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Comment on “Molecular dynamics simulation study of nonconcatenated ring polymers in a melt. I. Statics” [J. Chem. Phys.134, 204904 (2011)]
1. J. S. Higgins and H. C. Benoît, Polymers and Neutron Scattering (Oxford University Press, Oxford, 1996).
3.Assuming a self-similar scaling for chains and subchains it can be shown2,4,5 that quite generally θ is given by the contact exponent θ2 characterizing the size distribution of a subchain of arc-length s for small distances r with R(s) being the typical size of the subchain, x = r/R(s) and fc(x) a cutoff function. The well-known identity θ = θ2 is equivalent to the relation γ + β = 2 given in Ref. 5 relating the exponent γ ≡ 1 + νθ2 characterizing the return probability after s steps to the “roughness exponent” β ≡ 1 − νθ.
10.The convergence to the large-N asymptotics is difficult to demonstrate numerically even for the 2D SAWs where ds = 5/4 is known theoretically.4
11.Quite generally, a good scaling variable allows to collapse the data in an appropriate regime while the associated scaling function cannot be tuned to improve the collapse. Its task is to describe the data assuming the scaling.
12.We neglect the gradual crossover between the leading power laws below and above Ne. A better fit may be obtained (especially for shorter chains) using the measured subchain sizes R(s). This may allow to verify the assumed Gaussian subchain size distribution.
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Recent computational studies on melts of nonconcatenated rings suggest compact configurations of fractal dimension d f = 3. This begs the question of whether the irregular surfaces of these compact rings may be characterized by a fractal surface dimension d s < 3. We revisit the scaling analysis of the form factor by Halverson et al. [J. Chem. Phys.134, 204904 (2011)] implying d s ≈ 2.8. Our analysis suggests that this conclusion might be due to the application of the Generalized Porod Law at large wavevectors where length scales other than the total chain size do matter. We present an alternative “decorated Gaussian loop” model which does not require d s < 3.
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