Moment theory for the analytical determination of rate constants for solute permeation at the interface of spherical molecular aggregates

Abstract

Moment equations were developed on the basis of the Einstein equation for diffusion and the random walk model to analytically determine the rate constant for the interfacial solute permeation from a bulk solvent into molecular aggregates (kin) and the inverse rate constant from the molecular aggregates to the bulk solvent (kout). The moment equations were in good agreement with those derived in a different manner. To demonstrate their effectiveness in one concrete example, the moment equations were used to analytically determine the values of kin and kout of three electrically neutral solutes, i.e. resorcinol, phenol, and nitrobenzene, from the first absolute (μ1A) and second central (μ2C) moments of their elution peaks, as measured by electrokinetic chromatography (EKC), in which the sodium dodecyl sulfate (SDS) micelles were used as a pseudostationary phase. The values of kin and kout should be determined with no chemical modifications and no physical action with the molecular aggregates because they are dynamic systems formed through weak interactions between the components. The moment analysis of the elution peak profiles measured by EKC is effective to unambiguously determine kin, kout, and the partition equilibrium constant (kin/kout) under appropriate experimental conditions.

Kinetic study on solute permeation at the interface of molecular aggregates by partial filling capillary electrophoresis

“…The molecular weight of SDS micelle is calculated as ca. 1.4 × 10 4 -1.7 × 10 4 , because that of SDS monomer is 288 and the aggregation number of SDS micelle was reported as about 50-60 in water [1,24,[29][30][31][32][33][34][35]. It is about two orders of magnitude larger than the molecular weight of the solutes used in this study ranging from 94 to 123.…”

“…Electrophoretic behavior under partial filling conditions is considered in the five cases described above. In the following, however, the development of moment equations only in the case ( 1) is explained because it was reported in the previous study [24] that the migration direction of SDS micelles is opposite to that of electrically neutral solutes. Detailed explanations are provided in Supporting Information for the development of moment equations in the cases (1)-(5).…”

“…2 ) to μ 2C are analyzed on the basis of the moment theory. Moment equations for analyzing separation behavior in EKC have already been developed in previous studies [24]. The sum of the two contributions to μ 2C (σ B,L 0 ) 2 is calculated in the unit of m 2 as follows:…”

“…Equation (23) indicates that K p can be calculated from μ 1A because Z, M , and Z M 0 are experimental parameters, which are already known. The M value was calculated on the basis of information about cmc and the aggregation number of SDS micelles reported in the previous study [24]. Table 1 lists the K p values of the three solutes determined by the first approach from μ 1A data measured under various conditions of M and The results in the previous study (as in [24]).…”

“…We have applied the moment theory to the analysis of CE data to study intermolecular interactions and solute permeation at the interface of spherical molecular aggregates (MAs), for example, surfactant micelles and liposomes, from equilibrium and kinetic points of view. Moment equations were developed for various ACE [22,23] and EKC [24,25] systems on the basis of the combination of the Einstein equation for diffusion and the random walk model [26] and the combination of the Galilean principle of relativity and the Laplace transformation.…”