2873
Langmuir 1992,8, 2873-2876
A Self-Consistent Theory of Dressed Micelles John B. Hayter Solid State Division, Oak Ridge National Laboratory, P.O. Box 2009, Oak Ridge, Tennessee 37831-8218 Received May 5, 1992. In Final Form: July 9, 1992 The dressed micelle model of Evans, Mitchell, and Ninham is recast in self-consistentform and solved analytically. Considerationof the adsorption excess of ions about the micelle leads to a remarkablysimple expression for the degree of micellar ionization. Values calculated from this theory, which has no free parameters, generally agree well with experimental values measured by neutron scattering.
Introduction In the past decade theories of the structure of simple liquids have been extended to describe charged colloidal systems. One great success of such theories has been to show that the degree of ionization of charged micelles in solution may be determined throughout the spherical phase by small-angle neutron scattering (SANS) experiments. It has generally been found that the micelles are less than fully ionized at all concentrations, with a degree of ionization typically in the range 20-50 76. A number of researchers have addressed the question of why counterions should bind to the micelles,lg culminating in the seminal contribution of Evans, Mitchell, and Ninham4 (EMN)in which the "dressed micelle" theory of ion binding was developed. This theory establishes a method of calculating the degree of micellar ionization directly from measurable experimental parameters. Chao et al.6 undertook to test the dressed micelle theory against some of the large body of available experimental data, but they were unable to rederive the originaltheoreticalexpressions of EMN. They therefore assumed (incorrectly9 that the EMN expressions were erroneous and used an earlier, less accurate expansion method, leaving the EMN theory untested. In comparing the EMN results with experiment, I have found that while the theory is fairly good under conditions of high screening, it fails for small screening, a feature which was anticipated by the researchers. In this paper I shall show that this failure is not intrinsic to the thermodynamic approach used by EMN, but merely relates to an inconsistency in their approximate method of solution. A new, self-consistentapproach is presented here which leads to analytic results in good agreement with experiment, even for small Screening,justifying the EMN dressed micelle picture of ion binding. Dressed Micelle Theory Consider a spherical micelle of total radius R and aggregation number N immersed in a 1-1 electrolyte. In spherical symmetry, the nonlinear Poisson-Boltzmann (PB) equation which describesthe ionic distribution about (1) Loeb, A. L.; Overbeek, J. Th.G.; Wiersema, P. H. The Electrical
Double Layer Around a Spherical Colloidal Particle; MIT Preee: Cambridge, MA, 1961. (2) Ohshima, H.; Healy, T. W.; White, L. R. J . Colloid Interface Sci. 1982,90, 17. (3)E v m , D. F.; Ninham, B. W. J. Phys. Chem. 1983,87,5025. (4) Evans, D.F.; Mitchell, D. J.; Ninham, B. W. J.Phys. Chem. 1984, 88,6344. (6) Chao, Y-S.;Sheu, E. Y.; Chen, S-H. J. Phys. Chem. 1985,89,4882. (6)The equations in the Appendix of ref 4 are correct, apart from a eingle typographical error at the end of the first line in eq A.15, where the sign should be changed from negative to positive.
