Langmuir 1988,4,499-511 changing sign and varying from +lo4 (through zero) to -lo4 dyn. The positive branch of this dependence is separately c o n f i i e d by the indirect method of the critical bubble of microscopic r. This suggests that the values of K are reliable. The most sensitive indirect method used, namely, that of measuring the critical supersaturation for the condensation of water vapor on hexadecane, has led to a reliable determination of K in this system: K = -1.9 X lob dyn. The only uncertain point in this case is that, with nuclei (droplets) of molecular size, some effect of surface curva-
499
ture on surface tensions (or, for that matter, the effect of the curvature of the three-phase contact line on K ) cannot be excluded. As is known, this difficulty is typical for all cases of homogeneous or heterogeneous phase formation. The state of the line tension problem, thus outlined, shows that it emerges as a promising and important, although arduous, part of the physical chemistry of dispersed systems. Acknowledgment. The assistance of K. J. Mysels in editing this paper is acknowledged.
General Patterns of the Phase Behavior of Mixtures of H20, Nonpolar Solvents, Amphiphiles, and Electrolytes. 1 M. Kahlweit,* R. Strey, P. Firman, D. Haase, J. Jen, and R. Schomacker Max-Planck-Institut fur Biophysikalische Chemie, Postfach 2841,D-3400Gbttingen, FRG Received November 20, 1987 The phase behavior of mixtures of water, oils, nonionic amphiphiles, and electrolytes (microemulsions) follows general patterns that originate from the interplay between the lower miscibility gap of the binary mixture oil-amphiphile and the upper (closed) miscibility gap of the binary mixture water-amphiphile (section II). As a consequence,the critical line, connecting the plait pointa of the isothermal phase diagrams of the ternary mixtures, changes from the ail-rich to the water-rich side with rising temperature. If the chemical potentials of the components are appropriately changed, the critical line may be “broken” at a tricritical point, which gives rise to the evolution of a three-phase body (111). In the amphiphile-rich phase of the three-phase body (the microemulsion),one finds-for thermodynamic reasons-a maximum of the mutual solubility between water and oil combined with a minimum of the interfacial tension between the aqueous and the oil-rich phase (VII). Since the position and extensions of the three-phase bodies are strongly correlated with those of the binary miscibility gaps, they show the same systematic dependence on the nature of the oils and the amphiphiles (VI), on pressure (VIII), and on the concentration of an added electrolyte (1x1. In this review we summarize the general patterns of this dependence, the knowledge of which is indispensable for applying such mixtures in research and industry. I. Introduction Mixtures of water, nonpolar solvents (oils), amphiphiles, and inorganic electrolytes may separate into three liquid phases, an aqueous (a), an amphiphile-rich (c), and an oil-rich phase (b). In phase c, often referred to as “microemulsion”, one finds-for thermodynamic reasons-a maximum of the mutual solubility between water and oil. The three-phase body exists only within a well-defined temperature interval. Near the mean temperature of that interval one finds-again for thermodynamic reasons-a minimum of the interfacial tension between the aqueous and the oil-rich phase. The position of the three-phase body on the temperature scale as well as ita extensions depends sensitively on the chemical nature of the components and the concentration of the electrolyte. The detailed knowledge of this dependence is a prerequisite not only for performing meaningful experiments in order to clarify the properties of such mixtures but also for applying them in research and industry. In industry, in general, the temperature, the oil, and the composition of the aqueous solution (brine) are given. Wanted is the most efficient amphiphile, i.e., that amphiphile the addition of which leads to the highest mutual solubility between that particular oil and that particular brine at that particular temperature. That is why studying the dependence of the properties of the three-phase body on the chemical nature of the components of the mixture is so important. Experience has shown that the phase
behavior of such mixtures follows general patterns. The knowledge of these patterns permits predicting the mean temperature of the three phase body as well as the efficiency of the amphiphile and thus which amphiphile (or which combination of amphiphiles) to apply for solving a particular problem. In this paper we shall, therefore, summarize an empirical (!) description of these patterns. For reasons that will become clear later it is appropriate to distinguish between nonionic and ionic amphiphiles. Accordingly, we shall in part 1 of this series consider quaternary mixtures H20 (A)-“oil” (B)-nonionic amphiphile ((2)-inorganic electrolyte (E), first considering ternary mixtures A-B-C and then the quarternary mixtures. Mixtures including ionic amphiphiles will be presented in part 2. 11. Critical Line
A mixture of m components has-in
the absence of external fields-m + 1independent thermodynamic variables, namely, the temperature T, the external pressure p, and m - 1 composition variables, the choice of which is a matter of convenience. Since experience shows that the effect of pressure is weak compared with that of temperature, we shall, in general, dispense with p by keeping it constant at atmospheric pressure. The phase behavior of a ternary mixture may then be represented exactly in an upright phase prism with the Gibbs triangle A-B-C as the base and T as the ordinate (Figure 1). As for composition
Q743-7463/8a/2404-0499$01.50/0 1988 American Chemical Society
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500 Langmuir, Vol. 4, No. 3, 1988 critical line
,+, 10 b
S i
H,O
- phenylbutane - C E
-
Figure 1. Phase behavior of a ternary mixture A-B-C represented in a phase prism with isothermal phase diagrams (schematical). The “connected”critical line changes from the oil-rich to the water-rich side with rising temperature. variables, we found it convenient to introduce the mass fraction of the oil in the mixture of water and oil a = B / ( A + B) (1) and that of the amphiphile in the mixture of all three components y = C / ( A B C) (2)
