J. Phys. Chem. 1996, 100, 14991-14994
14991
Shape Changes of Globules in Nonionic Microemulsions M. Kahlweit,* G. Busse, B. Faulhaber, and J. Jen Max-Planck-Institut fu¨ r Biophysikalische Chemie, Postfach 2841, D-37018 Go¨ ttingen, Germany ReceiVed: May 3, 1996; In Final Form: June 18, 1996X
Assuming the globules at the (lower) stability limit of (diluted) nonionic o/w microemulsions in the waterrich corner of the phase prism to be spherical, and assuming, furthermore, the mean curvature of the (saturated) interfacial layers to decrease monotonically with rising temperature, it is shown that the size distribution of the globules must change to a lower number density of larger nonspherical globules upon raising temperature at fixed mean composition simply for volumetric reasons, irrespective of the explicit expression for the temperature dependence of the mean curvature. Applying Flory’s model, it is, furthermore, shown that this could explain the phase separation of o/w microemulsions at their upper stability limit in the water-rich corner. The same holds for w/o microemulsions in the oil-rich corner if the mean curvature is assumed to decrease with dropping temperature. The results are compared with the features of the phase behavior, and some experiments.
I. Introduction For an introduction to the problem consider Figure 1 (top) which shows a vertical section through the water-rich corner of the phase prism of a water-oil-nonionic amphiphile mixture at fixed amphiphile concentration, e.g., at that of the lower critical point of the “upper loop” of the oil-free wateramphiphile mixture, with the lower and upper stability limit of the (macroscopically) homogeneous o/w microemulsion. Upon isothermal addition of oil at temperatures well below Tl, the mixture remains homogeneous up to the binodal of the “central gap”, representing the lower stability limit at which the excess oil is expelled as molecularly disperse bulk phase (2). However, at temperatures just below Tl, the mixture first separates into a diluted and a concentrated dispersion (2h) and then becomes homogeneous again upon further addition of oil until, finally, the excess oil is again expelled at the binodal. Hence, if temperature is varied at fixed composition as indicated by the vertical arrow, there exists a lower as well as an upper stability limit of o/w microemulsions, the latter showing a distict minimum. If the (fixed) amphiphile concentration is that of the lower critical point cpβ of the upper loop, the upper stability limit runs close to the critical line clβ that terminates at Tl at the lower critical end point of the three-phase triangle. The problem we address in this paper is the change of shape of the oil globules in the homogeneous region between the lower and the upper stability limit. The origin of the upper stability limit is elucidated in Figure 18 in ref 1 which shows the evolution of the three-phase triangle with rising temperature in nonionic microemulsions with longchain amphiphiles. At low temperatures [(1) in that figure], the mixture shows a two-phase region (2), with a plait point on its oil-rich side. As one raises T, a loop appears with its lowest point on the water-rich side of the two-phase region (2). With further rising T, this point develops into a loop the tie-lines of which run parallel to the water-amphiphile side of the Gibbs triangle (3). As a consequence, this loop shows two plait points: one on its oil-rich side, the other one on its water-rich side. At T ) Tl, the plait point on the oil-rich side touches the central gap which leads to the formation of the three-phase triangle (4). The plait point on the water-rich side, on the other X
Abstract published in AdVance ACS Abstracts, August 15, 1996.
S0022-3654(96)01260-9 CCC: $12.00
Figure 1. Top: Vertical section through the water-rich corner of the phase prism at fixed amphiphile concentration with the homogeneous o/w microemulsion (1) bounded by its lower (2), and upper (2h) stability limit (schematic). Bottom: Isothermal section through the water-rich corner of the phase prism at T , Tl with the homogeneous o/w microemulsion (1), the lower stability limit (2), and the cmca tie-line (schematic).
