Theory of multicomponent micelles and microemulsions - The Journal

J . Phys. Chem. 1984, 88, 6519-6544

6519

Theory of Multicomponent Micelles and Microemulsions Mark M. Stecker and George B. Benedek* Department of Physics and Center for Materials Science and Engineering, Massachusetts Institute of Technology, and Harvard-MIT Division of Health Sciences and Technology, Cambridge, Massachusetts 02139 (Received: June 7, 1984)

A new theoretical approach to the analysis of micellar systems based upon the use of a variational principle is presented.

In this approach, we first express the free energy of a micellar system as a function of the micelle distribution function. Classes of trial distribution functions are chosen. That member of each class which minimizes the Gibbs free energy while simultaneously satisfying the conservation equations is selected as the optimum distribution function. This method determines the concentration of constituent monomers free in solution as well as the micelle size distribution and allows the prediction of phase boundaries. This theory is applied first to the case of single-component noninteracting micellar systems (e.g., SDS-water). The results obtained by this variational method for various classes of trial function are compared with exact results. This method shows the importance of polydispersity in that only trial distributions which have finite width give good results. The theory is then applied to two-component micelles of arbitrary dimensionality (e.g., disklike bile salt-lecithin micelles or spherical microemulsion droplets). In this case, we obtain predictions of the monomer concentration of each component, the micellar size distribution, and the detailed structure of the phase diagram for both two- and three-dimensionalmicelles. In the case of the two-dimensional lecithin-bile salt problem, our theory predicts a maximum micellar radius whose magnitude is determined by the composition of the system and the “formation energy” parameter 6’ in the Gibbs free energy. These theoretically obtained results are in good agreement with a variety of experimental measurements.

1. Introduction 1.I. Theoretical Approaches. Self-assembling micellar and microemulsion systems play a vital role in a broad variety of biological and industrial systems. For example, the essential processes of lipid solubilization and transport are mediated by mixed micelles of bile salt, lecithin, cholesterol, and dietary lipid in the gastrointestinal tract’ and by low- and high-density lipoprotein mixed micelles in the blood.* Furthermore, cell membranes, which are self-assembling bilayers of insoluble amphiphilic lipids and proteins, are crucial biological structures responsible for the compartmentalization and integration of cellular functions. Industrially, analogous micellar systems are of fundamental importance in processes as diverse as ore and oil r e ~ o v e r ydeter,~ gency, emulsion polymerization, catalysis, and food technology. A wide variety of empirical data has been obtained on the properties and phase behavior of these systems. This information has, in the past, provided an important qualitative framework for improving and expanding industrial and medical applications of micellar and microemulsion systems. Recently, much more detailed information on the microscopic structure of micellar systems has been gathered with such powerful methods as quasielastic light scattering s p e c t r o ~ c o p y ,nuclear ~ ~ ~ magnetic resonance,43 and neutron ~ c a t t e r i n g . ~ , ~ From the theoretical point of view, there now exist in the literature*-l3certain elements of the statistical mechanical and

(1) Carey, M. C.; Small, D. M. J . Clin. Invest. 1978, 61, 998. (2) Deckelbaum, R. J.; Shipley, G. G.; Small, D. M. J. Biol. Chem. 1977, 252, 744. (3) Morgan, J. C.; Schechter, R. S.; Wade, W. W. In “Solution Chemistry of Surfactants”; Mittal, K. L., Ed.; Plenum Press: New York, 1979; Vol. 2. (4) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y.; Carey, M. C. J . Phys. Chem. 1980,84, 1044. (5) Mazer, N. A.; Benedek, G. B.; Carey, M. C. Biochemistry 1980, 19, 601. (6) Dvolaitzky, M.; Guyot, M.; Lagues, M.; LePesant, J. P.; Ober, R.; Sautery, C.; Taupin, C. J. Chem. Phys. 1978, 19, 3279. (7) Ober, R.; Taupin, C. J . Phys. Chem. 1980, 84, 2418. (8) Hall, D. G.; Pethica, B. A. In “Nonionic Surfactants”; Schick, M. J., Ed.; Marcel Dekker: New York, 1967; Chapter 16. (9) Hill, T. L. J . Chem. Phys. 1962, 36, 3182. (10) Aranow, R. H. J . Phys. Chem. 1963, 67, 556. (1 1) Reiss, H. J . Colloid Interface Sci. 1975, 53, 61. (12) Ruckenstein, E.; Krishnan, R. J . Colloid Inte$ace Sci. 1979, 71, 321. (13) Israelachvilli, J. N.; Mitchell, D. J.; Ninham, B. W. J . Chem. SOC., Faraday Trans. 2 1976, 72, 1525.

thermodynamic basis for a molecular description of micelles.61 The fundamental theories of micellar structure have been applied successfully to certain simple special cases.4,13,14Of these, the most tractable have been those in which interactions between micelles are presumed small. However, even in the case of such weakly interacting systems, the analysis of multicomponent micellar systems has proved to be mathematically difficult. In the face of these mathematical difficulties, workers have been forced to resort to the ansatz of a monodisperse micellar size distribution and to ignore the essential role of equilibrium between the monomers and multicomponent micelles. The purpose of this paper is to explore a new mathematical method for the analysis of micellar systems. We will show that this method has mathematical advantages over other methods, that it provides insight into the physical factors which control the micellar size distribution, and that it can predict a class of phase transitions to which these weakly interacting systems are subject. Our mathematical method is based upon the use of a variational principle. This variational principle is an extension of a technique which has been used widely in such areas of physics as quantum mechanic^,^^ transport theory,16 and classical mechanic^.'^ In addition to developing the general theory, we shall apply our method to the analysis of a particularly interesting and important multicomponent gastroenterological system whose experimental properties have been well studied by quasielastic light s ~ a t t e r i n g : ~ , ~ the aqueous bile salt-lecithin system. In order to understand the necessity for the development of this new method, it will be useful to briefly summarize some of the current theoretical descriptions of micellar systems. These theories can naturally be divided into four categories based upon their different foci: geometry, surface tension, statistical mechanics, and multiple chemical equilibria. Perhaps the most intuitive theories of micelle structure are those that, making the assumption of a monodisperse size distribution, concentrate only on the geometric structure of a single micelle. A simple example of these theories is the initial description of the oil-in-water microemulsion by S ~ h u l r n a n . ~ *This J ~ description (14) Debye, P. Ann. N.Y.Acad. Sci. 1949, 51, 573. (15) Schiff, L. I. “Quantum Mechanics”; McGraw-Hill: New York, 1968. (16) Reif, F. “Fundamentals of Statistical and Thermal Physics”; McGraw-Hill: New York, 1965. (17) Konopinski, E. J. “Classical Descriptions of Motion”; Freeman: San Francisco, CA, 1969. (18) Hoar, T. P.; Schulman, J. H. Narure (London) 1943, 152, 102.

0022-3654/84/2088-65 19$01.50/0 0 1984 American Chemical Society

6520 The Journal of Physical Chemistry, Vol. 88, No. 26, 1984

is based upon the assumption that all of the oil present in the system is found inside of the microemulsion droplets and all of the surfactant is to be found at the interfaces between the oil and water. Thus, if we let X B be the mole fraction of surfactant molecules in the system and X A be the mole fraction of oil, it is possible, by assuming that the microemulsion particles are spheres, to use the matter conservation equations to calculate the mole fraction of micelles X o and the micellar radius R in terms of the experimentally controllable variables X A and XB:

Stecker and Benedek surfactant. According to his theory, the surfactant molecules would minimize their surface energy by curving concave toward the phase with which their surface tension is highest. Later, various authors including A d a m ~ o nPrince,z5 ,~~ and Okazawaz6 argued that, if the entropy of mixing of the microemulsion droplets could be ignored, microemulsions would be stable only when the net surface tension at the water-oil interface was equal to zero. In this scheme, the role of the surfactant (and cosurfactant) is to supply a surface pressure (a)so that the effective surface tension at the solvent-solubilizate interface can be written Y=Yo-a

