2186
J. Phys. Chem. 1980, 84, 2186-2190
Distribution of Solubilizates among Micelles and Kinetics of Micelle-Catalyzed Reactions Yoshikiyo Moroi DepaHment of Chemistry, Faculty of Science, Kyushu University, Higashi-ku, Fukuoka 8 12, Japan (Received:February 20, 1979; In Final Form: March 12, 1980)
This paper illustrates that a crucial factor in analysis of kinetic data from micelle-catalyzedreactions is the distribution of solubilizates (or reactants) among micelles, especially when reactant concentrations are comparable to or higher than micellar concentrations. The distribution of solubilizates among micelles in micelle-catalyzed reactions is treated as a thermodynamic problem. The random, Poisson, and Gaussian distributions of reactants were applied to evaluate the association constants between reactants and micelles, and kinetic equations in the presence of micelles were derived in an explicit form. The equation from the Poisson distribution was almost the same as that derived by Menger and Portnoy? (kapp - k b ) / ( k l - kaPP) = KJM,], where kapp is an apparent rate constant, k b a rate constant in the intermicellar phase, kl a rate constant in the micelle with one reactant, K1 an association constant of one reactant with the micelle with no reactant, and [Mt] a total micellar concentration. The kinetic equations from the random and Gaussian distributions were similar to the equation from the Poisson distribution, although other parameters such as an average number of reactant per micelle and the standard deviation of distribution had to be introduced.
Introduction Micelle-catalyzed reactions became an area of rapidly increasing interest in the last decade. Micelles were found to be very attractive reaction media for many kinds of The mechanisms of these reactions were investigated not only by analogy with the Michaelis-Menten equation for the enzymatic reactionl1~3~ but also from the perspective of volume fractions of the two-part reaction system: the micellar and intermicellar bulk solutions.6* Their reaction kinetics has been successfully used only when the micellar concentrations are much higher than reactant concentrations. However, the micellar concentrations just above the critical micelle concentration (cmc) are less than the reactant concentrations. Therefore, in studies of the kinetics of micelle-catalyzed reactions, in the case where reactant concentrations are comparable to or higher than micellar concentrations, the distribution of reactants among micelles must be taken into consideration. Indeed, studies have produced various curves of reaction rates plotted against the micellar concentration^,^>^ but their interpretations have been conventional in most respects and the distribution of reactants among micelles has not been considered. On the other hand, in photochemical processes in micellar solutions the distribution of photochemical probes among micelles was found to be a very important factor to elucidate the photochemical reactions, and a generally accepted concept concerning the distribution of probe molecules among micelles is their Poisson distribution.1°-12 However, the inevitability of the distribution has not yet been made clear in this case either. As a micellar solution is a thermodynamic system even in the presence of solubilizates or reactants, the equilibrium distribution is essentially a thermodynamic problem. The association constants of solubilizates with micelles which determine the state of the micellar solution system are too many to be determined precisely. Hence, the random, Poisson, and Gaussian distributions of reactants among micelles were adopted to evaluate the constants. The present study was directed toward the derivation of explicit kinetic equations using the above three distributions in the hope that the experimental results obtained from the micelle-catalyzed reactions would yield information on the reactant distribution, particularly in terms of which distribution is most preferable to analyze the re~u1ts.l~ 0022-3654/80/2084-2186$01 .OO/O
Equilibrium Distribution of Solubilizates among Micelles The major part of this paper is concerned with the distribution of solubilizates among micelles. First, the monodispersity of micelles in the absence of solubilizate is assumed in order to remove the difficulties arising from their polydispersity, which will be taken into account later. The ideality of chemical species in a solution is also assumed because of their low concentrations. The association equilibrium between surfactant monomers and micelles is presented by eq l. The stepwise association
