Aggregation number-based degrees of counterion dissociation in sodium n-alkyl sulfate micelles.

Values of the degree of counterion dissociation, alpha, for sodium n-alkyl sulfate micelles, denoted by SN(c)S, where N(c) is the number of carbon atoms in the alkyl chain, are defined by asserting that the aggregation number, N, is dependent only on the concentration, C(aq), of counterions in the aqueous pseudophase. By using different combinations of surfactant and added salt concentrations to yield the same value of N, alpha can be determined, independent of the experimental method. Electron paramagnetic resonance measurements of the hyperfine spacings of two nitroxide spin probes, 16- and 5-doxylstearic acid methyl ester (16DSE and 5DSE, respectively), are employed to determine whether micelles from two samples have the same value of N to high precision. The EPR spectra are different for the two spin probes, but the values of alpha are the same, within experimental error, as they must be. In agreement with recent work on S12S and with prevailing thought in the literature, values of alpha are constant as a function of N. This implies that the value of alpha is constant whether the surfactant or added electrolyte concentrations are varied. Interestingly, alpha varies with chain length as follows: N(c) = 8, alpha = 0.42 +/- 0.03; N(c) = 9, alpha = 0.41 +/- 0.03; N(c) = 10, alpha = 0.35 +/- 0.02; N(c) = 11, alpha = 0.30 +/- 0.02 at 25 degrees C and N(c) = 13, alpha = 0.22 +/- 0.02; and N(c) = 14, alpha = 0.19 +/- 0.01 at 40 degrees C. A simple electrostatic theoretical description, based on the nonlinear Poisson-Boltzmann equation for the ion distribution around a charged sphere, was compared with the experimental results. The theory predicts values of alpha that are in reasonable agreement with experiment, nicely predicting the decrease of alpha as N(c) increases. However, the theory also predicts that, for a given value of N(c), alpha decreases as N increases. Moreover, this decrease is predicted to be different if N is increased by adding salt or by increasing the surfactant concentration. A modification to the theory in which dissociated counterions contribute to the ionic strength while added co-ions (Cl(-)) do not, brings theory and experiment into closer accord. Assuming alpha to be constant versus N permits a direct application of the aggregation number-based definition of alpha using time-resolved fluorescence quenching to measure values of N as well as other experimental parameters that vary monotonically with N, such as the microviscosity measured with spin probes and the quenching rate constant. For S13S micelles at 40 degrees C, alpha = 0.20 +/- 0.02 is derived from N; alpha = 0.21 +/- 0.02 from the microviscosity, and alpha = 0.21 +/- 0.02 from the quenching rate constants, in agreement with the hyperfine spacing results. The aggregation numbers for S13S are well described by the power law N = N degrees (C(aq)/cmc(0))(gamma), where cmc(0) is the critical micelle concentration in the absence of added salt, N degrees = 67, and gamma = 0.26.