( 4 a R z / a o ) X o= X B

and hence

(3) (4)

where a. is the average surface area of a surfactant molecule and uo is the average volume of an oil molecule.67 Intuitively, we see from these equations that as the amount of surfactant increases relative to the amount of oil, the surface to volume ratio of the microemulsion particles, being inversely related to their radius, must increase. Shankland@ and Smallz0used a similar approach to predict the dependence of the size of disklike bile salt-lecithin micelles on the amounts of lecithin and bile salt in the aggregates. By modifying this description to allow bile salt molecules to be present within the lecithin interior as well as at the edge of the micelle, Mazer et ale5were able to achieve excellent agreement with experimental values for the micellar size. The theories mentioned above do not depend on the details of the molecular structure of the surfactant molecules, contrary to experimental evidence. Robbinszl refined the geometrical considerations presented above to account for the finite volume of the polar heads and hydrocarbon tails of the surfactant molecules. This modification, which adds additional constraints into the geometrical equations, produces a prediction of phase limits for the oil-in-water and water-in-oil microemulsions as functions of the structure of the surfactant molecules. I s r a e l a c h ~ i l l i and ’~~~~ Mitchellz3have also studied the effects of surfactant geometry on the formation of micelles. In particular, they showed that, for simple micelles, if u is the volume of a surfactant and I, its maximum length, there was a geometric quantity v/a,l, whose magnitude determines whether spherical, rodlike, bilayer, or other shaped aggregates will form spontaneously. Specifically, when this quantity is large ( > O S ) , inverted micelles and bilayer structures are favored while when this quantity is small (I

-(

A

....I

- no6 (31)

ii will be less than no. This result is unphysical, for, according to (13), states with ii C n, are associated with infinite energy. This is also inconsistent with the assumption ii > no used in the derivation of (30). Intuitively, this difficulty can be dealt with by requiring that rt = no whenever (31) applies, a procedure whose validity can be justified rigorously."1 With ii = n,, the conservation equation demands

when XT C X,*. This expression (32) can then be substituted into (27) to arrive at an expression for g as a function of XFwhich is minimized when

an expression that is similar in form to the exact result (19). Thus, the variational method provides us with an expression for X F both when XT > X p (29) and when XT C XT, (33). In Figure 2, we plot Xp exp(-6/kT), the fugacity of monomers, vs. X , exp(-6/kT) for various choices of trial function, using the following parameters consistent with the case of SDS: no = 60, exp[(A - no6)/kT] exp(b/kr) = 5000. The close quantitative agreement between the results obtained with the crude 6 function trial distribution and the exact results, as shown in this figure, is striking. In particular, we notice that our present approximation predicts a (41) Note that, by (26), when (31) is valid we must have X ,< exp[-(A - noS)/(kT)]. In this regime, it is not difficult to show that g is a decreasing function of X,: hence, g will be minimized with respect to X , (subject to the constraint i i 2 n), by making X, as large as possible. This is clearly accomplished by making ti as small as possible, Le., ii = no.

x:Rec+angulo; 60

0

with,firsl corrections in no/u

71

J-LA

0.4

08

1.2

xTe-S/kT

1.6

2.0

24

Figure 3. The dependence of the mean micellar aggregation number ( N ) on X , exp(-6/kT) as predicted with different distribution functions. The continuous line (-) indicates the results deduced from the exact solution (19). The dashed line (---) is the result obtained with the 6 function distribution and the dotted line is the result predicted with the rectangular distribution with first-order corrections in no/.. no = 60, exp[(A - no6)/kT] exp(6/kT) = 5000. (.e.)

critical micelle concentration (cmc) at X T = XT*above which X F exp(-b/kT) is equal to unity, which is just the limit approached by the exact value of X , exp(-6/kT) when XT >> XT* or equivalently when the average micellar aggregation number ( N ) ( ( N ) is equal to rt in the case of the 6 function distribution) is much greater than no (compare Table I columns 1 and 6). The limit X Fexp(-6/kT) = 1 is to be expected since, when the micelles are large, they can be regarded as a new bulk phase. The condition that this new phase be in equilibrium with the aqueous phase is simply that the free energy change associated with the transfer of one solute monomer from the aqueous phase to the micellar phase be zero. Now, the change in the entropic contribution to the free energy is -kT In X F (i.e., the loss of the free energy of mixing in the aqueous phase for one monomer), and the change in the standard part of the free energy of the monomer is, by (1 8), 6 so that we must have X F = exp(6/kT). This line of reasoning is, of course, independent of the trial function used. Also, in Figure 2, we see that the approximate and exact values of X , agree fairly well even when XT is significantly less than the cmc. In this limit, it is clear that the entropy cost of localizing no monomers into a micelle is large (due to the small number of free monomers and the large value of n,). As a result, few micelles form and hence, by the conservation equation, XT = X,. Again, this reasoning applies independently of the choice of trial function. The results of this variational calculation and the exact results for ( N ) are compared analytically in Table I columns 1 and 6 and graphically in Figure 3. Figure 3 demonstrates that the results for ( N ) obtained by the exact method are substantially different from those obtained by using the 6 function form of trial function. For XT above the cmc, the exact value of ( N )increases roughly as ((XT - XF) exp[(A - no6)/kT]]'/*while the result obtained with the 6 function is proportional to ( X , - X F )exp[(A - noS)/kT]. The origin of this result is the prediction of the 6 function distribution that for XT above the cmc the number of micelles is constant. Thus, the conservation equation written in the form (26) immediately forces ( N ) to rise more rapidly as XT increases than in the exact solution where the total mole fraction of micelles X,,, is allowed to increase with XT. In addition, we see that, when XT < XT*, the prediction of our method with the present trial function is that ( N ) = no, while the exact solution is that ( N ) decreases slowly to no as XT is decreased below the cmc. These same deficiencies of the 6 function are reflected in Figures 4 and 5 where X,,, = X o for a 6 function distribution is plotted as a function of XT. This deficiency must occur since the conservation equation states that the product of ( N )and X,, must be XT - X,. Later, we will demonstrate that most of the above-mentioned problems with the 6 function distribution are a direct result of the neglect of p~lydispersity.~


Stecker and Benedek XF

321

1

28

' X

16

8L

4

Delta function \

Figure 5. The dependence of the mole fraction of micelles X,, on X, exp(-6/kT) as predicted by the exact (-) equations and by the variational method with a 6 function trial distribution (- -). Dotted lines indicate the results obtained with the rectangular trial distribution.

-

2.3. Polydisperse Trial Functions. The simplest way of treating polydispersity is to conceptually widen the 6 function distribution into the rectangular distribution shown in Figure 1B:

I

0; N < n ,

X,[X,,u,N] = X,; n o t a > N > 11, 0; N > n ,

+a

1

(34)

In order to simplify the following calculations, we shall now assume that the mean micelle aggregation number is much larger than no and that, since the size is large, it will be permissible to regard N as essentially a continuous variable. Applying the conservation eq 10 to the rectangular distribution yields the following relation between the parameters Xo and u:

N

where the last transformation made use of both (38) and (36). In Figure 2, we plot, as a function of XT exp(-6/kT) the values of XFexp(-6/kT) which result from substituting (36) and (38b) into (39b) and solving the resulting nonlinear equation. In this figure, we also present the corresponding exact results and the results obtained with the other trial functions. Here, we see that an excellent fit to the values, substantially better than that of the 6 function, is obtained. This applies even in the regime XTexp(-6/kT) < 1 where the approximation 4 0 is the increase in the standard part of the free energy which, according to (18), is just A for each new micelle formed. Numerically, it is apparent that, in the limit C; no, we suppose that, under "standard" conditions, it requires an expenditure in free energy of el (analogous to the parameter A that we used in the discussion of SDS) to assemble the no B molecules and the 7(d)nOdld-lA molecules of the smallest allowed micelle. Then, if we let poAFbe the standard part of the free energy of a free monomer of A in the aqueous phase and poBFbe the standard part of the free energy of a free monomer of B, we can write the following relation describing the change in the standard part of the free energy that occurs when a micelle of NB B molecules is formed: p0NB€1

+

+

(NBhoBF 7(d)NBd/"'poAF) = Ns-1

[(p0n+1 -

(n + l)poBF - 7(d)(n

+ l)d/d-lWoAF) -

n=nO

(ban - npoBF - 7(d)nd'"'poAF)] (47)

In this expression [(pon+l

- (n + (n

-

dd)x

+ l)d/d-lpoAF)- (pori - npoBF- ?-(d)nd/"'HoAF)]

is just the change in the standard part of the free energy (Le., all entropy of mixing effects is ignored) associated with transferring one B molecule and ~ ( d ) ( ( n l)d/d-'- ndld-')A molecules from the aqueous phase to a micelle already having n B molecules. To a first approximation, valid when the micelles are very large, we can suppose that this energy of transfer is some constant A/.L'A for each A molecule and ApoB for each B molecule, independent of micelle size. Within the simplifying assumptions made above, we can use (47) to write

+

n Figure 10. Plots of the y distribution XflP exp(-n/u) for different values of P. We have fixed Xo and u by means of the equations .fo%K,nP exp(-n/u) dn = 100 and .fo"nzXop exp(-n/u) dn = 2000. These equations are similar to the actual conservation equations.