equilibria between micelles and solubilizates are presented by eq 2, where n is an arbitrary number as is shown in the K
M+R+MR, MR1 + R
2MR2
.... MR,-l
+ R & MR,
(2)
Glossary. The total number of components of this system is n + 4 including water molecules, and the number of phases is one. The n 1 equilibrium equations for the micellar system reduce the number of degrees of freedom of the phase rule by n + 1,resulting in 4 degrees of freedom. Hence, at constant temperature and pressure, two other intensive variables can be selected to prescribe the thermodynamic system. In the present paper, three sets of combinations of intensive variables will be examined to derive other intensive properties. (1)Monomer Concentrations of Surfactant, [SI, and Solubilizate, [R]. From monomer-micelle (eq 1) and micelle-solubilizate (eq 2) equilibria, the micelle concentrations without and with solubilizates are given by eq 3 and 4, respectively, where [MRJ is the concentration of
+
[MI = Krn[Slm
(3)
[MRJ = K r n ( f IKJ)[Slm[RIL
(4)
1-1
micelles associated with i solubilizate molecules. Thus, the 0 1980 American Chemical Society
The Journal of Physical Chemistry, Vol. 84, No. 17, 1980 2187
Kinetics of Micelle-Calalyzed Reactions
concentrations of any species can be determined by these two variables, if the values of K , and Kj are available from some suitable methods. The fraction of [MRi] to the total micellar concentration, F(MRi), and the average number of solubilizates per micelle are given by eq 5 and 6. F(MRi) = [MRi]/
5 [MRJ
i=O
(5) n
n
2 i[MRi]/C i=O i-0
R= =
[MRJ
depends on the thermodynamic system concerned.
Kinetic Equations of Micelle-Catalyzed Reactions In the case where reactant concentrations are comparable to or higher than micellar concentrations, we have to think about the distribution of reactants among micelles. The stepwise associations of monomer reactant with micelles lead to the distribution of reactants among micelles, and the association reactions can be assumed to be completed within a much shorter time as compared with each reaction time.14J5 Hence, the following reaction scheme is considered: k
M+R+MR--+P
5i ( h Kj)[FLli/{l + 'f (hKj)[RIi)
i=l
i = l j=1
js1
P
(2) Total ,Equivalent Concentrations of Surfactant, [S,], and Solubilizate, [R,]. This set of variables is the most commonly used. 'The total equivalent concentration of surfactant is
[S,l = [SI
4- m&
i=Q
n
i
+ C (uKj)[Rli) 1=1 js1
= f([Sl,[RI) (7) Hence, the total equivalent concentration can also be expressed by the function of [SI and [R], f([S],[R]). On the other hand, the total equivalent concentration of solubilizate is n
[R,] = [R] -t C i[MRi] i= 1
= [R] 9 K,[Slm
'f i ( h Kj)[RIi
i=l
i=O
2 (h Kj)[Rli]
i=l j=1
(9)
solubilizates per micelle, R , was given by eq 6. By rearranging eq 16, one obtains eq 10. The solution of eq 10
R
kn
P + MR,-,
-d[Rt]/dt = kaPP[Rt][XI
2'ki[MRil)[Xl
= (kb[RI +
+ i5 ( R - i)(fI Kj)[RIi = 0 =l j=l
should give the [Et] value in an association equilibrium. Substituting the [R] value into eq 9, one can have the equilibrium concentration of monomeric surfactant, [SI. Consequently, any intensive properties can be derived in the same way as before from this combination of variables, too. The three combinations of variables given above may be the most commonly used, but other combinations are also possible, wherein the suitability of each combination
(12)
i=l
where kaPP is the apparent rate constant and X may be hydroxide or hydrogen ions or any reactant other than R. Rearrangement of eq 12 gives the apparent rate constant:
5 [R] + 5 i[MRi]
kb[R]
[MRiI
= Km[Slrn(l+
(11)
MR,
MR,-, t R
kaPP
j=1
[RJ is a function of [SI and [R], too. Therefore, when [S,] and [R,] are' selected as two independent thermodynamic variables, [SI and [R] can be obtained as the solution of two simultaneous equations, f and g. Thus, the concentration of MRi species can be calculated in the same way as in the first case. In this case, too, the distribution of R among micelles depends on K , and Ki values. (3) Total Micelle Concentration, [MIt, and Average Number of Solubilizates per Micelle, R. The total micelle concentration is given by eq 9. The average number of [Mt] =
....
