J. Phy~.Chem. 1981, 85, 2511-2519
YDS/YDE
= [eRT/(8rWl[(t+ - t-)(l+ MI)+ BT] (A.8)
Appendix B In order to solve eq 3.3 for S and D, we write them in matrix form
PU = 0 (B.1) where P is the linear operator matrix and U is the solution vector. If W is defined such that DET(P) W = 0
(B.2)
2511
{A(a4/dx4) + B[d3/(dt ax2)] + C(d2/dx2) + ( d 2 / d t 2 ) + G(d/dt))W= 0 (B.5) where
A = YSSYDD - YSDYDS B = -(YSS + Y D D )
c = YSflDS - YDflSS G = YDE 03.6) Since W, in this case, must be an odd function about x = 0, we expand the solution in a Fourier sine series m
W(x,t)= C bk(t) sin ( k r x / a )
then it can be shown that
k=1
Vi = COFji(P) W
03.3)
where j can be any i (the cofactor can be expanded along any row). For our system
(a/at) - rss(az/ax2) -rsD(azlax2) + ~
S E
( a / a t ) - r D D ( a z l a x z )+ Y D E
(B.4)
The equation for W is
03.7)
Substitution of eq B.7 into eq B.5 produces the ordinary differential equation bk”(t) + [G - (kr/a)2B]b’k(t) + [ ( k r / ~ )-~(Ak ~ / a ) ~ C ] b k (=t )0 (B.8) The solutions of eq B.8 are bk(t) = L+k eXp(-R+kt)
+ L-k eXp(-&t)
(B.9) where R+k and R-k are given by eq 4.2, and L+k E d L-k are determined by using eq B.3 and the initial conditions. The result is eq 4.1.
Growth and Size Distributions of Cetylpyridinium Bromide Micelles In High Ionic Strength Aqueous Solutions GrQgolre Porte Centre de Dynamlque des Pheses Condens&s,t U.S.T.L. Montpllier, France
and Jacqueline Appell laboratolm de Spectrom6trie Rayb&h-Brlllouln, In Flnal Form: May 13, 1981)
* U.S.T.L. Montpelller, France (Received: February 13, 198 I;
The growth of cetylpyridinium micelles in high NaBr concentration aqueous solutions is observed by the use of quasi-elasticlight scattering. The variations of the mean hydrodynamic radius, R H , are measured over wide ranges of temperature (27 < T < 75 “C),detergent concentration (1.5 X 10-9-40X M), and added NaBr concentration (0.2,0.4,0.6, and 0.8 M). These results are interpreted in the frame of a multiple equilibrium description of the micellar elongation. This now classical description is here rewritten in order to incorporate the counterion influence through the use of the crude ion binding approximation. We then developed a quantitative geometrical model for the elongated micelles which, from our previous studies, are known to have the shape of long flexible spherocylinders. The obtained relation between RH and the mean aggregation number ( N ) allows us to compare quantitatively the experimental results to the predictions of the theory. The agreement is found to be very good in the range of concentration of the detergent where the ideality condition stands. Furthermore, the ion binding approximation appears to be unexpectedly good in predicting the variations of the micellar size with the temperature and the NaBr concentration. In the range explored in this work, the true N-micelle apparently behaves like a chemical species the enthalpy and entropy of formation and the chemical composition of which are independent of the temperature and of the added salt concentration. It is, however, concluded that this convenient equivalency is not necessarily significant and rather indicates that the functional dependence of the elongation is probably not discriminative between detailed a priori descriptions of the ionic environment of the micelles. Introduction In recent years, Mukerjee‘ initiated a simple theoretical description of the evolution of size and shape and partic‘Laboratoire Associ6 au Centre National de la Recherche Scientifique (LA 233). *Equipe de Recherche Associee au Centre National de la Recherche Scientifique (ERA 460). 0022-3654/81/2085-2511$01.25/0
ularly of the sphere-to-rod transition of micelles appearing in solutions of amphiphilic molecules. This description is given on the basis of a thermodynamic theory of multiple equilibrium (or equivalently of mass action law). Israelachvili et al.2 have &own that the sphere-to-rod transition (1)P. Mukerjee, J. Phys. Chem., 76, 565 (1972).
