Counterion Specificity in the Formation of Ionic Micelles - Size

Chem. , 1967, 71 (13), pp 4166–4175. DOI: 10.1021/j100872a702. Publication Date: December 1967 .... The Journal of Physical Chemistry B 0 (proofing)...
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P. MUKERJEE,K. MYSELS,AND P. KAPAUAN

4166

Counterion Specificity in the Formation of Ionic Micelles-Size, Hydration, and Hydrophobic Bonding Effects'

by Pasupati Mukerjee,2Karol J. Mysels,a and Paz Kapauan Chemistry Department, Unkw&y of Southern Cdifornk, Loa Angeles, Cdifornk

(Received August 8, 1966)

The effects of some monovalent counterions, Li+, K+, Cs+, (CH3)4N+,(C2H,),N+, and (n-C3H7),N+, on the critical micelle concentration (cmc) have been investigated by precision conductometry for the dodecyl sulfate system, and the slopes of the conductivityconcentration curves have been determined. The variation of the cmc with ionic strength has been studied for Li+, (CH&N+, and (C2H6),N+. The results are compared to other pertinent data. The concept of the degree of dissociation of micelles and methods for estimating their free energies of formation are reexamined. The over-all conclusion for the inorganic ions is that it is the interaction of the hydrated ion with the micelle that is important. Smaller hydrated radii favor greater interaction and produce lower degrees of dissociation and more compact double layers. This ionic size effect can be overshadowed for the quaternary ions, however, by the hydrophobic bonding between the micelle surface and the hydrocarbon exterior of the quaternary ions. Electrokinetic results and cmc values seem most meaningful in direct comparisons between counterions although their interpretation may not be rigorous. On the other hand, the slope of the log cmc us. log counterion concentration plot can be subjected to a formally rigorous analysis which also permits an improved calculation of free energies. Comparison of the estimates of degree of dissociation is less direct, however, and may be misleading unless corrections for various nonidealities are made. These are usually not available and not readily determined.

Forces and factors controlling one of the simplest examples of hydrophobic bonding, namely the formation of micelles, continue to be of interest and, particularly, the role of counterions in this process has been the subject of considerable Recently, specific ionic effects have been reported in and soap-film studies,'* with reference to preliminary announcements of Some of our early work,lS which has Of not been published because Of interpretation. This prompts us to report it at present with emphasis on the experimental results and on some problems which these raise. In the niIeguro and co-workersll-'s have studied the effect of counterions upon the cmc and have reached independently some conclusions similar to ours. The systems studied and the results obtained by us are, however, sufficiently different and more extensive to justify detailed reporting. The formation of an ionic micelle from monomeric The Journal of Physical Chemistry

ions results from a balance between hydrophobic interactions between the hydrophobic part of the amphipathic micelle-forming ions, electrostatic interactions between their hydrophilic charged parts, as well as with (1) Based in part on the Ph.D. diasertation of P. Mukerjee, University of Southern California, 1957; presented in part a t the Kendall Award Symposium, 131st National Meeting of the American Chemical Society, Miami, Fla., April 1957. (2) School of Pharmacy, T h e University of Wiseonsin, Madison, Wis. 53706. (3) Research Department, R. J. Reynolds Tobacco Co., WinstonSalem, N. C. 27102. (4) A. Lottermoser and F. Paschel, Kolloid-Z., 63, 175 (1933). (5) C . S. Samis and G. 6. Hartley, Trans. Faraday SOC.,34, 1288 (1938). (6) P. F. Grieger and C. A. Kraus, J . Am. Chem. Sac., 70, 3803 (1948);P. F. Grieger, Ann. N * Y.A d . 8Ci.t 51,827 (1949). (7) H.Lange, Kolloid-Z., 121, 66 (1951). (8) E.D.Goddard, 0. Harva, and T . G. Jones, Trane. Faraday Soc., 49, 980 (1953). (9) J. F. Voeks and H. v. Tartar, J. Phys. Chem., 59, 1190 (1955).

COUNTERION SPECIFICITY IN THE FORMATION OF IONIC MICELLES

and between the counterions. In addition, changes in hydration energies and specific interactions with counterions may also be important. The strength and importance of these various interactions depend, among others, upon externally controllable factors such as temperature and ionic strength, upon the properties of the particular ions involved, and also upon the concentration and structure of the resulting micelle (or more exactly, distribution of micelles), in particular, its association number, n, its shape, and the compactness of its electrical double layer. Needless to say, the actually existing micelles correspond to the lowest free-energy state of the system. Experimentally, certain parameters of the existing micelles can be determined more or less easily and accurately, but quantitative and sometimes qualitative explanations of these results are not readily available. The study of the effect of counterions eliminates some of the complications by leaving the properties of the amphipathic ion as a constant factor and thus simplifies some of the interpretation of the experimental results. On the other hand, it often leads to complications connected with limited solubility and preparative difficulties. That drastic changes in properties of counterions can have radical effects upon micellization is well known. When compared with Na+ counterions, polyvalent ions usually markedly reduce the cmc3-7~gJ4 and monovalent amphipathic ions may change the cmc by several orders of magnitude and lead to the formation of giant micelles.z0 These effects can be correlated with the well-known role of polyvalent ions in the double layer or with the incorporation of amphipathic counterions within the micelle to give nearly uncharged micelles. The latter effect is also involved in the lowering of the cmc by the addition of even small amounts of certain oppositely charged dyes.z1vz2 I n both cases, the salt formed by the surfactant with these counterionszztends to be insoluble in water and soluble in micelles. On the other hand, early experiments on monovalent ions giving water-soluble oil-insoluble salts suggested that there were no differences among them.4 Later work detected, however, quite substantial ones11-13J5-17 and, as these cannot be due to charge differences nor to solubilization in the body of the micelle, they should be related to the effects mentioned earlier. I n some cases, as with alkylpyridinium iodides, there is good evidence that charge-transfer interactions occur between pyridinium and iodide ions at the micelle surface, which explains the greater observed stability of iodide micelles than of chloride micelles.23 I n other cases, however, as with the alkali metal ions, there is some evidence from partial molal volume measurements that

