690
J. Phys. Chem. B 2001, 105, 690-695
Coupled Diffusion of Mixed Ionic Micelles in Aqueous Sodium Dodecyl Sulfate + Sodium Octanoate Solutions Derek G. Leaist* and Kimberley MacEwan Department of Chemistry, UniVersity of Western Ontario, London, Ontario, Canada N6A 5B7 ReceiVed: August 30, 2000; In Final Form: NoVember 9, 2000
Taylor dispersion and differential refractometry are used to measure ternary mutual diffusion coefficients for aqueous sodium dodecyl sulfate (NaDS) + sodium octanoate (NaOct) solutions at 25 different compositions at 25 °C. Diffusion of the mixed surfactants is strongly coupled. In particular, each mole of NaOct cotransports up to 2.5 mol of NaDS. The results are interpreted by using Nagarajan’s molecular thermodynamic model of mixed surfactant solutions to estimate average surfactant aggregation numbers for the mixed micelles. In addition, Nernst-Planck equations are used to relate the measured fluxes of the total NaDS and total NaOct components to the fluxes of free DS- and Oct- surfactant ions, free Na+ counterions, and micelles of local-average composition. The analysis shows that the large coupled flows of NaDS driven by NaOct gradients are caused by the migration of charged NaDS-rich micelles in the electric field generated by the diffusion free Na+ and Oct- ions.
Introduction Surfactant diffusion is important in emulsification, digestion, degreasing, foaming, and coating processes.1,2 Surfactant diffusion measurements can also provide basic information about extents of aggregation, solubilization, and counterion binding. Most studies in this area have dealt with solutions containing a single surfactant component.3-10 In many applications, however, mixed surfactants1,11-13 are used for improved performance. Surfactant mixtures are also employed inadvertently as a result of synthetic byproducts or unpurified raw materials used in manufacturing processes. In this paper, ternary mutual diffusion coefficients14 are reported for the mixed surfactant system sodium dodecyl sulfate (NaDS) + sodium octanoate (NaOct) + water. The measured diffusion coefficients (Dik) relate the coupled fluxes of the total NaDS(1) and total NaOct(2) components (J1 and J2) to the gradients in the concentrations of the total surfactant components (∇C1 and ∇C2).
J1 ) -D11∇C1 - D12∇C2
(1)
J2 ) -D21∇C1 - D22∇C2
(2)
This description of mixed surfactant diffusion is accurate, but phenomenological. To understand the results, it is important to relate the measured fluxes of the total surfactant components to the fluxes of the actual solution species, in this case polydisperse mixed micelles (DSnOctpNaq)q-n-p, free DS- and Oct- surfactant ions, and free Na+ counterions. In previous studies, Nernst-Planck (NP) equations have been used to analyze the coupled diffusion of weak electrolytes in terms of the fluxes of the constituent ions and associated species.15-17 The corresponding treatment of mixed surfactant diffusion is more complicated because of the large number of mixed micelle species. For example, aggregation numbers reported for binary NaDS + water3-5 and NaOct + water8,18 * Fax: (519) 661-3022. E-mail:
[email protected].