where kb is the rate constant of monomer reactant in the bulk phase, and kiis that of reactants in the form of MRi solubilized in micelle. If the reaction of R with X is assumed to obey the second order, the rate of disappearance of R, can be written in the form:
[MR,I)
= [SI t.mK,,I[S]"(l
k
MR+ R F Z M R * ~ P + M R
=
+ i=l ki[MRil
(13)
i=l
(1)From Random Distribution. Supposing that the way of solubilizing r reagents into q micelles corresponds to that of placing randomly r balls into q cells, one has the probability that a micelle is associated with i reagents in the form (Appendix):
R(i) =
-( -)1 +R R 1 1+R
i
where R is an average number of reactants per micelle, r/q. Accordingly, the concentration of MRi becomes (15) [MRiI = [MtIR(i) On the other hand, [M,] and [R,] can now be written in the case of infinite n [Mtl = [M1(1 + R )
(16)
[R,] = [R] + [MR,](l +
(17)
If it is assumed that the mass action law is applicable to the association equilibria between micelles and reactants, the association constant then becomes hc-1 = WRil K.= ' z i [MRi-lI[RI -b
= [R] I1 + (R R ) From eq 18 Ki is found to be independent on i; i.e.
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The Journal of Physical Chemistty, Vol. 84, No. 17, 1980
K1 = K2 = . . . = K,, (19) Turning now to the reaction kinetics in the presence of micelles in which reactants are solubilized in the fashion of the random distribution, the kivalues in eq 13 have to be made explicit in order to derive a clear kinetic equation. I t seems reasonable to assume that the kivalue is i times as large as kl,because the chance of the reactant R in MRi to react with X is i times as much as that of MR,; i.e. ki = ikl (20) Substituting k i of eq 20 and [R,] of eq 17 into eq 13 and summing the series up to infinite n yield the following expression for kapp:
or kapp
kl
- kb
- kaPP
= Kl(l
+ R)[M,]
-Moroi
reactants among micelles is not so definitive as those from the random and Poisson distributions, because the variable of the Gaussian distribution is continuous whereas the number of reactants in each micelle is an integer. In spite of this, it is interesting to derive the kinetic equations from the Gaussian distribution in view of its widespread use for distribution problems. When R is the value at which the distribution becomes maximum, namely, the mean value of the Gaussian distribution, the normalized Gaussian distribution function is in the form: G(i) = h/n1/2 exp{-h2(i- f E ) 2 ) (32) u = 1/2’I2h (33) where h is the Gaussian distribution constant relating to the standard deviation u. Then the concentration of the micelle having i reactants, [MRJ, can be given by [MRJ = [MtIG(i) (34) where i is not always an integer. By analogy with eq 13 the apparent rate constant in this case can be written in the form:
K1 is introduced to remove [R] and [MR,] from eq 21 by using eq 23. [MRiI/[Rl = Ki[MtINO) (23) (2) From Poisson Distribution. Returning to the first relationship of eq 18 and assuming that the ideas !o eq 20 are applicable to the association equilibria, hi = ikl, and that the raqe constant of the associating of monomer R with MRi, k,,can be Fade $0 remain the same regardless of any number of i, hi,= ko one has the following relationship concerning the association constant: K, = Kl/i (24)
If eq 24 is introduced into eq 4,8, and 9, then [MR,], [RJ, and [M,] become the following equations with infinite n: (25) [MRJ = K,z[R]z[M]/i! [Rtl = [RI + Kl[RI[Ml exp(K,[RI) (26) [Mtl = [MI exp(K~[Rl) (27) Hence, the average number of reactants per micelle, R , is given by eq 28. Thus, the probability for a micelle to be R = W t 1 - [Rl)/[M,I = Kl[RI (28) associated with i reactants which corresponds to eq 14 for the random distribution can be written P(i) = Ri exp(-R)/i! (29) This expression is exactly the same as the Poisson distribution. When eq 20 and 25 are introduced into eq 13, the equations corresponding to eq 21 and 22 are
kb[R] + [Mt]Jmk(i)G(i) -m di kaPP
=
(35)
[R] + [M,lJmiG(i) di -m
In the first place, if k(i) is assumed to be proportional to the number of reactants in a micelle in a similar manner as eq 20, k(i) = (i/R)kR (36) where kR is the rate constant in the micelle associated with R reactants. Introduction of eq 32 and 36 into eq 35 results in kb + k(l)K,R[M,] exp(h2(1- 2R)) kapp = (37) 1 K,R[M,] exp(h2(1- 2R)J or
+
kapp
--
- kb
- K1[Mt]R exp(h2(1- 2R)J