The assumption that the B molecules form a continuous monolayer surrounding the core is important for it enables us to relate the number of A molecules in a given micelle, NA, to the number of B molecules associated with that same micelle. This allows us to replace (as is rigorously demonstrated in Appendix SA) the two-variable distribution function X[NA,NB]representing the mole fraction of the micelles composed of NA A molecules and N B B molecules that would be required to describe the general two-component micelle by a much simpler single-variable dis-

(48a)

As discussed in Appendix SC, the contributions to p o N Bwhich vary with the micelle radius (henceforth called "curvature corrections") that have been neglected in (48a) can be approximately represented by

Multicomponent Micelles and Microemulsions

The Journal of Physical Chemistry, Vol. 88, No. 26, 1984

c(~~(d-2)/(d-l) - no(d-2)/(d-l)) where t is a constant. Adding this additional energy into our previous expression for the chemical potential of a mixed micelle containing NBB molecules we obtain p0jvB- (Ngp'gF f r ( d ) N B d / d - ' p o A F )= +

+

+

{ N ? A / L o ~ T(d)NBd/d-'A/LOA € N B ( ~ - ' ) / ( ~ -N' )B; > lZo

}

-, N B < ~ ,

(4 8b1

where e'

= el - noApOB- T(d)nod/d-lApoA- cno(d-z)/(d-l)

is a constant analogous to the term A - n,,S used in our discussion of the single-component micelle. As e' is just the difference between the energy expended to form the minimum-sized micelle and the energy expended to put these no molecules and 7(d)ngdld-l A molecules into an already formed micelle, we shall call terms proportional to e' "formation energy" terms. These terms will play an important part in determining the micellar size distribution since the sign of determines whether it is favorable to allow micelles that have already been formed to grow (e > 0) or to create new micelles. In order to better understand the expression 48b, we note as pointed out by Missel et aL4 that we can represent the dependence of pori for a simple micelle of aggregation number n on n by means of a simple energy level diagram like that shown in Figure 8a, based upon the assumptions of eq 18. Here, we see clearly, in the case of the one-component SDS-like micelle, that all of the energy levels are equally spaced (for n > n,,). In the case discussed above, however, we see in Figure 8b that, even when c = 0, the energy level spacing is not constant. 3.3, Exact Treatment. Equipped with the above expression for poNB,one could determine X[&] by first using the condition pAF of chemical equilibrium (13) pNB= NBpBF T(~)NB~/("'),,A', being the actual chemical potential of free A and pugFbeing the actual chemical potential of free B. Then, recalling that in dilute solution

+

+ k T In XFA pFB = poBF + k T In XFB

pFA

=

poAF

and pNg

= p o N ~+ kT In X[NBI

it is possible to write lo^, + kT In ~ [ N B=] NB(p0BF k T In XFB)

+

+ T ( ~ ) N B ~ / " ' ( ~ ' A+~kT In XFA)

complicate the evaluation of the sums C N B N B ( d / d - ' ) X [ N B ] , C N 8 [ N B ] N B ,etc. Despite this, these sums can be performed analytically when d = 2 , but, even in this "simple" case it is exceedingly difficult to solve the resulting nonlinear equations, even with numerical techniques. This problem is discussed in greater detail in Appendix SB where an approximate method for evaluating XAFand XBFaccording to the above procedure is briefly described. In spite of these difficulties with the "exact approach", it is still useful. In particular, we see that, since ( d / d - 1 ) > 1 > ( d - 2 / d - l), the convergence of all the sums required by the conservation equations is governed by the factor (PA e~p(-@Ap~~)]~(~~B~'~-l so that if XFAexp(-PApoA) C 1 the sums are convergent while if this same quantity is greater than unity the sums are divergent. This indicates that, when f l Aexp(-@ApoA)> 1 our model of the mixed micellar system breaks down. Physically, when X A exp(-@Ap0A) > 1 it becomes favorable to form a new bulk phase consisting of pure A. We can see that this is the case by realizing that the A molecules can potentially exist in a bulk A phase with chemical potential pAm. On the other hand, the chemical potential of the A monomers in the aqueous phase is p°F kT In XFA. Now, the change in free energy associated with a transfer of an A molecule from the aqueous phase to the bulk phase is -poAF - kT In X A pAm. Transfer to the bulk phase will be energetically favorable when this latter quantity is 0. Since poAF- pAb,,?-ApoA we see that the threshold condition for phase separation is XFA= exp(@poA) or XA exp(-PApoA) = 1. This condition is quite analogous to that used by Ruckenstein12 to determine the solubilization curve although he does not explicitly refer to the mole fraction of A free in solution. Although one might at first assume that we could carry out similar arguments for the quantity XBexp(-PApoB), an examination of (49b) will reveal that all of the conservation equations' sums will converge whether or not XB exp(-@ApoB)is greater than unity (except when XAexp(-@ApoA) = 1). This does not at all contradict the logic behind the physical argument presented above for the A molecules since there is no physically meaningful bulk B phase analogous to the bulk A phase. 3.4. Variational Theory-Monodisperse Trial Functions. Unable to find an exact solution for X[NB], we now resort to the variational method which will yield, without much more work than in the SDS case, approximate solutions for X[NB] and a simple explanation of the basic qualitative and quantitative features of the mixed micellar system. As in the SDS problem, the first step in using the variational method is to write an expression for g, the Gibbs free energy of the system divided by the number of particles in it (most of which are solvent molecules). Following the discussion presented in the Introduction (and Appendix SD), and using the above expressions for poNB,we arrive at the result (for NB L no):

+

+

+

X F A - XFA) k T ( f l B In XFB - P B ) + ~ T C ( X [ N BIn]X[NBI - ~ [ N B ]+) C X [ N B ] ( ~+'

g = kT(XFAIn

so that

6529

NB

+

+

NB

€ N B ( ~ - ~ ) / ( ~C - 'X ) )[ N B ](NBA/.L~B T(d)NBd/"'AyoA) NB

+ XApoAF

XBp0BF

+ (50a)

This expression can be greatly simplified by using the conservation eq 45 and 46 to replace the sums &.,X[NB]NB and E N E X [NB]7(d)NBdid-' by (XB - XBF) and (x, - xAF), respectively:

+

X A-~X A ~ ) kT(XBFIn X B -~X B ~+) ~ T C ( X [ N BIn]X[NBI -X[NBI) -k CX[NB](C'+

g = kT(XAFIn NB

NB

+

tNB(d-2)l(d-')) (XB - XBF)ApoB (49b)

In order to make use of this exact distribution function, it is first necessary to determine the unknown free mole fractions XBFand XAFby substituting (49b) into the conservation eq 45 and 46 and then solve the resulting nonlinear simultaneous equations. Unfortunately, this is very difficult because powers of NB other than the first occur in exponents in the expression for X[&] and

+ (XA - XAF)ApoA+

X B P ' B+ ~ XAp0AF (50b)

We now replace x [ N B ] by the variational trial function and attempt to minimize g with respect to XAFand XBFas well as any parameters in the trial distribution whose value is not completely determined by the conservation equations. The simplest possible trial function is the S function: xv[XO,ri,NB1

= x06N~3fi

6530

Stecker and Benedek


and so we begin our analysis with it. Experimentally,6s7 it is known in the case of microemulsions that the actual size distribution is quite monodisperse, having6 A R / R < 0.1 where R is the micelle radius and AR is the rootmean-square variance in the micelle radius. Thus, in this case we expect that the 6 function trial distribution will yield results much closer to reality than it did in the case of the simple micelles. Substituting the trial distribution into the conservation equations in place of x [ N B ] we obtain

Gamma (?=-0.5)1 Exponential-,

?a 0.8 La

x

0.6 0.4 0.2

or

0

0

(53) XO = [(XB - XBF)d/(XA- XAF)d-l]T(d)d-l

(54)

Notice that, in contrast to what we found in the SDS problem where there was only one conservation equation, the existence of two conservation equations is sufficient to determine both A and Xo in terms of XA, XB, XAF,and XBF. We now proceed to determine XAFand XBFby minimizing the free energy per particle g. With the trial function described above, g takes the form (for fi > no)

+

8

16

24

32

40

48

R(B)

Figure 12. Plots of XAFexp(-@AwoA) as a function of i? (in angstroms) for different trial functions. The following parameter values were se= 1.25 X c/kT = 0, ef/kT lected: aa = 8 A,ua = 60 A*,d = 2, = 20.

x,

18

'7

I

I

I

I

14

+

g kT(XAFIn X A -~X A ~ ) kT(XBFIn X B -~X B ~ ) kT(X0 In Xo - Xo) + (XA - XAF)ApoA (XB - XBF)Ap'B + xO(E~ ~fi("~)/("')) XBpoBF XApoAF (55)

+

+

+

+

Eliminating fi and Xo in favor of XA,XAF,XB, and XBFby using (53) and (54), we obtain g = kT(XAFIn X A -~X A ~4-) kT(XBFIn X B -~X B ~+) (XA - XAF)ApoA+ (XB - XBF)ApoB+

4l 2

Figure 13. Plots of XBFexp(-BAwoB) as a function of a (in angstroms) for different trial functions. Same parameter values as in Figure 12.

