Effect of counterion binding on micellization - Langmuir (ACS

Langmuir 1988,4,77-90

77

Effect of Counterion Binding on Micellization Eli Ruckenstein* and J. A. Beunen Department of Chemical Engineering, State University of New York a t Buffalo, Buffalo, New York 14260 Received April 15,1987.I n Final Form: June 16,1987 We develop a theory for the effect of counterion binding on the micellization of ionic surfactanta in aqueous electrolyte solutions. Whereas previous treatments of micellization have in general assumed a value of the surface charge a priori, the present theory determines the value from the requirement that the free energy o€the system be a minimum. This leads to a dissociation equilibrium the form of which depends on the structure adopted for the micelle-solution interface. A model for this interface is developed that incorporates inner and outer compact layers between the surface of the head group charge and the start of the diffuse double layer. In addition, the counterion-head group bond occurring at the interface is assumed to be of a general form, permitting some degree of covalency. The dissociation equation shows that the degree of binding is not the same for all aggregates but depends on their size as well as their shape, being smaller for spherical than for cylindrical micelles. Since account is also taken of nonideality of the surfactant monomers and the electrolyte, the theory should allow the properties of the system to be predicted up to high concentrations of added salt. The model is applied to the case of sodium dodecyl sulfate solutions in the presence of added sodium chloride. Good agreement with experimental cmc values and aggregation numbers is obtained. However, the degree of dissociation is too low, and varies too rapidly with salt concentration, most probably because we neglect the dependence on the electric field of the dielectricconstant in the inner layer of the micelle-solution interface. Small changes in the values of some parameters lead, however, to larger degrees of dissociation, but with somewhat poorer agreement for the average aggregation number. Significantly,the standard work of formation of a micelle almost equals the concentration-dependent part of the free energy of ita constituent amphiphiles in bulk solution. Since the size distribution depends upon the small difference between these two large quantities, the properties of the micellar system will be sensitive even to small contributions to the standard work of formation. A proper choice of micelle shape is thus seen as essential. It is further shown that the conformation of chains in the core should also be considered. At higher surfactant concentrations, volume exclusion effecta and van der Waals interactions, usually ignored, as well as the overlapping of electric double layers, become important and may lead to phase separation.

I. Introduction The formation of micelles in aqueous solutions of amphiphilic compounds has been the object of considerable investigation, both theoretical and experimental. The broad understanding that has emerged has been well summarized, most conspicuously by Tanford in his wellknown monograph.' Simple models have been constructed, embodying the main physical factors involved and capable of making some definite predictiom2 At the same time, effort has been devoted to removing some of the obvious shortcomings of earlier A question that has not so far received adequate attention is the effect of ion binding on the micellization of ionic amphiphiles. The electrostatic contribution to the energy of a micelle is generally evaluated in the DebyeHuckel approximation, assuming the micelle to be fully dissociated. Occasionally,5 the standard result is multiplied by a factor less than unity to make some allowance for incomplete dissociation. Where this is done, however, the factor is more an ad hoc modification than the result of a genuine theory of ion binding. Stigtere has developed (1) Tanford, C. The Hydrophobic Effect, 2nd ed.; Wiley: New York, 1980. (2) (a) Ruckenstein, E.;Nagarajan, R. J. Phys. C&n. 1975, 79, 2622. (b) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. SOC., Faraday Trans. 2 1976, 72, 1525. (3) Gunnarsson, G.; Jonsson, B.; Wennerstrom, H. J. Phys. Chem. 1980,84, 3114. (4) (a) Dill, K.A.; Flory, P. J. Proc. Natl. Acad. Sci. U S A . 1981, 78, 676. (b) Gruen, D.W. R. J. Colloid Interface Sci. 1981, 84, 281. (5) (a) Nagarajan, R.; Ruckenstein, E. J. Colloid Interface Sci. 1977, 60,221. (b) Nagarajan,R.;Ruckenstein,E. J . Colloid Interface Sci. 1979, 71, 580. (6) (a) Stigter, D.J. Phys. Chem. 1964, 68, 3603; (b) 1974, 78, 2480; (c) 1975, 79, 1008; (d) 1976, 79, 1015.

0743-7463I88 l2404-0077$01.50I 0

a model based on the cell and lattice theories of liquid mixtures, which takes into account the discrete nature of the surface and permits a general degree of ion binding. However, the theory does not yield a value for this quantity. Instead, experimental values are used, together with cmc and micelle size data, to draw some conclusions regarding the free energy of micellization and the properties of the interface. To date, the most detailed theoretical work on the micelle-solution interface is that carried out by Beunen and Ruckenstein.' Their model considers discrete ionic head groups and ionic head groups-bound counterion pairs and the interactions among them, which introduce a nonrandomness into their distribution over the interface. The effect of the double layer was included by an improved approximate solution to the nonlinear Poisson-Boltzmann equation. The approach allowed the calculation of the work of formation of the interface, from which an equation could be derived predicting the degree of dissociation of the surface groups. What is still needed is to employ such a model for predicting the aggregation characteristics of surfactant solutions, such as the cmc and the average aggregation number. The present work is aimed at providing the appropriate approach and its application to a specific surfactant, namely, sodium dodecyl sulfate. The model employed for the micelle-solution interface differs in some respects from that previously suggested. In considering the binding of ions to some substrate, such as a macromolecule or an aggregate of smaller molecules, one is immediately confronted by a question of definition. On what grounds does one regard an ion as (7) Beunen, J. A,; Ruckenstein, E. J . Colloid Interface Sci. 1983,96,

469. 0 1988 American

Chemical Societv

78 Langmuir, Vol. 4, No. 1, 1988

bound rather than free? Some ions may be attached to specific sites on the surface. Obviously, these are to be considered bound. However, even counterions not so attached are not completely free. On the contrary, the closer these counterions are to the substrate the more their electrostatic attraction for the substrate restricts them to moving parallel to the surface. These ions are therefore also bound to the extent that they move with the substrate and are thus not osmotically active. A number of experimental techniques are available for investigating the interaction between ions and substrates, e.g., NMR spectroscopy or transport property and osmotic pressure determinations. These enable some conclusions to be drawn concerning the nature of ion binding. In some cases results can be explained adequately on the basis of the diffuse double layer only. A recent example is the NMR quadrupole splitting work of Wennerstrom et al.8 However, results obtained by the same method for other systemsg can be given satisfactory interpretation only if one postulates the existence of specific binding sites. From the standpoint of theory, site-binding models allow a successful description of a variety of interfacial phenomena: particle deposition,l0adsorption of hydrolyzable metal ions at oxide-water interfaces,ll the binding of cations to membranes,12the titration of surface groups on ionizable latexes,13 and the behavior of the surface potential and interaction force during the approach of ionizable colloidal parti~1es.l~In view of this extensive use it is clearly of interest to examine the applicability of a site-binding model to a micellar solution. We will, therefore, develop the theory of micellization assuming that ions can bind to specific groups on the surface. All other ions are regarded as free. The plan of the paper is as follows. Section I1 presents a thermodynamic formulation of micellization, allowing for the occurrence of ion binding. Involved in this formulation is the standard work of formation of a micelle from its constituents, a quantity which is heavily influenced byevents occurring a t its interface with the solution. In section I11 we develop a model for this interface and derive an expression for the work of formation. In section IV we show how the results of the previous sections can be combined to determine the degree of dissociation and other micellar properties such as the cmc and the average aggregation number. Finally, in section V we apply the theory to the case of micellar solutions of sodium dodecyl sulfate. 11. Thermodynamic Formulation As stated in the Introduction, the main objective of this paper is to examine the consequences of allowing counterion binding a t the surface of the micelle. We then assume that the surfactant may exist in the neutral; associated form also in the bulk solution. The species present are thus surfactant ions R-, molecules MR, coun(8) Wennerstrom, H.; Lindman, B.; Lindblom, G.; Tiddy, G. J. T. J. Chem. Soc., Faraday Trans. 1 1979, 75,663. (9) (a) Lindblom, G.; Lindman, B.; Tiddy, G. J. T. J. Am. Chem. SOC. 1978, 100, 2299. (b) Lindman, B.; Lindblom, G.; Wennerstrom, H.; Gustavsson, H. In Micellitation, Solubilization and Microemulsions; Mittal, K. L., Ed.;Plenum: New York, 1977. (10) (a) Prieve, D. C.; Ruckenstein, E. J . Colloid Interface Sci. 1977, 60, 337. (b) Prieve, D. C.; Ruckenstein, E. In Colloid and Interface Science; Kerker, M . , Ed.; 1976; Vol. IV, p 73. (11) James, R. 0.; Healy, T. W. J.Colloid Interface Sci. 1972,40,65. (12) (a) Nir, S.; Newton, C.; Papahadjopoulos, D. Bioelectrochern. 1978,5, 116. (b) McLaughlin, S. G. A.; Szabo, G.; Eisenman, Bioenerg. J. J. Gen. Physiol. 1971,58, 667. (13) (a) James, R. 0.; Davis, J. A.; Leckie, J. 0. J . Colloid Interface Sci. 1978,65,331. (b) Healy, T. W.; White, L. R. Adu. Colloid Interface Sci. 1978, 9, 303. (14) Prieve, D. C.; Ruckenstein, E. J. Theor. Biol. 1976, 56, 205.

