Critical Micelle Concentration and the Size Distribution of Surfactant Aggregates

Transcription

1 2872 J. Phys. Che. 1980, 84, Critical Micelle Concentration and the Size Distribution of Surfactant Aggregates A. Ben-Nalt and F. H. Stllllnger Bell Laboratorles, Murray Hill, New Jersey (Recelved: June 8, 1980) A siple aggregation odel for aqueous surfactant solutions is introduced to exaine the relation between critical icelle concentrations and the size distribution of olecular aggregates. This odel assues ideal solution behavior for the aggregates and lends itself readily to nuerical study. For ranges of paraeter values that yield identifiable critical icelle behavior we find that the critical icelle concentration lies far above the concentration at which the size distribution changes fro onotonically decreasing to nononotonic. This provides a vivid counterexaple to the Ruckenstein-Nagarajan suggestion that these properties universally should occur together. 1. ntroduction The critical icelle concentration (cc) exhibited by any surfactants in water represents an abrupt change in the solution fro a state with ostly unassociated olecules to a state with large olecular aggregates or icelles. This transition can be onitored by corresponding abrupt changes in any of a wide variety of properties, including osotic pressure, electrical conductance, light scattering, viscosity, and dye solubilization. Micelles typically have sizes ranging fro twenty to several hundred surfactant olecules. Recently Ruckenstein and NagarajanZ4 proposed an intriguing connection between the occurrence of the cc on the one hand, and a qualitative change in the size distribution of aggregates on the other hand. At very low total surfactant concentration the aggregate size distribution decreases onotonically with increasing size, i.e., onoers are the ost frequently encountered species, diers are less frequent, triers even rarer, etc. However, as the total surfactant concentration increases a point is reached at which this onotonicity disappears. Subsequently, the aggregate size distribution continues to decline with size only for sall sizes,.passes through a subsequent iniu and axiu, and then reverts to onotonicity. Ruckenstein and Nagarajan identify the transition between these two regies (the concentration at which the size distribution displays a horizontal point of inflection) with the cc. t is our ai in this paper to test the Ruckenstein-Nagarajan proposal for an eleentary aggregation odel. For present purposes it is sufficient to suppose that the surfactant solution is ideal with respect to all aggregation species (this assuption is iplicit in ref 2-4). Then the concentration pi of i-olecule aggregates can be written Pi = Kip? (1.1) where p1 is the onoer concentration and Ki is an association constant related to the standard Gibbs free energy of foring the aggregate fro onoers: AGio = -RT n Ki (1.2) The total surfactant concentration is obviously given by the expression Pt =: ipi = C ikip: c=1 i=l while the ideal-solution osotic pressure is OD (1.3) f Peranent address: Departent of Physical Cheistry, The Hebrew University of Jerusale, Jerusale, srael ~4taoi2oa4-2a72$0.wo /RT = pi = Kip: (1.4) i-1 i=l The specific odel eployed by Ruckenstein and Nagarajan has its source in Tanford s about the various contributions to AGiO. Details are uniportant here. The generic for assigned to AGio is the following (at least for nonionic surfactants): AGiO ai413 - pi + yizf3 (1.5) where a, 6, and y are positive paraeters that depend on teperature and the olecular structure of the surfactant. t is the difference in sign between ters in AGio which perits occurrence of nononatonic behavior for the pi vs. i at sufficiently large pt. The specific calculations carried out by Ruckenstein and Nagarajan assign definite values to a, p, and y for a series of surfactants, and then copare experiental cc s for those substances with total concentrations pt at which the aggregate size distribution pi exhibits a horizontal point of inflection. The agreeent is quite rearkable. Unfortunately there was no siultaneous deonstration that any property such as the osotic pressure undergoes the required abrupt change at these crnc s. Nevertheless, the results seeed favorable enough to offer hope that a precise atheatical criterion for the cc in ters of the size distribution could aid in resolving odest discrepancies that arise in deterining this quantity fro distinct experiental techniques. 2. Eleentary Association Model One can view the foration of an i-olecule aggregate (Ai) as the result of a stepwise association, whereby onoers (M) are sequentially added. For the reversible step Aj-1 + M Aj (i 1 2) (2.1) the equilibriu constant ay be denoted by kj = P j / b j - d (2.2) then the previously introduced Ki are given by Ki = f kj (2.3) j=2 f the standard free energy were the sae for each aggregation step (2.1) then all kj would have a coon value, e.g., k, and so Ki = ki-1 (2.4) With such a unifor stepwise association it is not possible Aerican Cheical Society

