  We present a neutron spin echo study (NSE) of the nanosecond dynamics of
polyethylene glycol (PEG) functionalised nanosized gold particles dissolved in
D$_2$O at two temperatures and two different PEG molecular weights. The
analysis of the NSE data was performed by applying a Bayesian approach to the
description of time correlation function decays in terms of exponential terms,
recently proved to be theoretically rigorous. This approach, which addresses in
a direct way the fundamental issue of model choice in any dynamical analysis,
provides here a guide to the most statistically supported way to follow the
decay of the Intermediate Scattering Functions I(Q, t) by basing on statistical
grounds the choice of the number of terms required for the description of the
nanosecond dynamics of the studied systems. Then, the presented analysis avoids
from the start resorting to a pre-selected framework and can be considered as
model free. By comparing the results of PEG coated nanoparticles with those
obtained in PEG2000 solutions, we were able to disentangle the translational
diffusion of the nanoparticles from the internal dynamics of the polymer
grafted to them, and to show that the polymer corona relaxation follows a pure
exponential decay in agreement with the behavior predicted by coarse grained
molecular dynamics simulations and theoretical models. This methodology has one
further advantage: in the presence of a complex dynamical scenario I(Q,t) is
often described in terms of the Kohlrausch-Williams-Watts function that can
implicitly represent a distribution of relaxation times. By choosing to
describe the I(Q,t) as a sum of exponential functions and with the support of
the Bayesian approach, we can explicitly determine when a finer-structure
analysis of the dynamical complexity of the system exists according to the
available data without the risk of overparametrisation.
