2054
NOTES
NOTES
Hydrophobic and Electrostatic Interactions in Ionic Micelles. Problems in Calculating Monomer Contributions to the Free Energy
by Pasupati Mukerjee School of Pharmacy, University of Wisconsin, Madison, Wisconsin 63706 (Received May 13, 1968)
Emerson and H o l t ~ e r l - have ~ recently used a new approach for calculating the monomer contribution to the free energy of forming ionic micelles. At the critical micelle concentration (cmc), the equation derived is
RT In (cmc)
=
+ Noe$b(b,a,K,&)
AGHC
(1)
where R is the molar gas constant, T is the absolute temperature, AGHCis the hydrocarbon chain contribution to the standard free energy change on the addition of a detergent monomer t o the most probable micelle containing 8 monomers, No is Avogadro’s number, e is the electronic charge, $0 is the electrostatic potential at the micelle surface defined by the radius b, a is the distance of the closest approach of small ions to the micelle, and K is the Debye parameter. iv is equated to N , the value obtained from light scattering. Aranow4 pretously pointed out the similar significance of RT In (cm ) for nonionic systems. The present paper deals with several problems raised by the work of Emerson and H ~ l t z e r . ” ~The comments apply generally, but numerical examples will be confined to the system sodium lauryl sulfate (NaLS)-NaCl-HzO , frequently investigated in the pa~t.6-l~It will be shown that $b and -AGHc values are too high, the dependence of $b on the ionic strength p is unreasonable, the model of the electrical double layer used is probably inappropriate, the arbitrary fixation of the micellar radius r is unrealistic, and that eq 1 is not valid for spherical micelles. Table I summarizes the data on the NaLS-NaC1HzO system.2 Column 6 gives the values of -AGHc/ R T (eq 1). The values of e$,/?cT (k = R/No) were calculated from the tabulated values of $a, a t the surface defined by the radius a, and the equation.12
(e$b/kT) - (e$a/kT) = ( N e 2 / D k T ) ( ( 1 / b) ( l / a > ) (2) Here D is the dielectric constant. The $G.c. values are the Gouy-Chapman potentials,lO for which the counterions are assumed to be point c h a r g e ~ . ~ ~ The Journal of Physical Chemistry
Values of AGHc. For comparison with the - A G m / R T values for NaLS, the value for nonionic systems, 14.5, was calculated from R T In (cmc) for C I Z H Z S ( E O ) ~ ~ ~ M ) , where EO stands for an oxy(cmc = 8.7 X ethylene group, after applying a correction of 1.2RT for the self-interaction of the head groups at the micelle surface;6 14.5 is, therefore, an estimate of the contribution of a C12H25 chain alone. Emerson and Holtzer1p2estimated essentially the same value for nonionic micelles. The NaLS values2 are higher by about 5kT and would argue that the hydrophobic interaction of a Clz chain in micellar equilibria increases substantially when a charge is introduced a t one end, while the opposite result might be expected. Values of $6. One of several possible reasons for the high values of - AGHc/RT for NaLS is the high value of &, a quantity of independent interest. $b is much higher than 9G.c.. Experimental data on the dissociation of solubilized indicator dyes suggest, on the other hand, that the true surface potential, #, for micelles is substantially lower than #a ,c..15 A striking fact about $b for NaLS, in contrast to $G.c., is that it remains approximately constant as p increases by a factor of 10. If this trend is accepted, the observed lowering of the cmc with increasing p must be ascribed entirely to an increase in - AGHCrather than a decrease in charge effects. The constancy of $b with p is contrary t o all accepted ideas about the effect of p . Experimental estimates of $l6 suggest that although the absolute value of $ is lower than the theoretical $G.C., the changes in # and $G.c. with p are in fair agreement. (1) M.F.Emerson and A. Holtrer, J . Phys. Chem., 69,3718 (1965). (2) M.F. Emerson and A. Holtrer, ibid., 71, 1898 (1967). (3) M.F. Emerson and A. Holtzer, ibid., 71, 3320 (1967). (4) R. H. Aranow, ibid., 67, 556 (1963). (5) P. Mukerjee, Advan. Colloid Interfac. Sci., 1, 241 (1967). (6) D. Stigter, Rec. Trav. Chim., 73, 593 (1954). (7) J. Th. G. Overbeek and D. Stigter, ibid., 75, 1263 (1956). (8) D.Stigter and J. Th. G. Overbeek, Proc. Intern. Congr. Surface Acticity, Rnd, London, 1967, 1, 311 (1957). (9) P. Mukerjee, K. J. Mysels, and P. Kapauan, J. Phys. Chem., 71, 4166 (1967). (10) H.F. Huisman, Koninkl. Ned. Akad. Wetenschap., Proc., Ser. B’ 67, 407 (1964). (11) D. Stigter, J . Phys. Chem., 68, 3603 (1964). (12) M. Nagasawa and A. Holtzer, J . Amer. Chem. SOC.,86, 531 (1964). (13) A. L. Loeb, J. Th. G. Overbeek, and P. H. Wiersema, “The Electrical Double Layer Around a Spherical Colloid Particle,” M. I. T. Press, Cambridge, iMass., 1961. (14) J. M.Corkill, J. F. Goodman, and S. P. Harrold, Trans. Faraday SOC.,60, 202 (1964). (15) P. Mukerjee and K. Banerjee, J . Phys. Chem., 68,3567 (1964).
