1728
Gerson Kegeles
The Journal of Physical Chemistry, Vol. 83, No. 13, 1979
A Shell Model for Size Distribution in Micelles Gerson Kegeles Section of Biochemistry and Biophysics, Biological Sciences Group, The University of Connecticut, Storrs, Connecticut 06268 (Received January 8, 1979) Publication costs assisted by the University of Connecticut Research Foundation
The distribution of intermediates between monomer and very high polymers in micellar systems formed by dissolving surfactants in aqueous solvents has been of interest, although the equilibrium properties of many such systems may be approximated reasonably well by assuming a two-state system, monomers and very high polymers. For some micellar systems, in particular those formed from proteins, an appreciable population of intermediates may be present. A model predicting the population of all intermediates between monomer and high polymers is useful not only for describing the equilibrium properties of such systems, but also for the purpose of providing the description of an initial state from which size distribution alterations occur in kinetic processes. For this model, a hypothetical shell of limited surface area has been assumed, on which monomers can condense to form a micelle. The maximum possible number of monomers, n, added to one original monomer on this hypothetical shell, is treated as an experimentally accessible parameter. The rate of condensation and dissolution on this shell is assumed to be proportional to the number of monomer units already present. A property of this model is cooperativity, expressed specifically by the requirement that the propagation coefficient, which is the product of the intrinsic polymerization equilibrium constant times the free monomer concentration, is always greater than unity above the critical micelle concentration. This is a fundamental departure from previous models for micellization based on the law of mass action. The behavior of true micellar solutions which are thermodynamically ideal is approximated reasonably well by including a nucleation step in which the formation constant for the dimerization step only is reduced with a small scale factor.
Introduction A recent article1 presents a number of models for the description of the multiple equilibria involved in micelle formation, as well as some feasible experimental tests for their validity. A very early description of the equilibrium size distribution in micellar systems was published2 in 1937. The authors assumed that every step of polymerization had the same formation constant, K , and that the product of formation constant times free monomer concentration, here denoted by [A,], is less than unity. This theory, for ideal solutions, was developed in mathematical completeness, and was used to describe both the molar distribution and the weight distribution of oligomers. It predicted broad distribution functions and was not wholly successful in describing micellar behavior, because an enormous degree of polymerization seemed required to separate the micelle peak in the distribution from the region of monomer. Similar theory has been, in more recent years, usefully applied in the description of selfaggregating protein system^.^ Indeed, on a molar basis, this theory would actually yield a monotonically decreasing function.* Other descriptions of micellar equilibria are given in a number of recent studies.”1° some of which have attempted to define the energetics of polymerization in microscopic detail. In an attempt to introduce the concept of reactive intermediates in relaxation kinetics, Sams, Wyn-Jones, and Rassingll have provided a simple provocative interpretation which, although subject to modification, has stimulated other studies, including the present one. Formulation of the Distribution Function The first premise of the present model, unlike previous mass action law models for micelles, is that the product of the intrinsic formation constant for addition of a monomer to any oligomer, K , times the concentration of free monomer, [A,], must form a propagation coefficient larger than unity, in order to produce concerted or cooperative growth. It has been customarily taken1,2,8’11that 0022-365417912083-1728$01.OOlO
