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In Ref.6, the primitive path had been defined as the path with minimum elastic energy, rather than the one with shortest length. Larson discussed this topic and concluded that methods producing the shortest length paths were preferable. The geometric approaches Z1 and CReTA minimize total contour length rather than an elastic energy, and provide us with a shortest parameter-free path
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In Ref.6, the primitive path had been defined as the path with minimum elastic energy, rather than the one with shortest length. Larson discussed this topic and concluded that methods producing the shortest length paths were preferable. The geometric approaches Z1 and CReTA minimize total contour length rather than an elastic energy, and provide us with a shortest parameter-free path.
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For the case of wormlike (non-Gaussian) chains, C (N) is analytically related to the persistence length lp of short chains; C (N+1) =C (∞) -2α (1- αN) (1-α) -2 /N, where α=exp (- l0 / lp), and C (∞) = limN→0 C (N) = (α+1) / (α-1). Note that limα→1 C (N) =N and limα→0 C (N) =1. In the limit lp l0, the mean-square end-to-end distance of a wormlike chain consequently reads R ee 2 =2 lp L (1- [lp /L] [1-exp (-L/ lp)]) with contour length L (N-1) l0. In the further limit L lp, R ee 2 / (N-1) =2 lp l0 =C (∞) l02 lK l0, where lK denotes Kuhn length. These limits are not employed in this work, as we are seeking for corrections in the regime where these limits have not necessarily been reached. For freely rotating chains with a fixed bending angle θ the same result is obtained upon identifying α= cosθ.
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For the case of wormlike (non-Gaussian) chains, C (N) is analytically related to the persistence length lp of short chains; C (N+1) =C (∞) -2α (1- αN) (1-α) -2 /N, where α=exp (- l0 / lp), and C (∞) = limN→ C (N) = (α+1) / (α-1). Note that limα→1 C (N) =N and limα→0 C (N) =1. In the limit lp l0, the mean-square end-to-end distance of a wormlike chain consequently reads R ee 2 =2 lp L (1- [lp /L] [1-exp (-L/ lp)]) with contour length L (N-1) l0. In the further limit L lp, R ee 2 / (N-1) =2 lp l0 =C (∞) l02 lK l0, where lK denotes Kuhn length. These limits are not employed in this work, as we are seeking for corrections in the regime where these limits have not necessarily been reached. For freely rotating chains with a fixed bending angle θ the same result is obtained upon identifying α= cosθ. The above expressions for C (N) are directly obtained from a bond-vector correlation function which exponentially decreases with distance between bonds. However, simulations have shown that flexible bead spring chains in melts do not follow this form at small and moderate N. The ratio C (N) must sublinearly and monotonically increase with N in order to uniquely determine Ne from Eq. 15, which limits its use to systems which do not change their statistics on a length scale larger than entanglement length. For the wormlike chain as well as equilibrated bead spring chains, C (N)/N decreases monotonically with N.
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A. E. Likhtman (private communication). See also Refs..
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Reference reported Ne >40 and that Z became linear at N>24.
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Reference reported Ne >40 and that Z became linear at N>24.
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A counterexample would be LJ+FENE bead spring chains made semiflexible by the addition of a bending potential U=- kθ cos (θ); when kθ =2 LJ, C (Ne) is still well below C (θ).
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A counterexample would be LJ+FENE bead spring chains made semiflexible by the addition of a bending potential U=- kθ cos (θ); when kθ =2 LJ, C (Ne) is still well below C (θ).
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58
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Unfortunately, this approach can not be used with standard PPA because the tension in the primitive chains produces Lpp2 (n) 'b pp 2 n2, and L pp 2 (n) =A n2 +Cn cannot be fit. PPA runs using the linearized-FENE potential of Ref. were also performed, but the performance was poor. In contrast, Z1 and similar methods (e.g.,) produce a wide range of local fractional contour length reductions
-
Unfortunately, this approach can not be used with standard PPA because the tension in the primitive chains produces Lpp2 (n) 'b pp 2 n2, and L pp 2 (n) =A n2 +Cn cannot be fit. PPA runs using the linearized-FENE potential of Ref. were also performed, but the performance was poor. In contrast, Z1 and similar methods (e.g.,) produce a wide range of local fractional contour length reductions.
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