Ruckenstein and Beunen

terions M+, inert anions X- from the added salt, and micelles of various sizes, shapes, and states of charge. (We ignore the H+ and OH- ions that are of course also present in water. At intermediate pH any effect they may have is negligible compared with that of the far more numerous M+ and X-.The theory will be presented for the case of anionic surfactants but of course applies also, after obvious modifications, to those with positively charged head groups.) The properties of the system can be determined from the general thermodynamic'condition that the Gibbs free energy at constant temperature and pressure be a minimum. As usual, this requirement can be formulated in terms of the electrochemical potentials of the various components. For the monomeric species of interest we have

~ M = R ~MR'

+ kT In XMR 4- &&TI

(3)

where, as usual, It is Boltzmann's constant, T is the absolute temperature, x M , xR, and x m are mole fractions, and pMo, pRo, and p m o are the standard chemical potentials. The standard state of each of these species is that in which it is infinitely dilute in otherwise pure water. The form chosen for eq 1-3 is that appropriate for the "bulk" solution. This choice is allowable since we will assume that the micelles are sufficiently dilute for their double layers to occupy only a small fraction of the electrolyte volume. To date, most authors have assumed that the micellar system is a dilute solution in the thermodynamic sense. However, we wish to examine fully the effect of added salt on binding and so have included activity coefficients in the above equations. The third term on the right in (1)and (2) is the usual Debye-Hiickel expres~ion,'~ where q is the magnitude of the electronic charge, E the bulk dielectric constant, K the reciprocal of the screening length, and d a constant characteristic of the distance of closest approach of oppositely charged ions. The last term in each of the above equations is less familiar and warrants some explanation. It takes into account the so-called salting out of the particular solute that results from the modification of the structure of water by the ions present. If these include a structure-making species, e.g., one with small ionic radius, the water molecules will be more organized than they are in pure water. More work will, therefore, be required to form cavities to accommodate the given solute. In the case of the surfactant species, with their long alkyl chains, this work is the main component16 of the unfavorable hydrophobic interaction. Clearly, these effects raise the energy of the given solute, the increase being proportional to the ionic strength Z , except for high salt concentrations.'I Central to the development of the theory is the electrochemical potential of the micelles themselves. In a general treatment, micelles that are distinguishable by any intrinsic property are to be regarded as belonging to different chemical species. The distribution of micelles over (15) Bockris, J. O'M.; Reddy, A. K. N. Modern Electrochemistry; Plenum: New York, 1977; Vol. I. (16) (a) Shinoda, K. Principles of Solution and Solubility; Marcel Dekker: New York, 1978. (b) Ruckenstein, E. Progress in Microemulsion; Plenum: New York, in press. (17) (a) Wilcox, F. L.; Schrier, E. E. J. Phys. Chern. 1971, 75,3757. (b) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed.; Butterworths: London, 1965.

Langmuir, Vol. 4 , No. 1, 1988 79

Effect of Counterion Binding on Micellization the various aggregation numbers, shapes, state of charge, etc., is then determined by minimizing the free energy of the system. However, such an approach is hardly feasible, so two simplifications are commonly made. Firstly, micelles are assumed uniquely defined by their aggregation number j ; i.e., for any given j there is one shape and state of charge that is overwhelmingly more probable than the rest. Consequently, micelles can be characterized thermodynamically by electrochemical potentials of the form pl = p j o kT In xI (4)

+

where x j is the mole fraction of micelles containing j amphiphiles. The standard chemical potentials p j o include the interactions, e.g., the formation of the double layer, occurring between the micelle and the surrounding medium. Interactions among the micelles themselves are neglected, since we have already assumed their spacing to be much greater than the screening length. Thus there is no activity coefficient in ( 4 ) . The second simplification is that the shape of micelles of a given aggregation number is specified a priori rather than determined by minimizing their contribution to the total free energy. Small micelles will be taken to be spherical, with a core of radius rl given by

+

+

quantity 1.1.~ j a j p Mfor spherical micelles and plo (JaJ + 0'- J)&M for larger aggregates. A detailed thermodynamic derivatization of this result is possible (Appendix), but its plausibility can be seen from the following considerations. Of the chemical potential p j , p j o is that part which contains the micelle-solvent interactions and therefore depends on the degree of dissociation. The second term in each of the above quantities is a contribution from those counterions which may be thought of as belonging to the micelle, having originated from it by dissociation. The sum is therefore that part of the system free energy which changes when the charge on a particular micelle is altered. The condition that this sum be a minimum leads immediately to aPjo/affj

= -jPM

aPlo/aaJ

= -JpM

(6)

for j IJ and aPjo/aa

= -0'- a p M

(7)

for j > J. These equations have an obvious physical interpretation. If the free energy is to be stationary with respect to variations in micelle charge, then the change in pjo due to, say, a slight increase in dissociation above the rj = ( 3 ~ / 4 ? r ) ~ / ~ j ' / 3 (5) optimal value must be balanced by an opposite change in the energy of the solution resulting from the extra counwhere u is the volume of the surfactant alkyl chain. This terions released. Equations 6 and 7 are thus a statement radius becomes equal to the maximum chain length for of the dissociation equilibrium existing between the micelle some j = J , which thus defines the largest possible spherical surface and the aqueous medium. They are the general micelle. We will assume that for j > J the micelles are relations linking the micelle charge to the structure of the cylindrical with hemispherical end caps. Such a shape has interfacial region. been used in several theoretical a n a l y ~ e s and ~ ~ Jis~ conThe state of charge specifies the numbers of the sursistent with a growing body of experimental e v i d e n ~ e ' ~ ~ factant ~ ions and molecules forming a micelle of a given suggesting a rodlike shape for larger micelles, at least in aggregation number. This formation is governed by a the case of sodium dodecyl sulfate. condition of chemical equilibrium obtained by minimizing We come now to our main concern-the determination the free energy with respect to the amounts of various of micelle charge. The state of charge of the micelle can components: be described by giving the degree of dissociation of ionP j = jajPR + j ( 1 - ~ ] ) P M R ; j 5 J izable groups. At any point on the surface this is just the probability that a group located there is dissociated. For Pj = spherical micelles this will be constant over the surface but ( J ~ J+ 0 ' - J ) ~ P R + 0' - J a r Ci - J ) ~ ) P M R ; j > J (8) will vary with the aggregation number since the electric field a t the surface is affected by its curvature. AccordThe usual monomer-micelle equilibrium'v2 has thus been ingly, the degree of dissociation is denoted by a].For larger extended to allow for the two monomeric species taking micelles, with their less symmetrical shape,the dissociation part. A second condition resulting from the free energy will depend on the location of the group on the surface minimization is, as expected, the dissociation equilibrium since the curvature is not constant. Therefore, the defor the surfactant in the bulk: termination of the dissociation is in general a problem in (9) PMR = PM + PR the calculus of variations. To avoid such a task we assume that, over the uniform cylindrical section of large micelles, When explicit forms for the chemical potentials are the degree of dissociation is a constant, denoted by a,while substituted into eq 8, it becomes apparent that the comfor the end caps, which are just the hemispheres of the position of the system depends on the quantities p j o largest spherical micelle, it takes the value a> In this way - j ( 1 - a j ) p m o and klo - ( J a j 0' - J ) a ) p R 0 - 0' two quantities, rather than a function of position, are - JaJ - 0' - J)a)pmo.These are the standard free energies sufficient to describe the state of charge of larger micelles. of formation of spherical and rodlike micelles from their This simplification comes, however, at the price of an constituents. To evaluate them we need a detailed model unphysical abrupt change in dissociation at the junctions of the micelle. In the next section we develop such a model between the end caps and the cylindrical section. and calculate the corresponding standard work of formaAs stated above, the surface charge is to be determined tion. by requiring that the free energy of the system be a minimum. With the approximation that the micelles do not 111. Micellar Model and Standard Work of interact with one another, this reduces to minimizing the Formation The work done in forming the micelle from its molecules and ions may be broken up into contributions from the (18)Ruckenstein, E.; Nagarajan, R. In Micellization, Solubilization and Microemulsions; Mittal, K. L., Ed.; Plenum: New York, 1977. core and the interfacial region. Since our main concern (19)Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y.; Carey, is the effect of the ion binding which takes place at the M. C. J . Phys. Chem. 1980,84, 1044. surface, our treatment of the core will be along standard (20) Corti, M.; Degiorgio, V. In Solution Chemistry of Surfactants; Mittal, K. L., Ed.; Plenum: New York, 1978. lines. Its contribution derives from the energy released