solutions indicate binary micelles of approximate composition (DS60Na50)-10 and (Oct30Na20)-10, respectively. Interpolation of these results suggests that ternary NaOct + NaDS + water solutions will contain polydisperse (DSnOctpNaq)q-n-p mixed micelles with aggregation numbers in the approximate range 0 j n j 60, 0 j p j 30, and 20 j q j 50. The implementation of NP equations for so many different species would clearly be awkward. Moreover, precise diffusion coefficients for the numerous micelle species are not available. In this paper, a simplified treatment of mixed surfactant diffusion is developed based on NP equations for the fluxes of free surfactant monomers, free counterions, and a single micelle species of locally averaged composition. Predicted diffusion coefficients are compared with the measured Dik values for the aqueous NaDS + NaOct solutions. Critical micelle concentrations (cmc’s) for this system are reported to identify the solutions that contained micelles. Experimental Section Mutual diffusion coefficients were measured by the Taylor dispersion (peak-broadening) method. At the start of each run, a 0.020 cm3 sample of solution was injected into a carrier solution of slightly different composition at the entrance to a Teflon capillary tube (length 2590 cm, inner radius 0.04588 cm). The broadened distribution of the dispersed samples was monitored at the tube outlet by a liquid chromatography differential refractometer detector (Hewlett-Packard model 1047A, sensitivity 1 × 10-8 refractive index units). The refractometer output voltage was measured at 5 s intervals by a digital voltmeter. The dispersion profiles for the mixed surfactant solutions were analyzed by fitting the equation19
V(t) ) V0 + V1t + Vmax
()[ ( ( tR t
1/2
W1 exp -
(1 - W1) exp -
10.1021/jp003131v CCC: $20.00 © 2001 American Chemical Society Published on Web 01/03/2001
) )]
12D1(t - tR)2
r2t 12D2(t - tR)2 r2t
+ (3)
Diffusion Coefficients for NaDS + NaOct Solutions
J. Phys. Chem. B, Vol. 105, No. 3, 2001 691
TABLE 1: Critical Micelle Concentrations of Aqueous NaDS(C1) + NaOct(C2) Solutions at 25 °Ca
a
C1/(C1 + C2)
C1
C2
cmc
0.0000 0.0010 0.0025 0.0050 0.0100 0.0250 0.0400 0.0500 0.100 0.150 0.250 0.500 0.750 0.900 1.000
0.000 0.00016 0.0004 0.0006 0.0009 0.0015 0.0017 0.0018 0.0028 0.0033 0.0044 0.0061 0.0069 0.0079 0.0082
0.350 0.159 0.148 0.118 0.0856 0.0587 0.0404 0.0333 0.0232 0.0197 0.0132 0.0061 0.0023 0.0009 0.0000
0.350 0.159 0.148 0.119 0.0865 0.0602 0.0421 0.0351 0.0260 0.0230 0.0176 0.0122 0.0092 0.0088 0.0082
Units: mol dm-3.
to the refractometer voltages. Vmax is the peak height relative to the linear baseline voltage V0 + V1t, and tr is the retention time (about 8000 s). The Dik coefficients were evaluated from the normalized weights W1 and 1 - W1 of the preexponential factors and the eigenvalues D1 and D2 of the ternary diffusion coefficient matrix. Details of the experimental procedure and the analysis of the dispersion profiles have been reported.19,20 Solutions were prepared in calibrated volumetric flasks by dissolving weighed amounts of NaOct (Aldrich, >99% purity) and NaDS (BDH, “Specially Pure”) in distilled, deionized water. An Orion model 160 conductivity meter was used to measure the conductivities of solutions with fixed NaDS:NaOct ratios. Cmc’s were determined from the breaks in the specific conductivity plotted against the square root of the total surfactant concentration. Results Critical Micelle Concentrations. Cmc’s for aqueous NaDS(1) + NaOct(2) solutions were measured at 25 °C, and solute fractions of total NaDS in the range 0.001 e C1/(C1 + C2) e 0.900. The results are summarized in Table 1. Cmc’s were also measured for binary aqueous solutions of each surfactant. These results are consistent with previously reported binary cmc values:8,21 0.0079 to 0.0084 mol dm-3 for NaDS and 0.34 to 0.39 mol dm-3 for NaOct. In Figure 1, the measured cmc’s are plotted against the solute fraction of total NaDS. NaOct shows a weaker tendency to form micelles and therefore has a higher cmc because it has a shorter hydrocarbon tail than NaDS. Adding small amounts of NaDS to aqueous NaOct solutions produces a remarkably sharp drop in the cmc for the resulting mixed surfactant. Ternary Mutual Diffusion Coefficients. Diffusion of the mixed surfactants was measured over a wide composition range: total NaDS + NaOct concentrations of 0.020, 0.040, 0.050, 0.100, or 0.200 mol dm-3 and NaDS solute fractions 0.050, 0.250, 0.500, 0.750, or 0.950. Four to six replicate measurements for each carrier solution indicated that the diffusion coefficients were reproducible to within ( (0.01 to 0.02) × 10-5 cm2 s-1. Table 2 gives the average Dik coefficients for each composition. The results obtained below the cmc are italicized. Previous work has shown that the mutual diffusion coefficients of binary surfactant solutions drop abruptly in the region of the cmc.22,23 To avoid possible uncertainties in the present results caused by large changes in the Dik coefficients across
Figure 1. Critical micelle concentrations of aqueous NaDS + NaOct solutions at 25 °C plotted against the surfactant monomer fraction of DS- ions: O, measured values; s, predicted by Nagarajan’s model of mixed surfactant aggregation.