2055
NOTES Table I: Comparisons of Surface Potentials and ACHO NaCl concn, M
0,050 0.201 0,506
- AQHc/RT
- AQHc/RT
-In XOa
e&a/kT
e&alkT
e&GI.C./k T b
(NaLS)
(nonionic)
10.10 10.99 11.61
8.30 8.25 8.47
5.33 4.48 3.89
5.63 4.60 3.90
18.4 19.2 20.1
14.5
a X O equals the cmc expressed in mole fraction unitse2 * Interpolated and extrapolated from a plot of e ~ G . c . / k Tvs. log [Na+]cmo; data from ref 10, covering [Na+],,, concentrations from 0.0081 to 0.30 M . The plot is linear over the whole range.
Table I shows the dependence of $G..c. on p . Similar variations are shown by electrokinetic potentials also.16 As $a and $ G . C , values are obtained in essentially the same manner, the main difference between $o and $G.c. is due t o $h - $a (eq 2). It is concluded that the use of an exclusion radius of the counterion leads t o an overestimate of $ in micellar systems. For several cationic systems with a Clz chain,2 $h showed substantial variations with p , in contrast to the case of NaLS, and -AGHc/RT values were lower, although generally higher than 14.5. These differences are attributed (see eq 2) to the lower values of N for the cationic systems, much smaller variations in N with p , smaller values of a and b chosen, and the superior applicability of the constant b value assumed because both N and variations in N were much smaller. Micellar Radius Calculations. The customary procedure for calculating r from N for spherical micelles is to use an estimated d e n ~ i t y . 6 - 8 ~ ~ 0Thus, ~ ~ ~ ~implicitly ~5 or explicitly, it is assumed that r a I n contrast, Emerson and Holtzer assumed a constant radius for all micelles, irrespective of the monomer or the value of N , even though N varied from 84 to 126 for NaLS. This assumption has some importance for calculations of $ and a considerable bearing on the validity of eq 1. The true specific volume, V,, of micelles or any other solute is experimentally inaccessible. However, the experimental partial specific volume, V,, of micellar NaLS (including all counterions) is found to be constant within roughly the experimental error of about 0.275, over an NaCl concentration range of 0-0.12 M,17 as N varies from 57 to 98.’0 The major contributor to V, or V , is the hydrocarbon core of the micelle. In attempting to estimate V , for the “kinetic” micelle,17 assumed to carry 70% of the counterions, factors such as volumes of counterions, ionic electrostrictions, and the effect of the exposed hydrocarbon surface were examined .17 V , was estimated to be about 3% higher than V,. If similar corrections are estimated for the micelle without any counterion, possibly more appropriate for electrostatic calculations,l~V , for NaLS is slightly closer to V,. As the corrections are approximate, the calculation of r either from V, or V , involves some uncertainty in its absolute value, as a 3% error in 8, corresponds to 1% in r. The important question here
is the relative value of V , as N and p vary. I n view of the observed constancy of V, for several types of micelles17with variations in N and p and in the absence of any evidence to show that V , - V, has a pronounced dependence on N or p , the customary assumption that V 8is independent of N and p is preferred to the assump~ the purposes of the next tion of a constant 7 . l ~ For section, we will make the less restricted assumption that 8, is independent of N at constant p , and, therefore, r a “Is. The assumption of a constant r leads t o the unrealistic result that V , is inversely proportional to N , although V, is independent of N , and to a relative overestimate of $ at higher N values, because surface charge densities are overestimated. Inapplicability of Eq 1 to Spherical Micelles. The equation for the standard free energy (AGN) of the formation of a micelle (MN) containing N monomers (All), assuming unit activity coefficients for all species, can be written as6-* AGN/RT = -In
+
[luN]
N In W11 - (AGN,el/RT) (3) Here, AGN,,l/RT is a measure of the electrostatic contribution. For the micelle MN+l, we have AGN+I/RT = -In [MN+1] f
(N
+ 1) In [MI]
-
(AGN+I,~RT)
If now [MN] is equated to [MN+l], as suggested Emerson and Holtzer’ and previously discussed Stigter and Overbeek,s and [MI] is approximated the cmc, the difference of eq 4 and eq 3 gives the general expression for AGHc. We obtain R T In (cmc) where
AAGN*Bl
=
AGHC
+
AAGN,el
(5)
is the difference between AGN+l,el and
AGN,el-
When eq 5 is compared to eq 1, it is seen that A A G N , e l is replaced by Noe&. This is a good approximation for the addition of a small ion, if the change in r can be ignored, but not the addition of a monomer to a spherical micelle. (16) D. Stigter, J . Colloid Interfuc. Sci., 23, 379 (1967). (17) P. Mukerjee, J . Phuls. Chem., 66, 1733 (1962).
