 Smoluchowski association rate

For an absorbing spherical target of radius $a$ in an infinite uniform reservoir, steady diffusion gives $c(r)=c_{\infty }(1-a/r)$. Integrating the inward flux yields $k=4\pi D_{\mathrm{rel}}a$, where $D_{\mathrm{rel}}=D_A+D_B$ if both partners diffuse. Here $k$ multiplies number densities; its units are volume per time. Molecular orientation, finite-domain transients and imperfect capture require corrections.