+


\'i

\

Ruckenstein and Beunen \

€0

€

\

Figure 1. Schematic diagram of the surface region of a micelle of aggregation number j showing head groups (a), bound counterions (b),and counterions at the start of the double layer (c). Dimensions and dielectric constants of the various regions, as explained in the text, are indicated. by the removal of the surfactant alkyl chains from water to a hydrocarbon environment'

AWhc = -j(2OOO

+ 700(n, - 1)) cal mol-l

(10)

for a chain of n, carbon atoms. Equation 10 assumes that removal of the chains from water is complete. In actual fact, water molecules can penetrate between surfactant head groups, so some allowance must be made for residual contact between the core and the aqueous medium. This introduces the following positive, i.e., unfavorable, interfacial contribution to the work of formation:' AW,,,,

= s(aJ - 21)j; j 5 J

AWcOntact= s ( ~-J 2 1 ) J + s(Y3uJ- 21)(j - J); j

>J (11)

where s is a constant equal to 25 i 5 cal mol-' A-2, 21 A2 is the cross sectional area of a hydrocarbon chain, and a, (j 5 J) is the area per amphiphile at the surface of the core. This latter area is usually taken to be slightly greater than that predicted from purely geometrical considerations in order to allow for surface "roughness" and the finite distance of closest approach of a water molecule to the core surface.' Thus

+ 6)2/j;

uj = 47r(rJ

j 5J

(12)

where rj is the core radius given by (5) and 6 is a constant. For a rodlike micelle, the area per amphiphile over the end caps equals aJ. Over the cylindrical section it will be 2/3aJ since both regions have the same radius and must allow the same volume per amphiphile. Of great importance for a description of ion binding is the fact that, unlike the core, the interfacial region cannot be regarded as uniform. On the scale of the individual components the properties of this region undergo a dramatic variation as one moves outward from the micelle core. This will be represented in our model by a division of the region into several layers of different widths and dielectric constants chosen to reflect as far as possible the actual structure and composition of the interface. Our model for the interface is illustrated in Figure 1. Since residual contact between the alkyl chains and water molecules has already been allowed for, the remaining interfacial region is taken to start a t the surface of head group charge, located a distance rJ + dhd from the center or axis of the micelle. The constant dhdis a measure of the length of the head group, modified, as necessary, to take account of its orientation and charge distribution.

Next come the bound counterions, their centers lying on the surface of specific adsorption. Between the two surfaces so far defined lies the inner compact layer of thickness di and dielectric constant ti. The value of di will be determined by head group and counterion dimensions, while ti will include contributions from the polar, partly oriented water molecules as well as from the head group and counterion electron clouds. The electrical properties of this layer are very sensitive to the nature of the counterion head group bond. Experimental evidence seems to indicate that (positive) counterions generally retain at least their inner hydration sheaths on binding to a surface.gb This would appear to preclude any form of intimate contact between the counterion and head group. However, it is possible that some deformation of the hydration sheath may O C C U ~allowing , ' ~ ~ ~a closer approach of the two ions than would otherwise be the case. Moreover, for some surfactants the binding found to occur is more appropriately described as a type of ion pairing,22 and charge transfer between head group and counterion has also been observed.23 Such a transfer might take place even across an intervening layer of water molecules, by means of quantum mechanical tunneling. We will, therefore, consider a generalized form of ion binding, one which may have some covalent character. The formation of the bond thus involves some charge redistribution so that each ion has a fractional charge +pq where p may be thought of as a degree of ionicity. Beyond the surface of specific adsorption is the outer compact layer. Its width d, depends on the sizes of bound and mobile counterions, while its dielectric constant to will be determined largely by the degree of orientation of water molecules. On the outer boundary of this region lie the centers of the first ions in the diffuse double layer. Here the interface gradually gives way to bulk solution. We assume no further variation in the dielectric constant-throughout the double layer its value, e, will be just that of pure water. The present model of the interface is simpler than that employed in our previous paper' since the discreteness of the ionic head groups as well as the nonrandomness of the ionic head groups and of the ionic head groups bound to counterion pairs are neglected. This makes both the model and the calculations more transparent. Instead, some degree of covalency of the ion binding is included. Having specified the model for the interfacial region, we proceed to the calculation of its contribution, A W.interface, to the standard work of formation of the micelle. From this we can finally obtain the difference in standard chemical potentials, which enters into the thermodynamics:

It is through this quantity that the structure of the micelle determines the properties of the system. To proceed, we imagine that the interfacial region results from the release of counterions from a neutral interface previously formed by assembling dissociated and undissociated head groups together with the appropriate number of counterions. By considering it to be formed in this way we are able to decompose the work involved into contri(21) Robb, I. D. J . Colloid Interface Sci. 1971, 37, 521. (22) Lindman, B.; Wennerstrom, H.; Forsen, S.J. Phys. Chem. 1970, 74, 754. (23) (a) Ray, A.; Mukerjee, P. J. Phys. Chem. 1966, 70, 2138. (b) Mukerjee, P.; Ray, A. Zbid. 1966, 70, 2144; (c) 1966, 70, 2150.

Langmuir, Vol. 4, No. I, 1988 81

Effect of Counterion Binding on Micellization

butions for which well-known, if approximate, expressions are available. In our model the interface is represented by several different continuous layers, and the charge at the boundaries, or in the double layer, is assumed smoothly distributed. Within this approximation, the assembly of the neutral interface is equivalent to the formation of an electric capacitor of the appropriate shape (spherical or spherocylindrical) and charge, filled with a medium of dielectric constant ti. The work done to form this capacitor, A Wcap, represents the electrostatic interactions among the counterions and head groups of a real (neutral) interface. Through the value of ei it also allows to some extent for any change in the hydration of the components that may occur during the assembly process. From standard electrostatic theory24we have

For larger rodlike micelles the work is taken to be a simple combination of terms for spherical and cylindrical capacitor~:~~ AW- =

(15) where l j is the length of the cylindrical section of the micelle given by rrJ21j= 0' - J)v (16)

We now consider the release of counterions from the neutral interface. In doing so we must bear in mind that the standard state for the micelles, implied by the form assumed for bj, is that in which they are infinitely dilute in a solution of the same composition as the bulk solution of the actual system. As a result, the main part of the work done in this process is that associated with the formation of a diffuse double layer in the solution surrounding the micelle. In addition, the electric field generated by the charging of the interface modifies the dielectric constant of the aqueous medium nearby, thus forming the outer compact layer. The counterions released from the surface are returned to their standard state. For the total work done we make use of the usual charging formula:2s Awd-1 =

where II;.and rC, are the electric potentials in the interfacial regions of spherical and (infinite) cylindrical micelles, respectively. Since the counterions that dissociate originate from the boundary between the inner and outer compact (24) Jackson, J. D. Classical Electrodynamics, 2nd ed.; Wiley: New York, 1975. (25) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948.

layers, the potentials are evaluated at this surface. The integration variable d,specifying the point reached in the charging process, is the charge on the interface per unit area of this surface. Its final values are therefore related to the degrees of dissociation by bj

= -aJq/4?f(rj

+ dhd + di)2; j

when the micelles are spherical and a = -a(j - J)q/2r(rJ dhd

+

IJ

+ di)lj

(18) (19)

for the cylindrical geometry. The right-hand side of the last equation is of course independent of j as can be seen by using eq 16 for li. The effect of salt on the micelles is not limited to the formation of double layers. There is also some salting out of the aggregated amphiphiles. This effect of the salt on the micelles will contribute to their energy an amount A WE, = S$TI (20) where S. is a measure of the average salting out effect on an amphiphile in the micelle. Strictly speaking, Sj will depend on the size and degree of dissociation. However, if we ignore the difference in the salting out effects on neutral and charged head groups at the interface, then S, will simply be a constant. This salting out effect on the micelles will be smaller than the corresponding effect on the monomers since the hydrocarbon chains are essentially removed from contact with the aqueous medium. Thus, on balance salting out favors a g g r e g a t i ~ n , ~ and ~ ~we ~ ' may speak of a salting out from bulk solution into micelles. A final contribution remains to be considered, one that is of thermodynamic rather than electrostatic origin. It comes about because formation of the partially charged interface involves the mixing of dissociated and undissociated head groups. As a result there will be an entropic contribution to the (free) energy of formation. If the two different kinds of head group can be assumed to mix ideally on the surface, this contribution is given by AWmix= jkT[ajIn aj + (1- aj) In (1- aj)]; j IJ Awmix = JkT[aj In aj + (1- ( Y J ) In (1- aj)] + 0' - J)kT[aIn a + (1- a) In (1- a)]; j > J (21) We have now evaluated all the work done in forming the interface:

Combined with (13) this equation provides an expression for the difference in standard chemical potentials, which we need to investigate the behavior of our micellar solution. In the following section we discuss in greater detail how this is done.