0 1981 American Chemical Society
2512
The Journal of Physical Chemistry, Vol. 85, No. 17, 1981
was one among other possible evolutions of the shape and the size of micelles which can be described by this theory. This theoretical description applies rigourously to nonionic detergents, but it can easily be extended to the case of ionic surfactants in solutions of high ionic strengths1-’ In such cases, however, additional assumptions regarding the ionic environment of the micelles are required to interpret the variations of the micellar size with the concentration of added salt. Such a step has been taken by Mazer et aL5and Missel et al.? who developed a model for the growth of micelles from spheres to spherocylinders. These authors obtained complete experimental confirmation of this description from a systematic investigation on aqueous solutions of sodium dodecyl sulfate (SDS) + NaC1. Moreover, these authors also obtained a consistent interpretation of the influence of the added salt concentration on the micellar size by using the Gouy-Chapman double-layer approximation to account for the ionic environment of the SDS micelles. The particular weakness of this purely electrostatic description is that it only takes into account the valency of the counterions, disregarding their chemical nature. This is in contradiction with the experimental evidence on the striking specificity of the counterions to induce the elongation of micelles. An illustration of this specificity can be found in the light-scattering study of Anacker and Ghose8 on micellar solutions of cetylpyridinium salts with various monovalent counterions (P, C1-, NO,, C103-, etc.). As suggested by Mukerjee,l an alternative procedure to the Gouy-Chapman double-layer approximation is the counterion binding approximation. In this latter approximation the nature (electrical and/or chemical) of the interactions between the micelle and the counterions is not a priori stated and the approximation can include the experimentally observed specificity. In the course of our previous studiesgJOon the shape of cetylpyridinium bromide (CPBr) micelles (in aqueous solutions with added NaBr), we noted the large increase of micellar sizes with increasing concentrations of CPBr and NaBr and with decreasing temperature. This prompted us to undertake a systematic study in order to compare this evolution to the predictions of the theory. The results of this study are described herein. In the first part of the paper, we outline briefly the theory for ionic surfactants within the counterion binding approximation. We obtain a manageable algebraic relation between the mean aggregation number of the micelles and a pair of thermodynamic parameters. The general predictions of the initial theory are easily derived from this relation (e.g., the mean aggregation number is proportional to the square root of the detergent concentration, and the width of the micellar size distribution increases dramatically with increasing mean aggregation numbers). But, in addition, a term accounting for the strong influence of the (2) J. N. Israelachvili, D. J. Mitchell, and B. W. Ninham, J. Chem. Sac., Faraday Trans. 2, 72,1525 (1976). (3) C. Tanford, J. Phys. Chem., 78, 2469 (1974). (4) H. Wennerstrom and B. Lindman, Phys. Rep., 52, 1 (1979). (5) N. A. Mazer, M. C. Carey, and G. B. Benedek, J.Phys. Chem., 80, 1075 (1976); Micellization, Solubilization, Microemulsions, [Proc. Int. Symp.], 1976, 1, 359 (1977). (6) P. J. Missel, N. A. Mazer, G. B. Benedek, C. Y. Young, and M. C. Carey, J. Phys. Chem., 84, 1044 (1980). (7) R. Tausk and J. Th. G. Overbeek, Colloid Interface Sci., [Proc.Int. Conf.], 50th, 1976,2, 379 (1976). (8) E. W. Anacker and H. M. Ghose, J. Am. Chem. SOC.,90, 3161 (1968). (9) G. Porte, J. Appell, and Y. Poggi, J. Phys. Chem., 84,3105 (1980). (10) J. Appell and G . Porte, J. Colloid Interface Sci., 81, 85 (1981).
Porte and Appell
added salt concentration appears explicitly in the thermodynamic parameter. We follow the evolution in size of the micelles by using an experimental procedure similar to that used by Mazer et al.s and Missel et ala6 From the quasi-elastic light spectra (QELS) of the micellar solutions we deduce an average hydrodynamic radius, RH,which reflects the average size of the CPBr micelles. R H is measured as a function of the concentration in CPBr and NaBr and as a function of the temperature. The experimental procedure and the experimental results are described in the second part of this paper. In order to compare these results to the predictions of the theory, we must establish a quantitative relationship between the mean aggregation number and RH. The geometrical model assumed for the micelle and the details of the calculations are given at the end of the first section. As shown in the third section of the paper, good agreement is found between the theoretical predictions and the experimental results over a significantly large range of CPBr concentrations. Finally, we try to discuss the action of the added salt in terms of the counterion binding approximation.