+ +
both expressed in weight percent (wt %). Each point in the phase prism is then defined by a certain set of T, CY, and y. For systematically studying the dependence of the phase behavior of such mixtures on the chemical nature of the components, it is convenient to vary the oils as well as the amphiphiles within homologues series. As oils we chose, if not otherwise stated, n-alkanes, characterized by their carbon number k (e.g., Blo for n-decane), and as nonionic amphiphiles n-alkyl polyglycol ethers, characterized by CjEP This class of amphiphiles includes 1alcohols, denoted by CiEo, often referred to as “cosurfactants”. This choice permits discussion of the dependence of the phase behavior of the ternary mixture A-B-C on the chemical nature of its components in terms of the three parameters i, j , and k. H20 and oil are practically insoluble in each other. However, experience shows that adding a nonionic amphiphile increases their mutual solubility until, after having added a sufficient amount of C, one obtains a stable homogeneous mixture of the three components. On an isothermal (horizontal) section through the phase prism, this fact represents itself in a “central” miscibility gap that extends from the A-B side into the Gibbs triangle (Figure 1). The full lines represent the binodals, that is, sections through the surface of the body of heterogeneous phases at the corresponding temperatures. The thin lines represent the tie lies that connect two phases in equilibrium. These two phases become identical at a plait point (cp) on the binodal. If one varies the temperature, these plait points connect into a critical line (cl) that ascends along the surface of the body of heterogeneous phases with rising T. In general, the plait points will lie either on the oil-rich or on the water-rich of the binodal. In the first case, the ampliiphile is more soluble in water than in oil; in the latter it is more soluble in oil than in water. The position of the plait points can thus be determined qualitatively by preparing a mixture of equal volumes of A and B, adding an appropriate amount of C, and measuring the volume fraction 9, of the lower aqueous phase. If, after adding C, 9, > 0.5, the plait point lies on the oil-rich side of the binodal; if 9, < 0.5, it lies on the oil-rich side.
@o
I
-
ylwt%l
-
oImNm 1
Figure 2. Some features of the ternary mixture H20-phenylbutand4E&.Left: volume fraction 4, of the lower aqueous phase versus T indicating the change of distribution of the amphiphile between water and oil with rising temperature. Center: vertical section through the phase prism at a = 50 wt % showing a distinct maximum of the mutual solubility between water and oil at the inflection point of $a. Right: interfacial tension u between the aqueous and the oil-rich phase versus T showing a distinct minimum at the inflection point of 9,. If one performs such an experiment with the mixture A-B-C, one makes an important observation: at ambient T, one finds, in general, $a > 0.5; at elevated T, however, 4, < 0.5. From this result one concludes that at ambient temperatures C is more soluble in A than in B, whereas a t elevated T it is more soluble in B than in A. In order to symbolize this change of the distribution coefficient we shall follow a suggestion by Knickerbocker et al.’ by denoting a two-phase mixture with the amphiphile mainly dissolved in the lower phase with 2 and, accordingly, one with C mainly dissolved in the upper phase with 2. These symbols are easier to recognize than those introduced by Winsor? who suggeated “typeI” and “type 11”,respectively. The change of the distribution of C between A and B with rising temperature can thus be symbolized 2 2. At a particular temperature in between, the nonionic amphiphile must, consequently, be equally well soluble in both solvents, the tie lines must run parallel to the A-B side of the Gibbs triangle, and the plait point must lie on its center line. Figure 1 shows a schematic phase prism with the body of heterogeneous phases and the critical line as it ascends along the surface of that body, changing from the oil-rich side at ambient temperatures to the water-rich side at elevated temperatures. The second property of such mixtures important for theory and application is the minimum amount of amphiphile required to obtain a homogeneous solution of the three components. As Bancroft3 has established, the increase of the mutual solubility between two immiscible solvents by adding a third component is the strongest when the third component is equally well soluble in the two others. Accordingly, one expects the amphiphile to be most efficient at that temperature at which the distribution coefficient of the amphiphile between water and oil is unity, that is, at the temperature of the inflection point of the critical line. This is indeed the case and is demonstrated in Figure 2 for the mixture HzO-phenylbutane-C4&, the latter being a weak amphiphile. The left part of the figure shows the phase volume ratio 9, versus temperature determined at a = 50 wt % and y = 36.8 wt % and the center part a vertical section through the phase prism erected on the center line of the base, i.e., at CY = 50 wt % ,showing the profile of the body of heterogeneous phases with a distinct maximum of the mutual solubility
-
(1) Knickerbocker, B. M.; Pesheck, C. V.; Scriven, L. E.; Davis, H. T. J.Phys. Chem. 1979,83, 1984. (2) Winsor, P.A. Trans.Faraday SOC.1948,44, 376.
(3) Bancroft, W. D. J . Phys. Chem. 1986,I , 414,647.
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O / '
'
20
0 HP
"
'
LO
60
alwtV01
80
100 phenylbutone
Figure 4. Unfolded phase prism of the ternary mixture A-B-C showing the phase diagrams of the three corresponding binary
35,
i
0' 0
'
' 20
LO
I 60
80
100
a[wt%I Figure 3. Trajectory of the connected critical line in the ternary mixture H20-phenylbutan4& (top) projected onto the A-B-C base and (bottom) projected from the C edge onto the A-B-T plane of the phase prism.
mixtures (schematical).
