Langmuir 1990,6, 895-904 modynamic properties such as the osmotic compressibility. The proposed molecular-thermodynamic approach provides an excellent description of a wide range of experimental findings in aqueous solutions of nonionic surfactants belonging to the polyethylene glycol monoether and glucoside families. We are currently extending our theoretical framework to describe micellar solutions of ionic and zwitterionic surfactants, as well as micellar solutions of mixed surfactants. We believe that beyond its fundamental value the proposed molecular-thermodynamicapproach could become a valuable computational tool for the surfactant technologist. Indeed, using this approach, the surfactant technologist could identify, select, and possibly even tailor surfactants for a particular application without the need of performing routine measurements of a large number of equilibrium properties, thus making his work more efficient and productive. In a related study, we have examined2 the peculiari-
895
ties of osmotic pressure measurements of self-assembling micellar solutions. We have shown that under certain conditions these measurements c-an yield the weightaverage micellar molecular weight, M,, contrary to the generally accepted notion that they yield the numberaverage micellar molecular weight, M,. In view of this surprising new result, we have reevaluated osmotic pressure measurements of aqueous micellar solutions of the nonionic surfactant n-dodecyl hexaoxyethylene glycol monoether (C12Ee). We have found that, contrary to previous interpretations indicating the presence of monodisperse micelles, the measurements are consistent with the presence of polydisperse micelles which grow with increasing surfactant concentration and temperature. Registry No.
C12E8,3055-96-7.
(1) Puwada, S.; Blankschtein, D. J. Chem. Phys., in press. (2) Puwada, S.; Blankschtein, D. J. Phys. Chem. 1989, 93, 77537755.
Model Calculations on the Transitions between Surfactant Aggregates of Different Shapes? Jan Christer Eriksson* and Stig Ljunggren Department of Physical Chemistry, Royal Institute of Technology, S-100 44 Stockholm, Sweden Received May 19, 1989. In Final Form: November 1, 1989 In a series of papers published earlier, we presented theories about the formation of surfactant micelles which include detailed treatments of the mechanics, surface thermodynamics, and small system thermodynamics of spherical, rod-shaped, and disk-shaped aggregates. In the present paper, we develop a calculation scheme which yields the volume fractions of the various kinds of sodium dodecyl sulfate (SDS)micelles and also of SDS vesicles, at different solution states. The familiar hierarchy of surfactant aggregates of different shapes is generated automatically upon raising the salt concentration, i.e., without introducing any extraneous packing or geometrical constraints, and is due to the more rapid decrease of the electrostatic free energy for the less curved aggregate surfaces. Moreover, we find that when spherical and rod-shaped micelles coexist, the weight-average aggregation number as a function of the total surfactant concentration shows positive deviations from a linear relationship at high salt concentrations and negative deviations at low salt concentrations, in broad agreement with the available experimental evidence on the sphere-to-rod transition.
Introduction It has often been assumed in the past that large, rodshaped surfactant micelles are formed from small, spherical micelles by means of a straightforward elongation/ growth process. Accordingly, there would be a single peak in the size distribution function which would shift toward larger average sizes and, at the same time, broaden upon + Presented a t the symposium entitled 'Thermodynamics of Micellar Solutions", American Institute of Chemical Engineers Spring National Meeting, Houston, April 2-7, 1989.
changing the solution conditions so as to favor the formation of elongated, rod-shaped micelles. The models employed by Mukerjee,l Israelachvili et a1.,2 Nagarajan,3 and Missel et al.*v5 are essentially of this nature. (1) Mukerjee, P. J. Phys. Chem. 1972, 76, 565. (2) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. SOC., Faraday Trans. 2 1976, 72,1525. (3) Nagarajan, R. J. Colloid Interface Sci. 1982, 90,477. (4) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y.; Carey, M. C. J. Phys. Chem. 1980,84,1044. (5) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Carey, M. C. J. Phys. Chem. 1983,87, 1264.
0 1990 American Chemical Society
896 Langmuir, Vol. 6, No. 5, 1990
A few years ago, this kind of approach was questioned, however, by Porte et al.? who referred to the behavior experimentally observed about the so-called second cmc of surfactants in the absence of an added salt component and, on theoretical grounds, by the present authors.' We introduced the notion of a bimodal size distribution with a narrow peak centered at the average size of the small spherical micelles and a much broader peak at considerably larger micelle sizes. In between, there would be a region of very rare micelles of intermediate sizes. When the work required per monomer to form the central part of a rod-shaped micelle becomes small enough, in comparison with kT, the number of large micelles increases rapidly, and the broad peak gains in weight relative to the narrow, spherical micelle component. Small spherical micelles will, however, as a rule be present simultaneously with the rod-shaped micelles. The purpose of this work was to elaborate a quantitative theory of the transitions from spherical to rodshaped micelles and, moreover, to study the further transitions to disk-shaped micelles and vesicles. To a large extent, we shall rely upon the theoretical framework provided in our previous articles on the mechanics and thermodynamics of spherical (pure8 and mixedg), rod~ h a p e d ,and ~ disk-shaped micelles'O to which we frequently refer. The general feasibility of our approach for treating aggregated ionic surfactant systems has been further substantiated through more recent works on the Maxwell stress tensor formulation of the electrostatics of surfactantloaded interfacesll and on the molecular interpretation of surface tension data of surfactant solutions12and surface force data for interacting surfactant mon01ayers.l~ The factors which are of a prime thermodynamic importance in this scheme are the hydrophobic driving force as quantified by Tanford,14the Poisson-Boltzmann treatment of the electrostatics in different geometries, the hydrocarbon/water contact free energy which we throughout have assumed to be equal to its macroscopic value, 50 mJ m-2, and finally, the packing- and geometry-dependent configurational free energy of the aggregated hydrocarbon chains as calculated by Gruen and Lacey.15 In the present paper, we show that, on this basis and assuminga bimodal micelle size distribution, we can readily generate a rather complete theoretical description of the sphere-to-rod transition which agrees well with experimental findings as to how the average aggregation number depends on the total surfactant concentration, both at high and low salt concentrations. Moreover, we demonstrate that this kind of description can be extended, quite naturally, to cover also the formation of diskshaped micelles and bilayer vesicles. (6) Porte, G.;PoggY, J.; Appel, J.; Maret, G. J.Phys. Chem. 1984,88, 5713. (7) Eriksson, J. C.; Ljunggren, S. J. Chem. SOC.,Faraday Trans. 2 1985.81. 1209. (9) Ljunggren, S.;Eriksson 3X
(10) Ljunggren, S.; Eriksson, J. C. J. Chem. SOC.,Faraday Trans. 2 1986, 82, 913. (11)Ljunggren, S.; Eriksson, J. C. J. Chem. SOC.,Faraday Trans. 2 1988,84, 329. (12) Eriksson, J. C.;Ljunggren, S. Colloids Surf. 1989, 38, 179. (13) Eriksson, J. C.;Ljunggren, S. Progr. Colloid Polym. Sci. 1988,
.-(--".