TABLE 2: Ternary Mutual Diffusion Coefficients of Aqueous NaDS(C1) + NaOct(C2) Solutions at 25 °Ca C1
C2
0.001 0.005 0.010 0.015 0.019 0.002 0.010 0.020 0.030 0.038 0.0025 0.0125 0.0250 0.0375 0.0475 0.005 0.025 0.050 0.075 0.095 0.010 0.050 0.100 0.150 0.190
0.019 0.015 0.010 0.005 0.001 0.038 0.030 0.020 0.010 0.002 0.0475 0.0375 0.0250 0.0125 0.0025 0.095 0.075 0.050 0.025 0.005 0.190 0.150 0.100 0.050 0.010
D11
D12
D21
0.74 (0.62) 0.03 (0.01) -0.03 (0.22) 0.20 (0.12) -0.02 (-0.01) 0.09 (0.04) 0.18 (0.14) 0.03 (0.09) 0.07 (0.03) 0.22 (0.19) 0.12 (0.20) 0.07 (0.02) 0.26 (0.23) 0.22 (0.29) 0.02 (0.01) 0.34 (0.15) 0.00 (-0.01) -0.07 (0.03) 0.16 (0.12) 0.04 (0.09) 0.08 (0.02) 0.20 (0.17) 0.15 (0.24) 0.09 (0.03) 0.27 (0.25) 0.33 (0.48) 0.07 (0.02) 0.32 (0.34) 0.50 (0.70) 0.02 (0.01) 0.15 (0.11) 0.00 (0.00) 0.16 (0.00) 0.14 (0.13) 0.04 (0.11) 0.08 (0.01) 0.17 (0.18) 0.15 (0.29) 0.08 (0.02) 0.25 (0.27) 0.35 (0.55) 0.05 (0.02) 0.35 (0.38) 0.60 (0.84) 0.02 (0.01) 0.14 (0.10) 0.00 (0.02) 0.05 (-0.14) 0.13 (0.13) 0.08 (0.17) 0.08 (-0.06) 0.18 (0.20) 0.22 (0.40) 0.07 (-0.01) 0.28 (0.31) 0.52 (0.77) 0.06 (0.02) 0.45 (0.48) 0.88 (1.26) 0.02 (0.01) 0.12 (0.10) 0.00 (0.04) -0.05 (-0.58) 0.14 (0.12) 0.08 (0.22) 0.00 (-0.26) 0.19 (0.20) 0.21 (0.52) 0.07 (-0.08) 0.29 (0.34) 0.54 (0.93) 0.05 (0.00) 0.50 (0.54) 1.28 (1.54) 0.01 (0.01)
D22 0.90 (0.83) 0.90 (0.77) 0.88 (0.74) 0.84 (0.69) 0.80 (0.63) 0.80 (0.82) 0.90 (0.81) 0.86 (0.78) 0.80 (0.72) 0.63 (0.64) 0.81 (0.83) 0.79 (0.82) 0.76 (0.79) 0.70 (0.73) 0.64 (0.65) 0.90 (0.84) 0.85 (0.85) 0.81 (0.82) 0.75 (0.76) 0.68 (0.66) 0.84 (0.87) 0.79 (0.92) 0.74 (0.90) 0.66 (0.81) 0.55 (0.67)
a Submicellar results are italicized. Units: C in mol dm-3; D in i ik 10-5 cm2 s-1.
