SIZEDISTRIBUTION OF MICELLES
565
The Size Distribution of Small and Large Micelles: a Multiple Equilibrium Analysis1
by Pasupati Mukerjeea Chemistry Department, University of Southern California, Los Angeles, California 90007, and the School of Pharmacy, University of Wisconsin, Madison, Wisconsin 68706 (Received August 26, 1971) Publication costs borne completely by The Journal of Physical Chemistry
Some general procedures for dealing with multiple equilibria in stepwise self-association when all species are in rapid association-dissociation equilibrium are extended to and adapted for micellar systems. It is possible by this method to determine the number average degree of association,N,, from the concentration dependence of the frequently measured weight average value, N,. The size distribution index, N,/N,, is found to be close to unity for many systems, nonionic and ionic, when micelle sizes are small, corresponding to spherical or spheroidal micelles. For large micelles, the size distribution is wide. A step-wise self-association model is derived for such large micelles.which predicts an N , / N , value of 2 and a linear increase in N , with the square root of the concentration of micellized surfactant, in agreement with some experimental data for a nonionic and an ionic system in dilute solutions. In concentrated solutions of the large micelles, the nonideality effects become very large. The above self-association model, along with some approximate nonideality corrections, can successfully reproduce the shapes of some observed reduced turbidity data showing minima. Some other consequences of the self-association model are also examined.
Introduction Polydispersity in micellar systems has been discussed from a theoretical point of view by many investigators3-’ Experimental measures of the polydispersity have been difficult to obtain, however. While weight average molecular weights of micelles are known for numerous systems from light scattering and equilibrium ultracentrifugation, very few reliable numberaverage molecular weights have been determined.*pg The first part of this paper deals with a general method and its application for determining the polydispersity of micellar systems first presented several years ago.’ I n this method, the number average degree of association of the micelles, Nn, is determined from the variation of the corresponding weight average value, N,, with concentration, thus permitting the size distribution index, Nw/”n, to be determined for many systems for which N , data are available. The general procedures for handling multiple equilibria in step-wise self-associat ion when all species are in rapid association-dissociation equilibrium have usually been applied to systems showing small numbers of ~ligomers.’O-~~The extension and adaptations of these methods to micellar systems, as presented earlier,’ are different in procedure and details from subsequent publication^.^*^ The major part of this paper is concerned with the application of these methods to both nonionic and ionic micellar systems, the difficulties arising from nonideality effects due to intermicellar interactions, and the derivation and use of a simple step-wise self-association model for dealing with large micelles, both ionic and nonionic, which are shown to be polydisperse.
The Multiple Equilibrium Model and the Relation between Nn‘ and Nw‘. If the multimer b, containing q monomers has a molar concentration [b,] the equilibrium constant for association in an ideal nonionic system, P,, is given by where [bl] is the concentration of monomers. If we now define the quantities S, C,and G as
(1) Presented in part at the 152nd National Meeting of the American Chemical Society, New York, N. Y., Sept 1966. (2) School of Pharmacy, University of Wisconsin, Madison, Wisconsin 53706. (3) 0. Lamm, Arkiv. Kemi. Mineral Geol., 18A, No. 9 (1944). (4) M.J. Vold, J . Colloid Sei., 5,506 (1950). (5) D. Stigter and J. Th. G. Overbeek, Proc. Int. Congr. Surface Activ., 2nd, 1957, 1, 311 (1957). (6) D.G. Hall and B. A. Pethica, in “Nonionic Surfactants,” M . J. Shick, Ed., Marcel Dekker, New York, N. Y., 1967. (7) J. M. Corkill, J. F. Goodman, T. Walker, and J. Wyer, Proc. Roy. SOC.Ser. A , 312,243 (1969). (8) H.Coll, J . Phys. Chem., 74, 520 (1970). (9) D.Attwood, P. H. Elworthy, and 5. B. Kayne, ibid., 74, 3529 (1970). (lo) R. F. Steiner, Arch. Bwchem. Bwphys., 39, 333 (1952); 44, 120 (1953); 49, 400 (1954). (11) F.J. C. Rossotti and H. Rossotti, J . Phys. Chem., 65,926,930, 1376 (1961). (12) 1’. Mukerjee, ibid., 69,2821 (1965). (13) A. K. Ghosh and P. Mukerjee, J . Amer. Chem. Sac., 92, 6408 (1970). The Journal of Physical Chemistry, Vol. 76, N o . 4, I972
PASUPATI MUKERJEE
566
where S is the total molar concentration of all species including the monomer, and C the total equivalent concentration, then the average degrees of association of all species, including the monomer, Nn’ and N,’, at any equivalent concentration C, are given by
Nn’ = C/S
(5)
N,‘ = G / C (6) From the definitions of S , C , and G (eq 2-4), it can be shown by diff erentation that irrespective of how many or which multimers form dS/d[bil = C/[bil
(7)
dC/d [biI = G / [biI
(8)
d In C d In [bl] = N,’(C)
(9)
and, therefore
and
The symbol N,’(C) indicates that N,’ is a function of C . If this function is known or determined experimentally, [bl] and S can be determined by graphical integration and Nn’ can be determined. The estimate of [bl] is extremely valuable in any detailed analysis of self-association equilibria, as shown recently for some dye systems.13, l4 The above method for determining Nn’ from N,‘ data is generally applicable. For micellar equilibria, eq 11 below is more suitable.
