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Phase diagram near the three-phase temperature region of polymer solution
COLLOIDS AND SURFACES ELSEVIER
B
Colloids and Surfaces B: Biointerfaces6 (1996) 57--62
Phase diagram near the three-phase temperature region of polymer solution Toshiaki Dobashi a,,, Mitsuo Nakata
b
a Department oflBiological and Chemical Engineering, Faculty of Engineering, Gunma Universit£, Kiryu, Gunma 376, Japan b Department of Polymer Science, Faculty of Science, Hokkaido University, Sapporo 060. Japan Received 10 July 1995; accepted 22 August 1995
Abstract
Three-phase coexistence curves, a cloud point curve and a shadow curve were observed for the system polystyrene I (Mw= 1.73 x 104)+polystyrene II (Mw=7.19 x 10S)+methylcyclohexane with different overall volume fractions of polystyrene and a constant volume fraction of polystyrene II in total polystyrene of 0.0500. The observed cloud point curve and shadow curve intersected each other when extrapolated below the three-phase temperature region. The intersection was interpreted as a metastable critical point by comparing it with a phase diagram calculated from an empirically determined free energy. A simple method for determining the entire three-phase coexistence curve was demonstrated by the calculated three-phase behavior. Keywords: Coexistence curve; Free energy; Polymer solution; Stability of critical point; Three-phase equilibrium
I. Introduction
Tricritical phenomena and associated threephase equilibria are of particular interest because of their fundamental role in our understanding of the thermodynamics of multicomponent systems, which is involved in various problems in the fields of colloid science and biophysics: microemulsions in water + oil + amphiphile systems exhibit a threephase behavior similar to that evolving from tricritical points [1]. Equilibrium polymerization of actin and tubulin into large supramolecular structures was predicted to show the tricritical phenomena, examples of which are found in 3He-4He mixtures and sulphur solutions through the n--+0 vector model [2]. The tricritical phenomena have * Corresponding author. 0927-7765/96/$15.00 U*1996 Elsevier Science B.V. All rights reserved SSDI 0927-7765 (95)01240-0
been extensively investigated both theoretically and experimentally using model ternary and quaternary solutions [-3,4]. In previous studies, we investigated coexistence curves of homologous polymer solutions in the light of the tricritical phenomena [5,6]. The bimodal polymer solution polymer I + polymer II + solvent is a simple multicomponent system for which the three-phase equilibrium can be observed at atmospheric pressure and the free energy can be expressed by only one interaction parameter between segment and solvent. The Flory-Huggins theory for the free energy gives the three-phase equilibrium brought about via a heterogeneous double plait point from which metastable and unstable critical points appear [7]. For the ternary system polystyrene ( P S ) + PS+methylcyclohexane (MC), the observed three-phase equilibrium was consistent with
58
T. Dobashi, M. Nakata/Cblloids' Surfaces B." BiointerJaces 6 (1996) 57 62
the heterogeneous double plait point mechanism and was described by a symmetrical coexistence curve characteristic of the tricritical phenomena [6,8]. As an extension of Ref. 8, we will, in this study, elucidate coexistence curves, the cloud point curve and the shadow curve of the system PS + PS + MC near the three-phase region by comparing them with a phase diagram calculated from an empirically determined free energy. Another purpose of this paper is to clarify the relationship between the three-phase equilibrium and the overall composition of ternary systems. This relationship is obvious for the two-phase equilibrium in binary systems. Binary systems of the critical composition give rise to the entire twophase coexistence curve with the critical point at the top of the coexistence curve with variation in temperature. For ternary systems the entire threephase coexistence curve has two critical end points, at which two among the three coexisting phases become identical. The upper temperature limit T* and lower temperature limit T* of the three-phase region are given by the upper critical end point Pu and lower critical end point PL respectively. The two tie lines at the critical end point temperatures intersect each other at a point (IP) on a composition triangle. Only when the overall composition of a ternary system coincides with that at IP can the whole locus of the three-phase coexistence curve be monitored with changing temperature. The three-phase equilibrium starts from a twophase equilibrium with one phase being at the critical condition and the other phase being unchanged [8,9]. Thus, it is very important to determine IP in three-phase experiments. For ternary systems with the overall composition different from that at IP the three-phase separation is caused by an abrupt appearance of a third phase in the two-phase state. In experiments it is difficult to predict whether the third phase appears as an upper, a middle or a lower phase in the solution cell. Therefore, the relative position of the overall composition and the third phase on a composition triangle was investigated.
