Langmuir 1993,9, 113-116
113
Diffusion of Sodium Dodecyl Sulfate Studied by a Steady-State Technique Nabil Mikati' CEA AB,Box 174,645 23 StrlingnBs, Sweden
Staffan wall Department of Physical Chemistry, University of Gateborg and Chalmers University of Technology, 412 96 Gothenburg, Sweden Received June 5,1992. In Final Form: October 19, 1992
The mutual diffueion coefficient D of an aqueous SDS (sodium dodecyl sulfate) solution in the concentration range 1-27 mM has been measured using stationary diffusion. The dependence of D on the concentration was obtained in a single experiment. The slope of the curve in the premiceller region indicates the existence of attractive interactions between the amphiphiles leading to dimerization. From our data and with the help of diffusion theories,and attractive potential which accountsfor these interaction is formulated and the degree of dimerization is estimated. In the micellar region the monomer-micelle equilibrium causes D to increase dramatically as the critical micelle concentration is approached. Surfactant diffusion has been the subject of many experimental and theoreticalstudies.l-ll Above the critical micelle concentration (cmc), two features dominate the dependence of D on concentratin, namely, interparticle interaction and micelle/monomer equilibrium. In the absence of added salt, D decreases to a minimum just above the cmc and increases thereafter with concentration as a result of the electrostatic repulsive forces. The increase in D as the cmc is approached from the micellar side, is due to the monomedmicelle equlibrium.8JoJl Although micellar diffusion isfairly well understood, very few studies on the diffusion procew in the premicellar region are published, in spite of the fact that evidencefor the existence of premicellar aggregates has been fo~ndhJ~-14 and disputed.16 An exception is ref 3in which the self-diffusion coefficient is measured below and above the cmc. One of the purposes of this study is to present such data and to investigate the occurrence of dimerization. Our assumption is that this should lead to a deviation of the values of D from those of a 1:l strong electrolyte. The other purpose is to demonstrate the use of stationary diffusion in the study of complex or nonideal systems where the diffusion coefficient depends on concentration, a major advantage being the ability to determine the dependence of D on concentration with a single experiment. (1) Stigter, D.; Williams, J. R.; Mysels, K. S. J. Phys. Chem. 1955,59, 389. (2) Mazer, N. A.; Benedeck, C. B.; Carey, M. J. Phys. Chem. 1976,80, 1075. (3) (a) Kamenka, N. K.; Lindman, B.; Brun, B. Colloid Polym. Sci. 1974,252,144. (b)Lindman, B.; Puyal, M. C.; Kamenka, N.; Rymden, R.; Stilbe, B. J. Phys. Chem. 1984,88, 5048. (4) Kratohhvil, J. P.; Aminabohvi, T. M.J. Phys. Chem. 1981, 86, 1254. (6) Corti, M.; Degiorgio, V. J. Phys. Chem. 1981,85,711. (6) Philliea,G. D. J. ColloidlnterfoceSci. 1982,86,226;J. Phys. Chem. 1981,86,3540. (7) Weinheher, R. M.;Evans, D. F.; Cuseler, E.L. J. Colloidlnterface Sci. 1981, 80, 267.
(8) Evans,D. F.; Mukherjee, S.; Mitchell, D. J.; Ninham, B. W. J . Colloid Interfoce Sci. 1983,93,184. (9) Degiorgio, V.; Corti, M. J. Colloid Interface Sci. 1984, 101, 289. (10) E v m , D. F. J. Colloid and Interface Sci. 1984, 101, 292. (11) Mikati, N. Chem. Phye. Lett. 1986,123, 61. (12) Mukerjee, P.; Myeela, K. J.; Dulin, C. I. J. Phys. Chem. 1986,62, 1390. (13) Mukerjee, P. J. Phys. Chem. 1968,62, 1397. (14) Franks, F.; Smith, H. T. J. Phys. Chem. 1964,68,3681. (15) Porfitt, C. D.; Smith, A. L. J . Phyr. Chem. 1962,66,942.