76 lRR
(14) Tanford, C. The Hydrophobic Effect; Wiley: New York, 1980; Chapter I. (15) Gruen, D.W. R.; Lacey, E. H. B. Surfactants in Solution; Mittal, K., Lindman, B., Eds.; Plenum: New York, 1984; Vol. I, p 279.
Eriksson and Ljunggren Spherical Micelle State At the present stage, it is generally recognized that ordinary, spherical surfactant micelles undergo rather large size fluctuations or, otherwise expressed, that there is a considerable dispersion in the aggregation number. This is to be expected, of course, on general grounds since a spherical micelle constitutes a very small thermodynamic system. The neutron-scattering experiments by Cabane et a1.16 lend experimental support for the presence of these size fluctuations which, according to the thermodynamics of small systems derived by Hill,17give rise to a fluctuation entropy contribution. The related issue of shape fluctuations is not as yet finally resolved. The work of Cabane et al. (loc. cit.) on sodium dodecyl sulfate (SDS) micelles indicates the presence of significant shape fluctuations away from the spherical average shape but that the hydrocarbon/water interfacial zone is still rather narrow and well-defined. These results are definitely at variance with some recent molecular dynamics simulations of a sodium octanoate micelle by Jonsson et al.,ls which show that a rather substantial mixing between the water and the hydrocarbon chains might occur. However, the notorious difficulty with such simulations is the proper modeling of the hydrocarbon/ water contact. On the basis of theoretical calculations, Halle et al.19 have proposed that deformations of a spherical micelle core may occur quite easily due to thermal fluctuations or electrostatic interactions. The present authors20have shown, however, that it is likely that the effective surface tension counteracting the shape deformations is at least about 25 mN m-l for ionic surfactant micelles rather than about one-tenth of this value as was effectively assumed by Halle et al. Due to this circumstance, the amplitudes of the shape deformations should be fairly restricted in magnitude. Typically, the amplitudes of the thermally induced variations of the radius R of a closed micelle amount to about 0.1R.21 Toward this background, we may anticipate that the well-known Israelachvili picture of a spherical SDS micelle22actually constitutes a rather realistic physical model and that for many purposes it is approximately correct to consider the core of a small surfactant micelle as a very small hydrocarbon droplet of spherical average shape immersed in the water solution phase. Owing to the net effect of the hydrocarbon/water contact and the electrostatic interactions, there arises a comparatively large surface tension contribution y (25-35 mN m-l). Carrying through a complete surface-thermodynamic analysis based on employing the hydrocarbon core surface with radius R, as the dividing surface (rather than the Gibbs surface of tension), one finds that a generalized Y oung-Laplace equation holds where pi is the average value of the tangential pressure tensor component inside the hydrocarbon core and p e (16) Cabane, B.;Duplessix, R.; Zemb, T. J.Phys. (Les Ulis, Fr.) 1985,
-
46. - -, -2161. - -.
(17) Hill, T.L. Thermodynamics of Small Systems; Benjamin: New York, 1963, 1964; Vols. I, 11. (18)JBnsson, B.; Edholm, 0.; Teleman, 0. J. Chem. Phys. 1986,85, 2259. (19) Halle, B.;Landgren, M.; Jonsson, B. J.Phys. (Les Ulis,Fr.) 1988, 49, 1235. (20) Ljunggren, S.;Eriksson, J. C. J . Chem. SOC.,Faraday Trans. 2 1989,85, 1553. (21) Ljunggren, S.;Eriksson, J. C. J. Chem. SOC.,Faraday Trans. 2 1984,80, 489. (22) Israelachvili, J. N. Intermolecular and Surface Forces; Academic: New York, 1985.
Transitions of Surfactant Aggregates of Different Shapes
Langmuir, Vol. 6, No. 5, 1990 897
the external solution pressure.8 The curvature-related pressure xc in eq 1 is composed chiefly of two contributions which are both negative. These are associated with the electrostatic interactions and the hydrocarbon chain packing constraints, respectively. It follows from eq 1 that molecules which are solubilized inside the hydrocarbon cores of micelles are subjected to a high average pressure which results in an increase of their chemical potential and, thus, in a reduction of the extent of solubilization (cf. ref 23). Furthermore, it emerges that the work of formation at constant chemical potentials, NEAP(where e denotes the excess free energy per monomer), of a single spherical micelle of aggregation number N is given by the relations
ditions ( 5 ) are, in fact, equivalent to the usual set of chemical equilibria commonly introduced to account for micelle formation. However, the volume fractions ~ A should P be used here rather than the corresponding mole fraction^.^ In several respects, the hydrocarbon core of a spherical surfactant micelle resembles a very small hydrocarbon droplet. Nevertheless, it turns out to be quite essential for the surface-thermodynamicapproach that the pressure tensor is anisotropic everywhere inside the micellar core except at its center. By and large, this anisotropy is due to the circumstance that the core is composed of long hydrocarbon chains which are subject to certain packing and geometrical constraints. The molecular thermodynamics of spherical micelles can be formulated either in terms of the various tension/ pressure contributions to the effective surface tension y + aasor by means of the different free energy terms which determine the value of N e d . In the present context, it actually turns out to be more convenient to adopt the latter approach, and, hence, we next proceed to discuss the different contributions to the excess free energy, N e d , of a spherical micelle composed of N aggregated monomers.