(PS II) were characterized in Ref. [8]. Methylcyclohexane was double-distilled after passing through silica gel. P S I and PS II were put into a small flask, the volume fraction of PS II in total PS (~20) was adjusted to 0.0500, and dissolved in cyclohexane. The solution was subdivided into two fractions, followed by lyophilization with a vacuum pump. Final solutions with ~20=0.0500 were obtained by dissolving the dry samples in methylcyclohexane. The overall volume fractions of total PS, ~b~0,were 0.1837 (system C) and 0.2435 (system D). The composition of total PS in each coexisting phase was determined from refractive index measurements [10]. Cloud point temperatures Tp were determined by observing the flickering patterns of scattered laser light and the illuminating spot of the transmitted beam on a white screen. The shadow curve is the locus of the composition of the incipient phase, i.e. conjugate to the cloud point curve. The composition ~bs of the incipient phase was determined at a temperature just below Tp. Measurements were made in a water bath, the temperature of which was controlled within +0.003 K. Fig. 1 shows the observed coexistence curves for systems C and D together with the curve for system B (~2o=0.0500, ~b~o=0.2184) in Ref. [8]. The S-shaped three-phase coexistence curve, which is not clear on this temperature scale, appears in a narrow temperature range at 24.1 24.7°C. The
35
o~,, o
~°
x
\
3o o
¢-,
~
AR%
A D g ~o
25
o
~
,
",
Lx
, o o ., '~",Ng
2
20
rl
',:?,
2
ao
g 10 0.00
~o
, 0.I0
' 0.20
I
0,30
0,40
2. Experimental Well-fractionated polystyrenes of molecular weights M , = 1 . 7 3 x 10 4 ( P S I) and 7.19 x 105
Fig. 1. Observed coexistence curves for the systems C (O), B (&) [8] and DtD). The broken line and chain line are the cloud point curve and shadow curve respectively.
T. Dobashi, M. Nakata/Colloids Surfaces B: Biomterfaces 6 (1996) 57 62
three-phase coexistence curves obtained for the three systems coincide with each other. The agreement became excellent when each coexistence curve was shifted slightly ( ~ 0.2 ° C) along the temperature axis. In the following, the three coexisting phases are denoted by e, fi and ;~ from the top to the bottom of the solution cell. Each three-phase coexistence curve has an upper limiting temperature 7~ and a lower limiting temperature TL, where the curve is not closed. When the temperature is lowered, small amounts of phase /3 and phase e appear at Tu for systems C and D respectively, and phase /3 disappears at T~ for both systems. Above the three-phase temperature the two-phase region is developed depending on Go, while below the three-phase region both the dilute and concentrated branches of the coexistence curve appear to be independent of ~b~o. The cloud point temperatures Tp were determined as 35.29, 30.70 and 28.03°C for systems C, B and D respectively. The cloud points yield a cloud point curve shown by the broken line in Fig. 1. The points conjugated to the cloud point curve give a shadow curve, which is represented by the chain line.
3. Calculation
The following Gibbs free energy of mixing expresses the phase diagram of the system PS + MC semi-quantitatively [ 11 ]: AGmix/NRT
= ~bo In 4o + ~Y'~bimi-1 in ~ + g~bo~b~ (1)
g = a + h/( 1 - Cq~s)
(2)
where ~bo,~bi and ~b~are the volume fractions of the solvent, polymer homolog i and the total polymer respectively, mi is the chain length of the polymer, N is the total number of moles of lattice sites, and R and T have their usual meanings. The interaction parameter g is regarded as a phenomenological one that is to be determined from experiments, although Eqs. (1) and (2) are originally derived theoretically. Here, we used the coefficients for g determined from molecular weight dependence of the critical point: a = -0.1091, b = - 0 . 5 8 3 2 + 278.6/T+ 1.695 x 10-3T, and c=0.2481 [11]. The
59
coexistence curve was calculated by solving three simultaneous equations of A ~ = Zl~o~
(3a)
~
{3bt
= ~1 ~
Here, d/~ is the chemical potential of mixing, the subscripts 0, l and 2 denote MC, P S I and PS II respectively, and the superscripts a and /3 denote the coexisting phases. When we take ~b~= q~l + q~2 and ~bt =~b 1 -q~2 as the composition variables, instead of q~l and ~b2, we can rearrange Eq. (3a) so that ~b7 is an explicit function of three other composition variables q~', ~ and ~bt~. Substituting ~b~' into Eqs. (3b) and (3c), the above set of equations reduces to two independent equations. For a given ~b~', the other composition variables q~ and ~bt~ are calculated by the method of bisection with two variables at desired temperatures. The set of four composition variables q~, ~b~',~b~ and ~b~~ yields a tie line. The curve connecting the end of the tie line is the cross-section of the binodal surface at a particular temperature. The binodal surface is obtained by piling up such a cross-section over the whole temperature range. The cloud point is determined from the intersection of the tie line ~20 = q~2o/(q~lo+~b2o)=0.050 and the binodal curve drawn in the ~bl-~b2 diagram at each desired temperature. The shadow curve is obtained as the locus of the conjugate point of the cloud point. To determine the compositions of the coexisting phases for a given set of overall compositions ~blo and ~b2oanother equation which follows the lever rule is required: (~b2o - ~b~)(~b~- ~b~)= (~blO- ~b~)(~b~- ~b~)
{4)
The method of bisection was used again to find a set of compositions which satisfied Eq. (4) from among the solutions of Eq. (3). The critical points and their stability at given values of ~2o were calculated using the critical conditions for threecomponent systems [5]. The calculated coexistence curves, cloud point curve and shadow curve for ~2o = 0.0500 are shown in Fig. 2. Curves C, B and D are described for ~b~o=0.1837, 0.2184 and 0.2435 respectively. The three-phase equilibrium is obtained in a temper-
60
T. Dobashi, M. Nakata/Colloids Surfaces B." Biointerfaces 6 (1996) 57 62
36 C
!
