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Description of Phase and Emulsion Inversion Behavior for the Ethanol/Benzene/Water System by the Bancroft-Hand Transformation and Critical-Scaling Theory Kyung-Hee Lim,*,† Jong-Moon Lee,† and Duane H. Smith‡ Department of Chemical Engineering, Chung-Ang University, Seoul 156-756, Korea, National Energy Technology Laboratory, U.S. Department of Energy, Morgantown, West Virginia 26507-0880, and Department of Physics, West Virginia University, Morgantown, West Virginia 26506 Received January 15, 2002. In Final Form: May 23, 2002 The emulsion morphology diagram for the ethanol/benzene/water system is analyzed by a coordinate transformation of emulsion inversion data and by critical-scaling equations in the transformed coordinates. The transformation, which may be called the “Bancroft-Hand” transformation, converts the phase and inversion volume fractions into a new concentration variable. When this new concentration variable is used, the tie lines become parallel with each other. The parallel tie lines extend the validity of the criticalscaling theory to a large fraction of the binodal. The phase and emulsion inversion compositions obtained with these transformed concentration variables are compared to the corresponding phase and inversion compositions in polar coordinates (another coordinate transform for obtaining parallel tie lines). The interpretation of the behavior of this system in the Bancroft-Hand transformation also is compared to the behavior of the binary 2-butoxyethanol/water-temperature system (where the tie lines are rigorously parallel).
Introduction An emulsion is a colloidal dispersion in which droplets of one liquid phase are dispersed in another liquid phase.1 For the dispersed condition, two “simple” morphologies have long been recognized. They are phase I dispersed in phase J (designated I/J) and J dispersed in I (designated J/I). When I and J represent oil-rich (oleic) and waterrich (aqueous) phases, respectively, the morphologies are conventionally known as oil-in-water (O/W) and waterin-oil (W/O) emulsions. In these emulsions, the symbols O and W stand for the equilibrium conjugate phases, that is, O for the oleic phase and W for the aqueous phase, but not the components “oil” and “water”. More complicated types or “multiple” emulsions such as W/O/W, O/W/O, W/O/ W/O, and so forth also can appear. The physicochemical properties of emulsions and emulsion products are greatly affected by emulsion morphologies. For this reason, in many emulsion applications a particular morphology is required, and this morphology should be maintained for a certain period of time. However, emulsions experience an abrupt transition in morphology, for example, from O/W to W/O or vice versa at certain conditions.2 This transition is known as emulsion inversion and displays a hysteresis.2-5 Hence, it is imperative to understand at what conditions a morphology inverts and how large the hysteresis region is. Emulsion inversion has been known for almost 90 years, and many studies have been done during this period. As * To whom correspondence should be addressed. E-mail: khlim@ cau.ac.kr. † Chung-Ang University. ‡ U.S. Department of Energy and West Virginia University. (1) International Union of Pure and Applied Chemistry. Manual on Colloid and Surface Science; Butterworth: London, 1972. (2) Smith, D. H.; Lim, K.-H. J. Phys. Chem. 1990, 94, 3746. (3) Becher, P. J. Soc. Cosmet. Chem. 1958, 9, 141. (4) Salager, J.-L. Encyclopedia of Emulsion Technology; Becher, P., Ed.; Marcel Dekker: New York, 1988; Vol. 3, p 104. (5) Binks, B. P. Langmuir 1993, 9, 25.
Figure 1. Binodal and emulsion inversion cusp for the binary 2-butoxyethanol/water systems with parallel tie lines.
a result of these studies, we now understand emulsion inversion much better, but a quantitative theory of inversion is largely lacking. Hence, there has been an increasing interest in quantitative studies of emulsion inversion. As an effort toward this end, we performed a quantitative study of the dependence of emulsion hysteresis on temperature using electrical conductivity measurements and catastrophe and critical-scaling theories to interpret the data.6 Because the theories require that the tie lines be parallel with each other, the emulsion inversion measurements were done with the binary 2-butoxyethanol/ water-temperature system. Figure 1 shows the phase diagram of this system, along with the measured emulsion inversion hysteresis cusp. At temperatures above the lower consolute solution temperature (Tc), the system of Figure 1 has a miscibility gap, where it splits into a water-rich (i.e., aqueous) phase, AQ, and a conjugate amphiphilic phase, AM. Two inversion (6) Smith, D. H.; Lim, K.-H. Langmuir 1990, 6, 1071.