K[A,] is slightly less than, or equal to unity a t all concentrations above the critical micelle concentration. The law of mass action, ignoring thermodynamic nonideality, requires that [A11/[A,1 = K,1[-4ol [A,I/[All = K12[A,I [A31/[A21 = K,,[A,I
......... If, as in recent kinetic arguments,ll all of the various Ki,,+l are taken as equal to the intrinsic equilibrium constant, K , and if, moreover, the product K[A,] is taken as unity, then by the equation set 1, all of the intermediates, including dimer, as well as the largest micelle itself, have identical molar concentrations. This gives a horizontal straight line for the molar distribution function, at variance with experience. On the other hand, if the propagation coefficient KIAo] is taken as larger than unity to provide cooperativity in growth, then another factor must be present to limit growth, else the distribution function will increase montonically and without limit, yielding only a gel or solid polymer! The concept used to limit growth is a straightforward steric argument. It is assumed that the monomers arrange themselves upon a shell-like surface, which has only a limited space for the introduction of monomers. If the maximum possible number of monomers which can be added to one preexisting monomer on this shell is denoted by n, then if i additional monomers are already present, the probability factor for adding the next monomer is taken as (n - i)/n. Strictly speaking, we should be discussing the distribution of “head groups’’ on the shell surface,l but the same argument is still valid. In the present approach, we will consider n to be an experimentally accessible parameter. No attempt will be made to compute values of n for any particular system. 0 1979 American Chemlcal Soclety
Size Distribution in Micelles
The Journal of Physical Chemistry, Vol. 83, No. 13, 1979
Following Sams et al.,ll we take the statistical factor for adding a monomer to an oligomer to be equal to the number of monomers in the oligomer, and we take the statistical factor for dissociation of a monomer from an oligomer to be equal to the number of monomers in that oligomer. This latter attribution has been disputed12 in a more recent kinetic treatment, but it is an assumption of great utility in this and in a forthcoming paper.13 This assumption makes all of the details of micelle growth entropic in nature, rather than preassigning differential enthalpic effects to each level of polymerization. It does not, moreover, appreciably weight the equilibrium constant for growth except for the formation of the first few small oligomers. In place of eq 1, then, we have the following set:
[AiI/[Aol = ( ( n- 0)/4(1/2)K[Aol [&I/[AiI = (h- l)/nK2/3)K[AoI [&]/[A21 = ( ( n- 2)/nI(3/4)K[AoI
.............. n-(i-1) i -K[AoI (2) n i+l If we multiply the first of this set by the second, we obtain [A2]/[Ao]. This product multiplied by the third of the set gives [A3]/[Ao],etc., leading to
[Ail / lAi-11
[AlI/[AOl = ( ( n- 0)/4(1/2)K[AOl
The last equation in equation set 3 can now be written in the general form as
This is then the prediction of the molar distribution ratio of imer to monomer on the basis of the shell model. It requires for numerical evaluation the specification of n, K[A,,],and i. Since the molecular weights of Ai and A. are in the ratio i + 1, the subscript for an oligomer in the present notation indicating the number of monomers added to one original monomer, we obtain the weight distribution function from
Analytical Examination Equations 3 and 4 may be shown by algebraic manipulation to lead to distributions having' a maximum for i = 0 (monomer), and for 0 < i < n/2, for values of KIAo] between 1 and 2. The minimum value of the propagation coefficient K[A,,] required to just begin to show the second maximum is determined by the amount of cooperativity required to overcome the initial small drop in concentration predicted by eq 2 for the small oligomers. This minimum value of KIAo] decreases as the maximum possible degree of polymerization, n, increases. From eq 2, the ratio of concentration of two successive oligomers
1729
Flgure 1. Computed molar distribution functions, [A,]/[A,], eq 4, for shell model, n = 50, at successive K[A,] values (left to right and top to bottom) of 1.0, 1.1, 1.2, 1.22, 1.3, 1.5, and 2.0. Monomer, not shown, is at 1.0 relative concentration, and dimer = 0.5K[A0]*,in all plots.