~ I
30
26
20 B
15
0,00
i
i
i
0,10
0.20
0,30
Fig. 2. Calculated coexistence curves for the systems C, B [83 and D. The broken line and chain line are the cloud point curve and shadow curve respectively,for ~2o=0.050. The filled circle is the critical point for ~2o=0.050. Open and filled squares are the upper and lower critical end points.
ical point is obtained only in the temperature region above T*. Figs. 3(a) and 3(b) show typical cloud point curves and shadow curves with a metastable critical point (~2o = 0.020) and an unstable critical point (~2o=0.005) respectively. The cloud point curves exhibit a discontinuous change in slope near T*, while the shadow curve moves to the dilute branch of the coexistence curve at the same temperature. The discontinuity of the cloud point curves represents an intersection with the three-phase line. In Fig. 3(a), the metastable critical point is located at the intersection of the extrapolated cloud point curve and shadow curve, as indicated by the dotted lines. In Fig. 3(b), the unstable critical point does not fall on the intersection of the extrapolated lines. The above character(a) 36
ature range from 21.4 to 23.4°C. Both cloud point temperature and shadow point temperature linearly decrease with increasing ~bso. For ~2o = 0.0500 a critical point was found at the intersection of the cloud point curve and the shadow curve at T~=28.266°C and ~bsc=0.2293. To understand the entire phase diagram of this ternary system, we calculated the critical points defined as (~2/~m2)~l--~(~2[22/~2m2)#1=O and the sign of (6~3/,t2/633m2)~l to investigate the stability of each critical point for various values of ~20- Here, m2 is the molality of PS II. As a typical example we also calculated cloud point curves and shadow curves for the systems with metastable critical point ((~3~t2/~3m2)ul > 0 but the critical point exists in the two-phase region) and unstable critical point (((~3~t2/O3m2)t~l-~0). The critical point is stable for 420 > ~'0L =0-031 and located above the lower temperature limit T* of the three-phase region. The critical point is also stable at very low PS II concentration ¢2o<2 x 10-5 and is located between the upper temperature limit T~ of the three-phase region and the critical point T~1 for P S I + MC with T~ < Tel < T~. The critical point is metastable for 9.2 x 10 -3 < ~2o<0.031 and 2 x 10 - 5 < 4 2 o < 5 . 9 x 10 5 and unstable for 5.9 x 10 -5 < ~2o<9.2 x 10 -3. Thus, the stable crit-
\
\
3O
\
\
~. 25
,
\
I:.................~',N / \,', \
20 ['
\
15
I
0.'10
0.00
O. 20
0)30
(b) 35 \ \ \
30
\ \ \
P
\
26
\ \ \
f ...............
20
~,.\ \_ ° .
/
', \',
/ / 10 0.00
" "X i
t
~,i, ~
0,10
0.20
0,30
Fig. 3. Calculated cloud point curve (broken line) and shadow curve (chain line), together with the critical point (open circle) for ¢2o=0.020 (a) and ~2o=0.005 (b).
T. Dobashi, M. Nakata/Colloids Surfaces B: Biointerfaees 6 (1996) 57 62
61
istic behavior of the cloud point curve and shadow curve was also obtained for the Flory-Huggins solution [7].