Ne; = (y + 7r$,)A/3 (2) which replaces the ordinary expression yA/3 that is strictly valid (irrespective of size) only when adopting the surface of tension as the dividing surface. The large negative contribution to the "effective" surface tension, y + xcR,,is highly significant when it comes to accounting for the micelle formation by following the surfacethermodynamic approach. When considering an equilibrium assembly of non-interacting, spherical (anionic) surfactant micelles and making use of the general framework provided by small system thermodynamics derived by Hill," we have to require that a phase equilibrium condition of the kind
*aR,
= p(3) is fulfilled.s Here ( p ~ denotes ) the average chemical potential of the surfactant in the (spherical) micellar state and p- the corresponding surfactant ion chemical potential in the solution. The chemical potential p~ varies with the aggregation number N , or, otherwise expressed, the p value of each single micelle fluctuates during the course of time about the average value ( p ~ ) . There is, of course, also an aggregation (or dispersion) equilibriumcondition such that the following relation holds: (PN)
NmU
(Ne;)
+ k T x P N In PN + k T In 4' = 0
(4)
"in
where - k x P N In PN is the size fluctuation entropy and the last term accounts for the (partial) entropy of dispersion of the micelles in the solution, 4, denoting the overall volume fraction of spherical micelle^.^?^ The thermal shape fluctuations may likewise yield a minor fluctuation free energy contribution which, however, we shall take into account in an approximate manner when quantifying the value of N e d (cf. below). Alternatively, we may consider a set of dispersion equilibrium conditions
Ne,' + k T In 4', = 0 ( N = Nmin,...,N,,J (5) from which the size distribution of the micelles is readily obtained once e h a is known as a function of the aggregation number N . In order to be able to satisfy the phase equilibrium condition (3), we have to require that N e d / k T is large in the size ranges about N = "in and N = Nm=, respectively.8 Note that since €9 & - p-, where & is the (integral) free energy per monomer in the micellar state and p- the surfactant chemical potential in the solution with the monomer concentration c-, the above equilibrium con(23) Bolden, P. L.;Hoskins, J. C.;King, A. S C L 1983, 19, 454.
face
Contributions to Ne# for a Spherical Micelle It is well-known that the hydrophobic driving "force" which promotes surfactant aggregation can be quantified by utilizing the data presented by Tanford.14 Hence, the free energy gained upon removing a C12 hydrocarbon chain from a water solution to a dodecane hydrocarbon phase is given by -
ttan=
itan - p- = -(19.960 + In x-)kT
where x- is the mole fraction of surfactant (SDS)monomers in the solution. A counteracting electrostatic factor arises as a result of concentrating the ionic head groups and their counterions. Insofar as the solution is dilute in micelles and micellar interactions can be neglected, the expression derived by Evans and Ninham24 on the basis of the Poisson-Boltzmann approximation constitutes an excellent approximation for the electrostatic free energy
t,B/kT = (e@' + yelas)/kT= 2(ln [ S + (S2+ 1)1'2] +
(l/S)[(S2+ 1p211)4 In 1+ (1;s2"'2]
KS(R'+ 3.0 X lo-")
[
(7)
where the dimensionless, reduced charge parameter S is defined by S = u[8(c-
+ c , ) ~ ~ ~ P T ] - ~ / ~(8)
and where K, as usual, is K
= [ 2 ( c - + ~ , ) N ~ e ~ / ( t , r , k T ) ] ~ / ~ (9)
@O denotes the surface potential, e the electronic charge, yel the electrostatic interfacial tension contribution, and us the area per charge (the same as the area per head group). In eq 7,R, + 3.0 X 10-lo m is supposed to give the location of the smeared-out surface charges (sulfate groups). c- is the monomer concentration and cs the concentration of an added monovalent salt. The remaining symbols have all their conventional meanings. In passing, we may note that an expression similar to eq 7 is valid in the case of cylindrical geometry where
D.,Jr. J. Colloid Inter(24) Evans, D.F.;Ninham, B.W.J. Phys. Chem. 1983,87,5025.