(G - [bl])/(C - [bl]) represents the weight-average degree of association for all associated species, Le., excluding the monomer. Although the equation is still exact, the usc of experimental N , values in practical cases is facilitated by the use of a slight approximation inherent in the determination of N , itself. N , is determined above the critical micelle concentration (cmc) by assuming that the equivalent concentration of micelles is C - cmc, the corresponding approximation being that the monomer concentration above the cmc remains constant, and has the same value as the cmc. The validity of this approximation improves as the value of N , itself and the ratio C / [bl] or c/cmc increases in the experimentally accessible region of C. This is implicit in the exact equation
Within the range of the validity of the approximations, therefore, it is possible to write for concentrations above the cmc The Jotirnal of Physical Chemistry, Vol. 76, No. 4, 1978
N, = G
C
- cmc -C ~
$=3
C
(G (C
-
[bl]) [bl])
(13)
and, therefore d(S
- cmc)
=
d(C - cmc)
(14)
N4C)
By graphical integration, it is possible to determine Nn(C) defined as (C - cmc)/(S - cmc). As is to be expected14N , cannot decrease with increasing C, because, then Nn > N,, an impossible result. If N , is approximately constant with C , as is frequently observed, Nn = N,, and the micellar size-distribution is narrow. If N , increases with C as observed in many system^,'^-'^ N , > Nn, and there is a distribution in micelle sizes. For approximate purposes, the variation (C - cmc)a. of N , with C may be put in the form N , The size distribution index, N,/Nn, then, from equation 14, has the value 1/(1 --a).
Micelles with Narrow Size Distributions Nonionic Systems. The application of the above analysis to existing literature data suggests that for some nonionic micellar systems, including several zwitterionic ones,18-20 the size-distributions are quite narrow. These systems show characteristically horizontal lines in the usual plots of reduced turbidity against C - cmc, i.e., apparent N , values which do not change with C, or an apparent reduction in N , because of excluded volume interactions. Two points of caution must be mentioned, however. As the concentration of micelles increases, intermicellar interactions which lead to positive second or higher virial coefficients may compensate for or mask to some extent the decrease of the reduced turbidity curves with concentration corresponding to an increase in N,. Thus, the conclusion that size-distributions are narrow when reduced turbidities show small variations with concentration should be made only when nonideality effects are expected to be small. If no pronounced asymmctry of the micelles is expected from the N , values, i.e., the N , values are consistent with spherical or spheroidal micelles, and if the absolute concentrations are low, reasonably firm conclusions can be drawn for nonionic systems which do riot show long-range intermicellar interactions. The second point of caution arises from the fact that if highly polydisperse micelles form, N , increases rapidly with C, as shown later with examples. For such (14) P. Mukerjee and A. K. Ghosh, J . Amer. Chem. SOC.,92, 6403 (1970). (15) P. H. Elworthy and C. B. Macfarlane, J. Chem. Soc., 907 (1963). (16) R. R. Bdmbra, J. S. Clunie, J. iM.Corkill, and J. F. Goodman, Trans. Faraday Soc., 60, 979 (1964). (17) P. Debye and E . W. Anacker, J. Phys. Colloid Chem., 55, 644 (1951). (18) P. Becher, J . Colloid Sci., 16, 49 (1961). (19) K. W. Herrmann, J . Phys. Chem., 66, 295 (1962). (20) K. W. Herrmann, J . Colloid Interface Sci., 2 2 , 352 (1966) I
SIZEDISTRIBUTION OF MICELLES systems, the apparent cmc from the turbidity data may be too high.l6 If such high estimates of the cmc are used, spurious horizontal curves of the reduced turbidity vs. concentration may be obtained.17 I n general, consideratiohs to be discussed shortly indicate that if N , is large, requiring highly asymmetric micelles, the size-distribution is probably wide.