the dispersion profiles, the carrier solution and injected solution for each run were both above or both below the cmc. D11 gives the flux of the total NaDS component produced by its own concentration gradient. The results obtained at 0.020 mol dm-3 total NaDS + NaOct indicate a sharp drop in D11 as the concentration is raised through the cmc. Above the cmc, however, D11 increases as the concentration of NaDS is raised. Qualitatively similar behavior is observed for the binary mutual diffusion coefficient of aqueous NaDS solutions.22 NaOct concentration gradients can also produce significant flows of NaDS. Indeed, cross-coefficient D12 reaches a value 2.56 times larger than D11 for the most concentrated solution. NaOct concentration gradients can therefore drive coupled flows
692 J. Phys. Chem. B, Vol. 105, No. 3, 2001
Leaist and MacEwan
of NaDS that are more than twice as large as the main flow of NaDS driven by the same gradient in the concentration of NaDS. In the limit C1/(C1 + C2) f 0, however, D12 drops to zero because a gradient in NaOct cannot produce a coupled flow of NaDS in a solution free of NaDS. Similarly, D21 vanishes as the solute fraction of NaOct drops to zero. Table 2 shows that the composition dependence of D22, the main diffusion coefficient of the NaOct component, is weaker than that of D11 for the NaDS component. It is also evident that cross-coefficient D21 is significantly smaller than D12. Thus gradients in NaDS do not drive large coupled flows of NaOct. Discussion The analysis of mixed surfactant diffusion is complicated by the numerous polydisperse mixed micelle species. In many cases, however, the distribution of aggregation numbers is believed to be relatively narrow.11,12 For these systems it might be a good approximation to describe diffusion in terms of a restricted basis set of micelle species. In this section, a simplified treatment of the coupled diffusion of NaDS and NaOct components is developed using Nernst-Planck (NP) equations for the fluxes of only four species: free DS- and Oct- surfactant ions, free Na+ counterions, and a single micelle species (DSnOctpNaq)q-n-p of local average composition. Nernst-Planck Equations for the Solution Species. According to the NP model24 of diffusion, the flux js of solution species s is the sum of the pure-diffusion flux -Ds∇cs and the migrational flux (F/RT)zsDscsE driven by the electric field E generated by the diffusion of ionic species of different mobilities. In the notation used here zs, Ds, and ∇cs denote the charge number, diffusion coefficient, and concentration gradient for solute species s. F, R, and T are the Faraday constant, gas constant, and the temperature. NP equations, though accurate only for dilute solutions, provide a useful guide to the effects of ion association on the diffusion of weak electrolytes.15,16 NP equations can also be used to interpret multicomponent diffusion, including coupled transport, by resolving Dik coefficients into pure-diffusion contributions Dik(D) from the diffusion of species down concentration gradients and electrostatic contributions Dik(E) from the migration of charged species in the diffusion-induced electric field.17
Dik ) Dik(D) + Dik(E)
micellar Oct- ions.
J1(total NaOct) ) jOct + pjmic
D1k(D) ) DDS(∂cDS/∂Ck) + nDmic(∂cmic/∂Ck)
(11)
D2k(D) ) DOct(∂cOct/∂Ck) + pDmic(∂cmic/∂Ck)
(12)
for the pure-diffusion contributions to the ternary mutual diffusion coefficients and
[ [
(5)
jOct ) -DOct∇cOct - (cOctDOctF/RT)E
(6)
jNa ) -DNa∇cNa + (cNaDNaF/RT)E
(7)
jmic ) -Dmic∇cmic + [(q - n - p)cmicDmicF/RT)E] (8) where subscripts DS, Oct, Na, and mic designate the species DS-, Oct-, Na+, and (DSnOctpNaq)q-n-p, respectively. NP eqs 5-8 for the fluxes of the solution species are related to Fick eqs 1 and 2 for the ternary mutual diffusion of the total NaDS and NaOct components by noting that the total NaDS flux is the sum of the fluxes of free and micellar DS- ions
(9)
Also, the total NaOct flux is the sum of the fluxes of free and
] ]
D1k(E) ) Sk tDS +
ntmic n+p-q
D2k(E) ) Sk tOct +
ptmic n+p-q
(13) (14)
for the electrostatic contributions. Sk (k ) 1, 2) is an abbreviation for
Sk ) DNa
jDS ) -DDS∇cDS - (cDSDDSF/RT)E
(10)
Substituting eqs 5-8 into eqs 9 and 10 and collecting terms gives15,16
(4)
The NP equations for the species considered here are
J1(total NaDS) ) jDS + njmic
Figure 2. Concentrations cmic(n,p,q) of polydisperse (DSnOctpNaq)q-n-p micelles plotted against surfactant aggregation numbers n and q for C1 ) 0.0025 mol dm-3 total NaDS and C2 ) 0.0475 mol dm-3 total NaOct.
