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Liquid-liquid phase splitting—II
LIQUID-LIQUID TERNARY
SYSTEMS
PHASE AND
THE
SPLITTING-II SPINODAL
CURVE
JAIME WISNIAK Department of Chemi&l Engineering, Ben-Gurion University of the Negev, Beer Sheva, Israel (Accepted 28 March
1983)
Abstract-Solubility phenomena are analyzed for ternary nonideal liquid systems for which GE = ,T~~x,x~+01~~x,x~ + WSCZY~ + UXIXSI where a represents the ternary effect. Spinodal curves for this type of systems can generate any of the known miscibility curves. the temperature
produces an increase in the attractive interaction beiween like molecules which may lead to phase splitting. If a strong directional attraction is present, it will increase enough at lower temperatures and the system will again homogenize. Closed gap solubility curves have also been analyzed by Barker and Fock[E], Andersen and Wheeler[9], and Walker and Vause[lO] using decorated latticemodels. The following analysis will assume the presence of ternary effects and that the excess Gibbs function can be
In a previous publication{11 we have discussed the phenomena associated with phase segregation in binary liquid systems. The different analytic models used for describing nonideality have been tested for their ability to predict separation into two liquid phases. In this paper we will analyze the influence of a third component on the mutual solubility of two liquids; the discussion will also include the possibility of closed-loop solubility diagrams in ternary systems. Calculation of equilibrium phase diagrdms in ternary systems has been discussed by Tompa[2], Scott[3], Prigogine[4] and Meijering[S, 61. Both Tompa and Scott discussed the analytical relations present in polymer solutions making some severe assumptions regarding the value of the energy parameters and the nature of the species present in the system. Prigogine[4] has discussed the thermodynamic conditions that unstable phases must satisfy. A detailed analysis is given of a ternary system composed of strictly regular solution for which GE/RT = (Y,~x,x~ + a13x,x3 + (Y~~x~x~.On this basis Prigogine developed an equation that gives the variation of the critical solution temperature (CST) of the binary system 1 + 2 caused by the addition of a small amount of a third component. He concluded that if the third component is equally soluble in the first two components mutual solubility of 1+2 will increase. If the third component is much less soluble in one of the two components its addition will cause an increase in the CST. The analytical development includes also a calculation of the spinodal curve. Meijering[S, 63 has made a very thorough analysis of miscibility gaps and tie-lines in the ternary system that behave like strictly regular solutions. He classified the possible system in eight categories depending on the signs of binary energy parameters, Uij and determined
the necessary
described
by
GE/RT
= o~,zxIxz+ al3x,x,+
az3x2x3+
ax,xzx3.
Coefficients aq and 01 are assumed to be independent of temperature. Determination of coefficient OLwill require information on the ternary systems.
1. INFLUENCE
OF THE THlRD COMF’ONENT
ON THE MUTUAL
SOLUBILITY
In a previous publication [ l] it has been indicated that the line that separates stable states from nonstable states is determined by either of the following equations fQ,~ZZ-fi”:2=0 cL22Wz3- &
(2)
(3)
= 0
PXSlLL,-P:B=O
(4)
where the symbol pii or pi1’identifies the second derivative of G with respect to the number of moles given by the indices. Equations (2), (3) and (4) are equivalent to
conditions for the appearance of open or
closed miscibility gaps. Hirschfelder, Stevenson and Eyring[7] have discussed the theoretical implications of lower CST and closedloop existence curves. They suggested that there is a competition between strongly directional attractive interaction between tbe unlike components of the mixture, such as a hydrogen bond, and a weaker nondirectional repulsion. At high temperatures the entropy of mixing predominates and the system is homogeneous. Lowering
(1)
The critical point of the ternary by
111
system
is characterized
112 In
5. WISNIAK addition the Gibbs-Buhem
relation gives [4] = ; (-RT +(cr,r+ 7
Substitution into eqn (5) yields a more symmetrical equation for surface PI
+ 2~~23x3- 2az3~3~- 2~x1’ al3+LYZJ)XI -2(LY12-a13fa&1x31
(1- 2x2 - 2x3 + 6~x3).