898 Langmuir, Vol. 6, No. 5, 1990
the numerical factor in the last curvature-related term is 2 instead of 4. It appears from eq 7 that the electrostatic free energy depends on the surface charge density, the total electrolyte concentration (c- + cs), and the curvature. In the planar case, the last term in eq 7 drops out, of course, and we are left with an expression for the electrostatic free energy in terms of S, originally derived by Missel et alS4 An additional, quite important, free energy contribution is related to the hydrocarbonfwater contact. This term we assume to be equal to yhc/#, where us denotes the surface area ascribed to each surfactant head group which is given by the obvious geometric relationship a' = 3G'/R, (10) where G' is the hydrocarbon chain volume. It has often been surmised during the last decade that a comparatively low value of the hydrocarbon/water contact free energy would apply for surfactant aggregates. However, in a number of cases the present authors have successfully employed the macroscopic value of about 50 mJ m-2. As a matter of fact, a detailed analysis of the very precise surface tension data for dodecylammonium chloride solutions presented by Motomura et al.25 lends strong support for such a straightforward approach.12 Moreover, a marked curvature dependence of the contact free energy might be expected only at low total electrolyte concentrations, below about 0.1 mM, as may be invoked from recent studies of the hydrophobic attraction between hydrocarbon-covered surfaces.2&-2s Hence, for the time being, we judge that the proper, but not exactly known, Yhc/w value to insert when treating small spherical micelles is probably somewhat less than 50 but certainly larger than about 40 mJ m-2. Jonwon and W e n n e r ~ t r o mhave ~ ~ ?employed ~~ Yhc/w = 18 mJ md2 (or less in case of mixed surfactant systems). Their choice is open to criticism, however, since it is based on the ad hoc assumption that the hydrocarbon chain configurational free energy is always negligible, even when considering the state of tension of planar surfactant layers (cf. ref 20). Actually, the configurational restrictions on the aggregated hydrocarbon chains give rise to a further free energy contribution, which, accordingto the calculations of Gruen and Lacey,15 is of the order of l k T per each C12 chain. As was also shown by Szleifer, Ben-Shaul, and Gelbart,31 this free energy term varies rather rapidly with the hydrocarbon chain packing density and, hence, turns out to be of a major significance when accounting for the occurrence of micelle formation by taking the surface thermodynamic approach. On this point, it may suffice to invoke the relation8 aeNO/aas =y +~ p , (11) and the data published by Gruen and Lacey (loc. cit.), which can be reproduced by the expansion (ei/kT)conf = 7.612 - 0.9272 X lO'OR, + 0.03233 X 1020R: - 6.688 X lOZ5Rt(12) (25) Motomura, K.;Iwanaga, S.-I.; Hayami, T.; Uryu, S.; Matuura, R.J . Colloid Interface Sci. 1982, 151. (26)Eriksson, J. C.; Ljunggren, S.; Claesson, P. J. Chem. Soc.,
Faraday Trans. 2 1989,85, 163. (27) Christenson, H. K.; Claesson, P. M.; Berg, J.; Herder, P. C. J. Phys. Chem. 1989,93,1472. (28) Herder, P. C. Thesis, Stockholm, 1988. (29) Jonsson, B.: Wennerstrom, H. J. Colloid Interface Sci. 1981,80, 482. (30) Jonsson, B.; Wennerstrom, H. J. Phys. Chem. 1987,91,338. (31)Szleifer, I.; Ben-Shaul, A.; Gelbart, W. M. J. Chem. Phys. 1987, 86, 7094.
Eriksson and Ljunggren This yields (7~$,),~,~
= (kT/as)(0.9272X lO'OR, - 0.06466 X
1OZoR,2+ 20.064 X 1025R,3)(13)
Thus, for the most abundant (Le., equilibrium) spherical SDS micelle with N = 66, ( 7 ~ o R ~amounts ) ~ ~ ~ f to -20.5 mN m-l, which is sizable compared with y = 34 mN m-1. Finally, a surfactant head group parameter, epgs, is introduced in order to account for several less well known effects, such as (i) the reduction of the overall free energy of the hydrocarbonfwater contact because of the presence of the polar head groups (this is in part simply due to the circumstance that the head groups diminish the hydrocarbonfwater contact area), (ii) the unfavorable asymmetric hydration of the ionic species in close proximity to the hydrocarbon core and image charge effects, and (iii) specific interactions and correlationfion volume effects which are not taken into account when estimating the excess electrostatic free energy by means of the PB approximation assuming a smeared-out surface charge (provided the total contribution of this kind is independent of the head group surface densities). No quantitative theories of the above effects are as yet available. In the calculations to be presented below, we have used the value epgS = -1.310kT
(14)
which was derived by fitting available cmc data for SDS solutions with and without added salt.'#* Concerning the possible thermodynamic effects of the shape fluctuations, we may estimate the corresponding free energy contribution to be of the order of -0.lkT for the whole micelle, i.e., about -0.001kTfhead group.20 In the present context, this is a minor quantity (but still not always negligible since there is a very close balance of opposing forces) which is assumed to be included in the head group constant. Summing the different free energy contributions discussed above, we get ;e
+ In x-)kT + 2kT(ln [S + (S2+ 1)'/2]+ 4kT X (l/S)[(S2+ 1 p 2 - 11)-
= -(19.960
KS(R,+ 3.0 X lo-'')
The corresponding e 2 functions are plotted in Figure 1 v5 the aggregation number N with the monomer concentration c- as a parameter. The rising branch at high aggre gation numbers is due mainly to the configurational free energy term whereas the Yhc/,@ contact term is responsible for the increase at lower values of the aggregation number. Upon varying the monomer concentration, the whole of the N c g curve shifts position in an approximately parallel fashion due to changing both the Tanford and the electrostatic terms. Note that the head group constant epg* actually has the nature of a correction to the Tanford standard free of energy of transferring a C12 hydrocarbon chain from the water phase to a hydrocarbon phase, -19.960kT. Rod-Shaped Micelle By means of the surface tension approach, it can be demonstrated that a generalized Young-Laplace equation similar to eq l Pi-P,=Y/Rc-=c=2YlRc
(16)
Langmuir, Vol. 6, No. 5, 1990 899
Transitions of Surfactant Aggregates of Different Shapes
0 50
60
70
80
90
/
0.5;
Aggregation number
-I
Figure 1. Excess free energy of a spherical SDS micelle at 25 O C plotted against the aggregation number N . Values of the monomer concentration c- are shown on the curves in mM.