0743-7463/92/2408-2873$03.00/0
the micelle depends only on the radial distance, r, from the micelle center. In SI units
where +(r)is the potential, p(r) the spatial charge density, e the dielectric constant of the medium, and q,(-8.854 X 10-l2F0m-l) the permittivity of free space. Defining dimensionless units x = rr and y = @e+with @ = l/kBT, where K - ~is the Debye screening length, e is the charge on the electron, kg is Boltzmann's constant, and T is the absolute temperature, the PB equation may be written ae
-(d )
=[2sinhy-i*]* x d x dx With the boundary conditions y 0 and dyldx 0 ae x =, eq 2 may be integrated in the range ( x o , ~ )where , xo = KR,to yield *2
-
-
dx dx
-
where yo = y(x0). EMN give a convincing argument, followed here, that the main contribution to the integral in eq 3 is expected from the vicinity of y = yo, or for values of x near XO. Removing this factor from the integral then yields as a good approximation (4) where s is the scaled surface charge density X=Xo
€,era
and a = 4uR2/N is the area per surfactant headgroup. The crucial difference between EMN and the present paper now lies in how we evaluate the curvature integral in eq 4. The quite resonable ansatz used by EMN and others2 has been to approximate the curvature term by the planar result dyldx = - 2 sinh @/a) (5) at which point the whole problem may be carried through to an analytic conclusion. However, this approach is inconsistent; the value of s is taken in the theory to have the (unequal) values given simultaneously by the righthand side of eq 5, evaluated at y = yo, and the square root of eq 4. The key to the present calculation lies in using a self-consistent renormalization of the planar result to generate an approximation for dyldx which ala0 satisfies eq 4a, namely 0 1992 American Chemical Society
Hayter
2074 Langmuir, Vol. 8, No. 12, 1992 Table I. A Comparison of the Present Theory, Equation 11, with Some Other Theories for Surface Charge Density, s, as a Function of Surface Potential, yo, at Different Screening Values, XO. ~
xo
yo
SBInct
seq11
Sref4
Smf 2
Sref 1
0.1
1 2 4 6 8 10 1 2 4 6 8 10 1 2 4 6 8 10 1 2 4 6
11.1 22.09 44.89 10.86 110.33 201.06 2.03 4.25 10.53 23.96 58.68 152.49 1.24 2.72 1.88 20.17 55.36 149.20 1.09 2.44 1.41 20.22 54.11 148.60
19.65 31.12 61.18 11.59 105.39 193.03 2.41 4.84 10.91 23.98 58.51 152.41 1.26 2.15 7.89 20.17 55.36 149.20 1.09 2.44 7.41 20.22 54.71 148.60
4.64 9.61 22.24 43.04 84.18 183.68 1.77 3.11 9.84 23.38 58.31 152.30 1.22 2.10 1.84 20.15 55.35 149.20 1.09 2.44 1.40 20.22 54.77 148.60
10.98 21.84 43.39 61.15 106.90 198.85 2.03 4.25 10.52 23.96 58.61 152.49 1.24 2.72 1.88 20.11 55.36 149.20 1.09 2.44 1.41 20.22 54.17 148.60
10.84 20.84 31.12 56.24 93.14 181.81 2.02 4.20 10.30 23.66 58.44 152.35 1.24 2.12 1.86 20.16 55.35 149.20 1.09 2.44 1.41 20.22 54.77 148.60
1.0
5.0
20.0
8 10 a
The exact values are the numerical results of Loeb et al.’
Substituting eq 6 in eq 4 and performing the integration yield a self-consistent equation for s:
and interfacial tensions from the local surface slopes. For this purpose, the absolute value of g&a) is irrelevant (although one would like it to be reasonably accurate as a matter of principle); the primary requirement is to construct accurate deriuatiues. The present self-consistent formalismresults in electrostaticfree energyvalueswhich, although smaller, do not differ greatly from the EMN values. The appropriate derivatives, however, produce better agreement with experiment than the EMN values, and the results extend the utility of the EMN approach to much smaller values of XO. (In the opposite limit of xo m, both theories coincide.) The great advantage of the EMN formalism is retained, namely, that no further mathematical approximations will be required to derive all needed quantities in closed analytic form, using only elementary mathematical tools (albeit in some quantity). The electrostatic free energy is obtained by considering the energy required to charge a micelle to its final value in the presence of all the other components in the final (ionized) system. Per monomer, this is simply
