Langmuir 1990,6, 1071-1077
1071
An Experimental Test of Catastrophe and Critical-Scaling Theories of Emulsion Inversion Duane H. Smith**+and Kyung-Hee Limt US.Department
of Energy, Morgantown Energy Technology Center, Morgantown,
West Virginia 26507- 0880 Received August 22, 1989. In Final Form: November 17, 1989
Electrical conductivities of the conjugate phases and emulsions of the binary system 2-butoxyethanol/ water were measured over the temperature range T - T, = 4-21 K (Tc is the lower consolute solution temperature). From these data, T,,the emulsion morphologies, and the emulsion inversion and its hysteresis were determined. Equations from critical-scaling theory were fit to the phase conductivities, and the emulsion data were used to test catastrophe and critical-scaling theory equations for emulsion inversion.
Introduction Emulsions and other fluid-fluid dispersions have long been used in many different commercial products and processes, such as liquid-liquid extractions, foods, medicines, paints, cosmetics, and the production and transport of petroleum. One of the most novel applications is the use of dispersions of supercritical COZ to improve oil recoveries from miscible flood enhanced oil recovery.’ Each of these applications depends on the formation of a dispersion of the desired morphology, for example, an “oil-in-water” or a “water-in-oil” emulsion. Formation of the “wrong” morphology or inversion from the desired morphology to the “wrong” morphology causes the product or process to fail. The fractional phase volume at which inversion occurs is believed to depend on many different factors, such as the temperature, structure and concentration of the emulsifying amphiphile, salinity, and “oil” composition. However, the factors that control dispersion morphology and inversion are not well under~tood.~J Qualitatively, dispersion inversion has long seemed to be a catastrophic event. The properties of an A/B (A-in-B) dispersion are very different from the properties of a B/A (B-in-A) dispersion, and the inversion from one morphology to the other often is abrupt and irreversible. These and other common observations of dispersion behavior suggest that the mathematical catastrophe theory which Thorn4developed from algebraic topology might somehow be applied to dispersion inversion. Dickinson summarized several phenomena of emulsion inversion that seem characteristic of the so-called “cusp” catastrophe. These may be listed as f01lows:~ (1) Dispersion morphologies are bimodal; Le., a dis+ Adjunct, Institute for Applied Surfactant Research, and School of Chemical Engineering and Materials Science, University of Oklahoma, Norman, OK. 1 Present address: Department of Chemical Engineering, West Virginia University, Morgantown, WV 26506-6101. (1)Smith, D. H. In Surfactant-Based Mobility Control: Progress in Miscible-Flood Enhanced Oil Recouery; Smith, D. H., Ed.; American Chemical Society: Washington, DC, 1988; pp 2-37. (2) Becher, P. Emulsions: Theory and Practice; Rheinhold New York, 1965. (3) Smith, D. H.; Lim, K.-H. J . Phys. Chem., in press. (4) Thom, R. Structural Stability and Morphogenesis; Benjamin: Reading, MA, 1975. ( 5 ) Dickinson, E. J. Colloid Interface Sci. 1981,84, 284.
persed system of phases A and B can exist in either the A/B (A-in-B) or B/A (B-in-A) morphology. (2) Inversion is marked by a sudden change in the physical properties (e.g., electrical conductivity) of the dispersion, indicating an abrupt change in the structure of the di~persion.~ (3) Inversion exhibits hysteresis; i.e., the dispersion morphology sometimes depends on the experimental path, or history, of the di~persion.~ (4) Two dispersions of the same composition, prepared in ways that may seem negligibly different, sometimes show divergent Dickinson suggested that elementary catastrophe theory might provide a unifying framework to explain these phenomena in systems of variable oil/water ratio and amphiphile concentration at constant temperature and pressure.5 Salager incorporated this suggestion in a review of phase behavior, emulsion inversion, and the “cusp” and “butterfly” elementary catastrophe^.^ However, until now, no one (toour knowledge) has actually attempted to use catastrophe theory with real experimental data for emulsion inversion and hysteresis. In making such an attempt, we believed that the classical values of the exponents from catastrophe theory would prove to be theoretically incorrect (except by rare chance that the experiments accidentally approached a tricritical point), hence that critical-scaling theory equations with nonclassical exponents from group renormalization theory should be used in place of catastrophe theory.lOJ1 Moreover, the critical-scaling equations for the effects of compositionalchange at constant temperature and pressure are based on the chemical potentials of the components (which are hardly ever measured), whereas if temperature (or pressure) is the experimental variable, this parameter is readily measured and its experimental Values used in the catastrophe and critical-scaling equations. A further anticipated advantage of the latter exper(6) Sasaki, T.Bull. Chem. SOC.Jpn. 1939, 14, 63. (7) Ross, S.;Kornbrekke, R. E. J . Colloid Interface Sci. 1981,81,58. (8) (a) Smith, D. H.; Lim, K.-H. In Proc. SOC.Petrol. Eng. Intl. Symp. Oilfield Chemistry, Houston, TX, Feb 8-10,1989, SPE 18496. (b) Lim, K.-H.; Smith, D. H. J . Dispersion Sci. Technol., in press. (9) Salager, J.-L. In Encyclopedia of Emulsion Technology; Becher, P., Ed.; Marcel Dekker: New York, 1988. (10) Rowlinson, J. S.;Widom, B. Molecular Theory of Capillarity; Clarendon Press: Oxford, 1984; p 327. (11)Smith, D. H. AOSTRA J . Rsch. 1988, 4, 245.