Apparatus and Experimental Section The main features of the apparatus used are given below. A detailed description can be found Diffusion takes place in a vertical channel while keeping the concentrationat both ends of the channelconstant. This is done by pumping the solution under study past the lower end of the channel whereas the upper end communicates with a reservoir containing the solvent. The concentration gradient is observed interferometrically by placing the diffusion cell in a modified Michelson interferometer. The mirrors of the interferometer are adjusted such that the field of view is uniformly illuminated in the absenceof concentrationgradients. When, however, solute moleculesenter the diffusionchannel,par$lel fringesare formed. The fringe number,counting downward, is directly proportional to the concentration. At steady state D(c) is given by Fick's law J D(C) = dcldx By measuring dc/dx the dependence of D on concentration can be obtained. However, since the solute flux J is unknown, the values thus obtained are multiplied by an arbitrary constant. To relate the data to the absolute scale, the values of D should be known for at least one point in the proximity of the studied concentration range. In this work the value of D at infinite dilution, estimated from conductivity measurementa, is used. The experiment was performed at 25 O C and the SDS used was obtained from BDH.
Results and Discussion The distribution of the concentration in the diffusion channel is shown in Figure 1. The valuesaf D(c)obtained from the concentration gradient are plotted in Figure 2. These values were obtained as follows: In the premicellar region the data points were fitted to a second-order polynomial giving the following fit
N = 19.04- 1.073~+ 0.0664~~
(2) where N is the fringe number and x is the distance measured from the bottom of the diffusion channel where the concentration is highest. The inverse of the derivative was then calculated. In the micellar region Wldx was approximatad by AN) Ax, where AN was either 1 or 2.The (16) Mikati, N. Reo. Sci. Instrum. 1987,68,604.
0743-746319312409-0113$04.00/0 @I 1993 American Chemical Society
Mikati and Wall
114 Langmuir, Vol. 9,No. 1, 1993 c mM
Do =
2Dm0D,O
(3)
Dmo+ D,O
Dmo and Dco being the ionic diffusion coefficienta of the dodecyl and sodium ions at infinite dilution. These were assigned respectively the values 5.36 X 10-l0 m2 8-l and 12.43 X W0 m2 s-l calculated from conductivity data.ll Also in Figure 2 are plotted (1) in the micellar region experimental data from ref 7 and theoretical data from ref 11 and (2) in the premicellar region a curve calculated from
I \
D = D,(I+ c a ~n riac)
(4)
that is assuming the behavior of a 1:l strong electrolyte"J8 and self-diffusion coefficient data from ref 3a converted to mutual diffusion coefficient by the equation
I
0
0
10
20 x mm
Figam 1. Concentrationof SDS VB distance in diffusion channel at steady state. 10
2
-1
0'10 m 8
8
00
.t
0
X 0
O
0 0
5
10
15
20
25
30
35
c mM
+
calculated from (5)
0
Weinhelmer [71
x
where D, is the self-diffusion coeffi~ient.~ The activity coefficient factor has been introduced in eq 6 because the chemical potential is constant in self-diffusion measurementa and this should be taken into account when comparing with our regults.18Jg In the calculations the activity coefficient was aeeigned the valueg predicted by Debye-Huckel theory.18 As can be seen from Figure 2, our experimental pointa in the micellar region agree rather well with the published data. The minimum however is displaced to the right by roughly 2 mM in comparison with the theoretical calculations and the experimental point 6f Weinheimer et al. at 10 mM. A probable reawn for thisshiftie that themethod by which D was determined smoothed out the large variation in D between 8.5 and 13 mM. On the other hand there are very few experimental pointa just above the cmc to compare ours with. In the premicellarregionthe slope for the SDSsurfactant is considerably more negative than that expected for a completely dissociated salt. This euggesta the existence of strong attractive interactions between the monomers. Since however, there are no long range attractive forces acting on the hydrocarbon chaine,m the interaction is a short range one which, due to ita proximity, lea& to the formation of premicellar aggregatee. We describe below a short range potential well U(r) which in combination with Felderhof s diffusion theory2' accounta for the high value of the slope obtained below the cmc. U(r)is
calculated from ( 4)
U(r) = 03
Mikatl [ll]
Figure 2. Our valuee for D compared with theoretical and experimental data.
