ADBORPTI.ON O F COUNTERIONS AT THE SURFACE OF DNTERGENT
MICELLES
3603
On the Adsorption of Counterions at the Surface of Detergent Micelles’
by D. Stigter Western Regional Research Laboratory,a Albany, California
(Received August 36, 1066)
This paper presents a model of :an ionic micelle in which the electric double layer is dividedl into a Gouy-Chapman layer (outside and a Stern layer inside the shear surface. In thci Stern layer the discrete nature ,md the size of the counterions and of the ionic headgroups are introduced as well as a specific adsorption potential A p o of the counterions. The potential Apo is calculated as at residue from the equation for the adsorption equilibrium of counterions between the Stern layer and the bulk solution. The theory is applied to micelles of sodium dodecyl sulfate and of dodecyl ammonium chloride, both in aqueous sodium chloride solutions. Using the charge and size of kinetic micelles as determinedl from various experiments, we find Ap0 = 0.5kT. Therc is no significant trend of Ap0 withL ionic strength. The uncertainty in Apo is due mainly to uncertainties in the precise location of the shear surface, in the dimensions of ions, and to approximations in the ion distribution inside the Stern layer. The results, although developed for prolate ellipsoidal micelles, are not very sensitive to micelle shape. From the repulsion between the micello core with low dielectric constant and an ion in water, one expects for the sodium counterion Ap’ = 0.25kT and for the chloride counterion Apo = 0.45kT. The present results suggest that dehydration of counterions at the micelle surface is insignificant and that the distribu-. tion of small ions is governed almost wholly by electrostatic and dimensional factors.
Intraduction Micelles in ionic detergent solutions are interesting objects of research. On one hand, their structure is simple enough to be explored extensively with present experimental techniques. On the other hand, micelles have in common with more complex systems certain interesting features which may be studied fruitfully in the simpler system of micellar solutions. An example is the intoraction between small ions and charged colloidal particles, the subject of this paper. The most advanced theoretical model of a micelle discussed so far is a sphere with a uniform surface charge surroundled by a Gouy-Chapman diffuse double layer. The shortcomings of this model are at least twofold. First, it was found some time ago that the model is inconsistent with the low electrophoretic mobility of micelles and suggested that micelles have a rough rather than a smooth surfacea8 Second, the GouyChapman theory itself has some serious defects.4 For exarpple, it neglects the dimensions of small ions and recognizes only Coulomb interaction forces. On the basis of present experimental information,
we may introduce a more detailed model of ionic micelles. The new feature is a Stern layer4 at the micelle surface where correction of the Gouy-Chapman theory is most necessary. In the Stern layer we take into account the geometry of the micelle surface andl the size of the counterions. Furthermore, we allow for a specific adsoription potential of the counterioiicr in the Stern layer. The starting point of the theory is the equilibrium between the counterions in the bulk solution and in the Stern layer or, stated otherwise, the equality of the electrochemical potential of the counterions, 11, in the two regions. Indicating properties of counterions in the bulk solution and in the Stern layer with subscriptE (1) Presented in part a t the Kendall Award Symposium of the Divi. sion of Colloid and Surface Chemistry, 147th National Meeting of the American Chemical Society, Philadelphia, Pa., April, 1964. (2) A laboratory of the Western Utllization Research and Development Division, Agricultural Research Service, U. s. Department of Agriculture. (3) D. Stigter and K. J. Mysels, J . P h y s . Chem., 59, 45 (1955). (4) J. Th. G . Overbeek in “Colloid Science,” Vol. I, H. R. Kruyt, Ed., Elsevier Publishing Co., New York, S . Y., 1952.