hand, approaches the water-amphiphile side of the Gibbs triangle to touch it at T ) Tβ at the lower critical point of the upper loop of the binary water-amphiphile mixture. From then on one finds a two-phase region (2h) extending from the wateramphiphile side of the Gibbs triangle to the three-phase triangle (5). Vertical sections through the phase prism at fixed amphiphile concentration, hence, looks as shown on top of Figure 1 (for an experimentally determined section see Figure 4, left). On the bottom of Figure 1 (see also Figure 10, bottom, in ref 1) one can see an isothermal section through the phase prism at a temperature well below Tl, with the homogeneous o/w microemulsion (1), the two-phase region (2), and the cmca tieline. For spheres, the surface/volume ratio is a minimum, © 1996 American Chemical Society
14992 J. Phys. Chem., Vol. 100, No. 36, 1996
Kahlweit et al.
compared with any other shape. Because at a given amphiphile concentration the system tries to emulsify as much oil as possible, one expects the oil globules in (diluted) microemulsions at their lower stability limit, that is, along the initial part of the binodal above the cmca tie-line, to be spherical. Assuming saturated interfacial layers, the interfacial area AV per unit volume is proportional to the volume fraction ΦC of the amphiphile. Disregarding the thickness of the interfacial layer one, hence, expects for the dependence of ΦC on the volume fraction Φ of the dispersed oil along the initial part of the binodal, taking the cmc as origin
AV ) N‚4πr2
Φ ) N(4π/3)r3
ΦC/Φ ∝ AV/Φ ) 3/r (1)
Figure 2. For a binary mixture to show an upper loop (right), the dependence of the critical temperature Tc on temperature (left) must be such that (∂Tc/∂T) > 1 at the lower critical point of the loop (schematic).
where N is the number density of the droplets and r their (mean) radius. Experiments2 show that the initial part of the binodals is straight, from which one deduces that at fixed temperature T, the radius of the droplets along that part does not depend on ΦC or Φ. Experiments,2 furthermore, show that the slope of the binodals decreases with rising T, from which one deduces that r increases with rising T. We, therefore, set for the curvature of the droplets
of Tc on T must be such that ∂Tc/∂T > 1 at the lower critical point of the upper loop (Figure 2). Differentiation of eq 6 with respect to T gives
H ) 1/r ) Θ(T)
which demonstrates that even for ∂χ/∂T ) 0, a sufficiently strong increase of m with rising T
(2)
∂Tc/∂T ) (2/R)(1 + m-1/2)-2 (∂χ/∂T) + (2χ/R)(1 + m1/2)-3 (∂m/∂T) (7)
where Θ decreases monotonically with rising T and does not depend on ΦC or Φ. Hence,
ΦC/Φ ∝ Θ
(3)
Consider now some point on the binodal at T ) T0 and raise the temperature at fixed composition. If the globules remain spherical this would lead to
r/r0 ) Θ0/Θ ) (ΦC/Φ)0/(ΦC/Φ) > 1
(4)
in which the subscript 0 refers to T ) T0. The volume available for solubilizing the oil would, thus, become too large in spite of decreasing number density. Because both ΦC and Φ are fixed, the oil globules must, therefore, simply for volumetric reasons change their shape such that their surface/volume ratio (AV/Φ) becomes larger than for spheres, that is, change to larger nonspherical globules as, e.g., cylinders or ellipsoids, under the condition that (AV/Φ) remains that at T0
AV/Φ ) 3/r0 ) 3Θ0
II. Phase Separation at the Upper Stability Limit This change of the size distribution from a high number density of spheres to a lower number density of larger nonspherical globules may have an important consequence: Assume that the free energy of this “binary” mixture of oil globules and solvent can be described qualitatively by Flory’s model. In this model, the critical temperature Tc of the miscibility gap is given by
(6)
predicting Tc to rise with increasing interaction parameter χ (>0), as well as with increasing aggregation number m. Irrespective of any model, for a binary mixture to show a lower miscibility gap as well as an upper (closed) loop, the dependence
(8)
will result in ∂Tc/∂T > 1, which may lead to a phase separation into a diluted and a concentrated o/w microemulsion at a lower critical point between melting and boiling point, provided 2χ/R lies between the two. The mixture will then show an upper loop in the water-rich corner of the phase prism, with tie-lines running parallel to the H2O-amphiphile side, and rapidly widening with rising T, even more so if χ, too, increases with rising T as one would expect it to do. The problem is, thus, to model the shape change of the globules. III. Shape Change of the Globules Because cylinders are less favorable than spherical or nearly spherical shapes even in the absence of a natural mean curvature,3 we assume monodisperse ellipsoids with a and b as axis, a > b. Then, if a is the axis of rotation (prolates)
Φ ) N(4π/3)ab2
(5)
Evidently, if now oil is added at fixed T and ΦC, the globules will swell until, at the lower stability limit at that temperature, the spherical shape is restored.