will reach a stationary value when d In ([Ai]/[A;-11)/di = 0. This results in the condition that i = ic = (n 2)1/22. This result is put into the expression for the ratio [A,]/[Ai-l], eq 2, and the ratio itself is set equal to unity as a condition for a stationary value (two successive oligomers have the same molar concentration). On solution of this equation for KIAo], we obtain for the critical condition for the emergence of micelles
+
K[Ao] = n / [ ( n+ 2)'12 - 112
(6)
This condition on KIAo] replaces, for the case of the shell model distribution function, the usually accepted condition that KIAo] = 1. When KIAo] exceeds the value given by eq 6, for a given value of n, this implies that [A,] now exceeds somewhat its value under the critical conditions, and micelles will now grow relative to small oligomers. This will be discussed in further detail below, with the aid of Figure 1. A few illustrative values of the critical propagation coefficient from eq 6, together with the degree of polymerization of the incipient micelle, are shown in Table I for micellar systems of different maximum possible degrees of polymerization. Further justification for the use of eq 6 as a criterion for a critical condition will be made when detailed numerical calculations are discussed below. It is also clear that, for KIAo] values less than unity, a monotonically decreasing molar distribution function is predicted by eq 4. The total relative concentration, Co/ [A,,], in terms of monomer units in all oligomers is also given by summing of all terms in the form of eq 5 . The apparent critical micelle concentration then corresponds to the value of Cofor which a second maximum below n / 2 just emerges in the molar distribution. I t can be shown that essentially complete micellization occurs for K[A,] = 2, and that then the degree of polymerization at the maximum of the distribution function is equal to n/2. The discussion of the analytical examination of the predictions is abbreviated here, because the algebraic labor to visualize the situation is considerable, whereas direct numerical analysis with the aid of a digital computer is comparatively simple. Numerical Analysis As indicated following eq 2, we generate trimer concentration relative to monomer, [A,] / [A,,], by multiplying [AII/[AoI = (1/2)K[Aol by [&I/[A,I, etc. A Fortran program is written in which we specify numerical values for n and K[A,,]. In a DO loop we then generate [A2]/[Al], [A3]/[A2],etc., and the values are stored. We now use a DO loop to multiply [Al]/[Ao], which is specified as (1/
1730
The Journal of Physical Chemistry, Vol. 83, No. 13, 1979 I
I
I
Figure 2. Computed molar distribution functions, [A,]/[A,], eq 4, for shell model, n = 100, at successive K[A,] values (left to right and top to bottom) of 1.0, 1.1, 1.2, 1.22, 1.3, 1.5, and 2.0. Monomer, not shown, is at 1.0 relative concentration, and dlmer = 0.5K[A,]2, in all plots.
TABLE I : Critical Values of Propagation Coefficient K[A,], Eq 6 n 14 23 47 98 398 898
Gerson Kegeles
ic= ( n + 2)*'* - 2 2 3
5 8 18 28
K[Ao 1 1.556 1.438 1.306 1.210 1.102
1.068
2)K[Ao1, by [A2]/[Al];on the next passage the product is multiplied by [A3]/[A2]etc., thus generating [Ail/ [Ao] for i = 1 , 2 , n. These are output in tabular form and stored for computer plotting. The molar distribution function is simply plotted as [A,]/ [&I vs. degree of polymerization. In one additional DO loop, the weight distribution function is generated by multiplying [A,]/[Ao] by i + 1, according to eq 5. The sum of these terms is Co/[Ao].
...
Predictions In Figure 1 are shown the molar distribution function, eq 4, for n = 50 and various choices of KIAo]. In Figure 2 are shown the corresponding distribution plots for n = 100. It is seen that for KIAo] = 1,for both n = 50 and n = 100, the only maximum lies a t the monomer, and the distribution function decreases smoothly and monotonically, indicating, as expected, no cooperativity. When KIAo] = 1.1,a slight change in slope appears at the right, which is more pronounced for n = 100. When K[&] = 1.2, a decided alteration in slope is visible, and for n = 100, some resolution seems incipient. A t KIAo] = 1.22, the emergence of a second maximum is observed, for n = 100. Similar emergence does not occur for n = 50 until the value of KIAo] reaches 1.3. Reference to Table I indicates that these values for K[&] are predicted to be just at or slightly above critical values. For K[&] = 1.5, very little monomer remains for either n = 50 or n = 100. When K[A,] = 2 , no monomer is visible, a sharp distribution of micelles is produced, and the degree of polymerization at the peak of the distribution occurs at n / 2 . When KIAo] = 2, the ratio of molarity of oligomer at the peak to that of free monomer is 8.5(10)2for n = 50, and 6.8(1016for n = 100. The distribution functions are sharper if a much larger value for n is assumed, and the value of KIAo] required for the critical emergence of micelles lies closer to unity, as predicted by eq 6 and indicated in Table I. The weight distribution function plots according to eq 5 are shown in Figure 3 for n = 50 and n = 100, and for various values of KIAo]. These distributions appear generally sharper than the molar distributions. They show
Flgure 3. Computed mass distribution functions, ( i + l)[A,]/[A,], eq 5 , for shell model, n = 50, top row and n = 100, bottom row, at successive K[A,] values (left to rlght) of 1.1, 1.22, 1.3, and 2.0.
a polymer peak at all values of KIAo] greater than unity. They do not show the abrupt development of micelles as do the molar distribution plots in Figures 1and 2. It can be shown from eq 5 that the maximum in this distribution function occurs at i = n(K[&] - l)/KIAo]. As will be seen from the discussions that follow, the experimentally feasible tests for critical micelle formation generally agree with the predictions made on the basis of eq 6 and Table I for the molar distribution function. They do not reflect the existence of, or the position of any micelle peak, as suggested by the weight distribution function at all values of K[&] greater than unity, and would appear to discount the originally suggested use2 of the weight distribution as a criterion of micellization.
