Free Energy of Ionic Surfactants
1015
Micelle Formation by Ionic Surfactants. IV. Electrostatic and Hydrophobic Free Energy from Stern-Gouy Ionic Double Layer Dirk Stigter Western Regional Research Center, Agricultural Research Service, U.S. Department of Agrlcuiture, Berkeley, California 94 7 10 (Received September 9, 1974; Revised Manuscript Received February 21, 1975) Publication costs assisted by the U.S. Department of Agriculture
The charge effects and the hydrophobic interactions in micelle formation of ionic surfactants in aqueous salt solutions are reconsidered. The treatment is based on a Stern-Gouy model of the ionic double layer, with (1- a)n counterions in a regular distribution between the ionic head groups, and the remaining an counterions outside the hydrodynamic shear surface of the micelle in a diffuse Gouy-Chapman atmosphere. The model is developed in stages, starting from a Gouy-Chapman model used earlier. Results for the electrical free energy of the micelle, pnel, given a t each stage for representative solutions, show large effects of the discreteness-of-charge, and of the change of self-potential AC$ of the head groups upon micellization. In the final Stern-Gouy model results are given for the series of Cg to Cl2 sodium alkyl sulfates and for dodecyl ammonium chloride. The uncertainty in pnel/n is of the order of 0.5hT, due partly to the paucity of information about a. As long as n is constant the details of micelle shape and ionic radii have little influence on pnel. The residual free-energy change pn0 - nplo, obtained from the equilibrium equation, is attributed mainly to hydrophobic interactions. This free-energy change is related to the hydrocarbon/water contact area, A, for a micelle and A1 for a monomer. In the bilinear form pno/n - pl0 = aiAn/n + anA1 + u3, the coefficients are a1 = 33 ergs/cm2, a2 = -25 ergs/cm2 or -740 cal/mol per methylene group, and a3 1.6hT per monomer. A discrepancy in the Gouy-Chapman model between alkyl sulfate and dodecyl ammonium micelles is eliminated in the Stern-Gouy model, mainly due to the introduction of the self-potential term AC$.Corrections for salting-out effects on pl0 bring the calculated results in concentrated sodium chloride solutions, up to 1M , in line with those in more dilute solutions.
-
Introduction The first three papers of this series deal with free-energy calculations for the Gouy-Chapman model of the electrical double layer around ionic micelles,l and with improvements of the m 0 d e 1 . ~In~ the ~ preceding paper3 the ion distribution was discussed between the surface of the micelle core and the hydrodynamic shear surface of the micelle, the Stern layer which contains the head groups and about half the countercharge. It was concluded that a good approximation is the "low temperature" distribution which maximizes alternation of head groups and counterions on the sites of a two-dimensional hexagonal lattice. This distribution is used in the present paper to treat the electrical free energy of the micelle. We assume that the counterions in the Stern layer are at the average positions, with their center a short distance (2)further than the ionic radius from the core surface. The remaining counterions are outside the shear surface, in a diffuse, Gouy-Chapman type, atmosphere. On the basis of this model we evaluate the free-energy changes that accompany micellization. The thermodynamic framework has been developed ear1ier.l We recall that there are two different approaches to the equilibrium between monomers and micelles. The first one refers to the integral process, that is, the formation of a new micelle Dnn- from n monomers D1( 1)
nDi' S Dn"-
subject to'the equilibrium condition
kT In x,, - nkT lnfixi experiment
+
pnel
theory
< pno- n p i o = 0 residue ( 2)
where x, and x1 are the mole fractions of micelles and of monomeric surfactant, fine1 is the electrical and pno the nonelectrical part of the standard state chemical potential of the micelle, f l is the activity coefficient, and p l o the standard state chemical potential of the monomer. We shall deal with equilibria near the critical micelle concentration (cmc) where the activity coefficient of the micelles may be assumed equal to unity. The second approach considers the growth process D,"'