∂cNa ∂cDS ∂cOct - DDS - DOct ∂Ck ∂Ck ∂Ck (n + p - q)Dmic
∂cmic (15) ∂Ck
and ts is the transference number of species s.
ts )
z2s csDs 4
(16)
z2qcqDq ∑ q)1 Aggregation Numbers. Nagarajan11,12 has developed a detailed molecular thermodynamic model of surfactant aggregation. Using this model, surfactant aggregation numbers for mixed micelles can be evaluated as a function of the ratio of the concentrations of the free surfactant monomers. Figure 2 shows a representative distribution of surfactant aggregation numbers n and p for a solution of (DSnOctpNaq)q-n-p mixed micelles. For this system, the standard deviations in n and p are typically
Diffusion Coefficients for NaDS + NaOct Solutions
J. Phys. Chem. B, Vol. 105, No. 3, 2001 693
TABLE 3: Molecular Parameters for NaDS + NaOct Mixed Micelles at 25 °Ca,b parameter
NaDS
NaOct
Vs/nm3 ls/nm ao/nm2 ap/nm2 δ/nm δH/MPa1/2
0.350 1.67 0.17 0.17 0.545 16.76
0.217 1.04 0.11 0.11 0.600 16.31
predicted cmc’s are plotted against f1. Good agreement is obtained, including the correct prediction of the sharp drop in the cmc in the NaOct-rich composition region (f1 < 0.10). The aggregation numbers are sufficiently large to make the following excellent approximation22
K)
a Parameter values from ref 12. b Parameters: Vs, volume of the hydrocarbon tail; ls, surfactant tail extended length; ao, area per molecule of the core surface shielded from contact with water by polar headgroups; ap, cross-sectional area of the polar headgroup; δ, distance between the hydrocarbon core and the centers of the counterions; δH, Hildebrand solubility parameter for the hydrocarbon tail.
cmic n
cDS cOctpcNaq
=
1 cDSncOctpcNaq
(18)
for the equilibrium constant K for the association reaction nDS+ pOct- + qNa+ h (DSnOctpNaq)q-n-p. At a mixed cmc, cDS ) f1cmc, cOct ) (1 - f1)cmc, cNa ) cmc, and hence
K ) [fn1(1 - f1)p cmcn+p+q]-1
(19)
This relation and the cmc values plotted in Figure 1 are used to evaluate the equilibrium constant K as a function of the monomer fraction of DS-. Concentrations of the Diffusing Species. Given the concentrations C1 and C2 of total NaDS and NaOct components, the concentrations of the solution species are evaluated by solving the equations for mass balance,
C1(total NaDS) ) cDS + ncmic
(20)
C1(total NaOct) ) cOct + pcmic
(21)
electroneutrality,
cNa ) cDS + cOct + (n + p - q)cmic
(22)
and equilibrium of the association reaction
K ) cmic/(cDSn cOctp cNaq) Figure 3. Weight-average surfactant aggregation numbers n and p of (DSnOctpNaq)q-n-p plotted against the surfactant monomer fraction of DS- ions.