Volume 68, Number 13 December, 1964
D. STICTER
3604
b and s, respectively, we write conventionally for counterions in the bulk, outside the double layers = Pbo
+ kT In + kT 1nfb + ze#h
(1) f i b o is the standard chernical potential, p h is the concentration (number density), j b is the relevant single Ion activity coefficient, and ze$b is the electrostatic potential energy of a counterion with charge ze in the bulk solution. As #b is constant outside the double layers, we follow common practice and set $b = 0. We may express 9. formally in a similar way Tb
vS =
Pb
+ ICT In + x e h
(2) where ps is the effective concentration of the counterions in the Stern layer and ze& is their electrostatic potential energy in the field of the neighboring ions. There is no rigorous treatment for pa and for gS. The problems are similar to those in the theory of concentrated solutions. In fact, the entropy term kT In ps and the energy term xe& cannot be separated explicitly, as in eq. 2, except in certain approximate treatments. The approach in this paper is based on a cell model of the Stern layer that does allow separate evaluation of ps and As in most theories, our model is a compromise between tractability and reality. Although the present choice is not fully satisfactory, it allows us to decide which features of the model require refinement. The test quantity in the theory is the difference A p o = pso - p b o which is found as the final term in the equilibrium equation r)b = vs. With eq. 1and 2 we have AM’ =
pso
pso
- pClbo kT In
pS
(fb Ph
/PJ - ze$, (3)
The term Aho is generally called the specific adsorption energy of a counterion in the Stern layer. We note that this quantity is a free energy. We can identify one positive (repulsive) contribution to ApO. I n general, when an electrostatic charge approaches a region of lower dielectric constant, its self-energy increases. For this reason, po increases when a counterion approaches the hydrocarbon core of a micelle. This contribution to Apo can be calculated from a model. Comparison with the result of eq. 3 for Ap0 in various practical cases yields a rough test of the theory. A satisfactory model might then be used to treat other topics in micelle theory, e.g., the micelle-monomer equilibrium and the micelle size distribution. The applications of eq. 3 in this paper are restricted to two detergent systems for which sufficient experimental information is available : sodium dodecyl SUIfate and dodecyl ammonium chloride, both in aqueous sodium chloride solutions at 25”. We start with a The Journal of Physical Chemistry
discussion of the model and the derivation of some parameters from experiments. This part has been nearly completed in previous reports5J and a brief summary suffices.
Model of Micelle Figure 1 shows a partial cross section of the model of
a sodium dodecyl sulfate micelle. The interior of the micelle is formed by n associating hydrocarbon chains. The density of this core, as extrapolated from the density of liquid hydrocarbons a t 25”, is 0.802 g.jm1. Small micelles have a spherical core with a radius not exceeding the length of the straight dodecyl chain of 16.6 A. Larger micelles, with n 2 no = 54.7, have a prolate ellipsoidal core with a short semi-axis of 16.6 A. and a long semi-axis of 16.6 n/noA. Between the smooth surface of the core and the smooth shear surface is the aqueous Stern layer which contains the n ionic heads of the micellized detergent ions and (1 - a)n counterions. The core and the Stern layer together form the “kinetic micelle.” The charge of the kinetic micelle is neutralized by an excess of an counterions in the surrounding GouyChapman diffuse double layer. The reciprocal thickness, K , of the diffuse double layer is determined by the (effective) ionic strength of the solution, that is, by the concentration outside the double layer of nonmicellized (monomeric) detergent and of foreign salt. The model, as described so far, is essentially defined by three parameters: the association number n, the thickness s of the Stern layer, and the fraction Q of the total number of counterions located outside the shear Gouy-Chapman layer,
-S-
Figure 1. Partial cross section of sodium dodecyl sulfate micelle. Crosshatched area in Stern layer is available to the centers of sodium ions.
( 5 ) D. Stigter in “Electromagnetic Scattering,” ili. Kerker, Ed., Pergamon Press, N e w York, N. Y . , 1963, p. 303. (6) D. Stigter, presented at the IVth International Congress on Surface-Active Substances, Brussels, September, 1964.
ADSORPTION OF COUNTERIONS AT THE SURFACE OF DETERGENT MICELLES
3605
cause of this uncertainty, we shall inyestigate the effect on our results of a variation of 0.5 A. in s and in the ionic radii.