//
~
4. Discussion
The behavior of the observed coexistence curves in Fig. 1 can be compared with that of calculated ones as in Fig. 2. The observed cloud point curve and shadow curve in Fig. 1 intersect each other when extrapolated below the three-phase temperature region. In contrast, the corresponding calculated lines in Fig. 2 intersect each other above the three-phase region exhibiting a stable critical point. This difference in critical point is caused by the approximate expression of Eq. (2) for the present system. According to the analysis based on the free energy, the stable critical point can exist only above the lower limit of the three-phase temperature region as an intersection of the cloud point curve and shadow curve. For the present system the stable critical point can be expected for ~2o>0.050 and for very small values of ~2o [8]. As shown in Fig. 3 the intersection of the extrapolated cloud point curve and the shadow curve may give a metastable critical point. We have to confirm whether the observed intersection at T = 15.5°C and ~bs=0.34 gives a metastable critical point or not. To analyze the behavior of the critical point for the present system a more precise expression for the free energy is necessary. In experiments the transition between the twophase region and the three-phase region can be observed as an appearance and a disappearance of a phase among the upper c~, middle fl, and lower y phase in the solution cell. It depends on the overall composition which phase plays a main role at the transition. This problem can be easily solved with a calculated phase diagram. In Fig. 4 the threephase region projected on a composition triangle is divided into the four regions I, II, III and IV by the tie lines at T~ and T*. The three-phase line is divided into the three parts c~, fi and 7 by the critical end points Pu and PL. As regards the overall compositions in regions I, II, III and IV, phases :~, ~, fi and [3 newly appear at Tu respectively, and phases [L ?', fl and 7 disappear at TL respec-
o~
//
>
Ou
//
0. I0 0
O.OS
~2 Fig. 4. Calculated three phase region, together with the tie lines PvOu and PL RL at the upper and lower critical end temperatures respectively. Pu and PL are the upper and lower critical end points. The broken line denotes ~2=0.050. The filled circles are the concentrations of the systems A [8], B [8], C and D.
tively. The behaviors of systems C and D with the overall compositions indicated in Fig. 4 agree with the calculated one. This is also the case for systems A and B in Ref. [8] as shown in Fig. 4; that is, phase ~ appears at Tu for systems A and B and phase ? and phase fl disappear at TL respectively, for systems A and B. Since the system that corresponds to the intersection IP of the tie lines at T* and T* is found inside the three-phase composition triangle over the entire temperature range from T* to T*, we can determine the entire three-phase coexistence curve with this system by varying the temperature. If we measure ~bs values of the three coexisting phases just below T~ by varying Go at constant ~blo or q~2o, we will find that the composition difference of phase ~ and phase fl tends to zero when the point for q~o crosses the tie line at T[, on a composition triangle. Similarly, just above TL the composition difference of phase fi and phase y tends to zero when the overall composition crosses the tie line at T*. Therefore, using a set of such experimental data obtained at different values of ~blO or ~b2o, we obtain two sets of points on the
62
T. Dobashi, M. Nakata/Colloids Surfaces B." Biointerfaces 6 (1996) 57-62
0.15 -
Similarly, we have q~sl =0.249 and ~bsu=0.184 for q~2=0.0075. Using these data, the tie lines at T~, and T~ are represented by the equations ~b~= 20.8q~2 + 0.093 and ~b~= 2.6~b2 + 0.1645 respectively, and then I P is determined from the intersection as q~1o=0.175 and ~bzo=0.004. The present m e t h o d will be useful to obtain the entire three-phase equilibrium behavior, both in experiments and calculations.
%e
<
0.00°°5 0.10
~
. 0.15
~ 0.20
0.25
Fig. 5. Concentration difference in phase e and phase /3 just below the upper three-phase boundary A~bsuand that in phase/3 and phase I' just above the lower three-phase boundary A~bsL f o r ~2o = 0.0025. tie lines at T~ and T*. Thus, we can determine these tie lines, and then I P as the intersection of the two lines. The determination of I P was d e m o n strated by a calculation for the present system. Fig. 5 shows the difference Aq~u of ~bS of phase c~ and phase/~ just below Ttj and the difference A~b~L of ~bs of phase /~ and phase 7 just above TL calculated for the solution with 420=0.0025. The two compositions where concentration difference Aq~s is zero are ~b~r=0.145 and ~bsU=0.171.
References [1] [2] [3] [4] I-5]
M. Kahlweit, Science, 240 (1988) 617. J.C. Wheeler and P. Pfeuty, Phys. Rev. A, 24 (1981) 1050. R.L. Scott, Acc. Chem. Res., 20 (1987) 97. J.C. Lang, Jr. and B. Widom, Physica, 81A (1975) 190. T. Dobashi and M. Nakata, J. Chem. Phys., 99 (1993) 1419. 16] M. Nakata and T. Dobashi, J. Chem. Phys., 84 (1986) 5782. [7] K. Solc, J. Polym. Sci., Polym. Phys. Ed., 20 (1982) 1947. 1-8] T. Dobashi and M. Nakata, J. Chem. Phys., 84 (1986) 5775. [9] I.L. Pegg, Kinam, 6A (1984) 3. [10] M. Nakata, N. Kuwahara and M. Kaneko, J. Chem. Phys., 62 (1975) 4278. [11] T. Dobashi, M. Nakata and M. Kaneko, J. Chem. Phys., 72 (1980) 6692.