reason for using this approximation instead of fitting to a higher order polynomial and then takingthe derivative was that the valuea of the derivativea obtained at the ends of the curve, that is in the neighborhood of c = 9 mM and c = 27 mM, depended on the order of the polynomial, although the fit itaelf was satisfactory. At c < cmc, thie problem did not occur,and the derivatives and M l A x were also in that case quite similar. Inordertorelatethecurvetbueobtainedtothea~lute d e , the valw extrapolated to zeroconcentration, DOwas given the value 7.48 X 10-10 m2 8-1, calculated from the expreeeion for the m i o n coefficisnt of a strong single
U(r)= -2aAu
+ We,
U(r)= 0
r 2(Rb + 6)
(7) (8)
where R h is the hard sphere radius, 6 is the thickness of the water sheath surrounding the hydrocarbon chain, u is the surface energy per unit area at the hydrocarbon water interface, A is the surface area of a SDS chain, an a is the (17) Cwler, E. L. Multicomponent Diffusion;Wvier: Amabrdun, 1984;p 96. (18) Koryta, J.; Dorak, J.; Bohackova, V.Ebctmhcmirtry; Mettler 6 Co.: New York, 1870; Chapbr 3. (19) Crank, J. The Mathematic8 of Diffusion, 2nd d.; Oxford Univenity P m : Oxford, 1% p 212. (20) Tmford, C. The Hydrophobic Effect; Wdey-Inbncience: New
York, 1873.
(21) Felderhof, B.
U.J. Phya. A: Math. Gcn. 1978,II,924.
Longmuir, Vol. 9, No.1, 1983 116
Diffueion of SDS fraction of the total area involved in the interaction. We1 is the electrostatic potential which we estimate according to Debye-Huckel theory.l*a Although Wd is a long range effect, ita contribution at r > 2(Rb + 6) has been neglected for the sake of simplicity. We believe this approximation is reasonable because,as will be seen below, the surface energy term is considerably larger in magnitude. The model describingthe interaction between the amphiphilic molecules described above resembles Tanford's approach to calculate the free energy drop upon micellization.20 Whereas Tanford considers the energy change in transferring CH2 groups from hydrocarbon to aqueous environmenta, we have expressed the energy change in terms of surface energy. Also from the Felderhof theory we have D = Do(l + Xc)
(9)
The dynamic virial coefficient X arises from thermodynamic and hydrodynamic interactions. The explicit expression of X is given in Felderhof s paper.21 By substituting U(r) in Felderhof s equations and integrating we get
Xv = 8V0[l - (ev - 1)(36/R,
+ 3(6/R,J2+ (6/R,J3)] (10)
XA
= -1.76v0eq
A, = 0.29v0[l - ( R , , J ( R+ ~ 6))'1eq Xo
6v0[(ev
- 1)(26/R, + ( 6 / R d 2 - 13 X, = vo
(11) (12) (13) (14)
and e
XV
+ XA + + XO +
(15)
(-2aAu + Wel)/kTand vo is the partial molar volume. With the help of eqs 6-16, the value of a which satisfies the experimentally obtained value of X = -18 L/mol can be determined using the following data: volume/hydrocarbon chain V = 361 As, length of a hydrocarbon chain z = 17 A, radius of chain y = 2.6 A,l9 and Rb = (3V/4*)1/3 = 4.4 A, A = 2ryz = 278 A2,6 = 1.6 A, u = 60 erg cm-2, We] Ne2(1 - d h J ( 1 Ko))/2&h, 0.7kT,22 and Vo = 0.249 L/mol. We arrive at the value a = 0.06. This number should however not be taken too literally, the main idea being that a small portion, say 6-10 % ,of the total surface area of the two amphiphiles gets in intimate contact to produce the dimer. The electrostatic repulsion keeping the ionic head groups apart would of course sustain such a structure. Mukerjeell proposed a similar model. With the help of the experimental data, we now estimate the amount of dimers present below the cmc. To this end we make the following assumptions: (i) the diffusion coefficient of the monomer D behaves as a 1:l strong electrolyte, that is it follow eq 4; (ii) the diffusion coefficient of the dimer DZis related to D1 by DZ= Dl/f, where f is a friction factor arising from the increase in volume; (iii) the average diffusion takes the form q
