Langmuir 1994,10, 1410-1414
1410
Diffusion in a Micellar Solution P. Neogi Chemical Engineering Department, University of Missouri-Rolla,
Rolla, Missouri 65401
Received August 3, 1993. I n Final Form: January 20, 1994@ Difffusion in a micellar solution is characterized by structural changes. There has been some discussion on whether these changes affect diffusion. A successful model in this area makes an assumption that its effect is important, which goes against the very large rate constants in the micellization-demicellization kinetics which would suggest instantaneousequilibration (localequilibrium). Here,an asymptotic expansion has been put together to show that both local equilibrium and significant micellization-demicellization activity are compatible. Thus, if one looks at concentrations alone they appear to be governed by local equilibrium, but if one looks at the fluxes, then small lack of local equilibrium cannot at all be ignored. The exception to this occurs in the immediate vicinity of cmc. Some experimental results are introduced to show that the present approximationsare allowed only in the case of small perturbations from equilibrium. The case of tracer/NMR self-diffusion coefficient is solved successfully using this approach.
Introduction Diffusivities of micelles have been studied extensively as a means of determining their shape, size, and charge.14 One key feature is that the structure changes as a function of concentration, and the question that arises is if local equilibrium is satisfied. This is generally assumed to be the case, and the very high micellization and demicellization rates5 help to support this hypothesis. In contrast Weinheimmer et a1.6 measured the diffusivity of the surfactant as a whole in Taylor's dispersion experiment using small concentration changes. They explained their data by assuming a lack of local equilibrium (in that they never set their micellization-demicellizationreaction rate rm to zero) but later used the assumption of local equilibrium in order to calculate some of the terms. They were able to explain their data quantitatively using no adjustable parameters. However, a value of 50 for the aggregation number was used and when the published7 value of 57 is used, differences of at least 20 % are obtained. It should be noted in this connection that diffusivities of micelles obtained from the Langevin equation, where the charge effects are considered? when compared to experimental data lead to an apparent aggregation number of less than 30.9 All systems described in above are for sodium dodecyl sulfate (SDS). The correlations developed by Weinheimer et a1.6appear to predict quite well the effective diffusivities measured by other experimental methods.'0-12 Abstract published in Advance ACS Abstracts, April 1, 1994. (1) Anacker, E. W. In Cationic Surfactants; Jungerman, Ed.; Marcel Dekker, Inc.; New York, 1970; p 203. (2) Wennerstrom, H.; Lindman, B. Phys. Rep. 1979, 52, 1. (3) Lindman, B.; Wennerstrom, H. In Physical Chemistry of Colloids and Macromolecules (IUPAC); Ranby, B., Ed.; Blackwell Sci. Pub.: Oxford, 1984; p 67. (4) Lindman, B.; Stilbs, P. In Microemulsions: Structure and Dynamics;Friberg, S.E.,Bothorel, P.,Eds.;CRC Press, Inc.: BocaRaton, FL, 1987; p 119. (5) Aniansson, E. A. W.; Wall, S. N.; Almgren, M.; Hoffmann, H.; Kielmann, I.; Ulbricht, W.; Zana, R.;Lang, J.;Tondre, C. J.Phys. Chem. 1975. 80. 905. ( 6 ) Weinheimer, R. M.;Evans,D. F.;Cussler, E. L. J. Colloid Interface Sci. 1981. 80. 357. (7) H&nanT H. F. Proc. K. Ned. Akad. Wet., Ser. B 1964, 67, 367. (8) Stephen, M. J. J. Chem. Phys. 1971,55, 3878. (9) Rhode, A.; Sackmann, E. J . Colloid Interface Sci. 1979, 70,494; J. Phys. Chem. 1980,84, 1598. (10) Kratohvil, J. P.; Aminabhavi, T. M. J. Phys. Chem. 1981, 86, SI
___
1264_.
