Langmuir 1991, 7, 2928-2933
2928
On the Stability of Microemulsions M. Kahlweit*p+and H. Reissi Max-Planck-Institut fur biophysikalische Chemie, Postfach 2841, 0-3400 Gottingen, Germany, and Department of Chemistry, University of California, Los Angeles, California 90024 Received June 24, 1991. In Final Form: July 29, 1991 For a dispersion to form spontaneously,the Gibbs free energy of mixing, Ag,, must be negative. For the dispersion to be thermodynamically stable, Ag, must, furthermore, show a minimum. When applying these conditions to microemulsions with an amphiphilic monolayer separating the polar and the nonpolar solvent,it has been customary to attribute a natural curvature as well as a bending energy to the saturated monolayer, thereby making the interfacial tension u depend on the degree of dispersion. Both properties are difficult to measure. Although these effects are undoubtedly of great importance in the mechanism of phase separation, there is another effect that has received much less attention. This involves the reduction of amphiphile surface concentration below the saturation level, an effect that also makes the interfacial tension depend on the degree of dispersion. In this paper we give attention to this effect and examine its consequencesas if curvature and bending energy were not a factor. The resulting predictions for the stability limit of microemulsions and for the size of the droplets contain only measurable quantities and are, furthermore, in semiquantitative agreement with experiment. This suggests that further work should be done to determine the relative importance, in the case of particular systems, of the two sorts of effects.
I. Introduction
It is now well understood and generally accepted that the state of dispersion of a microemulsion, modeled as a heterogeneous array (on the intermediate length scale) of domains of solute (A) and solvent (B), delineated by boundaries consisting of a saturated monolayer of a nonionic amphiphile, is determined by the balance between the entropy of mixing of the domains and the interfacial energy associated with the boundaries. If the interfacial tension is regarded as constant when the monolayer remains saturated, then if enough amphiphile is supplied to the system to maintain saturation regardless of the size of the domains, the most stable dispersion is easily shown to be a molecular solution. If, on the other hand, the solute concentration is increased at fixed amphiphile content, the domain size increases until, a t a well-defined stability limit, the excess solute is expelled as a bulk phase in equilibrium with the microemulsion at its stability limit. From the mathematical point of view, the easiest resolution of this problem when the amphiphile content is fixed is to assume that the interfacial tension rises abruptly to a very large value as soon as a state of dispersion is reached such that the boundary layers become unsaturated. Then, even a small further decrease in domain size leads to an increase in free energy, and the microemulsion is stabilized almost exactly at the size at which the monolayer loses its saturation. Of course, this simple picture cannot be correct. For example, even though the interface remains saturated, e.g., before the amphiphile is exhausted, the necessary increase in mean curvature of the interface (bending) as the domains become smaller produces a variation in interfacial tension (usually an increase as the curvature departs from a “natural” value) so that the argument based on constant interfacial tension cannot be used. This particular aspect of the problem has been well studied, especially in connection with bilayers where the molecules + Max-Planck-Institut fur biophysikalische t University of California.
Chemie.
0743-7463/91/2407-2928$02.50/0
have little solubility in the ambient bulk phases.’ It was a natural step to carry over these ideas on curvature to the microemulsion problem and to argue that bending effects lead to an increase in free energy prior to the exhaustion of amphiphile, thereby accounting for the stability of the sy~tem.~-~ However, in spite of the elegant molecular theories on bending energy and the associated elastic coefficients,there is no assurance that the predictions are of enough quantitative reliability to ensure that, in every case, the point of “stability” due to curvature will have been reached before the exhaustion of amphiphile occurs. In summary, there is no firm argument for ignoring the effects of loss of saturation while focusing only on questions of curvature. One or both of these effects may be operative. The present paper is devoted to a study of the former effect. In particular a simple theory will be introduced that “softensn the “abrupt” increase in interfacial tension upon loss of saturation, assumed above, but which can still account for stability. Furthermore the theory contains only measurable quantities and leads to predictions in semiquantitative agreement with experiment regarding measurements where theories of curvature have not yet predicted anything. In the following sections we develop this simple theory, using an approximate method that contains enough of the physical essentials to provide a semiquantitative description of the observed phenomena.
