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A Triple Layer, Planar Coordinate Model for Describing Counterion Association to Micelles Ching-Chieh Lin and Chad T. Jafvert* School of Civil Engineering, Purdue University, West Lafayette, Indiana 47907-1284 Received April 2, 1999. In Final Form: November 10, 1999 A planar triple-layer model describing ion association to micelles composed of the anionic surfactant dodecylsulfate (DS-) was developed and evaluated. The governing Poisson-Boltzmann equation was solved (i) analytically and (ii) with a numerical shooting method. The second solution was developed to confirm the accuracy of the first and because the analytical solution is only valid for symmetrical electrolytes. The latter solution is applicable also to multiple valence counterion systems. Because the absolute value of the electrical potential at the micelle surface is so large, the classical Debye-Ηu¨ckle approximation is invalid and was not made. To evaluate the resulting algorithms, they were incorporated into an overall speciation model that describes surfactant and metal ion equilibria in aqueous solutions and includes an equation that describes DS- micelle-monomer phase distribution: log [DS-]w ) K1 - eψs/(2.3 kT). The Stern layer thickness (λs) and the Y-intercept of the cmc-electrical potential relationship (K1) were calculated by minimizing residuals between experimental and calculated cmc values over a range of experimental aqueous Na+ concentrations. For sodium dodecylsulfate in NaCl solutions, the analytical model and numerical model provided essentially the same result, the same optimum λs and K1 values, and accurate cmc predictions. For this system, λs ) 1 Å, and K1 ) -4.51.
1. Introduction The application of surfactants as agents to enhance environmental remediation or performance of membrane filtration systems has a recent history. An important element leading to process optimization is understanding the mass balances on the surfactant, on the chemicals that they affect, and on other ions that control speciation of the surfactant(s). In this work, focus is on the mass action equilibrium reactions between aqueous phase monomers and micelles and the influence that counterion (metal) association has on this equilibrium process. Many previous investigators have formulated models that describe surfactant-counterion equilibria in aqueous solutions.1-5 Variations of the Gouy-Chapman model6,7 are those most widely cited for characterizing surfactant micelle-counterion association. An illustration of the basic Gouy-Chapman, or double layer, model is provided in Figure 1. The charged micelle surface (the first layer) is assumed planar, no chemical-specific counterion binding or adsorption generally is considered, and all charges are assumed as point (i.e., infinitesimal) charges. The absolute value of the change in the potential gradient in the diffuse layer (the second layer) decreases exponentially as does the counterion concentration. However, in applying the most basic double-layer model to micelles under typical electrolyte conditions, it soon is discovered that the absolute value of the electrical potential, |ψ|, near the micelle surface is calculated to be so high that counterion * To whom correspondence should be addressed. Tel.: 765-4942196. Fax: 765-496-1107. E-mail:
[email protected]. (1) Helfferich, G. G. Ion Exchange; McGraw-Hill Book Co., Inc.: New York, 1962; p 624. (2) Hiemenz, P. C. Principles of Colloid and Surface Chemistry, 2nd ed.; Marcel Dekker: New York, 1986. (3) Kallay, N.; Bozio, J.; Krznario, I. Colloid Polymer Sci. 1979, 257, 201-205. (4) Kallay, N.; Pastuovic, M.; Matijevic, E. J. Colloid Interface Sci. 1985, 106(2), 452-458. (5) Rathman, J. F.; Scamehorn, J. F. J. Phys. Chem. 1984, 88, 58075816. (6) Gouy, G. J. Phys. Chem. 1910, 9, 457. (7) Chapman, D. L. Philos. Mag. 1913, 25, 457.