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the counterions exhibit mainly electrostatic interactionsz4or, as with tetraalkylammonium counterions, no chemical interaction is to be expected on structural grounds. It is in these systems, therefore, where the cmc differences are rather small, that the effect of ionic size and of physical adsorption can be best investigated. Our experiments deal with these classes of counterions and specifically with the dodecyl (lauryl) sulfates of lithium, potassium, cesium, and the symmetrical tetramethyl, -ethyl, and n-propyl ammonium. Experimental Section Most of the materials have been described previously: sodium dodecyl sulfate (NaLS) ,z5 lithium dodecyl sulfate (LiLS),ze tetramethyl (TMLS),z7 tetraethyl (TELS),z7and tetra-n-propyl (TnPLS)z7 dodecyl sulfates. The cesium dodecyl sulfate (CsLS) was prepared by careful turbidimetric titrationz7 of silver dodecyl sulfatez7with spectroscopic grade CsCl in water, followed by recrystallization from water. Conductivity measurements for the NaLS have been described,z8and values for the other systems were obtained by the same method except for LiLS in LiCl for (10) W. Dorst, W. Prins, and J. J. Hermans, Konink. Ned. Akad. Wetenschap. Proc., B59, 190 (1956). The summary should read 60% Na instead of 60% Li (personal communication). (11) K. Meguro, T. Kondo, 0. Yoda, T. Ino, and N. Ohba, Nippon Kagaku Zasshi, 77, 1236 (1956). (12) K. Meguro and T. Kondo, ibid., 80, 818 (1959). (13) K. Meguro and T. Kondo, ibid., 80, 823 (1959). (14) J. M. Corkill and J. F. Goodman, Trans. Faraday Soc., 58, 206 (1962). (15) E. W. Anacker and H. M. Ghose, J . Phys. Chem., 67, 1713 (1963). (16) A. Packter and M. Donbrow, J . Pharm. Pharmawl., 15, 317 (1963). (17) M. J. Schick, J . Phys. Chem., 68, 3585 (1964). (18) M. N. Jones, K. J. Mysels, and P. C. Scholten, Trans. Faraday Sac., 62, 1336 (1966). (19) K. J. Mysels, Final Report, Project NR 356-254, Office of Naval Research, Contract NONR-274 (004). (20) E. W. Anacker, J . CoEZoid Sci., 8, 402 (1953); H. W. Hoyer, A. Marmo, and M. Zoellner, J . Phys. Chem., 65, 1804 (1961); H. W. Hoyer and I. L. Doerr, ibid., 68, 3494 (1964); J. M. Corkill, J. F. Goodman, S. P. Harrold, and J. R. Tate, Trans. Faraday Soc., 62, 994 (1966). (21) E. D. Goddard and T. G. Jones, Res. Correspondence, 8, No. 8, A1 (1955). (22) P. Mukerjee and K. J. Mysels, J . Am. Chem. SOC.,77, 2937 (1955). (23) P. Mukerjee and A. Ray, J . Phys. Chem., 70, 2150 (1966). (24) P. Mukerjee, ibid., 66, 943 (1962). (25) K. J. Mysels and C. I. Dulin, J . CoZEoid Sci., 10, 461 (1955). (26) P. Mukerjee, K. 3. Mysels, and C. I. Dulin, J . Phys. Chem., 62, 1390 (1958). (27) P. Mukerjee and K. J. Mysels, ibid., 62, 1400 (1958). (28) R. J. Williams, J. N. Phillips, and K. J. Mysels, Trans. Faraday SOC.,51, 728 (1955).

VoEume 71, Number 13 Dccember 1967

P. MUKERJEE, K. MYSELS,AND P. KAPAUAN

4168

Table I : Results of Electrokinetic Measurements on Dodecyl Sulfates at 25" Counterion

Na+"

Added salt

IC

x

106,

ohm-'

...

0.139

NaCl 0 . 0 3 M NaCl 0 . 1 M

...

Li +

IdCl 0,0279 M I X l 0.0925 M cs TM

...

+ +

... TMBr 0 . 0 1 M TMBr 0.0284 M

TE +

... TEBr 0.015 M

TnP + K+

... ...

