1. Viscoelasticity of Entangled Rings with D. Vlassopoulos M. Kapnistos & M. Lang Outline • Introduction: Viscoelasticity of Polymers • Dynamics of Entangled Rings • Blends of Rings and Linear Polymers • Conclusions and Open Questions

2. Viscoelasticity of Polymers Unentangled polymers – self-similar dynamics G(t) = G0 (t/t0)-aexp(-t/trelax) Rouse a=1/2 Zimm a=1/(3n) Entangled polymers – plateau modulus Ge Linear chains - reptation Entangled stars – arm retraction “logarithmic” plateau What is the viscoelasticity of entangled rings?

3. Stress Relaxation Function of Entangled Rings is • Similar to entangled linear chains • with a plateau modulus (double-fold) J. Klein ‘86 • Similar to entangled branched polymers • with a logarithmic plateau • Similar to unentangled polymers with • a power law relaxation function G(t) = G0 (t/t0)-aexp(-t/trelax) 4. All of the above 5. None of the above

4. Dynamics of Entangled Rings A A B B A B Self-similar stress relaxation Effective friction coefficient of a monomer motion along AB is higher by the factor g/lAB≈ (g/Ne)1/2 Relaxation time S. Obukhov, MR, T. Duke, PRL 73, 1263, 1994

5. Stress Relaxation Functions of Entangled Rings melts of PS rings 160KDa 200KDa Tref = 170oC Kapnistos & Vlassopoulos

6. Dynamic Moduli of Linear and Rings 160kg/mol – PS rings 200kg/mol – PS rings 160kg/mol – PS linear 200kg/mol – PS linear entanglement plateau in linear melts no plateau in melts of entangled rings Kapnistos & Vlassopoulos Tref = 170oC

7. Mixtures of 200KDa Rings with 200KDa Linear PS plateau appears in mixtures of linear chains with rings Kapnistos & Vlassopoulos Tref = 170oC

8. Relaxation Times of Mixtures of 200KDa Rings with 200KDa Linear PS Melts 10 8 t (s) 6 4 melt of linear chains melt of rings 2 0 0 0.75 0.8 0.85 0.9 0.95 1.0 volume fraction of rings Mixtures have longer relaxation times than pure components.

9. Conclusions • Self-similar dynamics of entangled rings • leads to a power law stress relaxation • function without a plateau. 2. Entanglement plateau is recovered in mixtures of rings with linear chains. • Mixtures of rings with linear chains • have longer relaxation times than • pure components. Open Questions At what volume fractions of linear chains does the rubbery plateau disappear?

10. Mixtures of 160KDa Rings with 3% of 160KDa Linear PS Tref = 170oC Kapnistos & Vlassopoulos

11. Mixtures of 160KDa Rings with 0.8% of 160KDa Linear PS Tref = 170oC Kapnistos & Vlassopoulos

12. Mixtures of 200KDa Rings with 0.07% of 200KDa Linear PS Tref = 170oC Kapnistos & Vlassopoulos

13. Acknowledgements Collaborations D. Vlassopoulos M. Kapnistos M. Lang Financial Support National Science Foundation NASA URETI Bio-Inspired Materials