  Mixtures of polymers of varying topologies and stiffnesses display complex
emergent rheological properties that often cannot be predicted from their
single-component counterparts. For example, entangled blends of ring and linear
polymers have been shown to exhibit enhanced shear thinning and viscosity, as
well as prolonged relaxation timescales, compared to pure solutions of rings or
linear chains. These emergent properties arise in part from the synergistic
threading of rings by linear polymers. Topology has also been shown to play an
important role in composites of flexible (e.g., DNA) and stiff (e.g.,
microtubules) polymers, whereby rings promote mixing while linear polymers
induce de-mixing and flocculation of stiff polymers, with these
topology-dependent interactions giving rise to highly distinct rheological
signatures. To shed light on these intriguing phenomena, we use optical
tweezers microrheology to measure the linear and nonlinear rheological
properties of entangled ring-linear DNA blends and their composites with rigid
microtubules. We show that the linear viscoelasticity is primarily dictated by
the microtubules at lower frequencies, but their contributions become frozen
out at frequencies above the DNA entanglement rate. In the nonlinear regime, we
reveal that mechanical response features, such as shear thinning, stress
softening and multi-modal relaxation dynamics are mediated by entropic
stretching, threading, and flow alignment of entangled DNA, as well as forced
de-threading, disentanglement, and clustering. The contributions of each of
these mechanisms depend on the strain rate as well as the entanglement density
and stiffness of the polymers, leading to non-monotonic rate dependences of
mechanical properties that are most pronounced for highly concentrated
ring-linear blends rather than DNA-MT composites.
