  Entangled ring polymers, along with blends of ring and linear polymers,
continue to be a topic of great interest and debate due to the conflicting
experimental results in the literature as well as the difficulty of producing
entangled synthetic rings devoid of linear contaminants. Here, we create
blended solutions of entangled ring and linear DNA with varying mass fractions
of linear DNA. We use optical tweezers microrheology to measure the linear and
nonlinear viscoelastic response of these blends. Our measurements reveal a
strong non-monotonic dependence of linear viscoelastic properties on linear DNA
fraction, with a pronounced maximum when the mass fraction of rings and linear
chains are comparable, suggestive of pervasive threading of rings by linear
chains. We observe a similar non-monotonicity in the nonlinear regime; however,
a comparatively higher fraction of linear chains (0.5-0.7) is required for a
substantial increase in resisitive force and slowing of relaxation dynamics to
emerge. This nonlinear response also appears to be rate dependent, which we
argue arises from force-induced de-threading of rings at high strain rates. Our
results fill a longstanding gap in knowledge regarding the microrheology and
nonlinear response of ring-linear polymer blends. Moreover, the uniquely strong
mechanical response that ring-linear blends exhibit, along with the ability to
finely tune these blends by varying the blend composition, provides new
materials design principles.