(11) Leaist, D. G. J. Colloid Interface Sci. 1986, 111, 230. (12) Leaist, D. G. J. Colloid Interface Sci. 1986, 111, 240. (13) Evans, D. F.; Mukherjee, S.; Mitchell, D. J.; Ninham, B. W. J. Colloid Interface Sci. 1983, 93, 184.
0743-7463/94/2410-1410$04.50/0
Thus there is some consensus that apparent aggregation number is less than that at equilibrium, indicating some degree of demicellized state that a lack of local equilibrium would imply. In a later paper Evans et al.1° both stipulate the lack of local equilibrium in a lengthy Appendix (i.e., rm # 0) and also provide a description of mobility of the surfactant in terms of local equilibrium: When a slug of surfactant diffuses forward, the micelles at the rear demicellize and move ahead in form of singly dispersed amphiphiles, where they remicellize. The micelles and singly dispersed amphiphiles are at local equilibrium, according to this view. Data for SDS in the low concentration ranges have been given by Mikati and Wall.14J5 They also use models for diffusivities to bring in the mean field arguments, but problems arise with parameters to which unambiguous values cannot be given. Mikatil5 and Mikati and WalP4 provide a dissent in that they feel that the micellization-demicellization reaction does not affect diffusion. This is also the view of Corti and Digiorgio16 and Digiorgio and Corti'7 who measure diffusivities using light scattering and explain their data using local equilibrium, r, = 0, as a result of which their diffusivity is the same as that of nonassociating colloids such as globular proteins. It should be emphasized here that in the two cases, r m = 0 and r m # 0 will give effective diffusivities with qualitative and quantitative differences. Turq et a1.I8extend their treatment to include rm,but for nonionics. Later, Belloni et all9include ionics. One sophistication included is that the diffusional relaxation times can be allowed to approach the dynamic light scattering correlation times. It does not seem like a needed improvement. The latter is very high sincethe diffusivities are low. In contrast the charge effects are handled inadequately. In that, the derivation on light scattering has been treated best by Evans et al.l3 It is important to realize that all models find their origin as approximations to detailed governing equations. This has been explored here using a formal approximation process. Particular stress has been laid in uncovering the conditions under which the above models-in-words can be realized. (14) Mikati, N.; Wall, S. Langmuir 1993, 9, 113. (15) Mikati, N. Chem. Phys. Lett. 1986, 123, 51. (16) Corti, M.; Degiorgio, V. J. Phys. Chem. 1981, 85, 711. (17) Degiorgio, V.; Corti, M. J. Colloid Interface Sci. 1984, 101, 289. (18) Turq,P.;Drifford,M.;Hayoun,M.;Perera,A.;Tabony, J. J.Phys., Lett. 1983, 44, L-471. (19) Belloni, L.; Drifford, M.; Turq, P. J. Phys. Lett. 1985,46, L-207.
0 1994 American Chemical Society
Diffusion in a Micellar Solution
Langmuir, Vol. 10, No. 5, 1994 1411
Model Confining to 1:l anionic surfactant such as SDS and no added electrolyte, a reaction scheme may be proposed for the rate of micellization, following mass action kinetics
where the concentrations of singly dispersed surfactant ion, counterion, and the micelle ion bear subscripts 1,c, and m, respectively. Nand a are the aggregation number and the apparent degree of dissociation. On the other hand, one may not consider association at all. Investigators have suggested in a simple adsorption of a charged surfactant at the air-water interface that the adsorption rate is hindered by a term exp( - 1x1)where = +,/kT, where +, is the surface potential, It is the Boltzmann constant, and T is the absolute temperature.20121If this is included in micellization as well then for N ions and no counterion binding
jC= -Dc[ V C , -
e2 (7)
C,V+]
and
