  The mechanism behind the $^1$H NMR frequency dependence of $T_1$ and the
viscosity dependence of $T_2$ for polydisperse polymers and bitumen remains
elusive. We elucidate the matter through NMR relaxation measurements of
polydisperse polymers over an extended range of frequencies ($f_0 = 0.01
\leftrightarrow$ 400 MHz) and viscosities ($\eta = 385 \leftrightarrow 102,000$
cP) using $T_{1}$ and $T_2$ in static fields, $T_{1}$ field-cycling
relaxometry, and $T_{1\rho}$ in the rotating frame. We account for the
anomalous behavior of the log-mean relaxation times $T_{1LM} \propto f_0$ and
$T_{2LM} \propto (\eta/T)^{-1/2}$ with a phenomenological model of $^1$H-$^1$H
dipole-dipole relaxation which includes a distribution in molecular correlation
times and internal motions of the non-rigid polymer branches. We show that the
model also accounts for the anomalous $T_{1LM}$ and $T_{2LM}$ in previously
reported bitumen measurements. We find that molecular dynamics (MD) simulations
of the $T_{1} \propto f_0$ dispersion and $T_2$ of similar polymers simulated
over a range of viscosities ($\eta = 1 \leftrightarrow 1,000$ cP) are in good
agreement with measurements and the model. The $T_{1} \propto f_0$ dispersion
at high viscosities agrees with previously reported MD simulations of heptane
confined in a polymer matrix, which suggests a common NMR relaxation mechanism
between viscous polydisperse fluids and fluids under confinement, without the
need to invoke paramagnetism.