Modification of the Distribution Function While the successive diagrams in Figures 1 and 2 illustrating the development and emergence of micelles are very interesting, there is one characteristic of these distribution functions which makes them in their form as shown in Figures 1and 2 inadequate, at least to describe the behavior of systems showing a sharp critical micelle transition and a sharp micelle distribution. If one computes the ratio Co/ [Ao] which is characteristic for n = 50 at KIAo] = 1.3, or for n = 100 at KIAo] = 1.22, that is, according to Figures 1 and 2, near the apparent critical micelle conditions, then one finds for n = 50, Co/[Ao] = 83.6, and for n = 100, Co/[&] = 156.2. The characteristic behavior for the "two-state" system nB P B,, when n is very large, is that Co/[Ao] is approximately unity a t the critical micelle concentration, Le., when micelle first emerges in even trace amounts, everything else is essentially all monomer. Mukerjeel has pointed out that one can allow for minute amounts of intermediates by setting the formation constant for dimers equal to some extremely small number, and then allowing growth to take place on the vanishingly small amount of dimer as nucleus. This then generates a distribution function with appreciable monomer, followed by a very deep valley of all intermediates up to a level of polymerization large enough for a concerted effect to overcome the initial drop in concentration between monomer and dimer. Such a nucleation step has been a postulate in earlier developments,' especially in other fields of research.14-16 In order to have a distribution function capable of giving Co/[AoJ values not very large compared to unity, it is necessary to make a similar modification here. Thus, we might take, in place of eq 2 [Ail = f(1/2)K[Aol2
(7)
where f 1. For simplicity, we will assume that f does not vary with
Size Distribution in Micelles
The Journal of Physical Chemistry, Vol. 83, No. 13, 1979
1731
TABLE 11: Degrees of Polymerization for n = 100
1.0 1.2 1.22 1.3 1.35 1.5 2.0
4.29 12.02 13.35 19.16 22.80 31.91 49.95
8.57 18.38 19.56 24.21 26.98 34.33 51.00
2.00 1.53 1.47 1.26 1.18 1.08 1.02
1.19 2.82 3.38 8.00 13.45 30.55 49.95
2.61 13.22 15.13 22.42 26.15 34.29 51.00
Computed excluding free monomer. temperature or pressure, i.e., it is taken as an absolute constant. Equation 4 will then become, simply
2.18 4.70 4.47 2.81 1.94 1.12 1.02
1.27 1.17 1.06 1.02
1.00 1.02 1.03 1.11 1.28 6.44 49.95
1.02 1.40 1.56 3.44 6.96 30.07 51.00
1.02 1.37 1.52 3.10 5.43 4.67 1.02
1.20 1.14
1.07 1.02
a
[A,l/[AOl = f
n!
