J. Phys. Chem. 1982, 86, 393-400
rank tensorB and would therefore be the same. The numerator depends upon the structure and dimension of the aggregates and would be proportional to the square of the dimension of the cubic unit cell. A useful model, for which simplicity is the chief justification, for the translational motion in these isotropic liquid crystalline phases is that it occurs on a sphere inscribed in the cubic unit cell. In this model 2R, as given by eq 11,determines the dimension of the cubic unit cell. As described above, for a cubic phase consisting of rodlike aggregates, D L = 3 D , , d , and, for lamellar aggregates, DL = 3/@meas-+ Thus,the method gives different values of R depending on the structural model. Through a comparison with X-ray diffraction data on the dimension of the cubic unit cell, it is possible to discriminate between the two models. In Table TV calculated values of R are given for the cubic phase at 84% C12TACand the cubic phase in the C,,TAF system assuming rod- or lamellar-shaped aggregates. As
393
can be seen from Table IV, calculated values of R are in better agreement with X-ray investigations assuming a rodlike aggregate structure. This can be taken as further support for the assumption of ascribing T,S to the translational diffusion. A better description of the relaxation in terms of lateral diffusion from these continuous cubic phases should be given by a random-walk model of the diffusion on a surface equivalent to the proposed model for the aggregate structure. Structural information would thereby be obtained by-fitting the dimenjion of the cubic unit cell to the value of J,(O)+ J,(oo)- 2Js(2w0)in eq 5. Computer simulations along these lines are currently being carried out.
Acknowledgment. Thanks are due to Bertil Halle and to Hilkan Wennerstrom for valuable discussions. This work was supported by the Swedish Natural Science Research Council and by the Foundation of “Bengt Lundqvists Minne” (P.O.E.).
Patterns of Three-Liquid-Phase Behavior Illustrated by Aicohol-Hydrocarbon-Water-Satt Mixtures B. M. Knlckerbocker,t C. V. Perheck,’
H. T. Davls, and L. E. Scrlven
of Chmkal E n ~ h e r l n gend htatcwlele Sclence, University of Minnesota, Mlnneapoik, Minnesota 55455 ( R e c e W : June 18, 1981; I n Final F m : October 5, 1981)
Dep8-t
Ten salts were each dissolved in water and the solutions mixed with equal volumes of one of six hydrocarbons and one of ten monohydric alcohols. The resulting multiphase mixtures were examined for the number of coexisting liquid phases and, in wme cases, for the partitioning of alcohol among them. Several unusual patterns of phase behavior have been observed. For example, increasing concentrations of sodium chloride induce the widely observed 232 pattern of phase behavior in equal-volumemixtures of n-propyl alcohol, octane, and brine.’ However, with lithium chloride, calcium chloride, or magnesium chloride, the induced pattern is 2323. All of the patterns observed in the alcohol-hydrocarbon-brine systems chosen for this study can be summarized by two basic quaternary phase diagrams. It is conjectured that any three- or four-component system exhibiting three liquid phases in equilibrium can be described in terms of these two basic phase diagrams.
Introduction The phase behavior of alcohol-hydrocarbon-brine mixtures is made rich and varied by hydrogen bonding and other effects of polar intermolecular forces in water and salt solutions, nonpolar forces in hydrocarbons, and both types in alcohols, which are amphiphilic. The dynamic structures caused by hydrogen bonding and other polar interactions make these mixtures difficult to model thermodynamically with empirical equations of solution state, much less in terms of statistics of molecular mechanics. A useful step toward understanding the thermodynamics of such mixtures is to profile their phase behavior experimentally, as we did for certain systems with sodium ch1oride.l In this paper we take up other salts and the basic features of all of the phase diagrams. As amphiphiles, the lower molecular weight monohydric alcohols employed here can be considered to be protosurfactants even though they form neither micelles nor liquid crystals in aqueous solutions and so display neither property often taken to distinguish surfactants.2 Previously we reported’ that numerous alcohol-hydrocarbon‘Mobil Research and Development Corp. Paulsboro, NJ 08066.
wateraodium chloride mixtures mimic the phase behavior of surfactant-oil-watersalt systems, including those encountered in certain surfactant-based processes for enhancing petroleum recovery. At low salinities, certain mixtures of alcohol, hydrocarbon, and brine split into two phases with most of the alcohol in the lower water-rich phase. At intermediate salinities the split in many instances is into three phases, the alcohol residing chiefly in the middle phase, which also contains substantial fractions of water and hydrocarbon. At higher salinities the split is into two phases with most of the alcohol in the upper, hydrocarbon-rich phase. This so-called 232 pattern is typical of surfactant-oil-water-salt systems, but our results showed that the pattern is not restricted to amphiphiles which are highly surface active. In what follows we show that, whereas systems with potassium chloride, sodium sulfate, and sodium bromide also follow the 232 pattern, those with lithium chloride, magnesium chloride, and calcium chloride behave somewhat differently. Notwithstanding the differences, all (1)B. M.Knickerbocker, C. V. Pesheck, L. E. Scriven, and H. T. Davis, J. Phys. Chem., 83, 1984 (1979). (2) R. G. Laughlin, Adu. Lip. Cryst., 3,99 (1978).