This article not subject to U.S.Copyright. Published 1990 by the American Chemical Society
1072 Langmuir, Vol. 6, No. 6, 1990
imental design is that the tie lines (of the phases of the dispersions) are necessarily parallel-as opposed to isothermal, isobaric surfactant/oil/water compositionalchange experiments, where they are not. Because the present study appears to be the first experimental test of the applicability of catastrophe and critical-scaling theories to dispersion inversion, we have further simplified the emulsified system to two components (by omitting the oil of a surfactant/oil/water system). Thus, the present study reports electrical conductivity measurements of emulsion morphologies, inversion, and hysteresis in the two-(pseudo)component system 2-butoxyethanol/brine (aqueous 10 m M NaC1). (The small salt concentration was used to improve the sensitivity and accuracy of the conductivity measurements by increasing the conductivity differences between the conjugate phases of the emulsions.) This system has a lower consolute solution temperature (phase critical point) near 48 O C . Emulsions were formed from conjugate (preequilibrated) phases, and inversions from the A/B to B/A and B/A to A/B morphologies were determined as a function of phase volume fraction and temperature by means of isothermal experimental paths that moved back and forth along tie lines. Both catastrophe and critical-scalingtheories were tested by fits of respective sets of equations to the experimental data. Phase volume fractions were used in both theories, because of Ostwald's stereometric model and other historical antecedents, without any further theoretical justification.12J3 Experimental Section The amphiphile 2-butoxyethanol was from Aldrich and was used as received. It had a stated purity of 99%, which was confirmed by gas chromatography. The water was distilled. The experimental procedures were designed to ensure that the emulsions were formed from conjugate phases. The methods of sample preparation and the electrical conductivity apparatus were essentially the same as used in previous measurements on three-component system^,^^^ Briefly, for each experimental temperature, samples of known composition were prepared gravimetrically, placed in a thermostated separatory funnel, mixed thoroughly, and allowed to completely separate into their bulk, conjugate phases. The phases were separated and their conductivities measured. Emulsions for the electrical conductivity measurements were prepared by mixing known volumes of two conjugate phases in a thermostated beaker for 5.0 min, interrupting the mixing for 10.0 min (during which the emulsions partly separated), and then remixing for 5.0 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 was used throughout the present study, because in these and in other measurements it enabled us to measure the emulsion inversion with excellent reproducibility.3 Inversion was detected as a large change in the conductivity and was measured for each of the two directions along the tie line.
Results and Data Analysis Figure 1 shows the composition-temperature phase diagram of the 2-butoxyethanol/water system.14-ls The six (12) Ostwald, W. Kolloid 2.1910, 6, 103. (13) Ostwald, W . Kolloid 2.1910, 7, 64.
(14) Cox, H. L.; Cretcher, L. H. J. Am. Chern. SOC.1926, 48, 451. Poppe, G. Bull. SOC.Chim. Belg. 1935, 44, 640. (15) Elizalde, F.; Gracia, J.; Costas, M. J . Phys. Chern. 1988,92,3565. (16) Ellis, C. M. J . Chern. Ed. 1967, 44, 405. (17) Ito, N.; Fujiyama, T.;Udagawa, Y. Bull. Chern. SOC.Jpn. 1983, 56, 379.