T ["I
t
6ot LO
30
-
20 -
lot
a t the inflection point of 9,. If one performs the latter experiment at varying CY,one makes another important observation: each profile shows such a maximum of the mutual solubility at the plait point. In other words, along the critical line one finds a groove in the body of heterogeneous phases that ascends with rising temperature from the oil-rich side to the water-rich side and thus permits a quantitative determination of the position of the plait point at each temperature. Figure 3 shows on top the compositions of a few plait poinh of the above mixture projected onto the base of the phase prism-not to be mistaken for an isothermal phase diagram-and on bottom the corresponding temperatures projected from the C edge of the phase prism onto its A-B-T plane-not to be mistaken for the volume fraction 9,. As we shall see later, a similar groove is found along the trajectory of phase c of the three-phase body along the surface of the body of heterogeneous phases, although with an opposite inclination. The third property of such mixtures important for theory and application is the interfacial tension Uab between the aqueous and the oil-rich phase. The interfacial tension between pure water and oil is about 50 mN m-l. As is well-known, adding an amphiphile leads to a strong decrease of gab that vanishes at the plait point. The magnitude of the interfacial tension between the two phases is thus a measure for the distance between their tie l i e and.the plait point. Since the distance of the plait point from the A-B side of the Gibbs triangle reaches a minimum at the inflection point of the critical line, one, accordingly, expects gab to reach a minimum there too. This is indeed the case, as is demonstrated in the right part of Figure 2, which shows the dependence of gab on T, measured at the mean composition CY = 50 and y = 30 wt %.
The distinct change of the distribution coefficient of nonionic amphiphiles between water and oil with temperature that causes a maximum of the mutual solubility between water and oil combined with a minimum of the
0
10
12
1L
16
18
20
-k
Figure 5. Upper critical temperature T, of the binary mixture n-alkanes-CiE5 versus carbon number k of the oils with i as parameter.
interfacial tension between the aqueous and the oil-rich phase at the inflection point of the critical line is the most striking property of such ternary mixtures and is, as a matter of fact, the key to understanding so-called microemulsions. The origin of this phase behavior can be qualitatively understood by considering the phase diagrams of the three corresponding binary mixtures which are shown on the unfolded phase prism on Figure 4. Each of the three binary mixtures shows, for thermodynamic reasona, a lower miscibility gap with an upper critical point. The critical point of the A-B mixture lies well above its boiling point so that the very low mutual solubility between water and oil increases only slightly between melting and boiling point. The upper critical point of the B-C mixture lies, in general, close to its melting point. Its temperature T,depends on the chemical nature of both the oil and the amphiphile, that is, on i, j, and k, as is demonstrated in Figure 5 for n-alkane (Bk)-CiE6 mixtures. As one would expect, T, rises for a given amphiphile with increasing k, drops for a given oil with increasing i, and-not shown on the figure-rises with increasing j. The phase diagram of the A-C mixture is the most complicated of the three. Its lower miscibility gap lies, in general, below its melting point and plays no role in the further considerations. At ambient temperatures and low concentrations of the amphiphile, the mixture is molecular disperse. Above a critical concentration, the amphiphiles (medium and long chain) form association colloids (micelles). At even higher concentrations, these amphiphiles form lyotropic mesophases (liquid crystals) of various structures (not shown in Figure 4). As one raises the temperature, these will eventually dissolve. Instead,
502 Langmuir, Vol. 4, No.3, 1988 120,
,.....,.
I
I
I
I
I
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1
,
I
I
I
b
T
MI
t
M
T
30 20
A L
6
8
10 12
11
16
-i
F'igure 6. Lower critical temperature 'T8of the upper H,O-C,E( loops: (left)Tsversus i with j as parameter: (right) Tnversusi
with i as parameter.
however, the A-C mixture separates again into two phases with a lower critical point cp, on the water-rich side. This upper miscibility gap is, again for thermodynamic reasons, a cloaed 'loop", the upper critical point of which lies, in general, above the b o i i i point and plays no role in further considerations. The temperature T, of the lower critical point depends, evidently, on the nature of the amphiphile only, that is, on i and j. This is demonstrated in Figure 6, which shows on the left T8versus i with j as a parameter and on the right T, versus j with i as a parameter. As one would expect, T, drops with increasing i and rises with increasing j . For a given oil, T. and Td can thus not be changed independently of each other: the more hydrophobic the amphiphile, the lower both critical temperatures, the more hydrophilic the amphiphile, the higher. The shape of the T, curves on the right suggesta a description of the dependence of T8on j with i as a parameter by the empirical equation
T, = X ( i ) + Y(i) j[l
+ Zjl-'
(3)
If one fits this equation to the experimental data,one h d s (full lines) X ( i ) = -1055 + 4285/i, Y(i) = 1448 - 5152/i, and 2 = 1.2. Whether or not eq 3 can be substantiated by theory remains to be seen. The interplay between the lower miscibility gap of the B-C mixture and the upper loop of the A-C mixture determines the phase behavior of the ternary mixture. lta critical line starts at the upper critical point cp. of the lower B C gap. As one raises the temperature, it first ascends on the oil-rich side of the body of heterogeneous phases. With further rising temperature the influence of the upper A-C loop will eventually overcome that of the lower B X gap. At that temperature, the critical line will change to the water-rich side to further ascend on that side to terminate at cp,. The temperature of the inflection point depends, evidently, on both T. and T,. Now, T,and TBmay he varied in two ways: either by varying the amphiphile, which affects both critical temperatures, or by varying the oil, which affects T,, only. Let us perform the latter experiment first, that is, increase the carbon number k of the alkyl chain in the ternary mixtures HzOphenylalkane (k)-C,Ez. At thi8 point, the reader may argue that the binary mixture HzO-C,Ez does not show an upper loop, so that T, does not exist. For short-chain amphiphiles the tendency for phase separation is indeed too weak to enforce the formation of an upper loop but is still sufficiently strong to make the critical line of the
Figure 7. Phase prism of a ternary mixture A-B-c with a "broken" critical line. The three-phaee hcdy appears at TIby Beparation of the aqueous phaw into phaaes a and c a t the critical end point of el and disappears at Tuby merging of phasea c and b at the cep o{&. In between, the three-phase triangle changes ita shape such that phase c moves on an ascending trajectory
around the surface of the body of heterogeneous phases from cepe
to cep. (achematical).