10.1021/la0200515 CCC: $22.00 © 2002 American Chemical Society Published on Web 07/30/2002
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hysteresis lines (φ1 and φ2) inside the miscibility gap meet in a cusp at the critical point of the phase diagram and divide it into three regions. In the region bounded on one side by the aqueous ends of the tie lines, only AM/AQ (amphiphilic phase in aqueous phase) emulsions were observed; in the region bounded on one side by the amphiphilic ends of the tie lines, only AQ/AM emulsions were found. Separating these two regions is the hysteresis region bounded by the emulsion inversion lines φ1 and φ2, which meet at the critical point. For temperatures and phase volumes within the hysteresis region, the morphology depended on which experimental direction was taken along the tie line. When amphiphilic phase was added (left to right in Figure 1), the AM/AQ morphology persisted up to line φ2; when the volume fraction of the aqueous phase was increased (right to left in Figure 1), the AQ/AM morphology persisted up to line φ1. One of our findings for this binary system was that the hysteresis width, |φ2 φ1|, and the midpoint, (φ1 + φ2)/2, of the hysteresis varied with temperature as |T - Tc|1.26 and |T - Tc|0.528, respectively. In Figure 1, the emulsion morphologies were determined from measurements of emulsion conductivities and their fits to the equations for the respective morphologies. The inversion was detected as a large change in the conductivity. Emulsions for the conductivity measurements were prepared by mixing the conjugate phases (not the components), which were already separated after the mixed components were equilibrated. Known volumes of the phases were mixed in a thermostated beaker for 5 min, interrupted for 10 min, and then remixed for 5 min. The conductivity was then measured, while mixing was maintained. In subsequent steps, a known volume of one of the conjugate phases was added, and the mixing, separation, remixing, and conductivity measurement sequence was repeated. This three-step procedure enabled measurement of the emulsion inversion with excellent reproducibility.2 Emulsions used in practical applications are usually formed from multiple components, that is, three components or more. In these multicomponent systems, tie lines are almost never parallel with each other when the phase diagrams are represented in conventional concentrations. Hence, tie lines should be converted in some manner before critical-scaling theory is applied to these systems. Griffiths and Wheeler7 pointed out that common thermodynamic equations contain both density variables (e.g., physical density and composition), which have different values for coexisting phases, and field variables (e.g., temperature, pressure, and chemical potential), which have identical values for conjugate phases. The thermodynamic description of a mixture of phases is simpler if the description is entirely in terms of field variables. In particular, for systems of three or more components, field variables solve many difficult problems associated with the orientations of tie lines in the phase diagram, since tie lines are orthogonal to a field-variable dimension and thus automatically fall in parallel planes. As introduced by Fisher,8 “hidden” variables can sometimes be found, which also have the highly desirable property of making tie lines be parallel. Recently, we have shown that a hidden variable exists for ternary systems that have “polar” tie lines.9,10 In a ternary diagram, extensions of tie lines may meet at a common point, or (7) Griffiths, R. B.; Wheeler, J. C. Phys. Rev. A 1970, 2, 1047. (8) Fisher, M. E. Phys. Rev. 1968, 176, 257. (9) Lim, K.-H.; Smith, D. H. J. Colloid Interface Sci. 1991, 142, 278. (10) Smith, D. H.; Lim, K.-H.; Ferer, M. J. Chem. Phys. 1991, 95, 3649.
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Figure 2. Ethanol/benzene/water binodal with nonparallel tie lines, along with the emulsion inversion cusp, with concentrations plotted in volume-fraction units.