+ Di' e Dn+i(n+i)-
(3) which obeys, at the optimum micelle size, the differential equilibrium equation -kT lnfixi + ( a p n e ' / ~ n ) + , (aPno/an), - pi 0 = 0 experiment
theory
residue
(4) The subscript x in eq 4 indicates that the pressure, the temperature, and the composition of the solution are constant in the differentiation. For the purpose of this discussion we distinguish three different groups of terms, as indicated in eq 2 and 4.First, the concentration terms are determined from experiments, and so is the micelle number n. Secondly, the electrical terms are evaluated from the theoretical model summarized above. Finally, the nonelectrical parts of the standard free-energy changes are obtained as the remainder in the equilibrium equations. Current thinking is that the main contributions to these residual terms arise from hydrophobic interactions. It is interesting that, with the Gouy-Chapman model for the electrical terms, the residues for the Cs to Cl2 series of alkyl sulfates were found to vary linearly The Journal of Physical Chemistry, Vol. 79, No. 10, 1975
Dirk Stigter
1018
with the changes of the hydrocarbonlwater contact area in micelle f0rmation.l The main purpose of this paper is to investigate in how far this linearity persists when major shortcomings of the Gouy-Chapman theory are corrected. The previous work1 was based on some fairly accurate sets of experimental data on micellar solutions of sodium alkyl sulfates and of dodecyl ammonium chloride available in the literat~re.*-~These data are used again in the present paper. The quality of the results from eq 2 and 4 depends mainly on the treatment of the electrical free energy. Unless indicated otherwise, the Stern layer is defined with the parameter values used or established in the preceding paper.3 The discussion is given for micelles of cationic surfactants. The minor changes for anionic surfactants are obvious. A suitable expression for wnel is derived from an imaginary charging process of the micelle,l a t constant micelle size and constant ionic strength, with an additional term n A + for the change of self potential2 of the surfactant charge upon micellization. ne
=
0
&,(A) d(neh)
+ nA$
( 5)
The electrostatic potential of the head group $h is a function of the charging parameter X and takes into account the interionic interactions. The intraionic effects represented by A+ in eq 5 depend on the change in dielectric environment when a surfactant head group moves from the bulk of the aqueous solution to the vicinity of the apolar micelle core. When the head group is regarded as a point charge A+ is given3 by eq P23. Development of a Model for Electrical Free Energy wnel
The preferred model for pnel is developed in stages, starting from the Gouy-Chapman model used previous1y.l For this and for other intermediate models, values for wne1 are compared in Table I for a variety of micelles. The micellar solutions are characterized by the ionic strength and by the micelle number n. Further details are given e1~ewhere.l.~ The columns in Table I refer to models a to f described below. The change of self potential is introduced in model e. So A+ = 0 for models a to d. (a) We begin with the model of ref 1, spherical micelles whose radius R is calculated from the micellar weight and from the density of the micellized surfactant. The micelle has a uniform surface charge Xne, and is surrounded by a diffuse, Gouy-Chapman ionic atmosphere with a thickness or Debye length 1 / K related to the ionic strength in the bulk of the aqueous solution far from the micelles. We identify the head group potential with the surface potential of the micelle hne = p ~ , R ( l + KR) The parameter p is a function of KRand of e$hlkT, and has been represented as a double expansion in powers of these variables8 Results for wnel/nkT from eq 5 are presented in Table I under column a. The data for the alkyl sulfates do not differ significantly from those reported by Huisman7 for the same systems. For future reference we derive the contribution +d1 of the diffuse layer to $h. The potential of the surface charge is Xne/c,R. So by difference one finds with eq 6 Q 1 'dl = Q ( p ( 1 + KR) -