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Figure 1. Temperature-composition phase diagram of the
2-butoxyethanol/water system: (a)complete miscibility gap; (b) enlargement of the region near the lower consolute (critical) point16 and the tie lines used in this study. 80
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parallel lines are the tie lines along which emulsion morphology, inversion, and inversion hysteresis were measured. Depending on the various measurements and the way in which it was estimated from the data, the lower critical point of the miscibility gap is at about 48 "C and 26 wt % butoxyethanol. The electrical conductivities of the conjugate phases, as measured over the temperature range T = 52-68 "C, are shown in Figure 2. At all temperatures, the conductivity of the aqueous phase (Kaq, shown as diamonds) is greater than the conductivity of the amphiphilic phase (K-, shown as open circles). (Figure 2 is plotted with temperature as the vertical axis and with the amphiphilic phase on the right-hand side, corresponding to Figure 1.) Also shown (as triangles) in Figure 2 is the average of the conductivitiesof the aqueous and amphiphilic phases at each temperature. The U-shaped curve represents a fit of critical-scaling theory to the phase conductivities, and the central line shows a fit of critical-scaling theory to the average conductivities. As shown by Figure 2, the fits of the critical scaling equations to the conductivity data are excellent. To calculate these fits, we assume that the conductivities of the conjugate phases should be described by the (18)Quirion, F.; Magid, L. J.; Drifford, M. Langrnuir 1990, 6, 244.
Langmuir, Vol. 6, No. 6, 1990 1073
Theories of Emulsion Inversion 15
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Figure 3. Fit of eq 4 to the data of Figure 2; the extrapolation to K , - K., = 0 (filled circle) gives T,= 47.15 O C (correlation coefficient for the fit is r = 0.9886).
Figure 4. Plot illustrating calculation of K,, the electrical conductivity at the critical point; the fit of eq 5 to the data gives K , = 779.4 p S cm-1 (r = 0.9880).
equations K,, - K, K,,
+ K,
= K1t8
= 2K&"
(2) Here K1 and Kz are conductivity parameters that depend on the chemical system, K, is the electrical conductivity at the critical point, and a and /3 are universal criticalscaling exponents. The values of the exponents are /3 = 0.324 and a = 0.11.10 The parameter c is defined
.-
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( T - T,)/T,
(3) where T, is the critical temperature (for the miscibility gap of this paper, the lower consolute solution temperature). Equation 1 may be rewritten as t E
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(4) and elimination of t between eqs 1 and 2 yields (Kaq - K 0)(l-")/fl = K@,,
- K,)
(5) where K., = (Kaq+ Ka,)/2 and K3 is a simple function of KI and K2. Figure 3 shows a plot of (Kaq- Karn)l/oversus T and a linear regression of eq 4 to these points. The regression yields the value T, = 47.15 "C for the critical temperature, which agrees within about 2 "C with other reported value^.^^-^^ We attribute the differences among various researchers to a combination of factors, including different thermometers, chemical impurities, and methods of estimating T, from the experimental data. The value of the electrical conductivity at the critical point, K,, can be estimated from the data of Figure 2 with the aid of eq 5. Figure 4 shows a plot of (Kaq Karn)(l-#)/@ versus K., along with a fit of eq 5 to the data. The regression gives K, = 779.4 pS cm-l for the conductivity at the critical point. (The correlation coefficient for the fit is r = 0.9880.) Except very close to the critical point, the electrical conductivities of the conjugate phases differ by amounts that are very large compared to the experimental uncertainties. Hence, emulsion morphologies and inversions between them can be inferred from electrical conductivity measurements. Figure 5 shows typical experimental data for the electrical conductivity of emulsions of the 2-butoxyethanol/aqueous 10 mM NaCl system, plotted versus the volume fraction of the amphiphilic phase. The conductivities were measured for both directions along the tie line, that is, for additions of the amphiphilic phase (open circles in Figure 5) and for additions of the aque-
0.54
0.56
0.56
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Volume Fraction, Amphiphilic Phase
Figure 5. Example of electrical conductivity measurements on emulsions used to determine the emulsion morphology and inversion hysteresis: open circles, amphiphilic phase added to the emulsion; filled circles, aqueous phase added to the emuland sion (data at 65 "C). Phase volumes at inversion are @2.