Ionic Systems For ionic micelles, eq 1 is not directly applicable because charge effects are not included. The charge effects can be incorporated in the equilibrium constant either in terms of a physical picture based on the calculated electrostatic free energy of the electrical double layer6sZ1or in terms of a chemical mass-action picture, using the counterion binding approximation.21-22 At a constant ionic strength and a constant counterion concentration, either approach would lead to a constant value of P,*, however, where [b,l/[bll* (15) because the charge effects are constant. Thus all the equations derived above from 2 onward are applicable to ionic micelles as long as the effect of the changing concentration of micelles themselves on the counterion concentration-as also composition since counterions may show specific e f f e ~ t s ~ ~ ~ ~ be ~ - neglected. can This assumption is likely to hold true for many systems studied in presence of added salt, with common counterions. Numerous systems of this kind have been studied.24-26 When N , is not extremely high, they show similar characteristic features in that the reduced turbidity curves in the typical Debye plot are linear and increase with concentration, ie., the apparent molecular weight, Mw(s)ldecreases with concentration. This is ascribed to repulsive intermicellar interactions leading to positive second virial coefficients. Detailed theories of these virial interactions have been developed25and the theories give nearly quantitative predictions of the virial coeffi~ients.~~ When corrected for shows very little these intermicellar interactions, MWc,) change with concentration in many systcms a t high salt concentrations. T o take a typical example, sodium dodecyl sulfate in 0.1 M NaC126shows about a 25% decrease in Mw(*)as the micellar concentration C - cmc changes by a factor of about 30. The theoretically calculated virial coefficient is higher than the experimental value but the difference is less than 20%.25 Thus if there is a decrease in the molecular weight, M,, it is likely to be about 5% or less over the whole concentration range. The use of equation 14 here leads to a value of less than 1.03 for the size-distribution index Nw/Nn. Similar calculations at a higher NaCl concentration (0.3 M ) where the nonideality effects are even smaller show an Nw/Nn ratio even closer to 1.00, the value that characterizes a monodisperse system. It PP"
=
567 seems that micelle size-distributions are indeed narrow in many cases and the assumption of monodispersity in many theoretical treatments is well justified. At low salt concentrations the nonideality effects for ionic micelles are so large that even small uncertainties in the virial coefficients can hide appreciable changes in N,, and the above method is difficult to apply. It should be stressed that interactions between micelles are likely to be repulsive in most cases so that the nonideality effects in general oppose the trend produced by changes in N , which can either remain constant or increase with C. An apparently negative second virial coefficient is usually a good indication that the micellar system is polydisperse.
Polydispersity in Systems with Large Micelles When N , is very large, the micelles are asymmetric, and are probably rod-like17with some flexibilit~.~7The polydispersity of such micellar systems can be predicted on the basis of a simple model of self-association, assuming ideality, Le., negligible intermicellar interactions. I n nonionic micellar systems, the step-wise association reaction bg-1
+ bl =
bq
(16)
is governed by a step-wise association constant, K,, given by
Our previously defined ,8, then becomes Q PP
=
rIKQ 2
For ionic micelles, at constant ionic strengths and constant counterion concentration and composition, K , is still a constant as defined above but is labeled K,*, to emphasize the fact that it is not a true equilibrium constant. If we consider thc growth of micelles from small i clear that K , spherical ones to large spherical oncs, it R or K,* will vary with q. I n the case of ionic micelles, the free energy change associated with the introduction of a monomer into a micelle with q - 1 monomers is composed in part of a hydrophobic component, AGHc1 which is primarily rcsponsible for the formation and growth of micelles, and an electrical component, AGEL, (21) P. Mukerjee, Advan. Colloid Interface Sci., 1, 241 (1967). (22) P. Mukerjee, K. J. Mysels, and P. Kapauan, J . Phys. Chem., 71, 4166 (1967). (23) E. W. Anaeker and H . M. Chose, ibid., 67, 1713 (1963). (24) K. J. Mysels and L. H. Prineen, ihid., 63, 1696 (1959). (25) F. Huisman, Proc. K o n . Ned. Akad. Wetensch., Ser. B, 67, 367, 376, 388, 407 (1964). (26) M . F. Emerson and A. Holtzer, J . Phys. Chem., 71, 1898, 3320 (1967). (27) D. Stigter, ibid., 70, 1323 (1966).