11. The Random Mixing Model Consider a dispersion of isolated droplets with radius r of a solute (say A) in a solvent (B), stabilized by a saturated monolayer of a nonionic amphiphile (C). Let the volume fraction of the dispersed phase be CP @A, with 0 < 0 < 0.5. For the dispersion to form spontaneously, (1) Helfrich, W. Z. Naturforsch., C Biochem., Biophys., Biol., Virol. 1973,28C, 693. (2) De Gennes, P. G.; Taupin, C. J. Phys. Chem. 1982,86, 2294. (3) Widom, B. J. Chem. Phys. 1984,81, 1030.
(4) Andelman, D.; Catea, M. E.; Roux, D.; Safran, S. A. J. Chem. Phys.
1987,87, 7229.
0 1991 American Chemical Society
Langmuir, Vol. 7, No. 12, 1991 2929
Stability of Microemulsions
Figure 1. Dependence of Aggm(eq 11.6) on d at fixed T,9, and u
(schematic).
it is necessary that (11.1) where Agm is the Gibbs free energy of mixing per unit volume. For the dispersion to be thermodynamically stable, it is, furthermore, necessary that (Agm)T,p,*
50
*
{d(Agm)/drlTg,*= 0 and {a2(Agm)/dr21T,p,* > 0 (11.2) Considering only the contribution of the interfacial energy, we set Agm = -T As,
+ A,
u
(11.3)
where Asm is the entropy of mixing per unit volume, A , the interfacial area per unit volume, and u the interfacial tension. The problem is, evidently, to calculate Asm. For a semiquantitative calculation we follow previous by approximating the spheres by cubes of edge length d = 2r, and span the unit volume by a simple cubic lattice of cubic cells, each of volume d3,such that the number per unit volume is N = d-3. Applying the random-mixing approximation, one then has
Figure 2. Dependence of Agmb(eq 111.5) on d a t fixed T, 9, d,, and k b (schematic).
a natural curvature.’ This has led to the assumption that a saturated monolayer at the interface between a microemulsion droplet and the solvent also possess a natural curvature C,, disregarding the fact that the amphiphiles used for preparing microemulsions are, in general, considerably more soluble in the adjacent bulk phases than those used for preparing vesicles. As for the sign of C,, we define Cn < 0, if the monolayer is convex toward phase a, and C, > 0, if it is convex toward phase b. With r1 and r2 denoting the principal radii of curvature, any deviation of the mean curvature of the monolayer
(111.1) C (1/2)U/rJ + (1/r2)l from the natural curvature should then give rise to a contribution gbto the free energy of the dispersion where the superscript b indicates “bending”. Because for small deviations, the contribution gb per unit area must be a quadratic function of AC C - C,, we set gb = (1/2)kb(Ac)2 (111.2) where k b is a constant having the dimensions of an energy. The interfacial tension u in eq 11.3 must then be replaced by = 0) + (1/2)kb(Ac)2 (111.3) For spherical droplets one has C = l / r = 2/d, and C, = l/r, = 2/dn, thus U’ = U(kb
P (@ In @ + (1- @)In (1- a))< 0
(11.4)
where k~ is the Boltzmann constant, and5 Q E @(l - @) > 0 A, = 6Q/d; Equation 11.3 then becomes
(11.5)
(11.6) Agm/(kBT) = c3 {P+ 6 (Q/kBT) d2 4 For 6T = 0, A g m is then a function of d, @, and u. Consider now the dependence of &,/(kBT) on d a t fixed @ and u, that is, assuming the interfacial tension u not to depend on the degree of dispersion. A g m / ( k B T ) rises from -m for d = 0, changes sign at
-
do = {(1/6)(k~T/ u)(-P/Q)11’2
(11.7)
and passes a maximum to asymptotically approach zero for d m (Figure 1). From condition 11.1it then follows that only droplets of diameters d I do can form spontaneously. These droplets, however, are not stable but tend to dissolve until they reach molecular size. To obtain thermodynamically stable droplets, a third term has been added2-4to eq 11.3 by considering the contribution of a bending energy of the monolayer, thereby effectively making u depend on d.6 111. The Bending Energy In literature it has been assumed that a bilayer of a vesicle possesses, in the absence of all other constraints, (5) Debye et al. found for the correlation length 5 in a random two-
phasespongelike medium € = 4Q/A,. Debye, P.; Anderson, H. R.; Brumberger, H . J. Appl. Phys. 1957, 28, 679. (6)See, in particular, ref 4.