B
Colloids and Surfaces B: Biointerfaces6 (1996) 57--62
Phase diagram near the three-phase temperature region of polymer solution Toshiaki Dobashi a,,, Mitsuo Nakata
b
a Department oflBiological and Chemical Engineering, Faculty of Engineering, Gunma Universit£, Kiryu, Gunma 376, Japan b Department of Polymer Science, Faculty of Science, Hokkaido University, Sapporo 060. Japan Received 10 July 1995; accepted 22 August 1995
Abstract
Three-phase coexistence curves, a cloud point curve and a shadow curve were observed for the system polystyrene I (Mw= 1.73 x 104)+polystyrene II (Mw=7.19 x 10S)+methylcyclohexane with different overall volume fractions of polystyrene and a constant volume fraction of polystyrene II in total polystyrene of 0.0500. The observed cloud point curve and shadow curve intersected each other when extrapolated below the three-phase temperature region. The intersection was interpreted as a metastable critical point by comparing it with a phase diagram calculated from an empirically determined free energy. A simple method for determining the entire three-phase coexistence curve was demonstrated by the calculated three-phase behavior. Keywords: Coexistence curve; Free energy; Polymer solution; Stability of critical point; Three-phase equilibrium
I. Introduction
Tricritical phenomena and associated threephase equilibria are of particular interest because of their fundamental role in our understanding of the thermodynamics of multicomponent systems, which is involved in various problems in the fields of colloid science and biophysics: microemulsions in water + oil + amphiphile systems exhibit a threephase behavior similar to that evolving from tricritical points [1]. Equilibrium polymerization of actin and tubulin into large supramolecular structures was predicted to show the tricritical phenomena, examples of which are found in 3He-4He mixtures and sulphur solutions through the n--+0 vector model [2]. The tricritical phenomena have * Corresponding author. 0927-7765/96/$15.00 U*1996 Elsevier Science B.V. All rights reserved SSDI 0927-7765 (95)01240-0
been extensively investigated both theoretically and experimentally using model ternary and quaternary solutions [-3,4]. In previous studies, we investigated coexistence curves of homologous polymer solutions in the light of the tricritical phenomena [5,6]. The bimodal polymer solution polymer I + polymer II + solvent is a simple multicomponent system for which the three-phase equilibrium can be observed at atmospheric pressure and the free energy can be expressed by only one interaction parameter between segment and solvent. The Flory-Huggins theory for the free energy gives the three-phase equilibrium brought about via a heterogeneous double plait point from which metastable and unstable critical points appear [7]. For the ternary system polystyrene ( P S ) + PS+methylcyclohexane (MC), the observed three-phase equilibrium was consistent with
58
T. Dobashi, M. Nakata/Cblloids' Surfaces B." BiointerJaces 6 (1996) 57 62
the heterogeneous double plait point mechanism and was described by a symmetrical coexistence curve characteristic of the tricritical phenomena [6,8]. As an extension of Ref. 8, we will, in this study, elucidate coexistence curves, the cloud point curve and the shadow curve of the system PS + PS + MC near the three-phase region by comparing them with a phase diagram calculated from an empirically determined free energy. Another purpose of this paper is to clarify the relationship between the three-phase equilibrium and the overall composition of ternary systems. This relationship is obvious for the two-phase equilibrium in binary systems. Binary systems of the critical composition give rise to the entire twophase coexistence curve with the critical point at the top of the coexistence curve with variation in temperature. For ternary systems the entire threephase coexistence curve has two critical end points, at which two among the three coexisting phases become identical. The upper temperature limit T* and lower temperature limit T* of the three-phase region are given by the upper critical end point Pu and lower critical end point PL respectively. The two tie lines at the critical end point temperatures intersect each other at a point (IP) on a composition triangle. Only when the overall composition of a ternary system coincides with that at IP can the whole locus of the three-phase coexistence curve be monitored with changing temperature. The three-phase equilibrium starts from a twophase equilibrium with one phase being at the critical condition and the other phase being unchanged [8,9]. Thus, it is very important to determine IP in three-phase experiments. For ternary systems with the overall composition different from that at IP the three-phase separation is caused by an abrupt appearance of a third phase in the two-phase state. In experiments it is difficult to predict whether the third phase appears as an upper, a middle or a lower phase in the solution cell. Therefore, the relative position of the overall composition and the third phase on a composition triangle was investigated.