0 1982 American Chemical Society
KF NAA T44
T44 .*\A
IBA
IRA
? ? ? SB 4
3 2 2
_ 2- 2 2
TB4
-
TBA
- _ _
~
2 2 2
2 1 2 -
_
_
~
1 3 2
_ _
~?- 2 2
‘IPA
2 2 2 ~
2 3 2
~.
_2 3 3
E4
1 2 1 i
known patterns of liquid phase behavior can be generated from just two basic types of three-dimensional phase diagrams in which three coexisting fluid phases occur. The monohydric alcohols are ten: methyl alcohol (MA), ethyl alcohol (EA), n-propyl alcohol (NPA), isopropyl alcohol (IPA), n-butyl alcohol (NBA), sec-butyl alcohol (SBA), isobutyl alcohol (IBA), tert-butyl alcohol (TBA), n-amyl alcohol (NAA),and tert-amyl alcohol (TAA). The hydrocarbons are the six even-numbered n-alkanes having from 6 to 16 carbon atoms in the molecule.
Experimental Procedures Procedures were as described earlier,’ the hydrocarbons and alcohols were from the same sources as before. Lithium chloride, magnesium chloride, barium chloride, potassium bromide, lithium sulfate, and calcium chloride were from Baker Chemicals; sodium chloride, calcium fluoride, and lithium bromide were from Fisher Scientific; magnesium sulfate, sodium bromide, and potassium chloride were from Allied Chemical; sodium iodide and potassium sulfate were from Mallinckrodt; and sodium sulfate was from Merck. All chemicals were of 99% or higher purity except for their water content, and were of ACS or USP grade. Salt solutions were prepared by diluting saturated (at 25 OC) solutions of the salts with doubly distilled, deionized water. Lithium chloride solution concentrations were checked by using the Fiske Model 404 chlorocounter and by using the published saturated value3 as a standard. Lithium chloride solutions of 1 , 2 , 3 , ...,and 14 M and 1.4, 1.5, 1.7, and 2.3 M were mixed. The lithium chlorideoctane-n-propyl alcohol system was analyzed by gas chromatography and chlorocounter as described earlier.’
Experimental Results Phase behavior can be classified according to the number of liquid phases that form with increasing salinity up to the concentration at which solid salt precipitates. Our notation’ is a sequence of numbers of liquid phases, an underbar indicating that the amphiphile resides primarily in the lower, water-rich phase an an overbar that it resides mainly in the upper, hydrocarbon-rich phase. An underbar (3) H. Stephen and T. Stephen, Ed., ‘Solubilities of Inorganic and Organic Compounds”, Macmillan, New York, 1963.
~
-2 3
2
IPI
-2 _2 _2
11 4
_ 3 2 2
SB4
hB4
IP4
2 2 2
-
YBA
“UP4
-~
-
2 2 3 3
2 2 1
_ _
E4 2_ 2 2_
\I4 I
I
I
. I
I
and an overbar together signify nearly equal partitioning. The fmt number signifies the zero-salt phase behavior, the last number signifies the phases present when solid salt precipitates, and the middle number(s) respresent(s) the number of phases present a t any intermediate salinities. Phase boundaries are crossed only when three phases appear or disappear. Seven different phase behavior patterns were seen in mixtures of equal volumes of alcohol, hydrocarbon, and NaCl brine:’ 222,233,232,222,222,322 and 222. Table I shows the patterns of equal-volume alcoholhydrocarbon-KF brine and alcohol-hydrocarbon-NazS04 brine systems. These two salts induce sodium chloride like patterns, though a particular alcohol-hydrocarbon combination may display a different pattern with KF or Na2S04than with NaC1. For example, the EA-hexaneNaCl brine system gives 222 behavior, whereas the EAhexane-KF brine system gives 232 behavior. Some salts other than sodium chloride, e.g., lithium chloride, magnesium chloride, and calcium chloride, induce patterns of phase behavior not observed for any sodium chloride mixture. Table I1 shows the patterns for the equal-volume alcohol-hydrocarbon-LiCl brine mixtures and the salinity ranges (in wt %) over which three liquid phases occur. The patterns in Table I1 that do not occur in NaCl-brine systems’ are as follows: (a ) 2323 -- Mixtures (Figure l a ) . This pattern resembles the 232 pattern except at high salinities. At low salinities these systems are &phase mixtures. Over an intermediate salinity range an alcohol-rich phase is formed, and at somewhat higher salinities the alcohol resides primarily in the hydrocarbon-rich upper phase of a 2-phase mixture. However, at very high salinities an alcohol-rich phase again appears as one of three liquid phases. NPA shows this pattern in equal-volume mixtures of octane and LiC1, CaClZ,MgClZ,LiBr, and MgBrz brines. ( b )g23 Mixtures (Figure l b ) . Similarly, this pattern resembles the 222 pattern except at high salinities. At low salinities these systems are - p h a s e mixtures. Over a narrow intermediate range of increasing salinity, there is a gradual shift of alcohol from the lower to the upper phase without the appearance of a third liquid phase at the equal-volume mixing point. At very high salinities the upper phase splits into alcohol-rich and hydrocarbon-rich