ous phase (filled circles in Figure 5). Along the line lmn of Figure 5 the conductivity was large, implying that the aqueous phase was the continuous phase. I t is convenient to call this the AM/AQ (amphiphilic phase-in-aqueous phase) morphology. On the contrary, along the line pqr the conductivity was small, implying the morphology was AQ/AM (aqueous phase dispersed in amphiphilic phase). At points p and n, the conductivity suddenly changed, indicating that emulsion inversion occurred at these points. The volume fractions at these points, $1 or 4 2 , are inversion points, where the emulsion morphology changed from AQ/AM to AM/AQ, or vice versa, respectively. Usually the two volume fractions CPp1 and CP2 are different, and inversion hysteresis is exhibited, as shown in Figure 5. Figure 6 shows the emulsion morphology and inversion hysteresis determined at the temperatures 51, 53, 55, 60, 65, and 68 "C. (All of our measurements at 55 "C-for the individual phases, as well as the emulsion inversions-were anomolous and were omitted from all data treatments.) The morphology was AQ/AM for CP > CP2, AM/AQ for CP C @I, and history dependent for 91 I CP 5 @z. At all temperatures, CP1 and CPZ were larger than 0.5. Except for the data at the highest temperature (and the data at 55 "C), CP1 and CPZ decreased with decreasing temperature, presumably converging at the point defined by CP1 = CP2 = CPc and T = T,. The solid lines in Figure 6 are the fits of critical-scaling theory to the inversion data. (See eqs l l a and 12a, below.) Hysteresis and nonreversibility are familiar phenomena in the inversion of emulsions of amphiphile/oil/
1074 Langmuir, Vol. 6, No. 6, 1990
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sion of fixed T, P, and composition in its various morphologies or “states” of dispersion does not seem describable by a free energy function that is single-valued in the system composition, temperature, and pressure. In the literature, a new state variable, s, has been invoked to get around this problem, but “the precise physical meaning of s is as yet ~nspecified”.~ Thus, for the dispersion behavior one may write an equation of the form
+
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Figure 6. Emulsion inversion points (AQ/AM AM/AQ, circles) and (AM/AQ AQ/AM, diamonds) plotted as a function of temperature, along with fits of castastrophe (broken line) and critical-scaling (solid line) theories, respectively.
+
G(s;a,b)= s4/4 as2/2 bs (6) Here s is the state variable and a and b are control variables. This fourth-degree polynomial has the maximum and two minima required for the formation of two dispersion morphologies, e.g., AM/AQ and AQ/AM. These extrema are located at (dG/ds),,, = s3 + as + b = 0
(7) The free energy of eq 6 also has points where the number of extrema changes, given by
(d2G/ds2)a,b = 3s2+ a = 0
(8)
Elimination of s between eqs 7 and 8 gives the bifurcation, B, in a-b space:
Figure 7. Manifold and bifurcation for dispersion inversion, along with the inversion hysteresis produced by experimental paths of opposite directions along the tie line. water systems and indicate that inversion may be viewed as a catastrophic event.315p7-9 Figures 5 and 6 show that these phenomena also can occur for emulsion inversion in the miscibility gaps of two-component systems. Although use of the cusp catastrophe to describe emulsion inversion was suggested as early as 1981, no one (to our knowledge) has actually attempted to use the theory with experimental data.5~9The inversion process and free energy functions for emulsions are imperfectly under~tood;~ one J ~even may question whether the concept of free energy surfaces for dispersions, which is required by catastrophe theory, can be put on a firm theoretical basis. However, emulsions of some compositions can be stored for very long periods without detectable changes, and this fact implies that energy minima must exist for at least some emulsified systems.20 From the fact that some system compositions (those in the hysteresis region) can exist as A/B dispersions, B/A dispersions, or with neither phase dispersed in the other, it is clear that the dispersion diagram is related to, but not wholly described by, the phase diagram.3*7 For the same reason, the free energy function for an emul(19) Dickinson, E. J . Colloid Interface Sci. 1982.87, 416. (20) Friberg, S. E.; Mandell, L.; Larsson, M . J. Colloid Interface Sei. 1969, 29, 155.