k1 = k(1) (38)
k l - kapp
The three normalized distributions are illustrated in Figure 1. The random distribution is quite different from the other two in the sense that it decreases monotonously with the number of solubilizates per micelle, while the others are very similar in shape at the point of the distribution maximum. However, the maximum i value in the Poisson distribution is less than the R value, the average number of solubilizates per micelle. It is very important at this stage to consider which distribution is the best approximation to the real reactant distribution. For a simulation of the reactant distribution eq 21,30, and 37 are transformed so as to plot l/(kapp- k b ) against l/[Mt]: random
or
I
Poisson These two equations are the same as those derived by Menger and Portnoyl except for their derivation condition that micellar concentrations are much higher than reactant concentrations. Now it is very clear from the comparison of eq 22 with eq 31 that thege two equations are identical except for the factor of (1+ R). However, this factor makes no difference with increasing micellar concentration. (3) From Gaussian Distribution. The derivation of kinetic equations based on the Gaussian distribution of
Gaussian
-- 1
kb
--
l
+
1
1
Kl(kl - k b ) R exp{h2(1- 2R)) [Mt] (41) The conditions of their simulation curves (Figure 2) are kapp -
k,
-
kb
The Journal of Physical Chemistry, Vol. 84, No, 17, 1980 2189
Kinetics of Micelle-Catalyzed Reactions
0.3)-
0
2
6
9
8
10
NUMBER O F SOLUBILIZATESPER MICELLE
Flgure 1. Nclrinalized distributions of solubilizates among micelles in the case of R = 5: (a) random (eq 14); (b) Poisson (eq 29); (c) Gauss (eq 32) (a = 1.5).
given in the figure caption, wherein a = 0.3R for the Gaussian distribuition comes from (1.5/5)R in order to make the distribution curves like the figures. Now, it can be seen from this figure that the plots of l / ( k a P P - kb) against l/[Mt] give a straight line for the Poisson distribution, while those for the random and Gaussian distributions are concave and convex to the l/[M,] axis, respectively. From these simulation curves, it might be possible to predict which distribution is the most preferable. It is clear, of course, that application of the Gaussian distribution to the present case where R is as low as five is not appropriate, because the number of reactant molecules per micelle is of integral value. However, the Gaussian distribution becomes more adaptable when gegenions around the ionic surfactant micelle are reactants, because they are so many in number that they can be treated as a continuous variable. Interest therefore centers on whether the smooth Gaussian function can be applied to a discrete function and on how nicely the application can be made. Consider the Gaussian distribution in Figure 1 and c-ompare the value of the integration J-,"k(i)G(i) di in the numerator of eq 35 with the summation C,,o"k(i)G(i) using the identical conditions of the figure. The former value is kR while the latter is 0.995k~.Hence, it turned out that the summation is in excellent agreement with the integration. The position is, thLen, that the summation of a Gaussian function in a discrete form can be well replaced by its integration within an experimental error. What has been discussed thus far is the reaction kinetics only from the monodispersity of the micellar aggregation number. From here the discussion will consider the effect of polydispersity on the mathematical expression concerning micellar kinetics. The considerable effects of the polydispersity on the reaction kinetics are (1) a variation of the rate constant (k,) due to the variation of surface area per micelle and (2) a variation of the association constant of reactant molecule with micelles (KJ, which will be reflected in the variation of R value. Suppose the polydispersity of aggregation number of micelles ranges from a to P. The micellar concentration with no association with reactants then becomes !3
[MI = C Km[SIm m=o1
I
I
1
2
4
6
I
I
10
1/[Mt] (103mOl") Flgure 2. Simulation of reactant distributions among micelles from Variation of ll(kaPP- kb)values with micellar concentrations: K 1 = 5 X lo3 mol-' L, R = 5 at the micellar concentration of 2 X mol L-', and ll(ksf@ - kb)= 0.01 at an infinite micellar concentration: (a) random (eq 39); (b) Poisson (eq 40); (c,d) Gauss (eq 41, u = 1.5, u = 0.3R).