Patterns of Three-LlquM-Phase Behavlor
The Journal of Physical Chemistry, Vol. 86, No. 3, 1982 395
TABLE 11: Phase Behavior in Mixtures of Equal Volumes of Alcohol, Hydrocarbon, and LiCl Brineu
ri
TAAl
57.60
NAk
IPA
2
39-60 -
_ _ ?2
3
50.60
~
50.60
53.60
I
53-60
‘I4
’
‘16
I
45-60
222
’
‘6
I
c8
‘12
‘10
’
The bold-face labels indicate the patterns of phase behavior. Numerical ranges are the g of LiC1/100 mL of brine at which the mixture splits into three liquid phases. -SALINITY
-SALINITY-
2
2
(C )
-
__ S A L I N I T Y
-SALINITY-+
3
3
2
(d)
-
-SALINITY
3
2
2
+
3
(el
Figure 1. Patterns of phase behavior observed in mixtures of equal volumes of alcohol, hydrocarbon, and UCI brlne. Stippling qualitively Indicates the cmcentrationof alcohd In each phase. Numercai labels indicate the number of liquid phases: bars lndicate the phase(s) with a hlgh concentratlon of alcohol: upper (2), lower (2), or both (2).
phases to give a three-phase mixture. IPA shows this pattern with hexane and octane and LiCl brine. (c) 223 Mixtures (Figure I C ) . This pattern is the same as the 2 2 2 3 except that the alcohol is nearly equally partitioned in the absence of salt. NPA shows this pattern with hexane and LiC1, CaC12, MgC12, LiBr, and MgBr, brines. ( d ) 323 Mixtures (Figure I d ) . This pattern resembles the 322 pattern except at high salinities, where the upper phase splits into alcohol-rich and hydrocarbon-rich phases. NBA, SBA, and IBA show this pattern with hexadecane and LiCl, CaC12, MgC12, LiBr, and MgBrz brines. ( e ) 223 Mirtures (Figure l e ) . In these mixtures the alcohol partitions strongly into the hydrocarbon phase except at high salinities, where the upper phase splits into alcohol-rich and hydrocarbon-rich phases. With LiC1, CaClZ,MgC12,LiBr, and MgBr2brines, NBA and IBA show this pattern with dodecane and tetradecane, SBA does so with tetradecane, and NAA does so with hexadecane. (f) Mixtures Containing TBA and Very High Concentrations (above 55 wt 9%) of LiC1. All such equal-volume
TBA-hydrocarbon-LiC1 brine mixtures contain a solid, effectively frozen phase rich in TBA, indicated by the letter F in Table 11. In each case, this is the third phase, which makes the patterns analogous to the patterns a-c and e above except that the alcohol phase is’present as a solid rather than a liquid. The NPA-octane-LiC1 brine system was studied in detail. Phase compositions for the salinity range where three liquid phases coexist in equal-volume mixtures are plotted in Figure 2. The abscissas on each plot represent the salinity, in molar units, of the brine used in mixing. (The numerical data are tabulated in the supplementary material associated with this paper. See paragraph at end of text regarding supplementary material.) There are two salinity ranges of three-phase behavior for equal-volume mixtures, one from approximately 1 to 2.25 M LiCl, the other from 5.8 M to saturated LiC1. As overall salinity increases in the first three-phase region, the alcohol moves out of the lower phase into the middle and then the upper phase, the water becomes less soluble in the middle phase and moves to the lower phase, and more octane moves to the middle phase. The LiCl concentration increases in the lower phase, whereas it decreases in the middle phase. In the second three-phase region, as overall salinity climbs, the alcohol moves from the upper phase into a middle phase, the octane concentration in the middle phase decreases as the octane shifts into the upper phase, and the water concentration in the middle phase increases slightly. Meanwhile, the alcohol concentration in the lower phase falls, and the salt concentration in the middle phase rises dramatically. A set of 36 samples of NPA-octane-4.5 M LiCl were mixed, one at each interior intersection of lines of 10 vol % (80% brine, 10% NPA, 10% octane; 70% brine, 20% NPA, 10% octane; ...; 10% brine, 10% NPA, 80% octane). These mix points all lie on one plane in the tetrahedral phase diagram between the two regions that had been shown to contain three coexisting liquid phases. If those two three-phase regions were connected, there would have to be a three-phase region in this plane. All 36 of these samples contained only two phases, as far as we could tell by eye. Hence, we concluded that the two three-phase regions are probably unconnected, although a possibility remains that they are connected by a small three-phase region that passes between the mix points that we chose. Interpretation of Results Figure 3 shows the three-liquid-phase region of the TBA-tetradecane-water-NaC1 system at 25 “C in the triangular prismatic coordinates that we used previously.’ This system shows the 232 pattern in mixtures of equal volumes of alcohol, hydrocarbon, and brine. The threephase region, of course, consists of a sequence of tie triangles, of which just one (LMU) is drawn. The arrows at the points L, M, and U show the directions of change of the respective phase compositions caused by adding salt. Similar three-phase regions for other quaternary systems are described by Widom: Lang and Widom: Griffiths? and Hartwig et al.7 The line BD in Figure 3 is a tie line between stable phases of compositions B and D in equilibrium. It is a critical tie line, for the water-rich phase and the alchol-rich phme reach identical compositions at the critical end point (4)B.Widom, J. Phys. Chem., 77,2196 (1973). (5)J. C.Lang, Jr., and B. Widom, Physica A (Amsterdam), 81, 190 (1975). (6)R.B. Griffiths, J. Chem. Phys., 60,195 (1974). (7)G. M.Hartwig, G. C. Hood, and R. L. Maycock, J.Phys. Chem., 59, 52 (1955).
Knickerbocker et ai.
50 s
t
,
..,nn, c
LiCl CONCENTRATION. Molar
100
i
11
601
, , , , , , , , , ,
I
L i C l CONCENTRATION, Molar
, ,
I
12
14
I
0
MIDDLE
P
40
0
>
20 LOWER
' 0
2
4
LiCl
6
S
10
16
CONCENTRATION, Molar
LiCl CONCENTRATION, Molor
Figure 2. phase composltions for mixtures of equal volumes of n-propyl alcohol. octane, and LiCl brine at 25 "C. The culves have be%n mted to the data by visual inspection.
Alcohol
Fbus 3. Thdlquibphase region of the TEA-tet'adecane-watwNaCX system at 25 "C in tiangular pdsmatk madhates. Cuve ABCD is the Iocu8 of all three-phase equilibrium composltlons: AC and ED are the two crlticai tie lines.
Figure 4. Schematic representation of a 311 phase diagram. The zerwallnHy. three-phase region is thB triangle abd. AC i s h upper crlthxi tie line. Curves Aa and bcd represent all possible aWee-phase equillbrlum compositions.
B. S i m i i l y AC is a critical tie line; the alcohol-rich phase and the hydrocarbon-rich phase reach identical compositions at the critical end point C. The locations of the critical tie lines and critical end points are extrapolated from compositional data.' The curve BA is the locus of compositions of the water-rich phase L of the continuum of tie triangles. The curve BC is the locus of compositions of the alcohol-rich phases M, and the curve DC is the locus of compositions of the hydrocarhon-rich phase U. Altogether the space curve ABCD is the locus of all of the equilibrium phase compositions in the three-phase region. Because each point on line ABCD represents a vertex of a tie triangle and thus a stable phase, there must he adjacent to ABCD a one-phase region over its entire length. According to Gibbs' phase rule, the three-phase region must he enveloped by a region of twc-phase equilibria (not
shown), except along the curve ABCD. This two-phase region is simply connected it is possible to proceed by a suitable path of component additions from one two-phase point to any other without leaving the region. This twophase region reaches to the salt-free base; adding salt causes the alcohol amphiphile to partition from the aqueous into the hydrocarbon phase and to do so via the middle phase when it is present (i.e., the 232 pattern). Owing to the limited solubility of NaCl in this system, there must he a three-phase region of solid/liquid/liquid equilibria at high salinities, hut it is not shown. The three-phase region may he cut off by the salt-free base of the quaternary diagram to leave the 322 pattern, as in the NBA-hexadecanewatepNaC1 system. This type of diagram is illustrated in Figure 4. Its hallmarks are the tie triangle ahd in the base and the loci of tie-triangle
Patterns of ThresLiquWPhase Behavior
The Journal of PhVslcal chemistry. Vd. 66. No. 3, 1982 397
A I c i hol
ngun 5. Sdwmatic representah of a 233 phase diagam. The tie Mangle atd is ths lower face of a four-phase tetrahedron wim salt as ths fm-th PhaJe (off scale to me top of ths diagram). BD is ths lower cinkal tie Ihe. C w e s am ani d)repesent al ttrswhase equRkdun composnions.