0.011
0.03
0.1
0.3
Ionic strength, mol/l. Figure 2. Association number, 12, of micelles in aqueous sodium chloride Eiolutions. Circles: dodecyl ammonium chloride, experiments by Kusher, Hubbard, and Parker'; triangles: sodium dodecyl sulfate, experiments by Mysels and Princen.6
surface. In oi~derto determine these parameters wt: have carried out a concerted analysis of various experinients.5~~ The association number n is determined5 from light scattering data.7+3 As shown in Fig. 2, n increases with increasing ionic strength and, furthermore, n depends on the particular type of detergent. The thickness of tlhe Stern layer follows from the viscosity of micellar solutions and from the rate f : self-diffusion of micelles. We find6 that, within 1 AL., s equals the length of' the hydrated ionic head of the micellized ions. Information on niicelle shape is also obtained. Viscosity data indicate6 that large micelles behave like flexible rods a9 is consistent with the assumed fluid structure of micelles. Small angle X-ray scattering" has confirmed the liquid nature of micelles. The values of CY derived6 from electrophoresis and from electric conductance of micelles range from 0.4 to 0.6. The present treatment requires a rather detailed model of the Stern layer. We assume that the ions in the Stern layer remain hydrated and behave like hard spheres. A diameter of 4.6 b. is taken for both sodium and sulfate ions and 3.8 A. for chlorid? and ammonium ions. i n addition, we take 6 = 4.6 A. for the dodecyl sulfate micelle and s = 2.7 b. for the dodecyl ammonium micelle. Actually, there is considerable uncertainty about the size of hydrated ions.l0 Be-
Effective Concentration of Counterions in the Stern Layer In Fig. 1 the crosshatched sections indicate the free volume of the Stern layer, that is, the volume that is available to the centers of the counterions. Figure 3 shows a schematic top view of the Stern layer, again for sodium dodecyl sulfate, with a! = 0.5. The n fixed SO4- groups are arranged hexagonally, leaving n/2 face-centered sites or cells for the sodium ions. With this regular arrangement, elementary geometry suffices to make an estimate of the free volume v per sodium site, as a function of the micelle size. The results are presented in Fig. 4. It is apparent that the uncertainties in the values of s and of the ionic radiii may cause an error of a factor 2 in v. The concentration pa of the counterions in the Stern layer is, of course, connected with the free volume in the Stern layer. At a! = a crude estimate is pa = l / v . This estimate of ps is too high for two rea-
@"@ Figure 3. Top view of Stern layer of sodium dodecyl sulfate micelle a t the level of the centers of fixed sulfate groups: Crosshatched sections are available t o center of sodium ions. ( 7 ) L. M.Kushner, W. D. Hubbard, and R. -4.Parker, J . Res. Natl'. Bur. Std., 59, 113 (1957). ( 8 ) K. J. Mysels and L. 13. Prineen, J . Phus. Chem., 6 3 , 1696 (1959). (9) Fr. Reiss-Husson and V. Luzzati, ibid., 6 8 , 3504 (1964). (10) R. A. Robinson and R. H. Stokes, "Electrolyte Solutions,"
Butterworths, London, 1969.
Volume 68, Number 18 December, i9Ql
3606
D. STIGTER
provided that it can be carried through to sufficiently high terms. This problem was not studied in any dctail.
rNd = 2.3 A 0%
2’
I
Sodium dodecyl sulfate
I
100
0)
-5 e
501
,
/
,
,
\
S=2.7i
I I
.’:
Dodecyl ammonium chloride
0)
2 Y 60
100
80
120
140
160
Association number n Figure 4. Volume, v> per adsorption cell in Stern layer us. assoc.iation number, n, of detergent micelles, calculated with smoothed values from broken lines in Fig. 5 ofJef. 6: curve 1, sodium dodecyl sulfate, with s = 4.6 A. and, diameters of sodium ion and of sulfate group = 4.6 A . ; curve 2 , model as fp- curve 1, except that diameter of sodium ion = 5.6 A . ; curve 3, dodecyl ammonium chloride, with s = 2.7 A. and diameter of chloride ion and of ammonium group = 3.8 b.; curve 4, model as for curve 3 except that s = 3.2 A.