0.154 283.8 913.22 0.06 0.113 113.5 307.10 0.37 149.25 0.22

S

65.2 51.9 54.0 45.0 33.3 93.2 61.6 58.0 49.35 49.95 40.33 41.8

k' X 106, ohm-1

3.521

2.878 294.3 915.73 3.93 2.24 124.15 312.32 121.7 152.92 60.6

S'

CmoI x lo', M

24.6 21.3

8.10

23.7 21.2 19.5 29.8 23.1 21.7 19.62 19.3 15.4 15.7

8.77 5.94 5.41 3.64 2.19 7.OOb

CmoII

x IO',

S',

M

cor

x+

P

%

22.7 21.3 19.3 18.7 14.9 14.9

45.8 42.6 37.2 34.9 31.9 27.1 73.5 41.9 37.9 40.3 30.4 28.0 21.7

4.52 3.85 3.50 4.49 3.88 3.42 4.69 4.75 3.90 4.30 4.95 4.18 5.28

27.5 26.7 33.8 30.3 30.6 32.4 24.0 25.9 25.6 26.0 23.9 21.8 20.5

8.33 3.13 1.49 8.93 3.85 1.81 6.10 5.52 3.10 1.76 3.85 1.49 2.24 7.17*

Based on data of R. J. Williams, J. N. Phillips, and K. J. Mysels, Trans. Faraday SOC.,51, 728 (1955).

which the improved technique of doughnut-shaped cells2Qwas used. Over the range of interest the conductivity data. can be represented by two straight lines with a transition in the cmc region. Table I gives the coefficients of the straight-line equations of the form

K=k+8C

(1)

and K

=

k'

+ 8%

where K is the specific conductivity, k and k' are intercepts, c is the equivalent concentration, S and s' are the slopes, and the prime indicates the region above the cmc. The absolute values of these coefficients are meaningful in salt-free systems, but become less so as the concentration of added salt Nevertheless, the intersection of these lines defines well a cmc value, termed3I cmc 11, which is included in the table. For salt-free systems, a plot of equivalent conductivity os. also approximates two straight lines (with a transition region) whose intersection yields a slightly different crnc value, termed31cmc I, which is also listed. mole/l. at 25" CsLS has a solubility of 5.9 X which is close to its cmc. It gives, however, quite stable supersaturated solutions which permitted measurements a t 25". The solubility of potassium lauryl sulfate and the stability of its supersaturated solutions was not sufficient to permit measurements in the cmc region a t 25". A sample was prepared by metathesis of NaLS with KCI, and data at 32" were obtained but are not reported in detail because of internal inconsistencies. The Journal of Physical Chemistry

a,

At 32'.

As noted p r e v i o u ~ l ythe , ~ ~quaternaries probably contained small amounts of electrolytic impurities judging by their equivalent conductivities. These impurities (less than 0.1% of NaN03 or equivalent) are negligible except in the interpretation of the S' slopes above the cmc, and these have been corrected on the assumption that the impurities do not micellize.

Results and Interpretation Critical Micelle Concentration. Figure 1 shows the cmc of our salt-free systems as well as related values from the l i t e r a t ~ r e . ~ + ' ~ *The ' ~ * 'abscissa ~ should be a parameter measuring the interaction of the counterions with the micelle, and we have chosen to use the Stokes' law radius of the counterion for the following reasons. Measurements of partial molar volumes24and calculations of the hydration of micelles32indicate that there is lit,tle, if any, loss of hydration for these systems during micellization. Hence, the distance of closest approach will be limited by the tightly bound hydration shell, if any, of the charged groups. As a measure of the hydrated radius, we have used the Stokes' law radius of the counterion deduced from its limiting equivalent conductivity which is a precise though nonlinear measure of the true radius of the kinetic entity.33a This gives smaller values than the distance of closest approach, a, calculated from activity data so that it is a conservative (29) K.J. Mysels, J . Phys. Chem., 65, 1081 (1961). (30) E.K.Mysels and K. J. Mysels, J . CoEEoid Sci., 20, 315 (1965). (31) K.J. Mysels and R. J. Otter, ibid., 16, 462 (1961). (32) P. Mukerjee, ibid., 19, 722 (1964). (33) R. A. Robinson and R. H. Stokes, "Electrolyte Solutions," 2nd ed,Butterworth and Co. L a . , London, 1959: (a) p 124; (b) p 235.

COUNTERION SPECIFICITY IN

THE

FORMATION OF IONIC MICELLES

\

n- ALKYL TRIMETHYL

STOKES LAW HYDRATED R A O I U S , ~

Figure 1. Critical micelle concentrationsof dodecyl sulfates: this work: 25', 0 ; 32", 0;Goddard et al.? 40', A; Meguro and Kondo:l3 30', V; Packter and Donbrow:l6 25', 0 ; Schick:" 25", X.

estimate of the effect of hydration, and it is free of the uncertainties involved in the determination of a.3ab Figure 1shows that the data fall on t.wo very M e r e n t lines: a curve rising with the radius for the alkali metal ions and a descending straight line for the symmetrical quaternaries, where the cmc decreases with increasing radius. The rise for the alkali ion is apparent in our results as well as in those of every other report despite the imperfect agreement of cmc values for each counterion. The crnc corresponds to a concentration at which a very small but often clearly detectable concentration of micelles exist. At slightly below the cmc no micelles can be detected by present methods. Hence, the cmc is a good measure of the ease with which micelles form in the two-component system. Thus, if the concentration of the dodecyl sulfate is intermediate between that of the cmc's of NaLS and LiLS, micelles will be present in significant amounts when Na+ but not when Li+ is the counterion. Therefore, LiLS micelles form less readily than those of NaLS. This can be explained qualitatively in terms of the larger sine of the hydrated Li+ ion which cannot approach the highly charged surface of the micelle as closely, and therefore cannot screen its charge and reduce its surface potential as

