  The article presents a comprehensive study of counterion dynamics around a
generic linear polyelectrolyte (PE) chain with the help of coarse-grained
computer simulations. The ion-chain coupling is discussed in the form of
binding time, mean-square-displacement (MSD) relative to the chain, local ion
transport coefficient, and spatio-temporal correlations in the effective
charge. We have shown that a counterion exhibits sub-diffusive behavior
$\langle \delta R^2 \rangle \sim t^\delta$, $\delta\approx0.9$ w.r.t. chain's
centre of mass (COM). The MSD of ions perpendicularly outwards from the chain
segment exhibits a smaller sub-diffusive exponent compared to the one relative
to the chain's COM. Further, we confirm that the effective
diffusion-coefficient of counterions is strongly coupled with the chain. The
effective diffusivity of ion is the lowest in chain's close proximity,
extending up to length-scale of the radius of gyration Rg. Beyond Rg at larger
distances, they attain diffusivity of free ion with a smooth cross-over from
the adsorbed regime to the free ion regime. We have shown that the effective
diffusivity drastically decreases for the higher valent ions, while the
crossover length scale remains the same. Conversely, with increasing salt
concentration the coupling-length scale reduces, while the diffusivity remains
unaltered. The effective diffusivity of adsorbed-ion reveals an exponential
reduction with electrostatic interaction strength. We further corroborate this
from the binding time of ions on the chain, which also grows exponentially with
the coupling strength of the ion-polymer duo. Moreover, the binding time of
ions exhibits a weak dependence with salt concentration for the monovalent
salt, while for higher valent salts the binding time decreases dramatically
with concentration.