T h e Journal of Physical Chemistry, Vol. 76, No. 4, 197%'
568
PASUPATI MUKERJEE
which opposes micelle expressed as
-RT In K,*
K,* can thus be = AGHC iAGEL
=
AGHC
+ AGHG
=
Kz‘, K3 = Kq = K , = K
Kz’
X
[blI The Journal of Physical Chemistry, Vol. ‘76, N o . 4, 1972
-{
Kz’ X(2 K (1 - X)’
(23)
When Kz’ = K, i.e., all stepwise association constants are the same, the equations above pertain to the well known case of the most probable distribution.
(20)
(21)
This model of stepwise association” ha8 been previously investigated for the self-association of dyes.13 On summing the series for 8, C , and G and putting K [bl] = X , we obtain
X - [bl]
=
(19)
where AGHC is the head group self-interaction contribution. For spherical (or spheroidal) micelles, AGEL is known to vary with the radius r , and therefore q.*8 Since the addition of a monomer changes the surface density of the head groups and the volume available to each monomeric chain, and therefore its packing characteristics, AGHGand AGHCare also expected to depend upon q. For sufficiently large rod-like (or disk-like) micelles however, if the end effects can be neglected, and if the radius is assumed to remain constant, the quantities AGHGand AGEL become independent of q, because the surface group density and geometry do not change on the addition of a monomer, whereas AGHCbecomes independent of y because the local structure around a micellized monomer and the volume available to it do not change either. For very large micelles, therefore, K , and K,* become independent of q. The arguments above lead to a schemat,ic diagram of the variation of log /3, or log /3,* with q (Figure 1). Log p, here is a measure of the free energy of formation of a micelle from the monomers. At low p values, log 0, or log pq* is expected to increase with q nonlinearly, reflecting the fact that for small micelles the stability increases with size, and log K, or log K,* increases with q. After a transition region close to the fully formed spherical micelle, the details of which are very important for small micelles, asymmetric micelles form. When these become large, at high q values, log pq or log p,* increases linearly with q, with a constant slope, log K . When p values are very large, well outside of the transition region, the size distribution will be controlled mainly by the linear upper part of the log pp or log on* vs. y curve (Figure 1). If this linear curve is extended downwards to q = 2 to obtain log K2’ wherc Kz’ is a hypothetical dimcrization constant, then the stepwise association scheme, whcn q is very large, can bc represented by a two-parameter model, where
Kz
[b]
[blI
where R is the molar gas constant and T is the absolute temperature. For nonionic micelles, micelle formation and growth is opposed by the free energy change due to the self-interaction of the hydrophilic head groups of the monomers at the micelle surface.21 Thus
-RT In K,
C -
, 2
-Kx)
9
Figure 1. Schematic diagram of the variation of log pg or log pn* with q.
Equations 22-24 can be used to obtain N , and N , for micelles in terms of Kz’, K , and [bl]. When N , is large, X FS 1, so that X can be considered to be unity in all terms excepting 1 - X. For such a casc, we have ~
N,
=
_
_
_
2 1 / K / K z ‘ d ( C - [bil)/[bil
_
(25)
If now C >> [bl], [bl] = cmc, and [bl] is assumed to be constant above the cmc, equation 26 is obtained. _
N,
_
= 2dK/K2dC/cmc
~
(26)
The association model, therefore, predicts that for large micelles, N , or M , should be porportional to d C - [bl] or 4 2 when C >> [bl]. The size distribution index N,/N,, under these assumptions, is calculated to be 2.0. For testing the association model, it seems that few systems have been studied for which sufficient data are available at low enough concentrations to make intermicellar interactions in light-scattering studies negligible. Figures 2 and 3 show two suitable examples. The CleH33(EO)7systeml where EO stands for an oxyethylene group, was studied by Elworthy and Macfarlanel6 whereas the CleH3aN(CH3)3Brin 0.178 KBr was studied by Debye and A n a ~ k e r . ” *The ~~ molecular weights have been obtained from the turbidity data at eaqh concentration and are shown in Figures 3 and 4 without any corrections and when corrected for dissymmetry and second virial coefficients. Both were calculated using the rigid-rod model, the parameters used being given in the legends of Figures 2 and 3. (28) P. Mukerjee, J. Phys. Chem., 73, 2054 (1969). (29) E. W. Anacker, private communication.