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effectively, as the smaller Na+ ion, thus leaving a larger electrostatic energy to be overcome during micellization. The same reasoning applies to K + and Cs+ in accordance with the experimental results. It must also apply to the quaternaries but here the facts no longer agree. The hydrocarbon exterior of the tetraalkyl ions suggests that another effect, namely hydrophobic bonding with the exposed hydrocarbon on the micelle surface, overcomes the electrostatic contribution. I n a spherical micelle the area per sulfate group is about 60 A2, so that close to two-thirds of the micelle surface is hydrophobic and such interaction can be expected. The marked reduction in cmc as the size of the alkyl group in the quaternaries increases also supports this view. Our results are in qualitative agreement with other crnc reports for these and similar systems. Thus, Goddard, Harva, and Jones18 by extrapolating cmc's determined in mixed lauryl sulfate systems by a spectral-change method, found the same order of alkali and T M + cations. Their differences are, however, smaller than ours, which could be due to the 15" higher temperature of their experiments or perhaps, in part, to incomplete ion exchange in the mixed system and the inadequacy of the spectral-change method.21J2 These authors noted the effect of hydrated radii but preferred to use the distance of closest approach, a, which is discussed above. Meguro and Kondols avoided the complication of ion exchange and prepared individual tetraalkyl salts for their cmc determinations which included the conductivity method in addition to a spectral-change method. Unfortunately, if the conductivity values of their Figure 2 are correct, it appears that significant amounts of conducting impurities were present in their compounds since the infinite dilution values are more than 30% too high. Nevertheless, their cmc values showed clearly the trend and led them to point out the importance of hydrophobic character of the counterion. Study of Li, Na, and K systems" also showed the correct trend and the effect of hydrated radius, although here again the reported specific conductance data suggest conducting impurities. Recently, Schick" used surface-tension measurements and noted the relative magnitudes of Li-, Na-, and TMLS cmc's and also studied the effect of urea upon them. He found that 6 M urea affects the cmc of TMLS more than that of the others. This is to be expected since hydrophobic bonding is lessened by the presence of urea34 and plays a larger role in TMLS (34) M. Abu-Hamidyyah, J. Phy8. Chem., 69, 2720 (1965).

Volume 71. Number IS December 1967

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than in NaLS or LiLS according to the above picture. On the other hand, the reported difference between the effect of urea upon the Na+ and the Li+ systems is not directly interpretable until more is known about the effect of urea upon these ions. Packter and Donbrow16 found the cmc of TAILS in exact agreement with our value. This may be due to cancellation of errors since their sample is said to contain appreciable amounts (8%) of the tetradecyl homolog. Packter and DonbrowI6 also determined the effect of lengthening only one of the alkyl chains in this counterion. As may be seen in Figure 1, a typical result is that one c6 plus three C1 chains give the same cnic as four C4chains. Thus, a single long chain is not as effective as four of the same length in lowering the cmc, but the unsymmetrical ion is much more effective than a symmetrical one of same total number of carbons. Again, it appears that increasing the distance of closest approach (as is clearly the case in the symmetrical ion) tends to increase the cmc, whereas increasing the hydrophobic area (as is the case in both the symmetrical and the Unsymmetrical ones) lowers it. The longer hydrocarbon groups in unsymmetrical ions are certainly solubilized to some extent in the interior of the micelle, as already suggested by Goddard, et a1.8 However, it seems likely that the symmetrical ions, especially the smaller ones, are adsorbed on the surface of the micelle because of their residual affinity for mater due to their charge. The Degwe of Dissociation of Micelles. It is clear, on one hand, that ionic micelles are charged and have counterions and, on the other, that not all counterions are free in the sense of having the same activity or mobility as those of a simple electrolyte. Yet, there is at present no unambiguous method of measuring this degree of freedom or of binding of the counterions. Experimental measurements have been interpreted in simple and complicated ways giving different values ranging up to 50%35of free counterions but with very different meanings attached to the word free. The principal experimental measurements used in these interpretations are conductance slopes, alone or combined with transference data, light scattering, and cnic determination as a function of counterion concentration. For clarity, we can distinguish the degrees of ionization so determined by the symbols a, p/n, and s/n, respectively. Some data of each type are available for our systems and will now be discussed on a simple basis since we are interested mainly in comparisons and also because the size and specific effects in which we are interested are difficult or impossible to take into account in more complete approaches. The Slope of Conductivity above the Cmc. For NaLS The Journal of Physical Chemistry

P. MUKERJEE, K. MYSELS,AND P. KAPAUAN

the electrophoretic mobility of micelles has been determined36 as well as the average mobility of the counter ion^.^^ This, along with reasonable assumptions about the mobility of free counterions and the concentration of monomers, accounted for the total electrical conductivity of the system for the simplified model of a partially ionized micelle in which the counterions are divided into two categories: those completely free, and those attached to the micelle and moving with it. For the other systems here studied, the mobility of the micelles has not been determined, yet it would be interesting to establish if there exist differences in the electrical properties corresponding to different degrees of ionization calculated upon the same premises. This can be done to a sufficient accuracy using some reasonable extrapolations. For micelles that are similar to each other, such as those of NaLW or of dodecyl ammonium chloriden at various ionic strengths, and of several alkylammonium chlorides,n the mobilities (p) are a linear function of the logarithm of total counterion concentration. Hence, we estimate the mobility of our micelles by interpolation and extrapolation of the NaLS values.36 If the mobilities thus obtained are then combined with the conductivities, A+, of the respective counterions at the cmc (as calculated by applying the Onsager theory) and with the degree of ionization of the NaLS micelle,25 the calculated conductivities differ markedly from the experimental ones. This shows that the degrees of ionization, a, vary. Their values can be obtained by applying eq 3. S' = dK/dC =

a(A+

Fp)