-
&&,a)
= (l/s)KYo(s’)ds’
(12)
where the integration is performed at constant K and constant a; that is, we construct the final system with conceptually neutral electrons and then integrate the energy required to build up the electronic charge to its usual value. This process must be kept clearly in mind, because it will later inhibit us from taking derivatives with respect to K or a simply by formal differentiation under the integral sign in eq 12, requiring instead explicit differentiation procedures. The integral (eq 12) may be evaluated by integration by parts:
Settingz = cosh (y0/2),u = s2 + 4 , and t = 8S/xo, eq 7 may be rearranged to give a relation between z and s
- U ) 2 ( Z 4- 1)- t2(Z- 1) = 0
(8) which may be easily solved for z in terms of s, using straightforward Newton iteration with the EMN approximation (42’
+ x,O~)’/~]W~) -In ([l+ (1+ w ~ ) ” ~ ] x , ~(13) )]
(1
The EMN dressed micelle analysis now lets us determine the adsorption excess per monomer, FIN, of ions about the micelle:
as the starting estimate. The surface potential is then yo = 2 In [ z
+ (z2 - I ) ’ / ~ I
(10) Alternatively, eq 7 may be solved for the surface charge density, given yo, yielding
where w X O ( Z + 1)/2. At this point, some remarks on accuracy requirements are in order. The EMN estimate of s as a function of xo and yo is already in quite good agreement with the exact numerical values, and the present theory is not markedly better in the range ( X O 1 1and yo 1 3) typical of charged sphericalmicelles (seeTable I). In fact, if the only purpose of the theory were to provide arelationship between surface charge density and surface potential, there is no need to look further than eq 24 of Ohshima et al.,2particularly if values of xo < 1 should be of interest. The aim here, however, is to construct the electrostatic free energy surface, g,dK,a), and then to obtain adsorption excesses
As discussed above, this expression must be evaluated via explicit differentiation of eq 13. The result is
This remarkably simple expression is the central result of this paper. We first remark that it predicts a finite degree of ionization in the limit of small KR: 6, = lim 6 = 8/x@
(16)
xo+J
where the degree of ionization, 6, is given by 6 = 1- (I’/N). Using eq 11and noting that [(z
cosh (yd2) + 1 + l ) / ( -~ 1)]1’2 = coth (yd4) = sinh (yd2)
-
(17)
it may be shown that eq 15 coincideswith the EMN result as KR m. However, the values of 6 computed from the present theory differ substantially from those of EMN for
A Self-Consistent Theory of Dressed Micelles
Langmuir, Vol. 8, No. 12, 1992 2875
Table 11. Comparison of Degrees of Ionization, Calculated from EMN and from Equation 15, with Experimental Valuer' for Sodium Dodecyl Sulfate (SDS) Micelles at T = 313 K
[SDS](mo1.L-l)
[NaCl] (mol-L-l) 0.00
0.016 0.04
0.00 0.05 0.20 0.00 0.00 0.05 0.10 0.20 0.00 0.05 0.10 0.20 0.00 0.05
0.07 0.10
0.40
0.50 0.60
%O
a (nm2)
8
6EMN
6915
6t,
0.61 0.63 1.79 3.51 0.61 0.66 1.80 2.51 3.55 0.69 1.89 2.63 3.71 0.69 1.89
0.684 0.718 0.581 0.557 0.598 0.612 0.578 0.561 0.544 0.554 0.533 0.527 0.512 0.548 0.542
44.98 42.80 19.74 10.88 51.40 50.24 19.87 14.99 11.16 55.49 21.55 15.99 11.84 56.08 21.22
0.13 0.14 0.18 0.24 0.12 0.11 0.18 0.20 0.23 0.10 0.16 0.19 0.22 0.10 0.16
0.26 0.27 0.23 0.25 0.23 0.22 0.22 0.24 0.26 0.21 0.22 0.23 0.25 0.19 0.20
0.30 0.30 0.28 0.20 0.34 0.24 0.25 0.24 0.21 0.22 0.20 0.19 0.19 0.19 0.20
Table 111. Comparison of Degrees of Ionization, Calculated from EMN and from Equation 15,with Experimental Values' for DodecyltrimethylammoniumChloride ((C1ZTA)Cl)Micelles at T = 313 K [(C12TA)Cll (mol-L-l) 0.20
0.40 0.60
[NaCl] (mol-L-9 0.00 0.10 0.00 0.10 0.00 0.10
XO
a (nm2)
S
6EMN
6,
0.20 1.96 0.19 1.96 0.20 2.00
0.721 0.716 0.724 0.705 0.675 0.671
121.30 12.15 120.80 12.35 129.60 12.98
0.09 0.26 0.09 0.26 0.08 0.25
0.30 0.34 0.32 0.36 0.30 0.34