Figure 1. Comparison of the planar triple-layer model and the double-layer (or Gouy-Chapman model) for the specific case of anionic surfactant micelles with monovalent counterions.
concentrations at the micelle surface, in turn, are calculated to be unreasonably large. The calculated potential gradient near the surface of the micelle also is unreasonably steep. Despite this, most investigators make the “Debye-Hu¨ckel approximation”, which allows for considerable mathematical simplification; however, it is invalid for large negative or positive surface potentials. To avoid some of these disadvantages of the classic double-layer model, Rathman and Scamehorn5 evaluated two triple-layer models that each included a Stern layer. An illustration of a triple-layer model is provided in Figure 1 for the specific case where the surfactant is negatively charged. In triple-layer models, the charged micelle surface, again, is most often assumed planar and of uniform surface charge density. The second layer is the Stern layer, whose ion concentrations are homogeneous in all directions, whereas the diffuse (third) layer contains
10.1021/la990391z CCC: $19.00 © 2000 American Chemical Society Published on Web 01/11/2000
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Table 1. Constants and Parameters General Constants 10-19
e D o kT N F zi ni°
charge of an electron (1.602 × C/electron) dielectric constant (78.5 for water at 25 °C) permittivity of free space (8.854 × 10-14 C v-1 cm-1) ) o D product of Boltzmann’s constant and absolute temperature (4.11596 × 10-21 joules at 25 °C) Avogadro’s number (6.022 × 1023/mole) Faraday’s constant () e × N ) 96,472 C/mole) valence of ion i, e.g., for Na+, zi ) +1 number of i ions per unit volume bulk phase solution (1/cm3)
r η SDS
radius of a DS- micelle () 18.5 × 10-10 m) (from refs 12, 13) aggregation numbers ) 64.0 (ref 12) surface area of DS- micelle () (-4π×r2×N)/η m2/mole of hydrophilic group ) 404,682 m2/mole of hydrophilic group)
[]
species local concentration (mole/L), where the subscripts w, d, and s refer to the aqueous (or bulk) layer, diffuse layer, and Stern layer concentrations, respectively electrical potential (volts) distance from the micelle surface (Å) Debye-Hu¨ckle parameter for symmetric electrolytes (e2∑ni°zi2/kT)1/2 volume of the Stern layer (4πr2λs/1027) binding ratio ) molesNa in the Stern layer - molesDS in the micelle sum of the Stern and diffuse layer thicknesses (Å)
Constants Specific to Dodecylsulfate
Parameters
ψ x κ VSt βNa λsd
Adjustable Parameters λs Kl
Stern layer thickness (Å) intercept of the cmc-Stern layer potential relationship
cmc versus unbound Na+ concentration data. In this study, the cmc-Stern layer potential equation was derived avoiding the introduction of the empirical constant, Kg, in the cmc-potential relationship.11 In addition, the model is compared to results of a double-layer model for which all algorithms are identical except for the absence of a Stern layer.