X B W ' B+~X A P ' A ~( 5 6 )

of fi or alternately as functions of R the mean micellar radius where66

As the only variables in this equation are XAFand XBF,we now require simply dg/dxAFIXBF= 0

(57a)

This gives, on using eq 53

Analogously for X B ~ : ag/axBFIXAF=

o

This relationship between R and the mole fractions of A and B is plotted in Figure 11 (see supplementary material) for the case d = 2. Figure 12 shows X exp(-@ApoA)vs. R in the case where d = 2, a. = 8 A2, vo = 6 0 a 2 ,X A = 1.25 X e'/kT = 20, and c/kT = 0 using the 6 function as well as other variational trial functions. Figure 13 shows XBFexp(-@ApoB)vs. R for the same choice of parameters again using both the 6 function distribution and other trial functions. At this stage, we shall not undertake an explicit solution for XAFand X B in~terms of X A and XB.44 Rather, we shall first

(58a)

Equations 57b, 57c, and 58b containing intermediate steps in the calculation are to be found in the supplementary material.

Equations 57d and 58c represent two implicit simultaneous equations for XAFand X B F . Here, we must remember that, through eq 53, fi is a function of XAFand XBFas well as X A and XB. At this point, it is convenient to plot XAFand XBFas functions

(44) Note that it is, in principle, possible to solve the highly nonlinear eq 53, 57d, and 58c simultaneously in order to obtain an explicit solution for either XAFexp(-bAroA) or XBFexp(-pAwo ) However, we note that under most experimental circumstances X A >> X> and so we can find XAFand X,F as functions of ii (or equivalently R the mean micellar radius). In the limit XB>> XBF(which is not always satisfied experimentally) we can write i~ as an explicit function of X A and X,:

-

so that we can obtain XAFand XBFdirectly once X A and X are known without having to solve any equations. However, in the case X t X , we have no choice but to attempt a numerical solution of these equations although most of the interesting physics can be elucidated without recourse to such exact numerical calculations.

The Journal of Physical Chemistry, Vol. 88, No. 26, 1984 6531

Multicomponent Micelles and Microemulsions examine the manner in which XAFand XBFdepend on it when it is large. This shall provide considerable insight into the physical factors which determine X , and shall form the framework for further discussions. In this regard, we note that eq 57d and 58c predict that as n or, equivalently, the mean micellar radius becomes very large, both XAFeXp(-pAp”,) and XBFeXp(-@ApoB)approach unity. Physically, this result reflects the fact that, as the average micelle size becomes large, the mixed micelles act essentially as a new bulk phase. Consequently, since we have defined Ago, and ApoB as the free energy changes associated with the transfer of a single A or B molecule, respectively, from an aqueous phase to an infinitely large micelle (neglecting the entropy of mixing), we must have Ape, - kT In XAF* = 0 Ap0B -

-

kT In X B ~ = *0

-+

X A ~ eXp(-pApoA) * = 1 X B ~ eXp(-pApo,) * = 1

where XAF*and XBF*are the values of XAFand XBF,respectively, when R a. By studying the manner in which XAFexp(-pApoA) and XBF exp(-@Apo,) approach unity and by further considering the relationship between the bulk phase and the mixed micelles, we shall now be able to elucidate the origins of the various terms that occur in the exponents of the fundamental eq 57d and 58c. It is clear, by inspection of these equations, that XAFexp(-pApoA) approaches unity much more quickly than XBFexp(-@ApoB)a; R a. This behavior can be understood by first relating X , exp(-PApoA) to the effective chemical potential difference between an A molecule in a micellar phase and an A molecular in a bulk phase and then determining the dependence of this chemical potential as the actual chemical potential of on it. Defining pA,mic(R,Xo) an A molecule in a micellar phase composed of monodisperse micelles of radius R at a concentration Xo, we see that, in equilibrium, we must have

-

fiA,rnic(RJO)

-

+ kT In X A ~

=

(60)

In particular, for micelles of infinite size (and consequently X o 0) p A , , , i c ( ~ , O )= p o A F

+ kT In XAF*

(61)

so that if we let O(R,Xo)be the difference in the chemical potential of A molecules in these states we can write

where we have made use of the fact that XAF*= exp(PApoA). This equation demonstrates that the quantity k T In [XAFexp(-@ApoA)] can be interpreted as just the change in free energy accompanying the transfer of one A molecule from the interior of any micelle in a phase composed of micelles of radius R to a micelle of infinite radius. Observe further that the change in the entropy of mixing of a system of monodisperse micelles upon the addition of one A molecule is simply

a ax0 - k(Xo In Xo - Xo) = k- In Xo =

ax,

ax,

(d kfid/d-l

44

In xo

which is 1 / T times the first contribution to O(R,Xo) and is easily recognized as the first term in the exponent of (57d). Furthermore, we note that the ”formation energy” term in the expression for the Gibbs free energy (55) associates a constant energy with the formation of each micelle. Thus, the change in the total “formation energy” upon increasing the number of A molecules is the micellar phase by one is given by

which is the second term in the exponent on the right-hand side of (57d). A similar interpretation applies to the terms in the exponent in eq 58c. The only difference in the latter case is that In Xo - Xo) the terms in this exponent correspond to (a/aXB)k(Xo and (a/dXB)(e’Xo) both of which are proportional to aXo/aXB. However, since

it follows at once that XAFexp(-pApoA) approaches unity more quickly than does XBFeXp(-@ApoB)as we have already observed. In a more qualitative sense, the slower approach of XBFexp(@ApoB)to unity is a result of the fact that, when micelles are large, there are many more A molecules in the micellar phase than B molecules and hence adding one A molecule to this phase has proportionately less effect on the total number of micelles and consequently on both the entropy of mixing and “formation energy” than adding one B molecule. In this light, it is interesting to note that, making use of eq 57d and 58c, we can deduce the relation

where the quantity k(A) is a constant when either e = 0 or d = 2. Equation 63 is quite similar to the familiar solubility product (especially under those circumstances where k(A) is actually a constant) and so suggests the interesting interpretation that micelle formation can be thought of as the reaction of one A molecule with ( d - l ) / [ d f i ’ / ( d - l ) ~ (Bd )molecules ] to form an insoluble product, the micelle. This expression provides further expression of the fact that XBFexp(-@ApoB)must approach unity more slowly than XAFexp(-@ApoA)for large micelle sizes, because of the power to which XBFis raised in the reciprocal relationship of eq 63. 3.5. Phase Diagram ( d = 2 ) . Now that we have some qualitative understanding of the physical interpretation of the terms in eq 57d and 58c which specify XAFexp(-PApoA) and XBF exp(-@ApoB),respectively, as functions of the micellar size, we can use these equations to determine the phase diagram for our three-component (A-B-water) system. One simple way of depicting this phase diagram is that used by Mazer et a1.5 in which, as shown in Figure 14a, the total mole fraction of A added to the system is plotted on the x axis, the total mole fraction of B is plotted on they axis, and the regions over which the various phases can exist are labeled I, 11, and 111. With this figure as a guide, we shall now consider the transition between regions I1 and 111. Imagine first that we begin in region I1 where only mixed micelles are present and add more A to the solution at constant X B , a process which corresponds to following a horizontal line in the figure. As we travel along this path, each of the micelles in our system increases in size while their number decreases in accord with the geometrical relations 53 and 54. However, this monotonic increase in size cannot continue indefinitely. As the size of the which is positive when micelles increases, the quantity O(R,X0), the micelles are small, decreases until R reaches the point where O(8,Xo)= 0 or XAFeXp(-pApoA) = 1 . According to the definition of O(R,Xo) this is the point at which it becomes energetically favorable to transfer A molecules from the micellar phase to a bulk A phase. We also know from our considerations of the exact micellar distribution function that all of the moments of this distribution diverge when XAF exp(-@ApoA) > 1 . Thus, we conclude that at the point O(R,Xo) = 0 or XAFexp(-@ApoA) = 1 a phase separation must occur.45 Once this point is reached, (45) Although we have said that it was unphysical to have XAFexp(-