The condition of zero current
j, + Nm= 3, = j, where j, is the flux of the total surfactant, leads to
where
Y = C,(D,
+ D,) + NCm(NDm+ D,)
(11)
r , = k+e-NtmC,N- k-C,
Before obtaining the final form, a characteristic concentration based on the equilibrium
At equilibrium for rm = 0, one obtains exactly the conditions of equilibrium for micelles as given by Evans and Ninham.22 In there, they assume no association in the micellar ions and show that N -(1 -a)ln(C,). If the charge density is sufficiently high, the flat plate assumption can be which leads exactly to a = 0.5. Simple manipulations lead to eq 1. For convenience the EvansNinham model with no intrinsic counterion binding is used below; one could just as easily formulate the mass action kinetics model with an intrinsic binding. Conservation equations are for the singly dispersed amphiphiles
is introduced. If C is the characteristic concentration, C1-N(2-a).Thus, C is of the then according to eq 12, K order of cmc. Substituting eqs 10 and 11 into eqs 6 and 8, and then into eqs 3 and 4, one has on nondimensionalization
-
ao, - = v20, + T+,[
a7
d, - 1 -
-vo, +
z
Nq[(O,
dC1 _ - -V.j, - N[k+CINCcN('-"'- k - Cm 3 at
(3)
aom = dmV20m+ dmT*
a7
and for the micelle
N ( d c - dm) TO,]
z
(4)
The convective effects are being ignored. Because they are linear in concentration, their inclusion is straightforward as long as they are laminar and can be combined with the unsteady state term here. In no case do they affect the net diffusivity. The local electroneutrality leads to
C, = C, + N C , = C,
(5)
where C, is the total concentration of the surfactant. The fluxes are
e2
3, = -D1[ vc, + kT c,v+] where D1 is the diffusivity of the surfactant ion and 2 is its valence. is the diffusion potential. Further
+
(20) Chang, C.H.;Franses, E. I. Colloids Surf. 1992, 69, 189. (21) Langevin,D. Paper presented at Colloid and Surface Symposium, Toronto, June 1993; paper no. 80. (22) Evans, D.F.; Ninham, B. W. J. Phys. Chem. 1983,87,5025. (23) Loeb, A. L.;Overbeek,J. Th. G.; Wiersema, P. H. TheElectrical Double Layer Around a Spherical Colloid Particle; MIT Press: Cambridge, MA, 1961; p 37.
N ( d , - d,)
z
1
-
+ NOm)N(l-U)OIN - Om]
(13)
z
I+
q[(OlNOm)N~2"~O~Om] (14)
where Z = Oi(1 + d,) + NOm(Ndm + d,), 01 = C1/C, 7 = Dit/Lz,d, = DClDi, Om = Cm/C, V = L V , d , = Dm/D1, and q = kL2/D1. k+ has been expressed in terms of C. Here q , the ratio between the reaction rate to the diffusion rate or the Damkohler number, is a large number which makes it appropriate for one to construct an asymptotic expansion in small 9-1.
Solution The details of how asymptotic solutions are constructed are given elsewhere.24 A brief outline would be that a series is constructed such that the coefficientsare functions of q , such that the first coefficient is the largest and successive coefficients get smaller, and where the magnitudes are ascertained by how fast they go to zero in the limit q goes to infinity. The series are substituted in the original equations and divided by an appropriate function of q and the limit that q m is taken such that all but the dominant terms drop out. Such manipulations carried out successively give us the equations for all the terms in the series. The nature of the functions in q are brought out through internal constraints. The simplest solution is of the type
-
(24) Miller, C. A.; Neogi, P. Interfacial Phenomena; Marcel Dekker, Inc.; New York, 1985; p 310.
Neogi time
They lead to the equations
Figure 1. The geometry of the arrangement in the cuvette is shown on the left. The solution is layered up to 11 and the water is layered on the top between lI and 12. The beam measures absorbance between u1 and u2, corresponding to the mean concentration in that region. The concentration profiles are
shown on the right, including the initial (left) and final (right).