(15)
For the binary system in question 1 + 2 we have xg = 0 and a strictly regular behavior. Hence Intersection of P, with the surface Pz defined by the first relation in eqn (6) will generate the critical solubility line. Following the development of Prigogine[4] we obtain the influence of the third component on the critical temperature, dT/&, by the simultaneous solution of the following differential equations
(Xl)= = (x& = 0.5
T, = g
(16)
At the critical point eqns (13)-(15) will yield
(9) Making use of eqn (1) and
Hence
&lGE = RT In yi (-_)ani P. T,“j’“i
(10)
we obtain apI3
2 ( ) =-au12+a13+a,,+E
ndX3c
(18)
a a/.& -(YIz+(Yl,+%3+-. n ( ax, ) C= 2
(11) In addition In order to calculate pii and pii we recall that pi = p,O+ RT In 8, pi = pco + RT In xi + RT In yi
(12)
and since aii and (1 are not a function of the temperature
Hence (20)
= ; {-RT
Substitution of eqns (16), (18) and (19) into the derivative dP,IaT yields
+ 2a,2xz - 2a12xs2 - 2u,3x3*
+(a12+a,3-az~)x~-22Ia,2+a*3-(L13)x2x31 +5
= A {-RT
(1 - 2x, - 2x2 - 6x1~2)
+ 2u,,~,
zap,
+~(1-2xI-Zx~+6x,x~)
R
(21)
=Tfu12
which is the same value for the case when no ternary effects are present. Differentiation of eqn (8) with respect to x2 using the facts that x3 = 0 and proximity to the critical binary point yields
- 2~r,,~,’ - 2a,z~z2
+ (a12 + aa3 - a&z
( >
n -# (13)
- 2(a12 + a13 - a&2x31
(14)
ap,
nc-1 ax* 2
L
=
0
(22)
Liquid-liquid phase splitting-11
(23) Equation (22) permits immediate solution of system (9) aT aX,
( )L= -
ap,/axs aP,laT
(24)
113
cross-sections with inflection points the system will separate into two or more phases and their composition will be given by the contact points of a tangent plane. The curve that joins all these contact points is called the binodal; the boundary curve between the convex and concave-convex parts of the G surface is called the spinodal. It can be shown that at the CST the binodal and the spinodal curves are tangent to each other. In a previous paper [ I] we showed that the equation of the spinodal is given by eqn (3) which is equivalent to
Equation (25) may also be written The Gibbs function is given by aT
_
8x3 c C-J
1 r&z- (an - oL&l - 2cfa12. 2R (Y12
(26)
The following general consequences may be obtained: (a) If no ternary effects are present LI= 0 and equations 25-26 simplify to the relation given by Prigogine [4]. The sign of aT/ax3 will depend on the sign of OL&(o~,~-(Y& SO that aTlax,> for la,s-a23(>o,z and #r/ax, < 0 for 101,3- (Y*31<(Y,Z. (b) If the third component is about equally soluble in both 1 and 2 then
G=Gid+GE
(32)
where Gid = RTZx, In xi
and GE is given by eqn (1). We now proceed to calculate the second derivatives of G with respect to xi and obtain
and
aT
( ).=-a ax,
‘( Q,z--2a)
(Y will probably be a very small number so that we can expect that i~T/ax, < 0 and that mutual solubility for 1+ 2 will increase. (c) If the third component is very much more soluble in one of the components than the other, for example, if
(33)
- 2axx
(34)
-2ax z
(35)
-=fg ax+ax,
06) Substitution into eqn (31) yields the two following alternative equations for the spinodal curve (a)
1- 8x1~2~3- 2(A_x,x2+ Bx,xy + Cxzxx) - 6Dx,xzxx + ~D’x,x,~x,*
- D2(2x, - 1)%1x2xs
+ ~Dx,x~x~~(B + C -A) + 2Dx,x2*x3(A - B + C)
aT
1 a:9-u,la ax, .==a Q12
+ 2Dx,*xzxs(A + B - C) = 0
( )
Again we can expect that a z3> 2alza! so that aTI& > 0 and solubility of 1+ 2 will decrease. In the exceptional case that (Yis very large (very strong interactions in the molecular cluster 1 + 2 + 3) the opposite effect will be observed. We see that accounting for first-order ternary effects does not change substantially the consequences of using a strictly regular model.