> 1, our equations nevertheless predict that this will occur. Our solution is to simply disregard any results obtained in this region as unphysical. The situation here is reminiscent of that which occurs in the analysis of the nonideal gas. In that case, because the free energy of the system is calculated approximately, unphysical results are obtained in regions where phase separations occur yet these results, when interpreted properly, can yield important information about the phase separation. (3AfioA)


any further addition of A to the system cannot increase the micellar size or else X A Fexp(-PApoA) will become greater than unity. Hence, the added A must all go into the condensed phase rather than into the micelles. To obtain a more detailed physical interpretation of this phase separation, remember that (when e = 0 or d = 2) O(R,Xo)is made up of two terms: an entropic term and a positive (unfavorable) “formation energy”. The former is negative while the latter is positive. Thus, it may be said that the entropy of mixing of the micelles resists the formation of large micelles while the “formation energy” term (when e > 0) favors the reduction in the number of micelles and an increase in micellar size. As X A increases and the number of micelles decreases, the negative entropic term increases in magnitude and the positive “formation energy” decreases. Consequently (e’ > 0) at some point the sum of these two terms changes sign indicating that the balance of these factors has changed so that the entropic resistance to the accumulation of A molecules in micelles has completely overcome the energetic advantage of their dispersion. The value of X o at which this phase separation takes place is given by Xo, = exp( -

&)

( d = 2 or e = 0)

(64a)

and the corresponding value of ii which we call A, is given by

Using now the geometrical eq 53 relating A to the total amount of A and B present in our system, we see that (when X A >> XAF and X B >> X B F )the equation of the boundary between regions I1 and 111 of Figure 14a is given by

In likewise manner, we may also examine the bounds between regions I and 11. Considering X A as fixed and increasing XBso that we travel along a vertical line in Figure 14a, we see from the geometrical relations of eq 53 and 54 that the micelles decrease in size and increase monotonically in number. Two factors act to halt this decline in micelle size. The first is that a priori the micelles cannot have a size smaller than that of the minimum micelle characterized by A = no. Any effort to increase X B after the point ri = no is reached cannot produce any further reduction in micellar size. The relation between XB and Xo at this point is found from eq 53 to be

Stecker and Benedek @Aho,) which directly determine the micelle distribution function and the phase behavior of the A-B-water system. Thus, it is of interest to plot the phases accessible to our system as functions of X A Fexp(-PApoA) and XBFexp(-PApoB). We may construct such a plot using eq 63 (with t = 0 for simplicity) to predict the relationship between XAFexp(-PApoA) and XBF eXp(-(3ApoB) given A. The phase boundaries in this plot, shown in Figure 14b, ~ =1 are as follows. First, there is a bound at X A exp(-(3ApoA) and the corresponding bound at XBFexp(-@ApoB)= l 4 associated with the formation of a bulk phase.47 The other bounds occur at XA = 1, XB = 1, XBF = XB,cmc,and A = no. This latter boundary corresponds exactly to the line labeled R = 2.53 A in Figure 14b. All shaded regions beyond these boundaries are forbidden to a system of pure mixed micelles. Within the allowed region, we show possible states corresponding to fixed finite values of R rangin from 40 A near the toe of the allowed region to the value of 2.5 (which we have taken as the size of the minimum micelle) at the heel. As one example of the utility of this diagram, we now consider the behavior of X B Fas the region of coexistence with simple micelles is approached. Imagine beginning at the point labeled P in Figure 14b and adding more B to our system at constant X A so that we follow the curve PQ. At Q the radius of the micelles has reached the minimum value (assuming that it will reach this before reaching the cmc for simple micelle formation) and so any added B must go into the aqueous phase increasing XBFrapidly. Thus, our system follows the line QR until the cmc for the formation of simple micelles of B is reached. In the above discussion of the phase diagram, we have in fact made some implicit assumptions regarding the nature of the bulk phases which can exist in equilibrium with our dispersed micellar phases. The first assumption relates to the character of the “bulk” A phase. In the case that A is lecithin, as in the bile salt-lecithin mixed micelle, the bulk phase was thought of as a flat, multilamellar bilayer structure. However, as discussed by I~raelachvilli,~~ the chemical potential of lecithin molecules in a bilayer (or multilamellar) vesicle is less than that of similar lecithin molecules in an infinite bilayer. Therefore, the so-called bulk A phase actually should consist of lecithin molecules dispersed in large vesicles. This consideration will not substantively alter the location of the phase boundaries discussed above since the difference between the chemical potentials of lecithin molecules in the bilayer and vesicle phases is small13because of the large size of the vesicles. The second assumption that we have implicitly made is that there is, in fact, no separate bulk phase of B molecules. This is a natural assumption since B serves as a detergent for the A molecules. If the B molecules could form a bulk phase, then mixed micelles would not be stable under most experimental conditions. This is seen as follows. The energy go of a bulk phase of A coexisting with a presumptive separate bulk phase of B is by definition of Ap0A and ApoB just

i

or using (54) we find If one then uses eq 50 to evaluate g using our variational method, one obtains the results shown in Figure 15. This figure shows that g - go > 0 for a wide range of values of xA/xB. This (when X A >> XAFand XB >> XBF). There is a second physical inequality implies that the micellar phase will be unstable toward process which can determine the small-size phase boundary for the formation of separate bulk A and bulk B phases. Indeed, the the micellar system. Note that as X B increases X B Fwill also existence of the dispersed micellar phase over a large region of increase. If a critical micelle concentration XBFCmc exists for the the phase diagram is a consequence of the absence of the separate formation of simple micelles of B then, once X B Fexceeds X B ~ ~ bulk ~ ~B phase. , This is the case in the lecithin-bile salt system where simple micelles will coexist with mixed micelles. This condition the bile salt molecules can form simple micelles but no bulk phase. exp(-(3ApoB) = 7.5 as the dotted line is drawn in the case XBFcmc 3.6. Phase Behavior ( d = 3). Having discussed the case of in Figure 14a. From this we see that for the particular choice disklike micelles (d = 2) in detail, one finds it is now interesting of parameters used in that figure we will reach the limit imposed to consider the more complex case of spherical micelles. This by the cmc for the formation of simple micelles before the minproblem corresponds to the case d = 3 illustrated in Figure 9. At imum sized micelle is reached. Of course, with different choices first, it might seem that changing d from 2 to 3 would have very of the parameters, the location of the phase boundary can be reversed. (46) It is clear from (63) that if d = 2 or e = 0 then XAF(exp(-PAgoA) Further insight into the phase behavior of our micellar system will be greater than unity whenever XBFexp(-@ApDB),< 1. can be obtained by realizing that, although X A and X B are the (47) The entirety of what we called region 111 in Figure 14a is in Figure experimentally controllable variables, from the theoretical view14b contained in the vertical line XAFexp(-@ApaA) = 1 since only at this value of XAFcan any system be in equilibrium with a bulk A phase. point (e.g., 49b) it is really X A Fexp(-PApoA) and X B exp(~



- Mixed Micelles

4

0

8

12 ~

~

20

16 ~

1

0

24

4

C

A

E

n--, Mixed

(C) Figure 14. (a) Illustration of the different phases that can exist in our mixed micellar system ( d = 2) as a function of X A and XB. In region I mixed micelles and simple B micelles coexist, while in region I1 there are only mixed micelles. Region I11 consists of mixed micelles and a coexisting bulk phase. a. = 8 A,uo = 60 A2,d = 2, c’/kT = 12, e/kT = 0, no = 2, XAFexp(-PApoB) = 7.5. The boundary between regions I1 and I11 is given by (64c). (b) The different phases present in the system are shown as functions of XAFexp(-pApoA) and XBFexp(-pApoB). Only in the unshaded area is our system made purely of mixed micelles. Of the lines describing micelles of different radii, that labeled R = 2.53 A corresponds to N B = no. Parameter values are the same as in (a). (c) The five re ions of the (t’/kT, e l k 0 parameter space over which the phase behavior of the spherical micelles is qualitatively different. uo = 1000 A’,a. = 50 d = 3, and X A = 1.25 X lo-’. (d) Summary of the behavior of spherical micelles in each of the regions illustrated in part c. For regions A, B, and (2, we plot h(ri) which determines the stability of the system. For regions D and E, we plot XAFexp(-PApoA) vs. ri in order to determine when bulk phases can exist.

