Note: Improved boundary homogenization for a sphere with an absorbing cap of arbitrary size

Finding accurate approximations for the effective reactivity of a structured spherical target with a circular absorbing patch of arbitrary size is a long-standing problem in chemical physics. In this Note, we reveal the limitations of the empirical approximation proposed in Dagdug et al., [J. Chem. Phys. 145, 214101 (2016)]. We show that the original approximation fails at large patch surface fractions σ and propose a simple amendment. The improved approximation is validated against a semi-analytical solution and is shown to be accurate over the entire range of σ from 0 to 1. This approximation also determines the probability of reaction on the patch and the capacitance of such a structured target.