ternary mixture change from the oil-rich to the water-rich side, as is demonstrated in Figure 2. This can be visualized by assuming the upper miscibility gap of such binary mixtures to "lurk" behind the A G T plane of the phase prism like the nose of a jet aircraft. The critical line on the surface of that nose never enters the phase prism (at 1bar) but still affects the phase behavior of the ternary mixture in that the critical point on the tip of the nose plays the role of cp,. 111. Trieritieal Point and Evolution of the
Three-Phase Body
Aa one increaaes k, T. rises whereas T, (or, in the above mixture, the position of the lurking nose) remains uneff e d . Accordingly, one expects the inflection point of the critical line to rise temperaturewise, its tangent becoming increasingly more horizontal. The critical line can thus he looked at as an elastic spring, the bending tension of which increases with increasing k until it breaks a t a tricritical point (tcp),which gives rise to the formation of a three-phase body. The tricritical point is thus that plait point at which the homogeneous mixture separates into three coexisting liquid phases instead of two. In view of the above considerations, one may also define it as that point at which the tangent at the inflection point of the critical line becomes horizontal. In a ternary mixture, it represents an isolated point in T-p-a-y space for which reason it would be purely accidental to find a tcp at atmospheric pressure. To find a tcp at constant pressure one, therefore, has to replace the effect of pressure on the chemical potentials of the three components by that of an appropriate fourth component. The broken critical line shows two loose ends that terminate at critical endpointa (cep) as shown schematically in Figure 7. The three-phase body appears at the (lower) cep of cl, at which the water-rich phase separates into phases a and c and disappears at the (upper) cep of Cl.on the oil-rich side at which phases c and b merge. Let the two endpoints have the coordinates Tl,el, y1 and Tu,a", yy.respectively. From these coordinates one obtains the temperature interval between the two critical endpoints AT = Tu- Ti (4)
as well as the mean temperature T = (TI T J / 2
+
(5)
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Langmuir, Vol. 4, No. 3, 1988 503
As mentioned above, it would have been purely accidental
Figure 8. Evolution ofa three-phesebody. Lek vertical d o n a through the phase prisms (a= 50 wt %) of the mixture H20phenylalkanes (k)-C&. The lower homologues show conneded criticallines with a two-phase lmdy only. The critical line breaks at a tricritical point that lies between k = 6 and k = I. For the higher homologues one finds three-phase bodies that rise and widen with increasing k. Right: the three-phase temperature interval A T (vertical bars) versus k,shaping a cusp that terminates at the tcp. The mean temperature T (broken line) continues smoothly from the line connecting the plait points (at a = 50 w t %) of the mixtures without three-phase bodies (open points).
These two figures characterize position and extension of the three-phase body on the temperature scale. Close to the tcp, the two cep's lie close to each other. Accordingly, AT is narrow. As one increases k further, the two end points will move increasingly further apart, so that AT widens as T rises. The most convenient p d u r e for determining AT and ! I is ' to erect vertical sections through the phase prisms at a = 50 wt % by preparing mixtures of equal masses of A and B, adding various amounts of C, and observing the phase sequence with rising temperature. At sufficiently high y one will find a homogeneous solution of the three components between the melting and boiling point, at a somewhat lower y the sequence 2 1 2, at even lower y the sequence 2 3 2, and at very low y the sequence 2 2. Such an experiment yields the profile of the body i f heterogeneous phases as well as a section through the three-phase body, the latter resembling the shape of a fish The left part of Figure 8 shows such sections through the phase prisms of the mixtures HzO-phenylalkane (k) 40 O C (see Figure 6). For the binary mixture H,0-C12E4,however, we have T,< 40 O C . Accordingly, one finds a miscibility gap that extends from the A-Cl2E4 side into the upper-left triangle terminating at plait point cp,. The same gap is, of course, found in the lower triangle. Here, however, it merges with the A-BIZ gap so that this triangle shows a connected gap extending from its A-B,, to its A-Cl& side. As in the phase prism of a ternary mixture, one may now again prepare a mixture of equal masses of water and oil, add various amounts (y) of CI2E, (C), and observe the phase sequence with increasing 6 (instead of rising T),where 6 denotes the mass fraction of ClZE, (C’) in the mixtures of the two amphiphiles 6 = C‘/(C + C l (9) expressed in wt 5% , although in this particular case the mole fraction would be the more appropriate concentration variable. At sufficiently high y one will again find a homogeneous aolution of the four components between 6 = 0 and 6 = 100 w t %, at sufficiently low y the sequence 2 3 2, and at very low y the sequence 2 2. Such an experiment yields a vertical section through the phase tetrahedron of the quaternary mixture at fixed a and constant T equivalent to that through the phase prism of a ternary mixture at fixed (Y but varying T. The shape of the section through the three-phase body will again resemble the shape of a fish, in this case, however, tilted “head down” with respect to the base as shown schematically on the lower right in Figure 13,due to the fact that both amphiphiles are mainly dissolved in the middle phase (c). The evolution of the three-phase body follows the aame pattern as in a ternary mixture, with the plait point cp, playing the role of the critical point cpp A t low 6, the quaternary mixture separates into two phases only, with amphiphile C mainly dissolved in the lower aqueous phase (2). As one increases 6, the aqueous phase will eventually separate into phases a and c at the end point of the critical line c& that starts at cp, and p d into the tetrahedron. With further increase of 6, phase c moves along the surface of the body of heterogeneous phases from the water-rich to the oil-rich side to eventually merge with the oil-rich phase b at the end point of the critical line cl,, that enters
--
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20: 18
i 0
6
7
8
9
10
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12
13
1L
yIwt%l
0 001
001
01
1
o [mN ”‘1
Figure 14. p* trajectory for the quaternary mixture HzO-ndodecane-ClzE4-ClzEe, determined in a pseudoternary phase prism. The trajectory deviates only slightlyfrom that of the pure amphiphiles.