B(a,b) = (b/2)’ + ( ~ 1 3 =) 0~ (9) The surface characterized by eq 7 is called the manifold; it and the bifurcation are shown in Figure 7. The manifold is the set of (a,b,s)points at which G has extrema; between each minimum and the maximum is an inflection point. In general, these (a,b,s) inflection points differ from those of the extrema and do not lie on the manifold of Figure 7. However, changes of b in one direction (or the other) make one of the minima (or the other) become more and more shallow until it and the maximum coincide with the intermediate inflection point. The three points coincide for values of a and b given by the bifurcation; hence, for the values of a and b given by eq 9, the subset of the solutions to eq 8 that also satisfies eq 7 lies in the manifold.21 In thermodynamics, the inflection points of G are known as the spinodal. Dickinson has proposed a spinodal decomposition modellg as well as a catastrophe theory model5 for emulsions but did not discuss possible connections between the two. Although a detailed development of these connections is beyond the scope of the present study, it may be noted that the cusp catastrophe contains the spinodals required for such a theoretical development. The bifurcation, which is used in the present analysis, is the (a,b) limit for which the two spinodals lie in the manifold. The bifurcation has a cusp (which gives this particular elementary catastrophe its name); this cusp is the critical point for dispersion inversion. The straight line parallel to the b axis of Figure 7 is a tie line, representative of the tie lines along which our measurements were made. (See Figures 1 and 5.) The dispersion morphology on the line LMN is, in the present study, AM/AQ, and on the line PQR it is AQ/AM. At point N the value of the state variable, s, drops to its value at point Q; the dispersion inverts. The drop occurs because of the inaccessibility of the fold (in this case, the part of the surface between points P and Q ) . This inaccessibility is one of the characteristics of the cusp catastrophe.21.22 Thus, N is the dispersion inversion point for the direction of increasing b (cf. point n in Figure 5). Likewise, for the experi(21) Broecker, Th.;Lander, L. Differentiable Germs and Catastrophes; Cambridge University Press: Cambridge, 1975; pp 147-8.
Langmuir, Vol. 6, No. 6, 1990 1075
Theories of Emulsion Inversion 0.03 {
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Figure 8. Bifurcation for the dispersion inversions, along with the cusp (critical point) and representative tie lines (dashed lines), and illustration of changes of G along a tie line (eq 6).
mental path RQP, inversion occurs at point P (cf. point p in Figure 5). Hence, the inversion hysteresis MNQP of Figure 7 corresponds to the inversion hysteresis loop mnqp of the experimental data in Figure 5. In summary, the points L, M, N, P, Q, and R of Figure 7 correspond to the points 1, m, n, p, q, and r of Figure 5, respectively. Figure 8 shows the two lines of the bifurcation (which we associate with the inversion hysteresis lines), the tie lines of the conjugate phases of the dispersions, and representative dispersion free energy curves for various points along a tie line. From the elemental fact that a dispersion of two phases must disappear when the two phases become one, we expect that the critical point for the bulk A and B phases and the critical point for the emulsion inversions should Equation 6 assumes that parameters a, b, and s exist such that there is a dispersion free energy described by a fourth-order polynomial but gives no direct information on the definitions of these parameters. For the present study, s corresponds to the electrical conductivity of the emulsion. The choice a -(T- T c ) / T chas the properties required by the bifurcation that a be a measure of the distance from the critical point and that a = 0 at that point. Ostwald-type m o d e l ~ , in ~ ~which J ~ inversion occurs when the volume fraction of the dispersed phase becomes too large, suggest the choice of phase volume fraction to define the parameter b. We identify the width of the catastrophe theory cusp with the width of the inversion region. With these (tentative) identifications, eq 9 takes the form
-
A@ = a, - a,
- IT
- ~~13’2
(10) Here, is the volume fraction of the amphiphilic phase for the inversion AM/AQ AQ/AM, and CP1 is the volume fraction of the amphiphilic phase for the opposite inversion, AQ/ AM AM/ AQ. Figure 9 shows a plot of A@ versus [(T- Tc)/Tc]3/2 for the data of Figure 6 and the regression of eq 10 to the experimental points. The fit is good ( r = 0.9755), sug-
-
Figure 9. Plot of PO versus [(T - Tc)/Tc]3/2 for the data of Figure 6, and regression of eq 10 to the experimental points (r = 0.9755).