W
(37)
I- SXIXSC~ - 2(Ax,x2 + Bx,x~ + Cxg,) - ~Dx,x>x~ + 4D2x,x~zx,z - Dz(2x, - 1)zx,x2xs + 2Dx,x,x3[A(l-
+ C(l -2X,)]
2x3 + B(l - 2x3
=0
(38)
where
2. THIS SPINODAL CURVE
If the Gibbs function G(x,, x2, T) is plotted at constant temperature and the resulting surface is convex everywhere from below the system will remain homogeneous for all possible concentrations. If the surface has vertical
S=AZ+B2+CZ-2AB-2BC.
We will now discuss Meijering[6].
(3%
some of the cases analyzed by
114
WISNCAK
J.
2.1 Closed-gap solubility Equation (38) will yield a closed miscibility curve for the foIlowing value of the binary parameters A=-10
most of the concentration spinodal
curves
region. The projection of the around xj by the analy-
are symmetrical
tical nature of eqn (38).
B=C=O.
Figure 1 shows how the size of the immiscibility region is affected by the value of the ternary parameter 8. Negative values of 6 increase the relative solubility while positive values decrease it. For a value of 8 equal in size but opposite in sign to A the immiscibility region covers
2.2 Band solubility Here we consider the following values of the energy parameters A = 0.50
B = 2.25
C = 2.50
Figure 2 shows the appearance of an immiscibility band.
”
x2’
\
XI
Fig. I.
/ X2
”
Y
Fig. 2.
v
l/
Lf
”
XI
Liquid-liquid A change segregation ponent 3.
phase splitting--II
in sign and size in parameter 6 moves the area to solutions more concentrated in com-
2.3 Contribution of the fernury tern We assume A = B = C = 0 and DA 0. Hence S = 0 and
Equation (37a) becomes &
60 + D2(2x,x2 + 2x,x3 + 2X&
- Xl2 - x*2 - x32)= 0
(39)
This equation can be solved easily by taking advantage that it is symmetrical on x1, xs. Hence replacing x1 = xj = 0.5 (1 - x3 ,we get: 4
x2(1- x2)I - 60 t Dz[ 1 - (1 - x$ - 2xz2] = 0.
115 NOTATtON
A. B, C, D
G GE P PI R
T TC & a Qi YI s Jli
constants Gibbs function
excess Gibbs function pressure surface defined by eqn (5) gas constant temperature critical solution temperature molar fraction of component ternary constant energy interaction parameter activify coefficient ternary energy parameter chemical potential
i
(40) REFERENCES
The roots of this quadratic are
111Wisniak
J., Chem. Engng ScL 1983 38 %9.
121 Tompa H., Trans. Faraday
Sot. 1949 45 1142.
[31 Scott R. P., J Chem. Phys. 1949 17 268,279. (41) for either D>8 or that Figs. 1 and 2 to the spinodal curve and do not have the same significance as the phase equilibrium curve.
so that the system
will be unstable
D <28.5. It should be remembered relate
basic
I41 Prigogine I., Chemical Thermodynnmics. Longmans, London 1954. I51 Meijering J. L., Philips Res. Rep. 1950 5 333. [61 Meijering J. L., Philips Res. Rep. 1951 6 183. [71 Hirschfelder J. D., Stevenson D. and Eyring H., J. Chem. Phys. 1937 5 896. C81 Barber 3. A. and Fock W.. Disc. Faraday Sot. 1953 15 188. t91 Andersen G. R. and Wheeler I. C., J. Chem. Plays. 1978 69 3403. II01 Walker J. S. and Vause C. A., Phys. Lett. 1980 79A 421.