(PS II) were characterized in Ref. [8]. Methylcyclohexane was double-distilled after passing through silica gel. P S I and PS II were put into a small flask, the volume fraction of PS II in total PS (~20) was adjusted to 0.0500, and dissolved in cyclohexane. The solution was subdivided into two fractions, followed by lyophilization with a vacuum pump. Final solutions with ~20=0.0500 were obtained by dissolving the dry samples in methylcyclohexane. The overall volume fractions of total PS, ~b~0,were 0.1837 (system C) and 0.2435 (system D). The composition of total PS in each coexisting phase was determined from refractive index measurements [10]. Cloud point temperatures Tp were determined by observing the flickering patterns of scattered laser light and the illuminating spot of the transmitted beam on a white screen. The shadow curve is the locus of the composition of the incipient phase, i.e. conjugate to the cloud point curve. The composition ~bs of the incipient phase was determined at a temperature just below Tp. Measurements were made in a water bath, the temperature of which was controlled within +0.003 K. Fig. 1 shows the observed coexistence curves for systems C and D together with the curve for system B (~2o=0.0500, ~b~o=0.2184) in Ref. [8]. The S-shaped three-phase coexistence curve, which is not clear on this temperature scale, appears in a narrow temperature range at 24.1 24.7°C. The
35
o~,, o
~°
x
\
3o o
¢-,
~
AR%
A D g ~o
25
o
~
,
",
Lx
, o o ., '~",Ng
2
20
rl
',:?,
2
ao
g 10 0.00
~o
, 0.I0
' 0.20
I
0,30
0,40
2. Experimental Well-fractionated polystyrenes of molecular weights M , = 1 . 7 3 x 10 4 ( P S I) and 7.19 x 105
Fig. 1. Observed coexistence curves for the systems C (O), B (&) [8] and DtD). The broken line and chain line are the cloud point curve and shadow curve respectively.
T. Dobashi, M. Nakata/Colloids Surfaces B: Biomterfaces 6 (1996) 57 62
three-phase coexistence curves obtained for the three systems coincide with each other. The agreement became excellent when each coexistence curve was shifted slightly ( ~ 0.2 ° C) along the temperature axis. In the following, the three coexisting phases are denoted by e, fi and ;~ from the top to the bottom of the solution cell. Each three-phase coexistence curve has an upper limiting temperature 7~ and a lower limiting temperature TL, where the curve is not closed. When the temperature is lowered, small amounts of phase /3 and phase e appear at Tu for systems C and D respectively, and phase /3 disappears at T~ for both systems. Above the three-phase temperature the two-phase region is developed depending on Go, while below the three-phase region both the dilute and concentrated branches of the coexistence curve appear to be independent of ~b~o. The cloud point temperatures Tp were determined as 35.29, 30.70 and 28.03°C for systems C, B and D respectively. The cloud points yield a cloud point curve shown by the broken line in Fig. 1. The points conjugated to the cloud point curve give a shadow curve, which is represented by the chain line.
3. Calculation
The following Gibbs free energy of mixing expresses the phase diagram of the system PS + MC semi-quantitatively [ 11 ]: AGmix/NRT
= ~bo In 4o + ~Y'~bimi-1 in ~ + g~bo~b~ (1)
g = a + h/( 1 - Cq~s)
(2)
where ~bo,~bi and ~b~are the volume fractions of the solvent, polymer homolog i and the total polymer respectively, mi is the chain length of the polymer, N is the total number of moles of lattice sites, and R and T have their usual meanings. The interaction parameter g is regarded as a phenomenological one that is to be determined from experiments, although Eqs. (1) and (2) are originally derived theoretically. Here, we used the coefficients for g determined from molecular weight dependence of the critical point: a = -0.1091, b = - 0 . 5 8 3 2 + 278.6/T+ 1.695 x 10-3T, and c=0.2481 [11]. The
59
coexistence curve was calculated by solving three simultaneous equations of A ~ = Zl~o~
(3a)
~
{3bt
= ~1 ~
Here, d/~ is the chemical potential of mixing, the subscripts 0, l and 2 denote MC, P S I and PS II respectively, and the superscripts a and /3 denote the coexisting phases. When we take ~b~= q~l + q~2 and ~bt =~b 1 -q~2 as the composition variables, instead of q~l and ~b2, we can rearrange Eq. (3a) so that ~b7 is an explicit function of three other composition variables q~', ~ and ~bt~. Substituting ~b~' into Eqs. (3b) and (3c), the above set of equations reduces to two independent equations. For a given ~b~', the other composition variables q~ and ~bt~ are calculated by the method of bisection with two variables at desired temperatures. The set of four composition variables q~, ~b~',~b~ and ~b~~ yields a tie line. The curve connecting the end of the tie line is the cross-section of the binodal surface at a particular temperature. The binodal surface is obtained by piling up such a cross-section over the whole temperature range. The cloud point is determined from the intersection of the tie line ~20 = q~2o/(q~lo+~b2o)=0.050 and the binodal curve drawn in the ~bl-~b2 diagram at each desired temperature. The shadow curve is obtained as the locus of the conjugate point of the cloud point. To determine the compositions of the coexisting phases for a given set of overall compositions ~blo and ~b2oanother equation which follows the lever rule is required: (~b2o - ~b~)(~b~- ~b~)= (~blO- ~b~)(~b~- ~b~)
{4)
The method of bisection was used again to find a set of compositions which satisfied Eq. (4) from among the solutions of Eq. (3). The critical points and their stability at given values of ~2o were calculated using the critical conditions for threecomponent systems [5]. The calculated coexistence curves, cloud point curve and shadow curve for ~2o = 0.0500 are shown in Fig. 2. Curves C, B and D are described for ~b~o=0.1837, 0.2184 and 0.2435 respectively. The three-phase equilibrium is obtained in a temper-
60
T. Dobashi, M. Nakata/Colloids Surfaces B." Biointerfaces 6 (1996) 57 62
36 C
!