Figure 15. Interfacial tensions between the three phases of the mixture HZO-n-decane+E3versus 2’. The broken line represents the s u m of a, and uh, demonstratingthe violation of Antonow’s rule (for further discussion, see text).
the tetrahedron through its base on the oil-rich side. This procedure of searching for the “optimum amphiphile”, or rather for the “optimum mixture of amphiphiles”, for a given oil at fixed a and constant Tis in every respect equivalent to that of searching for the “optimum salinity” if temperature, oil, and amphiphile are given. The same procedure can be followed to search for the “optimum oil” if temperature and amphiphile (and brine) concentration are given. This is a neccesary consequence of the Gibbs-Duhem relation in that changing the chemical potentials of the components of a ternary mixture by varying the temperature at constant pressure is equivalent to changing the chemical potentials by adding further appropriate components a t constant T and p . The exact procedure for discussing the temperature dependence of the phase behavior of a quaternary mixture is to determine such sections through the phase tetrahedron a t various temperatures, which is Lather time consuming. If one is only interested in the T- trajectories, the procedure can be considerably simplified by reducing the number of composition variables from three to two. This can be done by combining two of the components at a fiied mean concentration ratio into a single “component” and then representing the phase behavior of the mixture in a pseudoternary phase prism with that combination placed in one of the corners of the Gibbs triangle. It should be emphasized, however, that such a combination will, in general, not behave like a pseudocomponent, that is, like a combination of components for which the concentration ratio in each of the phases is equal to the mean concentration ratio. When applying this procedure to the above mixture of water, an oil and two nonionic amphiphiles, the obvious choice is to combine the two amphiphiles into a single component (C + C’) at a fixed mean 6. Each point in the pseudoternary phase prism is then again defined by a certain set of T, a,and y, where y is now the fraction of the sum of the two amphiphiles in the mixture y = (C + C ? / ( A B C C’) (10)
These procedures may, of course, be applied to any ratio between water and oil other than a = 50 wt %, in particular to mixtures of equal volumes of water and oil ( 4 = 0.5) insgad of equal masses. Experience, however, shows that the T-7 trajectories at a = 50 wt % and at 4 = 0.5 differ only little. In both cases one looses the information about the temperature interval, AT, of the three-phase bodies. If required, one may chose the mean composition a = 50 wt % ’ and 7/2 and determine the lower and upper boundary of the three-phase body, which gives AT in sufficient precision. In view of eq 1 and the T 3 trajectories one can then estimate AT for the other mixtures of that series.
at fixed mean :6 One may then trace the coordinates and of point X as they change with 6. Figure 14 shows the T-4 trajectory of the quarternary mixture HzOB12-(C12E4+ C12E6).The empty points represent the T;i. values for the pure amphiphiles, the full points those for a few mixtures of C12E4 and CI2E6. As one can see, the trajectory of the mixture deviates only slightly from that of the pure amphiphiles. The same procedure can be followed when searching for the “optimum carbon number of the oil”, if temperature and amphiphile (and brine) concentration are given.
Experience, however, shows that for medium- and longchain amphiphiles Antonow’s rule does not hold but that instead aab < + abc (14) as is also demonstrated in Figure 15. The characteristic dependence of the various interfacial tensions on temperature as well as the inequality of eq 14 has important consequences for the preferential wetting
+ + +
VII. Interfacial Tensions The third property of the three-phase body important for theory and application is the minimum of the interfacial tention gab between the aqueous and the oil-rich phase at T. Since phases a and c separate at cep8, the interfacial tension aacbetween them must rise from zero at Tl to increase with further rising temperature, as shown in Figure 15 for the mixture H20-Bl,,-C8E3. Phases b and c, on the other hand, merge at cep,. Consequently, the interfacial tension abc must decrease with rising temperature to vanish at Tu.In view of Antonow’s rule12 aab = + (11) and of the temperature dependence of a,, and ah, the minimum of gab at T’is thus an inevitable consequence of the phase behavior. Below and above the three-phase body, the interfacial tension between the aqueous and the oil-rich phase is much higher. Theory12predicts that sufficiently close to a tcp Figure 15 should be symmetric, so that aa,(T = Tu)= ah(T = Tl) = a,. Theory, furthermore, predicts (12) a,,(T = r, = Ubc(T = 0 = a0/3
so that, if eq 11 holds U,b(T
=
= 2a0/3
(12)See, e.g.; Widom, B.Langmuir 1987, 3, 12.