(3)

The parameters used and the a values thus obtained are shown in Table I and the a values plotted in Figure 2. The results agree with the expectation that the larger Li+ ion is less firmly attached to the micelle than the Na+ ion and this one in turn less than the Cs+. The T M + ion, on the other hand, although larger than the Na+ one, is slightly more tightly bound, and the still larger tetraethyl and tetrapropyl ions are bound with increasing firmness. Thus, again, an additional binding increasing with available hydrophobic area is indicated for the quaternaries. The effect upon a of increasing the counterion concentration by adding a simple electrolyte, in which the experimental uncertainties become large, is less clear cut, but the TELS shows a marked decrease of ionization in contrast to the other com(35) D. Stigter, presented at the IVth International Congress of Surface-Active Substances, Brussels, 1964. (36) D. Stigter and K. J. Mysels, J . Phys. Chem., 59, 45 (1955). (37) H. W. Hoyer and A. Greenfield, ibid., 61, 735 (1957).

COUNTERION SPECIFICITY IN

Cd K*

.)e

NO'

TM'

e c

THE

fE*

Li'

log cmc = A

TM'

e

c

4171

FORMATION OF IONIC MICELLES

4

- (1 - s/n) log (c + cmc)

(4)

where c is the concentration of added salt,, s is the effective charge of micelles, n is the number of monomeric coions per micelle, and A is a constant. In accordance wit.h general experience, our experimental points define (where more than two are available) good straight lines. The degrees of ionization s/n (= 1 slope) thus obtained are listed in Table 11. They are higher than the electrokinetic a values, but agree with the previous deductions that the lithium micelle is more ionized than the sodium one. The quaternaries seem, however, to lie on the same line or give slightly higher s/n values as shown in Figure 2. The experimental error here for these compounds is very large and could obscure the actual trend. It is also possible that the effect of the quaternaries upon the structure of water could tend to increase the cmc and thus lower the slope, as suggested by Steigman, et aL140and in this way raise s/n. A closer examination of the premises involved in this approach suggests that this method must depend on a number of coincidences and may be misleading even in comparative results. Equilibrium between monomeric ions and a monodisperse micelle composed of n monomers and n - m counterions, can be written formally as

+

I

+f

t

f

7.

CiK' No* I I 1 2

t

f

TE*

Li' I

t

, I

TP: I

3

STOKE'S LAW HYDRATED RADIUS,

I 4

,

a

Figure 2. The apparent degrees of ionization as estimated by electrokinetics (a),cmc slope (s/n),light scattering ( p / n ) ,and the free energies of formation for dodecyl sulfates as a function of the counterion. (AG'of the figure is the same as AT of the text.)

K pounds for which data are available. It is also the one for which the strongest binding of counterions is expect ed . Alternative, not unreasonable, assumptions about the micellar mobility do not change the above conclusions. Thus, if one accepts the suggestion of Samis and Hartley5 that the conductivity ( F a p ) of all micelles is the same, the effect is greatly exaggerated. If, on the contrary, one assumes that the mobility varies with ionic strength, but is in addition proportional to the degree of ionization (p' = p a / a o ) ,the effect is somewhat reduced. The relative values of a thus obtained are in the same order as could be obtained by the method suggested by Evans,38which gives significantly lower absolute values of ionization. The low values are due to the fact that the mobility of micelles, as measured directly or as calculated from the diffuse double-layer theory, is less than calculated by the simple Stokes' law assumed by Evans. The Change of Cmc with Ionic Strength. The cmc of ionic surfactants decreases markedly upon addition of a simple salt of the counterion. By analogy with a solubility product, one can interpret the slope of the log cmc us. log total counterion concentration in terms of the degree of dissociation of the micelles according to39

= Cnifni/Csnfsn = CafaC~"f~"/Cl"fi"C~~f2~ (5)

where C denotes molar concentrations, f denotes the stoichiometric activity coefficients, and subscripts M and S refer to the micellar and unmicellized components, whereas 1, 2, and 3 refer to unmicellized (but not necessarily monomeric) coions, monomeric counterions, and micellar ions, respectively . By casting eq 5 into logarithmic form, it becomes apparent that if s is defined by s/n = m/n

-

[(l - m/n) log f2

+

log f1 - (log fdlnlllog C2 (6) this equation can be rewritten as an ideal equilibrium expression

K

(7)

C~/C1"C,"-s

In logarithmic form this gives log C1 = n-l log C3 - n-l log K

(1

- s/n) log CZ

(8)

In the cmc region, log Ca varies very rapidly with C1. (38) H. C. Evans, J . Chem. SOC.,579 (1956). (39) M.L. Corrin, J . Colloid Sei., 3, 333 (1948). (40) J. Steigman, I. Cohen, and F. Spignols, ibid., 20, 732 (1965).

Volume 71,Number IS December 1967

P. MUKERJEE,K. MYSELS,AND P. I ~ P A U A N

4172

Table I1 : Slopes of Log Cmc us. Log Counterion Concentration and Standard Free Energies of Formation of Dodecyl Sulfates a t 25' Cmc I1 x 10,

Counterions

Slope

Li Na K +( 32") cs + TI1 TE TnP

-0.67 -0.69

+

+

89.2 83.2 71.7 60.9 55.2 38.5 22.4

... ...

-0.67 -0.65

+

+

...