SIZEDISTRIBUTION OF MICELLES
569
I
C
o0 v
2
4
8
8
I
4-c xlop
Figure 2. Variation in micellar molecular weights with &. Molecular weights calculated from turbidity data for C16H,,(OCHzCHt)70H a t 25’16 (C in g/ml). Key: 0, uncorrected molecular weights; 0, corrected for dissymmetry; A, corrected for dissymmetry and second virial coefficients. Micelle dimensions calculated for the rigid cylinder model using a hydrocarbon core density of 0.775, a core radius of 20 A, and 10 A for the polyoxyethylene layer thickness.
8
Figure 3. Variation in micellar molecular weights with &. Molecular weights calculated from turbidity data for Cl6H3,N(CH&Br in 0.178 M KBr17**9(C in g/ml). Key: 0, uncorrected molecular weights; 0,corrected for dissymmetry; A, corrected for dissymmetry and second virial coefficients. Micelle dimensions calculated for the rigid cylinder model assuming density = 1.00, radius = 24 A, and 8 A for the effective thickness of the electrical double layer.
Figure 4. Variation in the monomer concentration, [bl], with the total concentration C from equation 23 assuming K = 1 for curve 3. and Kt’ = 1 for curve 1, 10-2 for curve 2, and
Prolate ellipsoids of r e v ~ l u t i o nlead ~ ’ ~t~o ~substantially higher corrections. The uncertainties in the corrections are several. For example, the virial coefficient itself should reflect a higher moment of the size distribution than the weight average and is thus underestimated. On the other hand, the rod-like micelles are probably fle~ible,~’ and thus the correction is over-estimated. However, the corrections are not very large even in the extreme cases, and Figures 2 and 3 show that the predicted linear variation of the molecular weight with dE holds for both systems, even at the lowest concentrations where the corrections are quite small. Thus, the simple self-association model developed for very large micelles fits both nonionic and ionic systems. Independently of the self-association model discussed above, the fact that the molecular weight varies linearly with itself leads to an N , / N , value of 2.0 through the use of eq 14. The size distribution is thus wide. It is interesting to note that whereas the most probable distribution model, where all K’s are the same, leads to a slow increase in the degree of association with concentration, with no sharp change that can be characterized as the cmc, the above distribution model, which can be described as a modified most probable distribution model and in which only the first constant Kz is allowed to vary, can lead to quite sharp changes characteristic of micellar systems. This is illustrated in Figure 4 in which log [bl] is plotted against>log C for an assumed value of K = 1 and different values of Kz’, the curve alusing eq 23. For a Kt’/K ratio of ready simulates a system with a sharp cmc. Figure 4 illustrates that only a mild cooperativjty is needed to produce micellar aggregation, considering that the value of K itself for many nonionic systems is in the range of
The corrections were applied assuming the micelles are monodisperse and using their weight average molecular weights. The corrections for the second virial coefficient were made using a reiterative procedure in which the apparent molecular weight at any concentration (30) A. Isihara and T. Hayasida, J . Phys. Soc., Jap., 6, 40 (1951); was first corrected for dissymmetry and then fitted to a T.Kihara, ibid., 8, 686 (1953). true molecular weight and the corresponding second (31) J. 0.Hirschfelder, C. F. Curtiss, and R. B. Bird, “Molecular Theory of Gases and Liquids,” Wiley, New York, N. Y.,1964. virial coefficient correction obtained by using Isihara’s (32) T. Kihara, J. Phys. Soc., Jap.,6, 289 (1951). e q u a t i o n ~ for ~ ~ ,rigid ~ ~ cylinders (see later). If the minor axis is the same, the use of Kihara’s e q u a t i ~ n ~ l s ~(33) ~ A. Isihara and T. Hayasida, ibid., 6,46 (1951). (34) C. Tanford, “Physical Chemistry of Macromolecules,” Wiley, for prolate sphero cylinders makes little difference. New York, N. Y., 1961. The Journal of Physical Chemistry, VoL 76, No. 4, 1972
PASUPATI MUKEPJEE
570
I
0
0.5
cx
I .O IO' (gm,/ml)
I
2.0
1.5
Figure 5. Simulated Debye plots using equation 26. cmc = g/ml;16 K/K2' = 2.5 X 108 for upper curves, 1.44 X 5X lo4 for lower curves; dashed lines represent uncorrected molecular weights, full lines after correction for second virial coefficients using the rigid cylinier model for micelles, assuming density = 1.00, radius = 26.7 A. Experimental data for C14H2s(OC€LCH~)60H estimated from Balmbra, et al.16 Key: 0, 25', A, 30".
