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

Robert S. Hoy, Katerina Foteinopoulou, and Martin Kröger

Phys. Rev. E 80, 031803 – Published 29 September 2009

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.

Received 11 March 2009

DOI:https://doi.org/10.1103/PhysRevE.80.031803

Authors & Affiliations

Robert S. Hoy^*

Materials Research Laboratory, University of California, Santa Barbara, California 93106, USA

Katerina Foteinopoulou

Institute for Optoelectronics and Microsystems (ISOM) and ETSII, Universidad Politécnica de Madrid (UPM), José Gutiérrez Abascal 2, E-28006 Madrid, Spain

Martin Kröger^†

Department of Materials, Polymer Physics, ETH Zürich, CH-8093 Zürich, Switzerland

^*Corresponding author. robertscotthoy@gmail.com
^†http://www.complexfluids.ethz.ch

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Vol. 80, Iss. 3 — September 2009

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Figure 1
(Color online) Performance of classical $N_{e}$ estimators $N_{e} (N)$ based on coils (4) (upper two curves) and kinks (5). Data are shown for the two model polymer melts studied in this manuscript. The trends with $N$ are in agreement with published results for other systems [7, 10, 11, 14, 31, 32, 33]. The convergence behavior is poor, as $N_{e} \equiv {lim}_{N \to \infty} N_{e} (N)$ obviously cannot be extrapolated studying chains with $N < 100$ , while $N_{e}$ turns out to stay well below 100 for both systems. An “ideal estimator,” as defined in Sec. 3, would converge when $N$ exceeds $N_{e}$ or earlier.Reuse & Permissions
Figure 2
(Color online) (a) Performance of modified S-kink estimator (6) (lower two curves) and the modified S-coil (7) (upper two curves). which approach $N_{e}$ from above. Data are for the same systems analyzed in Fig. 1. The single-configuration estimator for kinks exhibits an improved convergence behavior compared with Eq. (5). Under circumstances discussed in Sec. 5, application of both modified and original classical estimators allows one to obtain lower and upper bounds on $N_{e}$ which tighten with increasing $N$ . (b) Shown are the relative differences (“gap [%]”) between $N_{e} (N)$ values shown in Fig. 1 and the ones plotted in part (a) of the current graph. Differences are smaller for $N_{e}$ estimated from kinks (lower two curves).Reuse & Permissions
Figure 3
(a) Testing the applicability of Eq. (8) which predicts linear behavior (slope $\propto D$ , offset $\propto Y$ ) in this representation. We obtain $C (\infty) \approx 1.85$ and $C (\infty) \approx 8.3$ for the $LJ + FENE$ and PE models, respectively (cf. Table ). (b) Testing the validity of Eq. (9) for both types of melts (slope $\propto A$ , offset $\propto B$ ). The linear relationship is employed to derive estimator (15) in Sec. 4B. (a), (b) Data for larger $N$ are not shown but also agree to all displayed fit lines, to within statistical errors.Reuse & Permissions
Figure 4
Z1 results for the two model polymer melts. Testing the applicability of Eq. (11) which predicts linear behavior in this representation (slope $G$ , offset $H$ ). Clearly $⟨ Z ⟩ (N)$ becomes linear at an $N$ for which $⟨ Z ⟩ < 1$ . This implies $N_{1} < N_{e}$ and that Eq. (13) can be an ideal estimator. An interpretation for $N_{1}$ is given in Sec. 4A. Data for larger $N$ are not shown here, but the slope $d ⟨ Z ⟩ / d N$ does not change significantly with increasing $N$ .Reuse & Permissions
Figure 5
(Color online) (a) Performance of proposed estimators M-kink (13) (lower two curves with large symbols) and M-coil (15) (upper two curves with large symbols); see also Appendix . Data are for the same systems analyzed in Figs. 1, 2. Clearly, $N_{e} (N)$ has converged for $N ⪡ 100$ , and as shown by comparison to Fig. 4, $N_{e} (N)$ approaches $N_{e}$ before $⟨ Z ⟩$ exceeds unity. This allows us to estimate $N_{e}$ from mostly unentangled systems. (b) Same data as in (a) vs ${log}_{10} N$ , which allows the full range of $N$ to be presented. For comparison, blue broken and red dashed lines for PE and $LJ + FENE$ , respectively, show reference data for S estimators, already presented in Figs. 1, 2.Reuse & Permissions
Figure 6
(Color online) Graphical demonstration of evaluation of $N_{e} (N)$ using the M-coil estimator (15). Shown are both the lhs $C (N) / N$ , and rhs of Eq. (15) for both types of polymer melts. The dotted (red) path shows how to obtain $N_{e} (N) \approx 87$ for a given $N$ ; here $N = 48$ . As described in Appendix , this value is identical with both $N_{e}$ and $N_{e} (N_{e})$ , cf. Table . The ratio $C (N) / N$ (small points) monotonically decreases with increasing $N$ , while the rhs (large symbols) reaches a plateau when $N$ has approached $N_{e}$ (at the intersection of the curves), which is a distinguishing feature of an ideal estimator.Reuse & Permissions

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