Mesoscopic simulations of entangled polymers, blends, copolymers, and branched structures F. Greco, G. Ianniruberto, and G. Marrucci Naples, ITALY Y. Masubuchi Tokyo, JAPAN

Network of entangled polymers Actual chains have slack Primitive chains are shortest path

Microscopic simulations: Atomistic molecular dynamics (Theodorou, Mavrantzas, etc.) Coarse-grained molecular dynamics (Kremer, Grest, Everaers et al.; Briels et al.) Lattice Monte Carlo methods (Evans-Edwards, Binder, Shaffer, Larson et al.) Mesoscopic simulations: Brownian dynamics of primitive chains (Takimoto and Doi, Schieber et al.) Brownian dynamics of the primitive chain network (NAPLES)

Brownian dynamics of primitive chains along their contour Sliplinks move affinely Sliplinks are renewed at chain ends Each sliplink couples the test chain to a virtual companion

3D sliplink model Simulation box typically contains ca. 2 x 10 4 chain segments

Nodes of the rubberlike network are sliplinks (entanglements) instead of crosslinks Crucial difference: Monomers can slide through the sliplink

Primitive Chain Network Model J. Chem. Phys D motion of nodes 1D monomer sliding along primitive path Dynamic variables: node positions R monomer number in each segment n number of segments in each chain Z

Node motion Elastic springs Brownian force Chemical potential Relative velocity of node

Monomer sliding = local linear density of monomers = rate of change of monomers in i-th segment due to arrival from segment i-1 = sliding velocity of monomers from i-1 to i

Network topological rearrangement n i monomers at the end End if Unhooking (constraint release) else if Hooking (constraint creation) n 0 : average equilibrium value of n

Chemical potential of chain segment from free energy E The numerical parameter  was fixed at 0.5, which appears sufficient to avoid unphysical clustering. The average segment density is not a relevant parameter. We adopted a value of 10 chain segments in the volume a 3, where a is the entanglement distance.

Non-dimensional equations (units: length = a=b  n o, time = a 2  /6kT = , energy= kT) n=n/n o Stress tensor:  Relevant parameters: Nondimensional simulation: equilibrium value of (slightly different from initial value Z 0 ) Comparison with dimensional data: modulus G = kT =  RT/M e elementary time 

LVE prediction of linear polymer melts

Polybutadiene melt at 313K from Wang et al., Macromolecules 2003

Polyisoprene melt at 313K from Matsumiya et al., Macromolecules 2000

Polymethylmethacrylate melts at 463K from Fuchs et al., Macromolecules 1996

Polymers G (MPa)M e (kDa)M e literature M e  (s) PS (453K) PB (313K) x10 -6 PI (313K) x10 -5 PMMA (463K) G = kT =  RT/M e = M/M e

Polystyrene solution by Inoue et al., Macromolecules 2002 Simulations by Yaoita with the NAPLES code

Step strain relaxation modulus G(t,  )

Viscosity growth. Shear rates (s -1 ) are: , 0.049, 0.129, 0.392, 0.97, 4.9

Primary normal stress coefficient. Shear rates as before.

Polystyrene solution fitting parameters: Vertical shift, G = 210 Pa Horizontal shift,  = 0.55 s = 18.4 implying M e = 296

Blends and block copolymers

Phase separation kinetics in blends t=0 2.5 = 10 (  d ~ 40),  =0.5,  =

Block ratio = 0.5  = 0.5 = 40 BLOCK COPOLYMERS

Block ratio 0.1 Block ratio 0.3 = 40  = 2

Branched polymers Backbone-backbone entanglements cannot be renewed two entangled H-molecules Backbone chains have no chain ends

Sliplink Branch point End A star polymer with q=5 arms

Free arm If one of the arms happens to have no entanglements, … it has the chance to change topology

1/q Possible topological changes The free arm has q options, all equally probable (under equilbrium conditions)

Double-entanglement It can penetrate a sliplink of another arm, thus forming a …

If later another arm becomes entanglement-free, …

the topological options are … Enhanced probability for the double entanglement because the coherent pull of the 2 chains makes the branch point closer to double entanglement 2/q 1/q

If the multiple entanglement is “chosen”, … the branch point is “sucked” through the multiple entanglemet

The multiple entanglement has now the chance to be “destroyed” by arm fluctuations Similar topological changes would allow backbone-backbone entanglements in H polymers to be renewed

H-polymer simulations Click to play

Relaxation modulus for H-polymers With the topological change (liquid behavior) without (solid behavior)

Stress auto-correlation

Effect on diffusion of 3-arm star polymers

Diffusion coefficient For 3-arm stars For H’s having arms with Z a = 5

Backbone-backbone entanglement (BBE) cluster 10 5 The largest BBE cluster for H05 including 58 molecules

Size distribution of BBE cluster H05 H10 H20

Conclusions Mesoscopic simulations based on the entangled network of primitive chains describe many different aspects of the slow polymer dynamics For linear polymers, quantitative agreement is obtained with 2 (or at most 3) chemistry-and- temperature-dependent fitting parameters. More complex situations are being developed, and appear promising. A word of caution: Recent data by several authors (McKenna, Martinoty, Noirez) on thin films (nano or even micro) show that supramolecular structures can exist. These can hardly be captured by simulations.

Conclusion (social) to get the code & docs. NAPLES New Algorithm for Polymeric Liquids Entangled and Strained

Diffusion of H-polymers With the topological change Conventional (some still diffuse in the network)