The system of equations has the following characteristics: (i) It is valid only in the immediate vicinity of cmc. As 1/[N(2 - a ) - 11 in eq 15 is very close to zero, to a good approximation 81 is of the order of 1. However, N(2 a)/[N(2- a ) - 13 is close to 1, thus Bm in eq 16 is of ord(T-l), a very small quantity. That is, the amount of micelles is very small. (ii) Local equilibrium applies to the leading order, eq 18, correct to -0(7-'). (iii)The mechanism of diffusion of the micelles is exactly one of those discussed earlier: The micelles at the rear demicellize, diffuse to the front, and remicellize. Local equilibrium holds. (iv) Whereas micelles do not contribute much to the net movement, they do have a diffusivity, and in the presence of local equilibrium its diffusion will be that of nonassociative colloids. This is the view presented by Corti and DegiorgiolG and Degiorgio and Corti17and is seen to apply in the immediate vicinity of cmc. Consequently, the mechanism given above is restricted to a narrow region. To remedy the situation a different asymptotic solution is sought with a different scaling. For the solution to be valid over a wider range, both concentrations should be ord(1) to the leading order:
el
- elc0)+ 7-1el(1)+ ...
(20)
Further, it is stipulated that to the leading order local equilibrium is valid, or
Substituting eqs 20-22 into eqs 13 and 14, and taking the limit that 7 goes to infinity, one has a very lengthy equation for which also contains el(1),and similarly an equation for em(o)which also contains &(I). Such equations are insolvable. However, they show two important features. These are that in this solution the reaction term is not negligible,unlike in the previous solution, and that in spite of it local equilibrium still holds to the leading order, ord(1). These are of course Weinheimer's assumptions which are now shown to be internally consistent. Fortunately, Weinheimer et a1.6 also showed how to solve these at least partially. We summarize the results as follows: (1) (i) concentration range, negligible micelles, ord(r1-l) (ii) reaction much faster than diffusion, local equilibrium is accurate; ( 2 ) (i)
concentration range, the concentrations of singly dispersed amphiphiles and micelles are comparable, ord(1) (ii) reaction rate is comparable to diffusion, local equilibrium is a good approximation. In the next section we report some attempts at testing the limits of the present principles. Testing Those Limits Here we report a small result from the work of BhakhZ5 An attempt was made to measure diffusivities inside a spectrophotometer. Inside a cuvette, an aqueous solution of known strength and amount was layered, and then water was layered on top of it. Measured absorbance decreased with time, which was fitted to the solution to the boundary value problem to calculate the diffusivity. The geometry is illustrated in Figure 1. An innovative method was used to layer water on the top of the solution without significant mixing. Care had to be taken to eliminate natural convection. Calibration was done with KN03 and the results were found to be always within 5% of the value reported in literature. More details on the procedure are not being given here because they are being reported elsewhere at this time. For SDS below cmc, the value of diffusivity obtained was 8.9 X lo4 cm2/s. From the reported values of diffusivities of the sodium ion26 and the dodecyl sulfate i0n,27 one calculates an effective diffusivity2*of SDS below cmc at 8.02 X 10-6 cm2/s. This is quite good considering that SDS at such low concentrations has hardly any absorption peaks. A wavelength of 225 nm was used where Beer's law was valid. For SDS above cmc (starting concentration was 0.068 mol/L greater than cmc of 0.0082 mol/L), the situation is better, there is a small adsorption peak at 300 nm. For this case the effectivediffusivity was found to be 6.8 X 10-6 cm2/s, effective because the actual diffusivities are concentration dependent making the present value a weighted average. However, Weinheimer et a1.6 have provided the concentration-dependent diffusivities, and it should be possible to use their results in an appropriate boundary value problem to predict this effective value. The effort was made but it predicted only about 50% of this measured value. A few reasons could be cited for the discrepancy. It is to be noted in the concentration profiles drawn in Figure 1,that at first when the SDS diffuses to the top it must demicellize; that is, it crosses cmc. But in this case the final overall concentration is above cmc; hence it must eventually remicellize. Thus, (25) Bhakta, A. MS Thesis; Chemical Engineering Department, University of Missouri-Rolla, Rolla, 1992. (26) Clifford, J.; Pethica, B. A. Trans. Faraday SOC.1964,60, 216. (27) Kamenka, N.; Lindman, B.; Brun, B. Colloid Polym. Sci. 1974, 252, 144.