sons which are both connected with the liquid structure of the micelle. In the first place, the arrangement of the fixed charges is not a t all regular. Therefore, the average overlap of excluded volumes around fixed charges is larger than evaluated above and, consequently, the average value of Y is also larger. I n the second place, the cell theory assumes that each cell is occupied by one counterion which is restricted to movement inside the cell. Actually, this restriction is too severe. I n general, the relevant correction for communal entropy is difficult.ll I n a simple case a rigorous treatment shows that ps is a factor e’ less than predicted by the cell theory,ll e’ = 2.718... being the base of the natural logarithms. I n the present applications we distribute the (1 - a)n counterions in the total free volume of the Stern layer nv/2 and we add a factor e’ for conimunal entropy ps =
2(1 - a)/ue’
(4)
Expression 4 for ps is perhaps not very satisfactory. One might abandon the cell theory altogether and consider the ions in the Stern layer as an imperfect two-dimensional gas. However, it was found that the ion concentration is much too high for an application of the approximate van der Waals approach. The virial expansion method might be more practical, T h e Journal o j Physical Chemistry
Electrostatics of the Stern Layer We wish to derive the average electrostatic potential due to the interaction with the neighboring ions $a of a counterion in the Stern layer. To this end the ion distribution is divided into several parts and the contribution of each part to is evaluated separately. We have: (a) the potential $d due to the ions in the Gouy-Chapman diffuse double layer; (b) the potential $f due to the fixed ionic charges in the Stern layer; and (c) the potential caused by the counterions in the Stern layer, excepting the one counterion in the cell under consideration. The effect of micelle shape is introduced in an approximate way as follows. We first evaluate $s for spherical niicelles with a core radius a = 16.6(n/no)”3 A. Subsequently, we apply to $s a correction for the deviation from spherical shape of the micelle which will be estimated later. We assume that the dielectric constant is a step function with eo 2 inside a sphere with radius b and e,, 78.5 outside this sphere. We set b equal to the core radius a and, in addition, we shall investigate the effect on of a variation of b. We evaluate $s in the center of an adsorption cell and we shall consider, to some extent, the variations of $a within a cell. We now return to the shape factor in Let us consider a particle with volume V and electric charge e in a medium with dielectric constant e. The surface potential $of the particle nmy be written as
where ro is the radius of a sphere with volume V . So is a shape factor which equals unity for spherical particles. Standard texts on electricity show that the equipotential surface around a uniform line charge is the family of prolate ellipsoids with foci a t the ends of the line charge. Thus we find for the shape factor of prolate ellipsoidal particles with axis ratio p
so =
+
(1 - E2)”S 1 E 1 64 In= 1 - -E4 2E 1-E 45 283s
+
(11) See, e.g., T. L. Hill, “Introduction to Statistical Thermodynamics,” Addison Wesley, Reading, Mass., 1961, p. 290.
ADSORPTION OF
-.-
COUNTERIONS AT
THE SURFACE O F
3607
DETERGENT 3IICELLES
#
~.
41
where the eccentricity of the ellipsoid is E = - l/p2. Expression 5 has to be corrected for the effect of the electric double layer that is present around charged micelles in aqueous salt solutions. The presence of a double layer, with thickness 1 / ~ ,around a particle changes both the cofactor, e/ErO,and the shape factor. We are concerned with the change of the shape factor from So to S. I n limiting cases this change is obvious; for KTo
---f
KTO
--f
0, 8
---f
so
and for m,
8 --j 1
Furthermore, a reasonable guess is, for KTo
= 1, S = (So
+ 1)/2
A simple interpolation function satisfying the above three cases is
s = (so + Kro)/(l
+ Kro)
(7)
With equations 6 and 7 we shall estimate S for nilcelles in aqueous sodium chloride solutions. Contribution of Difuse Double Layer to Stern Potential. In order to evaluate $d we start with the tola,l electrostatic potential, [, in the spherical shear surface with radius a s. In the Gouy-Chapman model we have3
+
[:=
1
--
p %(a
+
one S)[l f
K(Q
+
ST1
(8)
The factor p, also denotedl2 I/I(DH), corrects the Debye-Huckel relation between the surface potential [ and the particle charge ane. Numerical values of p as a function of the parameters e{/kT and K ( U s) are available. 3, L 2 The desired potential $d is found by subtracting from [ the potential at the uniformly charged surface of the kinetic micelle without diffuse double layer
+
(9)
Contribution of Fixed Charges to Stern Potential. In the present model the Stern layer contains discrete ionic charges. It is of interest to establish the difference from the simpler, more conventional model which possesses a uniform surface charge instead of a Stern layer. The difference is demonstrated with the help of a planar, hexagonal array of point charges e as shown in Fig. 5. This prray is the same as that of the fixed charges in Fig. 3, provided that the curvature of the micelle surface is disregarded. I n order to evaluate
Figure 5. Plane hexagonal array of fixed charges, with repeat distance d. Potential in Q for discrete point charges is given by series 10, for uniformly smeared-out charges by series 11.