SYSTEMS
PHASE AND
THE
SPLITTING-II SPINODAL
CURVE
JAIME WISNIAK Department of Chemi&l Engineering, Ben-Gurion University of the Negev, Beer Sheva, Israel (Accepted 28 March
1983)
Abstract-Solubility phenomena are analyzed for ternary nonideal liquid systems for which GE = ,T~~x,x~+01~~x,x~ + WSCZY~ + UXIXSI where a represents the ternary effect. Spinodal curves for this type of systems can generate any of the known miscibility curves. the temperature
produces an increase in the attractive interaction beiween like molecules which may lead to phase splitting. If a strong directional attraction is present, it will increase enough at lower temperatures and the system will again homogenize. Closed gap solubility curves have also been analyzed by Barker and Fock[E], Andersen and Wheeler[9], and Walker and Vause[lO] using decorated latticemodels. The following analysis will assume the presence of ternary effects and that the excess Gibbs function can be
In a previous publication{11 we have discussed the phenomena associated with phase segregation in binary liquid systems. The different analytic models used for describing nonideality have been tested for their ability to predict separation into two liquid phases. In this paper we will analyze the influence of a third component on the mutual solubility of two liquids; the discussion will also include the possibility of closed-loop solubility diagrams in ternary systems. Calculation of equilibrium phase diagrdms in ternary systems has been discussed by Tompa[2], Scott[3], Prigogine[4] and Meijering[S, 61. Both Tompa and Scott discussed the analytical relations present in polymer solutions making some severe assumptions regarding the value of the energy parameters and the nature of the species present in the system. Prigogine[4] has discussed the thermodynamic conditions that unstable phases must satisfy. A detailed analysis is given of a ternary system composed of strictly regular solution for which GE/RT = (Y,~x,x~ + a13x,x3 + (Y~~x~x~.On this basis Prigogine developed an equation that gives the variation of the critical solution temperature (CST) of the binary system 1 + 2 caused by the addition of a small amount of a third component. He concluded that if the third component is equally soluble in the first two components mutual solubility of 1+2 will increase. If the third component is much less soluble in one of the two components its addition will cause an increase in the CST. The analytical development includes also a calculation of the spinodal curve. Meijering[S, 63 has made a very thorough analysis of miscibility gaps and tie-lines in the ternary system that behave like strictly regular solutions. He classified the possible system in eight categories depending on the signs of binary energy parameters, Uij and determined
the necessary
described
by
GE/RT
= o~,zxIxz+ al3x,x,+
az3x2x3+
ax,xzx3.
Coefficients aq and 01 are assumed to be independent of temperature. Determination of coefficient OLwill require information on the ternary systems.
1. INFLUENCE
OF THE THlRD COMF’ONENT
ON THE MUTUAL
SOLUBILITY
In a previous publication [ l] it has been indicated that the line that separates stable states from nonstable states is determined by either of the following equations fQ,~ZZ-fi”:2=0 cL22Wz3- &
(2)
(3)
= 0
PXSlLL,-P:B=O
(4)
where the symbol pii or pi1’identifies the second derivative of G with respect to the number of moles given by the indices. Equations (2), (3) and (4) are equivalent to
conditions for the appearance of open or
closed miscibility gaps. Hirschfelder, Stevenson and Eyring[7] have discussed the theoretical implications of lower CST and closedloop existence curves. They suggested that there is a competition between strongly directional attractive interaction between tbe unlike components of the mixture, such as a hydrogen bond, and a weaker nondirectional repulsion. At high temperatures the entropy of mixing predominates and the system is homogeneous. Lowering
(1)
The critical point of the ternary by
111
system
is characterized
112 In
5. WISNIAK addition the Gibbs-Buhem
relation gives [4] = ; (-RT +(cr,r+ 7
Substitution into eqn (5) yields a more symmetrical equation for surface PI
+ 2~~23x3- 2az3~3~- 2~x1’ al3+LYZJ)XI -2(LY12-a13fa&1x31
(1- 2x2 - 2x3 + 6~x3).
