Time dependent diffusion in a disordered medium with partially absorbing walls: A perturbative approach

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We present an analytical study of the time dependent diffusion coefficient in a dilute suspension of spheres with partially absorbing boundary condition. Following Kirkpatrick (J. Chem. Phys. 76, 4255) we obtain a perturbative expansion for the time dependent particle density using volume fraction f of spheres as an expansion parameter. The exact single particle t-operator for partially absorbing boundary condition is used to obtain a closed form time-dependent diffusion coefficient D(t) accurate to first order in the volume fraction f. Short and long time limits of D(t) are checked against the known short-time results for partially or fully absorbing boundary conditions and long-time results for reflecting boundary conditions. For fully absorbing boundary condition the long time diffusion coefficient is found to be D(t)=5 a²/(12 f D₀ t) +O((D₀t/a²)⁻²), to the first order of perturbation theory. Here f is small but non-zero, D₀ the diffusion coefficient in the absence of spheres, and a the radius of the spheres. The validity of this perturbative result is discussed.