vertices: aA for the water-rich phase, bC for the alcohol-rich phase, and dC for the hydrocarbon-rich one. The arrows at L, M, and U again show directions of change caused by adding salt. C is again a critical end point and AC a critical tie line. However, because there is no lower critical tie line-it has in effect disappeared into the physically unrealizable region of negative salinities-the locus of all three-phase equilibrium compositions is no longer a single m e . The two branches, aA and bCd, must be contacted by one-phase regions that plow inward to them through the two-phase regions of liquid/liquid equilibria that otherwise envelop the three-phase region above the base-two unconnected two-phase regions. The three-liquid-phaseregion of Figure 3 may be truncated at higher salinities by equilibria with a coexisting solid, salt-rich phase, Le., by a four-phase solid/liquid/ liquid/liquid tie tetrahedron, to leave the 233 pattern, as in the NPA-hexadecanewater-NaCl system. This type of diagram is shown in Figure 5. The tie triangle abd is the lower face of the four-phase tetrahedron; far off scale to the top is the vertex representing the salbrich solid. B is a lower (with respect to salinity) critical end point and BD a lower critical tie line; in this case it is the upper critical tie line that is missmgswallowed, in effect, by the tie tetrahedron. The locus of all three-liquid-phase equilibrium compositions is again divided, and each of the branches, aBb and dD, must be contacted by one-phase regions. Elsewhere, the threeliquid-phase region must be enveloped by two-liquid-phase regions, except for tie triangle abd, which faces into the tie tetrahedron. Incidentally, the other faces of the tie tetrahedron must join solid/liquid/liquid three-phase regions, and all six of its edges must make contact with two-phase, solid/liquid or liquid/liquid regions, none of which are shown. The three-liquid-phaseregion of Figure 5 can be further truncated by the salt-free base to give a 333 pattern, with three liquid phases occurring at all salinities. This pattern occurs in mixtures of water with NaCl, benzyl alcohol, and n-alkanes.' It will also occur in other mixtures of salt and water with two immiscible liquida that are both immiscible with water. The phase diagram of the NPA-octane-water-Licl system is more complicated than in Figure 3.4, or 5. Its pattern 2323 shows two three-phase regions. At very high salinities, a LiC1-rich solid coexists with the three liquid phases in a four-phase equilibrium like that shown in Figure 5. The lower three-phase region has phase-composition changes paralleling those found by Lind et al.8 in
npure 6. Idealized twwphase region in a threecomponent phase diagram wim an intensive varhble as ths vertical dkecfion. The upper c r b i sduhn point (UCSP)ani ths lower ddcal solumn point (LCSP) are connected by a pian-point Iwp (Ppl).
a system possessing two critical end points and suggests a three-phase region similar to that shown in Figure 3, with critical tie lines at approximately 1and 3 M LiC1. While a possibility remains that these two three-phase regions are connected, as explained above, the available evidence indicates a phase diagram that is qualitatively a combination of Figure 3 with a skewed version of Figure 5. In the combination are three critical tie lines, two associated with the lower three-phase region and one at the lower edge of the upper three-phase region. Generalization: Three-Liquid-Phase Behavior Patterns To understand all of the possible patterns of phase behavior in quaternary systems that can form three coexisting liquid phases, it is necessary to consider the most general cases of three-phase regions in a four-component system. As a prelude it is very instructive to study the most general cases of three-noncrystalline-phaseregions in a three-component system as a function of a field variable such as temperature or chemical potential or of a molecular parameter such as carbon number. Field variables have been introducedg to describe those thermodynamic quantities which are the same in all phases at equilibrium since many textbooks label as intensive variables these quantities plus specific quantities such as density, mole fraction, and the like. The triangular prismatic coordinate system of Figure 6 is appropriate, with salinity replaced by a field variable as the vertical axis. Because the field variable must be the same in all coexisting bulk phases at equilibrium, tie lines and tie triangles must now lie in horizontal planes of the stacked ternarydiagram version in Figure 6. In contrast, the concentration of a fourth component such as salt varies from phase to phase. Aa it does, the tie lines and tie triangles are tipped askew as in the isothermal quaternary diagrams, Figures 3-5. Nevertheless, the topological properties of the two versions are usefully equivalent, as brought out below. A three-phase region must be enveloped in a region of two-phase equilibria except along the loci of phase compositions. Moreover, the coexistence of noncrystalline, or (8) C. V. Pesheck. M. I. Lind, L. E. Serivsn, and H. T. Davis. manuscript to be submitted to J. Phya. Chem. (9) R. B. Griftiths and J. C. Wheeler, Phys. Re". A, 1041 (1970).