,f2,

little effect on our system. In fact, we shall see that this alteration dramatically changes the phase behavior of the mixed micelles. This is a result of the fact that the curvature energy (e) terms, and the “formation energy“ (e’) terms in the expression for the free energy (50) are different functions of the micelle: radius. In the special case d = 2 both of these terms depend upon the micellar radius in the same way and hence we were able to set e = 0 without loss of generality. On the other hand, in the case or = 3 the e’ term depends on the micellar radius in quite a different way than does the e term as we can see from (50). Moreover, the absolute magnitude and sign of both e and e’ are strong functions of the specific chemical identity of the A and B molecules. Thus, it follows that chemically different systems of spherical micelles will display a much greater variety of behavior than cheinically different systems forming disklike micelles. In order to describe the general phase behavior of systems forming spherical micelles, we need to specify the phases that are present for every value of x,, XB,e, and 6‘. We shall not endeavor to present a detailed analysis of such a four-dimensional phase diagram at this time. Instead, we will provide an alternate method

of classifying the phase diagrams that can be produced. In this analysis, we construct a parameter space with axes d / k T and e/kT as shown in Figure 14c and consider the behavior of the mixed micellar system corresponding to each point in the figure when X B is varied and X A is held constant. Equivalently and more conveniently, we can consider the effect of changing fi while X A is held constant ( f i = [1/7(d)] [(XA- X A F ) / ( X-~X B ~ ) ] ( ~ AS -’). we have indicated in Figure 14c, there are various regions in the (e’/kT, e / k T ) parameter space in which the phase behavior of the micelles is qualitatively different. In order to understand the factors differentiating the phase behavior in these different regions, we shall now discuss the stability of our mixed micellar system. It is first important t o show that, within the limitation of the 6 function trial distribution, our solutions (57d) and (58c) for XAF and XBFrepresent a true minimum for g rather than an inflection point or a maximum. In the limit where X A XAF). (55) These results are obtained under the assumption that XBFis dependent only on the mean micelle radius and not on the absolute values of the solute concentrations. Thus, these ex rimental results are necessarily approximate. (56) Since we can write XBrexp[-@A~o~ll= [XB' e x p [ - P A ~ ~ ~exp[-Pll (ApoB'- ApoB)]and we know that XBFexp(-PApoB) is always greater than unity (d = 2) when XAFexp(-PApoA) < 1 (6 function distribution), it is clear that XBFe-@*@'B will always be greater than unity as long as ApoB1- ApoB S 0. This latter inequality is just the condition that it be energetically more favorable for the bile salt molecules to be inside the micelle than on its surface. However, in order for our assumption that (at constant XA)a constant amount of bile salt is to be found inside the micelles regardless of the value of X , be valid, it must be more favorable for the bile salt molecules to be inside the micelle than on its rim (as long as the concentration of B within the micelle remains less than or equal to l/(a+ 1)). Thus, we can safely assume that XBFexp(-@ApoB)will always be greater than unity. (57) The theory of Ruckenstein'* predicts a maximum micelle radius that is independent of the temperature. (58) Schurtenberger, P.; Mazer, N. A.; Kanzig, W.; Preisig, R. In "Proceedings of the 4th International Symposium on Surfactants in Solution, June 1982, Lund, Sweden; Mittal, K. L.; Lindman, B., Eds.; Plenum Press: New York, 1983. (59) Only when the micelles are entirely specified by a single parameter can we rigorously use the term micelle size distribution. In general, when more than one parameter is needed to describe a micelle, we should use the term micellar distribution. (60) Of course, if the Gibbs free energy has multiple local minima, it is possible that our variational technique could lead us to consider an entirely wrong trial function (it., we may find a local minimum in g which is not near the global minimum). We do not believe that this occurs in the present calculations. First, it certainly does not occur in the case of SDS where our results reproduce the exact results very nicely. Second, only families of trial functions that were physically reasonable were considered in the variational process. Third, trial functions with very different shapes yield very similar results. Fourth, even in the case of the two-component micelles where we have no exact results with which to compare, approximate treatments of the problem based on the method of multiple chemical equilibria in Appendix SB confirm our results. Fifth, all of the results obtained are physically reasonable. Sixth, good agreement between the theoretical predictions and the experimental results are obtained. (61) Tanford has stated that there is no universally accepted definition of the term micelle. In view of this, we summarize our understanding of the term micelle as it will be used in this paper. The word micelle, derived from the Latin word mica meaning crumb or morsel, was originally used by Nageli to refer to macromolecular yet microscopic particles. Later, McBain used the term to refer to any stable aggregate of amphiphilic molecules. Schulman used the term more generally to refer also to structures composed of a surfactant and an otherwise insoluble component (the "oleopathic hydromicelle"). This latter structure is now commonly referred to as either a microemulsion or a swollen micelle. For the sake of simplicity, we shall use the term micelle to describe any thermodynamically stable aggregate of finite size whether it is composed of one or more than one component. We, however, exclude liquid crystalline structures such as the vesicle from this definition. Also, we shall refer to the solubilizate surfactant systems as both microemulsions and mixed micelles (rather than use the term swollen micelle). This is consistent with the terminology of Mazer et aL5 In addition, we will use the term simple micelle to refer to a micelle composed of only a single component. (62) Shankland, W. Chem. Phys. Lipids 1970, 4, 109. (63) From Table IVb we see that, within the approximation of the y distribution, the mean micellar radius increases as P increases (at constant X A and XB). This implies that the mean value of N B also increases with P. However, the conservation equations require that

-

(cX[NBI)(NB) = XB -XBF so that the number of micelles ( C X [ N B ] must ) decrease as P is increased.

Since P is inversely related to polydispersity, we see that as the polydispersity increases the number of micelles must increase. (64) Bancroft, W. D.; Tucker, C. W. J . Phys. Chem. 1927, 31, 1681. (65) In a dilute solution, to lowest order in the [X+] we can regard the entropy of mixing of the solvent as a constant. Thus, we need not consider it explicitly. (66) R is rigorously defined as follows:

r(d/z)U,

'=

cX[NB]NB1'(d-')

(x)

cX[NB] NB

I

1

-..-..

2 + 1.6-

Gamma ( p = 20)

1 CP v

I

Delta function ............... Gamma(p=2)

+ 2.4Y

-

I

---- Gamma(p=-0.5) -.-. Exponential ___-__ Gaussian -- Rectangular -

3.2 -

-

-

-

0.8

0 0

0.4

0.8

1.2

1.6

2.0

2.4

Figure 15. Plots of (g - go)/(XAkT)as a function of XA/XB as determined with different trial functions. go = XA(A/,LOA+ PA^) + XB(AP'B pBF),a. = 8 A,U O = 60 A2,&lkT = 20, c/kT e/kt = 0 , d = 2, XA = 1.25 x 10-3.

+

unity for all ft > no and so, as indicated in (E) of Figure 14d, minimum-sized mixed micelles will coexist with a bulk phase for all values of XB. Having demonstrated the dramatic effect that changing the values of the parameter c has on the phase behavior of a system of spherical mixed micelles, one finds it is useful to pause briefly to consider the physical factors which determine the sign of t. From Appendix SC we see that when t > 0 it is energetically unfavorable for the boundary between the A molecules and the solvent to be curved. Stated differently, when c > 0, the B molecules tend to orient themselves as parallel as possible to one another. Physically, a number of factors work to make t positive. Among these are the energy cost of exposing more of the hydrophobic surface of the B molecules to the aqueous phase as the curvature of the micelles is increased and the energetic disadvantage of bending the monolayer of the B molecules. On the other hand, factors such as the repulsion between the hydrophilic ends of the B molecules (be they the result of electrostatic or steric interactions) favor the formation of smaller highly curved micelles and tend to make t more negative. Let us next consider why t < make E leads to coexistence of the micelles with a bulk phase. Consider a situation in which e is negative and X B is continuously decreased. In the absence of a bulk A phase, the micelles must grow in size in accordance with the geometrical restriction (53). Such growth is energetically disadvantageous because it tends to decrease the angle between neighboring B molecules and consequently the distance between the head groups. As a result, the micelles, in effect, repel A molecules from their interiors into the bulk phase as X B decreases. In this way, the micelles avoid growth by reducing the amount of A that is contained within the micellar phase. We may thus understand the observation that bulk phases tend to form when c < 0. 3.7. Polydispersity. Having discussed above the consequences of taking the simple 6 function as our variational trial distribution, we shall now present the results which are obtained when more general forms of trial function are employed. Specifically, we shall consider two classes of polydisperse trial function. The first is the set of two-parameter distributions discussed earlier in our treatment of the single-component micelle. The second class that we will consider consists of a single trial function having three (67) In assuming that no and uo are constant independent of the micelle radius, we have implicitly assured that all micellar components are incompressible. (68) Generalizing the variational approach described in this paper, we may in addition determine the shape of the micelles formed. To do this it is necessary to introduce some variable which affects the shape of the micelles. The simplest of such variables is the scaling dimensionality d of the micelle. Although fractional dimensionalities are not explicitly discussed here, the use of arbitrary dimensionality adds flexibility to the theory and facilitates its use.