ions whose concentrations are homogeneous parallel to the micelle plane. In Rathman and Scamehorn’s “mobile adsorption” model Stern layer ion concentrations are calculated with the Boltzmann equation,8,9 whereas in the “localized adsorption” model, specific binding of Stern layer ions to the micelle surface is assumed.10 In both models, however, no specific distributions of ions between the bulk phase and the micelle were considered, and the diffuse layer potential and ion concentrations were represented as average values without formal calculation of the length or other properties of this layer. Hence, no formal charge balance or mass balances exist on the micelle or aqueous phases. Despite these shortcomings, these previous models are very illustrative in furthering our understanding of these complex systems. In this work, many of the simplifying assumptions mentioned above are examined by constructing a triplelayer model that does not employ them and by applying this model to the case of sodium dodecylsulfate (NaDS) with added NaCl. Specifically, a planar coordinate triplelayer model is developed that includes both a Stern layer and a diffuse layer, where the diffuse layer ion concentrations are calculated at distances from the surface with the original Poisson-Boltzmann equation. Both analytical and numerical solutions were developed with the potential, ψ, and the ion concentrations within the Stern layer considered uniform. After solving for the potential and ion concentration profiles (with either an analytical and numerical solution to the Poison-Boltzmann eq), a cmcStern layer potential equation is employed to calculate the critical micelle concentration (cmc). The Stern layer thickness (λs) and Y-intercept of the cmc-electrical potential relationship are calculated with experimental
2.1. The Planar Triple-Layer Model. The system considered, composed of NaDS and NaCl in water, is relatively simple and previously has been investigated extensively. Our intention, however, is to present equations for this simple case which, without much modification, can be expanded to include systems that contain multiple counterion species and/or divalent or trivalent counterion species. Additionally, our intention is to provide a basic mass action-mass balance equilibrium model in which all species concentrations may be calculated from knowing only total component concentrations and basic thermodynamic constants, which may be evaluated through application of the model to phenomenological data. All model terms and definitions are provided in Table 1,12,13 and all equations are provided in Table 2. Equations T1-T3 of Table 2 are the mass balance equations on the three components, Na+, Cl-, and DS-, respectively. Note that unlike similar equations for homogeneous solutions, these equations include integrals that sum species concentrations over the Stern and diffuse layers. The left- and right-hand sides of eqs T1-T3 are the total moles of ions in the solution. The subscripts, w, s, and d on concentration terms on the right-hand side refer to water, Stern, and diffuse layer concentrations, respectively. The last term in eq T3 is the total moles of
(8) Stigter, D. Rec. Trav. Chim. 1954, 73, 593-610. (9) Hirasaki, G. J.; Lawson, J. B. SPE 10921, 57 Annual Fall Conference and Exhibition of the Society of Petroleum Engineers of AIME; New Orleans, LA, 1982. (10) Adamson, A. W. Physical Chemistry of Surfaces, 4th ed.; John Wiley & Sons: New York, 1982.
(11) Shinoda, K. In Colloidal Surfactants; Shinoda, K., Tamamushi, B., Nakagawa, T., Isemura, T., Eds.; Academic Press: New York, 1963; pp 39-42. (12) Woolley, E. M.; Burchfield, T. E. J. Phys. Chem. 1984, 88, 21552163. (13) Doughty, D. A. J. Phys. Chem. 1981, 85, 3545-3546.
2. Theory
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Table 2. Equations of the Planar Triple-Layer Model for the NaDS/NaCl System Mass Balance Equations
∫[Na ] d V + ∫[Cl ] d V