''
The Journal of Physical Chemistry, Vol. 79, No. 10, 1975
where Q = h e . The diffuse layer has spherical symmetry, and its charge inside the micelle vanishes. Therefore, the potential of the diffuse layer is constant and equal to $dl everywhere within the surface R = constant. (b) In the second model eq 6 is used again, but the charged micelle surface is now shifted to the surface of the head group charges in the Stern layer, compare Figure P1 in ref 3. According to Table I this choice, R = t, reduces pnel somewhat compared with model a. (c) The distance of closest approach between head groups and counterions is u = rh rc. For this reason the diffuse layer is now moved outward and its potential $dl is obtained from eq 7 with R = t u, see Figure P1 in ref 3. The head group charges are smeared out in the surface t = constant as before in model (b), and contribute Xnele,t to +h. The results in Table I indicate that the shift of the diffuse layer, column b to c, increases pnel/n by amounts varying from 0.92kT to 2.36kT. In a discussion of a very similar model Mukerjeeg has pointed out that results for the surface potential of the micelle are too high and the dependence on ionic strength is unreasonable. The main reason is the assumption that a = 1, that is, no counterions in the Stern layer. In the final model more realistic values, about a = 0.5, will be taken. However, in order to show the various discrete ion effects, we shall assume a = 1 in the intermediate models d, e, and f. (d) This model introduces the largest discrete ion correction. Around a head group one assumes a charge-free circular surface with area 4*t2/n, the average area per head group. In the rest of the surface t = constant the uniform charge density Xne/4nt2 is unchanged. A schematic cross section of the micelle is given in Figure 1, with the head group charge at P for r = t. Looking ahead to the interaction with counterions in the Stern layer, we need also the more general case r # t , when the potential in P is given bylo
+
+
2x cos 0
s,
Xi/(l+W)(l
-
2y"W
+
x41/2
+
xwf(l+*) x
cos (3 + y 2 + 2 w ) 1 / 2 YltW
-
+
yl+w d4 ( 8)
where x and w have the same meaning as before3 in eq P17. Following spherical geometry the charge-free surface area associated with the angle 0 in Figure 1 is equal to 2at2(1 - cos e). So the contributions of the smeared-out -neighboring head groups to $h are obtained from eq 8 with cos 0 = 1 - 2/n and r = t. The diffuse layer contribution remains the same as before. The comparison of columns c and d in Table I shows that the discrete ion correction for the central head group lowers wnel/n by an appreciable amount, from 1.51kT to 2.59kT. The present discrete ion effect was first introduced by Overbeek and Stigterll whose approximate corrections for dodecyl sulfate micelles ranged from 1.76kT to 1.89kT. This compares well with the results in Table I, 1.69kT to 1.941zT for essentially the same micelles. A similar correction has been applied by Grahame12 in the "cut-off disk" model for counterions adsorbed a t the mercurylwater interface. (e) The self-potential change A4 of the central head
Free Energy of ionic Surfactants
1017
TABLE I: Electrical Free Energy for the Different Micelle Models a to f as Described in the Text pne’/nkT C,*t,a
C ,H,,S04Na C12H25S04Na
C12H25”3C1 a
cmc
+
144
n
a
b
C
d
e
f
0.134 1.03 0.0081 0.3 0.0156 0.066
23.7 47.8 57.3 123 100.7 163
2.44 1.67 4.72 2.62 4.84 4.06
2.05 1.40 4.44 2.40 4 .BO 3.90
2.97 3.17 5.71 4.69 5.69 6.26
1.46 1.41 4.02 2.75 4.26 3.67
1.78 1.78 4.41 3.19 6.20 5.67
1.72 1.68 4.31 3.06 6.01 5.46
CN~CI.
+ +
terions are at distance r = b r, ( 2 ) from the micelle center, as indicated3 in Figure P1. The potential of the diffuse layer is obtained from eq 7 , with R = t u as before, and with Q = aXne. In this way the calculation of the head group potential $h( A), and hence of p n e l , is straightforward, provided that a is known as a function of A.