(15)
For the binary system in question 1 + 2 we have xg = 0 and a strictly regular behavior. Hence Intersection of P, with the surface Pz defined by the first relation in eqn (6) will generate the critical solubility line. Following the development of Prigogine[4] we obtain the influence of the third component on the critical temperature, dT/&, by the simultaneous solution of the following differential equations
(Xl)= = (x& = 0.5
T, = g
(16)
At the critical point eqns (13)-(15) will yield
(9) Making use of eqn (1) and
Hence
&lGE = RT In yi (-_)ani P. T,“j’“i
(10)
we obtain apI3
2 ( ) =-au12+a13+a,,+E
ndX3c
(18)
a a/.& -(YIz+(Yl,+%3+-. n ( ax, ) C= 2
(11) In addition In order to calculate pii and pii we recall that pi = p,O+ RT In 8, pi = pco + RT In xi + RT In yi
(12)
and since aii and (1 are not a function of the temperature
Hence (20)
= ; {-RT
Substitution of eqns (16), (18) and (19) into the derivative dP,IaT yields
+ 2a,2xz - 2a12xs2 - 2u,3x3*
+(a12+a,3-az~)x~-22Ia,2+a*3-(L13)x2x31 +5
= A {-RT
(1 - 2x, - 2x2 - 6x1~2)
+ 2u,,~,
zap,
+~(1-2xI-Zx~+6x,x~)
R
(21)
=Tfu12
which is the same value for the case when no ternary effects are present. Differentiation of eqn (8) with respect to x2 using the facts that x3 = 0 and proximity to the critical binary point yields
- 2~r,,~,’ - 2a,z~z2
+ (a12 + aa3 - a&z
( >
n -# (13)
- 2(a12 + a13 - a&2x31
(14)
ap,
nc-1 ax* 2
L
=
0
(22)
Liquid-liquid phase splitting-11
(23) Equation (22) permits immediate solution of system (9) aT aX,
( )L= -
ap,/axs aP,laT
(24)
113
cross-sections with inflection points the system will separate into two or more phases and their composition will be given by the contact points of a tangent plane. The curve that joins all these contact points is called the binodal; the boundary curve between the convex and concave-convex parts of the G surface is called the spinodal. It can be shown that at the CST the binodal and the spinodal curves are tangent to each other. In a previous paper [ I] we showed that the equation of the spinodal is given by eqn (3) which is equivalent to
Equation (25) may also be written The Gibbs function is given by aT
_
8x3 c C-J
1 r&z- (an - oL&l - 2cfa12. 2R (Y12
(26)
The following general consequences may be obtained: (a) If no ternary effects are present LI= 0 and equations 25-26 simplify to the relation given by Prigogine [4]. The sign of aT/ax3 will depend on the sign of OL&(o~,~-(Y& SO that aTlax,> for la,s-a23(>o,z and #r/ax, < 0 for 101,3- (Y*31<(Y,Z. (b) If the third component is about equally soluble in both 1 and 2 then
G=Gid+GE
(32)
where Gid = RTZx, In xi
and GE is given by eqn (1). We now proceed to calculate the second derivatives of G with respect to xi and obtain
and
aT
( ).=-a ax,
‘( Q,z--2a)
(Y will probably be a very small number so that we can expect that i~T/ax, < 0 and that mutual solubility for 1+ 2 will increase. (c) If the third component is very much more soluble in one of the components than the other, for example, if
(33)
- 2axx
(34)
-2ax z
(35)
-=fg ax+ax,
06) Substitution into eqn (31) yields the two following alternative equations for the spinodal curve (a)
1- 8x1~2~3- 2(A_x,x2+ Bx,xy + Cxzxx) - 6Dx,xzxx + ~D’x,x,~x,*
- D2(2x, - 1)%1x2xs
+ ~Dx,x~x~~(B + C -A) + 2Dx,x2*x3(A - B + C)
aT
1 a:9-u,la ax, .==a Q12
+ 2Dx,*xzxs(A + B - C) = 0
( )
Again we can expect that a z3> 2alza! so that aTI& > 0 and solubility of 1+ 2 will decrease. In the exceptional case that (Yis very large (very strong interactions in the molecular cluster 1 + 2 + 3) the opposite effect will be observed. We see that accounting for first-order ternary effects does not change substantially the consequences of using a strictly regular model.