10 to 15% of the average values, indicating relatively narrow distributions of aggregation numbers. Details of the evaluation of the aggregation numbers are given in refs 11 and 12. Table 3 gives the parameter values that were used. In Figure 3, the weight-average surfactant aggregation numbers n and p are plotted against f1 ) cDS/(cDS + cOct), the surfactant monomer fraction of DS- ions in the aqueous phase. Over most of the composition range, the surfactant fraction n/(n + p) of DS- ions in the micelles is considerably larger than the corresponding fraction cDS/(cDS + cOct) of DS- ions in the aqueous solution phase. The mixed micelles are therefore enriched in DS- ions, reflecting the stronger micelle-forming tendency of the longer-chain surfactant ion. It is difficult to estimate precise values of q, the number of Na+ counterions bound to the micelles. Previous work3-5,8,18 suggests the approximate values q/n ) 0.83 and q/p ) 0.60 for the fraction of Na+ ions bound to binary aqueous NaDS and NaOct micelles, respectively. We will use the simple linear combination of these binary values
q ) 0.83n + 0.60p
(17)
to estimate the number of Na+ ions bound to the (DSnOctpNaq)q-n-p mixed micelles. Critical Micelle Concentrations. Nagarajan’s model11,12 of surfactant aggregation can also be used to predict mixed cmc’s for NaDS + NaOct solutions. In Figure 1, the measured and
(23)
Species Diffusion Coefficients. It is necessary to obtain values for the diffusion coefficients of the various solution species to complete the model of mixed surfactant diffusion. At 25 °C, the limiting ionic conductivities25-27 (λos ) of the aqueous DS-, Oct-, and Na+ ions are 22.9, 23.1, and 50.1 S cm2 mol-1, respectively. The conductivity data and the identity Ds ) RTλos /z2s F2 give 0.610 × 10-5, 0.615 × 10-5, and 1.334 × 10-5 cm2 s-1 for DDS, DOct, and DNa. Previous studies3-5,22 of diffusion in binary aqueous NaDS solutions suggest that 0.10 × 10-5 cm2 s-1 is a reliable value for the diffusion coefficient of micelles of composition (DS60Na50)-10. The diffusion coefficients of the other micelle species can be estimated by assuming that Dmic is inversely proportional to the micelle radius and hence the cube-root of the micelle volume. Using 0.390, 0.227, and 0.004 nm3 for the volumes11,12 of the DS-, Oct-, and Na+ ions gives
Dmic(n,p,q) )
23.6 (0.390n + 0.227p + 0.004q)
1/3
(0.10 × 10-5 cm2 s-1) (24) for the diffusion coefficient of mixed micelles with aggregation numbers n, p, and q. Comparison of Measured and Predicted Ternary Mutual Diffusion Coefficients. Starting with trial values for the concentrations of the surfactant monomers (cDS, cOct), the curves plotted in Figures 1 and 3 were used to evaluate n, p, q, the cmc, and K. Equations 20-23 were then solved for cDS, cOct, cNa, and cmic. The cycle was repeated until convergence was
694 J. Phys. Chem. B, Vol. 105, No. 3, 2001
Leaist and MacEwan
Figure 4. Main-diffusion coefficients D11 and D22 of aqueous NaDS(C1) + NaOct(C2) solutions for C1 + C2 ) 0.100 mol dm-3 total surfactant plotted against the solute fraction of total NaDS: O, 0, measured values; s, predicted values; - - -, predicted electrostatic contributions D11(E) and D22(E); ‚‚‚, predicted D11 values with the number of bound counterions increased by 5% at each composition.
