Diffusion of copolymer surfactant to a polymer/polymer interface

I consider a situation in which a homogeneous concentration of diblock copolymer greater than the critical micelle concentration (cmc) is initially added to phase I of a system of two immiscible polymer liquids (I and II), and diffuses to the interface. I consider only diffusion-limited transport, in which micelle dissolution and interfacial adsorption and desorption '' reactions '' are assumed to be rapid, while allowing for both free molecule and micelle diffusion. In an early time regime during which surfactant accumulates on the interface, an '' exclusion zone '' with no micelles is created near the interface. If the surfactant has a non-negligible solubility in phase II, there also exists an intermediate time regime during which copolymer diffuses through the interface without further interfacial accumulation, leading to a time-independent interfacial coverage and interfacial tension while transport continues. The exclusion zone may either close at intermediate times, which leads to an interfacial tension equal to the equilibrium value for a micellar solution, or it may persist, leading to a higher nonequilibrium tension, depending on the rate of diffusion into phase II. Asymptotically exact solutions to the diffusion problem are given for both early and intermediate times, in which the width of the exclusion zone (when present) increases with time t as /t.