molesNa+ ) [Na+]wVw + [Na+]sVs + molesCl- ) [Cl-]wVw + [Cl-]sVs molesDS- ) [DS-]wVw + [DS-]sVs +
+
d
(T1)
d
-
d
∫[DS ] -
(T2)
d
d Vd + molesDSmic -
d
(T3)
Charge Balance Equations
[Na+]w ) [Cl-]w + [DS-]w [Na+]sVs +
∫[Na ] +
d
dVd ) [Cl-]sVs +
∫[Cl ] -
d
(T4)
dVd + [DS-]sVs +
∫[DS ] -
d
d Vd + molesDSmic -
(T5)
Stern Layer Potential, ψs (a boundary condition (B.C. 1))
σs ) σ2s 2000kTN
)
∑C i
i,w
(
( ))
VsN F 1 + ([DS-]s + [Cl-]s - [Na+]s) SDS η
( ( ) ) { exp
-zieψs kT
- 1 ) [Na+]w exp
( ) -eψs kT
+ ([DS-]w + [Cl-]w) exp
( )} eψs kT
(T6)
- {[Na+]w + ([Cl-]w + [DS-]w)} (T7)
Boltzmann Equation (to calculate the Stern or diffuse layer ion concentrations, Ci, where i ) Na+, Cl-, or DS-)
Ci at x ) Ci,w exp
( ) -eψx kT
(T8)
Diffuse Layer Potential Distributions (Governing Equation for Numerical Shooting Method) Poisson-Boltzmann Equations: for x ) λs to ∞; B.C. 2: at x f ∞, Ψ ) dΨ/dx ≈ 0
d2ψx dx2
{
)
-e
∑n °z exp i
i
i
( ) -zieψx kT
( )
( )
( )}
d2ψx -eψx eψx eψx -e (volts/Å2) ) [Na+]w exp - [DS-]w exp - [Cl-]w exp 2 kT kT kT dx
(T9)
Analytical Solution for Symmetric Electrolyte (zi ) 1 for 1:1 monovalent electrolytes)
tanh(zieψ/4kT) ) tanh(zieψs/4kT) exp(-κx)
(T10)
where
n°(cm-3) )
[Na+]w (mol/L)N(mol-1) 1000(cm3/L)
)
([Cl-]w + [DS-]w)(mol/L)N(mol-1) 1000(cm3/L)
(T11)
cmc-Potential Relationship (where [DS-]w ) cmc)
log[DS-]w ) Kl -
DS- in the micelle control volume. Equations T4 and T5 are the charge balances for aqueous bulk phase and micelle pseudo-phase, respectively. Concentrations in eq T4 are normalized to the total solution volume, whereas concentrations in eq T5 are based on the Stern and diffuse layer volumes, with calculations based on the Stern and diffuse layer volumes around one micelle. Ions in the diffuse layer experience an electrostatic force due to the net difference between area densities of DSions in the micelle and ions (predominately Na+) in the Stern layer. Calculation of the surface charge density at the plane between the Stern layer and the diffuse layer, σs, requires an estimate of the specific surface area of the micelle, SDS. The value of SDS may be estimated by assuming a spherical geometry and an aggregation number of 64 or equal to (4π×r2×N)/η ) 404,682 m2/mole of anion charge, where r is the radius of the micelle, η is the aggregation number, and N is Avogadro’s number. This value agrees with that originally calculated by Rathman and Scamehorn14 using a method described by
eψs 2.3kT
(T12)
Tanford.15 The surface charge density contributed by DSat the micelle plane equals -F/SDS. This value also is known as the “unbound” surface charge density. The “total” surface charge density may be defined as the surface areanormalized sum of all micellar ions (-F/SDS) and all Stern layer ions, as calculated with eq T6 of Table 2. All bracketed concentration terms in eq T6 are in units of mol/L of Stern layer volume. Equations T7-T8 are used with eq T6 to evaluate the Stern layer potential, ψs, and will be discussed after eq T9 as eq T7 is derived from eq T9. 2.2. Diffuse Layer Governing Equation. The planar Poisson-Boltzmann equation is eq T9 of Table 2, with all terms defined in Table 1. This equation defines the change in the potential gradient at any distance from the micelle surface as a function of the potential at x and the bulk phase ion concentrations. The potential distribution is calculated from the Stern layer to a distance where the (14) Rathman, J. F.; Scamehorn, J. F. Langmuir 1987, 3, 372-377. (15) Tanford, C. J. Phys. Chem. 1972, 76, 3020.