“pole”, and thus define a set of polar coordinates. Since the polar coordinates, r and θ, are orthogonal to each other, the tie lines become parallel when they are represented in terms of r and θ. However, in order for the tie lines in the ternary diagram to be parallel, the ratio of the difference of the equilibrium phase compositions of the two components should be constant, or (x2 - x1)/(y2 - y1) ) constant where xi and yi are the concentrations of any two components X and Y in the equilibrium phase i (i ) 1, 2). For real ternary systems at constant T and P, such special behavior is unexpected and hardly ever observed for all tie lines of a system. Therefore, this requirement limits the theory to tie lines extremely close to the plait point, where the ratio remains approximately constant. Moreover, in the polar coordinates, the tie line end points are characterized by the quantities in terms of r and θ, and these quantities are far different from conventional concentrations. In the work presented here, we have analyzed the phase and emulsion inversion compositions for the ternary ethanol/benzene/water system using another set of hidden variables, obtained by the “Bancroft-Hand transformation”. Figure 2 shows the phase behavior and emulsion morphologies of this ternary system.9 Analogous to the 2-butoxyethanol/water binary emulsion morphology diagram, the two-phase region of Figure 2 is divided into three distinct parts: W/O emulsions, hysteresis region, and O/W emulsions. However, unlike the binary-temperature diagram (Figure 1), the tie lines are not parallel in Figure 2. The Bancroft-Hand transformation makes the tie lines whose extensions intersect at a common point in any conventional concentration units be parallel with each other. Moreover, the transformation converts the phase compositions to new concentration variables that are similar to the conventional concentrations. After the transformation, we analyzed the transformed phase and emulsion data of Figure 2 using the critical-scaling equations. We found that the emulsion inversion hysteresis width and the curvature varied with temperature as |T - Tc|0.11 and |T - Tc|0.69, respectively, which are quite different from |T - Tc|1.26 and |T - Tc|0.528 for the binary system. We also found that the Bancroft-Hand transformation was as effective as the transformation to the polar coordinates but that it was more convenient and easier to use in the analysis for the phase and emulsion inversion data.
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Bancroft-Hand Transformation10
and therefore
The distribution of a solute between two phases was studied by Nernst in 1891.11 The so-called distribution law of Nernst states that “if the molecular weight of the solute is the same in both solvents, the ratio in which the solute distributes itself between the two solvents is constant for a given temperature.”12 That is,
A1/A2 ) K
(1)
where A1 and A2 are the concentrations or solubilities of the solute A in phases 1 and 2, and K is a constant. Equation 1 is valid only when the solubility is very small and when the solute exists as simple molecules. When eq 1 was applied to the distribution of acetic acid between water and benzene, K was found to increase with A1. The failure of eq 1 for this system was that acetic acid molecules formed dimers by self-association and A12/A2 was found to be much more constant than A1/A2. Further, Dawson13 found that if dissociation of acetic acid dimers was taken into consideration, the constancy of the values of K was made better. In Dawson’s method, A1 and A2 were replaced by A1(1 - aD)/r, the concentration of acetic monomer, and A2 - A1(1 - aD)/r, the concentration of associated molecules, respectively. Here aD is the degree of dissociation of the acid and r is the ratio of distribution of the acetic monomers between the two phases. Hence, the distribution law of Nernst may be extended to systems in which association and dissociation take place. In this case, the following equation may describe the distribution of the solute between the phases 1 and 2:
[A1(1 - aD)/r]n A2 - A1(1 - aD)/r
)K
(2)
Here n is the association number, that is, the number of molecules participating in the association. The validity of eq 2 is still limited to the distribution of consolute liquid in exceedingly low concentrations. Hand12 developed an equation which would describe the distribution in concentrated solutions up to the point at which the three components become entirely miscible. He made a simple assumption that “the consolute component (A) is divided between the other two components (B and C), according to a simple ratio, so that in each liquid phase the ratio of the weights of the consolute component held by the other components is a constant.” If this assumption is stated mathematically, then for each phase
A1C/C1 A1B/B1
)K)
A2C/C2 A2B/B2
(3)
where A1B, A1C, A2B, and A2C are the amounts of consolute component A held by B and C in phases 1 and 2, respectively. Similarly, B1, C1, B2, and C2 are the amounts of B and C in phases 1 and 2. Rearranging eq 3 yields the equations
A1B/B1 ) A1C/KC1 ) A2B/B2 ) A2C/KC2 (11) Nernst, W. Z. Physik. Chem. 1891, 8, 110. (12) Hand, D. V. J. Phys. Chem. 1930, 34, 1961. (13) Dawson, H. M. J. Chem. Soc. 1902, 81, 521.