Tc ) (2χ/R)(1 + m-1/2)-2
∂m/∂T > (R/2χ)(1 + m1/2)3
AV ) N‚2πb[b + (a/) arc sin ]
≡ e/a
(9) e2 ≡ a2 - b2 (10)
Hence
AV/Φ ) (3/2)(1/a)[1 + (a/b)(1/) arc sin ]
(11)
with the two unknowns a and b. The mean curvature at the two apexes of a prolate is
Hapex ) a/b2
(12)
that around the “equator” perpendicular to the a axis
Heq ) (1/2)[(b/a2) + (1/b)]
(13)
while that at all other points on the surface lie between these two extremes. For simplicity we, therefore, assume
H ) (1/2)[(1/a) + (1/b)] ) Θ
(14)
Globules in Nonionic Microemulsions
J. Phys. Chem., Vol. 100, No. 36, 1996 14993 as shown on the right of Figure 3, demonstrating that as T rises it becomes increasingly more difficult to experimentally distinguish between prolates and cylinders. The “aggregation number” m of the prolates is, again disregarding the thickness of the interfacial layer,
m ) (1/V)(4π/3)ab2 ) (1/V)(π/6)r03 (1 + F)3/R3F2
(23)
where V is the volume per oil molecule. Or, for F . 1
m ≈ (1/V)(π/6)r03F/R3
(24)
which yields, observing eq 2 and the definition of R, eq 15,
∂m/∂T ∝ (1/Θ3)(∂F/∂T) - 3(F/Θ4)(∂Θ/∂T) Figure 3. Left: Dependence of F (full line, right ordinate, eq 17), and of N/N0 (broken line, left ordinate, eq 21) on R. Right: Dependence of a/r0 and b/r0 (eq 22) on R. Decreasing R corresponds to rising T.
which reduces to that of a sphere for a ) b, and to that of a cylinder for a . b. Then from eqs 5, 11, and 14
R ) [1 + (a/b)]/[1 + (a/b) (1/) arc sin ] R ≡ Θ/Θ0 e 1 (15) or, after introducing
i.e., 2 ) 1 - F-2
(16)
R ) [1 + F]/[1 + F(1/) arc sin ]
(17)
F ≡ a/b g 1
The full line on the left of Figure 3 shows F vs R as evaluated from eq 17. First, F increases weakly with decreasing R, that is, rising T, to then increase strongly. Equation 17 holds irrespective of the mean composition and the explicit form of Θ, as long as the globules along the binodals are droplets, and Θ decreases monotonically with rising T. The mean composition enters via eq 5 in that it determines T0 which, however, can be evaluated only if the explicit dependence of Θ on T was known. Equation 17, furthermore, holds for the shape changes in w/o emulsions (T > Tu) as well, if Θ is assumed to decrease monotonically with dropping T.4 From F ) F(R) one may now evaluate N, as well as a and b, as they depend on R. For evaluating N, one writes eq 9 as
Φ2 ) N2(4π/3)2F2b6
(18)
whereas eq 10 gives in combination with eq 17
AV3 ) N3(2π)3b6(1 + F)3/R3
(19)
This gives
AV3/Θ2 ) N(9π/2)(1 + F)3/R3F2
(20)
or, observing eq 1,
N/N0 ) 8R3F2/(1 + F)3
(21)
which yields the broken line on the left of Figure 3. For evaluating a, one starts from eq 11 which gives, observing eq 17,
a/r0 ) (1/2)(1 + F)/R hence
b/r0 ) (1/2)(1 + F)/FR
(22)
(25)