(28) Nernst, W. Z.Phys. Chem. 1888,2, 613.
Langmuir, Vol. 10, No. 5, 1994 1413
Diffusion in a Micellar Solution it crosses cmc again but now from below. First, the correlation of Weinheimer et a1.6 needs an equilibrium relation, and both phase separation model and mass action kinetics model are not expected to be any good near cmc. Thus, their model is not adequate near cmc (see also the results of Evans et al.13). Consequently, where the system crosses cmc twice, the ability to predict the outcome using this approach falls. Second, this model is inadequate near cmc for the reasons that the assumptions/scaling arguments used by them are invalid as shown earlier. Consequently, systems which cross over from one regime to another are difficult to quantify. In discussing the different experimental methods for measuring the micellization-demicellization kinetics, Kahlweit and Teubner29 argue that the stopped flow experiments should highlight these effects the most. The present system is asimplification of the stopped flow experiment, and these authors come to their conclusion for exactly the reasons discussed earlier, that is, significant activity across cmc.
NMR/Tracer Self-Diffusion Coefficients The conclusion so far is that in a system with small perturbations, such as in the experiments of Weinheimer et a1.6, it is possible to quantify the results following the scheme given by them. The procedure has been shown here to be consistent. It follows then that the results of tracer or NMR self-diffusion coefficients are best treated by this method. In an SDS solution, one would have six species, 1 (dodecyl sulfate ion), 1* (tagged dodecyl sulfate ion), m (micelle), m* (micelle with a tagged dodecyl sulfate ion in it), c (counterion), and w (water). The concentration of the tagged dodecyl sulfate ion is so small that a tagged micelle is assumed to have no more than one such ion. All species can be assumed to have sufficiently small concentrations such that coupled fluxes can be ignored. The fluxes of tagged species are = -Dl,[ VC,,
+
e2
C,*V$
1
The untagged species are assumed to be equilibrated, but the coupling to the counterions through zero current condition cannot be ignored. That is
jl* + ”,
=
3, = p*
where the last term is the overall flux of the chargedspecies. The flux of the counterion is given by eq 7. Now, the counterions can be resolved into
c, = c, + c,‘ where the term with the overbar indicates the part which balances the untagged species and is hence equilibrated V C , = 0. And the term with prime is the part which balances the tagged species, - hence when derivatives are not being taken C, N C,. Thus eq 7 becomes
3,
e2 -
= -D,[ VC,‘-
kT C,v$]
Condition of zero current leads to (29) Kahlweit, M.; Teubner, M. Adu. Colloid Interface Sci. 1980,13, 1.
-Dl,VCl, - D,*NVC,,
+ D,VC,‘
(28)
On the left-hand side the last term within the bradkets is far higher than the other two, and C,’ = Clt + NC,,. Thus, one has
- e2 D C -V $ ,kT
N
(D, - D,,)VC,,
+ (D, - ND,,)VC,,
(29)
If this expressionfor the term (eUkT)V$ is now substituted into the two fluxes, eqs 23 and 24, it is seen that the electrostatic term is dependent on the square of the concentrations of the tagged species, and hence negligible. On discarding those terms, the fluxes with the conservation equations become
ac,, -= v.[D,,VC,,I at
ac,, - V*[D,*VC,*l -at
- r*
(30)
+ T*
where T* is the rate of micellization with only one tagged molecule in a tagged micelle. Adding eqs 30 and 31, with the total concentration of the tagged species as
c * = c,, + c,,
(32)
one has
a*‘ - V.[D,,VC,, + D,,VC,J --
at
This is rewritten as
Cl* Cm* = V.[D,.V -C* + D,,V -C*] (33b) at C* C* Now, C1*/C*is the fraction of the charged molecules that are singly dispersed and C,*/C* is the fraction in micelles. If there is no difference between the tagged and untagged molecules in any of the physicochemical phenomena that are present, then C1*/C* = p1 and Cm*/C* = Pm, where p1 is the fraction of surfactant molecules singly dispersed and p m is the fraction in the micelles. Note that these are effectively mass fractions. More important is the fact that they do not vary spatially. Hence eq 33b becomes