A2 A1 ) B1 + KC1 B2 + KC2
(4)
in which A1 ) A1B + A1C and A2 ) A2B + A2C are the total amounts of A in phases 1 and 2, respectively. In obtaining eq 4, use is made of the mathematical relation of ratios x/(x + y) ) w/(w + z) for x/y ) w/z. Equation 4 is a general equation for distribution of a consolute liquid, which was discovered by Hand following Bancroft’s advice.14 In the limiting case in which the two immiscible liquids remain practically immiscible on addition of consolute liquid, eq 4 becomes (A2/C2)/(A1/B1) ) K, which is similar to the Nernst equation, eq 1. With another application of the relation of ratios to eq 4, the equation can be written in a different form,
A2 A1 ) A1 + B1 + KC1 A2 + B2 + KC2
(5)
It is this equation that we call the Bancroft-Hand transformation, because it suggests a new set of concentrations which can be defined by
a)
A A + B + KC
(6a)
b)
B A + B + KC
(6b)
c)
KC A + B + KC
(6c)
These equations satisfy the condition
a+b+c)1
(7)
and make tie lines parallel in the transformed coordinate system. Consider a tie line with end-point component concentrations (A1, B1, C1) and (A2, B2, C2). Then one obtains a1 ) a2 after the transformation by eq 5. This implies that the tie lines are parallel in the new triangular coordinate system (a, b, c). That is, the tie line compositions (A1, B1, C1)-(A2, B2, C2) are converted to (a, b1, c1)-(a, b2, c2) in which a ) a1 ) a2. In the transformation of eq 5, K is calculated from the tie line compositions as
K)
A2B1 - A1B1 A1C2 - A2C1
(8)
If a point (a, b, c) is known, the corresponding compositions in the conventional concentration diagram can be calculated with the equations
A)
aK c(1 - K) + K
(9a)
B)
bK c(1 - K) + K
(9b)
C)
cK c(1 - K) + K
(9c)
The Bancroft-Hand transformation, eq 5, converts one of the compositions to the hidden variable of Fisher, in that it is a “concentration” which is in equilibrium with the system, which is equal valued in conjugate phases, (14) Hand, D. V. J. Phys. Chem. 1930, 34, 1973.
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and which is a measure of the “temperature-like” distance from the plait (critical) point. Because a is a variable that measures the temperature-like distance to the critical point along a direction that is orthogonal to parallel tie lines, it will be convenient to use the reduced variable aR ) |a - ac|/a, where (ac, bc, cc) is the critical (plait) point, in the critical-scaling equations. Critical-Scaling Equations with a Hidden Variable According to renormalization group theory, the concentrations, X+ and X-, of any of the three components in the two conjugate phases should behave as
X( ) Xc ( Lβ ( Mβ+∆ + N1-R
(10) Figure 3. K values for the tie lines.
where Xc is the critical concentration and the subscripts plus (+) and minus (-) denote the tie line end points of the concentrations which are larger and smaller, respectively, than Xc. The values of the parameters L, M, and N depend on the chemical system, but β, R, and ∆ are universal scaling exponents whose values 0.324,15-20 0.112,16-20 and 0.5,15,16,20 respectively, are the same for all systems. The quantity is the reduced temperature defined as ≡ |T - Tc|/Tc. From eq 10, one obtains
β/(1 - R). Hence
| |
b2 - b1 ) b′aβ/(1-R) ) b′
|
c2 - c1 ) c′
|
a - ac ac
β/(1-R)
) -b′
| |
a - ac ac
β/(1-R)
a - ac ac
β/(1-R)
(15a)
(15b)
X+ - X- ) 2Lβ + 2Mβ+∆
(11)
Similarly, in the equation for the average concentration of the two conjugate phases (eq 12), the 1 - R exponent is renormalized to 1. Thus the equations become
X+ + X≡ Xavg ) Xc + 2N1-R 2
(12)
a - ac b 1 + b2 ) bc + b′′a ) bc + b′′ 2 ac
On the right-hand side of eq 11, the first term predominates in the vicinity of the critical point, while the second term is more significant away from the critical point, because of the larger exponent value (β + ∆ ) 0.824 versus β ) 0.324). Without the second term on the right-hand side, eq 11 is rewritten as
(X+ - X-)1/β ) L′|T - Tc|
(13)
where L′ ) (2B)1/β/Tc. A principal advantage of eq 13 is that it allows simple fits to experimental values of the conjugate-phase properties, X+ and X-, without measurement of . For systems in which one of the compositions is made equal for the conjugate phases, the dependence of the hidden compositional variable a must be used,8 that is,
a )
|
|
a - ac ac
1/(1-R)
(14)