KR I3; the self-consistencyrequirement imposes a subtly different shape on the free energy surface, even though the absolute free energy values from the two theories do not differ greatly. EMN propose that "the ideal dressed micelle picture will be justified if an ab initio calculation shows that I'lN is constant as a function of salt concentration and if the value so computed agrees with the measured value." The results of the next section will demonstrate that the second criterion is generally true. With respect to the first criterion,evaluationof eq 15shows that the self-consistent ab initio calculation predicts fairly constant ionization in the range 0 < KR I4, which is the range applicableto most spherical charged micellar phases. The validity of the dressed micelle picture is thus substantially reinforced by requiring self-consistencyin the theory. The approximate constancy of I'/N at small xo provides a posteriori justification of the use of the present theory to compute I'lN at small values of KR, but we can actually make a stronger statement. In the limit of small yo, eq 7 reduces to
lim s = (1+ l/xo)y0 YO+
which is the exact limiting value;2the EMN theory is not valid in this limit. The attractive contribution to the interfacialfree energy due to the surface tension of the hydrocarbon-water interface also follows from the dressed micelle model: Yo = @ g e , / W , (18) Again, this expression cannot be evaluated by the apparently attractive route of formal differentiation under the integral sign in eq 12, since that integration is performed at constant a. (Applying such a procedure would yield @YOU= yo- @gd,a result which has appeared in the literature but which is wrong.) Explicit differentiation of eq 13gives (after some remarkably lengthy manipulation)
= (8/~@)[(1+w2)'l2- (1+ ~ 2 )+" 2~In (11 + (1+ X,2)'/2IW2) - 2 In (11 + (1 + w2)1/21x,2)l (19) Inspection of eqs 13,15, and 19 shows that the surface potential is characterized by the first-order local topology
16
L
t
0.31 0.33 0.36 0.36 0.30 0.28
of the electrostatic free energy surface:
This result should be viewed with some caution, since it is derived on the basis of an approximation. However, it also holds true for the EMN theory and may be more general.
Comparison With Experiment A variety of micellar systems have been analyzed by small-angleneutron scatteringexperimentsusing the rough micelle model, which yields values for the aggregation number,N, core radius,R1,total radius, Rz,and the degree of i o n i ~ a t i o n Since . ~ ~ the headgroups lie between R1 and R2 in a shell which may be thicker than a single headgroup, the value of a is computed assuminga random distribution of headgroups in the she& that is, R is chosen to partition the shell into equal volumes. The experimental results provide all of the parameters necessary to calculate the theoretical degree of ionization from eq 15 without any fitting, so that a very direct test of the dressed micelle model is provided by comparison between theory and experiment. Tables 11-VI show values of X O , a , and s derived from SANS data on solutionsof sodium dodecyl sulfate' (SDS), dodecyltrimethylammoniumchloride and bromide7(CW TA)Cl and (C12TA)Br), hexadecyltrimethylammonium chloride' ((CIGTA)Cl),and sodium octanoate8(NaCs),with varying concentrations of added NaCl or NaBr. The experimentaldata were analyzed using the one-component model correctedfor the effect of a penetrating background! Counterions from the micelles were included in the calculation of XO. Figure 1,which summarizes the data, showsthat the agreementis generallyvery good, especially since the theory has no free parameters. The circled points are for systems with a high concentration of added Br-, (7) Hayter, J. B.; Penfold, J. Colloid Polym. Sci. 1983,261, 1022. (8) Haytar, J. B.; Zemb, T. N. Chem. phy8. Lett. 1982,93, 91. (9) Hayter, J. B. Solid State Division Progress Report. Oak Ridge
National Laboratory Report ORNL-6306, Green, P. H.,Wataon, D. M., Us.; Oak Ridge National Laboratory: Oak Ridge, TN, 1988; p 154.