which is equivalent to eq 3. The micellar concentration associated with i reactant molecules takes a similar form as eq 4: P
[MRi] = (I? Kj)[RIi C Km[SIm m=a
j=1
, (43) Hence, the apparent expressions become the same as those from the monodispersity. The problem is, however, that the effect of the polydispersity must be introduced into the mathematical expressions of the reaction kinetics. Let it be supposed that the rate constant ki does not remain the same but changes with the aggregation number. In terms of k,i, ki on the micelles whose aggregation number is m, the summation of the numerator of eq 13 becomes for the random l
n
P
- C ki[MRil = K1 (1 + 8) C km,[MmtI [RI i=l m=a
(44)
for the Poisson (45)
for the Gaussian B
= KIR exp(h2(1 - 2R)l
C m=a
kml[Mmt] (46)
where [M,,] refers to the total concentrations of micelles whose aggregation number is m. When the right-hand sides of eq 44-46 are compared with the second terms of the numerators of eq 21, 30, and 37, respectively, the difference in the mathematical expression between the corresponding equations will disappear, if the following
2190
The Journal of Physical Chemistry, Vol, 84, No. 17, 1980
Moroi
operation for averaging is made:
r
C Wq-l(r - i)
i=O
(47)
As is clear in the above consideration, the reaction kinetics can be normalized by the reaction parameter kml. Now, consider the microscopic aspect of the micellar surface where one reactant molecule sits. The micelle-catalyzed reaction takes place between the solubilized molecule and another reactant which are equally distributed all over the micellar surface. Then, the surroundings around the solubilized molecule would not be chemically affected by the aggregation number. In other words, the reaction parameter kml should remain the same, kl, irrespective of the micellar surface area. Starting from another point of view, an alteration in the surface area is related to an alteration of surface concentration of the reactants. Therefore, the apparent rate constant (k,)would vary with the aggregation number. Micelles being assumed to be spherical, the differential of micellar surface area (S) with respect to the aggregation number ( m )becomes dS = -2 - 1 dm (48) S 3 m The distribution of the micellar aggregation number has been discussed already,17and the standard deviation of the aggregation number is less than 10% against the mean aggregation number. Hence, the standard deviation of kmi due to the polydispersity remains within less than 7% against its mean value. From this point of view, too, the polydispersity can be said not to have so much effect on the reaction kinetics in micellar systems. Taking into consideration that the variation of the R value is also possible from the polydispersity, the second effect mentioned before, one can rewrite eq 47 for the random distribution in the form: P
E
(1 + RAkmi[MmtI
(49) where R, corresponds to R for the micelles whose aggregation number is m. Thus, in spite of the introduction of the polydispersity of micellar aggregates, the apparent kinetic expressions for the reactions still remain the same. On the basis on the above discussion, it is concluded that the mathematical expressions concerning reaction kinetics in micellar systems can be well expressed by eq 39,40, and 41 for the random, Poisson, and Gaussian distributions of reactant molecules among micelles, respectively.