-
(22) (a) Poston, T.;Stewart, I. Catastrophe Theory and Its Applications; Pitman: London, 1978. (b) Gilmore, R. Catastrophe Theory f o r Scientists and Engineers; Wiley: New York, 1981.
gesting that eq 10 may provide reasonable fits to the inversion hysteresis width in other chemical systems. However, other choices for a, b, and/or the value of the critical exponent may be better. Figure 6 shows the inversion hysteresis lines a1 and aPz(broken lines), as calculated with catastrophe theory. An examination of Figure 6 quickly reveals that these theoretical lines are not at all satisfactory. Catastrophe theory assumes that the average value of b at which inversion occurs (in the present case, bav a a v = (+I + @2)/2) is constant. (See Figure 8 and eq 9.) Thus, in Figure 6 the catastrophe theory value of a a v equals the experimental value only at one point, the critical point, where = At all other values of a , catastrophe theory fails to provide a satisfactory description of where inversion occurs. Unlike catastrophe theory, critical-scaling theory predicts that the average value of a parameter (e.g., changes with distance from a critical point. Moreover, the cusp catastrophe gives classical values of the scaling exponents (that is, the exponents are multiples of 1/2). However, it is well-known that these values are incorrect for the physical properties of bulk fluids and that the exponents must be renormalized to nonclassical value^.^^^^^ Thus (above), we have fit only equations with nonclassical exponents to the electrical conductivity data. Although the relatively simple equations of catastrophe theory have many heuristic uses, Figure 6 suggests that, in the description of dispersions by catastrophe or critical-scaling theory, equations with nonclassical exponents eventually will prove preferable to the corresponding classical exponent expressions from catastrophe theory. One alternative to the catastrophe theory treatment of Figure 6 is to retain the definitions of a and b but to use the critical-scaling theoretically correct exponents. However, more generally, while retaining the use of T and used in catastrophe theory (eq lo), we may write
-
A@ = a, - a, = W e p ’
(11)
+
log A@ = p’log [ ( T -T,)/T,] log a’
(lla)
a, + CP, = 2CP’‘tPi1 + 2ac (12) log laav-acl= p” log [(T- T c ) / T c+] log a’’ (12a) Here aav= (ap2 + @1)/2,aCis the volume fraction of the
amphiphilic phase at the (dispersion) critical point, and a‘ and a‘’ are system-dependent parameters. (23) Wilson, K.G.Phys. Reu. B 1971,4, 3174.
1076 Langmuir, Vol. 6, No. 6, 1990
Smith and Lim
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Figure 11. Plot of log (Oav - aC)versus log t for the data of Figure 6; the regression gives p" = 0.528 i 0.047 for the value of the critical exponent ( r = 0.9923).
Equations 11 and 12 correspond to critical-scaling eqs 1 and 2, respectively. However, instead of assuming that
For comparisons of catastrophe and critical scaling theories (below), we note that the critical-scaling equations (eqs 1 and 2) are not written in terms of the individual phase conductivities but in terms of the sum and the difference of the conjugate conductivities. Thus, Figure 2 shows that both the average phase conductivities and the difference between conjugate phase conductivities are accurately described by the equations. As shown by a comparison of Figure 5 (experimental) and Figure 7 (theoretical), one of the advantages of catastrophe theory is its ability to predict inversion hysteresis. Moreover, as shown by Figure 9, catastrophe theory exhibits some degree of success in quantitatively predicting how the width of the hysteresis should increase with the distance from the critical point. The value of the exponent in catastrophe theory, 3/2, is not greatly different from the measured value of the exponent, p' = 1.26 f 0.10 (Figure 10). This semiquantitative agreement (3/2 vs 1.26) is in accord with our expectation that catastrophe theory's classical values for the exponents would prove to be approximately correct. In Figure 7, between points P' and Q' (i.e., within the hysteresis region), the manifold is triple-valued; the free energy surface has a double minimum (shown in Figure 8). The depth of each minimum is a function of the distance between P' and Q'. At P', the minimum for the A/B morphology degenerates to an inflection point; at Q , the minimum for the B/A morphology becomes an inflection point. This double minimum suggests that between P' and Q' both A/B and B/A dispersions might simultaneously form, with the relative amounts of the two morphologies depending on the location of the system between points P' and 8'. Indeed, the simultaneous observation of both morphologies in a single sample is not uncommon. This simultaneous formation conforms to the Maxwell c o n ~ e n t i o n . 9 ? ~ ~ . ~ ~ However, if the height of the maximum between the two minima is very large compared to the available "activation" free energy needed to cross it, the system may be trapped by its previous history within one of the two morphologies. As the amount of the internal phase is increased (say, phase B in Figures 7 and 8),the minimum free energy of the B/A morphology increases, and this increase lowers the activation free energy needed to escape the B/A trap. In the limit that the available free energy for crossing the barrier goes to zero, the dispersion remains trapped in the B/A minimum until the minimum disappears (i.e., becomes an inflection point). (In Figure 8, such a free energy curve is shown for point Q'.) This limiting behav-