IV. Determination of System Behavior We show here how the results of the previous sections may be used to determine the properties of the micellar system. Chief among these is the micelle charge. In section 11, a general equation was obtained relating charge to the details of micelle structure contained in the standard chemical potential bjo. The specific model of the micelle introduced in the foregoing section relates structure to bj0 in an explicit fashion by providing an expression for the standard work of formation. Thus, by substituting the expressions for the various terms in (22) and (13) and using the result in (6), we obtain (26) Beunen, J. A.; Ruckenatein, E. Adv. Colloid Interface Sci. 1982, 16, 201.

(27) Nishikido, N.; Matuura, R. Bull. Chern. SOC.Jpn. 1977,50,1690.


82 Langmuir, Vol. 4,No. 1, 1988

8 and 91, and the expression for the work of formation of section I11 is used, one obtains equations which together with eq 23 and 24 can be solved for the composition variables xM, xR, x m , and xj once the amount of salt and the total amount of surfactant are specified. By including activity coefficients, one thus obtains X j = (fRXR)j"j(f~XMR)"'-*j' exp(-AWj/kT); j 5 J

L

q2 __--

2tkT 1

K

+ Kd

xj =

where K-, defined as exp[(pmo - KM'.- pR")/kfl, is just the equilibrium constant for the bulk dissociation reaction MR M+ R- governed by (9). Similarly, from the second member of eq 7, we have

-

a l-a

-XM

+

= Kdiss exP q$'(rJ+dh,j+di,.)/kT

q2 K 2ckT 1 + Kd

-

SMI-I-

aa

1

(24)

which gives the dissociation over the cylindrical section of the larger micelles. Equations 23 and 24 are the essential results to emerge from our model of micelle structure. The details of the model enter via the exponents on the right, so we examine briefly the physical content of the various terms. The first is the electrostatic potential occurring in the surface dissociation equilibria derived previously for planar, macroscopic interfaces; it functions to convert the bulk concentration of counterions to that at the surface of the micelle. In the present model it depends not only on the structure of the double layer but also implicitly on that of the outer compact layer (via to and do)and of the inner layer as well (via the surface charge itself). In addition, however, eq 23 and 24 incorporate the structure of the inner layer explicitly in the capacitor term. This is a consequence of the partially covalent nature of the counterion head group bond. In essence it represents the energy change caused by the redistribution of charge that accompanies bond formation. This redistribution reduces the repulsion among the ion pairs assembled at the surface; i.e., it decreases the energy of the inner-layer capacitor (eq 14 and 15). Hence it favors the formation of ion pairs at the interface. Correspondingly, this term reduces the value of the exponents in the above equations so that lower degrees of dissociation are predicted. The next two terms in the exponents are corrections for the nonideal behavior of the counterions. For salt concentrations that are not too high the Debye-Hiickel term dominates. Its inclusion in the above equations raises the dissociation, as expected, since it takes account of the screening of the counterions in the bulk, which lowers their energy there. At high salt concentrations the salting out term becomes dominant. This favors binding because the water can be structured to such an extent that it no longer readily accommodates ions that dissociate. The find term in each exponent is related to the unfavorable salting out effect on the aggregate. If this is, in fact, not independent of the micelle charge but, say, decreases as the charge increases, then a higher dissociation is predicted. With eq 23 and 24 for the degrees of dissociation we have the basic relations needed to determine the remaining properties of the system, e.g., composition, cmc, and micelle size. If the explicit forms for the chemical potentials (eq 1-4) are substituted into the equilibrium conditions (eq

C~RXR)~~J+C~J)~(~MRXMR)~-~~ exp(-AW,/kT); CC~-J)"

j

>J (25)

where xi, xR, and XMR are the mole fractions of micelles of size j, of surfactant ions R-, and of undissociated surfactant molecules MR, fR and fm are activity coefficients; a, is the degree of dissociation for a spherical micelle, and a is the degree of dissociation for the cylindrical part of larger aggregates. The mole fractions xR and xMR are related to the mole fraction, XM,of counterions M+ by the following condition of dissociation equilibrium: XMXR/XMR = (fMR/fMfRlKdiss (26) where Kdissis the dissociation constant. The mole fractions xM, xR, and xMR, and the corresponding activity coefficients, are those in the "bulk solution", Le., those far from the nearest micelle where the electric field of the latter is negligible. The concept of bulk solution is meaningful as long as the micelles are sufficiently dilute, a condition easily satisfied for surfactant concentrations near the cmc. For sodium dodecyl sulfate in the absence of salt the cmc is 8.1 X mol dm-3,and the micelles have an aggregation number of about 60. If we assume that at the cmc 5% of the surfactant is aggregated, the micelle separation is greater than 600 A, which is large compared with their radius (25 A) and the Debye length (35 A at an ionic strength of 8 X mol dm-3). For this reason also there is no activity coefficient corresponding to micelle-micelle interactions in (25). The quantity AWj also appearing in (25) is the standard work of formation of a micelle from its constituent ions and molecules. This quantity depends on the structure assumed for the micelle, especially that of its interface with the solution. In the model used here A w j = -jaw,, + S ( U j - 21)j +

47F(rj + dhd

+ di)2Sou'ICj(rj+dhd+di,u') da' + jkT(aj In aj + (1in (1 - a,))+ jSjkTI; j I J cyj)

(27)

when the micelle is spherical and AWj =

+ dhd + di)ljSo'IC(r~dh,+d,a') du' + 0'- J)kT(a In a + (1 - a) In (1 - a))+ 0' - J)SjkTI + AW,; j > J (28) for larger micelles. Before the above equations can be used, the electric potentials ICj and must be calculated. Consider first $j. Since we have assumed the micelles to be in effect infinitely dilute, the potential in the double layer around a


Effect of Counterion Binding on Micellization spherical micelle depends only on the radial coordinate r. I t will satisfy the spherically symmetric Poisson-Boltzmann equation:

\

Inside each of the compact layers there is no charge density, and so the potential satisfies Laplace's equation, obtained from (29) by replacing the right-hand side by zero. The required solution to (29) is that satisfying the boundary conditions d$j

dr

+

0 as r

+

w

(30)

The second of these equations is, in effect, the relation given by Gauss's law between the electric field at the start of the double layer and the micelle charge. It involves the as yet undetermined degree of dissociation aj. This quantity is related to the potential at the surface of specific adsorption, $j(rj dhd+ di), by (23). To obtain a closed set of boundary conditions, we must relate this potential to that at the start of the double layer. This can be done by solving Laplace's equation, which holds in the outer compact layer, with the result $j(rj+dhd+di) = $j(rj+dhd+di+do) Pj + dhd + di + do d$j

+

d o -€0

rj

+ dhd + di

(

%)rj+di,d+di+d,,

(32)

where t is the bulk dielectric constant. The potential, $, around a cylindrical micelle satisfies the cylindrically symmetrical Poisson-Boltzmann equation

with the same boundary condition at infinity and the following at the start of the double layer:

Corresponding to (32) is the following equation:

1.52;

Figure 2. Geometry assumed for the sulfate head group (and first carbon atom of the alkyl chain). Numerical values are taken from ref 29.

available solution. Accordingly, Cdt may be regarded as the bulk concentration of anions X-,and (37) is then simply a statement of the charge neutrality of the bulk solution. Similarly, we may omit from the surfactant total (36), the double-layer excess of surfactant ions. The equations for the potentials and degrees of dissociation constitute a two-point boundary value problem, which must in general be solved numerically. To avoid the lengthy computation required, we will make use of approximate solutions of the nonlinear Poisson-Boltzmann equation in spherical and cylindrical geometries developed by White.% These provide explicit relations between the potential at the start of the double layer and its derivative there. Combined with (23), (311, and (32), or (24), (34), and (35), these approximate solutions of the nonlinear Poisson-Boltzmann equation enable the degrees of dissociation to be determined. In addition, expressions for the "charging integrals" appearing in (27) and (28) are also available. Equations 25-28 and 36 and 37 can then be solved for the mole fractions xR, xMR, and xj. From the mole fractions x j the various quantities characteristic of the micellar solution can be evaluated, e.g., number or weight-average micelle sizes Cijxj/C;xj and c;jzxj/ C;!xjand the mole fraction of aggregated surfactant C ~ J XSince ~ . the latter quantity depends on the monomer concentration, the cmc can be determined as that total amount of surfactant for which some small fraction, e.g., 5%, is aggregated. To implement this program of calculation for a specific case, values need to be fixed for the various parameters and the electrical potentials must be determined as functions of surface charge and ionic strength.