(13)
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508 Langmuir, Vol. 4, No.3, 1988
m @ @
@
Fmre 16. Referential wetting betwean two p h e a in a stirred macroemulsion (for further discussion, we text).
between the three phases in a stirred macroemulsion. In the absence of gravity and convection, the nature of the inkfaces as well as the shape of the droplets is determined by the equilibrium condition KEu,r4,,)Ta+*? = 0
(15)
where A,, denote the area of the interface between phases i and j and u,, the corresponding interfacial tension. In a convection-free mixture, gravity enforces a separation of the three phases into layers with plane interfaces on top of each other irrespective of the fact that because of eq 14 phases a and b would prefer to be in contact with each other than to be separated by phase c. If, however, one removes phase c between phases a and b until only a drop of c is left, one observes that this drop does not spread a u w the a/b interface but shapes a lens floating on that interface. The shape of that lens is determined by the relation between the interfacial tensions as well as by that between the densities. If, in a mixture with comparable volume fractions of the three phases, the effect of gravity is counteracted by steady stirring, the relations between the interfacial tensions may lead to various types of emulsions. As an example let us consider a mixtwe within a three-phase body with a mean composition close to the a comer of the three-phase triangle. Let us further simplify the problem by neglecting the effect of gravity and assuming phases c and b to be present each in a single droplet of equal volume. We now ask the question: will the two droplets stay apart from each other or will they form a dimer, as is shown schematically in Figure 16? If the two droplets are separated, the free energy of the internal interface is GA(singlets)= 4r?(uab + u,J (16) If the two droplets form a dimer with radius
R = 2113~
(1’0
the free energy of the internal interface is GA(dmer) = 2&W(uab
+ a,) + &/3r?7h
(18)
For the difference between the two situations one then finds GA(sing1ets) - GA(dimer)= r22/3mu.b + u~2121/3- 1) (19)
If this difference is negative, the “emulsion” will consist of singlets, if positive, of dimers. Accordingly, if 0.52(uab + u d / u h (20) is smaller then unity, one will find singlets, if it is larger than unity,dimers are found. Close to TI,one has ( F i i 1_5) urbiiuh >> u,, so that one expects singlets. Close to T,however, one has u . ~> uk a u-, so that one expects dimers. These considerations are supported by experiment at temperatures close to Tb phases a and c separate very slowly after the stirrer has been turned off,due to the small density difference and low interfacial tension between near-critical phases. As one raises T , one observes a sudden transition from slow to rapid phase separation. Above that temperature one observes dimers of phases b and c resembling the shape of -Russian dolls” rising in the
lower aqueous phase until they break up at the a/c interface, the oil droplets continuing their way through phase c to the upper oil-rich phase. Correspondingly, phases b and c separate very slowly near Tu. As T drops, one observes a transition from slow to rapid phase separation, with dimers of phases a and c raining out of the upper oil-rich phase. In addition, one observea dimers of phases a and b floating in phase c, the “Russian dolls” now resembling free balloons, the buoyance of which is determined by the volume ratio of the two half spheres. This slow rapid slow transition in the dynamics of phase separation,apparently c a d by the change of preferential contacts between the various phases, should be of importance upon application because in practice one may be interested in either slow or rapid phase separation, depending on the problem to be solved. It reflects itself in sudden changes of the viscosity of the stirred solution as well in changes of the electric conductivity (if traces of an electrolyte are added). Ita velocity depends sensitively on the mean composition of the mixture, that is, on the volume fractions of the three phases that change with temperature according to the levers rule. Equivalent transitions are found if one scans through the three-phase body at constant temperature by adding a fourth component. Of the three relevant properties of a three-phase body, namely ‘?, T, and uab(T= lp), temperature is a field variable, the mass fraction 7 a density variable, and 0.b an order parameter. Accordingly, one expects a relation between these three properties which permits evaluating one of them, if the other two are known. As such we have recently suggestedi3 +
-.
(uab/T21r)fl =
1
(21)
with u expressed in mN m-’
fl = 2 ( M , / ~ a , ) ~
(22)
where M , is the molecular weight of the amphiphile. P = (1 + PJ/2 (23)
p is the mean density of phase c, and a, is the mean area occupied by each amphiphile molecule in a monolayer, being of the order of 0.5 nm2. Whether relation 21 can be substantiated by theory remains to be seen. As an empirical relation it may serve for estimating the interfacial tension uabbetween the aqueous and the oil-rich phase if ’?and 4 are known,the two latter quantities being much easier to measure than uab VIII. Effect of Pressure Before discussing the effect of inorganic electrolytes on the phase behavior, let us briefly discuss the effect of preesure. Since the phase behavior of the ternary mixture is mainly determined by that of the two binary mixtures B-C and A-C, one may predict the effect of pressure on the properties of the three-phase body by first considering its effect on the lower B-C gap and the uppet A-C loop. This effect is determined by the sign a2V,/ay2 at the critical point, with V, denoting the mean molar volume of the correspndhg mixtureJ4 For an upper critical point ( T J , aT,/ap bas the opposite sign of a2V,/ayz, whereas for a lower critical point (To),aT,/ap has the same sign. Experience sbom that the excess volume of l3-C mixtures is, in general, positive (i.e., (a2V,/-3y2),,1W < O), whereas that of A-C mixtures is, in general, negative (Le., (13)Kahlweit, M.;Shy,R:Haase,D.; Firman, P. Langmuir,in presa. (14)See, e.&: Prigogine, I,; Defay, R. Chemcal Thermodynamics; Translated by Everett, D. H.,Longmans: London, 1954; p 288ff.