d = u(kb = 0) + 2(kb/d2){1- (d/d,))2 (111.4) In applying this result, it has been assumed U ( k b = 0) is negligible compared with the second term. Equation 11.6 then becomes
U m b / ( k ~ T=) 6 3 [ p 12(kb/k~T)Q(1-(d/dn))21 (111.6) where again the superscript b indicates the inclusion of bending effects. Consider now the dependence of Agmb/(kBT) on d a t fixed @ and kb. It shows three zeros, namely at d = a, and at (dld,) = 1f {(1/12)(kBT/kb)(-P/8)11/2 (111.6) For both solutions of eq 111.6 to be positive, the square root must be smaller than unity. This requires (kBT/kb)(-P/Q) < 12 (111.7) Agmb then drops from +m for d = 0, changes sign, passes a minimum, changes sign again, and passes a maximum to asymptotically approach zero for d m (Figure 2). The minimum, d,, lies at the smaller of the two roots of
-
0
(d/d,I2 - 4(d/d,) + (3 - (1/4)(k~T/kb)(-P/Q))
that is (dm/d,) = 2 - (1+ (1/4)(k~T/kb)(-p/Q)l”~ = f(@)
(111.8) Because eq 111.8holds for droplets withsaturated monolayers only, there is an additional relation between the
Kahlweit and Reiss
2930 Langmuir, Vol. 7,No. 12, 1991 r
L
interfacial concentration I's in a saturated monolayer can be evaluated from the Gibbs adsorption equation, namely
rs= (NAuJ1= -(RT)-'(au/d In Nc)T,p
A
cmc tte-line
-In
N,
Figure 3. Dependence of the interfacial tension u of a planar interface between the aqueous and the oil-rich phase on the amphiphile concentration N , at fixed T (schematic).
diameters d of such droplets and the number density N , of amphiphile molecules. Disregarding the solubility of the amphiphile in the two solvents, the relation between A, and N , reads (111.9) A , = Nca, = (*,/u,)a, where a, is the area per amphiphile molecule in the saturated monolayer, uc is its volume, and 3,is the volume fraction of the amphiphile. Inserted into eq 11.5,this yields (111.10) d = 6(u,/a,)Q/@, = d*(@,@,) where the last equality indicates that d corresponding to a saturated monolayer is a function d* of @ and a,. Henceforth the asterisk will be used to denote values of d corresponding to saturated monolayers. Setting d* = d, yields the relation d*(@,@,)= f ( @ ) dn, which shows that at the free energy minimum, associated with a saturated monolayer, 3,is a function of 3. Denoting this value of 3,by appending an asterisk, we find = 6(u,/aCd,)Q[2 - (1i(1/4)(k~T/kb)(-P/Q))"~l-' (111.11) which specifies the volume fraction of amphiphile that produces a stable dispersion. If for 6T = 0, both dn and k b are assumed to be constant, this result implies that for each 3,there exists only a single 3,for which the dispersion becomes stable. However, experiments indicate that for 3,> a,*, the size of the droplets decreases as 1/3,. This constraint is, however, only apparent because theoretical estimates of kb show it not to be constant, but rather a function of d.
(IV.l) where N Ais Avogadro's number, and the derivative is the slope in Figure 3 as the cmc is approached from the left. Consider now a dispersion at fixed N , and fixed 3. If the amphiphile concentration is such that the interfacial layers are saturated, an attempted decrease of A,, that is, an increase of d at the expense of the number density N , of droplets, will make r exceed rs. Because this is energetically unfavorable, the excess amphiphile would leave the interface to form additional droplets with saturated interfaces, i.e., such a decrease would not occur, at least not without a decrease of stability. An increase of A,, on the other hand, would cause I' to decrease and thus u to increase sharply. The latter effect is of considerable interest, and like bending energy, could presumably be estimated from molecular theory. However, for the purpose of a semiquantitative characterization of microemulsion stability phenomena, it is probable that the correct order of magnitude can be extracted from the following linear approximation = go- (go- us)(r/rs)