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potential and potential gradient approach zero. Once the potential distribution is known, the ion distributions are calculated with the Boltzmann equation (eq T8). 2.3. Analytical Solution for the Diffuse Layer Potential Distribution. Analytical solutions to eq T9 exist (i) if only symmetrical electrolytes are present in the bulk phase, or (ii) if the potential at the surface has a low absolute value (i.e., the Debye-Hu¨ckle approximation where |ψs or o| , (25.7 mV/|z|) at 25 °C). In our case, the first condition is satisfied. For 1:1 symmetrical electrolytes, eq T9 can be simplified through the following identity:
sinh (zeψ/kT) )
(e(zeψ/kT) - e-(zeψ/kT)) 2
(1)
Substituting eq 1 into eq T9 (for the Na+, DS-, Cl- system),
n°ze d2 ψ ) sinh (zeψ/kT) 2 dx
( )
(2)
where n° ) n+° ) n-°, z ) z+ ) z- ) +1, and ) 0 D The analytical solution to eq 2 is presented in Table 2 as eq T10, where n° is the number of ions/cm3 of bulk phase and is defined by eq T11, and the Debye-Hu¨ckle parameter, κ, for symmetric electrolytes is defined in Table 1. Note that the value of ψs needs to be resolved by numerically solving eqs T6-T8, prior to application of eq T10. Equation T7 is derived from eq T9 and the charge balance equation,
σs ) -
x)∞ F dx ∫x)λ s
(3)
where F (C/m3) is the charge density in the diffuse layer, and (eq T7) defines the relationship between σs, ψs, and the aqueous phase ion concentrations. To solve for the specific case examined in this study, eqs T6-T8 were solved simultaneously with a numerical grid search method. Next, the concentration distributions of counterand co-ions in the diffuse layer are calculated with the Boltzmann equation (eq T8). The charge balance over the entire micelle system was confirmed by integrating the concentration distributions of all ions (Na+, DS-, and Cl-) in the Stern and diffuse layers. 2.4. Numerical Solution for the Diffuse Layer Potential Distribution. Because we wish to examine systems containing nonsymmetrical and mixed electrolytes in the future, a numerical solution for the diffuse layer potential distribution (i.e., Poisson-Boltzmann eq) was developed and compared to the analytical solution for accuracy. As in the case of the analytical solution, ψs and σs were calculated with eqs T6-T8 with a numerical grid search method. Knowing the value of ψs, the potential distribution was calculated with eq T9, solving this equation with a numerical shooting method. In this method, one of the boundary conditions is ψ ≈ dψ/dx ≈ 0 at x f ∞, and the other boundary condition is an initial guess of the potential gradient at the Stern-diffuse layer interface (dψ/dx|xs). In the shooting method, a guess on the initial gradient (dψ/dx|xs) is made and the potential is calculated with distance. Over or under shooting the other boundary condition (ψ ≈ dψ/dx ≈ 0 at x f ∞) provides information for a new guess on the initial gradient. The initial gradient continues to be adjusted until the calculated potential and potential gradient at x f ∞ approach zero. After satisfying these conditions, all ion concentration
distributions in the Stern and diffuse layers are calculated with eq T8. Integration of ion concentrations with distance is performed to calculate mass balances and confirm the charge balance over the entire micelle. 2.5. Cmc-Potential Relationship. For any monovalent surfactant, the logarithm of the critical micelle concentration (cmc) is a linear function of the logarithm of the total concentration of an oppositely charged counterion.16 This relationship cannot be derived without making numerous simplifying assumptions, including constant surface charge density at any electrolyte concentration. A thermodynamically more rigorous equation relating the cmc to the “surface” potential can be derived by equating the electrochemical potential of the surfactant in the bulk phase to that in the micellar phase.11 The electrical work required to transport a monovalent surfactant anion to the micelle surface should equal -∆zeψ, where ∆z equals 1. However, due to the adsorption of counterions, Hobbs16 proposed that on average, the work required is less than this, or approximately -Kgeψ. The empirical constant, Kg, is often referred to as “the effective coefficient of electrical energy” and is necessary if the “surface” potential, ψ, is calculated without considering the contribution to ψ of associating counterions at the micelle surface. Hence, ψ is a hypothetical potential of a completely ionized or “unbound” micelle, ψunb. Moroi et al.17 define Kg more quantitatively as “the ratio of the real surface electrical potential to the completely ionized hypothetical surface electrical potential”. In our case, by invoking a Stern layer around the micelle, Moroi’s “real surface electrical potential” may be