must hold for a large cylindrically shaped micelle which is in mechanical equilibrium.7 Furthermore, it can be shown that no net work has to be expended when reversibly forming the cylindrical part of a rod-shaped micelle out of monomers supplied from the surrounding water solution. However, surfactant aggregation into rod-shapedmicelles occurs in a somewhat different regime that definitely belongs to the realm of small system thermodynamics. We may describe the thermodynamic situation in broad terms as follows. Suppose we have a distribution of micellar rods of different lengths. The chemical potential p~~ of the surfactant monomers in an aggregate of size N is defined by the partial derivative p;
=
a(~;;)/a~
(17)
where & is the (integral) free energy per monomer. Thus, th,e phase equilibrium condition ( 5 ) can be satisfied if Np# varies in such a way with N that ppf is less than the solution chemical potential p- for low N values and larger than p- at high N values. This obviously requires that some repulsive mechanisms are operating which prevent the micellar rods from becoming too short. The equilibrium micelle for which p$ = p- is of a," intermediate size determined by the condition that Nppf has a minimum. We may have large fluctuations away from the size of the equilibrium micelle. The size distribution becomes particularly broad when the free energy difference emc = ;LmC- p- = PCkT
(18)
corresponding to the work needed per monomer to form the middle part of a long rod out of surfactant ions present in the solution at the concentration c- is made very minute (/3c < 0.01) by means of adjusting the solution condition~.~ For the infinite cylinder part of a rod-shaped micelle, we can easily carry out calculations based on an expression analogous to eq 15. We merely need to replace the numerical factor 4 in the curvature-dependent electrostatic term by 2, insert the appropriate Gruen function (cf. Figure 2)) and change the head group constant to ePB
= -0.5897kT
(19)
This value was chosen by comparing in a rather crude manner with available data on the micellar growth in SDS solutions caused by raising the salt concentration.7 In Figure 2, the results of such calculations at a particular solution state are depicted. It is seen that, even in this case, the two opposing trends producing the minimum at about ac = 48 hiz (where p" = 0) are due to the (ec)conf term and the sum of the electrostatic free energy
44
45
46
47
48
49
50
51
52
Molecular a r e a l A 2
Figure 2. Example of a 8' calculation for SDS at 25 O C at c= 8.68 mM and cs = 5.0 mM. The variable contributions due to the micelle/water interactions, eelc and Yhc/&', and the counteracting hydrocarbon chain conformationalcontribution, tcOnfC, generate the minimum in co,c at a, = 48 Az where, in this case, p" = t,C/kT = 0 (a) t , f / k T - 4; (b) econf/kT; (c) t,c/kT; (d) yhc/&/kT - 5.
and the yhc/&c terms. In analogy with the Ntha curves in Figure 1,one finds that the whole p" function in Figure 2 is shifted in its horizontal position upon varying the salt concentration or the monomer concentration. Note that p" values I 0 are unacceptable as /3" = 0 would correspond to phase separation into infinitely long, noninteracting micelles. In Figure 3, we show where in the c-, cs diagram the p" = 0 condition is fulfilled. In addition, a cmc condition and a similar /3 condition for an infinite bilayer, Ob = 0, are also demarked. More precisely, this cmc condition corresponds to where the volume fraction attains the M. was computed by means of the value 2.48 X expression (cf. eq 4)
4' = C exp[-(Nc,B)/kT] (20) where C = 20.7 is the estimated size fluctuation entropy factor corresponding to a standard deviation u = 5 of the aggregation number. We may consider a rod-shaped micelle as a kind of composite aggregate consisting of a straight, cylindrically shaped part with two end caps of an approximately spherical shape. There should be a junction zone of an intermediate geometry (Figure 4) for the following reason. The revious calculations of N e 9 as a function of N = 4~(3!')2/(a~)3 and emc as a function of a c indicate that the hydrocarbon core radius of the spherical equilibrium micelle (Rs = 17.7 A, us = 59 A2) is significantly larger than the corresponding radius of the cylindrical micelle at the minimum in Figure 2 (Rc = 14.6 8,ac = 48 Az). Hence, it is reasonable to assume that the end caps are actually somewhat swollen when equilibrium is established throughout a rod-shaped micelle. In order to proceed, we next make the following ansatz as to how the excess free energy of a rod-shaped micelle varies with the aggregation number N. NeNc/kT= CY+ ' PcN + 6(N) Here the end cap contribution
CY^
(21)
is defined by
The function 6(N) is added to account for the end-to-
900 Langmuir, Vol. 6, No. 5 , 1990 2.0
Eriksson and Ljunggren I
I
I
0.01
I
I
-
0.008
I
I
I
I
i
1
0.5
s 0.006 -
e
spheres
3 f
i
0.004 -
11
0.002
i
i,
0
0
rods -
,
200
I 400
I 600
1 800
I
1000
Aggregation number
0.005
I
I
I
I
I
I
I
Figure 5. Micelle size distributionscalculated at the salt concentration cs = 0.2 M where we obtained ar = 5.695. The lower curve was generated at c- = 0.902 mM yielding p" = 1.7596 X 10-2 and $ = 0.051 14. The corresponding values for the middle curve are c- = 0.904 mM, p" = 1.5374 X 10-2,and $ = 0.063 84, and for the upper curve, c- = 0.906 mM, p" = 1.3156 X 10-2, and 4 = 0.082 25. XIO-4
i
Figure 4. Modified spherocylindrical SDS micelle model with R, z R, and a junction zone between the end cap and the cylindrical part, where the head group concentration gradient is counterbalanced by stresses in the hydrocarbon core.