in which, in view of ∂Θ/∂T < 0, both terms are positive. One can, hence, very well imagine a sufficiently strong increase of m with rising T such that at some temperature Tc > T0, ∂m/∂T meets condition eq 8. If this was so, then it would, furthermore, follow from eq 25 that, because of ∂r0/∂T > 0, hence, ∂m0/∂T > 0, the temperature interval Tc - T0 should shrink with rising T0, that is, with increasing volume fraction of the oil, Φ, at fixed ΦC (see Figure 1, top). If this simple model was an approach to reality, the critical end point at Tl at which the mixture separates into three coexisting phases would be that point at which Tc - T0 vanishes. Evidently, the model may now be improved by taking into account the thickness of the interfacial layer, etc. This, however, should not change its qualitative predictions. IV. A Previous Model Instead of ellipsoids, Menes, Safran, and Strey 5 assumed cylinders of varying length but uniform radii coexisting with monodisperse spheres. As for the radii rc of the cylinders they assumed them to be determined by
1/rc ≈ Kξ/T
(26)
where K is the (temperature dependent) bending modulus of the interfacial layer and ξ is the persistence length of (long) cylinders which they identified with the size of the “monomers”. By applying a Flory-type expression for the free energy they were able to show that upon increasing the ratio Φ/ΦC at fixed T and ΦC, with T and ΦC as parameters, the mixture should separate into two phases if the parameters were appropriately chosen. V. Some Experiments Leaver and Olsson6 measured the temperature dependence of the dynamic viscosity η of o/w microemulsions in H2Odecane-C12E5 mixtures at four compositions on the water-rich side of a vertical section through the phase prism at fixed amphiphile/oil ratio. At the two lower oil concentrations, the relative viscosity ηr ≡ η/η0, with η0 denoting the viscosity of water at the corresponding T, remains practically constant with rising T, whereas at the two higher oil concentrations, it remains constant at low T, to then (see Figure 3 (left)) increase rather steeply which the authors interpreted as a transition from spheres to spheroids. In order to distinguish between prolates and oblates, they measured the temperature dependence of the selfdiffusion coefficients of the three components by NMR7 of which those for the amphiphile and the oil should increase as the globules grow. The experiment confirmed the prediction. However, while the data for the amphiphile and the oil did not permit distinguishing between prolates and oblates, those for
14994 J. Phys. Chem., Vol. 100, No. 36, 1996
Figure 4. Left: Vertical section through the phase prism of a H2Ooctane-C12E5 mixture at fixed C12E5/H2O ) 1.4/98.6 (by weight). The point at zero oil concentration represents the lower critical point of the H2O-C12E5 loop. Right: Isotherms of the relative viscositity ηr ) η/η0 along that path, with η0 being the viscosity of the oil-free micellar solution. ηr shows a peak at the composition of the lowest point of the upper stability limit, with the peak decreasing upon lowering T.