+

-

AGO/~T

It is, therefore, legitimate to assume that the first righthand term is constant when C1is taken as equal to the cnic since minute adjustments in the latter value would make this assumption exactly true. Hence, we can rewrite eq 7 for this particular condition log cnic =

(n-l

log C3 - n-l log K ) (1 - s/n) log CZ (9)

A linearity of the log cmc us. log CZgraph corresponds to a constancy of sln. By light scattering and other methods, n can be determined independently, and s is therefore an experimentally measurable quantity. Equations 6 and 9 also define it rigorously as an equivalent charge which takes into account all the nonidealities of the system. In reality, the system must be polydisperse and the above calculation has to be generalized. Let us consider r micellar species each having nt coions and mi counterions per micelle. Here i varies from 1 to r and some of the nl's and some of the m,'s may be the same, but all the nt, m,pairs differ. If the corresponding concentrations are denoted by C,, we can write for each species equations corresponding to eq 5 and define a si value by s,/n, = m,/n, - [(I - mi/na) logf2 logf1

-

+

(logfi)/n,l/log

cz

(10)

Rhich leads to

Kt = CJC~naCZn2-Sa

(11)

for each species. We have then r corresponding logarithmic equations log C1 = (log Ct - log Ka)/ni (1 - s,/nt) log

cz

(12)

If each of these is multiplied by C1nt,which is proportional to the weight concentration of each micellar species, summtttion over the r equations and division by I;Clnt gives The Journal of Physical Chemistry

-14.4 -14.7 -14.8 -15.4 -15.4 -16.1 -17.2

0.33 0.31 0.32 f0.03 0.29 f0.03 0 . 3 1 i 0.05 0.30 f0.05 0.28 f0.05

log c1 =

(n-1

log C),

-

(n-1

log K ) ,

f 0.3 f0 . 3 i0 . 5 & 0.5 rk 0 . 5

-

11 - (s/n>wl 1%

cz

(13)

where X, indicates the weight average of XI. This equation is the equivalent of eq 8, and the same argument can be given about the constancy of the first righthand term so that a linearity of the logarithmic plot of cmc against Cz shows a constancy of (sin), the weightaverage degree of ionization of polydisperse micelles. Thus far, each st is defined in terms of activity coefficients of the individual species by eq 10. The corresponding definition of the weight average is obtained by multiplying eq 10 by Cint, summing over all r species and dividing by ZC,n, which gives (sin), = (m/n>w - [(I - (m/n),)

kfi

+

log f1 - (n-l log f)wlllogCZ (14) Equation 14 thus defines rigorously (sin), as an effective degree of ionization for a polydisperse system which takes into account all the nonidealities. Equation 13 shows how this quantity can be determined experimentally. Even if we assume the simpler monodisperse model, the above analysis shows that in order for s/n, eq 6, to be constant as it is experimentally, several terms including the actual degree of ionization and all the activity coefficients must either remain constant or vary in a compensating manner over a wide range of ionic strengths. Since the former is extremely unlikely, the latter must be true, and the reasons for this are far from clear. In order to show the probable magnitude of these compensating effects, some of the estimated variations due to individual terms will now be given. Thus, it has been suggested that the LS- ion undergoes dimerization with an equilibrium constant of the ~ ~ * in ~ ~the above formulation order of 100 l . / r n ~ l e ,which means a change in its activity coefficient to about 0.53 (41) F. Franks and H. T.Smith, J. P h y s . Chem., 68, 3581 (1964).

COUNTERION SPECIFICITY IN

THE

FORMATION OF IONIC MICELLES

for NaLS in water and 0.81 for NaLS in 0.1 M NaCl. By itself this would cause a marked curvature in a log-log plot of the monomer concentration a t the cmc vs. the total counterion concentration, as shown in Figure 7 of Mukerjee, et aLZ6 Mean activity coefficients, f*, due to interionic interactions, which affect both $1 and fz, are 0.90-0.91 for NaLS in water, depending on whether dimerization is taken into account or not, and 0.78 in 0.1 M XaC1. The f* are estimated from the equivalent ionic strength of the supporting electrolyte, which is justified in mixed systems by Harned’s rulea3and in salt-free systems because it involves a small effect. For the quaternary counterions there is also evidence for ion-pair formation with the LS- ion (Le., a weakness of the salt)27which can introduce another change of up to 25% in fl in the presence of salt. Other effects, such as the salting-out, of the hydrocarbon chains and perhaps also the pola,r heads, must come in. A rough estimate of the salting-out of the hydrocarbon chain42suggests an increase in the activity coefficient of the chain with salt concentration up to 12% in 0.1 M NaCl. The water-structure-forming effect of the quaternaries40 is another factor which may be significant in affecting fi but cannot be estimated at present. Thus, in general, the slopes of the log cmc vs. log counterion plots or their intercomparison should not provide reliable information about the interaction of ions with micelles as reflected in the true degree of dissociation, m/n. However, when the counterions are very similar, it is likely that the effect of ionic strength upon the various factors affecting activity coefficients will also be similar, and an intercomparison can give significant results. This should be the case for Li+ and Na+ counterions here. If the m/n is calculated either neglecting all corrections when it equals s/n, or by introducing, individually or collectively, correctionsdue to dimerization, due to interionic interactions, and due to salting-out of the chains, the value for the Li+ salt remains about 10% higher than that for the Na+ salt, in approximate agreement with the estimate from conductance data. The comparison for Li+ and Xa+ is further justified by the close similarity of their micellar molecular weights in water.43 On the other hand, an intercomparison of quaternaries with the alkali ions or of quaternaries among themselves involves differences in micellar weights, in activity coefficients, particularly due to weakness, and in ionic strengths, which render it very uncertain. Light Scattering. The above results are confirmed at least in part by light-scattering data where these are available, namely for NaLS, LiLS, and T,MLS, which were studied by Rrincen and M y ~ e l s . As ~ ~ the molec-

