Surface-reaction-controlled particle growth ($\dot{a}\simeq k_s(c_{\infty }-c_{\mathrm{eq}})/c_p$)

When attachment is much slower than exterior diffusion, the surface concentration stays close to the reservoir value. An attachment coefficient $k_s$ with units of velocity gives nearly constant radial growth in the dilute-feed approximation. For a sphere the ratio $k_{s}a/D$ distinguishes attachment control from transport control. A finite attachment rate leads to a Robin boundary condition rather than a fixed equilibrium surface concentration.