e n t r a t i o n
fluctuations
down the value of the principal
peak of S11(q), component,
size difference.
are r e f l e c t e d both by the p r e p e a k
b e t w e e n number
while
30
of
the atoms
tendencies
I
2O
of the principal
peaks m i r r o r i n g
exhibited b y S11(q)
I
10
qz,
tendenc-
ies are r e f l e c t e d b y the large low q values Sij(q),
0
Here we define
the number
that,
0.5
the partial
for the smaller
increases
the value of the
Same as in Fig.
FIGURE 3 2 but for c = 0.8
D. J. Gonzalez et a l . / T h e structure of model liquid binary alloys
134
principal peak of $22(q) , without affecting
0.60
their relative positions. We know turn to the thermodynamic properti
G _
ies, which have been evaluated via the internal
o,~s
energy route, and using the expressions developed by H~ye and Stell 7.
The two properties
030
presented in this work are the volume of mixing,
0.15 0.075 ¢
|
i
/
0.2
0.4
0.6
O.8
0.050 C oncentrotion .x_
0.025 FIGURE 5 Scc(0 ) as a function of concentration for the thermodynamic states as in Fig. 4. Ideal mixture result Scc(0 ) = c(l-c) is also shown.
-0.025
0
0.2
0,4 0.6 Concentration
0.8 out effecting its qualitative behaviour.
Fig.
5 shows Scc(0) for the same thermodynamic states as in Fig. 4, where we have also inFIGURE 4 Volume of mixing, in units of ~ i , at T* = 1.05 solid line: P* 3.675; broken line: P* = 5.775
cluded the ideal mixture behaviour for comparison.
Scc(0) also follows closely the
behaviour of v(r), with phase separating tendency at c = 0.5 and weak compound forming tendVmi x - V - cIV I - c2V2, and the long-wavelength
encies at c = 0.2 and 0.8, independently of the
limit of the Bhatia-Thornton concentration-
size difference.
concentration partial structure factor,
surprisingly,
Scc(0) - NkBT/(82G/aC2)T,p, N.
in units of ~ I ,
at two different pressures.
is
The results are
shown in Figures 4 and 5 respectively. shows Vmix,
The affect of pressure,
to enhance these features.
Fig. 4
at T* = 1.05 and We note that the
3. CONCLUSIONS The calculation presented above support the proposition that the ordering potential deter-
behaviour of Vmi x follows that of the ordering
mines the qualitative behavfour of both the
potential, with a maximum at c = 0.5 and minima
structure and thermodynamics of liquid binary
at 0.2 and 0.8 irrespective of the size
alloys.
difference.
actual liquid binary alloys depends on the
It is not yet clear why Vmi x
However,
quantitative agreement with
changes sign again at both concentration ends
development of reliable effective interionic
but,
potentials.
from preliminary calculations,
we tenta-
tively attribute this to size difference. effect due to pressure
The
is to depress Vmi x with-
In this respect,
the work by
Hasegawa et al 8 reported in these Proceedings are very promising.
D. J. Gonzalez et al./ The structure of model liquid binary alloys
135
REFERENCES
ACKNOWLEDGEMENTS We thank S Tamaki and W H Young for helpful
i. H. Hoshino and H. Endo, Phys. Chem. Liq. ii (1982) 327.
discussions and valuable advice concerning this work.
This work has been supported by the
DGICYT, Spain (Grant PB-86-0654-C02),
the Junta
de Castilla y Leon, and the SERC, U K (Grant
2. D. J. Gonzalez and M. Silbert, Z. Phys. Chemie 156 (1988) 657; J. Phys. F 18 (1988) 2353. 3. L.Blum and J. S. H~ye, J. Stat. Phys. 19 (1978) 317.
GR/E84174). 4. E. Arrieta, C. Jedrejek and K. N. Marsh, J. Chem. Phys. 86 (1987) 3607. 5. G. Pastore, Mol. Phys. 63 (1988) 731. 6. W. H. Young, Can. J. Phys. 65 (1987) 241. 7. J. H~ye and G. Stell, J. Chem. Phys. 67 (1977) 439. 8. M. Hasegawa, K. Hoshino, M. Watabe and W. H. Young. These Proceedings.