2 Surfactant Aggregates cc and Size Distribution to produce n nononotonic size distribution for aggregates. Furtherore, calculations of the type described below show that this unifor stepwise association produces no abrupt changes in physical properties which could reasonably be identified with a critical icelle phenoenon. n order to odel the behavior of surfactants realistically, the kl ust evidently display a strong nonunifority with respect to j, so that the resulting Ki will deviate arkedly (and in the proper anner) fro the for shown in eq 2.4. n particular, the Ki should exhibit a preferential range of icelle sizes over which these constants are substantially larger than ight be indicated by the trend established by stepwise association of sall nubers of surfactant olecules. The physical reason is well-known qualitatively, naely, that after achieving a certain critical size the aggregates are structurally capable of hiding their hydrophobic units copletely fro the water in the icellar interior. This advantage is siply not available to sall aggregates of, e.g., two to ten onoers. The Ruckenstein-Nagarajan Ki)s, based on free energy expression 1.5, iplicitly incorporate this sort of nonunifor association behavior, as the resulting size distributions deonstrate at sufficiently high overall concentration. We ight note in passing that other calculations of aggregate size distributions, such as those by Hoeve and Benson and by Poland and Scheraga,8 also incorporate nonunifor stepwise association which, like that of the Ruckenstein- Nagarajan study, produces nononotonic size distributions with a single axiu at a ost probable icelle size. t has not been our ai in the present study to develop a coplete olecular theory of surfactant solutions. nstead, we have been content to exaine the iplications of a phenoenological, but physically reasonable, faily of nonunifor association constants. n particular we have investigated the case where Ki = k lf(i) (2.5) with the following odulation function F: F(i) = 1 (i = ) (1 < i 5 n,) = exp[a - (i - n*)2/u2] + 1 = 0 (i > naj (2.6) Here A, n*, u, and nar. are paraeters which deterine - - the character of the odulation. The previous case of unifor association is recovered in the liit A -a, nax +OD. For finite values of these paraeters, however, there will be an enhanceent of the tendency to for aggregates of sizes near to n*. The quantity u controls the range of sizes over which enhanced aggregation occurs, while A controls the strength of the enhanceent. No aggregation is peritted to occur beyond size n=; while this absolute cutoff ay see arbitrary it is coputationally convenient below, and only is peritted to occur beyond the size range of physical relevance. While no fundaental significance fro the standpoint of surfactant scieiice can be attached to the Gaussian function invoked in eq 2.6, its general qualitative properties are easily understood. Furtherore, we clai that the resulting Kj can (with proper choice of paraeters A, n*, u, and n) closely iic the association constants that would follow fro a ore detailed olecular theory such as that which yields AGio in eq 1.5. For our odest purposes eq 2.5 and 2.6 are quite sufficient. f the Ruckenstein-Nagarajan hypothesis has the universality claied, then in particular it ust be valid for the odel association process represented by eq 2.5 and 2.6. t is this presuption that we set out to test quantitatively. The Journal of Physical Cheistty, Voi. 84, No. 22, Nuerical Procedure n order to siplify nuerical analysis of our association odel we found it advisable to use the following reduced variables in place of pl, pt, and osotic pressure n: x =kp1 y=kpt f =kr/rt (3.1) As a result the basic equations becoe nnln y = i=l C i F(i)xi (3.2) f = F(i)xi (3.3) i=l For a given set of paraeters A, n*, u, and nar our procedure involved first selecting a value for y and then solving the resulting polynoial equation (3.2) for that real positive root x1 which has the