At fixed 3,the diameter d* of droplets with saturated interfaces is unambiguously determined by the amphiphile concentration 3, = 3,* (eq 111.10). If we start with N,* Ydrops"of size d* corresponding to a fixed 3 and a,*, the total surface area is determined by eq 11.5 to be A,* = 63(1- 3)/d* (IV.3) If 3 remains fixed and 3,remains at a,* while the size of the drops is changed to d, the new surface area will be A , = 63(1- 3 ) / d (IV.4) Since aCremains fixed r = N c / A v= (3,/uc)/Av is inversely proportional to A,. Thus F/r, = A,*/A, = (d/d*)
@*,
IV. Dependence of Q on t h e Interfacial Concentration 'l As we have already indicated, a dependence of u on the degree of dispersion can be introduced when it is recognized that the interfacial tension of a planar interface between a water-rich (a) and an oil-rich phase (b) is a sensitive function of the amphiphile concentration N,. The interfacial tension in the amphiphile-free mixture is uo = a(I'=O) 50" m-I. As one adds amphiphile, it decreases steeply to us 5 0.1 mN m-l at the critical micelle concentration (cmc)tie-line, where the u/ln N , curve shows a distinct discontinuity of slope. As N , is increased further, u remains practically constant at the very low level until it eventually decreases further to vanish at the plait point (Figure 3). Making the conventional assumption that the interfacial layer achieves saturation at the cmc, the
(IV.2)
(IV.5)
which, when inserted into eq IV.2 yields for the dependence of u on d u = u0 - (ao- a,)
(d/d*) (IV.6) This is to be compared with eq 111.4. Ford 2 d* one may set, in view of Figure 3, dld* (IV.7) where usdoes not depend on N C ( g 3 Jup to concentrations near the plait point. u=ag
V. The Stability Limit Inserting eq IV.6 into eq 11.6 then gives &?,'/(kBT)
= d-3[P + 6(Q/k~T)d~{uo - ( 6 0 - u,)(d/d*))l
(V.1) where the prime is used to indicate that in Agm' the surface concentration may vary. Qualitatively, the dependence of Ag,'/(kBT) on d (at fixed 3)is similar to that of Agm/ ( ~ B T(eq ) 11.6): it rises from -m for d = 0, changes sign at the smaller of the two positive roots of the cubic equation 0 = P + 6(Q/kBT)d2{uo- (u0- us) (d/d*)} (V.2) passes a maximum, changes sign again at the larger of the two positive roots of eq V.2, and asymptotically approaches -6(Q/hT) (u,~- u,)/d* for d m.
-
Langmuir, Vol. 7, No. 12, 1991 2931
Stability of Microemulsions
whereas for 9,< a,*, the dispersion becomes unstable: the excess solute is expelled as a bulk phase in equilibrium with the dispersion at its stability limit. By applying the hard-sphere model for evaluating the entropy of mixing, Reiss7found (for dilute solutions where aggregation could be ignored) instead of eq V.5
+ r2Us/(kBT)= (4~)-'[(5/2) - (3/2)/(1 In ((16r3/u)3iz(l- *)/@]I (V.7) where u is the volume of a solute molecule. For sufficiently low 9, eq V.7 simplifies to -d
r2as/(kBT)is: ( 4 ~ ) - ' [ 1 -In ( ( ~ / 1 6 r ~ )9 1 ~1" (V.8)
to be compared with eq V.6. For semiquantitative calculations one may, therefore, iet for the radii of the droplets at the stability limit r2us/(kBT) = 1
Figure 4. Dependence of Ag,' (eq V.l) and Ag, (u=ug) (eq 11.6) on d at fixed T and @ (schematic). Point d* represents the intersection of the two curves.
The Gibbs free energy per unit volume of the dispersion Agm'/(kBT) may then be constructed by setting Agm+
E
Agm'
d I d*
where Agm* is given by eq V.l and
did* where Agm is given by eq 11.6 with u = us. The resulting curve is shown schematically in Figure 4. From eqs V.l and 11.6 one finds that the intersection of Agm' and A g m occurs at the minimum of Ag,+/(kBT) Agm+=Agm
Agm* =&, for d = d * (V.3) while the zero of Agm, do, is given by eq 11.7 with u = us. We recall that at fixed 9,d* is inversely proportional to the amphiphile concentration 9,(eq 111.10). It then follows that if ip, is chosen such that (i) d* < do, Agm+ < 0 at its minimum (Figure 4, top), whereas if aCis chosen such that (ii) d* > do, Agm+ > 0 at its minimum (Figure 4, bottom). In case i, the dispersion is thermodynamically stable, whereas in case ii it is metastable. The stability limit lies at d* = do (Figure 4, center). Equating eqs 11.7 and 111.10 then yields for the dependence of a,* = 9,(d* =do) on the volume fraction 0 of the solute at the stability limit of the dispersion