the potential $Q at, the center of a cell, we divide the array into groups of equivalent neighbors of the centr,al charge, separated by concentric circles in Fig. 5. The potential in Q is written as the sun1 of successive group contributions, starting with that of the six nearest neighbors $Q
e Ed
= - (10.39
+ 5.20 + 7.86 + 5.77 + . . . )
(10)
E is the dielectric constant of the medium and d is the repeat distance of the array along a trigonal axis. We now smear out the charges over the appropriate rings so that the plane of the array has a uniform charge density 4e/dz1/3. Similar to eq. 10 the potential in Q can be written as the sum of contributions froin successive charged rings and we find
$Q
=
e (13.20 + 5.47 + 7.73 + 5.93 + . . . )
Ed
(11)
The comparison between corresponding terms in eq. 10 and 11 reveals that only with the nearest neighbors the discreteness of the charges is importani. The more distant charges may be smeared out without changing $Q significantly. Consequently, in the evaluation of $f we treat the six nearest ionic heads as discrete charges and we smear out the remainder of the fixed charges over the appropriate surface. This procedure is also, at least to some degree, in accord with a fluid nature of the Stern layer. (12) A. L. Loeb, J. Th. G. Overbeek. and P. H. M’iersema, “The Electrical Double Layer Around a Spherical Colloid Particle,’’ Massachusetts Institute of Technology Press, Cambridge, Mass.,
1960.
Volume 68.Number 18 December, ius/,
3608
D. STIGTER
Q
T"
' f P
n
I
I
Figure 6. Cross section through dielectric sphere with charge e in P. Potential in Q is given by eq. 13.
The introduction of discrete charges in the Stern layer destroys the spherical symmetry of the charge distribution. To obtain the proper solution of Laplace's equation, A* = 0, we first look a t the potential field of a single charge in the Stern layer. Let the charge e in Fig. 6 be in P at distance t from the center 0 of the micelle. The position vector r makes an angle 8 with OP. Modification of the field around an electric charge near a dielectric sphere under vacuum13gives for the present situation, in the region r>b
Figure 7 . Cross section through dielectric sphere with uniform surface charge of density ne/4& in concentric spherical surface with radius t between e = eo and 0 = 7. Pot,ential in Q is given by eq. 15.
parameters t, r , and x are derived from the micelle model as specified above and 0 is obtained from an approximate expression 0 = 41/.rr/,?n/q/3
(14)
which is exact for e -+ 0. It remains to find the potential of the (n - 6) smeared-out, fixed charges. This requires the potential in a point Q near a partially charged dielectric sphere (Fig. 7). I n the surface t = constant, between 0 = eo and 8 = T , the charge density is ne/4at2. This surde. The face is divided into rings between e and 0 contribution of a charged ring to the potential in Q is obtained with eq. 13. The total potential is found by integrating with respect to e between the limits 00 and T . The potential in Q is
+
+ 1) + e d (")'t (p)' + l]P,(cos -
e&
e) (12)
where P l (cos e) are Legendre functions. The summation in eq. 12 is carried out similarly to one by IGrkwood and Westheimer,14 who derived the potential for a point charge inside a dielectric sphere, t < b in Fig. 6. With the abbreviations b2/tr = x and ec/ew = a,eq. 12 is converted into
5
(1 - 22 cos
e
+ x2)"2
-
zw/(l
+
0)
t
(1 - 2; cos I30
+r2
ne(1 - a) X 2e,b(l w)
+
X
With eq. 13 we evaluate the contribution to t,bf of the six nearest neighbor fixed ionic charges. The The Journal of Physical Chemistry
t
+=---I+-2nEt,e { r
(13) See, e.g., C. J. F. Bottcher, "Theory of Electric Polarization," Elsevier Publishing Co., R'ew York, N. Y., 1952, p. 102. (14) J . G. Kirkwood and F. H. W7estheimer,J . Chem. Phuls., 6,506 (1938).