rods
0
end repulsion resulting from the overlap of the two junction zones. We next introduce a set of simplifying assumptions. (i) The equilibrium (=average) spherical micelle always has the aggregation number N , = 66, irrespective of the detailed solution conditions. This does not hold strictly, because (N,) must necessarily increase upon raising the surfactant concentration, as follows in a rather general way from the equivalence of the energetic and surface tension approaches (cf. ref 8). Hence, by fixing the average aggregation number to ( N , ) = 66 (which is correct at c- = 8.4 mM and c. = 0), we will deliberately introduce a source of error which to some small extent might influence the quantitative estimation of the volume fraction 4, at other solution states. (ii) The corresponding surface density on the cylindrical part is fixed a t 48 A2/head group. Our previous calculations show, in fact, that there is a variation of ac amounting to 1.5 A2, only, within the whole concentration range along the p" = 0 curve in Figure 3. (iii) The rod-shaped micelle is modeled as a composite particle composed basically of a straight cylindrical part of equilibrium thickness, R, = 2v/ac, and one whole spherical equilibrium micelle with R , = 3v/aB. A relaxation process of this imaginary aggregate results in the contour depicted in Figure 4. (iv) The misfit free energy arising for a multitude of reasons in the junction zones is estimated to be roughly 66 X 0.01 = 0.66kT per micelle. The introduction of this small extra term will have a rather minor influence on the resulting size distribution of rod-shaped micelles. However, it is a largely unknown factor which might be of
0
I
200
400
800
800
1000
Aggregation number
Figure 6. Micelle size distributionscalculated at the salt concentration c, = 0.3 M where we obtained 1yr = 9.1362. The lower curve was generated at c- = 0.6750 mM yielding fl = 3.7170 X and 4 = 0.018 64. The corresponding values for the middle curve are c- = 0.6760 mM, p" = 2.2342 X 10-3, and $ = 0.036 47, and for the upper curve c- = 0.6765 mM, p" = 1.4936 X 10-3, and 4 = 0.059 57.
interest to evaluate by comparing with experimental results. (v) The end-to-end repulsion function b ( N ) is taken care of in an approximate manner by means of cutting the micelle size distribution a t N,O = 175. On this basis, we can readily derive a*= (t,'//kT
- pc + 0.01)66
(23)
which implies that
NcNr/kT= (C,'/kT - p'
+ 0.01)66 + pcN
( N = N,o, ..., m) (24) and that the volume fraction of rod-shaped micelles is given by 4; = exp(-NCNr/k T ) (25) Since €66' as well as PC are theoretically known as functions of the state of the surfactant solution (c-, c,), $2 distributions are now easily generated. Examples of such distributions are shown in Figures 5-7, where the corresponding spherical micelle peaks are also included. It is evident that rod-shaped micelles are present in
Langmuir, Vol. 6, No. 5, 1990 901
Transitions of Surfactant Aggregates of Different Shapes x10-5 61
I
I
I
I
I
I
I
1
I
I
spheres
4 1
iI
il
4
rods 0 0
2000
I
I
4000
6000
8000
B
5
0.2
L
E
y
o 0
10000
Aggregation number
Figure 7. Micelle size distributions calculated at the salt concentration c,, = 0.4 mM where we obtained CY' = 11.6226. The lower curve was generated at c- = 0.5495 mM yielding p" = 1.3488 X 10-3 and C#I = 0.005 67. The corresponding values for the middle curve are c- = 0.5501 mM, p" = 0.25661 X 10-3,and q5 = 0.033 39,and for the upper curve c- = 0.550 15,p" = 0.165 63 X 10-3,and 6 = 0.053 03.
0.02
0.04
0.06
Volume fractlon 01 surfactant
Figure 8. Fraction &/(4 of aggregated surfactant in the rod state vs the total volume fraction of surfactant 4 at the salt concentrations c. = 0.2,0.3,and 0.4 M.
appreciable amounts only when 8"is less than about 0.01. In addition, the parameter p" gears the width of the distribution as well as the total volume fraction of rods in agreement with the relations'
( N , ) = N,o + l/pc
(26)
which hold for the exponential distribution given by eqs 24 and 25. ( N , ) denotes the weight-average aggregation number of rod-shaped micelles. Note that the overall volume fraction, @, depends on ar as well as p" whereas the width of the distribution is determined solely by p". This is in harmony with Hill's treatment of linear aggregates (ref 17, Vol. 11, p 77) as well as the work of Israelachvili et al. (ref 2, eq 2.14, where Y = e-@). However, these authors used mole fractions instead of volume fractions in the dispersion equilibrium conditions. It is worth noting that the parameter ' a increases substantially with raising the salt concentration. This implies that ever smaller p" values are required in order to attain a certain volume fraction of micellar rods (eq 27). Simultaneously, the size distribution becomes very broad. Thus, the (indirect) cause of the elongation of rod-shapedmicelles upon adding the salt to a surfactant solution is the more rapid decrease of the electrostatic free energy for the cylindrical surface than for the hemispherical end cap surfaces. The fraction of micelles in the rod state is shown in Figure 8 at different salt concentrations. It is seen that rod-shaped micelles become strongly favored upon raising the salt concentration, as is in agreement with a wealth of experimental evidence. In fact, it is proper to call cs = 0.20 M a threshold salt concentration above which rods predominate over spheres. The corresponding overall weight-average aggregation numbers computed from the relation
W) = (664' + Wr)4*)/4
(28) are drawn in Figures 9-11. It is seen that the courses of these functions are qualitatively different at high and low salt concentrations. At cs = 0.4 M, the ( N ) vs function resembles a $l/z function for low surfactant concentrations, but at c. = 0.2 M, there is initially a concentration range where ( N ) grows slowly.
0
0.02
0.06
0.04
Volume fraction of surfactant
Figure 9. Weight-averageaggregation number, ( N ) ,as a function of the total volume fraction of surfactant, 4, computed at the salt concentration c, = 0.2 M where crr = 5.695.
0
I
I
I
0.02
0.04
0.08
Volume fraction of surfactant
Figure 10. Weight-averageaggregation number, ( N ) ,as a function of the total volume fraction of surfactant, 4, computed at the salt concentration cs = 0.3 M where ar = 9.1362. Assuming a single peak distribution of micelles with Nro= 66 corresponding to a spherocylindrical micelle model resembling the one proposed by Missel et al.,4 one finds that the ( N ) vs 4 function is practically a straight line with a slightly more rapid growth at low aggregation numbers (Figure 12). In other words, as was first pointed out by Porte et a1.F such a model cannot be reconciled with the notion of a second cmc that is inferred from viscosity and magnetic birefringence measurements. From the results presented here, it is evident that a bimodal size distribution which implies a coexistence of small spher-
902 Langmuir, Vol. 6, No. 5, 1990
0.02
0
Eriksson and Ljunggren
0.04
0.06
Volume fraction of surfactant
Figure 11. Weight-averageaggregation number, ( N ) ,as a function of the total volume fraction of surfactant, 4, at the salt concentration ca = 0.4 M where ar = 11.6226. 1
4
0
0
d
i
i 0
0.02 0.04 Volume fraction of surfactanl
ical and large rod-shaped micelles provides a theoretically sounder, as well as a more general and useful, basis when accounting for experimentally observed (N) vs 4 behaviors of ionic surfactant solutions. In particular, it becomes possible to account for such experimental data which indicate a much more rapid increase of (N) after a certain critical concentration is reached (cf. Figure 9).