(37)

V. The Case of Sodium Dodecyl Sulfate Initially, we present what is known about the various parameters involved in our model of the micellar solution. In general the information available is insufficient to define their values with the precision necessitated by the sensitivity of the results. We are, therefore, forced to consider different values for some of the parameters, adopting those which bring predicted results into closest agreement with experiment. On the basis of the values thus required we are able to draw some conclusions regarding the adequacy of the model. Reasonable values for the various size parameters can be deduced from the known structure of the sulfate head illustrated in Figure 2, together with the bare and and hydrated radii of the sodium ion (0.98 and 2.76 the covalent radius of an oxygen atom (0.66 A).29 Thus dhd,the distance from the core to the surface of head group

where u, is the molar volume of pure water. In writing these equations we have neglected the mole fractions of the various components compared with unity. We have assumed earlier that the micelles are dilute, i.e., their double layers extend over only a small portion of the

(28) White, L. R. J. Chem. SOC.,Faraday Trans. 2 1977, 73, 577. (29) Cotton, F.A.;Wilkinson, G. Advanced Inorganic Chemistry, 4th ed.; Interscience: New York, 1980. (30) Anacker, E. W. In Solution Chemistry of Surfactants; Mittal, K., Ed.; Plenum: New York, 1979; Vol. 1.

The final equations needed relate the unknown mole fractions to the independent. variables specifying the composition of the system, namely, the total amount of surfactant and the amount of added salt. If these are given as numbers of moles per unit volume of water, CEWfand Cdt, then m

GUrf= (XR + X M R + CjXj)/u, 2 Csalt

= (XM - x d / u w

(36)


charge, equals 3.6 A. Since we allow the counterion-head group bond to be partially covalent, we assume that no water molecules intervene between a bound sodium ion and the oxygen atoms of the head group. Assuming further that the former is symmetrically located with respect to the latter, and as close to them as possible, we arrive at a value of 0.79 A for the inner-layer thickness di. To determine the width do of the outer compact layer we assume that the counterions of the double layer are fully hydrated and that even bound counterions retain a (distorted) part of their hydration sheaths, as is generally considered to be the case,21 The resulting value for do is 5.52 A. Finally, for the term taking into account ion size in the Debye-Huckel activity coefficient, we need an estimate of the distance of closest approach of opposite charges in solution. The exact value is unimportant since it has little effect on the results, and so we take simply 3.42 A, the sum of the radii of an oxygen atom (in the sulfate group) and a hydrated sodium ion. For the dielectric constant of the inner layer, ei, a value of 6 is sometimes quoted.I5 This assumes that any water present, in the vicinity of dissociated head groups, for example, is fully oriented by the strong electric field. If the degree of orientation is less than complete, as may be the case when there is a sufficient proportion of neutral head groups, a higher value is more appropriate. The dielectric constant of the outer compact layer, eo, will be intermediate between ei and the bulk value (78.3 at 25 OC). We will use the value 40.15 The salting out constant for the cation, SM,Le., the constant in the term in the chemical potential proportional to I, may be deduced from mean ionic activity coefficients measured for solutions in which the cation is Na+.17bIf we assume that, compared with the sodium ion, the anions in these solutions, being less strongly hydrated, are not salted out, we obtain S M = 0.256 d m mol-'. As regards the surfactant species, the contribution due to the salting out of the hydrocarbon chains can be estimated on the basis of a group contribution rule.17a Unfortunately, no data are available on the effect of the head group. To bypass this difficulty we assume that the associated and dissociated head groups are salted out to the same extent, which is not unreasonable at the high salt concentrations for which salting out is important, since there will then be a high degree of pairing between charged head groups and neighboring cations in the solution. Thus we take S M R = SR= 1.486 dm mol-'. Since the hydrocarbon chains of aggregated molecules are removed from contact with water, they are not salted out; therefore, ignoring the head group contribution, we have simply Sj = 0. The remaining parameters are not known with any certainty, and the values available should be regarded as a guide only. The constant s, related to the interaction between the micelle core and the aqueous medium, is consideredl to be in the range 20-33 cal mol-l A-2. As regards Kdiss,the only relevant information seems to be the estimate of mol dm-3 for the (molar) dissociation constant of the ion pair NaS04-,31equivalent to a value of 10-0,7u,for K h . The apolar hydrocarbon chain, which is, of course, also present in the molecule of concern, probably makes dissociation less easy, so we win consider lower values for Kdissas well. A value is also needed for the ratio 0 of the average chain length in the core to the fully extended length, which will be taken equal to unity. Finally, little is known about the degree of ionicity P other than that for a strongly elec(31) Davies, C. W. Ion Association; Butterworths: London, 1962.


1

1

I

150

A X

I

+ /

50

0.0

0.4

0.2

0.6

(MI Figure 3. cmc (mol dm-3) and weight-average aggregation number, j-, at the cmc versus the concentration of added salt, C ,, (mol dm-9. Theoretical curves are for ti = 8, K h = s = 35.8 cal mol-1A-2, and fi = 0.657 (values of the other parameters are as given in section V). Experimental pointa are those ~ ~Emerson ~ and H o l t ~ e r(+), ~~~ of Mysels and P r i n ~ e n(X), H u i ~ m a n(o), ~ ~Corti ~ and D e g i o r g i ~(m), ~ ~and Hayashi and Ikeda33(A). Csalt

Figure 4. Degree of dissociation versus concentration of added salt, Cdt (mol dm-3),for the same arameters as Figure 3 (ti = 8,K h = 10'%,,,, s = 35.8 cal mol-PA-2,and @ = 0.657). Curve a is the dissociation,aJ, at the surface of the largest spherical micelle; curve c is the dissociation, a,over the cylindrical section of large micelles. Curve b is the average djssociation over-the surfac_eof a micelle of aggregation number j-, Le., (JaJ+ (i44/je tropositive element such as sodium it should be close to unity. The cmc and the (weight-average)aggregation number a t this concentration are the quantities which have been most reliably determined for sodium dodecyl sulfate solutions. Their variation with the concentration of added sodium chloride has been investigated by several worke r ~ ,and ~ it ~ is* mainly ~ ~ with these observations that we will compare the predictions of our model. In our calculation the cmc is defined as the total concentration of surfactant at which 5% is aggregated. Also available are some data on the size variation for a fixed concentration of surfactant (6.9 X mol dm-3).33J4 (32) (a) Mysels, K.J.; Princen, L. H. J. Phys. Chem. 1959,63,1696. (b) Emerson, M. F.; Holtzer, A. J. Phys. Chem. 1967, 71, 1898. (c) Huisman, J. F. aa quoted by Spamaay, M. J. The Electrical Double Layer; Pergamon: New York, 1972. (d) Corti, M.;Degiorgio, V. J.Phys. ~. Chem. 19Si,86, 711. (33) Hayashi, S.;Ikeda, S. J. Phys. Chem. 1980,84,744. (34) Mazer, N.A.; Benedek, G. B.; Carey, M. C. J.Phys. Chem. 1976, 80,1075.