4173

ular weight of LiLS is within experimental error of that for NaLS, the slopes of the reduced turbidity can be compared directly, and that of LiLS is markedly higher (37.6 vs. 20.6 X despite a slightly higher ionic strength. Thus, its effective charge, p , and the second virial coefficient are definitely higher. For TMLS the comparison is less direct as the molecular-weight changes appreciably. The effective degree of ionization, p / n , as calculated by ref 43 is larger than for NaLS but much lower than for LiLS as shown in Figure 2. The molecular virial coefficient is similarly much closer to the extrapolated value for NaLS than is that of LiLS. Hence, again LiLS shows less binding than NaLS, and TAILS binds somewhat more than expected on ionic size basis alone, although the position of the latter two is not reversed. It is interesting to note the appreciable effect of counterions on the second virial coefficients, which, for the highly charged micelles, are primarily determined by the micellar interactions a t long distances, and therefore in the low potential regions.44 Different counterions thus give rise to different potentials in this region, far from the micelle. Indications of such effects have been obtained from soap-film work also.18 Standard Free Energy. There are several approaches to the problem of transforming the cnic values into standard free energies. For ionic systems the free energy includes a contribution due to the involvement of the counterion with the micelle. This contribution can be taken into account either by a chemical approach based on binding of some counterions to the micelle, as will be discussed later, or by a physical approach using an electrostatic calculation based on a fully ionized model. The latter approach was begun by Overbeek and Stigter in 195645 and led to a separation of the electrostatic from what was then called the van der Waals and is now referred to as the hydrophobic component. This was based on the detailed analysis of the free energy involved in the buildup of a charged micelle. Some of the approximations involved were examined later and since then much work investigating the effect of the detailed structure of the micelle surface on the electrostatic surface potential of the micelle and on the problems of the doublelayer theory a t high potentials has been done particularly by Stigter46and by L e ~ i n e . ~ ’We feel, how(42) P. Mukerjee, J. P h y s . Chem., 69, 4038 (1965). (43) K. J. Mysels and L. H. Princen, ibid., 63, 1699 (1959). (44) D. Stigter in “Electromagnetic Scattering,” M. Kerker, Ed., Pergamon Press, New Tork, N. Y,, 1963, p 303; A. Vrij, Thesis, Utrecht, 1959. (45) 3. Th. G. Overbeek and D. Stigter, Ree. Trac. Chim., 75, 1263 (1956).

Volume 71,Xumber 1.9

December 1967

4174

P. MUKERJEE, K. MYSELS,AND P. KAPAUAN

ever, that much work remains to be done before definitive results can be obtained by this method. It has been pointed out recently by Emerson and H01tzer~~ that in a distribution of micelles there will be two substantially equal concentrations of micelles differing only by one monomer at the most probable number of monomers per micelle or micelle number, fi. Then, the standard free energy of adding one monomer to one of these micelles to form the other, AG"g, is given by R T l n (cmc) = AGm&

(15)

This approach has been treated in considerable detail earlier by A r a n o ~ . * For ~ ionic micelles, the use of this equation implies a standard state in which the coions have the standard concentration whereas the counterions have the same concentration as at the c ~ c . ~Thus, * the reference state is not electrically neutral and, in effect, uses single-ion activities. A plot of these standard free energies has exactly the appearance of Figure 1 provided that the ordinate is multiplied by the proper factor and thus gives no additional information. The qualitative discussion which we gave in terms of crnc values can therefore be exactly reproduced in terms of AG-3. Using eq 15 and calculations of electrostatic potentials based on assumptions which neglect the complications due to the topography of the micelle surface and those of the double-layer theory at the high potentials involved, Emerson and H 0 l t z e r ~ ~have ~ ~ 0 obtained the hydrophobic contribution to the free energies for a series of micelles. Unfortunately, these authors made the additional simplifying assumption that all the micelles, despite widely differing fi, have the same radius. This runs counter to the well-known fact that the density of the micelles (their partial specific volume) is essentially constants1 with ionic strength, although A varies, .md to the generally accepted liquid hydrocarbon model for the interior of micelles, thus clearly invalidating their results. Application of the chemical approach is based on the phase-separation or mass-action models. The phaseseparation approach, although simple, has the weakness that micelles are not in fact a separate phase in the normal sense of the word. As a result, this approach cannot account for a large number of experimental facts as pointed out elsewhere.52 The mass-action approach, on the other hand, has given the first rationale of the experimental peculiarities connected with the existence of a critical micelle concentration in terms of a micelle as summarized in Hartley's 1936 book.63 Phillips, in 1955,54used it to calculate for the first time the standard free energy of micelle formation for an The Journal of Physical Chemistry

ionic system. This approach has been used often since, and our treatment of the meaning of the charge s above has given a more rigorous physical meaning to the terms used. We shall therefore apply it to our systems. For a monodisperse system described by the equilibrium eq 7 we can write

AG"/nRT

= -In

K/n

=

+ In C1 + (1 - s / n ) In Cz

-(l,/n) In C8

(16)