~ I
30
26
20 B
15
0,00
i
i
i
0,10
0.20
0,30
Fig. 2. Calculated coexistence curves for the systems C, B [83 and D. The broken line and chain line are the cloud point curve and shadow curve respectively,for ~2o=0.050. The filled circle is the critical point for ~2o=0.050. Open and filled squares are the upper and lower critical end points.
ical point is obtained only in the temperature region above T*. Figs. 3(a) and 3(b) show typical cloud point curves and shadow curves with a metastable critical point (~2o = 0.020) and an unstable critical point (~2o=0.005) respectively. The cloud point curves exhibit a discontinuous change in slope near T*, while the shadow curve moves to the dilute branch of the coexistence curve at the same temperature. The discontinuity of the cloud point curves represents an intersection with the three-phase line. In Fig. 3(a), the metastable critical point is located at the intersection of the extrapolated cloud point curve and shadow curve, as indicated by the dotted lines. In Fig. 3(b), the unstable critical point does not fall on the intersection of the extrapolated lines. The above character(a) 36
ature range from 21.4 to 23.4°C. Both cloud point temperature and shadow point temperature linearly decrease with increasing ~bso. For ~2o = 0.0500 a critical point was found at the intersection of the cloud point curve and the shadow curve at T~=28.266°C and ~bsc=0.2293. To understand the entire phase diagram of this ternary system, we calculated the critical points defined as (~2/~m2)~l--~(~2[22/~2m2)#1=O and the sign of (6~3/,t2/633m2)~l to investigate the stability of each critical point for various values of ~20- Here, m2 is the molality of PS II. As a typical example we also calculated cloud point curves and shadow curves for the systems with metastable critical point ((~3~t2/~3m2)ul > 0 but the critical point exists in the two-phase region) and unstable critical point (((~3~t2/O3m2)t~l-~0). The critical point is stable for 420 > ~'0L =0-031 and located above the lower temperature limit T* of the three-phase region. The critical point is also stable at very low PS II concentration ¢2o<2 x 10-5 and is located between the upper temperature limit T~ of the three-phase region and the critical point T~1 for P S I + MC with T~ < Tel < T~. The critical point is metastable for 9.2 x 10 -3 < ~2o<0.031 and 2 x 10 - 5 < 4 2 o < 5 . 9 x 10 5 and unstable for 5.9 x 10 -5 < ~2o<9.2 x 10 -3. Thus, the stable crit-
\
\
3O
\
\
~. 25
,
\
I:.................~',N / \,', \
20 ['
\
15
I
0.'10
0.00
O. 20
0)30
(b) 35 \ \ \
30
\ \ \
P
\
26
\ \ \
f ...............
20
~,.\ \_ ° .
/
', \',
/ / 10 0.00
" "X i
t
~,i, ~
0,10
0.20
0,30
Fig. 3. Calculated cloud point curve (broken line) and shadow curve (chain line), together with the critical point (open circle) for ¢2o=0.020 (a) and ~2o=0.005 (b).
T. Dobashi, M. Nakata/Colloids Surfaces B: Biointerfaees 6 (1996) 57 62
61
istic behavior of the cloud point curve and shadow curve was also obtained for the Flory-Huggins solution [7].