+
Table I. Ertimated Monomer and Dimer Concentrationr
~~
0.96 1.43 2.39 4.31 6.21 7.17
7.35 7.32 7.28 7.22 7.18 7.16
6.02 6.00 5.97 5.92 5.88 5.87
0.94 1.30 1.98 3.23 4.01 4.18
7.3 7.1 6.9 6.7 6.5 6.4
0.01
0.07 0.20 0.54
1.10 1.49
0.01 0.05 0.10 0.17 0.27 0.36
10 41 51 52 68 85
Since however we lack sufficient knowledge about the structure of the dimer, the value 1.22 will be used in our computations as in ref 12. Equation 14 together with the condition c = c1 2c2, led to the following expressions
+
c1 =
2(D - D,)C D + Dl - 20,
The results of computing c1 and c2 are presented in Table I. As can be seen from the table the ratio of dimers to monomers is rather smallat low concentrations,increasing to about 30% at 7 mM. The values of the equilibrium constant obtained also show an increasing tendency. According to Mukerjeels this should occur because increasing the ionic strength results in more effective shielding of the ionic head groups, thus promoting the formation of dimers. The values of the equilibrium constant Kd reported in the literature are Kd = 100 M-l and Kd = 260 M-' obtained by Frank and Smith" and Mukerjee.12 Although we obtained smaller values, it should be noticed that the calculations above are very sensitive to variations in the input data. For example a 1% change in D causes a 20% change in Kd. Similarly putting f = 1.17 instead of 1.22 as has been done leads to Kd = 86 M-l at c = 4.3 M. In summary, the slope of the D(c) w c curve obtained below the cmc has been explained by the presence of a short range interaction leading to the formation of premicellar aggregates. We have considered the case of dimers only because in the absence of further information it is not possible, with the method used, to differentiate between dimers and higher order premicellar aggregates. Moreover it has been argued by Mukerjee12that the dimer is the only species involved in the equilibrium. Our results show that in the dimer, the ionic head groups are kept far apart by electrostatic repulsion and that a small portion of the total area of the hydrocarbon chains is in intimate contact. We were able to calculate the concentration of the dimers and the equilibrium constant. The values of the equilibrium constant obtained by us are smaller than those reported,12J4probably as a result of the simplifications inherent in our model. In the micellar region our results are similar to reported data. The method of measurement used enabled us to obtain a relatively large number of points in the neighborhood of the cmc. The increase in the value of the diffusion coefficient, due to monomer/micelle equilibrium, as the cmc is approached is also demonstrated. Appendix
as derived in the Appendix. The friction factor f becomes = 1.26 if the volume of a spherical particle is doubled. (22)Tanford,C.physical Chemktry ofMacromolecule8; John Wiley ib SOM:New York, 1967;p 461.
Consider the following reaction 2c1
c2
(A.1)
The equilibrium constant Kd = cdc12 = K1/K2. Also the
116 Langmuir, Vol. 9, No.1, 1993
Mikati and Wall
total concentrationc, and the flux of matter are given by c = c, + 2c, J = D, ac,/as + 20, a q a x
(A.2) (A.3)
ale0
J = D ac/ax = D(ac,/ax + 2ac2/a~)
(A.4)
Since K1c12 = Kg2, we get by partial differentiating with respect to x ac,/ax = u(,c1 ac,/az
(A.5)
substituting in eqa A.3 and A.4 we get
J = (D, + 4D&dc1) a q a x
(A.6)
and J = D ( I + m d c 1 ) ac,/ax (A.7) Equating eqs A.6 and A.7 and substituting by Kd = c2/c12, we finally obtain
D = DIG, + 4D2c2 c1+
4c,