which leads to the exponent “renormalization” of Fisher. Therefore, the differences between concentrations in the two conjugate phases, which scale as β without the second term in eq 11, will be renormalized to scale with exponent (15) Baker, G. A.; Nickel, B. G.; Green, M. S.; Meiron, D. I. Phys. Rev. Lett. 1976, 36, 1351. (16) Baker, G. A.; Nickel, B. G.; Meiron, D. I. Phys. Rev. B 1978, 17, 1365. (17) Beysens, D. In Phase Transitions: Carge` se 1980; Levy, M., Ed.; Plenum Press: New York, 1982; pp 25-62. (18) Greer, S. C.; Moldover, M. R. Annu. Rev. Phys. Chem. 1981, 32, 233. (19) Le Guillou, J. C.; Zinn-Justin, J. Phys. Rev. B 1980, 21, 3976. (20) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon: Oxford, 1982.
|
| |
a - ac c 1 + c2 ) cc + c′′ 2 ac
|
(16a)
(16b)
Equations 15 and 16 replace eqs 11 and 12, respectively. These equations summarize the results of our substituting the parallel tie lines and hidden-variable dependence of the temperature-like distance from the critical point, a, for the field dependence of . For real systems, these substitutions replace the difficult measurement of chemical potentials with the much simpler measurement of concentrations. Theoretically, these substitutions avoid the thermodynamically unsatisfactory assumption that different components have identical chemical potentials. Results and Discussion The values of K were found from the data in Figure 2 and eq 8, and the results are shown in Figure 3. The volume fractions were used for the calculation of K and also for the new concentration variables (a, b, c) and emulsion inversion points throughout the text. For five tie lines closer to the plait point (or for the tie lines with larger a values), K was nearly constant and its average was 13.0 ( 0.3. However, as the tie lines were farther away from the critical (plait) point, K increased rapidly with the distance from the plait point; K became as large as 29. For these tie lines, the contents of water and ethanol were small in the benzene-rich phase and a small error in the concentration measurements of these components could be responsible for the large error in evaluation of K; or the behavior of K could reflect the thermodynamic behavior of the system, with the value of K becoming constant because of the approach to the critical point. The new concentration variables, ai, bi, ci (i ) 1,2), were calculated from eqs 6a, 6b, and 6c. The Bancroft-Hand transformation requires that a1 be equal to a2, that is, a1
Behavior of the Ethanol/Benzene/Water System
Figure 4. Differences between the transformed ethanol concentrations a1 and a2 vs the average value of a, i.e., (a1 a2)/(a1 + a2) vs (a1 + a2), testing how parallel the transformed tie lines are.
Figure 5. Phase and emulsion morphology diagram after the Bancroft-Hand transformation. (a) Tie line end points (filled circles) and measured emulsion inversion compositions (triangles). The parallel lines denote the tie lines. (b) Expanded view of the emulsion inversion cusp. For clarity, the tie lines have been omitted, but connecting conjugate pairs of end points with their tie lines illustrates how parallel the lines are.
) a2 ()a) for the tie lines to be made parallel. This requirement of a1 ) a2 was evaluated by examining the error |a1 - a2| with respect to the average a, that is, aavg ) (a1 + a2)/2. For the five tie lines closer to the plait point, this error was 0.4% at most, as shown in Figure 4. Like K values, the error became larger for the three tie lines which were far away from the plait point but still remained under 0.9%. This may imply that the Bancroft-Hand transformation was quite effective in making the tie lines parallel. Even far from the critical point, where there appears to be a systematic nonconstancy in K, the tie line parallelization is quite good. The phase and emulsion inversion data of Figure 2 were converted by the Bancroft-Hand transformation. That is, the phase and emulsion inversion data were expressed using eqs 6 and K ) 13. The transformed data are shown as points in Figure 5. (The curves in Figure 5, which are fits to the transformed points, are discussed following eq 20.) In the transformed coordinates, both the binodal and the emulsion inversion cusp shrank substantially. They spanned up to 57% of the ethanol axis in the triangular coordinates in terms of the conventional concentrations (Figure 2) but only up to 19% of the ethanol axis in terms of transformed concentrations. This shrinkage arises from the fact that the water concentration was weighted by K, that is, C became KC. Since the K value was larger than 1, the weighting made the denominators in eqs 6 also
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Figure 6. Determination of the plait point ethanol concentration, ac, in the transformed coordinate system.