3609
ADSORPTION OF COUNTERIONS AT THE SURFACE OF DIETERGEKT MICELLES
--
-.
*t
2
0.01
ammonium chloride
L
0.03
cmc + cNaCl,mol/l
Figure 8. Plane trigona,l array of counterions with repeat distance d . Hexagonal cells indicate arrangement of fixed ionhi. Potential in Q for discrete counterions is given by series 16, for uniformly smeared-out counterions by series 17.
The contribution of the (n - 6 ) smeared out fixed charges to $f is derived with eq. 15, where cos BO == 1 - 12/n as follows from solid geometry. Contribution of Counterions in Stern Layer to Stern Potential. The discussion of $e follows closely that of We start with the case of a = l/Z. I n the regular $f. plane array, as shomn in Fig. 8, the counterions form a trigonal lattice with repeat distance d. Some of the n/2 hexagonal cells formed by the fixed charges are also indicated in Fig. 8. The charges are divided Into equivalent groups, separated by the circles in Fig. 8, and the potential in the central site Q is evaluated as the sum of group contributions. Counting all counterions discretely we find $Q
=
e ed
(a
+ G + 3.46 + 3 + 4.54 + . . )
(16)
For the uniformly smeared-out charges the equivalent series is =
Ed
(3.81.
+ 6.27 + 3.83 + 2.70 + 4.60 + . . . ) (17)
The main diflerence between the series 16 and 17 is in the lea$ing term, that is, the contribution of the central ion. But this term refers to the self-potential of the central counterion which should be omitted altogether in t h i evaluation of the Stern potential. Indeed, the Stern potential, as defined in this treatment, deals with the interionic interaction only. Possible changes of the self-potential of an adsorbed counterion are
-
d 20.06 0 01
01
0.03
cmc t c N a C lmol/l ,
Figure 9. Absolute values of surface potentials us. ionic strength for detergent micelles in aqueous sodium chloride solutions a t 25". Comparison of the Stern potential $s calculated for va,rious models (solid lines) with the r-potential (broken lines). In all cases the smoothed a-values have been used from the broken lines in Fig. 5 of ref. 6 : curve 1, regular, face-centered hexagonal array of charges in Stern layer; change from etu = 78.5 to ec = 2 a t surface of micelle core; shape factor S = 1; curve 2, model as for curve 1, except that one nearest-neighbor sulfate group is as close as possible t o the central sodium ion; curve 3, model as for curve 1, except that the change of e occurs a t 2.3 A. from the core, at the level of the centers of the fixed ions; curve 4, model as for curve 1, but including shape factor S for prolate ellipsoids according to eq. 7 .