the water favored prolates. Strey, Glatter, Schubert, and Kaler8 studied D2O-octane-C12E5 mixtures by SANS and found on the water-rich side a sequence of spherical, cylindrical, and planar structures (the latter as one enters the lamellar mesophase) with rising T, and on the oil-rich side the same sequence with dropping T. From their data for the lowest oil concentration (sample 55xx), they evaluated for the ratio between the radii of cylinders and spheres (R(Fmax) in their Table 2) ≈ 2/3, to be compared with b/r0 in Figure 3. Using a microviscosimeter (Haake), we measured the temperature dependence of the dynamic viscosity η of homogeneous o/w microemulsions in H2O-octane-C12E5 mixtures at 10 oil concentrations between 0 and 2 wt % octane at fixed C12E5/H2O ratio ) 1.4/98.6 (by weight), which is the composition at the lower critical point of the upper loop in the oil-free H2O-C12E5 mixture. On the left of Figure 4 one can see the phase diagram along this path with the two stability limits. From the data we obtained isotherms of the relative viscosity ηr ) (η/η0)T, with η0 denoting the viscosity of the oil-free micellar solution (!) which increases practically linearly from 2 to 3 Pa‚s in the considered temperature range. The results are shown on the right of Figure 4. Upon adding oil to the micellar solution at a temperature just below the lowest point of the upper loop, ηr rises steeply, passes through a narrow maximum, drops to values < 1, and passes through a wide minimum to then rise weakly with further increasing oil concentration. As one can see, the peak of ηr lies at the same composition as the lowest point of the upper stability limit but decreases rapidly upon lowering T, that is, increasing the “distance” to the stability limit. Although this behavior agrees qualitatively with the model presented above, a more detailed interpretation of the ηr isotherms is difficult
Kahlweit et al. for the following reason: Adding a little oil to the micellar solution will first lead to the formation of “swollen micelles” in which every solubilized oil molecule is in direct contact with amphiphile molecules. As one adds more oil, these swollen micelles turn into globules in which an increasing number of oil molecules is no longer in direct contact with the interfacial layer. According to the considerations presented above, these globules must be nonspherical, possibly ellipsoidal. As further oil is added, these globules swell to spheres until, at the binodal, the excess oil is expelled as bulk phase. Hence, the number density as well as the shape of the globules varies considerably along this path which makes even a semiquatitative interpretation difficult, the more as the globules cannot be viewed as being nondeformable. We remark that if the experiment is performed with short-chain amphiphiles such as C4E1, ηr shows no peak which supports our suggestion to draw the border line between weakly and strongly structured mixtures at medium-chain amphiphiles.1 At present, we are measuring the temperature dependence of the turbidity of homogeneous o/w dispersions at fixed compositions along the same path. The turbidity of diluted dispersions of (monodisperse) isotropic particles with diameters small compared with the wave length of light is given by
τ ∝ NV2
(27)
with V denoting the volume of a globule, or, in view of NV ) Φ,
τ ∝ Φ2/N
(28)
Because our model predicts N to drop with rising T (Figure 3, left), τ should increase proportional to 1/N. In addition, we are performing condenser-discharge T-jump relaxation experiments along that path in the presence of a trace of salt, using turbidity for detection. The (inverse) relaxation time constant drops monotonically with increasing oil concentration, whereas the (relative) relaxation amplitude, being a measure for the relative change of V, shows two peaks, a first one at the composition of the lowest point of the upper stability limit, decreasing upon lowering T, and a second one at a somewhat higher oil concentration. The results as well as an analysis of the data will be published in forthcoming papers. References and Notes (1) Kahlweit, M.; Strey, R.; Busse, G. Phys. ReV. E 1993, 47, 4197. (2) See Figure 12 (bottom) in ref 1. (3) Golubovic, L.; Lubensky, T. C. Phys. ReV. A 1990, 41, 4343. Golubovic, L. Phys. ReV. E. 1994, 50, R2419. Morse, D. C. Phys. ReV. E. 1994, 50, R2423. (4) See Figure 12 (top) in ref 1. (5) Menes, R.; Safran, S. A.; Strey, R. Phys. ReV. Lett. 1995, 74, 3399. (6) Leaver, M. S.; Olsson, U. Langmuir 1994, 10, 3449. (7) Leaver, M. S.; Furo, I.; Olsson, U. Langmuir 1995, 11, 1524. (8) Strey, R.; Glatter, O.; Schubert, K.-V.; Kaler, E. W. J. Chem. Phys., submitted for publication.
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