larger than 1, and such larger denominators were responsible for the shrinkage. The emulsion inversion cusp was shifted toward the water apex of the diagram, and this was also due to the K values being larger than 1. Notably, the Bancroft-Hand transformation brought the plait point to the maximum of the binodal and made the tie lines parallel, as can be seen clearly by connecting conjugate pairs of end points in Figure 5. This enabled the critical-scaling equations to be used for the transformed quantities. With the K values determined and the new concentration variables a, b1, b2, c1, and c2 calculated, the plait point composition (ac, bc, cc) was determined from the criticalscaling equations for the hidden variable. For the determination of ac, eqs 15a and 15b were changed to eqs 17a and 17b.
| | | | | | | |
|b2 - b1|(1-R)/β ) b′(1-R)/β |c2 - c1|(1-R)/β ) c′(1-R)/β
a - ac a - ac ) b˜ ′ ac ac
a - ac a - ac ) c˜ ′ ac ac
(17a)
(17b)
Figure 6 shows the fit of eq 17a to the data of Figure 2. The fit was excellent; the correlation coefficients were 0.9988 and 0.9987, respectively, for the three or five points closest to the plait point. The quantity ac is the x-axis intercept in the plot; the values obtained were 0.196 ( 0.007 or 0.200 ( 0.010, from fits to three or five points, respectively. The two values are identical within the uncertainties of Figure 6. Since the K value used was obtained from the five closest points, ac might also be determined using these points. However, ac ) 0.196 was used for the subsequent calculations, because an excellent critical-scaling fit to the binodal curve was obtained with ac ) 0.196 whereas the fit was inferior with the value 0.200. (See Figure 9.) Similarly, bc was obtained as the y-axis intercept from the fit of eq 16a to the data and found to be 0.271 ( 0.006; the value cc ) 0.533 ( 0.006 could be found from the fit to eq 16b, or from the relation cc ) 1 - ac - bc. Hence, the plait point was determined to be (ac, bc, cc) ) (0.196, 0.271, 0.533) in the transformed coordinates. These values of the new concentration variables were converted to the usual concentrations Ac, Bc, and Cc through eqs 9; for the plait point in volume fraction units, this gave (Ac, Bc, Cc) ) (0.386, 0.533, 0.080). These values are in good agreement
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Figure 7. Inversion width vs distance from the plait point in the transformed coordinates.
with the measured (0.413, 0.501, 0.085),21,22 and this agreement appears to support the validity of the fitting equations with the new concentration variables converted by the Bancroft-Hand transformation. The measured and the calculated plait points are denoted in Figure 5 as the open square and the open circle, respectively. For the emulsion inversion cusp, where emulsion inversion points were denoted by φ1 and φ2, the width and curvature of the inversion hysteresis were analyzed for the transformed quantities using the critical-scaling equations. We assumed that the width, |φ2 - φ1|, and the curvature, φavg - φc, varied with distance from the critical point according to the following equations:
|φ2 - φ1| ) Ad
| | | |
|φavg - φc| ) Av
a - ac ac
a - ac ac
nd
(18) nv
(19)
Here φavg ) (φ1 + φ2)/2 and φc denote the midpoint of the inversion cusp and the inversion point at the critical (plait) point, respectively. Figure 7 shows a plot of the emulsion width |φ2 - φ1| versus reduced distance from the critical point, |ac - a|/ac. There was considerable scatter in the data, and the fit of the first three data points to eq 18 yielded nd ) 0.11 ( 0.06 (correlation coefficient 0.734). This value is significantly smaller than the corresponding value of 1.26 for the binary 2-butoxyethanol/watertemperature system.6 In contrast to the inversion width, the curvature was described well by the power law, eq 19. Figure 8 illustrates a plot of ln |φavg - φc| versus ln |(a ac)/ac| and the excellent fit obtained (correlation coefficient 0.999). In this plot, φc ) 0.5 was used, as expected from theory and found for this system.6,22 The fit yielded the exponent nv ) 0.69 ( 0.01, which is larger than the corresponding 0.528 for the binary-temperature system.6 For phase and other truly thermodynamic behavior, the universality of critical-scaling exponent values is theoretically and experimentally well established. When unexpected values are found, these values indicate some problem in the samples, measurements, or interpretation, although hidden variables have received considerably less attention. For emulsion inversion, our attempts to apply extensions of critical-scaling theory must be considered exploratory. Inversion appears to be a catastrophic (21) Solubilities of Inorganic and Organic Compounds, Supplement to the 3rd ed.; Seidell, A., Linke, W., Eds.; Van Nostrand: New York, 1952; p 921. (22) Ross, S.; Kornbrekke, R. E. J. Colloid Interface Sci. 1981, 81, 58.