7’
+
Figure 1. Cross section through the dielectric sphere. Part of concentric surface with radius t, heavy curve, has uniform charge density ne/4&. Potential in P is given by eq 8. group, calculated3 with r = t in eq P23, is now added to Inspection of columns c, d, and e of Table I shows that allowing for A 4 partially compensates the previous correction in model d for the smeared-out central charge. Both corrections are strongly dependent on the distance between head group charge and core surface, t - b in Figure 1. (f) We now introduce the lattice model of the Stern layer, with discrete head groups on the six nearest neighbor sites of the central head group. Their contribution to $h(X) is evaluated3 with eq P17, with r = t and with Xe instead of e, and with M = n in eq P3 and P4 for 0. The n - 7 more distant head groups remain smeared out as before. For these ions we use eq 8, again with r = t. In order to account for the larger charge free area corresponding to 7 out of n sites, we now have in eq 8 cos 0 = 1 - 14/n. Following columns e and f in Table I the present changes do not decrease p n e l / n a great deal, only 0.06kT to 0.23kT. This confirms the earlier findinglo that the smearing out of the nearest neighbor ions does not change the results very much. At this point we are ready for the final model of the series in which counterions are admitted in the Stern layer, and the equilibrium of the charge distribution comes into play. The n head groups and n, = (1 - a ) X n counterions are distributed over A4 = n n, lattice points. Following the low temperature a p p r ~ x i m a t i o n ,eq ~ P21, each head group has as nearest neighbors f h = 6 ( 1 - a)X counterions with charge -e and 6 - f h head groups with charge +Xe. The contributions of these ions to $h are obtained3 with eq P17, P3, and P4.Outside the area allocated to these seven discrete ions the remaining charges in the Stern layer are smeared out uniformly and their potential is calculated with eq 8 where cos 0 = 1 - 14lM. It is observed in eq P17 and 8 that head groups are a t distance t = b h, and counpnel/n.
+
+
Variable Parameters in the Evaluation of pnel As mentioned already, experiments suggest a roughly equal distribution of counterions between the Stern layer and the diffuse layer. This means that for fully charged micelles, X = l, the equilibrium is around a = 0.5. The main problem now is to choose a function a(X)such that an equilibrium distribution of counterions, consistent with the model, is maintained during the entire charging process. Strictly speaking, one should require that the electrochemical potential of the counterions in the Stern layer be independent of A, and always equal to that in the bulk solution. The application of this condition entails the evaluation of the adsorption isotherm of the counterions, Le., n,(X),from a model of the Stern layer. In the earlier worklo the adsorption of counterions in the fully charged Stern layer, X = 1, was treated with a cell model, consistent with our present assumptions of the Stern layer. Unfortunately, further, exploratory calculations have shown that the function n,(X) is very sensitive to some uncertain details of the model, such as the cell size as a function of M ( X ) = n .,(A). Such details do not otherwise enter into the evaluation of wnel. Moreover, the cell model for counterion adsorption is by no means exact. For these reasons the attempts to derive n,( X) were abandoned in favor of a comparison between results of pnel for two different, assumed functions for n,(X) = (1 a)Xn,that is, two functions for a(X) which are intuitively reasonable. We choose the two functions a(X) by considering the diffuse layer. For very low potentials, e$h(X) ni (14) The surface area of a bent cylinder, such as a torus, does not depend on the curvature of the cylinder axis. Consequently, eq 14 is valid for wormlike micelles with arbitrary curvature of the central axis. Since the electrostatic calculations are made only for The Journal of Physical Chemistry, Vol. 79. No. 10, 1975
Flgure 3. Electrical free energy of micelle, per monomer, in various micellar solutions defined in Table I, vs. fraction of countercharge in Gouy-Chapman layer. Curves a and b correspond to different charge distributions in the charging process, eq 9 and 10, respectively.
