Viscoelastic Properties of Polymer Melts from Equilibrium Molecular Dynamics Simulations

Abstract

Introduction. The dynamics of polymer chains are now generally very well understood in the pioneering frameworks of the Rouse model for short chains, and the reptation model for longer chains. [1][2][3][4][5] While these theories yield well-known expressions for the variation of the diffusivity D and viscosity η with the chain length N, an open question is the chain length at which one crosses over from Rouse-like behavior to reptation. This length, termed the entanglement length, N e , is relatively easy to estimate experimentally, but has been hard to access in a simulation due to the fact that, up until recently, there was no satisfactory definition of an entanglement. 6,7 In numerical studies, most estimates of the entanglement length involve conducting simulations and examining dynamic quantities, such as the frequency dependent storage modulus. Extensive work has been done by Kremer and Grest 8 to identify the onset of chain entanglement, with somewhat mixed results. Examination of the chain length dependence of chain diffusion yields a cross over from the Rouse scaling (N -1 ) to reptation scaling (N -2 ) behavior in the vicinity of chains of length 35. In contrast, estimates of storage modulus from nonequilibrium molecular dynamics simulations yield an entanglement chain length in the vicinity of 80.We approach this problem with the tools of equilibrium molecular dynamics (MD). 9 Equilibrium stress fluctuations which result naturally from these simulations directly provide estimates of the viscosity through the Green-Kubo equation, as suggested by Smith et al. 10 The storage and loss modulus are also computed from the stress autocorrelation function, and the hints of a plateau are seen in the storage modulus. The plateau is the strongest indication of onset of reptation dynamics, and is seen for chain lengths of 80 and higher. Estimates of the entanglement length N e , provide a number closer to 30, consistent with the first estimate provided by Kremer and Grest.Simulation Model and Methods. The MD simulation employs a standard chain model. Interaction between nonbonded monomers are described by a shifted, purely repulsive Lennard-Jones (LJ) potential: U(r) ) 4 [(σ/r) 12 -(σ/r) 6 ] + for r < 2 1/6 σ, and U(r) ) 0 for r > 2 1/6 σ. Adjacent bonded monomers interact via a stiff FENE potential, in the form of V FENE ) -k(R 0 2 /2) ln(1

Determination of viscoelastic properties by analysis of probe-particle motion in molecular simulations

“…2, the dynamic moduli of the polymer melt estimated from the bead's MSD via the IGSER and GSER are compared with literature values for ( ) * G ω that were obtained from NEMD [7,9] and equilibrium MD using the Green-Kubo relation [8]. The dynamic modulus derived from the IGSER agrees with the results from NEMD up to 1 ω < , whereas that from the inertia-less GSER cannot capture the high frequency behavior of ( ) …”

“…Previously molecular simulations in conjunction with Green-Kubo or nonequilibrium MD (NEMD) techniques have been used to determine the overall viscoelastic properties of the bulk polymer melt [7][8][9]. A technique for determining the local elastic modulus tensor from the second derivative of free energy with respect to strain has also been applied to polymeric systems [10].…”

Direct Measurement of the Confining Forces Imposed on a Single Molecule in a Concentrated Solution of Circular Polymers

“…On the other hand, efforts have recently been made to define and quantify entanglements directly through molecular dynamics simulations. [29][30][31][32][33] Zhou and Larson have recently used such an approach to directly calculate the tube confining potential by analyzing an ensemble of simulated molecular trajectories, 13 and our measurements on linear DNA compare favorably with some of their predictions. 12 According to the PPR model proposed by Iyer et al, the relative size of the tube radius for circular polymers compared with linear polymers depends on N as well as the ratio of the corresponding N e values for both topologies (i.e., N eC /N eL , where the C and L subscripts refer to the values for linear and circular polymers, respectively).…”

Static and dynamic properties of large polymer melts in equilibrium

“…Using the Green-Kubo relationship [51], the stress relaxation modulus where the off-diagonal element…”