Hiroshi Maeda, Masa-aki Tsunoda, and Shoichi lkeda
1086
lonelectric Interactions of a Nonionic-Cationic Micelle aecia,” Masa-aki Tsunoda, and Shoichi lkeda De,oartmeiii of Chemistry, f a c u l t y of Science. Nagoya University. Nagoya. Japan
(Received August 70. 1973)
The standard free-energy change accompanying the micelle formation of a nonionic-cationic surfactant is re!n.ted to the critical micelle concentration after a treatment developed by Aranow, Emerson, and Holtzer. It is shown that electric and nonelectric contributions are experimentally separable. Respective contributions consist of various interactions which can not be separately treated in the case of ionic micelles, such as the effect of charge on nonelectric interaction or the variation of electric free energy due to the add tion of a nonionic monomer. An experimental method of the approximate evaluation of these interactions is suggested and applied to the experimental data on aqueous solutions of dimethyldodecylamine oxide. The results can be reasonably interpreted in terms of the effect of charge on the solvent structure around hydrocarbon chains.
Introduction Stability of micelles can be measured in terms of the standard free.energy change for the reaction of the addition of one more surfactant monomer to the most probable micelle and hence ii; can be related to the critical micelle concentration (cmc).I For ionic micelles electric and nonelectric interactions contribute to the stability. When effects of environmental factors on the stability are considered, such :as ionic strength, pH, and temperature, it is desirable to separate each contribution. For ionic surfactants of strong electrolyte, however, the separation is generally difficult to carry out experimentally, because both hydrocarbo:n part and charge of a monomer can not be added independently to the most probable micelle. Enierson and lt-Tolt.zer2calculated the electric part of the standard free-energy change and estimated the hydrocarbon part as a (difference of the former from the total which was obtained from the values of cmc. In their calculation shape and volume of the most probable micelle were assumed to remain constant even though aggregation number of the .micelle and ionic strength might change. The assumption has been criticized on theoretical ground^.^ Aside from this as;sumption, results of the calculation were affected by a slight change in the value assigned to the distance of closest approach of counterions. In the case of nonionic-ionic surfactants, two ways of addition of a monorner to the most probable micelle are possible except for two extreme cases (full charge and no charge): the addition of either a nonionic species or an ionic one. Heme we may expect that in the case of nonionic-ionic surfactants separate evaluation of electric and nonelectric parts can be achieved experimentally. In the first part of the present paper we show how the standard free-energy change for respective additions of a monomer can be re1,sted t o the cmc after a reasoning developed General expressions are obtained including those for both nonionic and ionic micelles as oreover: we can understand more clearly the nature of the interactions involved in the parameters describing the stability by means of the obtained general expressions. Titration property of micelles is also discussed, which is one of the most charact,eristic properties of nonionic--cationic surfactants. In the second part application of the treatment presented in the first part to the experimental data is given, which were obtained on aqueThe Journal of Physical Chemistry. Val. 78, No. 11, 1974
ous solutions of dimethyldodecylamine oxide in the presence of sodium chloride. Potentiometric titrations were carried out on the system and the data are anaIyzed in combination with those of the cmc. Theoretical Consider a solution containing four components: water, simple salt, surfactant molecule (nonionic species), and neutralizing agent (acid). For the sake of brevity, we consider the case of a nonionic-cationic surfactant. Conversion to the case of a nonionic-anionic one can be achieved formally. At a concentration higher than cmc various micelles are present in solution, each characterized by the aggregation number m‘ (the total number of surfactant monomers involved in a micelle, either ionic or nonionic) and the number of charges n’. Chemical potential of a micelle ion, p m , , specified by a set of values (m’,n’) can be written according to a standard treatment of polyelectrolyte^.^ ,LLmf
= porn‘ n l
4- RT Ira C&
+ n’(RT In K , + pOH)$. R‘T m‘ - rt’ + n’) + GeX(rn’,n’) - n’) In m 7
(1)
pori; ,,r = porn’ (m’
1
Gel(,’,
(2)
For both monomers, denoted by D and DE$ for nonionic and ionic species, respectively pD = poD
+ RT
In Cl(1 -
CY,)
13)
and ( 4) ,LLDH = poDn C RT In C,~U, Here the superscript zero represents that the signed quantity refers to the standard state. The standard state for each micellar species as well as for each kind of the monomers is defined as a state of unit concentration ( M ) in an aqueous salt solution with respect to each species under consideration, in which the interaction between the species is absent as in the state of infinite We assume that the ionic strength is high enough that inter. actions between micelles can be ignored. Accordingly, activity coefficients for micelles and mommers are assumed to arise solely from their interactions with small ions. These activity coefficients are considered to be constant irrespective of pH and the total surfactant concentration