Figure 8. Inversion curvature vs distance from the plait point in the transformed coordinates.
phenomenon; catastrophe theory yields polynomial equations with “classical” exponents (values that are multiples of 1/2); various thermodynamic models of emulsions and their inversion have been formulated by different researchers; renormalization-group theory definitively establishes for thermodynamics that classical values of the universal exponents obtain for systems of four dimensions but not when the number of dimensions is fewer. These considerations motivated us to hypothesize that inversion might be modeled by scaling theory and to seek to obtain experimental values of the exponents obtained with this assumption. For the ethanol/benzene/water system, nd ()0.11) is significantly smaller and nv ()0.69) is slightly larger in the Bancroft-Hand transformation than in the polar tie line transformation (nd ) 0.48 ( 0.04 and nv ) 0.62 ( 0.06).9 These two transformations appear to yield different nd values but similar nv values for the same data. According to our model, the two transformations should yield the same exponent values.10 At least some of the discrepancies may be attributed to the scatter of the measured inversion data and the sensitivity of the exponents to the critical compositions. The scatter in the data is significant, particularly for the emulsion inversion width, |φ2 - φ1|, which is the difference between nearly equal numbers. This scatter is clearly seen in Figure 7. Furthermore, the exponents nd and nv are quite sensitive to the critical compositions. (Such sensitivity is further considered below, in the discussion of Figure 9. For the phase behavior, sensitivities of this type are described in the paragraph of eqs 17.) Hence, a slight difference between the critical (plait) point compositions (ac, bc, cc) determined by the two methods would give different exponent values. Accurate determination of the exponents for the inversion width and curvature may require accurate and precise measurements of inversion data around the critical (plait) point. Since the width of the inversion hysteresis becomes smaller as the tie lines get closer to the critical point and inversion itself seems to be a probabilistic phenomenon,22 reducing scatter in inversion experiments around the critical point seems to be a formidable task. Equation 10 was fit to the transformed binodal of Figure 5a and to the transformed emulsion inversion data of Figure 5b. For these fits, X was either the transformed composition b or c, Xc was either bc or cc, and was a defined by eq 14. As already stated, bc and cc were determined with eqs 15 and the data close to the plait point. However, neither fit was fully satisfactory, because eq 10 is more symmetrical than the experimental data. This situation is known to be common for phase behavior, even when one of the variables is pressure or temperature,
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Table 1. Values of Fitting Parameters When Equation 10 Was Fit to the Tie Line Data, with Concentrations X Defined by Equation 20, and Three, Four, or Five Experimental Tie Lines Were Used in the Fits no. of tie lines
ac
L
M
N
f
3 4 5
0.196 ( 0.007 0.199 ( 0.008 0.200 ( 0.010
-0.728 -0.128 -0.154
0.248 -0.249 -0.253
0.037 0.309 0.322
1.97 0.206 0.251
so that the tie lines are rigorously parallel without any transformation. The critical-scaling equations, eqs 10-12, describe basically the critical behavior in terms of the order parameter. Fractional concentrations such as mole, weight, and volume fractions approximate the order parameter but are not fully satisfactory replacements for the correct, but unknown, expression for the true order parameter. Liquid-liquid equilibria in real fluids usually display asymmetries in the common fractional concentrations, and these asymmetries limit the applicability of eqs 10-12 to the region close to the critical point. One way of successfully incorporating these asymmetries into the critical-scaling equations is to define a new order parameter OP and use it in place of X in eqs 10-12. In this method, OP is defined as
OP( )
fX( 1 + (f - 1)X(
Figure 9. Sensitivity of critical-scaling fits to the location of the critical points for the binodal.