Figure 5. Cross-diffusion coefficients D12 and D21 of aqueous NaDS(C1) + NaOct(C2) solutions for C1 + C2 ) 0.100 mol dm-3 total surfactant plotted against the solute fraction of total NaDS: O, 0, measured values; s, predicted values; - - -, predicted electrostatic contributions D11(E) and D22(E); ‚‚‚, predicted D12 values with the number of bound counterions increased by 5% at each composition.
achieved for each overall solution composition, C1 (total NaDS) and C2(total NaOct). In Table 2 the predicted ternary diffusion coefficients are listed (in parentheses) for comparison with the measured coefficients. The average magnitude of the difference between the measured and predicted D11, D12, D21, D22 values are 0.04 × 10-5, 0.14 × 10-5, 0.10 × 10-5, and 0.07 × 10-5 cm2 s-1, respectively. Though far from quantitative, the agreement obtained is reasonably good considering the complexity of the system and the necessary approximations. The ability to predict coupled diffusion is a sensitive test of multicomponent diffusion models. In this case the large positive values predicted for crosscoefficient D12 and the relatively small values for D21 are qualitatively correct and encouraging. In Figures 4 and 5, the Dik values for the 0.100 mol dm-3 NaDS + NaOct solutions are plotted against the solute fraction of total NaDS to illustrate the composition dependence of the mixed surfactant diffusion. The electrostatic contributions Dik(E) are also plotted to help interpret the results. The mobility of the Na+ counterions is considerably larger than the mobilities of the other solute species. To prevent charge separation, the electric field generated by NaDS or NaOct gradients slows down the Na+ ions and speeds up the DS-, Oct-, and micellar species. The positive electrostatic contributions D11(E) and D22(E) shown in Figure 4 indicate the enhanced transport of the free and micellar surfactant ions driven by the electric field. The aggregation number and the transference number of the DS- ion both vanish as the solute fraction of NaDS drops to zero. In this limit the tracer diffusion of free and micellar DSions is unaffected by the diffusion-induced electric field, and hence D11(E) is zero. Similarly, D22(E) vanishes in the limit C1/ (C1 + C2) f 1. As shown in Figure 4, the changes in the electrostatic contributions account for the composition dependence of main-diffusion coefficients D11 and D22. The cross-diffusion coefficients for the 0.100 mol dm-3 surfactant solutions are plotted in Figure 5. Notice that the large positive values predicted for D12 and D12(E) are nearly identical.
Therefore, at the NP level of approximation, the large coupled flows of NaDS are driven primarily by the electric field produced by the NaOct concentration gradients. Combining NP eqs 5-8 and the electroneutrality condition jNa - jDS - jOct (n + p - q)jmic ) 0 gives the expression
E) RT DNa∇cNa - DDS∇cDS - DOct∇cOct - (n + p - q)Dmic∇cmic F DNacNa + DDScDS + DOctcOct + (n + p - q)2Dmiccmic (25) for the diffusion-induced electric field. A study of the magnitude of the terms in eq 25 shows that the electric field produced by NaOct gradients is generated primarily by the diffusion of free Na+ and Oct- ions. Most of the coupled flow of the NaDS component is transported by the (DSnOctpNaq)q-n-p micelles, which is more efficient than the transport of free Na+ + DSions because the frictional resistance per mole of NaDS is reduced significantly by the formation of compact micelles.22 In sharp contrast, cross-coefficient D21 and the coupled flows of NaOct are very small. As a consequence of the strong micelleforming tendency of NaDS, NaDS concentration gradients produce relatively small gradients in free Na+ and DS- ions, and hence weaker electric fields than those produced by NaOct gradients. Moreover, because the micelles are highly enriched in NaDS, only small amounts of the NaOct component are transported in micellar form. For these two reasons the values of D21 and D21(E) are