Free-Energy Change Ac;c~m~anying Micelle Formation
1087
and invoived in the chemical potentials of respective standard states. Concentration of a micelle species consisting of m' monomers and n' charges is expressed by C,,,,,. Concentration of salt and of the total monomer are denoted by C, and Cl,respectively. The latter is considered to coincide with cmc when the total surfactant concentration is slightly higher than cmc Degrees of ionization of monomers and micehles are denoted by 011 and or, respectively, and the latter is defined as n'/m'. Gel represents the work necessary to charge a micelle from no charge to a final value against the electric repulsion alone. Electric and nonelect,ric f r e e ~ e ~ e r gchanges y accompanying the introduction of a proton from its standard state to a fully discharged micellt? in a given salt solution are given by RT In M,, where K,, denotes the intrinsic proton dissociation constant of the micelle. The latter is assumed to be independent of the aggregation number. GeX represents any other nonelectric contributions when more than two charges age present on a inlcelle. It should be noted that since the electiic free energy Gel is defined as above, any contributions a ~ c ~ ~ r ~ionization ~ ~ i ~ yother ~ n than g Gel and taken into account by the term K , is to be involved in the term Gex. Equilibria fo,. Proton Dissociation. Consider a series of ~ ~ ~equilibria t i for ~ proton ~ ~exchange e among those micellar species which have the same aggregation number m'. he distribution curve of charges n' will be characterized with a single m ~ located~ at n. Hence ~ ~ ~
dm' I0. Since n(m') never decreases as m? increases, the variation of n with m' should obey the following inequalities except for the case that n = m'
i'lP , i[a(Ge' -t c")/anl, -I- RT in (a,/aJ (16)
+
for the addition of an ionic monomer. ID the above AGHc is defined as d~Om,o/ana - p o i~ (dGexlam), and represents the most important part of nonelectric contribution to the standard free-energy change associated with the adpH = (AuO,, q, / ~ T Z " ) , , ~ < for np = n(m') (6) dition of a nonionic monomer to the micelle. The term (dGe'/arn). represents a change in electric free energy of Introduction of eq 2 into ey 6 yields a well-known expression for potentiometric titration of multicharged species. the micelle due to a change of i t s dimmsions caused by the addition of a nonionic monomer at a constant number pK = pH log(u,/l - a,) = pK, of charges. The last terms of both equajions represent re(0.434/RT)[d(C1 -t G"")/dn],t ( 7 ) spective changes in mixing free energy o i charged and unEquation 7 shows how the degree of ionization of a micelle charged species both in a micelle and in solution, caused is determined by pH. Similarly the degree of ionization of by the addition of either species. When intrsnsic tendency monomer a1 is reiateld to pH in terms of the dissociation for protonation of the ionizable group rditfers in the two constant for monomer K1. states (monomer or micelle), the term RT In (Km/Ki) will appear, &hi& will be discussed later. If the degree of (8;) ionization of the most probable micelle 1s assunned very pH 4- l o g ( a , / l - cy1) = pK, close to the average degree of ionization of micelles in soThe Most Probable Aggregation Number.6 A distribulution, then am as well as 011 can be e x ~ e r ~ m e ~ toball~ tion curve with respect to the aggregation number m' can tained. be drawn if th1.e concentration of micelles that have a Equations 15 and 16 are general expressions relating specified value of m' is plotted against m'. The distribucmc to the stabilits of micelles. For nonionic micelles eq tion curve is alrd,o expected to have a single maximum lo15 reduces to cated a t m. Theit
-+
cnT.pfw
c'm
- C m - l . m m - 11 + i . n i m + 1) -
( 9)
The most probable aggregation number m is determined by the differentiation of eq 1 with respect to m' where n' is replaced by n(m'). ~ 8 p m ~ , j z L m ~= J &(d~',: d) n(m~,/8m') for m'
=
m (10)
Behavior of n(rrr') should be briefly examined here. Generally 'we can assume that degree of ionization is independent of m' or it decreases as m' Increases, i.e., a(n/m')/
RT in (cmc),
=L;
AGHc(0) = [ape,, n / d m ~ m = , ~ .-o , poll (17)
Here (cmc)o, rn(0). and bG,,,(O) denote that they are evaluated at zero ionization. For ionic micelles formed by strong electrolytes eq 16 becomes
RT In (cmc)
=
AG'EIc -I(dG"'/dm),, i(;~G"I%z),, == AG'EI~4- (dGe'/3m),= ,, (18)
(As to the notations appearing in eq 18 see Discussion section.) In this case only the sum of electric and nonelectric The Journal of Physical Chemistfy Vol. 78, No. 7 1. 1974
Hiroshi Maeda, Masa-aki Psunoda, and Shoichi i keda