a,* = 6(u,/ac)Q((l/6)( k ~ yus) / (-p/Q)]-li2 W.4) to be compared with eq 111.11. We note that eq V.4 contains only measurable quantities. For 9,= a,*, the size of the droplets is determined by eq 11.7 which can be written as 07.5) d2as/ (kBT) = ( 1/61 ( - p / Q ) For sufficiently low 9one has (-P/Q) = 1 -In 9. Equation V.5 then simplifies to d2uS/(kBT)% (1/6)(1- 1n 9) (V.6) For 9, > @,*, d decreases according to eq 111.10 as l/ipc,
(V.9)
Equation V.8 holds for isolated droplets only, that is, for low ip and, accordingly,low 9,.As mentioned in section IV, usremains practically constant above the cmc tie-line on its low level before it eventually decreases further as one approaches the plait point. In this concentration range, the size of the droplets is thus independent of act and only a rather weak function of 0. Along the initial portion of the binodal, the size d of the droplets should, accordingly, change only little, whereas their number density N , should increase almost linearly with 9 until the mean distance, N,-1/3,between their centers approaches d. With further increasing 9,the droplets start aggregating. The treatment of aggregated doublets is dealt with implicitely in the "random cube" approach, eqs 11.4 and 11.5, albeit crudely. A more rigorous treatment as well as the phase separation of the dispersion a t the endpoint of a critical line will be considered in a forthcoming paper.8
VI. Distribution of the Amphiphile Consider now a dispersion of solute (A) in solvent (B), in equilibrium with an excess bulk phase of A. Let the distribution coefficient of monomers of the amphiphile between the two solvents (at the cmc tie-line) be
K, = N l b / N l a
(VI.1)
and that of the total concentration of the amphiphile
K = N,b/N,"
(VI.2)
where N a denotes the number density in phase a. For the number, m, of amphiphile molecules in the saturated interface of a droplet at the stability limit one has
m = 6d,2/aC = a,-'(k,T/a,)(-P/&)
(VI.3)
Assuming that concentration can be used in place of activity, the law of mass action then requires
Km= Nv/(N,b)m
(VI.4)
For the total concentration of the amphiphile in the dispersion, one thus has
N,b = Nlb+ mN, (7) Reise, H. J . Colloid Interface Sci. 1975, 53, 61. (8) Reiss, H.; Kahlweit, M. In preparation.
(VI.5)
2932 Langmuir,
Kahlweit and Reiss
Vol. 7, No. 12,1991
With rising temperature, the cmcb surface turns faster than the cmca surface. Near T', the two surfaces intersect so that the cmca and the cmcb tie-line coincide, being identical with the a-b tie-line of the isosceles three-phase triangle. At temperatures above T', the cmcb surface lies at a lower amphiphile concentration than the cmc? surface. At mean concentrations between the two surfaces one, therefore, finds a water dispersion in oil with a positive curvature of the interfacial layers in equilibrium with the water-rich phase. The sign of the curvature of the interfacial layers at each temperature is thus determined by that one of the two cmc surfaces that lies at the lower amphiphile concentration at that temperature.
nonionic amphiphile (CI
H2O
(AI 01 I
(B)
T
VIII. Conclusion Making use of eq IV.l, and uc = (M,/Nu,), eq V.4 can be written as Figure5. Phase prism of a Hphil-nonionic amphiphile mixture with cmca and cmcb surface (schematic).
and for that in the excess phase of solvent A N: = N,"
(VI.6)
Combination of VI.l, VI.2, VI.4, VI.5, and VI.6 gives
N,b = K,N?
+ mKmKlm(N?)m
(VI.7)
where m is a weak function of 9. The apparent solubility of the amphiphile in the dispersion thus increases effectively with the mth power of the monomer concentration of the amphiphile in the excess bulk phase. Consequently, the plait point of the (central) miscibility gap lies on the side of the excess phase. As the dispersion separates into two phases on the opposite, amphiphile-rich side of the miscibility gap, this, necessarily, leads to the formation of a three-phase triangle.