the exponents p' and p" have theoretical critical-scaling values, we treat these parameters as unknowns, the values of which are to be found from the inversion data. Figure 10 shows the data of Figure 6 plotted in the form log A 0 versus log [(T- T c ) / T c ] .Here T , = 47.15 " C from the fit above, of critical-scaling theory to the phase conductivity data. Figure 10 also shows a fit of eq l l a to the data. The fit of Figure 10 is somewhat improved over that of Figure 9 ( r = 0.9935 vs r = 0.9755) and gives the value p' = 1.26 f 0.10. This exponent is equal, within the experimental uncertainty, to the critical-scaling exponents y = 1.24, p = 1.26, and 2v = 1.26.1° We note, in particular, that p is the exponent for the temperature dependence of the interfacial tension. Figure 11 shows a plot of log (aav- aC)versus log [(T - T c ) / T c ]for the data of Figure 6. The plot assumes that aC= 0.5, as expected from theory and found for the ethanol/benzene/water system.' Also shown is the regression of eq 12a to the experimental values. This fit yields p" = 0.528 f 0.047 for the exponent (r = 0.9923). By combining the fit of eq 11 (for 0 2 - 01)and the fit of eq 12 (for 02 @I), we can calculate 01 and 02 separately and thus test the ability of eqs 11 and 12 to fit the experimental data. The results of this test are shown in Figure 6 (solid lines), which shows both the experimental inversion data and the fits from eqs 11 and 12. Except for the data at the highest temperature studied (65 " C ) , the agreement between the experiments and the fitting equations is excellent.
+
Discussion As shown by Figures 2-4, critical scaling theory provided excellent fits to the electrical conductivity data, for distances up to 20 " C from the critical point (the limit of the range of the measurements). In these fits, the exponents were forced to have their theoretical values. These fits are comparable to those made by Fleming and Vinatieri, who studied three-phase conductivities in chemically complex systems of surfactant, alcohol, oil, salt, and water.24 (In their measurements, those researchers varied the salinity, instead of the temperature, as we have done.) The fit to the conductivity data gives the critical point temperature, for use in fits to the emulsion inversion data. Because of the experimental design, the critical point composition is not needed to interpret the conductivity or the emulsion inversion data. ~~~
(24) Fleming, P. D.; Vinatieri, J. E. J. Colloid Interface Sei. 1981,81, 319.
Langmuir, Vol. 6, No. 6,1990 1077
Theories of Emulsion Inversion ior is called the "perfect-delay convention".21y22It is shown in Figure 7, in which, for example, the dispersion cannot escape the path L-M-N until it reaches point N, where the A/B minimum becomes an inflection point, and the AJB dispersion is forced to fall to point Q and the B/A morphology. The perfect-delay convention would appear to be favored by relatively large interfacial tensions, because larger tensions increase the free energy needed to flatten two droplets when they collide as part of the coalescence and inversion process.25 The abrupt conductivity changes and highly reproducible inversion points that we observed can be rationalized by the assumption that the emulsion morphologies and inversions closely approached the perfect-delay limit. Figure 6 shows the phase-volume fractions at which inversions occurred, measured as a function of the temperature (i,e., distance from the critical point). It may be significant that the inversion lines bend toward the phase of the greater amphiphile concentration (rather than toward the aqueous phase). Similar bending of the inversion lines toward the phase of the greater amphiphile concentration has been observed for emulsions of an amphiphile/oil/water system at constant temperat~re.