contributions are experimentally determinable. For general nonionic -ionic micelles, however, both terms, [a( Gel Gex)/dn], an3 AGiic -t (dGel/dm),, can be experimentally determinecl, which demonstrates electric and nonelectric contribu. ions are separable. When correlation between electric and rionelectric interactions is insignificant, as has been postulated previously,2 then respective main contributions, (aGel/an),, and AGHc, can be experimen tally evaluated. Efject of C*Zarg.gP on Solvent Structure Further separation of electric and nonelectric contributions may be posstble if we examine the dependence of the term AGI-lC -t iXP/ldm), or, ionization and ionic strength. In general, the term (aGe'/am), is negative and increases in r n a ~ ~ i t ~ as ~ cionization le increases. However, a t low loniratior, ihe magnitude will be negligibly small, for the effect is siqnificant only when charge density on a micelle is considerabl) high. The term becomes less negative when ionic strength s raised a t a constant ionization. e assume that difference of solvent structure around charged a r d discharged micelles is the largest contribution to 1he t e m G e < since the experimental data are well interpreted i f the contribution is assumed to be of primary importance iri the formation of nonionic m i c e l i e ~ .When ~ a polar head 19 ionized, the structure of water around hy drocarbnn chams ncw the charge will be considerably altered in a di ~ c t i o othat the contact between water and hychcarhons cbaiw3 is less favored. At low ionization the amount of d v o n t structure altered by the introduction of charges will he nearly proportional to the number of charges introihced. The effect is thus additive but it is already taken mto account lhrough the term RT In R, in eq 2. Accordingly T e x is expected to be approximately zero a t low ~ o w ~ a t i oAs n ionization proceeds, the structure of water irounti hydrocarbons becomes destroyed to a different degrw h z n what is expected when the effect i s YO or more charges can destroy the same qtiucture In other words, the amount of altered by the introduction of a charge IS less than 13mt sccuunted in the term RT In K,. This leads to a negative 1Gex~When ionization becomes so large that strucfur+h of water around micelle surface is completely destroyed. t'lien CeX will be the most negative and remain const ani. upem furlher ionization. Since major contnl"ution to G X assumed to come from charges, the term (dG@X/arr,b, will ha approximately zero fer the entire region of ioniFation, irrespective of ionic strength. On the other hand, the aggregation number may also change with charge denstty. When electric interaction a i o m is considered, the aggregation number is expected to decrease as t i e cheurge increases. However, when ionization exceeds :m e x t m t that Gex becomes appreciably negative, then vie siippose that the aggregation number zncreases 50 as 1 0 redlire the contact area between hydrocar.turdess water. Equilibrium aggregation en I-ce determined by c? balance between 11 arid extent of the contact When micelle surface is completely covered with polar heads, the aggregation nurnbt~i~ beLornes maximum It should be noted that the described 6 arialion of Gex with ionization will not be affected if the dggregacron number changes. For, GeX IS t structure around micelle ndenl of whether a change perturbation due to charges 07 by the vr,duc,tion of the contact area resulting from mice [le grom tli
+
? ';
The j o u r n a l
ol Ph fsicsl Chemistry, Vol. 78, No. i i, 1974
The term d p Q m 3 0 / d r n - POD, which is equivalent to AGHCin the present context, depends on degree of ionization only through the variation in aggregation number. If aggregation number goes through the m i n h u m as predicted above, the term AGHC - AGHc(0) becomes positive a t low ionization, reaches the maximum, an.d then decreases as ionization proceeds further. The variation of the term, AGHC - AGHC(0) -i-(dGe'/arn),, with ionization is expected to resemble in shape what is just described about the term AGHe - AGiie(0). However, an essentially identical. shape will result for the dependence of the term, QGcic - ..?iGi~c(Q) -t- (dGel/ am),, even if we suppose the opposite situation to be the case that solvent contribution is insignificant. Under this simple situation the term Gex will be negligible a t any degree of ionization and aggregation number continuously decreases as ionization proceeds. Accordingly the term AGHC - AGHc(0) will continue to increase in positive di(8Ge1/8rn),, rection but the total, AGHc - AGHc(Oj will resemble in shape as is described above. At high ionic strength the term ( i J G e ' / a m ) , will disappear. If we extrapolate to infinite ionic strength the values obtained a t varof RT In [(cmc)(l - a ~ ~ ) / ( c m c ) o-.( l ious ionic strengths, we can evaluate the term AGiic AGHc(0) under the limiting condition. The limiting values will differ from those expected a t res strengths where experimental data are obtained. The difference may arise from both changes in perturbation of solvent structure due to small ions in ionic a t ~ o s p ~ e r e and in aggregation number caused by a change in ionic strength, The dual dependence on ionic strength can not be treated quantitatively a t present. However, we assume the difference to be raegligible. The approximation may be partly justified if we not,ice that the quantities under discussion have magnitude of second order as compared with main contributions, (dGel/lan), and :lGtic. Further, the dual. dependence is in the opposite direction and is expected to cancel out each other to some extent. By means of this approximatjon we can separately evaluate each term.