W
(37)
I- SXIXSC~ - 2(Ax,x2 + Bx,x~ + Cxg,) - ~Dx,x>x~ + 4D2x,x~zx,z - Dz(2x, - 1)zx,x2xs + 2Dx,x,x3[A(l-
+ C(l -2X,)]
2x3 + B(l - 2x3
=0
(38)
where
2. THIS SPINODAL CURVE
If the Gibbs function G(x,, x2, T) is plotted at constant temperature and the resulting surface is convex everywhere from below the system will remain homogeneous for all possible concentrations. If the surface has vertical
S=AZ+B2+CZ-2AB-2BC.
We will now discuss Meijering[6].
(3%
some of the cases analyzed by
114
WISNCAK
J.
2.1 Closed-gap solubility Equation (38) will yield a closed miscibility curve for the foIlowing value of the binary parameters A=-10
most of the concentration spinodal
curves
region. The projection of the around xj by the analy-
are symmetrical
tical nature of eqn (38).
B=C=O.
Figure 1 shows how the size of the immiscibility region is affected by the value of the ternary parameter 8. Negative values of 6 increase the relative solubility while positive values decrease it. For a value of 8 equal in size but opposite in sign to A the immiscibility region covers
2.2 Band solubility Here we consider the following values of the energy parameters A = 0.50
B = 2.25
C = 2.50
Figure 2 shows the appearance of an immiscibility band.
”
x2’
\
XI
Fig. I.
/ X2
”
Y
Fig. 2.
v
l/
Lf
”
XI
Liquid-liquid A change segregation ponent 3.
phase splitting--II
in sign and size in parameter 6 moves the area to solutions more concentrated in com-
2.3 Contribution of the fernury tern We assume A = B = C = 0 and DA 0. Hence S = 0 and
Equation (37a) becomes &
60 + D2(2x,x2 + 2x,x3 + 2X&
- Xl2 - x*2 - x32)= 0
(39)
This equation can be solved easily by taking advantage that it is symmetrical on x1, xs. Hence replacing x1 = xj = 0.5 (1 - x3 ,we get: 4
x2(1- x2)I - 60 t Dz[ 1 - (1 - x$ - 2xz2] = 0.
115 NOTATtON
A. B, C, D
G GE P PI R
T TC & a Qi YI s Jli
constants Gibbs function
excess Gibbs function pressure surface defined by eqn (5) gas constant temperature critical solution temperature molar fraction of component ternary constant energy interaction parameter activify coefficient ternary energy parameter chemical potential
i
(40) REFERENCES
The roots of this quadratic are
111Wisniak
J., Chem. Engng ScL 1983 38 %9.
121 Tompa H., Trans. Faraday
Sot. 1949 45 1142.
[31 Scott R. P., J Chem. Phys. 1949 17 268,279. (41) for either D>8 or that Figs. 1 and 2 to the spinodal curve and do not have the same significance as the phase equilibrium curve.
so that the system
will be unstable
D <28.5. It should be remembered relate
basic
I41 Prigogine I., Chemical Thermodynnmics. Longmans, London 1954. I51 Meijering J. L., Philips Res. Rep. 1950 5 333. [61 Meijering J. L., Philips Res. Rep. 1951 6 183. [71 Hirschfelder J. D., Stevenson D. and Eyring H., J. Chem. Phys. 1937 5 896. C81 Barber 3. A. and Fock W.. Disc. Faraday Sot. 1953 15 188. t91 Andersen G. R. and Wheeler I. C., J. Chem. Plays. 1978 69 3403. II01 Walker J. S. and Vause C. A., Phys. Lett. 1980 79A 421.