spheres, the wormlike micelle is replaced by an equivalent sphere. We compare the results for two different substitutions. One is based on constant volume of the core, in which case eq 12 is used for n > nl. In a second method the surface area of the core is kept constant and, in view of eq 14, the core radius becomes
b = li(2n 4- T Z ~ ) i ~ 2 ( 3 n i ) ~n* ~>2
(15)
The comparison of results for wnel based on eq 12 and 15 gives an idea of the effect of micelle shape on K ~ In ~fact, ~ Table 11, columns a and c, shows no more than minor differences, a t most 0.17kT between the two calculations. It is clear that the effect of micelle shape on hnelis not very important. In the rest of this paper we shall use the wormlike model, eq 15 for n > nl. Finally, we consider variations of the corelwater interface where the dielectric constant changes from tC = 2 to tw = 78.5. This interface was found2 a t d = 0.8 f 0.4 8, above the a-carbon atoms of the micellized surfactants. Table I1 gives results of Knel/n and of A$ for the limiting values d = 0.4 and 1.2 8, in columns d and e. I t is observed that the changes of hnel/n are close to those in A$. Between d = 0.4 and 1.2 A Hnel/n varies up to 3.68kT, while pnelln - A$ changes a t most 0.29kT. This means that only the self potential A$ of the head groups is sensitive to the exact change of the dielectric constant through the micelle surface. However this change is not of major influence on the interaction between the ions. This result, expected from the nearly spherical symmetry of the charge distribution, has been used already in the previous paper.2 The discussion of the model for hnel is now concluded. The rest of the paper deals with the application of the theory to the experimental data used in ref 1. Evaluation of Residual Free Energy in Equilibrium Equation The analysis of the residual terms in eq 2 follows largely the procedure developed earlier for the Gouy-Chapman model.1 We take advantage of the outstanding feature of ionic interactions in water, their great sensitivity to the ionic strength. As we add salt to a micellar solution, the equilibrium shifts and all thermodynamic terms, including K~~ - npl0, sweep through a range of values. This magnifies the experimental output and is of considerable help in testing any kind of micelle theory. We start with a cor-
.
Free Energy of Ionic Surfactants
1019
TABLE 11: Influence of Model Parameters on Electrical Free Energy of Micelles, pnel, and on Self Energy of Head Groups, A$ p ,"/nkT
n
A $ / kT
a
Q
p;'/nkT b
pne'/nkT A $ / k T
pne'/nkT A $ / k T
d
e
C
C ,Hi ,S04Na
23.7 0.50 0.78 0.32 0.74 0.78 0.73 0.24 0.86 0.43 47.8 0.50 0.69 0.36 0.61 0.71 0.65 0.28 0.76 0.49 i2H25S04Na 57.3 0.50 0.39 2.37 2.41 2.41 2.36 0.31 2.48 0.52 123 0.50 1.45 0.44 1.36 1.54 1.42 0.35 1.52 0.58 Ci2H25"3C1 100.7 0.41 3.59 1.94 3.53 3.64 2.88 1.12 6.56 5.05 163 0.41 3.14 2 .oo 3.05 3.31 2.45 1.17 6.12 5.13 a Reference values. b Test of ionic radii, with u 0.5 A smaller than in a. Test of micelle shape: spheres, eq 12, compared with wormlike micelles, eq 15 for n > nl, in a. d , e Test of dielectric constant, with d = 0.4 A in d and 1.2 A in e cofnpared with d = 0.8A in a.