+-
Note that the term ( d G e x / a n ) , can no? he separated from (t?Gei/Ctn), in the titration data. Hence, separation of electric and nonelectric contributions is still insufficient in this respect. Nevertheless, the procedure to extrapolate to infinite ionic strength is usefd because variation of the limiting values with ionization will provide a measure of the extent of solvent eontri.bution mentioned; whether the limiting values continue to increase or they become negative after passing through t h e maximum. Comparison with Experiment Potentiometric titrations of aqueous solutions of dirnethyldodecylamine oxide (DDAO) were carried out at 25" with hydrochloric acid a t various sodium chloride concentrations. Titrations a t high ionic strengths (0.10-1.0 M, were reported by Tokiwa and O k ~ k i Titrations .~ of the solutions below cmc enabled us to determine and pK1. The degree of ionization of soiution as a whole? denoted by a , was determined experimentally with the aid of titrations of the solvent, i.e., salt solutions containing no surfactants, Details of the procedure and underlying assumptions have been. given.g The degree of ionization of micelles am,,equivalently, the titration curve for the micelle,
Free-Energy Change Accompanying Micelle Formation 0.2
0
-0.2
-0.4
-0.6
-0.8 0
0.i
0.4
0.6
o
1.0
0.8
0.2
fin
TABLE I: Intrinsic ]DissociationConstants for Monomer (KJof DDAO Micelle (Km)
e,, M
0
5.01
0.55
0.1
5.63 4.7’8 1.16
5.76 4.85 1.2‘4
5.82 4.88 1.28
5.89 4.88 1.37
0.2 6.01
4.95 1.44
(KdKrJa
‘Expressed in kcal/rnok, was evaluated in terms of the following relation
aC
=
LYJC
- CJ
+ QIC,
0.6
om
1.0
am
Figure 1. Titration curves of the micelle of DDAO. Data of the cmc are also indicated by dotted lines: salt concentrations ( M ) O( Q,a),0.01 (O,b), 0 05 ( 8 , c ) ,0.1 (ad),and 0.2( .,e),
PK,, PK1 RT I n
0.4
(20)
Were C denohes the total surfactant concentration, Titration curves for the miceile were independent of C and are shown in Figure 1 for different ionic strengths. Extrapolation of pK to x r o ionization is made in such a way that pK, may depend on ionic strength, since the standard states are defined in a salt solution. The values of pK, are given in Table 1 together with those of pK1. Data of the cnic are obtained from the measurement of surface tension of the solutions10 and are also shown in Figure 1. In Table I1 are cited the dat,a necessary to discuss the stability of micelles in terms of eq 7 and Values of RT In [(cmc)(l - a~),’(cmc)o(l- a,)] are plotted in Figure 2 against a,. At luw ionization the value is slightly positive and increases as a,, increases. When ionization proceeds further, a doavn~vartltrend appears and the value becomes more negative ais a m increases further. The general feature of the curves drawn In Figure 2 is quite consistent with that described in the preceding section. We tried to extrapolate the data to infinite ionic strength as suggested. Extrapolations could be carried out only when the data obtained a t 0.2 M WaCl were not taken into consideration.11 The limiting value:.; t!nus obtained are shown in Figure 2 by a dashed line;. Variation of the limiting values with a m reveals that solvent contribution is significant since they do not increase monotonously. The behavior implies that the aggregation numh,er decreases or remains almost constant at low ionization (below about 0.3) and then increases as ionization proceeds. Herrmann found that the aggregation number o:f fully protonated micelle of DDAO was lmger than that of fully discharged species.12 Moreover he fou.nci that the fully protonated micelle is rod like in shape,I3 which con:firmerl that t,he aggregation number was larger than t h e maximum value expected for spherical
Figure 2. An example of the analysis of experimental data on aqueous solutions of DDAO: salt concentrations j M ) O ( Q ) , 0.01(@), 0.05(9),0 . 1 ( 0 ) , and 0.2(@@). Limiting values of ordinate at infinite ionic strength are shown by a dashed line.
