Modeling diffusion in miscible polymer blend films

Recent experiments designed to probe polymer transport in the bulk and in the vicinity of surfaces have examined the interdiffusion of multilayer sandwiches of isotopically labeled polymers. The measured time dependent concentration profiles normal to the surface are typically fit to Fick's law, with a single fitting parameter, the mutual binary diffusion coefficient (MBDC). The resulting MBDCs are found to vary over a broad range of film thicknesses and time, with the time dependence being viewed as a unique signature of the reptation mechanism of long chain motion, and the thickness dependence being attributed to the slowing down of chain dynamics near surfaces. Since the experiments are conducted at finite concentration, the MBDC, which is a product of the bare mobility and the concentration derivative of the chemical potential, could be dominated by the time and thickness dependence of this second term (which is ignored in Fick's law). To quantify this conjecture we consider the more rigorous Cahn formulation of the diffusion problem in terms of chemical potential gradients. We use square gradient theory to evaluate chemical potentials, and fit the resulting time dependent concentration profiles to the analytical solution of Fick's law. By thus mimicking the experimental analysis we find that the apparent MBDCs vary with time as t(-1/2) at short times, in good agreement with existing experiments. We show that this time dependence reflects the system's desire to minimize concentration gradients, a fact ignored in Fick's law. Since these arguments make no reference to the mechanism of chain motion, we argue that the time dependence of MBDC derived from interdiffusion experiments does not provide unequivocal support for the reptation mechanism of long chain transport. The MBDC values, which also vary with the degree of confinement, are predicted to increase with decreasing thickness for model parameters corresponding to experimental systems. In contrast, since the experimental fits yield an opposite trend, we suggest that the bare mobility of the chains decreases strongly with decreasing thickness. These findings strongly support the idea that the chains are "pinned" irreversibly to the surfaces, in good agreement with other, independent experiments.