at
= V.[(D,,p,
+ D,,p,)VC*l
(34)
With no differences between the properties of the tagged and untagged species D1* = D1 and Dm* = D,, and eq 34 leads to the effective diffusivity of the charged species as (35)
This equation was first arrived at by Lindman and B r ~ n , ~ ~ based on qualitative arguments. It has also been tested extensively.24 It is very important to note that as derived here the constituents of D* are all “mutual” diffusion coefficients. Thus, Clifford and Pethica26 (Na+), Kamenka, et ale2’(DS-),and Stigter et aL31(SDS),all measure “self”-diffusion coefficients, yet Weinheimer, et a1.16 who very clearly measure mutual diffusion coefficients explain (30) Lindman, B.; Brun, B. J . Colloid Interface Sci. 1973, 42, 388. (31) Stigter, D.; Williams, R. J.; Mysels, K. J. Phys. Chem. 1955,59, 330.
1414 Langmuir, Vol. 10, No. 5, 1994
their data perfectly well using these "self"-diffusion coefficients. The analysis here shows that this is possible because in the self-diffusion coefficient studies actually the mutual diffusion coefficients are measured. In the present developments a diffusivity such as Dp is the binary D I ~ ,A~typical . self-diffusion coefficient such as Dp,1 does not appear, at least under the approximations used. Lindman and Brun30also show that if in eq 35 a constant value of D, is used, then it overpredicts the diffusivity D* at large surfactant concentrations. They suggest that in this region hindrance effects due to the presence of other micelles bring down the effective value of D,. The concentration effect was introduced using the theory of Anderson and Reed32for charged colloids. It lowered the predicted values but not enough. It is possible to further improve the match by considering coupled fluxes in multicomponent diffusion. This is done conveniently by using the generalized Stefan-Maxwell equation33but was not carried out, as it gave rise to coefficients that cannot be evaluated (such as DmJ. In any case the discrepancy at this level is about 5 7%.
Discussion The main conclusion from above is that as long as the perturbations are small, the diffusion of surfactants in a micellar solution is tractable. In spite of the large rate constants in the micellization-demicellization process, the assumption of local equilibrium is not exact. (32) Anderson, J. L.; Reed, C. C. J. Chem. Phys. 1976,64, 3240. (33) Mason, E. A.; Viehland, L. A. J. Chem. Phys. 1978,68, 3562.
Neogi
The obvious problem in trying to quantify diffusion far from equilibrium is that it opens up an unending number of possible reactions. It is easy to argue that the one reaction scheme which has been used here is enough for case8 close to equilibrium. This is based on the fact that eq 12 provides a good description at equilibrium; hence its extension into nonequilibrium situations alone should suffice when close t o equilibrium. Beyond that is a region of many reactions and polydispersivity that needs to be sorted out. Obviously if we ignore these, there should be some independent way of forecasting how the average aggregationnumber changeswith time. This is unavailable at present. The present analysis also helps to put various diffusivities in order. It is found that the self-diffusivitystudies lead to diffusivities without the need to know the values of N nor a beforehand. On the other hand the mutual diffusivity studies put enough pressure on the system such that one wonders if these studies can also be used to determine the reaction rate constants. It is apparent that this is only possible if the detailed profiles of the species are known. The profiles for micelles can be separated from those of the overall surfactant using tracers but would at least need theoretically predicted solutions to back out additional information. Unfortunately the asymptotic scheme cannot be solved and a numerical problem with the very large 7 in it would be very stiff and very difficult to solve. The main problem still remains that the data from the experiments that give us the mutual diffusion Coefficients are very few. What data are available have been put into order.
