Langmuir 1999, 15, 3483-3485
3483
Polydispersity of Sodium Dodecyl Sulfate (SDS) Micelles Denver G. Hall Northeast Wales Institute, Plas Coch, Mold Road, Wrexham, LL11 2AW, UK Received November 18, 1997. In Final Form: January 28, 1999 It has been argued previously that the effect of NaCl on the aggregation number of sodium dodecyl sulfate (SDS) micelles is inconsistent with the observation that SDS micelles are fairly monodisperse. This inconsistency is resolved by arguing that the “effective micellar degree of dissociation”, R, decreases with increasing aggregation number to an extent that cannot be detected from the effect of electrolyte on the critical micelle concentration.
Introduction Work based on a variety of techniques indicates that the aggregation number of sodium dodecyl sulfate (SDS) micelles increases significantly with increasing surfactant concentration and with addition of electrolyte.1-4 In particular the following expression has been forwarded4
N ) Kmn2
(1)
where N denotes the average aggregation number, m2 is the “so-called” free counterion concentration, (vide ultra), and K and η are constants. For SDS, η ∼ 0.25 and K ∼ 162. Other work5-7 indicates that despite this increase the micelles are fairly monodisperse. It has also been argued8 that, taken together, these two conclusions are inconsistent with thermodynamic arguments and that a critical evaluation of the issues involved is required.2 The aim of this paper is to analyze and resolve this alleged inconsistency. In particular, it is shown that the behavior summarized in eq 1 and the extent of polydispersity cited in refs 5-7 can both be accommodated within the thermodynamic framework described in refs 9 and 10. To do this it is argued that the effective micellar degree of dissociation, R, can be expected to decrease with increasing aggregation number. The changes in R required are shown to be sufficiently small that they are unlikely to show up in estimates of R obtained from the effect of electrolyte on the critical micelle concentration (CMC).
is easier to use, that in ref 10 is preferable for other reasons. In the present context, this difference is unimportant. The version used by Nagajaran8 is that given in ref 9. It should be emphasized that both versions are less soundly based than the treatment of nonionic surfactant micellar solutions using small systems thermodynamics,11 or the theory of multiple equilibria.12 Although some theoretical justification for the approach in refs 9 and 10 has been provided,9 its credibility is enhanced by the good agreement between theory and experiment for both ionic micellar solutions with only a single counterion and dilute solutions of polyelectrolyte plus salt. The key equations we require for the discussion which follows are eq 2-117 and 2-124 of ref 10 which, at constant T and p, may be written as
( )
d ln N ) 1 -
N2 N
2
(
d ln Ncm + β -
(1) Huisman, H. F. Proc. K. Ned. Akad. Wet. 1964, B67, 367. (2) Bales, B. L.; Almgren, M. J. Phys. Chem. 1995, 99, 15153. (3) Bezzobotnov, V. Y.; Borbely, S.; Czer, L.; Farago, B.; Gladkih, I. A.; Ostanevich, Y. M. J. Phys. Chem. 1988, 92, 5738. (4) Quina, F. H.; Nassar, P. M.; Bonilha, J. B. S.; Bales, B. L. J. Phys. Chem. 1995, 99, 17028. (5) Aniansson, E. A. G.; Wall, S. N.; Almgren, M.; Hoffmann, H.; Kielmann, I.; Ulbricht, W.; Zana, R.; Lang, J.; Tondre, C. J. Phys. Chem. 1976, 80, 905. (6) Siemiarczuk, A.; Ware, W. R.; Liu, Y. S. J. Phys. Chem. 1993, 97, 8082. (7) Dutt, G. B.; van Stam, J.; De schryver, F. C. Langmuir 1997, 101, 1957. (8) Nagajaran, R. Langmuir 1994, 10, 2028. (9) Hall, D. G. J. Chem. Soc., Faraday Trans. 2 1981, 77, 1121. (10) Hall, D. G. In Aggregation Processes in Solution; Wyn-Jones, E., Gormally, J., Eds.; Elsevier: Amsterdam, 1983; Chapter 2.