property li (y/xl) = 1 (3.4) Xl-9 The substitution of this root xl into eq 3.3 then yields the osotic pressure (reduced). The separate ters in the f expression are in fact the concentrations of aggregates of different sizes, so we obtain the size distribution fro the set of quantities xi = kpi = F(i)x) (3.5) Our decision to treaty rather than x as the independent variable stes fro the experiental fact that the total surfactant concentration pt is under control, not the onoer concentration. The reverse procedure (x the independent variable, y dependent) can inadvertently lead one to consider states with absurdly large total surfactant concentrations. n all of the calculations to be reported here we have set n* = 40 u = 3 nax = 50 (3.6) The paraeter A was given five distinct values: A = -a, 50,60,70,80 (3.7) For each A the reduced total density y was assigned 100 equally spaced values in the range 0 < y 0.5. A series of exploratory calculations with alternative n*, u, and nax choices has also been carried out. No behavior essentially distinct fro that to be reported below for the set in (3.6) was uncovered. 4. Results Figures 1-5 show logarithic plots of the aggregate size distributions for each of the A choices in eq 3.7. The first of these choices produces straight lines since this case has F 1 for all sizes below the cutoff. But as we switch on the Gaussian function by increasing A the influence of the preferred aggregation size becoes obvious. And indeed we then observe the typical behavior that has been reported by previous a~thors,~~~j*~ naely, that onotonic distributions at low total concentration convert to nononotonic distributions at higher total concentration. We note that when a axiu in the size distribution first occurs it does so at a size substantially less than n*. But this axiu position shifts closer to n* as the total concentration increases. Figure 6 shows how the reduced onoer concentration x1 varies with y. When A = -a there is only a sooth featureless increase. But as A increases toward the axiu value 80 a region of large curvature develops, separating regies with distinctly different slopes. We

3 2874 The Journal of Physical Cheistty, Vol. 84, No. 22, 1080 Ben-Nai and Stillinger 01 x C P Figure 1. Aggregate size distributions for A = -03. The curves are labeled by the values of reduced total concentration y. For this case the curves are strictly linear for n nu = 50. Figure 3. Aggregate size distributions for A = 60, labeled with the Figure 2. Aggregate size distributions for A = 50. The labels are the identify this abrupt change in slope with icelle foration, and following PhillipsQ it will be convenient to define the cc as that y value for which i ld2xl/dy21 = axiu (4.1) Notice that as A increases the sharper the transition becoes, and the lower the y value at which it appears. The cc values obtained by nuerical differentiation of xl(y) Figure 4. Aggregate size distributions for A = 70, labeled with the TABLE : Paraeters Coputed for Micelle Aggregation Model A y (cc) x, (cc) d2x,/dyz icc (RN) are listed in Table, first row, where they are denoted y(cc). Figure 7 presents curves for the reduced osotic pressure quantity f plotted vs. y. Once again we see the 1

4 Surfactant Aggregates cc and Size Distribution The Journal of Physlcal Chestty, Vol. 84, No. 22, " Figure 5. Aggregate size distributions for A = 80, labeled with the Flgwe 7. Reduced osotic pressure f vs. reduced total concentration. "'" Figure 6. Monoer concentratlon x, vs. total concentration y, each in reduced units. eergence of an increasingly abrupt transition as A increases. The positions of axiu curvature in these curves stand in substantial agreeent with the cc's listed in Table fro criterion 4.1. Fro eq 3.2 and 3.3 it is easy to see that the ean aggregation nuber is given by (i) =./f (4.2) This quantity is displayed as a function of y in Figure 8. As before, increasing A tends to sharpen the transition Figure 8. Average aggregation nuber ( ) vs. reduced total concentration y. between regions of distinctly different slopes, and the position of axiu curvature once again corresponds very closely to the cc values listed in Table. n addition to the cc values of y, Table also lists the values of y [denoted by y(rn)] at which the size distribution satisfies the Ruckenstein-Nagarajan criterion of a horizontal inflection point. These were obtained by siple interpolation fro our nuerical results at preassigned y values. The table also gives values of x1 and of d2xl/dy2 at y (crnc). 0