Volume 78, Number 6
June 1969
2056
NOTES
To illustrate this difference, we note that the procedure for the calculation of AG,a-8 involves the hypothetical processes of a reversible discharging of micelles and single ions and association of N monomeric ions in the uncharged state to form a micelle, followed by the charging of all ions and the micelles. For the analytically soluble Debye-Hiickel model (D.H.) of the electrical double layers, used here as an example, it has been that
elC/N,D.H./kT= Ne2/DrkT(1 f K T )
+
AGN,eI,D.H./RT = N2e2/2DrkT(1
KT)
5.07
(6)
(7)
If we obtain A A G N , e ~ , ~from . H . eq 7 by differentiation, at constant r 2.2
[(~AGN ,el,~.~./dN)/RT]7=oonsta =n t
Ne2/DrkT(1 f
I
I
0
0.8
I
1.6
I
2.4
I
3.2
I
4.0
I
I
4.8 5.6
B # N , D . H . / ~ T( 8 )
Concentraiion of p - bromophenol, lO*M.
However, if 1" a N1'3,as assumed by Stigter and Overbeekg for a similar differentiation
Figure 1. The effect of p-bromophenol on G(HaO,,+) in neutral solutions: a, diffusion model predictions; b, experimental results.
Kr) =
[ ( ~ A G,N e l , ~ . ~ . / d N ) / R T ] r a ~=' j s Ne2(5 4- 4 ~ r ) / B D r k T ( 4l K ? " ) ~ (9)
Thus [(dAG,,
el ,D .H
./dN) / R T ],a ~
1 / 3
=
[ e d " , D , H . / k T ] [ j-/ ~( ~ ~ / 6f ( 1K r ) ) ]
(10) Depending on the value of K P , the use of e+N,D.H./kT leads to an overestimate of A A G N , e ~ , ~ . ~by . / Ra Tfactor of to 3/2. The inequality e+,/kT > AAGAr,,l/RT is expected to be shown by all models of the electrical double layer for spherical micelles as long as r depends on N . For NaLS in water, for example, the Gouy-Chapman model, for N = 80, leads to values of e+c.c.lkT = 7.42 and AAG,1/ R T = 6.3.8 We conclude, therefore, that eq 1 is inapplicable to spherical micelles and that its apparent simplicity is misleading. Potential calculations alone are not sufficient to obtain the electrical free eneigy contributions in the growth of spherical micelles. Acknotdedgment. I am grateful to Dr. Karol J. Rlysels for several helpful discussions.
Hydrogen Ion Yields in the Radiolysis of Neutral Aqueous Solutions
by B. Cercek and fig. Kongshaugl Paterson Laboratories, Christie Hospital and Holt Radium fnstitute, Manchester 20, England (Receined July 19,1968)
Extensive investigations of the radiolytic decomposition of water have shown that the species eaq-, H, OH, The Journal of Physical Chemistry
I
6.4
Hz, and HzOz are being produced, Since positively charged species must be produced as well, it has been postulated that these are H30as+, No attempts to measure G(H3Oaq+)were reported; its value has been assumed to be between 3.62ta and 3.ga4 Herein, we report the results of a radiolysis study of deaerated aqueous p-bromophenol solutions in which the hydrogen ion yields were measured. All solutions were prepared using distilled water redistilled from alkaline permanganate. The reagents were used without further purification. 2-Propanol and p-bromophenol were Hopkin and Williams Ltd. "Analar" grade and "General Purpose" reagent, respectively . An underwater 6oCoy source and a linear accelerator giving 5-psec pulses of 8-MeV electrons were used for irradiations. Details of these irradiation facilities and dosimetry were extensively described elsewhere.6j6 All solutions were deaerated by bubbling with argon for at least 1 hr before irradiation. Doses in the range from 10,000 to 36,000 rads were used. The concentration of hydrogen ions was measured with a PYE pH meter. Its expandable scale allowed accurate readings of 0.01 pH unit. The pH of the unirradiated solutions was around 6.50, and 30 min after the irradiation with 'j0Co y rays (or 8-MeV electrons) changes in p H of at least 2 pH units were mea(1) Norsk Hydro's Institute for Cancer Research, Montebello, Oslo 3, Norway. (2) L. M. Dorfman and M. S. Matheson, Progr. Reaction Kinetics, 3, 258 (1965). (3) K.H.Schmidt and W. L. Buck, Science, 151, 70 (1966). (4) M. Anbar and E. J. Hart, J. Phys. Chem., 71,4163 (1967). (6) J. P. Keene and J. Law, Phys. Med. Biol., 8,83 (1963). (6) J. P.Keene, S.Sei. Instr., 41, 493 (1964).