990
Knickerbocker et al.
The JownaI of Physical Chemlsby, Vol. 88, No. 3. 1982
Flgwa 7. schematic rqwesentatlon oi a -phase ternary system.
regbn
88811 in
a
disordered, phases can be thought of as terminating, in principle a t least, in critical points. Therefore, we start with the nonisothermal case of two-phase equilibrium in a ternary system pictured in Figure 6. This representation goes back to the pioneering work of Sehreinemakers.'O The two-phase region is a free-standingsolid ovoid, a more or less deformed ellipsoid, composed of a stack of horizontal disks, each of which is a planar ruled surface of tie lines. On the surface of each disk are two tie lines of zero length corresponding to ordinary critical points, or plait points, and these on successive disks define a closed space curve, the plait-point loop (PPL), which at its highest and lowest points defines the upper critical solution point (UCSP) and the lower critical solution point (LCSP), respectively. From this archetypal diagram a whole catalog of phase diagrams with two-phase regions can be generated by (a) deformations of the ovoid that leave tie lines as rulings and disks horizontal and convex with respect to tie lines and (b) horizontal and vertical translations of the resulting body. (such deformations and translations do not tear or puncture the surface and so define a class of topological transformations that preserve the essential thermodynamic features.) Only those parts of the ovoid that remain within the triangular prism are physically significant, however. Figure 7 is an example of a section of an ovoid truncated on two vertical sides by the compositional edges of the prism. Other examples in a tetrahedral coordinate system have been reported." The catalog can be extended to diagrams with more than one two-phase region by starting with multiple ovoids but keeping them out of contact. The complications brought by the existence of ordered, crystalline phases are unimportant for present purposes, except that such a phase may truncate an ovoid in a fan of solid/liquid/liquid threephase tie triangles between a pair of critical tie lines, each running between the crystalline state and a plait point. There are only a few ways that a three-phase region can occur in a free-standing two-phase ovoid. Figure 8 illustrates one way, which we call achiral. At a lower critical end point B, the @ phase bifurcates to and phases, and, at an upper critical end point C, these fuse again into the 0 phaae. B and C are linked by a plait-point segment PPS that is unconnected with the plait-point loop PPL that belts the altered ovoid. Because this loop closes on the ovoid, it divides the hinodal surface into two distinct (10) F.A. SEhrsinSmaLara,"Die Hetercgeaen GleichgewickW, VoL 3, part 2, H. W. B. Rooseboom, Ed., Viswsg, Braunschweig, 1913. (11) K. P. Myasnikova, N. I. Nikurashina, and R.V. Mertslin, Ruas. J. Phyr. Chom. (Engl. Trannl.), 43, 223 (1969).
Rgun 8. Achiral eruptim of a threephase regbn In a two-phase regbn. AC and BD are critical tie lines. The two phases @, and & are kwdved h both dtlcald pdntsC and D. The~nguiardiaqams show ths sequence of phase behavior in the ovoid.
N
ICTT i NE
9. Schematic representation of ths water-nioctlnemercuy phase diagram. This phase diagram is a truncatbn of the type of diagram shown in Figure 8.
*re
regions. The situation is analogous to an island micibility gap in a binary system. What is significant is that the a phase is not involved in either critical end point, although it is connected to each by a critical tie line, BD and AC, respectively. Each of these amounts to a tie triangle of infinitesimal width, just as the critical end points amount to tie lines of infinitesimal length. Between the two lies the three-phase region. The locus of three-phase equilibrium compitions is divided into segment AD and the loop B@lC&,At each level (value of intensive variable or molecular parameter) of three-phase equilibria, there are three plait points, each one corresponding to a different side of the tie triangle-each side is necessarily bordered by a two-phase region. There are two unconnected regions of two phases, one connected with the plait-point loop PPL, the other with the plait-point segment PPS. A truncated version of this type of diagram would be the nicotine water-mercury diagram sketched in Figure 9;the closed miscibility gap in the nicotine-water binary system is well-known.'* By imagining the critical end points B and C in Figure 8 moving together (as pressure changes, or as molecular structures are varied) and coalescing in a critical double (12) W. J. Moore, "Physical Chemistry", 3rd ed.. Prentice-Hall, Englewwd Cliffs, NJ. 1962. p 114.