approximated by the Stern layer potential, ψs, and the electrical work required to transport a monovalent surfactant anion from the bulk phase to the micelle surface now equals -eψs. Through this interpretation, one commonly invoked parameter (Kg) is eliminated, and another onesthe Stern layer thickness (λs)sis introduced. From λs and r, the Stern layer volume, Vs, is calculated. The final equation relating the cmc to ψs is provided as eq T12 (in Table 2) and, again, is derived by equating the electrochemical potential of the surfactant in the bulk phase to that in the micellar phase. To summarize, the final “numerical” solution consists of all equations listed in Table 2 except for eq T10-T11. The “analytical” solution replaces eq T9 with eqs T10T11, however, both solutions require a numerical grid search method (applied to eqs T6-T8) to solve for the potential and ion distributions. 3. Results and Discussion 3.1. Parameter Estimation. The only unknown constants in the model are the Stern layer thickness, λs, and the Y-intercept of the cmc-electrical potential relationship, K1. These two terms were used as fitting parameters to minimize residuals between experimental and model calculated dodecylsulfate cmc values reported by Stellner and Scamehorn.18 The data are shown in Figure 2, plotted as log(cmc) versus log [Na+]w. These data were obtained from surface tension measurements, where the cmc is located at the inflection point on the surface tension versus log [NaDS] curve. Singular values for λs and for K1 were determined by finding the minimum residual over all cmc, [Na+]w data pairs reported. If Kg is invoked as a parameter, its value is explicitly proportional to the slope of the log cmc-“unbound” potential relationship, whereas the value (16) Hobbs, M. E. J. Phys. Colloid Chem. 1951, 55, 675. (17) Moroi, Y.; Nishikido, N.; Uehara, H.; Matuura, R. J. Colloid Interface Sci. 1974, 50, 255-264. (18) Stellner, K. L.; Scamehorn, J. F. Langmuir 1989, 5, 70-77.
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Figure 2. Experimental and calculated dodecylsulfate cmc (M) values as a function of aqueous phase Na+ (M). The experimental data are from Stellner and Scamehorn.18
of λs implicitly affects the slope. One of the reasons for minimizing residuals on cmc data is that, at the cmc, Na+ associated with micelles is negligible compared to [Na+]w, so it may be assumed that [Na+]w equals the NaCl dose plus the cmc, and the cmc equals the total DS concentration. In practice, upon substituting eq T8 into eq T6 for each ion, the new eq T6 plus eqs T7 and T12 compose a set of three sufficient equations with three unknown values (ψs, σs, and [DS-]w) for which the residuals are found by adjusting Vs and K1. From Vs, λs is calculated directly. For the data, the optimum λs and K1 values are 1.0 Å and -4.51 (unitless), respectively. With these values, the model predicts a straight-line function between log cmc and log [Na+]w as shown in Figure 2,
log(cmc) ) -0.695 log [Na+]w - 0.369
Figure 3. Potential (ψ) distributions calculated with the triplelayer and double-layer models from the plane of the DS- micelle surface.
(4)
(R2 ) 1.00) with an average error of 5.78% between experimental and calculated cmc values. The value of K1 is nearly the same as that calculated by Stellner and Scamehorn18 from the thermodynamic data reported by Moroi et al.17 of -4.58. The calculated value of λs also seems reasonable, as the hydrated radius of the aqueous Na+ ion is about 3.6 Å.19 Many underlying assumptions affect the magnitude of λs and include (i) the value used for the average radius of the micelles () 18.5 Å), and (ii) all assumptions inherent in Gouy-Chapman theory, such as point charge ions and constant dielectric properties through the diffuse and Stern layers. Certainly, spatial overlapping of surfactant headgroups with Stern layer counterions occurs, resulting in this distance representing an “effective” Stern layer thickness. This interpretation is consistent with Rathman and Scamehorn’s5 proposed “mobile” adsorption model. If the Stern layer thickness and volume are set to zero, the calculated results are those of a double-layer or GouyChapman model. Assuming the same value for K1, the calculated log cmc versus log [Na+] line for this condition is also provided in Figure 2. 3.2. The Potential and Ion Distributions around the Micelles. Figures 3-5 show the calculated potential and ion concentration distributions of both the triple- and the double-layer models for the specific case of 0.1 M [Na+]w. From these figures, it can be seen that the Na+ concentration near the surface of the micelle calculated with the double-layer model (λs ) 0) is unreasonably large (19) Morel, F. M. M.; Hering, J. G. Principles and Applications of Aquatic Chemistry; John Wiley & Sons: New York, 1993; p 71.