Langmuir, Vol. 4, No. 1, 1988 85

Effect of Counterion Binding on Micellization Figure 3 presents the best results of our model for the cmc and weight-average aggregation number, j W , as functions of salt concentration. To obtain them we have assumed that there is no restriction on chain extension (0 = 1) and have varied the parameters ti, 0, s, and Kdiss somewhat so as to optimize the fit to the experimental data. Since our main concern is to examine the effect of counterion binding on micellization, we also give, in Figure 4,the degree of dissociation, cyJ, of the largest spherical micelle, the corresponding quantity, a,for the cylindrical section of larger micelles, and thirdly the averag? degree of dissociation over the surface of a micelle of size j,. This last quantity is more closely related to the experimentally measured dissociation than are the other two. As can be seen, the observed cmc and micelle size behavior are quite well reproduced. Concerning the degree of dissociation, this is a difficult property to study experimentally. In general, considerable analysis is required to extract a value from the quantity that is actually measured, e.g., solution conductivity or the coefficient of self-diffusion, and so the results are to some extent model dependent. Some early reported degrees of dissociation as low as 0.14, but values twice as high were obtained when a different method was used.35 More recent values are even higher.30v32ds36In spite of the uncertainty surrounding the data, it is apparent that the calculated values are too low. Moreover, they vary significantly with salt concentration while experiment seems to show that the opposite is the case.37 However, we should make mention here of a point discussed more fully later: viz., the degree of dissociation is affected by the inner-layer dielectric constant. The low value used for the latter (ei = 8), as well as the neglect of its dependence on the electric field, may explain at least in part the above shortcomings. Indeed, as shown later, the calculated degree of dissociation becomes comparable to the experimental values if a larger value is taken for the above dielectric constant. Let us also note that the degree of dissociation is smaller for cylindrical than for spherical micelles. This happens because the distance between two neighboring head groups is greater on a spherical micelle and, as a result, the repulsive energy is smaller. The free energy contains contributions from the entropy and repulsion, and both are greater when a is larger. Its minimum leads, therefore, to a greater value of a for the sphere. Considerable insight into the behavior of the model may be gained by examining the magnitudes of the various contributions to the standard work of formation AW. (eq 27 and 28). Therefore, in Table I we present a breakdown of the standard work of formation per amphiphile of the largest spherical micelle, and of the cylindrical section of larger micelles, for two concentrations of added salt (0 and 0.5 mol dm-3). Also given are the quantities ApJbdk and Allbulk,which we define implicitly by rewriting the monomer-micelle equilibrium in the form xj

= exp[GApjbulk- AWj)/kT]; j IJ

+ 0'- J)Apbuk- AWj)/kT];

x j = exp[(JApJbUk

j

>J (38)

Thus, the combination of these quantities with the standard work of formation determines the aggregate size distribution and hence the properties of the system such (35)Stigter, D.; Mysels, K. J. J.Phys. Chem. 1965, 59,45. (36)Frahm, J.; Diekmann, S.; Haase, A. Ber. Bunsenges. Phys. Chem. 1980, 84, 566. (37)Lindman, B.;Wennerstrom, H. Top. Curr. Chem. 1980, 87, 1.

Table I. The Standard Work of Formation per Amphiphile (in Units of kT)and Its Various Components" no salt 0.5 mol dm-3 salt sphere cylinder sphere cylinder work of formation -10.767 -11.176 -10.863 -11.229 hydrocarbon chain -16.377 -16.377 -16.377 -16.377 2.278 4.052 core-water contact 4.052 2.278 inner layer 1.801 2.920 3.047 1.596 double layer 0.184 0.001 0.075 0.005 -0.199 -0.050 entropy of mixing -0.426 -0.138 0 0 0 0 micelle salting out A/LJ~'~,AhbUU' -11.093 -11.339 -11.283 -11.250

nComponents in the order in which they occur in (27) and (28) for spheres of the maximum allowed aggregation number, J , and for the cylindrical section of large micelles, for two different concentrations of added salt. The last row contains the quantities defined by (38). The values given are for the same parameters as in Figures 3 and 4.

as cmc, average micelle size, and the fraction of surfactant that is aggregated. By comparing (25) and (38) we see that ApJbdk and Apbdk are given by

In words, JApJbulkis thus the composition-dependent part of the free energy in bulk solution of the amphiphiles making up a spherical micelle of size J; 0'- J)Apbulk has a similar interpretation for the cylindrical section of larger micelles. If the activity coefficients are ignored, these quantities are the free energies corresponding to the bulk entropy of the monomers, which is lost on aggregation. The micelle size distribution can be expected to be very sensitive to the values of the exponents in (38). Given this fact, the results presented in Table I have some significant implications for theories of micellization. For example, Table I shows that the standard work of formation per amphiphile almost exactly cancels the free energies involving &Jb"" and &bulk; i.e., each of the exponents in (38) is the small difference of two large quantities. The main component in the work of formation is the energy released by the removal of hydrocarbon chains from water to the micelle core (the hydrophobic interaction energy), so Table I bears out the common notion that micellization involves a competition between monomer entropy and hydrophobic interactions. However, the outcome of this basic antagonism is controlled by the remaining effects, which are therefore of great importance even though they may be much smaller in magnitude. Table I implies that an adequate theory of micellization must take into account not only the "main" physical factors but also those contributing less than 1kT per amphiphile to the energy of a micelle. Such factors will presumably be very dependent on the basic assumptions made regarding the shape of the micelle and the structure of its interface with the aqueous medium. As a corollary, one expects results to be sensitive to the values of the various parameters involved in a given micellar model. This is illustrated for the present case by Figure 5. A feature of Table I worth noting is the minor contribution made by the double layer of the micelle to the standard work of formation. However, this must be considered in the light of the small value obtained for the degree of dissociation. For more reasonable a, and a values, as obtained later in the paper, the double-layer terms would be much larger and could be expected to play a significant role. In addition, let us note that the activity coefficients of the monomeric species play a role in the dependence of the degree of dissociation, critical micelle



lo-zA

CMC

150

I I

150

I

A

1

X

-

b

j wt

C

d

100 -

+ A

0.0

I

I

I

0.2

0.4

0.6

""

0.0

0.2

Figure k. cmc (mol dm-3) and weight-average aggregation number, j, at the cmc versus the concentration of added salt, Cdt (mol dm-3),for various seta of parameter values: curve a, the same values as for Figures 3 and 4 (ti = 8, K b = 10-l%w,s = 35.8 cal mol-' A-2, and p = 0.657); curve b, as for curve a but with Kdiss= 10-l.ouwinstead; curve c, as for curve a but with s = 35.6 cal mol-' A-z; curve d, as for curve a but with B = 0.662. For clarity the cmc for the parameters of Figures 3 and 4 is not shown; it lies between curves b and c. concentration, and average aggregation number on the ionic strength. After the large hydrophobic term, the most important contributions to the work of formation are those from the inner layer and from the contact between the micelle core and the aqueous medium. Each of these has a rather different value for spherical and cylindrical geometries, a fact of some significance since many of the micelle sizes determined experimentally are in the range for which the numbers of amphiphiles in the hemispherical end caps and the cylindrical central section are comparable. (For 8 = 1,the largest spherical micelle has an aggregation number of 55.) Because of the charge redistribution that occurs on formation of the counterion-head group bond, the inner-layer term favors binding (see (23) and (24)). Now, as discussed earlier, the values obtained for the degree of dissociation are too low. Partly responsible for this are the low values used for the dissociation constant Kdiseand the degree of ionicity @, the effect of a lower value for the latter being to increase the redistribution of charge. One may well ask why values for these parameters that are too low are required to obtain agreement with experimental crnc values and micelle sizes. The explanation is most likely that for more reasonable values of the degree of dissociation and degree of ionicity the present model overestimates the inner-layer contribution to the energy of a micelle. As can be seen from (27) and (28), reducing aj,a,and @ decreases the magnitude of the capacitor term. This term may be an overestimate because of the value taken for the dielectric constant of the inner layer, ep When the dissociation is low the electric field strength may be insufficient for dielectric saturation. Although our value of 8 for the dielectric constant corresponds to less than complete orientation of water molecules, it may still be too low. A higher value would reduce the magnitude of the capacitor term, allowing more reasonable values for aj,a, K h , and j3. By way of illustration, Figures 6 and 7 present the best resulta possible for 9 equal to 15. The dissociation is indeed higher, and the values of j3 and Kdiss(0.882 and

0.4

0.6

Csolt (M)

Csolt (MI

Figure S, cmc (mol dmT3)and weight-average aggregation number, j, at the cmc versus the concentration of added salt Cdt (mol dm-3),for ci = 15,Kh = 104.9uw,s = 35.4 cal mol-' A-< and 0= 0.882. The experimental points are the same as in Figure 3.