Langmuir, Vol. 4. No. 3,1988 509
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E 5011
q - , 0
,
-
m
LOO
,
wo 800 p[borl
.. IWO
Figure 17. Effectof pressure on the phase behavior of a temary mixture. Top: achematical representation of the effect on the position and extension of the three-phase bodies, assuming 7’. rises faster than T,with increasing pressure. Bottom: effect of pressure on the three-phase bodies of the mixture HzOphenylalkanes (k)-C,E,.
(8Vm/ay2)_o > 0). As a conwquence, both T. and Torise with increasing p as if increasing the pressure makes nonionic amphiphiles effectively more hydrophilic. The effectof this rise on the properties of the three-phase body depends on the outcome of the race between T, and T,. In view of the considerations presented in section I1 one expects the three-phase bodies to rise and widen with increasing p if T, rises faster than T, but to shrink if T. rises slower than T,. As an example we consider the mixtures H,O-phenylalkanes (k)-C,E,. At atmospheric pressure, the tcp lies between k = 6 and k = 7 (Figure 8). In these mixtures T. rises faster than T,. Consequently, one expeds the cusp to rise and widen with increasing p as is shown schematidy in the top part of Figure 17. Let the cusp at p 1 represent that at 1bar. As one increases the pressure, the three-phase bodies rise and widen, as will the cusp until its tip-the tep-will eventually reach k = 6 at p,. With further increasing pressure, the tcp will move gradually down the k scale until the tcp reaches k = 5 at p3and so forth, thus forming a trimitid line (tcl) in T-p-k space, each point on it representing a tcp being defined by a particular composition at the corresponding T and p . In the bottom part of Figure 17 one can see the cusps for k = 6 and k = 7 projected onto the T - p plane, the first one determined at a = 38 and y = 42.9 wt % and the latter a t a = 50 and y = 44 wt %. With these particular mixtures, one may thus search for a tcp either by mixing k = 6 and k = 7 at atmospheric pressure or hy increasing the pressure on the ternary mixture with k = 6. For the latter mixture one expects the tcp to lie close to 50 OC at about 200 bars. The shape of the p cusp of the mixture with k = 7 (bottom of Figure 17) on the other hand, suggests considering the phase behavior of this ternary mixture as evolving from a tcp at a “negative” pressure.
IX. Quaternary Mixtures H,O-Oil-Nonionic Amphiphile-Salt Another way of searching for a tcp in the mixture H,OphenylhexaneC4E, is to add an appropriate inorganic electrolyte. Again we first consider the effect of salts on the phase behavior of the lower B-C gap and the upper A-C loop. Due to the very low solubility of such salts in
Figure 18. Phase tetrahedron of the quaternary mixture HzOoil-nonionic amphiphilelyotmpic electrolyte(schematical). The diagram on the lower right shows a vertical section through the tetrahedron at m = 50 wt %. oil and water free nonionic amphiphiles one expects the major effect on the A-C loop. The effect of an inorganic electrolyte on the mutual solubility between HzO and nonionic amphiphiles is known since 1888, but has, unfortunately, not attracted the attention it deserves.16 If one applies the hard-oft-acid-base (HSAB) principle,’6 the facta may be summanzed ’ asfollows: saltsofhardacids and hard bases widen the upper A-C loop (for which reason they are called “lyotropic”). Salta of soft acids and hard bases or of hard acids and soft bases make the loop shrink (for which reason they are called “hydrotropic”). Accordingly, salts like NaC1, CaCl,, or NazS04decrease the mutual solubility between A and C, whereas ionic amphiphiles increase the mutual solubility. A major exception is the ClO; ion which, although clansifled as a hard base, increases the mutual solubility when combined with a hard acid (NaC10,). With respect to the efficiency of lyotropic sodium salts in decreasing the mutual solubility, one fmds the following order: SO4%> Cr04* > Cl- > NO; > C10,. This series is quoted either as a “Hofmeister”or a “lyotropic”series. At constant salt concentration, the effect on the mutual solubility decreases with increasing amphiphilicity, that is, increasing chain length of the amphiphile. Accordingly, one expects the strongest effect with weak amphiphiles, in particular with those that do not show an upper loop like C,Eo, C3Eo,tert-C,E,, or C4E,. Indeed, as one adds an appropriate lyotropic electrolyte the loop appears; that is, the “lurking nosen of the binary mixture is pushed into the phase prism. Which salt to apply for “salting out” the amphiphile depends, of course, on the hydrophilicity of the latter, that is, on the ‘distance” between the lurking nose and the A-C-T plane at 1 bar. If the upper loop is allready present, it widens so that T, drops as if adding a lyotropic salt makes the nonionic amphiphileg effectively more hydrophobic. In so far, adding a lyotropic salt has the opposite effect of increasing the pressure. We shall now apply these considerations for interpreting the following experiment on the quaternary mixture HzO-phenylhexane-C4Ez-NazS04on the basis of Figure 18. As base of the phase tetrahedron we chose the A-B-C triangle and placed NazS04(E) on top. At ambient temperatures, the base shows the phase diagram 2. In the upper right W E triangle one fmds complete miscibility along the B-C side and a two-phase region with solid E as the second phase. In the lower A-B-E triangle, the central A-B gap is bounded by homogeneous mixtures. (16) See,e&: Firman. P.;Haase, D.;Jen,J.; Kahlweit, llz;Strey, R Longmuir 1986,Z. 718. (16)See, e.g.: Pearson, R. G. Sura h o g . Chem. 1969,5, 1.
