Surface Cooper-Mann theory

The complementary application of film balance and electron spin resonance techniques has given information regarding molecular motion in monolayers. The molecules used, 3-nitroxide androstan 17fi-ol and 3-nitroxide cholestane, have the molecular geometry required for a test of the Cooper-Mann theory of surface viscosity. These ellipsoidal molecules show distinctly different surface behavior as shown by the complete collapse of the hyperfine lines in the case of the cholestane derivative and the apparent lack of exchange broadening for the androstane derivative. These results are modeled in terms of transitional surface regions. 

The Cooper-Mann theory of monolayer transport was based on the model of a sharply localized interfacial region in which ellipsoidal molecules were constrained to move. The surfactant molecules were assumed to be massive compared with the solvent molecules that made up the substrate and a proportionate part of the interfacial region. It was assumed that the surfactant molecules had many collisions with solvent molecules for each collision between surfactant molecules. A Boltzmann equation for the singlet distribution function of the surfactant molecules was proposed in which the interactions between the massive surfactant molecules and the substrate molecules were included in a Fokker-Planck term that involved a friction coefficient. This two-dimensional Boltzmann equation was solved using the documented techniques of kinetic theory. Surface viscosities were then calculated as a function of the relevant molecular parameters of the surfactant and the friction coefficient. Clearly the formalism considers the effect of collisions on the momentum transport of the surfactant molecules. 

Theoretical predictions of surface viscosity have been made by Blank and Britten (6) who constructed a formalism based on fluctuations. Cooper and Mann (7) posed a dilute gas kinetic theory of monolayer transport that involved a modified Boltzmann equation which could be solved in detail. The substrate surface coupling could be easily accounted for quantitatively. The results of these calculations were similar in the sense that surface viscosities of the order of 10"10 g/sec were computed for typical monolayer systems. Even when the extensions involving the dense gas region were formulated and computed, the expected surface viscosity numbers were around 10"8 g/sec. 

The construction of Cooper and Mann (7) for the surface viscosity includes the substrate effect by a model that represents the result of very frequent molecular collisions between the small substrate molecules and the larger molecules of the monolayer. This was done by adding a term to the Boltzmann equation for the 2D singlet distribution function that is equivalent to the friction coefficient term of the Fokker-Planck equation from which Equations 24 and 25 can be constructed. Thus a Brownian motion aspect was introduced into the kinetic theory of surface viscosity. It would be interesting to derive the collision frequency of Equation 19 using the better model (7) and observe how the T/rj variable of Equation 26 emerges. 

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