0.0

0.2

0.4 Csait

0.6

(M)

Figure 7. Degree of dissociation versus concentration of added salt, Cdt (mol dm-9, for the same parameters as Figure 6 (ti = 15, K& = 104%w,s = 35.4 cal mol-' A-z, and p = 0.882). Curve a is the dissociation CUJfor the largest spherical micelle; curve c is the dissociation, a,over the cylindricalsection of large micelles; curve b is the average dissociation over the surface-of a micelle of aggregation number j, i.e., (JCQ+ Gw - J)a)/j& 104.9u,, respectively) are closer to those expected, though the fit to the micelle size data is not as good as previously obtained. The dissociation once again decreases rapidly with salt concentration, but clearly this would be more moderate if the dependence of ti on electric field strength were included in the calculation. The above discussion reveals how important the inner layer of head groups and bound counterions is in determining the properties of the micellar solution. A more refined model of this region would have to include the discreteness of head group and counterion charge, electrical images, and the motion of bound ions as discussed by Stigter6 and Beunen and Ruckenstein' as well as the dependence of the dielectric constant on the electric field.? Overriding these details, however, is a more basic question that needs to be addressed, viz., that of the shape of the micelle as a function of ita aggregation number. It seems inevitable that small micelles should be spherical. Larger micelles have been shown e ~ p e r i m e n t a l l yto~be ~ prolate rather than oblate and so if they are not the result of secondary aggregation of spherical micelles, as was suggested,32band if they are sufficiently large, they must have a cylindrical central region. It is the shape of the ends, and hence of micelles of intermediate size, that is less certain, as is evidenced by the variety of assumptions

Langmuir, Vol. 4, No. 1, 1988 87


_ _ - _ _ _ - - - - --1-"O

7w io2 0.0

0.2

0.4

Csolt 0.0 I

30

-

90

60

7 120

0

150

i Figure 8. Degree of dissociation and electrostatic potential (mV)

at the surface of specific adsorption versus micelle aggregation number j in the absence of salt. For j > J = 55,the quantity plotted as the dissociation is the average over the surface of the micelle, i.e., (JaJ+ 0' - J)a)/j.Also shown are the values of the for the cylindrical section of potential (- - -) and dissociation large micelles. The average dissociation approaches the latter quantity as j increases. The parameter values used are the same as for Figures 6 and 7 (ti = 15,Kdb= 10-0.9uw,s = 35.4 cal mol-' A-2, and B = 0.882). (-e)

made in this regard.1*34s38Moreover, there are indications that the ends have an important bearing on the properties of the system. As mentioned earlier, unless the surfactant concentration is high, the values obtained for the aggregation number place the micelles in the transition region between spherical and cylindrical aggregates. Secondly, Table I shows that most of the contributions to the work of formation per amphiphile are different for the hemispherical and cylindrical sections of large aggregates. Because of the cancellation occurring in the exponents of (39),these differences are highly significant. Thus,we have been able to quantify the general observation of some earlier workers39that the size distribution is likely to be sensitive to that part of the micelle free energy coming from the amphiphiles at the ends. It also follows that the shape assumed for the micelles will be of great importance. Properties of the micelle surface that are accessible to experiment include the degree of dissociationmand, more recently, the surface p ~ t e n t i a l .In ~ ~the measurement of these quantities it is commonly assumed not only that they are constant over the surface of the molecules but also that they are independent of its size. However, the above results would lead us to question these assumptions. Some idea of the variation to be expected in the surface properties can already be gained from the present model in spite of its crude treatment of micelle shape. This is illustrated by Figure 8, which gives the degree of dissociation and the potential at the surface of bound counterions as functions of aggregation number when there is no added salt. As noted earlier, the degree of dissociation determined experimentally does not appear to vary significantly as salt concentration, and with it the average micelle size, increases. The results of Figure 8 are not necessarily inconsistent with this since the effect of salt concentration on the degree of dissociation is a combination of several opposing trends which may largely compensate each other. Thus, for a fixed salt concentration a larger micelle has a more negative surface potential, because of the smaller area per head group, and so a lower degree of dissociation (see (23)). On the other hand, increasing the salt con(38) (a) Ruckenstein, E.;Nagarajan, R. In Micellization, Solubilization and Microemulsions; Mittal, K., Ed.;Plenum: New York, 1977; Vol 1. (b) Nagarajan, R.; Ruckenstein, E. J. Colloid Interface Sci. 1979, 71, 580. (39) Mukerjee, P.J . Phys. Chem. 1972, 76, 566.

0.6

(MI

Figure 9. Weight-average aggregation number at a fixed concentration of surfactant (6.9X mol dm-3) versus the concentration of added salt, Cdt (mol d m 9 The parameter values are as for Figures 3 and 4 (ti = 8, K b = l(rl%w, s = 35.8cal mol-' A-z, and @ = 0.657). The experimental points are those of Mazer et aLU As explained in the text, the neglect of micellar interaction leads to computed aggregation numbers greatly in excess of ex-

perimental values.

centration makes the potential less negative at the surface of a micelle of a given size and charge density. Acting alone, this effect would lead to a greater dissociation, but an increase in salt concentration of course also provides more counterions for binding. While the interactions occurring at the micelle-solution interface are of prime importance in determining the micelle shape and size distribution, and thereby the properties of the system, the state of the hydrocarbon core may also play a significant role, a fact which is not widely recognized. Since the early work on micellar models, it has been assumed that the core is in a fluid state resembling bulk hydrocarbon. However, more recent work on chain conformation in micelles4reveals that the requirement that the chains be continuous and fill the available volume imposes on the core a considerable degree of structure. Moreover, the constraints of geometry are clearly different for cylinders and spheres. Thus, the distributions of chains over allowed conformations will be different in bulk hydrocarbon and in cores of different shapes. That such differences will affect micelle formation becomes evident when one considers the energetics of chain conformation.40 The work needed to introduce a gauche rotation into an initially all-trans chain is about 400 cal mol-l (0.675kTl molecule), while that involved in producing two consecutive gauche rotations of opposite sense is 2200 cal mol-' (3.72kT/molecule). Clearly such energies are significant compared with the contributions we have considered. In addition to chain conformation,an improved theory should also take into account the translational and rotational motion of amphiphiles within the aggregate.41 A final question we consider is that of micelle interactions. Several authors42have treated electrostatic interactions, and some previously unexplained effects can now be understood, e.g., the variation in monomer concentration above the cmc. A t higher salt concentrations electrostatic effects are reduced and the van der Waals component of micellar interaction can be determined?%* The effect of volume exclusion, not usually considered, may then also become evident. At high concentrations of surfactant, micelle size may be large, as indicated by the (40) Flory, P. J. Statistical Mechanics of Chain Molecules; Interscience: New York, 1969. (41) Nagarajan, R.; Ruckenstein,E. J. Colloid Interface Sci. 1977,60, 221. (42) (a) Gunnarsson,G.;Jonaeon, B.; Wennerstrom,H. J. Phys. Chem. 1980, 84, 3114. (b) Mille, M.; Vanderkooi, G. J . Colloid Interface Sci. 1977, 59, 211. (43) Corti, M.; Degiorgio, V. In Light Scattering in Liquids and Macromolecular Solutions; Degiorgio, V., Corti, M., Giglio, M., Eds.; Plenum: New York, 1980.