1
(“;I);
-
When this done, all intermediates and the largest micelles are depressed in molar concentration by the identical scale factor, f , compared to their previous values, while monomer remains unchanged. The complete effect of this modification is to reduce the vertical scale for the plots of Figures 1,2, and 3 everywhere except at the left hand axis (representing monomer), while monomer is unchanged. Moreover, the value of the propagation coefficient KIAo], for which emergence of micelles just becomes critical, still follows eq 6 and Table I, remaining completely unchanged, as does the degree of polymerization at the emerging micelle peak. However, the value of the relative total concentration at the apparent critical micelle condition or under any other condition is Co’/[Ao] = 1 + (Co/[Ao] - l)f (9) Here Co/[Ao] is the original value from the unaltered distribution function and Co’/[Ao]is the new value. It is clear from eq 9 that we can make Cw/[A,] as close to unity as desired, by arbitrarily decreasing the factor f, while still using the shell model distribution function to generate a concerted formation of micelles. With eq 4, 5, 7 , 8, and 9, we can construct a prediction of what the experimenter might find with direct methods for measuring molecular weights. Table I1 shows the values of number average and weight average degrees of polymerization and their ratios, for a variety of KIAo] values below and above the critical values shown in Table I. These DP values would come from experimental ratios of number average and weight average molecular weights to monomer molecular weight. Also shown are ratios of weight average to number average degree of polymerization above the apparent cmc at f values of 0.01 and 0.0001, for micelles alone, excluding the monomer. The criteria of Mukerjeel state that the ratio of the weight average degree of polymerization to the number average degree of polymerization for systems above the cmc, when computed excluding monomer, is equal to 1 for sharp micelle distributions and is equal to 2 for broad micelle distributions. If one includes monomer, the sharpest distributions occur overall for f = 1, the original shell model distribution function, even though the ratio of total concentration to free monomer concentration for this distribution is unrepresentative of most micellar systems. On the other hand, when monomers are excluded, the distributions for f = 0.01 and f = 0.0001 are just as sharp, for the micelles alone. It is further noted that all the distributions for micelles become extremely sharp, with a ratio N w / N n= 1, a t K[Ao] = 2, but this is predicted in general for impractically high total concentrations. It would be possible for the experimenter to plot directly
the weight average molecular weight of the whole micellizing solute system against its total weight concentration. In Figure 4 are shown predictions of the plots which would be obtained for three systems with n = 100; in Figure 4a is a system following the original shell model distribution, in Figure 4b is a system following a modified shell model distribution with f = 0.01, and in Figure 4c is the plot for a system following a modified shell model distribution with f = 0.0001. Although the change in slope in Figure 4a and 4b is fairly abrupt, we are obligated to designate the intersection as an apparent cmcl because the points do not all fall on the two straight lines, and none of the straight line portions has a zero slope. However, we note that the value of KCO at the intersections agrees remarkably with the predictions of eq 6 and Table I for the location of the critical emergence of a micelle peak in the molar distribution function. If we now focus attention on Figure 4c, for an f value of 0.0001, we note that the entire qualitative character of the M, vs. KCO plot has changed. Not only is it true that almost every point falls on the straight lines, but at concentrations below that for the intersection, the value of M , remains very close to that of monomer (one of the two straight lines has almost zero slope). Moreover, just above the intersection, at K[&] = 1.22, the computed ratio of total concentration to that of free monomer is only 1.023. Hence this modified distribution function satisfies reasonably well the requirements for true micellar behavior. Once again, the point of intersection gives a value of KCO in very close agreement with that from Figure 2, Table I, and eq 6, for the emergence of a micelle peak in the molar distribution function. Consequently, the appearance of a polymer peak in the weight distribution function, Figure 3, is not, as has been assumed in the case of continuous self-aggregation,2satisfactory evidence that the concentration lies above a cmc value, or even above an apparent cmc value. We conclude that the modified shell model distribution function, with an appropriate value for the small nucleation factor f , may adequately express the behavior of many true micellar systems, which do not have infinitely sharp micellar distributions, and for which thermodynamic nonideality effects may be neglected. In the case of some micellar systems containing large amounts of intermediates, such as some self-aggregating protein systems, the factor f , if it needs to be considered, is probably much closer to unity than for most detergents. It should be added that the inclusion of an f factor different from unity is a nonstatistical assertion; it tacitly assumes a very specific local free energy effect between monomer and dimer which is different energetically from all the remaining reaction steps.
Summary A shell model distribution function for micelles has been developed on the basis of the following assumptions: (1) The intrinsic formation constant for each monomer-oligomer bond is identical, and equal to K .
