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PHASE SEPARATION AND THERMODYNAMIC
PROPERTIES OF BINARY SOFT-CORE MIXTURE
T. Shiotani, T. Ichimura, A. Ueda
To cite this version:
PHASE S E P A R A T I O N AND THERMODYNAMIC P R O P E R T I E S OF B I N A R Y SOFT-CORE M I X T U R E ,
T. ~hiotani*, T; Ichimura and A. Ueda
'Chubu I n s t i t u t e o f Technology, Kasugai, Nagoya, Japan. Dept. AppZied Math. and Phys. Kyoto, Japan.
Abstract.- Phase separation of a binary mixture caused by repulsive forces is studied for a binary soft-core mixture with non-additive core-sizes
@ij(r) = ~ ( o . . ~ r ) ~ , i,j = A or B, oAB = (Q + sBB)(I+A)/2.
1 3
The system is simulated by a Monte Carlo method with use of the fine-grained lattice model which is made by successive binary-division of the unit cell. The result for case that different species of atoms have the same core size is presented. The critical solution curve on the P-T plane is determined.
91. Introduction (or A). We present the results for aBB
Fluid-fluid phase separation caused by repulsive forces is studied with a binary soft-core mixture of non-additive diame- ters. Pair potentials are given by
The model'is a binary mixture version of the one-component soft-core model which was studied in detai11%5)and satisfactorily described the properties of rare gases, alkaline metals and alkaline earth metals
6) according to the softness parameter n
.
According to a recent high-pressure exper- iment7) a He-Xr mixture causes phaseseparation at pressure of more than 10 3 2
kg/m where repulsive forces may become significant rather than attractive forces. Among possible models which cause phase separation, the soft-core mixture is the
-
-a*.
52. Method of computation
The system is simulated by a Flonte Carlo method with use of a fine-grained lattice model which consists of 32%32%32 lattice points per unit cell and which aggroximates a corresponding continuum system in good accuracy?) Computations are carried on for a system of 108 parti- cles with n = 1 2 and A = 0.2 at concentration x A = 0.5, 0.667 (=72/108) and 0.883 (=9
7'
/108). Since a m = uBBr the model has symmetric properties about x A = 0.5. As a denkity parameter we introduceAccording to the scaling properties, pres- sure, for instance, is expressed as
simplest owing to the scaling properties. 3 , where
3
(p*,
xA) = p* (PV/NkT) and depends In the present model, parameters whichcharacterize the phase separation are the ratia of core sizes uBg/uAA and uAB/uAA
Table I
JOURNAL DE PHYSIQUE
which is given by
only on p* and xA. Computations were made for 10000 SIC time steps. For the most of the runs to generate initial configura- tions the end configurations of the previ- ous runs of lower densities were used, but p~
B molecules.
53. Determination of the critical curve The compressibility factors are summa- rized in Table I. Though the values for phase-separating states contain someerrors
8 due to the snallness of the system size
,
the values for homogeneous states are believed to be as accurate as those for aone-component system. Figure 1 shows the excess compressibility factor (PV/NkT) E
r I
(PV/NkT) = (PV/NkT)
-
(PV/NkT) where the second term in the r.h.s. is the same as that of the one-component soft-core system at the same values of p*4'5! Mate- rial instabilities appear at p*c 0.3, '~0.4 and 20.5 for xA= 54/108, 72/108 and 90/108, respectively. The initial and end config- urations for p * = 0.375 at x = 0.5 are shownA
in Fig.2. Phase separation is observed in the end configurations for p* > 0.3. Exam- ples of static correlation functions g
A A r
gAB, nNN and nCC for p*= 0.2, 0.3 and 0.4 at x A = 0.5 are shown in Figs. 3 and 4, respectively, where the latter two repre- sent the nunber-number and concentration- concentration correlations. Since the critical density is p**0.3 (see below),
C
the curve for p*= 0.4 has some errors due
Initial configuration
Y Fig. 4
2
0
0 '/%A 2 3
to the same reason mentioned above. How- ever, as far as the first peaks are con- cerned, the errors may not be large even for p* 0.3. Figure 5 shows the height of the first peaks as functions of p*. Some changes in slopes of g AA,, gAB and nCC are
observed at about p*= 0.3.
Since om= uBB, the critical point appears at equal mole fractions. Thecrit- ical density p; is determined as follows. First, from the results mentioned above, p; is found to be very close to 0.3. Next we calculated the mixing Gibbs energy at several values of
3
( = p* (PV/NkT) ) and looked for p; graphically using the ther-' Height - 2 . 0 A A A A
1
G N
0 0 0 0 ° 01
A9
t + & 0-
1.0+
++.
.
+i
+
*cc 0.1 0 . 2 ~ * 0 . 3 0.ymodynamic condition of the critical point. To derive the Helmholtz free energy the equation of states which are approximated by polynomials are integrated. The free energies of phase-separating states are obtained by extrapolating the equation of
stable states. As a result we obtain %
p;= 0.3t0.02 and hence P C = 0.87k0.14. Therefore the critical curve is given by
which lie on the plane xA=0.5 in the P-T- xA space.
54. Phase boundary and excess properties % %
The states for P < P C are one-phase
E
states. The excess volume v* ( = l/p*) and %
the excess Gibbs free energy at P = 0.42, 0.60 and 0.80 are shown in Fig.6. Figure 7 shows the phase boundary where the ordi-
% 3
nate is l/P = (kT/Pom) (~T/E) The crosses represent our result and the curve
JOURNAL DE PHYSIQUE
Fig. 7
is that for n = m
,
that is, for the hard sphere system with non-additive core sizes( A = 0.2) 8114! The latter is obtained
with use of the Carnahan-Starling equation of State12) together with the perturbation 13) theory of one-component reference fluid
.
Though our points have rather largeerrors, all these points are within the curve, which suggests that the phase-separating2,
region in the 1/P-xA plane increases with increasing n.
4 5 . Summary and discussions
The critical density of the present soft-core system of a m = o is given by
BB
pf2=0.30t0.02, which is tp be compared with p:= 0.42 8' of the hard sphere system with non-additive core sizes of A = 0.2. The critical curve is simply given by
2,
P o3 /E = bC(k~C/~)
'I4
with pC = 0.87t0.14 C AAand TC is the same as the upper critical
2,
solution temperature. If A decreases, PC increases. The shape of the critical curve has a resemblance to critical curves of the first kind. 15) The phase-separating region in the l/$- xA plane increases with the softness parameter n.
So far we present the result for aAA = a B B . It is qualitatively understood that, if the ratio aBB/aAA decreases, the phase-separating region shown in Fig.7 contracts towards the right and the one-
phase region at small xA widens. Computa- tions are now being undertaken for oBB/oAA
= 0.5.
Acknoledgements
The authors are indebted to Institute of Plasma Physics, Nagoya University, at which numerical computations were perform- ed as a research project.
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