properly counted with the specijk adsorption potential as discussed later. In the actual case of micelles, when a # l / z , an - 1 counterions are smeared out over (n/2 - 1) cells, that is, over (1 - 2/n)th part of the total surface of the sphere. The desired potential $G is now obtained from eq. 15 in which n is replaced by [ ( l - a)n - 1]/[1 2/n]and with cos O0 = 1 - 4 / n . The Stern Potential. We are now in a position to evaluate the Stern potential $s = $d $f In Fig. 9 we compare the {-potential with the results of $a for various models in two micellar systems. I t appears that { is 50 to 75 mv., that is, about 2 t o 3 units in ep/lcl', lower than $@. This difference has but one major cause. The [-potential is derived for a uniformly charged sphere. This means that { is an average surface potential to which all ions contribute. On the other hand, in evaluating we specifically exclude the central counterion because we wish to incorporate in only the interactions with neighboring ions. Now, the potential of the central, smeared-out counterion is quite significant. It is represented, in the case of a plane surface, by the leading term of
+
+
Volume 68,Number 18 December, iQ6.4
D. STIGTER
3610
series 17. For spherical micelles we find from eq. 15 that the omission of the central smeared ion lowers e h / k T by 2.2 to 3.0 units in all cases. Further details of the model may change +s only by small amounts. The present model of the ion distribution in the Stern layer is very schematic. I n all cases \L, was calculated for the center of a hexagonal cell. There are, of course, potential variations within a cell. The potential a t the center is minimal. A crude estimate is that variations up to 0.4 unit in e+,/kT occur within a cell. Another estimate is obtained when we allow variation in the position of a neighboring fixed ionic group. Curves 1 in Fig. 9 are for regular arrays of ions. However, & is raised to curve 2 when one neighboring sulfate group is moved toward the central sodium ion. It is likely that curves 1 represent a low estimate of the Stern potential. Another difficulty is the change of the dielectric constant near the micelle surface. An increase of the radius of the dielectric sphere by 2.3 A. lowers e&/lcT from curve 1 to curve 3. We have also tested the effect of micelle shape on &. The introduction of the shape factor S from eq. 7 lowers IC., from curve 1 to curve 4 for dodecyl ammonium chloride. It is evident that this shape effect is hardly significant. For micelles of sodium dodecyl sulfate the correction for shape is even smaller. I n further studies one might introduce an intermediate dielectric constant for the Stern layer. I n addition, one might improve the ion distribution in the Stern layer by using, say, a two-dimensional DebyeHuckel approach. The Specific Adsorption Potential A p o of Counterions in the Stern Layer With the information collected so far we can now evaluate Apo by means of eq. 3. The activity COefficient f b is identified with the average activity coefficient in an aqueous sodium chloride ~ o l u t i o n ' ~ at concentration cNac1 c.m.c. The charge of the kinetic micelle may be evaluated from micellar electrophoresis and also from the conductance of micelles. We have used both sets of datae for LY in the present calculations. I n Fig. 10 the results for A p 0 are plotted us. ionic strength. In each of the micelle systems the considerable spread of Ap0 is due mostly to the uncertainty in 01 which gives rise to errors in ps and $sa The micelle charges derived from micellar conductanceG are by no means definite, A thorough study of this method, both theoretical and experimental , would be very helpful. There are additional errors in ps and in & which are reflected in Ape, Some refinements in the theory are
+
The Journal of Physical Chemistry
l2
I
- z
i
Sodium dodecyl sulfate
0
O 1
Figure 10. Specific adsorption potential Ap' of counterions in the Stern layer of micelles in aqueous sodium chloride solutions us. ionic strength a t 25': open circles, micelle charge derived from micelle electrophoresis6; solid circles, micelle charge derived from micelle conductance.6
possible as pointed out before. However, a large error in u , and hence in AM', may be due to our inaccurate knowledge of the thickness of the Stern layer and of ionic radii. Direct experimental determination of these parameters with sufficient accuracy does not seem feasible. The comparison of micelle systems differing only in, say, the type of counterion presents itself as an indirect way of testing the geometric assumptions of the Stern layer model. Before further discussion of the data in Fig. 10, me evaluate an electrostatic contribution to Aho. The electrostatic energy (self-energy), 6, of a counterion increases when the ion moves from the bulk of the aqueous solution to the micelle surface. This change, A+, increases the adsorption potential Apo of the counterion. Let the counterion be a sphere with radius 6 and charge e , located in point Q of Fig. 6. The surface potential of the ion, qS,is derived with eq. 13. The desired change of self energy is A+ = ('/s)e[*s
(T)
-
*!5
(r
=
O0)
1
(18)
When we treat the ion as a point charge, that Is, assuming 6 / ( r - b ) + 0, eq. 18 and 13 yield A+ =
e 2 ( 1 - w)b
+ w ) ( r 2 - b2)
2t,(l
e2(1 2e,b(l
+
(
w)
-
/j)zm/(l+m) J (b/r)2/(1+w)
1 -dy y'+"
(19)
I n the case of the flat interface, b / ( r - b ) + 03 , eq. 19 is converted into
~~~
(15) H. 9. Harned.nnd B. B. Owen, "The Physical Chemistry of Electrolytic Solutions," 2nd Ed., Reinhold Publishing Corp., New York, N . Y., 1950,pp. 360, 362.