indicate that a power-law description of the emulsion inversion phenomenon requires data on a multitude of tie lines close to the critical (plait) point. Conclusions
(20)
(where X( ) b( or c(). Here f is a fitting parameter, chosen to make the best fit of the inherently symmetric criticalscaling equations to the asymmetric data. This method was shown to significantly extend the range of validity of the critical-scaling equations for the binary 2-butoxyethanol/water-temperature system, where the tie lines are “automatically” parallel without any transformation.23 Hence, to improve the accuracy of the fits, the BancroftHand transformed emulsion and binodal data sets were each fitted by the critical-scaling equation, eq 10, with X and replaced by OP and a, respectively. The results are represented by the solid lines in Figure 5a (for the phase data) and Figure 5b (for the emulsion inversions). The fits were excellent (correlation coefficients greater than 0.99). For the emulsion data, the fitting parameters were found to be L ) 0.0254, M ) 0.0307, N ) -0.498, and f ) 5.82. The corresponding values for the binodal data are listed in Table 1. Note that OP ) X when f ) 1. The fitting results were converted to the conventional concentrations by eqs 9. Figure 9 shows the binodal and the fits in terms of component volume fractions. The solid line corresponds to the fitting curve in Figure 5a, and the fit was in excellent agreement with the measured data. In this fit, the three experimental tie lines closest to the plait point were used for the determination of the plait point composition. However, when the four or five closest tie lines were used, the critical-scaling equations did not fit the data on the aqueous side, as shown in Figure 9 (broken line with four tie lines and dotted line with five). As the number of tie lines used increased from three to four, the plait point composition, ac, changed slightly; it increased by 1.6% from 0.1960 to 0.1992 (Table 1). Even for this small difference in ac, the fits to the binodals yielded significantly different results, reflecting the well-known high sensitivity of such fits to the location of the critical point. These results (23) Kim, K.-Y.; Lim, K.-H. J. Chem. Eng. Data 2001, 46, 967.
The method of using critical-scaling equations with a new order parameter was extended to a ternary system for fits to binodal and emulsion inversion hysteresis experimental data. Because the experimental tie lines for the ternary system were nonparallel, they were made parallel by the Bancroft-Hand transformation, so that critical-scaling equations could be used. It was found that the Bancroft-Hand transformation was quite effective in making the tie lines parallel. Fits of critical-scaling equations to the binodal and emulsion inversion hysteresis were quite successful even for the large values of the reduced field variable a. Thus, the fits could be extended practically over the entire binodal. This suggests that the combination of criticalscaling equations and Bancroft-Hand transformation may be useful for the liquid-liquid equilibria of many ternary systems.24 The fits to the emulsion inversion hysteresis yielded the exponents nd ) 0.11 ( 0.06 for the inversion width and nv ) 0.69 ( 0.01 for the hysteresis curvature. These values of nd and nv for the three-component two-phase emulsions were smaller than the corresponding values of 1.26 and 0.528 for the two-component two-phase emulsions. For the same emulsion inversion data, the polar tie line method yielded a similar nv ()0.62 ( 0.06) value but a significantly larger nd ()0.48 ( 0.04)10 than the present method. This discrepancy may be due to the sensitivity of these exponents to the location of the critical point. A small difference between the predicted critical points may give rise to a large difference in the exponents. This may imply that accurate and precise determination of the exponent values for the emulsion inversion hysteresis requires a multitude of phase and emulsion inversion data close to the critical point. LA0200515 (24) Lee, J.-M.; Kim, K.-Y.; Lim, K.-H. Proceedings of CKCSST ’01, Hangzhou, China, 2001; pp 53-59. Lee, J.-M.; Lim, K.-H. Manuscript in preparation.
