## Applications of Thermodynamic Linkage

B. 1. Diffusion Coefficients of Micelles by Using Electrochemical Probes

Micelles are spherical or rod-shaped aggregates of surfactant molecules in water that can be characterized by their diffusion coefficients (D) [3], Electrochemical measurements of current or charge in an electrochemical cell under an applied voltage can be used to estimate diffusion coefficients of the species being electrolyzed. A variety of specific electrochemical methods can be utilized for this purpose [3],

If D is available for a spherical micelle, its average radius (r) can be obtained from the well-known Stokes-Einstein equation:

where k is Boltzmann's constant, rj is the viscosity of the solution, and T is temperature in Kelvins.

For a reversibly electrolyzed probe with a large binding affinity for the micelle, the electrochemically measured diffusion coefficient will reflect the equilibrium in eq. (6.3), with the overall equilibrium constant in eq. (6.4). Values measured in such systems are called apparent diffusion coefficients (D') because they depend on the equilibrium of the probe (X) with the micelle (M) [3, 4]. That is, D' depends on the amount of free probe in the solution.

The Single-Micelle Model Micelles actually have a distribution of sizes in aqueous solutions [5]. For a solution with a single narrow size distribution of micelles, where the probe-micelle equilibrium is rapid with respect to the measurement time, D' is given by

where the fractions are those defined in Table 6.1. D0 is the diffusion coefficient of the free probe, and D, is the diffusion coefficient of the probe bound to the micelle. The binding (on) rates and dissociation (off) rates (cf. eq. (6.3) for small solutes in micellar systems are on the time scale of a few milliseconds to microseconds. The electrochemical methods used to measure D' are in the 10 millisecond to seconds time scale [3]. Therefore, eq. (6.26) is expected to hold for the majority of electrochemical probe experiments in micellar solutions with a single size distribution.

Insertion of fractions/x,0 and/x,i from Table 6.1 into eq. (6.26) provides a model appropriate for analysis of the data (Table 6.3). If the goal of the data analysis is to obtain the diffusion coefficient of the micelle (£>,), D' vs. Cx data obtained at constant surfactant concentration (i.e., constant CM) is the best choice. This is because micelle size generally depends on the surfactant concentration, so if surfactant concentration, and consequently CM, is varied, the value of D] is no longer constant but depends on CM. Only an average value of £>i over the range of CM used can be obtained when CM is the independent variable.

An example of regression analysis onto the single micelle model in Table 6.3 using D' vs. Cx data for ferrocene as the probe shows an excellent fit for n = 3 (Figure 6.1). In these analyses, a series of nonlinear regressions

Table 6.3 Model for Single Size Distribution of Micelles

Assumptions: [A"] <i C, (probe tightly bound); K" is an apparent equilibrium constant Regression equation:

1 + K"CmC£-' 1 + K"CuCrl Regression parameters: Data:

Special instructions: Run a series of nonlinear regressions with successively increasing fixed integer n values beginning with n = 2 until the best standard deviation of regression is achieved are run with n fixed at an integer value, beginning at n = 2. The n value at which the standard deviation of the regression is a minimum is taken as the best fit. In this case, for n = 3 the deviation plot was random. Values of n = 4 or 5 gave larger standard deviations.

The characteristic decrease in D' as Cx increases is explained by the single micelle model in Table 6.3 for 0.15 M hexadecyltrimethylammonium bromide in 0.1 M tetraethylammonium bromide (Figure 6.1), a system that most probably features rod-shaped micelles. A micelle diffusion coefficient of 0.33 X 10 6 cmV1 was obtained from this analysis, which was similar to the value of 0.38 X 10"6 cmV1 found for rod-shaped micelles of the same surfactant in 0.1 M KBr solution [3, 6],

The Two-Micelle Model In applications to a series of surfactant solutions of concentrations above 0.05 M, modifications to the single micelle model were required to account for polydispersity caused by the presence of two size distributions of micelles. These systems typically contain spherical and rod-shaped micelles in the same solution [6], with the rod-shaped micelle having the smaller diffusion coefficient.

The following equilibria involving micelle 1 (Mi) and micelle 2 (M2) need to be considered in the two-micelle model, and and

M2 + mX ^ M2Xm with overall concentration equilibrium constants

PROBE CONC., mM

Figure 6.1 Influence of concentration of ferrocene (probe) on D' measured by cyclic voltammetry in 0.15 M hexadecyltrimethylammonium bromide/0.1 M tetraethylammonium bromide. Points are experimental data; line is the best fit to the data (n = 3) by the single micelle model (Table 6.3). (Adapted with permission from J. F. Rusling et al, Colloids and Surfaces, 1990, 48, 173-184, copyright by Elsevier.)