results of Hayashi and Ikeda33 and Mazer et al.34 (see mol dms. If Figure 9) for a concentration of 6.9 X large micelles are rigid rods, their numbers at this concentration are such that there is appreciable steric interaction among them. If we assume for the purposes of illustration that the micelles are monodisperse with an aggregation number of 500, then the spacing between their centers at the above concentration is about 240 A. From the known dimensions of the surfactant molecule it follows that the micelle length is already greater than 210 A. An aggregation number of 500 is high but typical of those found at high salt concentrations. In the present calculations, we take into account neither volume exclusion nor van der Waals and double-layer interactions among micelles. As is evident from Figure 9, the actual micelle size at a high surfactant concentration is much smalIer than that predicted by our model, suggesting that the above factors play a role. Indeed, at higher surfactant concentrations the decrease in entropy generated by the volume exclusion of the large micelles can be minimized by the formation of a larger number of smaller aggregates. The effect of volume exclusion among micelles is to reduce their translational and orientational entropy. If the reduction is great enough, phase separation may take place, forming a more dilute micellar solution in equilibrium with a concentrated surfactant phase. The greater entropy of the former outweighs the lower entropy and the less favorable interactions in the latter, and a lower free energy for the system as a whole is thereby achieved. Indeed, as soon as the aggregation number becomes sufficiently large a surfactant phase is observed to form.34 The volume exclusion of rodlike micelles recalls the much-studied Onsager transition in which fluids containing anisotropic particles interacting via repulsive forces separate into two phases: one concentrated and ordered and the other dilute and disordered.M Here again, the greater entropy of the dilute phase is sufficient to offset both the greater repulsion and the lower entropy in the concentrated phase, and so the system attains a lower free energy. It is interesting to speculate whether the phase separation occurring in the micellar solution may not be a modified form of the Onsager transition. For example, large micelles are anisotropic, and it appears that the new phase forms when the volume exclusion among them becomes significant. On the other hand, the systems for which Onsager transitions have been observed contain particles which retain their identity under all conditions while micelles, being aggregates of smaller molecules, may break up. Furthermore, attractive and repulsive interactions among the surfactant molecules, as well as their unfavorable hydrophobic and favorable hydration interactions with water, play a role in determining the nature of the new phase that forms. Indeed, this appears to be a crystalline hydrate rather than an ordered liquid crystalline structure. Let us also note that the degree of dissociation of the micellar surface is decreased by the presence of intermicellar repulsive interactions. Indeed, this minimizes the free energy of the system since, while the smaller number of independent particles involved decreases the entropy, the repulsive interactions are decreased as well. VI. Conclusion In this paper we have applied a theory for the effect of ion binding on micellization to the case of sodium dodecyl sulfate solutions containing added sodium chloride. Agreement with experiment is good for both the cmc and ~~~

~~

(44) Onsager, L. Ann. N . Y . Acad. Sci. 1949, 51, 627.

the aggregation number at the cmc as functions of the concentration of added salt. The values for the degree of dissociation, however, are too low. Small changes in the values of the parameters lead, however, to reasonable values for the degree of dissociation, but the values predicted for the average aggregation number are not as good. Our calculation reveals that the (negative) standard work of formation of the micelle almost exactly compensates the free energy corresponding to the entropy in bulk solution of its constituent monomers, which is lost on aggregation. As a result, the micelle size distribution is sensitive even to small contributions to the work of formation. The values of the various contributions for cylinders and spheres of the largest size allowed are compared and found to be significantly different given the closeness of the compensation referred to above. From these results we may draw some important conclusions for theories of micellization. Firstly, the shape of the micelles must be properly chosen since it significantly affects their energy and hence the size distribution and ultimately all the properties of the system. Secondly, it is necessary to determine accurately the various contributions to the work of formation. The present calculation provides an illustration in that overestimation of the inner-layer contribution is the likely c a m of the low values obtained for the dissociation. Partly responsible for this overestimation is the neglect in our model of the dependence of the dielectric constant ei of the inner region on the electric field strength. Properties of the micelle depend appreciably on its size and shape, and this must be taken into account. Furthermore, effects which have previously been considered negligible may have to be included. We show, for example, that the conformation of chains in the core should be taken into account in calculating the energy of the micelle. Finally, we consider interactions among the micelles and demonstrate that at higher surfactant concentrations volume exclusion effects become appreciable and may be responsible for the formation of a new phase.

effective area per amphiphile in a spherical micelle of aggregation number j 5 J, ai = 4r(r, S 2 / j A2, where 6 (‘3 A)’ is an allowance for surface roughness and the finite size of water molecules concentration of added salt (mol dm+ of water) total concentration of surfactant (mol dm” of water) length of the surfactant head group, i.e., distance from the first alkyl carbon atom to the surface of head group charge thickness of the inner and outer compact layers, respectively distance of closest approach of opposite charges in solution activity coefficient of the counterion; In f M = -(q2/ ( 2 € k n ) ( K / ( 1+ Kd)) + SMI activity coefficient of the surfactant molecule; In f R = -(q2/(2tkT))(K/(1 Kd)) + &I activity coefficient of the surfactant molecule; In f&fR = SI, ionic strength; I = X M / V , maximum aggregation number for a spherical micelle, for which rJ = e(1.5 + 1.265nc) A Boltzmann’s constant bulk dissociation constant for the surfactant length of the cylindrical section of a micelle of size j > J; 1, = 0’ - a ( 2 7 . 4 26.9nC)/dJA number of carbon atoms in the surfactant chain magnitude of the electronic charge radius of a spherical micelle of aggregation number j 5 J; r, = E(27.4 26.9n,)3/4~]’/~ 8,

+

+

+



a

K

e

3

energy per unit area of core-water contact salting out constants for the counterion, surfactant ion and molecule, and micelles of aggregation number j , respectively absolute temperature (298 K) molar volume of water (0.018dm,/mol) decrease in energy on removal of a hydrocarbon chain from water to micelle core standard work of formation of a micelle of aggregation number j mole fractions of counterions, surfactant ions and molecules, and micelles of aggregation number j , respectively degree of dissociation a t the surface of a spherical micelle (i IJ) degree of dissociation over the surface of the cylindrical section of large micelles (i > J) degree of ionicity of the counterion-head group bond dielectric constants of the inner and outer compact layers and the bulk solution composition-dependent part of the free energy in bulk solution of the amphiphiles in a spherical micelle of aggregation number J and in the cylindrical section of large micelles, respectively reciprocal of the Debye-Huckel screening length; K~ = 8rq21/ekT ratio of the maximum allowed length of an alkyl chain in the micelle core to the fully extended length electrostatic potential as a function of distance from the center of a spherical micelle of aggregation number j electrostatic potential as a function of distance from the axis of a cylindrical micelle charge on a spherical micelle of aggregation number j per unit area of the surface of counterion charge; u, = -a,jq/4r(r, + dhd + di)l the corresponding quantity for the cylindrical section of large micelles; a = -a(i - J)q/2n(rJ+ dhd + dill,

Appendix We present a thermodynamic derivation of the general dissociation equilibria, (6) and (7). We consider a system consisting of nswf,nx, and nw,moles of surfactant MR, added salt MX, and water at a pressure p and temperature T,respectively. The surfactant will be in the form of ions R-, neutral molecules MR, and aggregates of (in principle) any size j (j = 2,3, ...). AIso present are counterions M+ from the added salt and the dissociation of surfactant. The numbers of moles of these species are denoted by nR, n m , n. and nM,respectively. The aggregates are, of course, ccarged as a result of dissociation of some of their constituent molecules. Our treatment will be specific to aggregates of the form assumed in the text. Thus, small aggregates, being spherical, can be characterized by a degree of dissociation, a,.For those too large to be spherical 0’ > J),the state of ckarge is specified by two degrees of dissociation: one, aJ,for the hemispherical end caps and a second, a,for the cylindrical section. These last quantities, aJ and a,are assumed to he independent of aggregate size j . The quantities nR,nm, nj,aj,and a cannot be arbitrarily specified. On the contrary, they are constrained to satisfy conditions of surfactant and charge conservation: m

nsurf= nMR + nR + Cjnj 2

J

m

2

J+l

nx = nM - nR- C a j n j -

[ a d+ a(j - J)]nj

(AI) (A2)

For such a constrained system we can calculate a free energy from the corresponding partition function G = G(nM,nXinR,nMR,nl,nw,P,T,aj,cr) (A3)

This free energy, and the related electrochemical potentials, will equal the “real” free energy and electrochemical potentials of the system when nR,nm, nJ,nM,aj, and LY assume the values that actually occur. By the usual methods of statistical mechanics, it follows that the latter will be the values that minimize the form (A3) for the given p and T subject to the constraints (Al) and (A2). We carry out this minimization in two stages. Firstly, ajand a are held fixed, and G is minimized with respect to n,, nm, nl, and nM.This leads to the reaction equilibria, (8) and (9) of the text. These, together with the conditions (Al) and (A2), determine nM,nR,nm, and nJas functions of the independent variables nsd, nx, nw,p , and T for the given values of aj and a. In the second stage, the free energy resulting from the substitution of these functions in (A3) is minimized with respect to the choice of aj and a. It is this step that will lead to the equations we seek. If G is to be a minimum over all values of aJ,its derivative with respect to aj at constant nsd, nx, n,, T , and p must be zero: o = -dG -

-

daj

where aG/aaj is present due to the explicit dependence of G on the degree of dissociation. Using (8) and (9) we have -dGda;

-

an,

ani +-an,, + C (1- a&--daj + ~ =CJ + I[i - a d J

(A5) From the constraints on the minimization, (Al) and (A2), we see that the term in the first set of parentheses equals inj while that in the second must vanish. So the condition becomes

o = -aaj aG + jnjp,;

j

Effect of counterion binding on micellization - Langmuir (ACS

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