Figure 4. Cation concentration distributions calculated with the triple-layer and double-layer models from the plane of the DS- micelle surface.
Figure 5. Anion concentration distributions calculated with the triple-layer and double-layer models from the plane of the DS- micelle surface.
at over 16 M. This high concentration results from the large negative value of the surface potential of less than -130 mV at the surface of a DS- micelle, which is devoid of counterions. Including a Stern layer decreased the Na+
Model for Counterion Association to Micelles
Figure 6. ψs (V) as a function of [Na+]w (M).
Langmuir, Vol. 16, No. 6, 2000 2455
Figure 8. The decrease in λsd (Å) with increasing [Na+]w (M).
data reported in Figure 2. The difference, (1 - βNa), equals all terms in parentheses on the right-hand side of eq T6, where the sum [DS-]s + [Cl-]s is negligible. Figure 7 shows that βNa increases from 0.31 to 0.35 as [Na+]w increases from 0.01 to 0.94 M and that a nearly linear relation exists between βNa and [Na+]w,
βNa ) 0.0382 [Na+]w + 0.314
Figure 7. The calculated binding fraction, βNa, with increasing [Na+]w.
concentration at the surface by one-half and results in more reasonable potential and ion concentration gradients over distances of the scale of atomic radii. Figure 5 indicates that anion concentrations and mole numbers in the Stern and diffuse layer are negligible compared to those in the bulk phase and, for all practical purposes, may be ignored. Figure 6 shows the calculated value of ψs as a function of [Na+]w. Recall that the relationship between these two parameters is defined through eq T6-T8 and T12. The model provides a straight-line correlation between ψs and log [Na+]w,
ψs ) 0.041 log [Na+]w - 0.0485
(R2 ) 1.00) (5)
The symbols in Figure 6 correspond to the 12 values of [Na+]w reported in Figure 2. Because both eqs T12 and 4 are predicted to be straight-line functions by the model, eq 5 also must be a linear relationship and may be derived from these other two. 3.3. Calculated Binding Fractions. Although the model assumes no specific ion association occurs at the micelle surface, the ratio of the number of Na+ ions in the Stern layer to DS- ions in the micelle provides a good estimate for the fraction of ions that may be considered associated or bound. This ratio, referred to as the “binding fraction”, or for Na+ as βNa, is reported in Figure 7, calculated at the same values of [Na+]w as the experimental
(R2 ) 1.00)
(6)
Rathman and Scamehorn14 have calculated βNa ) 0.65 for dodecylsulfate micelles in the absence of added electrolyte. The major reason for discrepancy between values is the way in which βNa is calculated. In their case, βNa is calculated based on sodium ion-selective electrode measurements, which provide estimates on [Na+]w. For stirred solutions, this value may include some diffuselayer Na+ due to shear forces around the micelles. Values of βNa calculated through their method, along with the calculated distribution profile of Na+ in the diffuse layer, however, may provide a reasonable estimate of the location of the shear plane in the diffuse layer and the hydrodynamic radius of the micelle. 3.4. Diffuse Layer Thickness. Recall that the length of the Stern layer is a fitting parameter and is assumed constant for the specific case examined, whereas the length of the diffuse layer is a calculated boundary condition and decreases with increasing [Na+]w. To calculate the diffuse layer thickness, convergence criteria were required for ψ and dψ/dx at the limit of the diffuse layer. The convergence criteria (or residual errors on these terms) themselves, were adjusted to smaller and smaller values until no appreciable change in the calculated diffuse layer thickness was observed. The point at which this occurred was |ψ| < 10-6 V and |d ψ/dx|