shape.14 These findings are quite consistent with that predicted by the present analysis about the thermodynamic data. On the other hand there are no experimental data indicating the predicted decrease or invariance in aggregation number at low ionization. It is to be stated here that although the extrapolation was made in a highly approximate manner, the general trend of the limiting values just described i s by no means altered. For example, if the limiting values were assumed to increase continuously with a,, in a consistent manner with the data corresponding to a m below about 0.3 in Figure 2, then resulting absolute values of (.1GE1/dm), would be so large that the term [ a ( W Gex)/c?a],, (dGel/ am), decreased as a m increased. Values of the term (8Ge1/8m), can be evaluated by means of the limiting values and are given in Table 11. They should be taken as representing the minimum values. We can see that the fraction of the term (JGel/ am), to the term [a(Gel GeX)/an], i s about 10% or more. Accordingly, neglection of the term (dGel/am)n seems to be not necessarily a good approxinralron. However, quantitative discussions are impossible about the magnitude of the term (aGel/am), or of the limitvng value shown in Figure 2, if the highly approximate nature of the employed extrapolation is taken into consideration. Therefore, in so far as the micelles of DDAO are concerned, the assumption of Emerson and Hoitzer i s not necessarily permissible that a charged monomer can be added to a micelle without altering micellar volume.
+
+
+
Discussion The standard free-energy change associated with the addition of a nonionic monomer to a discharged micelle is AGHc(0), as given in eq 17. When a charged moiiomer is added to a discharged micelle the corresponding free-energy change is AGHc(0) + RT In (Km/K1). Although the term RT In (K,/Kl) may include electric interaction, it is to be regarded as representing a part of nonelectric standard free-energy change in question according to the definition introduced. Equation 15 shows thst the corresponding change is AGFrC (aGel/c?m), &f a nonionic monomer is added to a charged micelle. On addition of a charged monomer to a charged micelle the corresponding result is what is given in eq 16. Though the free-energy
+
The Journal of Physical Chemistry. Vol. 78, No. 7 1. 1974
Hiroshi Maeda, Masa-aki Tsunocla, and Shoichi 1 keda
1 a90
hesgy of Micelle Formation of DDAOa pH
C, = 0 30 M , AGHc(O) a ,n
RT In ( c m c )
=
6.5
-4.09 0.12 -4.09 -0 07
6.0
5.5
5.0
0.21 -4.09
0.32 -4.09
0.44 -4.09
-0.10
-0.11
-0.01
-0.04
-0.08 1.36
0.67
0.99
0.19
0.29 -4.00 -0.07
-4.00 -0.08
0.16 -3.91 -0.06 0.65
-4.06 0.26
0.74 -4.03 0.76
-0.13
.-0.16
0.56
2.07
-4.00
0.53 -3.94
-0.01
0.29
0.66 -3.79 0.83
0.41
0.99
1.35
0.25
-3.89
0.36 -3.80
-0.05
0.05
1.02
4.0
1.73
-0.10 0.67
4.5
-0.05 1.35
.-0.21
1.68
0.48
-3.69 0.35 --0.16
1.66
-0.18 2.02 0.62 -3.50 0.98 -0.23 1.99
Expressed 111 ~cal/mole
+
change m this case has been simply written as AGHC Ne$,2 each term contains various contributions, iL being electric potential at micelle surface. For example, AGHC in this case may generally differ from that for nonionic species by what art> symbolized by the term RT In ( K m / Klj and (?Gl'x/an)7 1 . Hence a slightly different notation AG'rir is used for the case in eq 16. The term RT In (K,/Kl) should be interpreted as representing an effect of charge on nonelectric interaction, which is clearly present in the case of strong electrolytes although respective dissociation constants, K , and K1, have no physical meaning, Sinuiarly the electric part will be (8Ge1/dm), = which can be divided, at least conceptually, into two terms as illustrated in eq 18. The term (aGei/an), is exactly identical with We$. The difference hetween pKm and pK1, experimentally found in the case of DDAO, should be briefly discussed. The differenve IS considered to provide a quantitative measure for ihe effwt of charge on nonelectric interaction (hydrophobic interaction in the present context). This interpretation I E justified if electric interaction between a charged