104-106 so that plo0, for example, is in the range of 10400-600. When molecular weights are high, and the micelles highly asymmetric, excluded volumc interactions alone can be so strong as to completely obliterate the effect of increasing M , with C. This is illustrated in Figure 5 in which same Debye plots have been simulated by using the sclf-association model described above (eq 26) to calculate M , and eq 27 below to calculate the corrcctions for intermicellar interactions.
II(C - cmc) (7
-
romo)
1 Mw(a)
-
4-
'"
(27)
M w
points were estimated from the Debye plot given for CUH~C,(EO)~ a t 25" and 3O0.I6 In view of the numerous sources of uncertainty in the calculations and the use of the turbidity data, exact agreements are not to be expected. The deviations are qualitatively in line with over-estimates of the second virial coefficients on using the rigid-rod model and neglect of the higher order interaction terms. The important points to be made here are that the shapes of the experimental curves are reproduced well as also the positions of the minima. The nonideality corrections can clearly be important enough to change the shapes of the curves. The apparent molecular weights obtained from the minima of experimental curves of the kind shown in Figure 5 should not be used for comparative or theoretical purp o s e ~ 'because ~ ' ~ ~ the true M , may be very different and may remain a strong function of the concentration. Experimental molecular weight data of the kind shown in Figures 3 and 5 have sometimes been explained by invoking a reversible aggregation of the micelles t h e m s e l v e ~ . ' ~This ~ ~ ~procedure does not seem to be justified. For rigid cylinders of constant radius, the shape factor f becomes proportional to 1Ww when M , is high. Equation 27, therefore, predicts a constant value of r when 84f >> 1. For C14H29(E0)6 micelles, the parameters of Figure 5 lead to a maximum r value of 6.4 X lo2which appears to be of the right order of magnitude from the data presented by Balmbra, et al.16 Even an approximate agreement here may be fortuitous, particularly because third and higher virial interactions must be important for high concentrations. Qualitatively, however, the introduction of these higher order interaction terms in equation 27 would suggest that r can go through a maximum, Such maxima have been Thus, even this apparexperimentally observed. ently unusual behavior is not incompatible with the model of multiple equilibria presented in which M , continues to increase with C. l6t3'
H hcrc is the optical constant in light scattering, r is the turbidity, and 4 is the volume fraction of micelles. The concentration C and cmc arc expressed in g/ml units. f is the shape factor (f = 1 for spheres) which was obtained by using Isihara's equation for rigid cyli n d e r ~31 ~ ~ 7
f=--
Tr2
+ + 3)rl + l2 (T
8rl
(28)
where 1 is the length and is the radius, and the parameters are given in the legend of Figure 5. The diffcrence bctwecn the upper and lower curves is produced by changing only one constant which is truly disposable, namely the ratio K / K a ' . The experimental
The Journal of Physical Chemistry, Vol. 76, N o . 4, 1972
Acknowledgment. This work was supported in part by the P.H.S. Research Grant GM 10961-02 from the Division of General Medical Services, U. S. Public Health Service, administered by Karol J. Mysels at the University of Southern California. (35) D. C. Poland and H. A. Soheraga, J. Phys. Chem., 69, 2431 (1565). (36) J. M. Corkill, J. F.Goodman, and T. Walker, Trans. Faruday SOC.,63, 758 (1567). (37) J. M. Corkill and K. W. Herrmann, J . Phys. Chem., 67, 934 (1563).