VII. The Cmc Surfaces As was recently shown: a mixture of water (A), an oil (B), and a nonionic amphiphile (C) possesses two cmc surfaces that ascend almost vertically in the phase prism (Figure 5). The cmcasurface starts at the (almost vertical) cmc curve of the binary A-C mixture, passes through the narrow homogeneous water-rich phase toward the binodal surface of the central miscibility gap where it shapes the cmca curve. From there it proceeds along the corresponding tie-lines to terminate at the oil-rich side of the binodal surface. The cmcb surface, on the other hand, may be somewhat diffuse in the water-free B-C mixture but becomes well-defined on the oil-rich side of the binodal surface where it shapes the (almost vertical) cmcb curve. From there it proceeds along the corresponding tie-lines to terminate a t the water-rich side of the binodal surface. Because the energy of formation of (normal) micelles in the water-rich phase will differ, in general, from that of (inverse) micelles in the oil-rich phase, the two surfaces will, in general, not coincide. Experiment shows that both cmc surfaces turn counterclockwise with rising temperature (if looked at from above), the cmcb surface, however, somewhat faster than the cmca surface. At temperatures below the mean temperature T' of the threephase body, the cmca surface lies a t a lower amphiphile concentration than the cmcb surface. At mean concentrations between the two surfaces, one, therefore, finds an oil dispersion in water with a negative curvature of the interfacial layers in equilibrium with the oil-rich phase. (9) Kahlweit, M.; Strey, R.; Busse, G. J. Phys. Chem. 1990, 94,3881.
where M, is the molar mass of the amphiphile and pc its density. In view of the considerations presented in the preceding section, eq VIII.l holds for oil in water dispersions for T I1'7 and for water in oil dispersions for T I
Tu.
If 1/9* is taken as a measure for the uefficiency" of an amphiphile in solubilizing either water in oil, or vice versa, eq VIII.l predicts that the efficiency of an amphiphile is proportional to the inverse of the square root of the interfacial tension a, between the two solvents in the presence of a saturated monolayer of the amphiphile. Experiment shows that the interfacial tension a, can be approximated by a parabola with its minimum near the mean temperature T' of the three-phase body as = a,(T=T)
+ O(T- n2
(VIII.2)
Accordingly, the efficiency reaches its maximum at 7'. For a given amphiphile, that is, fixed M,, both 7' and a,( T=0 increase with increasing carbon number k of the oil. The efficiency of an amphiphile thus decreases with increasing k . For fixed k , a,( T = n decreases strongly with increasing amphiphilicity,that is, with increasingM,, so that (M,a,1/2) decreases with increasingM,. The efficiencythus increases with increasing amphiphilicity. All these predictions are in qualitative agreement with experiment.1° Alternatively, eq VIII.l can be written as
where the right side is a function of 9 only and thus represents a universal expression for the stability limit, that is, the shape of the binodal expressed in terms of the left side. The right side is 0.15 at its maximum a t @ = 0.5. Experiment shows that for medium- and long-chain amphiphiles I', (=3 X mol cm-2) varies only little with the amphiphilicity of CiEj and the carbon number k of the oil. For a semiquantitative comparison with experiment, ~ , =4 X erg, one may, therefore, set pc = 1g ~ m -kBT and replace 9, by the mass fraction y (in wt %) of the amphiphile y
lo2@,
(VIII.4)
I,f one then measures-for fixed oil-the minimum amount (T=T', 9=0.5) required for homogenizing equal volumes of water and oil at the mean temperature T of the y = y*
(10) See ref 9, and further references therein.
Langmuir, Vol. 7, No. 12, 1991 2933
Stability of Microemulsions Table I. Comparison of Equation VIIId with Experiment CiE; M, 7,"C 4,w t % ull,mNm-' ~/(Mcus1/2) C6E3 CsE4 C10E5 ClzEs
234 306 379 451
44.4 42.3 44.5 48.7
39.5 24.4 14.7 7.6
0.151 0.041 0.013 0.00367
0.43 0.39 0.28 0.28
three-phase body, as well as the interfacial tension u,(T=T) at the lower tie-line of the isosceles three-phase triangle as a function of M,, eq VIII.3 predicts ;/(Mcu8"')
0.3
(VIII.5)
Table I shows the results of such experiments wi,th n-octane as oil. The amphiphiles were chosen such that T was about the same for all mixtures. Although the apparent agreement between the values in the last column of Table I and the prediction of eq VIII.3 should not be exaggerated, the
result suggests that further experiments and theory (on the molecular level) should address the question of the dependence of Q on I?, as well as its dependence on temperature, the amphiphilicity in terms of i and j , and the carbon number k of the oil-all this in addition to the focus on curvature and bending energy. In closing, we would like to draw attention to a very interesting recent paper by M. Borkovec" that does address some aspects of the question of desaturation. We feel that more work of this type should be done.
Acknowledgment. We are indebted to Mr. G. Busse and L. M. Trejo for performing the experiments summarized in Table I. This work was in part supported by a NATO Travel Grant (No.0425/88). (11) Borkovec, M. J. Chem. Phys. 1989,91, 6268.
