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)-a exp(-t/trelax)
Rouse a=1/ 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)-a exp(-t/trelax)
4. All of the above
5. None of the above

4. Dynamics of Entangled RingsB B A A B
Effective friction coefficient of a monomer motion along AB
is higher by the factor g/lAB ≈ (g/Ne)1/2
Relaxation time
Self-similar stress relaxation
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 2
melt of rings
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 Open Questions
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