Disk-Shaped Micelles and Vesicles It is fairly easy to extend the above calculation scheme so as to encompass, in addition, disk-shaped surfactant micelles and vesicles. Referring to the idealized contour of a disk-shaped micelle in Figure 13, we may consider a disk-shaped micelle as being composed of a planar bilayer part joined with a toroidal, semicylindrical rim part. Hence, the essential novel feature here is, of course, the bilayer part for which we can carry out calculations based on eq 15 upon dropping the curvature-dependent electrostatic term and inserting the appropriate Gruen term
(cb/knconf = 3.12630 - 1.15534Rb + 0.18273Rt -
+
1.40763 X 10-'R; 4.77423 X 10-4Rb4(29) where Rb is the half-thickness of the bilayer. In this case, cpgb was adjusted so as to yield a Bb = 0 line in the c-, cs plane crossing the p" = 0 line at c, = 1.38 M. The value of tpgb determined in this way was = 0.5962kT
try shown.
Our calculations of
pb = (imb - M-)/kT = tmb/kT
(31) show that ab is always equal to 34.0 A2 (Rb = 10.3 A) to a rather good approximation a t the minimum of the Ob vs ab curves corresponding to the p c vs a, curve in Figure 2. Hence, we have to anticipate that the rim part of a disk-shaped micelle in reality is somewhat swollen in comparison with the bilayer part. Due to this circumstance, we have tentatively assumed that the number of surfactant monomers in the semicylindrical rim part is twice the number of monomers pertaining to the idealized contour in Figure 13
# = 2(?r2RJ?t+ 4?rR2/3)/
0.06
Figure 12. Weight-averageaggregation number, ( N ) ,as a function of the total volume fraction of micelles obtained at the salt concentration c, = 0.3 M assuming a continuous exponential size distribution with NoC = 66 and ar = 10.
€Pg
Figure 13. Contour of the hydrocarbon core of an idealized disk-shaped micelle. In the present paper, we assume that the toroidal part at equilibrium is swollen compared with the bilayer part and contains twice as many monomers as for the geome-
(30)
(32)
Consequently, by writing
+
+
Nt,d/kT = atNt pbN 6(N)
(33)
where
N = Nt + 2?rR;/ab
(34) and assuming at and 6(N) to be given by the relations at = p" - pb
+ 0.05
(35) 6(N) = 5 0 / ( N- 100) (36) it is straightforward to compute the volume fraction of disk-shaped aggregates by means of the usual expression @* = exp(-Nt,d/kT)
(37) In the above equation for at,we have added 0.05kT units as a rough estimate of the misfit free energy that inevitably arises in the junction zones as a consequence of a relaxation/stretching process which leads to a smooth geometrical transition as depicted in Figure 4. The b ( N ) function (36)is included in eq 33 in order to account for the repulsive overlap of the junction zones and the bending free energy required when attaching the toroidal rim. Interestingly, at can be positive or negative depending on the salt concentration. Below the at = 0 line in Figure 16 at cB = 1.314 M, at < 0 and above this line at > 0. Figure 14 shows a size distribution obtained at c. = 0.5 M where spherical micelles, rods, and disk actually coexist according to our calculations. A t even higher salt concentrations, the toroidal rim becomes progressively more and more disadvantageous, and we will have to consider the option that closed bilayer vesicles are formed instead. The vesicle is readily treated along the same lines as those employed above for rods
Langmuir, Vol. 6, No. 5, 1990 903
Transitions of Surfactant Aggregates of Different Shapes x10-8 8,
I
I
I
I
I
I
I
I
I
I
0.05
I
I 0.2
I
0.1
200
0
400
800
800
1000
5.0
Figure 16. Diagram showing approximately where in the c-, cs plane surfactant aggregates of different shapes predominate. The threshold salt concentrations are as follows: sphere/rod transition, c, = 0.20 M; rod/disk transition, cs = 0.7 M; disk/vesicle transition, c, = 1.75 M. at of the toroidal rim part of the diskshaped micelles is zero at cs = 1.314 M.
I
\
\ I
2.0
Monomer ConcentrationImM
Aggregation number
Figure 14. Size distribution of surfactant aggregates calculated at the salt concentration c, = 0.5 M and the monomer concentration c- = 0.47 mM where we obtained a' = 13.5698, at = -0.153 299, p" = 1.15394 X 10-3, and /3b = 0.204 45.
\\
1 1.0
I 0.5
\
vesicles
i01 0
discs
I
\
I
\
1
400 800 Aggregation number
200
1000
0.1
Figure 15. Size distribution of surfactant aggregates calculated at the salt concentration cs = 2.0 M and the monomer concentration c- = 0.193 mM where we obtained ar = 25.5779, at = 7.2958 X 10-2, p" = 3.838 33 X 10-2, and /3b = 1.542 55 X 10-2.
and disks resulting in
Ne,'/kT = pbN+ 6(N)
(38)
Taking care of the repulsive interaction function 6 ( N ) in an approximate fashion by cutting off the distribution below Nvo = 500, we can write
( N = N:, ..., a) (39) In Figure 15, we display an aggregate size distribution generated at c, = 2.0 M where vebicles predominate but where there is still a peak of small, disk-shaped micelles. It is worth noting that the salt concentration cBby and large determines the hierarchy observed of surfactant aggregates. In Figure 16, horizontal lines are drawn to demark approximately where the sphere-to-rod, rod-todisk, and disk-to-vesicle transitions do occur. More precisely, this diagram is to be interpreted in the following way. When the surfactant concentration is increased from zero at a given constant salt concentration, the following will happen: (i) In the range of salt concentrations below 0.4 M, the cmc curve is first reached and spherical micelles begin to form. The condition p" < 0.01, necessary for the formation of rod-shaped micelles, can only be reached near the crossing point of the cmc and p c = 0 curves a t c, = 0.5 M, more exactly, at salt concentrations above 0.20 M $' = exp(-Npb)
10-31
,
I
I
i
I
I
I
/"!