rection for the salting-out effect on the alkyl group of the unassociated surfactant. When pno - nwl0 is obtained as the residue in eq 2, p l 0 refers to the particular salt solution in which the micellar equilibrium has been studied. In general this value, plo(salt), differs from the value in pure water, plo(water). In the following discussion of pno - npl0 we shall use water at 25' as a reference solvent for @lo,after conversion of the data with
plO(water)= piO(salt)- (0.129
+
4
.6!-
0 01
0.0645~?kTc,,,~
(16) In this expression csalt is the total molarity of cmc and added sodium chloride, and i is the number of carbon atoms in the alkyl chain of the surfactant. Equation 16 is based on the solubilities of ethane, propane, and n-butane in water and in 1 M sodium chloride solutions, reported by Morrison and Billett.15 As explained above, a major problem is the function a(X) for the distribution of the counterions between the GouyChapman and the Stern layer, that is, outside and inside the shear surface of the micelle. For the fully charged micelle this function, a ( l ) , can be evaluated from transport measurements.16 Figure 4 shows the data available for sodium dodecyl sulfate (SDS) as open points, and for dodecyl ammonium chloride (DACl) as solid points. As stated earlier,16 it is likely that the conductivity results (triangles) are more reliable a t low ionic strength, and that the electrophoretic data (circles) are more accurate a t high ionic strength. On the basis of Figure 4 we select a(X) = a(1) = 0.5 for SDS and a(X) = a ( 1 ) = 0.41 for DACl to evaluate the electrostatic terms in eq 2 and 4. The residues are plotted vs. n in Figure 5a. The integral and the differential data are compared using the method of ref 1 as follows. The open points for pn0/n - y l o from eq 2 are fitted with a second degree polynomial in In n (least-squares test), the solid curves in Figure 5 . The corresponding differential polynomials for (apno/an)x- p1O are the broken curves which for good experiments and a good theory should match the filled points for the residues from eq 4. It is obvious from the large discrepancy between broken lines and filled points in Figure 5a that something is wrong, either in the experiments or in the theory. Now, for SDS the circles and the triangles have been derived from two different sets of light scattering experiments on different SDS preparations. The difference between the two sets of results is relatively small, so it is not the experiments but the theory that fails the test of the differential residues in Figure 5a.
01
CMC t CNaC,MOLE/L
Figure 4. Fraction of countercharge outside shear layer of micelles vs. ionic strength: open points, sodium dodecyl sulfate: filled points, dodecyl ammonium chloride; circles, from electrophoresis;triangles, from electric conductance.
There is considerable latitude in the choice of a(1). For instance, the straight lines in Figure 4 are well within the uncertainty of the experimental points. The solid line for SDS is given by ~ ( l = ) 0.5 - 0.0174 log (c,,,t/O.031) The broken line for DACl is ( ~ ( 1 )= 0 . 4 1
- 0.0087 log (csa,t/O.031)
(17) (18)
Equations 17 and 18 for a(1) have been used with the two functions a(X) of eq 9 and 10. Figure 5b is based on eq 9 for constant a(X),and Figure 5c on the variable function a(X) in eq 10. In both Figure 5b and 5c the broken lines are close to the filled points for the differential residues. This means that such self-consistency depends mainly on a(1) as a function of ionic strength, or of micelle size, but not much on the particular function a(X). As explained above, there are theoretical grounds to prefer a variable function a@). Equation 10 is a simple, linear example of such a function. Unfortunately, a t the present time there is no objective criterion by which to narrow the choice of a(X) any further. The difference between the results in Figure 5b and 5c is indicative of this uncertainty in cy(X). Changes of Free Energy a n d of Interfacial Area In this section we consider the correlation between p,O nMl0 and the change of hydrocarbon/water contact area, A , - nA1, in micellization. For the monomer, a cylindrical alkyl group with a rounded end of radius r1 = 2.615 A and length 1, from eq 11, the contact area with water is A, = 2i~Zi~i (19) For micelles the surface area of the core is 4.rrb2, with b from eq 12 and 15 for spherical and wormlike micelles, reThe Journal of Physical Chemistry. Vol. 79, No. 10, 1975
Dirk Stigter
1020
n
n
n
Figure 5. Comparison of average and differential free energy changes as a function of micelle number n in aqueous sodium chloride solutions: circles and triangles, sodium dodecyl sulfate (SDS); squares, dodecyl ammonium chloride (DACI); open points and solid curves, ( p n o l n p l o ) / k T ;filled points and broken curves, ((apno/an),- plo)/kT.Points in 5a, 5b, and 5c are based on different distributions for counterions between the Stern layer and Gouy-Chapman layer from eq 9, 10, 17, and 18. 1
