  This papers addresses the connection between two classical models of phase
transition phenomena describing different stages of the growth of clusters. The
Becker-D\"oring model (BD) describes discrete-sized clusters through an
infinite set of ordinary differential equations. The Lifshitz-Slyozov equation
(LS) is a transport partial differential equation on the continuous half-line
$x\in (0,+\infty)$. We introduce a scaling parameter $\varepsilon>0$, which
accounts for the grid size of the state space in the BD model, and recover the
LS model in the limit $\varepsilon\to 0$. The connection has been already
proven in the context of outgoing characteristic at the boundary $x=0$ for the
LS model, when small clusters tend to shrink. The main novelty of this work
resides in a new estimate on the growth of small clusters, which behave at a
fast time scale. Through a rigorous quasi steady state approximation, we derive
boundary conditions for the incoming characteristic case, when small clusters
tend to grow.