The Journal of PhVJlcal Chemktry. Vol. 66, No. 3. 1982 399
pattsms of Three-Liqu&F%aseBehavlor
UCSP
PPC 'pc
(a)
(b)
Chlral eruption of a three-phase region In a twwphase region: (a) Um levo fwm: (b) the dextro form. AC and BD are the two dtlcal Ue Ilnes. The LII and 6 phases bewme ldentlcal at 8, and Um 0 and y phases at C. The triangular diagrams give the seqtmnce of phase behavior as the lntenslve variable Is Increased for (a). Figure IO.
point at a particular temperature, the reader can picture the reverse of the proem by which this type of three-phase region erupts from a tie line at one end of which there is an incipient pair of plait points at infinitesimally different values of temperatures or analogous variable. This is not a tricritical point, since only two phases become identical. A special case is to suppose that both B and C migrate to and fuse into either the upper or lower critical solution point, thereby shrinking the whole three-phase region to iniiiteaimal size that can, in effect,fold up into the critical solution point turned tricritical point. This, however, requires two additional variable parameters, such as pressure and salinity, as well as the three components and temperature or analogous variable. The alcohol-hydrocarbon-brine three-phase regions do not resemble the type just described, but there is a second way that a threephase region can occur in the ovoid in the prismatic ternary diagram (see Figure 10). Here the third phase that appears is comparatively rich in an amphiphilic component, the relative affinity of which for each of the other two components shifts regularly as temperature or analogous field variable or parameter rises or falls. Again the three-phase region lies between a lower critical tie line, BD, B being the lower critical end point where a single phase bifurcates into a and @ phases, and an upper critical tie line, AC, C being the upper critical end point where the @ and y phases fuse into a single phase. The locus of tie-triangle vertices, i.e., three-phase equilibrium compositions, is a single curve ABCD. The plaibpoint curve PPC is an open curve that conneds the two critical end points but in doing so passes through the lower and upper critical solution points (LCSP and UCSP) and swings around each of the critical tie lines, in such a way as to provide three plait points, one for each side of the tie triangle, at each level of three-phase equilibrium. Because the locus of three-phase compositions is not closed, the surface of the ovoid has only one region (in contrast with Figure 8,whose surface is divided into two regions), and the two-phase region is thus simply connected. Another consequence of the open curve, or equivalently of the involvement of the @ phase with the y phase in the lower critical end point and the a phase in the upper one, is that the chiral ovoid of Figure 10 exists in two enantiomorphic forms, levo (Figure loa) and dextro (Figure lob). Hence, there me two poasible sequences of triangular diagrams as temperature or analogous field variable or
LCSP
A A A
mum 11. Schematlc representation of Um completely symmetrlc regular solution model showing three trlcrklcal points (TCPs) and a four-phase equlllbrlum plane. The trlcrklcal points arise from the symmew 01 tha model. but the four-phase equlllbrlum plane remains In Um phase diagram wRh a move away from tha symmetrical system.
molecular parameter is changed. The sequence of triangular diagrams shown corresponds to Figure loa. While the experimental results reported earlier in this paper a!J show alcohol shifting from the aqueous phase to an alcchol-rich phase and on to the hydrocarbon phase as salinity increases, the progression with respect to some parametera is the opposite in certain amphiphilehydrocarbon-brine ~ystems.'~ If one imagines the critical end points Band C in Figure 10 moving together, the critical tie lines AC and BD shorten to zero length, and the whole three-phase region with its progreeaion of tie triangles shrinks to infinitesimal size,flattens, and folds up into the fusion of B and C, which is a tricritical point. This type of tricritical point has been carefully studied experimentally by Lang and Widoms and compared to a Landau model developed by Griffiths! Note that, when the tricritical point is reached, there is no three-phase region in the freestanding egg. The reader can imagine the converse unfolding by which the threephase region achieves detectable proportions in the free standing ovoid. There is a particular ternary model, the regular solution model examined by Meijering," which has a three-phase region that terminates in a trimitical point in the same composition-temperature space. However, this tricritical point appears only when the regular solution model is symmetrical with respect to two of the binary interaction parameters; furthermore, in this model, if the parameters are equal at one temperature, they are equal at all temperatures. Were the two parameters to vary differently with temperature, the tricritical point would be isolated, unconnected to a three-phase region. But were the two parameters to depend on pressure as well, the tricritical point would terminate a three-phase region in the fourdimensional composition-temperaturepressure space. Figure 11 shows the sequence of the phase diagrams obtained from regular solution theory when all three interaction parameters are exactly equal. This diagram shows three ovoids, each containing a tricritical point and (13) M. L. Robbina in 'Micellization, Solubilization. and Micmemulsiom", K. L. Mittal, Ed., Plenum Press, New York, 1977, pp 713-44, R. N. H d y and R. L. Reed. Soe. Pet. Eng. J., 16, 147 (1976). (14) J. L. Meijering, Philips Res. Rep., 6,183 (1951).