I
800
.
i
CMC I
I
I
I
I
1
I
1.0 Salt concentrationhi
0.5
5
I
>
10
Figure 17. Diagram showing at what total surfactant concenis satisfied (lower trations the cmc condition 4 = 2.48 x set of curves) for the various surfactant aggregates at different salt concentrations. The upper curves correspond to aggregate concentrationswhich are 10 times the cmc. By and large, relevant experimental data are lacking that would enable a reliable determination of cpgb, the misfit free energies and the cutoff functions for disks and vesicles. For this reason, the courses of the dashed curves relating to disk-shaped micelles and vesicles are rather uncertain. and at high enough surfactant concentrations where rodshaped micelles may, in fact, predominate. (ii) For salt concentrations between about 0.4 and 1.4 M, the p" = 0 curve is first approached, and rod-shaped micelles or, alternatively, disk-shaped micelles (with a semicylindrical rim part) begin to form. Spherical micelles cannot reach high volume fractions in this range. The number of rod-shaped micelles tends to become suppressed upon raising the salt concentration because a' will then grow in magnitude. (iii) For salt concentrations above 1.4 M, the curve Ob = 0 is first approached. Since the at = 0 line is situated at a salt concentration of 1.3 M, vesicles are primarily formed. The reason for this is that vesicles are energetically favored when at > 0. Small disk-shaped micelles will coexist near the crossing point of the p" = 0 and pb = 0 curves, in particular a t a salt concentration below 1.9 M. Figure 17 further illuminates how the genesis of surfactant aggregates of different shapes depend upon the salt concentration. Note that the volume fraction of each of the different coexisting surfactant aggregates can be
904 Langmuir, Vol. 6, No. 5, 1990
computed theoretically a t any solution state as specified by the salt and total surfactant concentrations.
Discussion In summary, we have presented a straightforward thermodynamic calculation scheme which is capable of predicting the relative occurrence of surfactant aggregates of all the familiar shapes a t different salt and surfactant concentrations. Since aggregate interactions are neglected, our approach is limited to surfactant solutions which are dilute with respect to surfactant aggregates and/or contain enough of an added salt component. The modeling involved assumes that the different structural units, which are joined to form composite aggregates, retain in essence their respective equilibrium sizes and that junction zones develop to yield smooth geometrical transitions. Furthermore, in order to gain simplicity, we have neglected those small free energy changes which are related to the rather minor variations in surfactant packing density which are brought about by changing the monomer and salt concentrations. More precisely, we have presupposed that these contributions are small in comparison with the variations of the Tanford and electrostatic terms resulting from concentration changes alone. The only unknown parameters invoked are related to the repulsive interactions which must be present in order for separate ensembles of rods, disks, and vesicles to exist in the sense of small system thermodynamics, i.e., each of them obeying eq 3. These include misfit free energies and cutoff aggregation numbers, No. Moreover, the head group constants, tpg, have to be determined by comparison with experiments. The key role of the electrostatics in our treatment is hardly amazing but deserves nevertheless to be appreciated. The differing slopes of the @ and cmc lines in Figure 16 can be traced back to the effect of geometrical shape on the electrostatic free energy (eq 7). A closer analysis shows, in fact, that the succession of various surfactant aggregates generated upon raising the salt concentration is essentially caused by the concomitant decrease of the electrostatic surface potential which is more rapid
Eriksson and Ljunggren the less an aggregate surface is curved. The remaining free energy terms are important mainly because they balance in such ways that the critical condition lines in Figure 16 cross at attainable salt concentrations. Furthermore, an interesting point to discuss is that the magnitude of ar is found to be considerably larger at high than at low salt concentrations. Along the cmc line in Figure 16, (Ntha) is actully a constant equal to 11.3322. To the left of this line, (Nt9)will be larger than 11.3322, and to the right it will be smaller. Referring to the definition of ar, we also note that there will be a positive contribution to a' because of the p" term when the c-, cB state considered is shifted further to the left of the p' = 0 line. That is to say that the ar value derived is strongly dependent upon how closely and in what order the cmc and p" lines are located. When they are much apart, as is the case at low salt concentrations, the spherical micelles will predominate because the work needed to form the middle part of a rod always stays prohibitively large and is not fully compensated by a small enough ar value before excessively high volume fractions p are attained.
Conclusions In this paper, we have shown how a quantitative description can be developed of the successive formation of surfactant aggregates of different shapes by employing the theories of surfactant micelles which the present authors published a few years ago. The curvature-dependent electrostatic interactions are found to be of a major significance in this context. By adding quantitative estimates to our scheme of the attractive and repulsive interactions between the aggregates, an even more complete theoretical account of ionic surfactant systems would result, covering also the formation of hexagonal and lamellar liquid crystals. This problem has not yet received a satisfactory treatment in all respects, but the previous work of Jonsson and W e n n e r ~ t r o mrepresents, ~ ~ ~ ~ ~ in our opinion, a major achievement in that direction. The calculations described in this paper are complementary to theirs as regards the genesis of the basic surfactant aggregate structures. Registry No. SDS, 151-21-3.
