  We report observations of a remarkable scaling behavior with respect to
concentration in the passive microbead rheology of two highly entangled
polymeric solutions, polyethylene oxide (PEO) and hyaluronic acid (HA). This
behavior was reported previously [Hill et al., PLOS ONE (2014)] for human lung
mucus, a complex biological hydrogel, motivating the current study for
synthetic polymeric solutions PEO and HA. The strategy is to identify, and
focus within, a wide range of lag times ${\tau}$ for which passive micron
diameter beads exhibit self-similar (fractional, power law)
mean-squared-displacement (MSD) statistics. For lung mucus, PEO at three
different molecular weights (Mw), and HA at one Mw, we find ensemble-averaged
MSDs of the form ${\langle}{\Delta}r^{2}({\tau}){\rangle} =
4D_{\alpha}{\tau}^{\alpha}$, all within a common band, [1/60 sec, 3 sec], of
lag times ${\tau}$. We employ the MSD power law parameters
$(D_{\alpha},{\alpha})$ to classify each polymeric solution over a range of
highly entangled concentrations. By the generalized Stokes-Einstein relation,
power law MSD implies power law elastic $G'({\omega})$ and viscous
$G''({\omega})$ moduli for frequencies $1/{\tau}$, [0.33 sec$^{-1}$, 60
sec$^{-1}$]. A natural question surrounds the polymeric properties that dictate
$D_{\alpha}$ and ${\alpha}$, e.g. polymer concentration c, Mw, and stiffness
(persistence length). In [Hill et al., PLOS ONE (2014)], we showed the MSD
exponent ${\alpha}$ varies linearly, while the pre-factor $D_{\alpha}$ varies
exponentially, with concentration, i.e. the semi-log plot,
$(log(D_{\alpha}),{\alpha})(c)$ of the classifier data is collinear. Here we
show the same result for three distinct Mw PEO and HA at a single Mw. Future
studies are required to explore the generality of these results for polymeric
solutions, and to understand this scaling behavior with polymer concentration.
