Topological analysis of polymeric melts: Chain-length effects and fast-converging estimators for entanglement length

Abstract

Primitive path analyses of entanglements are performed over a wide range of chain lengths for both bead spring and atomistic polyethylene polymer melts. Estimators for the entanglement length N_{e} which operate on results for a single chain length N are shown to produce systematic O(1/N) errors. The mathematical roots of these errors are identified as (a) treating chain ends as entanglements and (b) neglecting non-Gaussian corrections to chain and primitive path dimensions. The prefactors for the O(1/N) errors may be large; in general their magnitude depends both on the polymer model and the method used to obtain primitive paths. We propose, derive, and test new estimators which eliminate these systematic errors using information obtainable from the variation in entanglement characteristics with chain length. The new estimators produce accurate results for N_{e} from marginally entangled systems. Formulas based on direct enumeration of entanglements appear to converge faster and are simpler to apply.

Monte Carlo simulations of densely-packed athermal polymers in the bulk and under confinement

“…It has been established, that subject to small statistical fluctuations, both the Zl and CReTA methods provide similar results with respect to the topology and average contour length of the primitive paths (Foteinopoulou et al, 2006;Tzoumanekas and Theodorou, 2006). Moreover the PPA and Zl, when applied to the same configuration of polymer chains, give similar results for the mean primitive path contour length, the error being less than 10% (Hoy et al, 2009).…”

“…The derivation of Mi <ink is based on measuring directly the interior kinks in the PP instead of adopting a random coil assumption for the primitive paths; it is defined as the inverse of the slope in the linear regime of the (Z)-versus-N curve. In order to apply this measure, chain lengths in system configurations are required to span a spectrum of different N values with N > Ni, JVi being the threshold value above which (Z) becomes linear with N. It has been shown for atomistic and bead-spring polymer systems Hoy et al, 2009) and further verified for systems of hard-sphere chains, that (Z) scales linearly with N even when (Z) is less than unity. Therefore the proposed estimator is readily applicable to the majority of the simulated polydisperse athermal systems.…”

“…For the calculation of the entanglement distance we adopt the simple M kink ideal estimator proposed in Hoy et al (2009). Through such estimators the PP statistics in the (longchain) polymeric regime can be predicted through application to significantly smaller (short-chain) and computationally-expedient systems.…”

Molecular-Scale Understanding of Cohesion and Fracture in P3HT:Fullerene Blends

“…The average numbers of entanglements <Z> were determined using the primitive path analysis with the Z1 software. [48][49][50][51] Results & Discussion.…”

Nanomanufacturing : nano-structured materials made layer-by-layer.

“…The polymer chain length of N=500 is above the entanglement length $N_e =85 +--7 (Hoy et al 2009). The number of chains for all simulations was 1200 while the number of atoms in the stamp varied from 298925 to 379250 depending on the number of stamps.…”