Diffusion in Inhomogeneous Systems

Single-particle diffusion, also known as Brownian motion, is the random motion of a particle in space due to collisions with its surrounding medium. The theory of diffusion in a homogeneous medium, one with a constant diffusion coefficient, is well established and recognized. The theoretical basis of diffusion in heterogeneous environments, where the diffusion coefficient varies in space, is far less developed. In this thesis, we employ underdamped Langevin dynamics simulations and analytical methods for studying these systems, which are common in many fields, including biomedical engineering. We investigate two model systems: The first is the dynamics of Brownian particles in a heterogeneous one-dimensional medium with a spatially-dependent diffusion coefficient of a power-law form, at constant temperature. At long times, provided that the friction coefficient drops slowly enough (or grows at any rate), both approaches yield identical results, corresponding to subdiffusion or superdiffusion. In the opposite case, where the friction coefficient decays more quickly, the diffusion equation foresees that the particles accelerate, while the Langevin equation predicts ballistic motion at long times. We argue that the phenomenon of particle acceleration in an isothermal medium is unphysical, and demonstrate that in this case underdamped Langevin dynamics simulations must be used. In the second case study, we use a one-dimensional two layer model with a semi-permeable membrane to model the diffusion of a therapeutic drug delivered from a drug-eluting stent (DES). The rate of drug transfer from the stent coating to the arterial wall is calculated by using underdamped Langevin dynamics simulations. Our results reveal that the membrane has virtually no delay effect on the rate of delivery from the DES.