monomex and its ionic atmosphere is approximately identical w t h that encountered when the monomer is added to a discharged micelle, and if transfer of a monomer from aqueous solution to a micelle is a proper procedure for measuring the hydrophobic interaction. When we compare the values of AGHC(0) from Table I1 with those ol RT In (K,/K1) from Table I, we can conclude that a chargcd monomer is more stabilized than a nonionic one by about 30% when they are transferred to a discharged mirelle, which is equivalent to say that introduction of a charga? destabilizes a monomer by a corresponding amoLmt. A picture iiitroduced rn this paper concerning the effect of charge on .iolveiit contribution may differ from "the electrostatic d v e n t effect" proposed by Poland and Scheraga.lJ 7 hey introduced the idea to explain that heat of micelle formation of ionic micelles is generally less positive than 1 h a t of nonionic oncs.7 They ascribed the negative contribution t o a change in the solvent structure around the charged head. According to our picture, however, it IS thi. contact area between hydrocarbon chains The Joilrnal of Physical Chemistry. Voi 78. No. 1 1 , 7974
and the solvent that is supposed to be largely diminished when a monomer is transferred to a micelle, and hence accessibility of the charged heads to the solvent is considered not so largely altered by the process, contrary to their supposition. Instead, the effect of charge is to reduce the positive solvent contribution to the enthalpy change which plays a central role for the change in the case of nonionic micelle, since more solvent structure is destroyed by the introduction of a charge in the state of monomer than in micelles.
References and Notes (1) R. H. Aranow,J. Phys. Chem.. 67, 556 (1963). (2) M. F. Emerson and A . Holtzer, J. Phys. Chem.. 69, 3718 (1965); 71, 1898 (1967). (3) P. Mukerjee. J. Phys. Ghem.. 73, 2054 (1969). (4) A. Katchalsky and J . Gillis, Recl. Trav. Chim. Pays-Bas. 68, 879 (1949); F. E. Harris and S. A. Rice, J. Phys. Chem., 58, 725 (1954): R. A. Marcus, J . Chem Phys.. 23, 1057 (1955) (5) The use of the term "reference state" will be more appropriate in place of the term "standard state" here employed. rlowever, we use the latter for the sake of consistency, since .the term -RT In (cmc) or LGHC has been conventionally termed as "standard" free-energy change. (6) I f the most probable aggregation number m is defined as sych that m'Cm,.n(m,) instead of C,(,' is the maximum at m = in. then corresponding aggregation numbers should be multiplied to respective concentrations in eq 9 and a term RT/m' should be added to left-hand side of eq 10. However, the difference will be unimportant for the range of values of m usually encountered. (7) L. Benjamin, J. Phys. Chem.. 68, 3575 (1964) (8) I. Reich, J. Phys. Chem.. 60, 257 (1956); D. C. Poland and H, A. Scheraga, ibid.. 69, 2431 (1965). (9) F. Tokiwa and K , Ohki, J. Phys. Chem.. 70, 3437 (1966) (10) M. lsunoda, H.Maeda, and S. ikeda, manuscript in preparation. (11) When C, is smaller than about 0.1 U . values of RT In [(cmc) (1 a,)/(crnc)o(l a,)] graduaily increase as Ch-' decreases, if they are plotted against the latter at constant a,. Steep increase is observed, however, when C, is larger than about 0.1 M . and extrapolation of the curve to infinite ionic strength becomes difficult. The observed trend to tend to infinity as ionic strength approaches infinity is quite Inconsistent with the expectation from our theoretical treatment. The cause for this discrepancy will be considered to arise from the limitation of our treatment rather than to arise from experimental error. A possible limitation will be that lamellar micelles are present at high ionic strength which are quite different from the assumed model that they are dispersed In a solution as independent particles. (12) K.W . Herrmann, J. Phys. Chem.. 66, 295 (1962). (13) K . W. Herrmann,J. Phys. Chem,. 68, 1540 (1964). (14) C.Tanford,J. Phys. Chem , 76, 3020 (1972) (15) D. C. Poland and H. A. Scheraga, J. Colloid interface Sci. 21, 273 (1966)
-