N2
)
d ln a2 (2)
and
1 d ln cm ) d ln a1 + (1 - R) d ln a2 N
(3)
where the notation used is as follows
∑r cr
(4a)
yr ) cr/cm
(4b)
cm )
Theory and Notation The thermodynamic theory which underlies the alleged inconsistency is developed in refs 9 and 10. The two versions differ slightly in the way in which activity coefficients are included. Although the approach in ref 9
Nβ N
N)
∑r yrNr
(4c)
N2 )
∑r yrN2r
(4d)
∑r yrβr
(4e)
∑r yrNrβr
(4f)
β) Nβ )
R ) β/N
(4g)
where cr denotes the concentration of micellar species r, Nr is the number of surfactant ions this species contains, and βr/Nr is its “effective degree of dissociation”. Also
10.1021/la971267n CCC: $18.00 © 1999 American Chemical Society Published on Web 04/17/1999
3484 Langmuir, Vol. 15, No. 10, 1999
a1 ) m1γ
(5a)
a2 ) m2γ
(5b)
m 2 ) m 1 + c3 +
∑r βrcr
(5c)
AxI 1 + xI
(5d)
m 1 + m 2 + c3 2
(5e)
ln γ ) I)
Hall
where m1 is the surfactant monomer concentration, c3 is the concentration of added 1:1 electrolyte, and A is the appropriate constant given by Debye-Huckel theory. Discussion If the micelles are strictly monodisperse, it is apparent from eq 2 that N can depend on neither Ncm nor a2. Also, if βr/Nr is the same for all r, then N cannot depend on a2 at constant Ncm. Obviously strict monodispersity is inconsistent with the kinetic process of micelle-monomer exchange. Direct evidence concerning micelle polydispersity is allegedly obtainable from studies of micellization kinetics via the intercept of graphs of 1/τ1 vs Ncm as Ncm f 0 where τ1 is the fast relaxation time.5 Quoted values of σ for SDS, where σ2 ) N2 - N2, are of order 13, which is too small to account for the reported dependence of N on SDS concentration. The original theory on which this estimate is based5 is strictly applicable only to nonionic surfactants. More elaborate theories13-15 which allow for the decrease in m1 with increasing surfactant concentration and for solution nonideality lead to the same expression for the intercept. Nevertheless, when the different theories are applied to the same data, the values of σ obtained do differ,16 and it is clear that estimates of micelle polydispersity obtained in this way are prone to substantial error. Despite this, the general consensus seems to be that SDS micelles are fairly monodisperse at moderate concentrations of surfactant and added salt. When N is large, the left-hand side (LHS) of eq 3 is negligible compared with the individual terms on the righthand side (RHS). Therefore, a good approximation to write for solutions both at and above the CMC is
(∂ ln a2/∂ ln a1)T,p ) -(1 - R)
(6)
It is found in practice, within experimental error, that graphs of ln a1 versus ln a2 are linear and are uninfluenced by surfactant concentration.17 For SDS, the value of R obtained from CMC data18 is 0.2. This appears to provide strong evidence, in support of the viewpoint expressed by (11) Hall, D. G.; Pethica, B. A. In Nonionic Surfactants; Schick, M., Ed.; Marcel Dekker: New York, 1967; Chapter 16. (12) Corkill, J. M.; Goodman, J. F.; Walker, T.; Wyer, J. Proc. R. Soc. London, Ser. A 1969, 312, 243. (13) Hall, D. G. J. Chem. Soc., Faraday Trans. 2 1981, 77, 1973. (14) Lessner, E.; Teubner, M.; Kahlweit, M. J. Phys. Chem. 1981, 85, 1529. (15) Elvingson, C.; Wall, S. N. J. Colloid Interface Sci. 1988, 121, 414. (16) Gharibi, H.; Takisawa, N.; Brown, P.; Thomason, M. A.; Painter, D. M.; Bloor, D. M.; Hall, D. G.; Wyn-Jones, E. J. Chem. Soc., Faraday Trans. 1991, 87, 707. (17) Palepu, R.; Hall, D. G.; Wyn-Jones, E. J. Chem. Soc., Faraday Trans 1990, 86, 1535. (18) Hall, D. G.; Huddleston, R. W. Colloids Surf. 1985, 13, 209.