Knickerbocker et al.
Flpn 12. Fusion of twolhree-pkasa owl&. Fl@re 10. a and b. wiW7 accounts fa2323 phase behavior. This figure has two mresphase regbns. two plan-point segments. and four crnid tka lines.
a three-phase region, fusing in a collapsed four-phase tetrahedron, which could inflate if there were more than three components. The three ovoids containing tricritical points suffer from the same model-dependent objections as above, but the appearance of the four-phase plane does not. A change in the interadion parameters hetween the three components will turn the tricritical points into critical tie lines, while maintaining the existence of a plane which contains four phases. A less symmetric case is presented in Figure 2 of ref 15, showing isothermal slices of a nonisothermal ternary phase diagram, with the ovoids truncated by the composition edges of the prism and, at low temperatures, by a solid phase. Concluding Remarks Figure 8 and Figure 10a (with its mirror image Figure lob) are the basic ways-achiral and chiral-that three coexisting liquid phases can occur in phase diagrams of the sort used here, whether the vertical axis is an intensive variable or molecular parameter common to all phases, or a compositional variable such as salt concentration, which generally differs from phase to phase. In this latter situation tie lines remain straight but are tilted out of horizontal; two-phase regions become twisted bundles of tie lines, upper and lower critical solution points need no longer be uppermost or lowermost; and tie triangles, though still planar, are tipped askew within the diagram. Still, all of the contact requirements of Gibbs’ phase rule are preserved, and more complicated diagrams can be assembled from the basic ones by deformations, now including certain modes of tipping and twisting, by translations, by truncations, and by fusion of more than one (15) A. W. Francis, J . P h p . Chem., 60.20 (1956).
basic two-phase or three-phase body. The skewing effect of salinity is obvious in the comparison of Figures 3 and 10. Nonetheless, each of the pattern of phase behavior reported in this paper or earlier‘ 232, 233, 322, etc., can he represented by starting with Figure 10, truncating it a t high salinity with a solid, salbrich phase, and otherwise manipulating it in allowable ways. For example, fusing the solid ovoid of Figure 10a onto the top of its mirror image, Figure lob, with due regard for joining plait-point loops and segments, leads to the body shown in the prismatic ternary diagram in Figure 12. Plainly its topology accounts for 2323 phase count pattern, and only moderate topology-preserving manipulations can convert it to the combinations of Figures 3 and 5 called for above. Moreover, truncating and skewing the resulting diagram would produce diagrams for 233 and 323 systems that have been reported. Figure 11also shows the possibilities of fusing ovoids to form more complicated diagrams, including a four-phase region (which must be planar when a field variable is used as the vertical axis, hut will inflate to a tetrahedron when salinity is the vertical axis.) An intriguing possibiity worth pursuing where, as in the science and technology of enhancing petroleum recovery, there is justification, is to use computer methods to generate the basic ternary and quaternary phase diagrams,’+” to manipulate them systematicallyto fit experimental data for interpolation and extrapolation, and to display them for research and development purposes. Tricritical points are of particular interest in computations of multiple fluid-phase equilibria from equations of state. The reason is that by parameter variation it is often possible to trace ordinary critical points from the tricritical points out of which they have unfolded, the tricritical points being easier to locate in the first place. Understanding the thermodynamic principles behind patterns of three-phase behavior will lead to deeper, molecular-level research as well as more efficient representations of multiphase equilibria. Acknowledgment. This research was largely supported by the Fossil Energy Division of the Department of Energy. B.M.K. started working as an NSF Undergraduate Research Participant. W.R. Rossen’s insights into the achiral diagram and P. K. Kilpatrick’s into tricritical points were very helpful. Supplementary Material Available: Compositions of the three equilibrium phases in the two three-phase regions of the water-NPA-octaneLiCI phase diagram (3 pages). Ordering information is availahle on any current masthead page. (18) R. G. B m m . M. S.Bidner, H. T. Davis,S. Prager, and L. E. Scriven. SPE Paper No. 6122, Bakersfield, CA, April 14, 1977. (17) D.Y. Peng and D. B. Rabinson, Ind. Eng. Chem. Fundom., IS, 59 (1976). (18) J. M. Soreman, T. Msgnuaaen, P. Rasrnuasen and A. Fredenelund. Fluid Phoge Equilib. 2. 297 (1979); 3. 47 (1979); 4, 151j1980). (19) W. R. Rossen. Ph.D. Thesis. University of Minnesota, Minneap olis, MN, 1982.