This definition of the standard free energy implies an ideal standard state in which the activities are unity, and the structural components involved in defining s are the same as at the cmc. The absolute and even relative values of AGO depend also on the choice of the standard concentration as always is the case with reactions involving a change in the number of particles. We are using the unit mole fraction standard state. When the system is polydisperse, eq 11 applies to each species, for which we obtain

AG",/n,RT = -In K J n l = -(l/nJ In Ct In CI

+

+ (1 - sl/ni) In CZ

(17)

Averaging over all species in the manner used to derive eq 13, we have

(AGo/n),/RT = m / R T = -(n-l In C), In C1 [l - (s/n),] In CZ (18)

+

+

Here aG" is the average quantity per monomer defined above. For actual calculations, we cannot estimate the first quantity on the right-hand side precisely, because the polydispersity of micelles is not known. The term, however, is small in comparison with the others, being of the order of 2 or 3% of the total. We therefore calculate this term on the basis of a monodisperse system using the weight-average micellar weight in water assuming 4% micelles at the cmc. The value of n is not critical here and has been chosen rather arbitrarily when not available experimentally. If the concentration of (46) D. Stigter, J . Phys. Chem., 68, 3603 (1964). (47) 8. Levine and G. M. Bell, Discussions Faraday SOC., 42, 69 (1967). (48) M. F. Emerson and A. Holtzer, J . Phys. Chem., 69, 3718 (1965). (49) R. M. Aranow, ibid., 67, 556 (1963). (50) M. F. Emerson and A. Holtzer, ibid., 71, 1898 (1967). (51) P. Mukerjee, ibid., 66, 1733 (1962). (52) P. Mukerjee, %%id., 66, 1375 (1962); K. J. Mysels, P. Mukerjee, and M. Abu-Hamdiyyah, ibid., 67, 1943 (1963); P. H. Elworthy and K. J. Mysels, J . Colloid Interface Sci., 21, 331 (1966). (53) G. 5. Hartley, "Aqueous Solutions of Paraffin Chain Salts," Hermann et Cie, Paris, 1936. (54) J. N. Phillips, Trans. Faraday soc., 51, 561 (1955).

COUNTERION SPECIFICITY IN THE FORMATION OF IONIC MICELLES

the average micelle to which aG“ applies exactly is even only l/lo0 of the concentration used assuming monodispersity, the error in is less than O.lkT, which is small compared to the other uncertainties. Values of (sln), are obtained from the slopes of cmc’s supplemented by the relative values of a from electrokinetic measurement,s. This introduces an uncertainty which is not significant in the end result. The parameters used and the estimated relative uncertainties are shown in Table 11. Because of the complications involved in the definition of s, comparison of these free energy values has to be done with caution and is most justified when both ionic strengths and s/n values are similar which is the case of low-salt systems. For these systems, the order of the aG” values is also the same as that of the cmc’s. This is related to the fact that s/n tends to increase with the cmc in our system (and would have to change substantially in the opposite direction to reverse the trend). Figure 2 shows these AG” values. Again, the alkali ions fall on one line, the quaternaries on another. As the hydrated simple ions become smaller, the micelle becomes more stable. The quaternaries give somewhat stabler micelles, and this stability increases in the opposite direction, Le., with increasing size of the ion. Extrapolation suggests a stabilizing contribution as high as 4kT per micelle-forming ion for the tetra-n-propyl system.

Discussion A direct consideration of the cmc values in salt-free systems and the free-energy calculations show that micelles form more easily as the size of the hydrated ion becomes smaller. This effect can be overshadowed by hydrophobic interactions for organic counterions. Micelle formation is facilitated as the double layer becomes more compact, which corresponds to a reduction in the effective degree of ionization shown by conductance and also indicated by light-scattering and the cmc slope, especially for the alkali metals (Figure 2). These general conclusions are strengthened by similar results obtained for the two-dimensional analog of

4175

micelles, the surface monolayer. Thus, decreasing the hydrated radius has been shown to facilitate adsorption at the air-water interface by surface-tension measurem e n t ~ . In ~ ~soap-film studies, the different compactness of the double layer of the alkali metals results in different equilibrium thicknesses, and the specific adsorption of the quaternaries markedly affects transitions between first and second black films.18 Some details of the picture are, however, not clear a t present. We can mention two. One concerns the effect of the counterion on micellar size. In going from Na+ to Li+ there is no significant change, whereas from Na+ to T M + there is a marked increase.43 Since the polar heads are more tightly packed on a larger micelle, this cannot be explained in terms of hydrophobic bonding, favoring larger mutually exposed hydrophobic areas. In fact, we are not in a position to offer any simple explanation. Another difficulty arises if one asks the question of the effect of counterions not on the property of the average existing micelle, but upon a micelle of a given structure. Thus, for example, we have seen that the free energy of micelle formation is less negative for LiLS than for NaLS. This pertains to some averages, to highly stable, or perhaps even most stable micellar structures. It is therefore permissible to draw some generalizations such as that the stablest NaLS micelle will be more stable than any LiLS micelle (because it is more stable than the stablest of these). However, it is not possible to conclude that an NaLS micelle having the same size and charge as the stablest LiLS micelle will be either less stable or more stable than the latter. Not enough is known about the variation of stability with these factors to know how the two distributions of stability of micelles with size and charge overlap. Acknowledgment. This work was supported in part by the Office of Naval Research and some of the interpretation by P.H.S. Research Grant GM 10961-02-03 from the Division of General Medical Services, Public Health Service. (55) I. Weil, J . Phya. Chem., 70, 133 (1966).

Volume 71, Number 19 December 1967