ADSORPTIOXOF COUNTERIONS
AT THE SURFACE O F DETERGENT
1.0 I
0.3 -
d kT 0.1 5
--
0.03I-
0.01
0
2
4
6
8
10
r-b, Figure 11. Chmge of self-energy A@ of monovalent ion in water vs. distance r - b from surface of sphere with dielectric constant e, = 2 and radius b = 16 k , b = 20 A., and b = m .
It is observed that according to eq. 20 the influence of the dielectricuin on 4) may be represented by the action of a virtual point charge (1 - w)e/(l w ) situated in the mirror pomt, that is, a t distance 2(r - b ) across the interface from the real charge e. For this reason A$, in eq. 20, is often called an image term. The corresponding term in eq. 19 is the first one on the righthand side. The other term in eq. 19 is the correotion for curvature of the interface. In Fig. 11 we have plotted A$ us. the distance between the charge and the interface for dielectric spheres and also for a flat interface. The data show that curvature of the interface m?y reduce A$ considerably. A radius of 16 to 20 A. is representative for micelles. For the counterions in the Stern layer, A$ may vary because of the finite thickness of the Stern layer. I n the present model we estimate Ad, = 0.15 to 0.3 kT for sodiuni dodeqyl sulfate and A$ = 0.3 to 0.55kT for dodecyl amnioniiim chloride, These values compare favorably with the results of Ap" in Fig. 10, but the agreement might be fortuitous. The un-
+
n'f ICELLES
3611
certainty in A p o is perhaps of the order of 1kT. Moreover, there might be other contributions than Ad, to Abo. One possibdity would be a significant decrease of the free energy of hydration of the counterions due to disarrangement of water molecules around the ions in the Stern layer. This effect would add a positwe (repulsive) term in Abo. We have no theoretical reason to anticipate a pronounced trend of Apo with ionic strength. Therefore, the absence of a special trend of A p o in Fig. 10 supports the present model, in particular the assumption that the distance s between core surface and shear surface does not depend on the ionic strength (compare Fig. 4). This reinforces the conclusion of a previous paper that for micelles a dependence of s on ionic strength due to the viscoeleetric effect is insignificant.16 Strictly speaking, a factor exp( - Ad,/kT) should be introduced into the Poisson-Boltzmann equation that gives the ion distribution in the diffuse double layer. Such a factor would tend to lower the ion concentration near the interface and, consequently, imply a more extended diffuse double layer around the micelles. However, Fig. 11 indicates that a t a distance larger than 3 or 4 A. from the core surface, Ad, drops to a small fraction of kT. Thus it is a good approximation to neglect the image term in the region outside the shear surface of micelles. In conclusion, it should be mentioned that there in a considerable body of literature on the Stern layer in various double layer systems including detergent monolayers at a dielectric jnterface.17-19 Now a micelle might be viewed as a detergent monolayer adsorbed onto itself. Therefore, one expects a similarity between the theory o'f micelles and of monolayers. However, there are considerable differences. One reason is that the experimental information of the two systems is quite different. Another reason is that in monolayer theory the statistical treatments of the Stern layer are essentially based on dilute solution theories whereas for the Stern layer of micelles we have used approximations devised to deal with concentrated solutions. (16) D. Stigter, J . Phys. Chem., 68, 3600 (1964). (17) D. A. Haydon and F. H. Taylor, Phil. Trans. Roy. Soc. (London), A252, 225 (1960); A253, 255 (1960). (18) S. Levine, J. Mingins, and G. M.Bell, J . Phys. Chem., 67, 2095 (1963). (19) S. Levine, G. M.Bell, and B. A. Pethica, J . Chem. Phys., 40, 2304 (1964).
Volume 68, h'umber 18 December, 1364