6536 The Journal of Physical Chemistry, Vol. 88, No. 26, 1984 parameters, the y distribution:

Stecker and Benedek 1.5

I

I

I

I

I

0.4

sco/e+

This distribution has the advantage that, in the case of twocomponent micelles where there are two conservation equations rather than one, it allows us a greater flexibility in selecting the precise shape of the distribution. Thus, with this class of functions, it should be possible to more closely approach the rigorous micelle size distribution than with the two parameter trial functions. That the y distribution can assume many different shapes is illustrated in Figure 10. In this figure, we plot xv[X0,u,P,NB]vs. NB for values of P ranging from 0 to 60. The parameters Xo and u are determined for each value of P in such a way that the first and second moments of the distribution are set equal to 100 and 2000, respectively. (These conditions are similar in form to the conservation equations for A and B in the problem of mixed micelles.) Clearly, the parameter P controls the polydispersity of the distribution function, even when the parameters Xo and u are fixed by the conservation conditions. In the case of the first type of trial function, the two parameters of the distribution are fixed as functions of XA,X B ,XAF,and X B F by the conservation equations. XAFand XBFin turn are determined by the conditions

In this way, we obtain the best values for the height and width of the distribution consistent with the restricted choice of trial function. In the case of the y distribution, the values of Xo and u are determined as functions of XA, XB, XAF,XBF,and P by the two conservation equations. We first obtain XAFand X B Ffor arbitrary P from dg/axAFl&Pp= ag/axBFIXAfp = 0. The best value of P is then finally obtained by finding the value of P which gives the minimum value of g. At this point, there is no need to go into the details of the variational calculation with any of the polydisperse trial function^.^' Suffice it to say that the calculation can be performed readily for the rectangular, Gaussian, and exponential functions as well as with the y distribution defined above. The results of these calculations are shown in Tables 11-IV. Line l of table I1 shows the general form of the relationship between XAFeXp(-fiAjLoA) and R obtained with the 6 function distribution. For comparison, the analogous relationship between XAFexp(-/3ApoA) and R as calculated with any of the polydisperse trial functions is illustrated in the second line of the table. This relationship has the same form for any polydisperse trial function. Only the values of the coefficients dl), d2),and d3)are dependent on the exact choice of distribution. The form for these coefficients as a function of the dimensionality d appropriate to each of the different trial distributions is shown below the formulae in Table I1 and their numerical value is tabulated for both d = 2 and d = 3. In addition, in the case of the y distribution, we show the values of these coefficients for various values of the parameter P. Table I11 shows in similar format the general relationship between XBFexp(1 , To @A/L'B)and R in terms of the coefficients ~ ( l ) ~, ( ~ and supplement these abstract formulae, we have plotted XAFexp(-PA~L'A) and XBFexp(-fiAjLoB) vs. R in Figures 12 and 13 using each of the trial functions. The general expression for g (in the limit X A >> XAFand X B >> XBFis illustrated in part A of Table IV. The first line in this figure shows the expression for the g that is obtained when one employs the 6 function trial distribution. The relationship between g and R as calculated with any of the polydisperse trial functions is contained in the second line of Table IVa. As in Tables I1 and 111, when presenting the results obtained with the y distribution, the expressions for g are calculated as general functions of P and not only for that value of P which minimizes the Gibbs free energy. From the expressions for g contained in Table IVa, it is now possible to determine which of the trial functions is associated with the lowest value of g. Consider Figure 15 where we plot ( g - g o ) / ( X A k T vs. ) xA/xB for fixed X A and the parameter values shown in the figure legend. Here, we see that, although all of

......................................

.................................... c scale

0.7 - '

**..a

I

I

I

I

I

I

I

I

I

I

I

0

(68) In the case of a y distribution, however, we can show that (68) becomes a very simple function of P S 3

Thus, once we know P as a function of XA/& it is straightforward to plot V a s a function of xA/xB. Such a plot is contained in Figure 17, where we see that Vis rather small and increases slowly as the mean micellar radius increases. Previous analyses of this problem often start with the assumption that the micelles are monodisperse. This calculation provides the theoretical justification as well as an understanding of the limitations of this widely used approximation. As we might expect, the width of the micelle size distribution is determined by a balance of energetic and entropic effects. We turn first to the effects of the entropy of mixing of the micelles. It is intuitively clear that these effects must favor the broadening of the micelle size distribution. This will allow for the presence of more distinguishable micelles and will thus increase the entropy of the system. On the other hand, the effects (in the case d = 2 or E = 0) of the formation energy terms (when 'E > 0) is to increase the Gibbs free energy by an amount proportional to the



Y 1 I

-

Y 1

4

--.

g h

F

?

N

k n 8

F

i7

9

gS

I S

P

ij.

a

I

3

I N

f.

II

-c

3 v

a

-

Stecker and Benedek


6538

H

111

-

5

II m

ll

a c

II h

i

+

-

h

l

h

P

-1c YI



TABLE I V (A) Expressiona for the Cibbs Free Energy as a &unctionof a in the Limits XA>> XAFand XB>> XBFand (B) Expression for the Mean Micellar Radius as a Functionbof XAand XBfor Each of the Trial Functions

Della Function 2d

Rectangular

cp=

Gamma

The first line contains the expression obtained with the 8 function trial distribution while the second line contains the result appropriate to any polydisperse trial function. The coefficients d'),d2),and d3)are the same as in Table 11. is a numerical coefficient whose value changes with the choice of trial function.

I

-T

Figure 17. Plot of the variance V of the micelle size distribution as a The calculation was performed with the y distribufunction of XA/Xs. tion. Only integral values of P were considered during the minimization process. The error bars represent the estimated error associated with this calculational simplification. Parameters are the same as in Figure 16.

total number of midelles in the system. Thus, the energetic effects tend to oppose any changes in the distribution function which would tend to increase the total number of micelles. In particular, CN$U[NB] increases as the polydispersity of the micellar system increases.63 For this reason the formation energy terms (when e' > 0) act to make the micelle size distribution more monodisperse. It is the balance between these two competing effects which sets the variance a t the values calculated above. To reiterate, in this section we have employed the variational method to discuss the basic physical principles governing the formation of two-component mixed micelles with the shape of a d-dimensional sphere. Although no exact solutions to this problem exist, we were able to produce approximate expressions for the concentrations of both micelle components in the aqueous phase. We were also able to write an expression for the Gibbs free energy of such a system, study the conditions under which such a system is stable, and discuss its phase behavior. By considering a number of variational trial functions, it was not difficult to determine the best approximation to the true micelle distribution function and

6540


TABLE V Comparison of the Values of (g - g , ) / ( X A k T )Obtained with Various Trial Functions Listed in Order of Decreasing Free Energy" trial function (E- d l W n k T ) P = -0.5 P=O

2.25 1.32 0.940 0.835 0.817 0.744 0.715 0.710 0.694 0.683 0.6754 0.6751

Gaussian 6 function P = 2

rectangular P = 5 P = 99 P = 60 P = 10 P = 20 P = 17

Stecker and Benedek including the redistribution of B molecules into the interior of the micelles, we are led to the following choice for g:

+

+

g = kT(XAF In XAF- XAF) kT(XBFIn XBF - XBF) ~ T ~ ( X [ NInB~ ][ N B- ]~ [ N B + ] )( X A- X A ~ ) A P ' A+ NB

1 1 (XB - XBF- -(XA - XAF) Ap'B -(XA - XAF)ApoB1 a > f f x x [ N B ] ( & tNB(d-Z)/(d-l))XBIoBF X#'AF (74)

+

+ +

+

+

NE

"The values of P refer to the parameter in the y distribution. P = 0 is just the exponential trial function. to show how the width of this distribution depends on the composition of the system.