I. Theoretical Model We describe in this section how the ion binding approximation can be introduced in the thermodynamic theory for micellar growth from spheres to spherocylinders which has been developed by Tausk and Overbeek,’ Mazer et al.,5 and Missel et a1.6 We first derive from the thermodynamic theory an expression for the distribution of micellar size. Then, assuming a reasonable geometric structure for the individual micelles, we will derive their hydrodynamic radius and the intensity of light which they scatter. We will then be able to compute the average values to be compared to the experimental results. ( A )Distribution of Micellar Sizes. In a thermodynamic theory devised to describe the formation of the micelles of ionic surfactants, one should, in principle, consider the aggregation number (N) and the number (a)of associated counterions to be the unknown variables. The equilibrium distributions of these numbers are then simultaneously determined by the equilibrium relations. However, with such a starting point, heavy calculations are necessary with a detailed knowledge of the interactions between micelles and counterions. Thus, we make the simplifying assumption that, a t equilibrium, the number of counterions bound to a micelle of aggregation number N is narrowly dispersed around its mean value a(N);the formation of such a micelle can then be described through the “quasi-chemical” reaction involving counterion binding: KN
Nbl + a(N)cl (mic)N (1) Here bl denotes the free detergent ion, c1 is the counterion, and KN is the equilibrium constant of reaction 1. Classically, one can consider reaction 1as the result of a series of stepwise association reactions. Consistent with the experimental evidence for a sharp cmc for long-chain surfactants (see, e.g., ref 4),we assume that there are no premicellar aggregates, and the first step of the series is
nobl
+ a(no)cl
Km
(mic),
(2)
where no is the aggregation number of the minimum-size spherical micelles. The following steps are obtained by successive additions of one single monomer until the aggregation number N is obtained (mic)q-l + bl
+ aqcl
k,
(mic),
(3)
The Journal of Physical Chemistry, Vol. 85, No. 17, 1981 2513
Cetylpyrldlnlum Bromide Micelles
with KN = K,&k, N
a(N) = a(no) +
Ca,
(6)
no
In agreement with our previous studies?'O we assume the large micelle to be a long flexible spherocylinder containing n monomers in the cylindrical part and nomonomers in the hemispherical ends:
N=no+n
(7)
Following the arguments of Mukerjee,' it is clear that, when N is sufficiently large, the k , as well as the aq)s become independent of N . To account for this asymptotic behavior together with the existence of the initiating reaction 2, we write KN = K,k," = ksWkcn
(8)
+ na, = noa, + na,
a(N) = a(n0)
x
(IO-~M
)
Figure 1. Illustration of the theoretical predictions for (1) (bl), (2) X - (bl), and (3) ( n ) as a function of Xfrom relations 18 and 19 with (bl),, = lo-' M, K 4.8 X lo8, and no = 100. This corresponds roughly for CPBr to a solution with CbBr = 0.2 M and T = 30 OC.
Then X o and (bl) can also be expressed as functions of ( n ) , and we get the following algebraic relations:
(9)
where the subscripts s and c refer to the hemispherical or cylindrical position of the monomers in the micelle. Indeed, the differences between k, and k, as well as between a,and a, are expected to be determinant for the mean elongation of the micelles.1~2~6~6 Assuming no intermicellar interactions, the mass action law applied to reaction 1 can be written as
X, = [k~(~~)~~(bi)l~[k,(ci)"~(bi)l" (10) where X, is the concentration of micelles of aggregation number N , (b,) is the concentration of free monomers, and (cl) is the activity of the counterions. And the two equations of conservation for the detergent and the counterions are
X = (bJ ( ~ 1 )N
(c,)
+ C(n + no)X, n
(11)
+ (1 - a ) [ X - Wl + (bi)
(12)
(c,)
when X
