  We study the first passage time for a polymer, that we call the narrow
encounter time (NETP), to reach a small target located on the surface of a
microdomain. The polymer is modeled as a Freely Joint Chain (beads connected by
springs with a resting non zero length) and we use Brownian simulations to
study two cases: when (i) any of the monomer or (ii) only one can be absorbed
at the target window. Interestingly, we find that {in the first case} the NETP
is an increasing function of the polymer length until a critical length, after
which it decreases. Moreover, in the long polymer regime, we identified an
exponential scaling law for the NETP as a function of the polymer length. {In
the second case, the position of the absorbed monomer along the polymer chain
strongly influences the NETP}. Our analysis can be applied to estimate the mean
first time of a DNA fragment to a small target in the chromatin structure or
for mRNA to find a small target.