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510 Langmuir, Vol. 4, No. 3, 1988
Between this gap and the E comer, one finds a three-phase triangle hounded by two two-phase regions, all with solid E as the second and third phase, respectively. The upper-left A-C-E triangle is the important one. Along the A-C side one finds complete miscibility. Since the lyotropic salt enforces a phase separation between A and C, one finds a two-phase region at sufficiently high salt concentrations with two liquid phases. These two phases merge a t plait point cp,. Between this two-phase region and the E comer one finds a three-phase triangle hounded hy two two-phase regions, all with solid E as the second and third phase, respectively. The evolution of the three-phase body within the phase tetrahedron follows the same pattern as in the quarternary mixture discussed in Figure 13, with the plait point cp. playing the role of plait point cpT. A t low salt concentrations, the quatemary mixture separates into two phases only, with amphiphile C mainly dissolved in the lower aqueous phase. As one increases the salt concentration, the aqueous phase will eventually separate into phases a and c at the end point of cl, that starts at cp. and descends into the tetrahedron. With further increasing salt concentration, phase c moves along the surface of the body of beterogeneous phases from the water-rich to the oil-rich side to eventually merge with phase b at the end point of cl, that enters the tetrahedron through its base on the oil-rich side. As in the phase tetrahedron of quaternary mixture with two amphiphiles (see Figure 13). one may again erect a vertical section through the phase tetrahedron and determine the position and extensions of the three-phase body. Again the shape of the section through the three-phase body will resemble that of a fish, now, however, tilted “head up” with respect to the base, as shown schematically in the lower right part of Figure 17, due to the fact that the salt is mainly dissolved in the aqueous phase a. For discwing the temperature dependence of this phase behavior, we recall Figure 1. The ternary mixture shows a connected cl that changes from the oil-rich to the water-rich side with rising temperature. When adding a more hydrophobic oil (e.g., k = 7), one raises T, (at fixed To)until the critical line breaks at a tcp. In this experiment, however, one keeps T,,fixed hut decreases Toby adding the salt. Accordingly, the tension on the cl will again increase, and the tangent at its inflection pointe will become increasingly more horizontal until it again breaks at a tcp. This is, in fact, what Lang and Widom” have done when searching for the tcp in the quatemary mixture HzO-benzeneCz~-(NIiz)zSOo The following experiment is less ambitious. We were only interested in demonstrating the equivalence of the phase behavior of A-BX mixtures with a second oil, a second amphiphile, or an electrolyte as the fourth component. For this purpose we combine the salt and the water into one component (the brine), expressing ita composition by the fraction of the salt in the brine t
= E / ( A + E)
(24)
in wt % and placing the brine in the A corner of a pseudotemary phase prism. Figure 19 shows some 01 = 50 wt % sections through the pseudoternary phase prism of the quatemary mixture HzO-phenylhexane-C,E~Na$304with t as parameter. The section at t = 0 represents the one through the phase prism of the truly ternary mixture with a connected critical line (see Figure 8). As t is increased, the plait points drop temperaturewise, the groove hecoming sharper. Between e = 0.25 and t = 0.40, the critical (17) Lang,J. C.; Widom, B.Phyaica A (Amstenlam) 1975,8Z. 190.
t
20 0 ; ’ 0 35
LO
L5
50
55
y Iwt %I
Figure 19. Evolution of a three-phase body in the quaternary mixture HzO-phenylhexane-C4E2-Na2S0,, represented in a peudotemary phaae prism. The salt free mixture (e = 0)shows a connected critical line. The cl breaks between e = 0.25 and 6 = 0.4 w t % which givea rise to the evolution of a t h r e e - p h M y . The three-phase bodies drop and widen with increasing e.
line breaks. From then on, one finds three-phase bodies that drop and widen with increasing t as long as one is looking sufficiently close to a tcp. A t high brine concentrations, the three-phase bodies may shrink again. We repeat that the three-phase triangles do not lie on isothermal sections through the pseusoternary phase prism hut are tilted with respect to the base,more or less parallel to the center line of the fishes. The procedure of breaking the connected critical line a t a tcp by adding a lyotropic salt is thus in principle analogous to that of breaking the cl by adding an appropriate second oil, by adding an appropriate second amphiphile, or hy increasing the pressure. In each case the tcp can he looked at as a pivot point from which the three-phase bodies evolve, with AT widening with increasing distance from the tcp. With nonionic amphiphiles, T rises with increasing carbon number k hut drops with increasing brine concentration t. If the ternary mixture A-B< shows a connected cl at 1 bar, the tcp lies at a positive t. If, however, the ternary mixture allready shows a three-phase body at e = 0, one may consider the phase behavior of the quaternary mixture as evolving from a tcp at a “negative” salt concentration. The general pattern is thus quite similar to that shown in Figure 16, if one replaces p by t (setting t = 0 for p = pl) and takes either k (for a given amphiphile) or the amphiphilicity (for a given oil) as the second abcissa, the only difference being that for nonionic amphiphiles the cusps drop temperaturewise with increasing e. With respect to the dependence of the efficiency of the amphiphile on e, experience shows that for short-chain amphiphiles 7 increases with increasing t as can he seen on Figure 19. For medium- and long-chain amphiphiles, however, 4 decreases so that these amphiphiles become effectively more efficient with inmasing e. Figure 20 (top) shows the effect of NaCl on the upper H,0-CJZ4 loop in a pseudohinary representation and (bottom) the Ti. trajectory of the quaternary mixture H,0-Blo