~,' Also shown in Figure 6 are the fit of catastrophe theory to the experimental data (broken lines) and the fit of two equations (eqs 11 and 12) based on critical-scaling theory (solid lines). It is immediately obvious from Figure 6 that eqs 11 and 12 give the better fit. In catastrophe theory (as shown in Figure 8), the average volume fraction for the two inversions A/B B/A + @2)/2,is constant and and B/A A/B, i.e., a, = (91 independent of the distance from the critical point. Thus we can define the catastrophe theory parameter b so that aaVhas the correct value (0.5) in the limit at the critical point, but catastrophe theory does not provide any simple description for the dependence of 9 , on distance from the critical point. Because ( 9 2 - 91)< (@a" - a,.,),the ability of the theory to describe 9 2 - 91 does not compensate for its inability to describe 9 a v - 9,. As shown by Figure 6, eqs 11 and 12 fit the data within the experimental reproducibility. Equations 11 and 12 are taken from critical-scaling theory; but we have defined only 9 and t and let the exponents be fitting parameters. A major reason for taking this approach is that while t seems naturally defined as ( T - T,)/T,, the choice of 9 is suggested by hard-sphere, stereometric models, and the use of such models is based more on convenience and historical precedent than on any definitive proof of their quantitative applicability to emulsions. The experimental value for the exponent in eq 11, p' = 1.26 f 0.10, is equal to the value of several different exponents of critical scaling theory. These exponents include y = 1.24, p = 1.26, and 2v = 1.26.1° Although we do not yet know if dispersions prepared from other compounds and/or by other techniques will give the same value for p', we note, in particular, that 1.26 is the theoretical value of the exponent for the temperature dependence of the interfacial tension. The agreement between p' and p suggests that a surface free energy model may be more applicable to the emulsions of this study than a hard-sphere model. The simplest such model neglects the entropy of dispersing the droplets, as well as interactions among droplets, which are known to be much, much smaller than the interfacial free energy.15 Hence, the free energy of the emul-
-
-
(25) Friberg, S.
E.;Solans, C. Langrnuir 1986,2, 121.
sion may be written as
-
AG uA (13) where u is the interfacial tension and A is the total surface area of the droplets. If the surface area at inversion varies little with temperature, then eq 13 suggests that the experimental temperature dependence of the inversion hysteresis width might indeed vary with the critical scaling exponent p = 1.26. As shown by Figures 6 and 11, the exponent p" = 0.528 f 0.047 provides an accurate fit to the temperature dependence of the experimental data of Figure 6. As for p', there are not yet any other data to indicate whether the value measured for p" will prove to be general. This value is, of course, completely different from the predictions of catastrophe theory but in agreement with the classical exponent 1/2. The measured value of p" is not particularly close to the theoretical values of any of the common (nonclassical) critical-scaling exponents. (The exponents L3 = 0.325 and u = 0.63 are the closest.lO) This fact (along with the experimental value of p' and eq 13) may be another indication that a hard-sphere, phase volume-fraction model may be inappropriate and that surface area may be a better parameter than phase volume for critical-scaling and other theoretical treatments of emulsion inversion. The width of the hysteresis measured in this study may be compared to the hysteresis width measured for emulsions of aqueous and oleic phases in the 2-butoxyethanol/ decane/brine-temperature dispersion morphology diagram.3 For the ternary system, widths of 0.024, 0.005, and 0.031 were measured at 20, 35, and 50 "C,respectively (ref 3, Table 11). The smallest of these widths was for phases near the optimum point, Le., the point on the aqueous oleic side of the stack of aqueous oleic middle phase tie triangles where the interfacial tension passes through a minimum. Because the optimum point is not a critical point, along a path through this point the hysteresis width theoretically should pass through a minimum, but not go to zero. To within the resolution of the experiment, this is exactly the behavior that was ~ b s e r v e d .On ~ the other hand, the phases used in the present study approached a critical point, and the hysteresis width (like the interfacial tension) went to zero as the critical point was approached. The latter finding supports the approach to zero hysteresis width for isothermal approaches to plait points previously shown for ternary systems (ref 3, Figures 8-10). Finally, we wish to point out that the experiments of this study were designed with the intent of maximizing the success of catastrophe theory and/or critical-scaling theory in fitting the experimental results. Specifically, we varied the temperature, rather than the composition at constant temperature, so that the tie lines would remain parallel as we increased the distance from the critical point. Both theories encounter difficulties in fitting nonparallel tie lines; therefore, we anticipate that both theories may encounter problems in describing the concentration dependence of dispersion inversion at constant temperature and pressure.
Acknowledgment. This research was supported in part by an appointment of K.-H. Lim to the US.Department of Energy, Fossil Energy, Post-Graduate Research Program administered by Oak Ridge Associated Universities. We thank Peter Shifflet for helping to set up the conductivity apparatus. Registry No. 2-Butoxyethanol, 111-76-2.