Nagajaran,8 that the βr/Nr for different r are sufficiently close that the second term on the RHS of eq 2 is negligible compared with the first. According to the theory in refs 9 and 10, βr is 2 times the negative adsorption of coions (including surfactant monomer) around an isolated micelle of type r. Hence, if the distribution of counterions and coions around a micelle conforms qualitatively with the predictions of the PoissonBoltzmann equation, it is reasonable to expect βr/Nr to decrease as Nr increases. When a2 > 0.06 M, the slow relaxation time of SDS, τ2, is governed by the rates of micellar association and disproportionation,19 rather than micelle-monomer exchange. Under these conditions, it has been shown that the dependence of τ2 on a2 can be understood in terms of the dependence of βr on Nr.20 This dependence, together with Huisman’s light scattering data,1 can be fitted well by supposing that
βr ) β0 + R′Nr
(7)
where β0 and R′ are constants. This equation is, of course, an approximation. From the kinetic data it was found that β0 ≈ 5. Together with eq 2, eq 7 gives
(∂ ln N/∂ ln a2)Ncm ) β0σ2/N2
(8)
If (Nr - βr) is regarded as the number of counterions “bound” by a micelle of type r, then β0 may be regarded as the number of bound counterions released when a large micelle disproportionates into two smaller micelles. It is clear from eqs 2 and 8 that (∂ ln N/∂ ln a2)Ncm is about 5 times bigger than (∂ ln N/∂ ln Ncm)a2. Consequently, if eqs 7 and 8 are correct, then apart from concentrations very close to the CMC, N is much more sensitive to added electrolyte than to increasing surfactant concentration. This accords well with the conclusions of ref 4. Taking the LHS of eq 8 as 0.25, as eq 1 suggests, gives N2/σ2 ≈ 20. Because, in the absence of added salt, N ≈ 60 at the CMC,1 it follows that N2 ≈ N2 ≈ 3600 so that σ2 ∼ 180, which agrees quite well with the estimate of 169 given in ref 5. The remaining issue is whether eq 7 with β0 ) 5 is compatible with the observed dependence of ln a1 on ln a2. To check this, we integrate eq 6. When eq 7 is valid, eq 6 becomes
( ) ∂ ln a1 ∂ ln a2
(
) - 1 - R′ -
T,p
)
β0 N
(9)
Substituting for N as given by eq 1 yields
( ) ( ∂ ln a1 ∂ ln a2
) 1 - R′ -
T,p
β0
)
K(a2)η
(10)
where the concentration in eq 1 has been replaced by a2. On integration between two states I and II, eq 10 gives (19) Hall, D. G. In Organised Solutions; Friberg, S. E., Lindman, B., Eds.; Surfactant Science Series No. 44; Marcel Dekker: New York, 1992. (20) Lessner, E.; Teubner, M.; Kahlweit, M. J. Phys. Chem. 1981, 85, 3167.
Polydispersity of SDS Micelles
Langmuir, Vol. 15, No. 10, 1999 3485
II [ln a1]II I ) -(1 - R′)[ln a2]I +
[ ]
β0 (a2)-η K η
II I
(11)
It is also apparent from eq 1 that when m2 therein is replaced by a2, K ) N(CMC)/[a2(CMC)]η. Hence eq 11 becomes
ln
[ ] a1
a1(CMC)
[ ]
) -(1 - R′) ln
a2
a2(CMC)
+
β0(a2(CMC))η -η {a2 - [a2(CMC)]-η} (12) ηN(CMC)
Taking the CMC of SDS as 0.008 M gives a1(CMC) ) a2(CMC) ) 0.007 29. Choosing R′ ) 0.1467 gives a slope of ln a1 versus ln a2 of 0.8, as obtained experimentally, and leads to the behavior depicted in Figure 1. Although strictly the graph is nonlinear, the deviations from linearity are very small and are likely to be hidden by experimental error. A possible exception is the point at largest a2. The corresponding concentration for this point is ∼0.64 M and is of similar order to the salt concentration at which the micelle size distribution is alleged to become binodal.6 It appears therefore that the observed constancy of R which appears to result from plotting ln a1 versus ln a2 can conceal a dependence of βr on Nr sufficiently large to account for the observed increases in aggregation number with increasing concentrations of surfactant and of added salt.
Figure 1. ln a1 vs ln a2 calculated from eq 12 with ln a1(CMC) ) ln a2(CMC) ) -4.921, β0 ) 5, R′ ) 0.1467, η ) 0.25, and NCMC ) 60.
Finally, it should be noted that replacing the concentration in eq 1 with the counterion activity may influence the exponent. Also, using activities rather than concentrations in ref 18 may alter the value of β0. However, the changes concerned can be expected to be fairly trivial, and it is very unlikely that they will alter significantly the main conclusions stated above. LA971267N