4. Variational Theory of Mixed Disks 4.1. Basic Theory. In the previous section of this paper we considered model micelles in which A molecules are confined to the interiors of the micelles and the B molecules are confined exclusively to their surfaces. However, in many micellar systems, the A and B molecules mix either within the micelle interior or on its surface. In particular, in the case of the lecithin-bile salt system, Mazer et aL5 have shown that a more accurate model is the so-called mixed disk model. In this model, the A molecules are still present only in the interiors of the micelles although the B molecules are allowed to be in the interior of a micelle as well as on its surface. In order to simplify the analysis of this problem, Mazer et al. assumed5 that the mole fraction of B contained within the micelles is proportional to the number of A molecules in the interior of the micelles. If we call the constant of proportionality 1 / a we can write the conservation equation for B molecules as

In this expression, ApOB is the change in the standard part of the free energy that accompanies the transfer of a B molecule from the aqueous phase to the surface of an infinite sized micelle and ApoB1is the change in chemical potential of a B molecule, including entropy of mixing effects, transferred from the aqueous phase to the interior of a bilayer ( d = 2) or a bulk phase ( d = 3 ) containing A and B molecules in the ratio l/a. In order to make the above expression for g as similar as possible to that of (50b), we define 1 Aji'A = AM'A + -(ApoB' - Ap'B) a

(75)

so that we can rewrite (74) as

g = kT(XAFIn X A -~X A ~+) k T ( x In ~ X~ B -~X B ~+) k T C ( X [ N , ] In ~ [ N B- x][ N B I ) + Cx[NBl(e' + NB

+

+

NB

+

C N B ( ~ - ~ ) ~ ( ~(XA - ' )-) XAF)AfioA (XB - XBF)ApoB XBp'BF + XAp'AF (76) 4.2. Theoretical Results and Comparison with Experiment. Now that we have formulated the conservation equations and developed an expression for the Gibbs free energy, we can apply the variational method as in the previous section of this paper. We begin first by discussing the results that can be obtained with the 6 function trial distribution: Xv[xO,%NBI = XO~NB,A When this ansatz is substituted into the conservation eq 70b and 71b, we obtain

where ( X B- X B F ) (1 / a ) ( X A - X A F )is the mole fraction of B molecules on the surfaces of micelles and x [ N B ] is the mole fraction of micelles having NB B molecules on their ~ u r f a c e . ' ~ Letting uB be the effective volume of each B molecule in the interior of a micelle, we see that the conservation equation for A molecules becomes

where ( l / a ) v B ( X A- X A F )is the total volume taken up by B molecules in the micelle interiors. In order to simplify matters, we now define

In terms of these new quantities, we can write the conservation equations for B and A in the form c x [ N B ] N B= 3

B

- TBF

(77)

These expressions are quite similar to expressions 53, 54, and 59 which apply in the case 1 / a = 0. There is, however, one important ) as difference. The mean micellar size will (XB>> X B F diverge X A approaches axB(unless a phase boundary is reached before this point is attained) for at this point there are essentially no B molecules free to occupy sites on the surface of micelles. In Figure 18 we-show the dependence of R on XA/XB for a = 2 and the parameter values mentioned in the figure legends. Making use of the extensive similarities between the equations describing the mixed disk and those describing the simple twocomponent mixed micelles, one can easily see that the condition ag/dXBFIxAF= 0 implies that

(70b)

NE

x x [ N ~ ] T ( d ) N g ~ /= ( ~3-Al )- 3A

(71b)

NB

which are identical in form with the conservation eq 45 and 46. Our next task is to write an expression for the Gibbs free energy of this system. Following the arguments we made in the previous section with regard to the simple two-component micelle, and

Not surprisingly, this expression is the sarne as (58c) except that XA is replaced by XA, X B is replaced by X B ,etc. Analogously,


Multicomponent Micelles and Microemulsions I

m

m

XA/ Xe Figure 18. Plot of the mean micellar radius R as a function of XA/.XB in the mixed disk model when a = 2. According to the conservation equations, R would become infinite a t XA/XB = 2 when XB >> XB'. a0 = 8 A, u,, = 60 A2, uB = uo, and the 6 function distribution was used.

b 4.0

I

I

I

I

Data of Mazer et.al.

we can determine XAFexp(-pApoA) by requiring that 8g/8XAF(&F = 0, with the result

10

Mathematically, this result is much more complex than that for XBFexp(-pAp'B) since XAFenters into the expression for X B( ~X B ~ does not enter into the expression for XAF). The physical content of (81) is easily appreciated if we rewrite this expression in the form

Figure 19. (a) Plots of XA' exp(-(3hpoA) (-) and XB' exp(-@ApoB) (---) vs. XA/XB for the mixed disk model with a = 2. a. = 8 A2, e'/kT = 20, e/kT = 0. The total concentration of lipids (bile salt plus lecithin) was held constant at 5 g/dL. (The molecular weights of lecithin and bile salt were taken as 747 and 519, respectively.) (b) Plot of XBFexp(papo,) vs. R under the same conditions as in (a). The error bars represent the approximate" results of Mazer et al.' (XB>> &'). 0.6

I

I

I

I

1

0.5

0

0.I

0 0

(0) 1

'

20

1

1

40

'

1

60

'

1

Data of Mazer et a1

80

1

1

100

1

120

R(iQ

This indicates that the fundamental component of the micelle interior is not just A since XAFexp(-@poA) depends on other values of XBFexp(-PApoB). Furthermore, since it is the combination [XAFexp(-pApoA)] [XBFexp(-@poB)] 'iU which is simply related to the variables describing the composition of our system, we may assume that the interiors of the micelles, instead of being viewed as composed separately of A and B molecules, are better thought of as composed of "quasiparticles" made up of 1 A and l / a B molecules. The effective volume of each quasiparticle is uo + ( l/a)uB, which is exactly the quantity which appears in the exponent of (82) in place of uo. I n order to illustrate these results, we plot in Figure 19a the ~ B) (---) and X AeXP(-P&' ~ A) dependence of both X BeXp(-PAp' (-) as functions of XAlXB where a = 2, uB = uo, d = 2, t = 0, d / R T = 20 and the total amount of lipid (bile and lecithin) is held fixed at a concentration of 5 g/dL. In Figure 19b where we have plotted XBFexp(-PApoB) as a function of R , we have also

Figure 20. Plot of the variance Vof the micelle size distribution in the case of the mixed disk model (since d = 2, this is also just the variance in micellar radius) in the case where ct = 2. The same parameter values as in Figure 19a are used here. The dots represent the experimental results of Mazer et al.'

shown the experimental estimateP of XBFexp(-@ApoB)obtained by Mazer et ale5as error bars. It is also straightforward to obtain the expressions for XBF exp(-PApoB) and xAF exp(-PAKoA) when polydisperse trial functions are employed. In particular, the expression for XBF exp(-PApoB) is obtained by replacing XAby XAin Table I11 and by computing the mean micellar radius from Table IIIB by replacing XA by XAand XB by XB. The expressions for XAFexp(-PAp'A) are obtained from TableJI by first carrying out the XA,XB XBdiscussed above and then same substitution X, multiplying the resulting expression for XAFeXp(-@&'A) by [XBF eXp(-fiApo~')]-'/". In addition it is possible t o use the y distribution as described previously to compute the variance of the micelle size distribution as a function of R. Such a plot is found

- -

Stecker and Benedek


> 1,56the mixed bulk phase will be energetically favored over the pure A bulk phase and will thus form at lower values of XAF exp(-PApoA). Using the expression for XAFexp(-PApoA) (81), we see that the condition 83 is equivalent to

a

80

-3

60

E

(84)

5 40 0

or 20

'0

20

40

60

80

100

120

The point at which the mixed micelles come into equilibrium with a bulk phase defines the boundary between region I1 where only mixed micelles are present and region I11 where mixed micelles and a bulk phase coexist. From ( 8 5 ) and (79) we can see that the equatioh of this boundary is ( X A >> XAF)

or in terms of CA and C , 1 ( c , - -e, - 5.5 x a

104x,~

5.5 x 104( 1 20

40

60

80

100

120

Figure 21. The phase diagram of the micellar system in the mixed disk model with a = 2. CBis the concentration of bile salt in millimolar and C, is the concentration of lecithin in millimolar. Mixed micelles and simple micelles coexist in region I. The locus of points for which NB = q,= 20 and the curve defined by XBFeXp(-pAp"B) = 7.5 (which we have assumed is the cmc for the formation of simple micelles) are shown as

solid lines. In region I1 only mixed disks are present. In region I11 both mixed disks and a bulk lecithin-bile salt phase are present. a. = 8 A, uB = uo, U B = 60 A2,c'/kT = 20, e / k T = 0. In (a) we assume that Here, W e exp(@p"B)

Theory of multicomponent micelles and microemulsions - The Journal

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