significantly smaller than the corresponding values of D12 and D12(E). The present model of mixed surfactant diffusion can be elaborated by including a set of NP equations for a distribution of micellar species. Before this refinement is attempted, however, the errors from other approximations should be considered. For example, activity coefficients, which are not available for NaDS + NaOct solutions, were omitted from the analysis. At 0.200 mol dm-3, the highest concentration used in the present study, the assumption of ideal solutions can produce errors of about 10% in the predicted diffusion coefficients for
Diffusion Coefficients for NaDS + NaOct Solutions “simple” electrolytes, such as aqueous NaCl. A smaller but possibly significant source of error in the predictions is the neglect of changes in the diffusion coefficients caused by changes in solution viscosity with increasing surfactant concentration. Of the various parameters used in the NP equations, the number of bound counterions (q) has the largest uncertainty. Check calculations showed that the predicted values of D21 and D22 are relatively insensitive to changes in q. As shown in Figures 4 and 5, however, a 5% change in value of q at each composition changes the predicted values of D11 and D12 by up to 20% and 5%, respectively. Despite these limitations, the model developed in this paper provides considerable insight regarding the composition dependence, strong coupling, and the importance of electrostatic forces in the diffusion of mixed ionic surfactants. Coupled diffusion in solutions of ionic + nonionic and nonionic + nonionic mixed surfactants will be the subject of future studies. Acknowledgment. We thank the Natural Sciences and Engineering Research Council for the financial support for this research. References and Notes (1) Flick, E. W. Industrial Surfactants; Noyes: Park Ridge, NJ, 1988. (2) Myers, D. Surfactant Science and Technology; VCH: New York, 1992. (3) Evans, D. F.; Mukherjee, S.; Mitchell, D. J.; Ninham, B. W. J. Colloid Interface Sci. 1983, 93, 184. (4) Weinheimer, R. M.; Evans, D. F.; Cussler, E. L. J. Colloid Interface Sci. 1981, 80, 357.
J. Phys. Chem. B, Vol. 105, No. 3, 2001 695 (5) Mitchell, D. J.; Ninham, B. W.; Evans, D. F. J. Colloid Interface Sci. 1984, 101, 292. (6) Kratohvil, J. P.; Aminabhavi, T. M. J. Phys. Chem. 1982, 86, 1256. (7) Langevin, D. Annu. ReV. Phys. Chem. 1992, 43, 341. (8) Lindman, B.; Brun, B. J. Colloid Interface Sci. 1973, 43, 388. (9) Fabre, H.; Kamenka, N.; Lindman, B. J. Phys. Chem. 1981, 85, 3493. (10) Stigter, D.; Williams, R. J.; Mysels, K. J. J. Phys. Chem. 1955, 59, 330. (11) Nagarajan, N. Langmuir 1985, 1, 331. (12) Nagarajan, N. In Mixed Surfactant Systems; Holland, P. M., Rubingh, D. N., Eds.; ACS Symp. Ser. 501; American Chemical Society: Washington, DC, 1992. (13) Rosen, M. J. Surfactants and Interfacial Phenomena, 2nd ed.; Wiley: New York, 1989. (14) Tyrell, H. J. V.; Harris, K. R. Diffusion in Liquids; Butterworth: London, 1984. (15) Leaist, D. G.; Hao, L. J. Chem. Soc., Faraday Trans. 1993, 89, 2775. (16) Leaist, D. G.; Curtis, N. J. Phys. Chem. 1998, 102, 2820. (17) Leaist, D. G.; Curtis, N. J. Solution Chem. 1999, 28, 341. (18) Danielsson, I.; Stenius, P. J. Colloid Interface Sci. 1971, 37, 264. (19) Deng. Z.; Leaist, D. G. Can. J. Chem. 1991, 69, 1548. (20) Leaist, D. G. J. Chem. Soc., Faraday Trans. 1991, 76, 597. (21) Mukerjee, P.; Mysels, K. J. Critical Micelle Concentrations of Aqueous Surfactant Systems; Natl. Stand. Ref. Data Ser.; Natl. Bur. Stand. (U.S.): Washington, DC, 1971. (22) Leaist, D. G. J. Colloid Interface Sci. 1986, 111, 230. (23) Leaist, D. G. Ber. Bunsen-Ges. Phys. Chem. 1991, 95, 113. (24) Newman, J. S. Electrochemical Systems; Prentice Hall: Englewood Cliffs, NJ, 1973. (25) Parfitt, G. D.; Smith, A. L. J. Phys. Chem. 1962, 66, 942. (26) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed.; Academic: New York, 1959. (27) Campbell, A. N.; Lakshminarayanan, G. R. Can. J. Chem. 1965, 43, 1729.