5 2870 J. Phys, Che. 1980, 84, a Figure-9. Logarithlc plots of cc and the Ruckenstein-Nagarajan concentration vs. the aggregation strength paraeter A. 5. Discussion and Conclusion t is obvious fro the first two rows in Table that the Ruckenstein-Nagarajan criterion fails to locate the cc s for the present aggregation odel. Furtherore, the failure is quite draatic since y(cc)/y(rn) tends to be about 20. Evidently, the desired logical connection between the cc and the occurrence of a horizontal inflection point in the aggregation size distribution cannot universally be realized. Ruckenstein and Nagarajan have found agreeent fro their theoretical calculations (predicated upon their inflection point criterion) and experiental cc s for hoologous series of surfactants (hexoxyethylene glycol onoethers,2 and alkyl glucosides and sodiu alkyl sulfates3). n view of the present work it appears that the agreeent ay have been fortuitous. t is not at all clear that their odel would exhibit conventional cc behavior (abrupt breaks in properties vs. total concentration) any- where close to the occurrence of horizontal inflection points. The fact that apparent agreeent in the Ruckenstein- Nagarajan studies extends over hoologous series of copounds is not in itself persuasive. Our calculations show that while y(cc) and y(rn) ay substantially disagree they nevertheless tend to behave in siilar fashion with respect to strength of aggregation. n hoologous series of Surfactants this strength is priarily deterined by the chain length n, of nonpolar hydrocarbon oieties. The corresponding quantity in our siple odel is A, which should roughly be proportional to n,. Figure 9 shows plots of ln y(cc) and n y(rn) vs. A, prepared fro entries in Table. The curves are linear (as are plots of experiental values for n pt(cc) vs. n,1o) and close to parallel. f this parallelis is generally the case for all aggregation odels, then fortuitious agreeent for one eber of a hoologous series would iply agreeent for the reaining ebers. We are forced to conclude that if the cc is to be universally related to soe specific attribute of the aggregate size distribution then that attribute ust be ore subtle than ere occurrence of a horizontal inflection point. References and Notes (1) 0. S. Kresheck, Sufactants n Water, A Coprehensive TreaW, Vol. 4, F. Franks, Ed., Plenu, New ork, (2) E. Ruckenstein and R. Nagarajan, J. Phys. Che., 79, 2622 (1975). (3) R. Nagarajan and E. Ruckenstein, J. Colbkf nterface Sci., 80, 221 (1977). (4) E. Ruckenstein, Proceeding of Syposiu on Surface Active Agents, Nottingha, U.K. Sept 1979, Society of Chelcal ndustry, to be published. 5) C. Tanford, The Hydrophobic Effect, Why, New ork, ) C. Tanford, J. Phys. Che., 78, 2469 (1974). (7) C. A. Hoeve and 0. C. Benson, J. Phys. Che., 81, 1149 (1957). (8) D. C. Poland and H. A. Scheraga, J. Phys. Che., 89, 2431 (1965). (9) J. N. Phllllps, Trans. Faraday Soc., 51, 561 (1955). (10) K. S. Blrdl, n Micelllzation, Solubilization, and Mlcroeulslons, Vol. 1, K. L. Mktal, Ed., Plenu, New ork, 1977, p 151. Direct Observation of the Kinetic Behavior of a Charge-Transfer Reaction on the Cation-Scavenging Reaction for Cation Radicals of N-Vinylcarbazole and N-Ethylcarbazole by Using a Pulse Radiolysis Technique M. Washio, S. Tagawa, and. Tabata Nuclear Engineerlng Research Laboratory, FacuRy of Engineering, University of Tokyo, Shirakata Shlrane 2-22, Tokal-ura, barakl , Japan (Received: Deceber 3, 1979; n Final For: May 8, 1980) The cation-scavenging reaction hi been directly observed by using the pulse radiolysis ethod for cation radicals of both N-vinylcarbazole (VCZ+.; the initiating species of radiation-induced cationic polyerization of VCZ) and N-ethylcarbazole (EtCZ+., nonpolyerizing odel copound) in order to elucidate the role of inhibitors in radiation-induced cationic polyerization. The rate constants for the reactions of VCZ. with triethylaine, diphenylaine, and diethylaniline in nitrobenzene solution have been deterined to be 1.3 X lo9, 2.0 X lo9, and 3.9 X lo9 M- s-l, r espectively, which are alost the sae as those for EtCZ+. with these three scavengers. ntroduction t is well-known that pulse radiolysis is a very powerful ethod to observe directly the initial species fored by irradiation. ts first application to a polyerizing syste was ade in the polyerization of a-ethylstyrene by Katayaa et a1.l Cationic polyerization of N-vinylcabazole (VCZ) was studied by eans of the puise radiolysis technique, and /80/ $01.OO/O Aerican Cheical Society