PROBE CONC., mM

Figure 6.1 Influence of concentration of ferrocene (probe) on D' measured by cyclic voltammetry in 0.15 M hexadecyltrimethylammonium bromide/0.1 M tetraethylammonium bromide. Points are experimental data; line is the best fit to the data (n = 3) by the single micelle model (Table 6.3). (Adapted with permission from J. F. Rusling et al, Colloids and Surfaces, 1990, 48, 173-184, copyright by Elsevier.)

Now, three relevant species contribute to the apparent diffusion coefficient D'. The fractions of free probe (fx.o), probe bound to M\ (fx. 1), and probe bound to M2 (fx.2) must all be represented in the model. The expression for D' is

The model for use in nonlinear regression was obtained by inserting the correct forms of the/*,,- into eq. (6.31), and it is summarized in Table 6.4. When using this model, a series of fixed integer values of n and m must be tested until the n, m combination giving the minimum standard deviation of regression is found.

As with the single-micelle model, the best choice for experimental data when the micellar diffusion coefficients D, and D2 are the desired parameters is D' vs. Cx- Regression analysis using the one- and two-micelle models is illustrated for a micellar solution 0.1 M in sodium dodecylsulfate (SDS) containing 0.1 M NaCl using methylviologen as the electroactive probe. As in any situation in which successively more complex models are suspected, the simple model should be applied first. Therefore, an initial series of regressions onto the single-micelle model gave the best standard deviation for n = 3. However, a graph of the regression line does not indicate a good fit to the data (Figure 6.2), and the deviation plot was distinctly nonrandom [6]. On the other hand, regressions of the same data onto the two-micelle model (Table 6.4) gave the best standard deviation for n = 4, m = 8. The graph of the regression line for the two-micelle model passed through all the data points (Figure 6.3) and gave a random deviation plot.

The two-micelle model contains five parameters and the one-micelle model has three parameters, so these two models will have different degrees of freedom for a given data set. It is necessary to use the extra sum of

Table 6.4 Model for the Diffusion of Two Different Sized Micelles in the Same Solution

Assumptions: [A-] <§ C, (probe tightly bound): K" and K'" are apparent equilibrium constants Regression equation:

1 + K7CM|C? 1 + K'"CM2C'x ' 1 + K'iCM|C? 1 + K'{<CM2CH 1

DIK'ICmiC'X 1

Regression parameters: Data:

D„ D, D2 (K'ICli")" (ICVC'^y D' vs. C, or D' vs. CM

Special instructions: Run a series of nonlinear regressions with successively increasing fixed integer n and m values beginning with n = 2, m = 2 until the best standard deviation of regression is achieved which indicates the best values of n and m

Figure 6.2 Influence of concentration of methyl viologen (probe) on D' measured by cyclic voltammetry in 0.1 M sodium dodecylsulfate/0.1 M NaCl. Points are experimental data; line is the best fit to the data (n = 3) by the single-micelle model (Table 6.3) (adapted with permission from [6], copyright by the American Chemical Society).

squares F test (Section 3.C.1) to examine the probability of best fit of the five-parameter vs. the three-parameter model. For the set of data in Figure 6.3, the error sum was 0.242 X 10~6 cmV1 for the three-parameter model and 0.005 X 10"6 cmV for the five-parameter model. Analysis using eq.

Figure 6.3 Same data as in Figure 6.2. Line is the best fit from regression onto the two-micelle model with n = 4, m = 8 (Table 6.4) (adapted with permission from [6], copyright by the American Chemical Society).

'in rvl

Figure 6.3 Same data as in Figure 6.2. Line is the best fit from regression onto the two-micelle model with n = 4, m = 8 (Table 6.4) (adapted with permission from [6], copyright by the American Chemical Society).

(3.15) with statistical F tables showed that the two-micelle model gave the best fit to the data with a probability of 99%.

Electrochemically determined diffusion data for number of micellar and microemulsion systems have been analyzed by the models in Tables 6.3 and 6.4. Results are in excellent agreement with diffusion coefficients of the same micelles estimated by alternative methods not requiring probes [3, 6]. Some of these data are summarized in Table 6.5. An advantage of the electrochemical probe method over many alternatives is that analysis for the presence of two micelles is relatively easy. The values of D0 in these systems were similar to or a bit smaller than the D values of the probes in water. Slightly smaller values of D0 than in water are expected because of the obstruction effect from the other micelles in solution [3, 4], In some cases, D0 may be influenced by association of the probe with submicellar aggregates [6].

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