//
~
4. Discussion
The behavior of the observed coexistence curves in Fig. 1 can be compared with that of calculated ones as in Fig. 2. The observed cloud point curve and shadow curve in Fig. 1 intersect each other when extrapolated below the three-phase temperature region. In contrast, the corresponding calculated lines in Fig. 2 intersect each other above the three-phase region exhibiting a stable critical point. This difference in critical point is caused by the approximate expression of Eq. (2) for the present system. According to the analysis based on the free energy, the stable critical point can exist only above the lower limit of the three-phase temperature region as an intersection of the cloud point curve and shadow curve. For the present system the stable critical point can be expected for ~2o>0.050 and for very small values of ~2o [8]. As shown in Fig. 3 the intersection of the extrapolated cloud point curve and the shadow curve may give a metastable critical point. We have to confirm whether the observed intersection at T = 15.5°C and ~bs=0.34 gives a metastable critical point or not. To analyze the behavior of the critical point for the present system a more precise expression for the free energy is necessary. In experiments the transition between the twophase region and the three-phase region can be observed as an appearance and a disappearance of a phase among the upper c~, middle fl, and lower y phase in the solution cell. It depends on the overall composition which phase plays a main role at the transition. This problem can be easily solved with a calculated phase diagram. In Fig. 4 the threephase region projected on a composition triangle is divided into the four regions I, II, III and IV by the tie lines at T~ and T*. The three-phase line is divided into the three parts c~, fi and 7 by the critical end points Pu and PL. As regards the overall compositions in regions I, II, III and IV, phases :~, ~, fi and [3 newly appear at Tu respectively, and phases [L ?', fl and 7 disappear at TL respec-
o~
//
>
Ou
//
0. I0 0
O.OS
~2 Fig. 4. Calculated three phase region, together with the tie lines PvOu and PL RL at the upper and lower critical end temperatures respectively. Pu and PL are the upper and lower critical end points. The broken line denotes ~2=0.050. The filled circles are the concentrations of the systems A [8], B [8], C and D.
tively. The behaviors of systems C and D with the overall compositions indicated in Fig. 4 agree with the calculated one. This is also the case for systems A and B in Ref. [8] as shown in Fig. 4; that is, phase ~ appears at Tu for systems A and B and phase ? and phase fl disappear at TL respectively, for systems A and B. Since the system that corresponds to the intersection IP of the tie lines at T* and T* is found inside the three-phase composition triangle over the entire temperature range from T* to T*, we can determine the entire three-phase coexistence curve with this system by varying the temperature. If we measure ~bs values of the three coexisting phases just below T~ by varying Go at constant ~blo or q~2o, we will find that the composition difference of phase ~ and phase fl tends to zero when the point for q~o crosses the tie line at T[, on a composition triangle. Similarly, just above TL the composition difference of phase fi and phase y tends to zero when the overall composition crosses the tie line at T*. Therefore, using a set of such experimental data obtained at different values of ~blO or ~b2o, we obtain two sets of points on the
62
T. Dobashi, M. Nakata/Colloids Surfaces B." Biointerfaces 6 (1996) 57-62
0.15 -
Similarly, we have q~sl =0.249 and ~bsu=0.184 for q~2=0.0075. Using these data, the tie lines at T~, and T~ are represented by the equations ~b~= 20.8q~2 + 0.093 and ~b~= 2.6~b2 + 0.1645 respectively, and then I P is determined from the intersection as q~1o=0.175 and ~bzo=0.004. The present m e t h o d will be useful to obtain the entire three-phase equilibrium behavior, both in experiments and calculations.
%e
<
0.00°°5 0.10
~
. 0.15
~ 0.20
0.25
Fig. 5. Concentration difference in phase e and phase /3 just below the upper three-phase boundary A~bsuand that in phase/3 and phase I' just above the lower three-phase boundary A~bsL f o r ~2o = 0.0025. tie lines at T~ and T*. Thus, we can determine these tie lines, and then I P as the intersection of the two lines. The determination of I P was d e m o n strated by a calculation for the present system. Fig. 5 shows the difference Aq~u of ~bS of phase c~ and phase/~ just below Ttj and the difference A~b~L of ~bs of phase /~ and phase 7 just above TL calculated for the solution with 420=0.0025. The two compositions where concentration difference Aq~s is zero are ~b~r=0.145 and ~bsU=0.171.
References [1] [2] [3] [4] I-5]
M. Kahlweit, Science, 240 (1988) 617. J.C. Wheeler and P. Pfeuty, Phys. Rev. A, 24 (1981) 1050. R.L. Scott, Acc. Chem. Res., 20 (1987) 97. J.C. Lang, Jr. and B. Widom, Physica, 81A (1975) 190. T. Dobashi and M. Nakata, J. Chem. Phys., 99 (1993) 1419. 16] M. Nakata and T. Dobashi, J. Chem. Phys., 84 (1986) 5782. [7] K. Solc, J. Polym. Sci., Polym. Phys. Ed., 20 (1982) 1947. 1-8] T. Dobashi and M. Nakata, J. Chem. Phys., 84 (1986) 5775. [9] I.L. Pegg, Kinam, 6A (1984) 3. [10] M. Nakata, N. Kuwahara and M. Kaneko, J. Chem. Phys., 62 (1975) 4278. [11] T. Dobashi, M. Nakata and M. Kaneko, J. Chem. Phys., 72 (1980) 6692.