2876 Langmuir, Vol. 8, No.12,1992
Huyter
Table IV. Comparison of Degrees of Ionization, Calculated from EMN and from Equation 18, with Experimental Value" for Dodecyltrimethylammonium Bromide ((C12TA)Br) Micelles at T = 313 K [(ClzTA)Br] (mo1.L-l) 0.05 0.10 0.20
[NaBr] (mol-L-l) 0.00 0.00
0.00 0.10 0.00 0.10 0.00 0.10
0.40 0.60
XO
a (nm*)
5
~EhfN
6, 1s
0.19 0.20 0.20 2.09 0.21 2.11 0.21 2.13
0.730 0.682 0.672 0.646 0.647 0.638 0.647 0.622
119.80 128.30 130.20 13.47 135.30 13.70 135.30 14.00
0.09 0.08 0.08 0.23 0.08 0.23 0.08 0.22
0.33 0.29 0.29 0.29 0.27 0.28 0.27 0.27
6m 0.33 0.24 0.29 0.17 0.24 0.16 0.24 0.15 ~
~~
.
Table V. Comparison of Degrees of Ionizat-Jn, Calculated from EMN a n i Prom Equation 15, with Experimental dlueP for Hexadecyltrimethylammonium Chloride ((C16TA)Cl) Micelles at T = 313 K ~~
[(CleTA)Cll (mo1-L-l)
0.03 0.05 0.10 0.20 0.40 0.60
[NaCll (mol-L-l) 0.00 0.00 0.10 0.00 0.10 0.00 0.10 0.00 0.10 0.00 0.10
XO
0.24 0.24 2.58 0.24 2.58 0.25 2.60 0.26 2.60 0.26 2.63
a
(nm9
0.724 0.718 0.625 0.716 0.642 0.666 0.635 0.651 0.627 0.629 0.619
S
6EMN
120.90 121.90 13.92 122.20 13.57 131.30 13.72 134.40 13.88 139.00 14.06
0.08 0.08 0.21 0.08 0.22 0.07 0.21 0.07 0.21 0.07 0.21
6,15 0.26 0.26 0.25 0.26 0.25 0.23 0.25 0.22 0.25 0.21 0.24
6qt
0.28 0.28 0.34 0.33 0.27 0.24 0.20 0.20 0.21 0.23 0.19
Table VI. Comparison of Degrees of Ionization, Calculated from EMN and from Equation 16, with Experimental Valuess for Sodium Octanoate (NaCd Micelles at T = 301 K ~~
[NaCd (mol-L-l) 0.60 0.75
0.90 1.05 1.20
xo
a (nm2)
s
2.20 2.33 2.41 2.47 2.54
0.909 0.904 0.862 0.821 0.792
4.81 4.84 5.07 5.33 5.52
~EMN
0.50 0.49 0.47 0.45 0.44
,6 is
Lpt
0.59 0.57 0.55 0.53 0.51
0.67 0.59 0.53 0.48 0.56
for which polarizabilityeffects such as specific adsorption12 (not included in the present theory) are expected to be important. In conclusion,the self-consistentsolution of the dressed micelle model presented here provides a simple analytic form for the degree of micellar ionization which overcomes the deficiency of the original theory4 at low screening. Agreement with experiment is sufficientlygood to provide strongevidence for the validityof the EMN dressed micelle approach to ion binding. The theory providesa foundation for attackingthe many problems in which surfactant charge plays a role.lOJ1 (10)Chevalier, Y.;Belloni, L.; Haykr, J. B.; Zemb, T. N. J. Phys. (Paris) 1985,46,749. (11) Brackman, J. C.; Engberta, J. B. F. N. Langmuir 1992,4424. (12)Hunter, R.J. Foundations of Colloid Science: Clarendon Press: Oxford, 1989;Vol. 11, Chapter 12. '
I
'
0.00'8
o C&TACI I
'
1
'
I
'
1
0.00
1
I
I
I
0.65 'Experiment
Figure 1. Theoretical vs experimental degrees of ionization, 6, for micelles formed by the indicated surfactants at several ionic strengths (see text).
Acknowledgment. This work was supported by the Division of Materials Sciences,U.S.Department of Energy. Oak Ridge National Laboratory is managed by Martin Marietta Energy Systems, Inc., for the U.S.Department of Energy under Contract No. DE-AC05-840R21400.