Appendix The number of distinguishable distributions of placing randomly r balls into q cells id6 (50) When one cell has i balls in it, that of placing randomly r - i balls into q - 1 cells is W9-l(r - i), whereby Wq-l(r i) satisfies the following relationship with W&):
= Wq(r)
(51)
Thus, the probability that a cell is associated with i balls becomes R(i) = Wq-l(r - i)/Wq(r) (52) When i is much smaller than r and q is much larger than one, eq 52 then becomes
L(
i
R(i) = q + r i q+r,
(53)
where R(i) satisfies the following equation:
2 R(i) = 1
i=O
(54)
Glossary S
[Stl M MRi R LRt] R P Km Ki
-
kL-1
Ki n
monomer molecule of surfactant total equivalent amount of surfactant micelle without any solubilizate micelle associated with i molecules of solubilizate monomer molecule of solubilizate or reactant total equivalent amount of solubilizate average number of solubilizates per micelle product equilibrium constant of micelle formation stepwise association constant between MR,-I and R rate constant of MR, formation reaction of MRi-l with R rate constant of MR, decomposition reaction to MR,-l and R arbitrary number
References and Notes (1) F. M. Menger and C. E. P a y , J. Am. Chem. Soc., 89,4698 (1967). (2) E. H. Cordes and R. B. Dunlap, Acc. Chem. Res., 2, 329 (1969); E. H. Cordes, “Reactlon Kinetics in Micelles”, Plenum Press, New York, 1973. (3) E. J. Fendler and J. H. Fendler, A&. Phys. Org. Chem.,8, 271 (1970); J. H. Fendler and E. J. Fendier, ”Catalysis in Mlceliar and Macromolecular Systems”, Academic Press, New York, 1975. (4) C. A. Bunton, E. J. Fendler, L. Sepuhreda, and K. Yang, J. Am. Chem. Soc., 90, 5512 (1968); C. A. Bunton and L. Robinson, ibM., 90, 5972 (1968); J. H. Fendler, E. J. Fendler, R. T. Medary, and V. A. Woods, ibld., 94, 7288 (1972); J. H. Fendler, J. Chem. Soc.,Perkln Trans. 2 , 1041 (1972). (5) F. M. Menger and M. J. McCreery, J. Am. Chem. Soc., 96, 121 (1974). (6) P. Heitmann, fur. J. Biochem., 5 , 305 (1968). 17) A. K. Yatsimirski. K. Martinek. and I. V. Berezin. Tetrahedron.27. 2855 (1979); K. Mahnek, A. K. Yatsimirskl, A. P. Oslpov, and I. V: Berezin, ibM., 29, 963 (1973). (8) S. J. Dougherty and J. C. Berg, J. Co//oidInterface Scl., 49, 135 11974). 191 K. Shirahama. Bull. Chem. Soc. JDn.. 48. 2673 11975). ( i O j K. Kalyanasundaram, M. Gratzel, and J. K. Thomas,’J. Am. Chem. Soc., 07, 3915 (1975). 111) . . M. Maestri, P. P. Infelta, and M. Gratzei, J. Chem. Phys., 69, 1522 (1978). (12) M. Almgren, F. Grleser, and J. K. Thomas, J. Am. Chem. Soc., 101, 279 (1979). (13) T. Harada, N. Nishikdo, Y. Moroi, and R. Matuura, unpublished work. (14) B. C. Bennlon and E. M. Eyring, J . Colloid Interface Sci., 32, 286 (1970). (15) T. Yasunaga, K. Takeda, and S. Harada, J. Colloid Interface Sci., 42, 457 (1973). (16) W. Feller, “Ah Introduction to Probability Theory and Its Applications”, Wiley, New York, 1960. (17) P. MukerJee in “Mlcellization, Solubillzatlon, and Microemulslons”, Vol. 1, K. L. Mittal, Ed., Plenum Press, New York, 1977, pp 171-94. ?
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