1732
The Journal of Physical Chemistty, Voi. 83, No. 13, 1979
Gerson Kegeles
K expresses the steric restriction that the shell over which the monomers (or their head groups) arrange themselves can only hold n 1 monomers. (4) The product of the intrinsic polymerization constant and the free monomer concentration, termed here a propagation coefficient, KIAo], is greater than unity, at concentrations above the critical micelle concentration. Actually, assumption 4 is not made independently; it is a direct result of the first three assumptions. Examples of this shell model distribution are shown, in order to illustrate the predictions of the abrupt emergence of micelles in the molar distribution a t critical values of KIAo], and the eventual complete micellization centered around a degree of polymerization equal to n/2, when KIAo] = 2. While the original distribution function gave very high ratios for apparent critical micelle concentration to free monomer concentration, these ratios can be brought as close to unity as desired by a modification in which the formation constant for dimer only is multiplied by a small scale factor f . This modification allows the model to simulate true micellar behavior, with reasonably sharp micelle distributions and satisfactory cmc behavior, Mukerjee' has pointed out that at high concentrations where globular structures become saturated, many micellar systems can keep growing in extended rodlike structures. The present treatment is not designed in its present form to take such behavior into account. It is demonstrated that feasible experimental tests for a cmc agree with the predictions of the molar distribution function, and that the existence of a polymer peak in the weight distribution is not satisfactory evidence that the concentration lies above a cmc. Application of this distribution function to micellization kinetics is treated in a forthcoming pub1i~ation.l~
+
0
w:/ 12
IO 20 IO-' K Co
0
30
28
20 Mw/ M
12
I
A
P 20
40
60
80
K C0
Acknowledgment. The author was stimulated in the pursuit of this research as a result of the experimental study in our laboratory by Dr. Cinnia Huang of the relaxation kinetics of @-caseinmicelles, which was supported by a grant from the National Science Foundation. This material was kindly furnished by Dr. T. A. J. Payens of the National Institute for Dairy Research, Ede, Netherlands, through whose courtesy and that of the National Academy of Science of the Netherlands the author also had the opportunity to present a seminar on Dr. Huang's experimental work. He thanks Dr. Payens, Dr. Schmidt, Dr. Vreeman, Dr. Jeffrey, and others in the audience for provocative discussion which led to this further work.
References and Notes 05
IO
I .5
K Co
20
Flgure 4. Predicted relative welght average molecular welght M,l M vs. reduced total concentratbn of monomer In all forms, KCO, for micellar systems with n = 100. In a, f = 1, In b, f = 0.01, and In c, f = 0.0001 (see text). The straight llne Intersections for the three plots are at KCo values of 170, 6.0, and 1.25 for a, b, and c, whereas the computed values of KCo corresponding to KIAo] = 1.22 are 190.8, 5.007, and 1.258, respectlvely.
(2) A statistical factor equal to (i + l)/(i + 2) multiplying K assumes that the probability for adding one monomer
+
to an oligomer Ai containing i 1 monomers is proportional to the number i 1, while the probability of dissociating a monomer from the product Ai+l containing i 2 monomers is proportional to i + 2. (3) An additional statistical factor ( n- i ) / nmultiplying
+
+
(1) P. Mukerjee, J . Phys. Chem., 76, 565 (1972). (2) K. H. Meyer and A. van der Wyk, Helv. Chlm. Acta, 20, 1321 (1937). (3) E. T. Adams, Jr., P. J. Wan, and E. F. Crawford, Methods Enzymol., 48, 69 (1978). (4) P. Dessen, Biochlmle, 55, 405 (1973). (5) I . Relch, J . Phys. Chem., 60, 257 (1956). (6) C.A. Hoeve and G. C. Benson, J . Phys. Chem., 61, 1149 (1957). (7) F. J. C. Rossottl and H. Rossottl, J . Phys. Chem., 65, 1376 (1961). (8) T. Nakagawa, Collold Po/ym. Scl., 252, 56 (1974). (9) J. Lang, C. Tondre, R. Zana, R. Bauer, H. Hoffmann, and W. Ulbrlcht, J . Phys. Chem., 70, 276 (1975). (10) C. Tanford, J . Phys. Chem., 78, 2469 (1974). (11) P. J. Sams, E. Wyn-Jones, and J. Rasslng, Chem. Phys. Lett., 13, 233 (1972). (12) E. A. G. Anlansson, S.N. Wall, M. Almgren, H. Hoffmann, I. Klelmann, W. Ulbrlcht, R. Zana, J. Lang, and C.Tondre, J. Phys. Chem., 80, 905 (1976). (13) 0. Kegeles, to be published. (14) 8. H. Zlmm and J. K. Bragg, J . Chem. Phys., 31, 526 (1959). (15) M. Saunders and P. D. Ross, Blochem. Blopbys. Res. Comm., 3, 314 (1960). (16) F. Garland and S. D. Christlan, J . Phys. Chem., 70, 1247 (1975).
