Smoluchowski association rate ($k=4\pi D_{\mathrm{rel}}a$)

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.

If $N$ independent well-mixed searchers each encounter a small absorbing target with approximate exponential hazard $k_a/V$, their minimum waiting time has hazard $N{k}_a/V$. The Smoluchowski association rate is $k_a=4\pi D_{\mathrm{rel}}{r}_0$. Cell geometry, orientational capture and nonequilibrium spatial distributions can change this idealized estimate; an arbitrary nonexponential first-passage mean does not in general divide by $N$ exactly.