2020/06/12 by Suematsu, Kazumi, Ogura, Haruo, Inayama, Seiichi +1
#FOS: Physical sciences #Soft Condensed Matter (cond-mat.soft)
paper · doi:10.48550/arxiv.2006.10130
Mathematical expressions for mass distributions around the center of gravity are derived for branched polymers with the help of the Isihara formula. We introduce the Gaussian approximation for the end-to-end vector, r_Gνi, from the center of gravity to the ith mass point on the νth arm. Then, for star polymers, the result is φstar(s)=(1)/(N)∑ν=1f∑i=1Nν(\fracd2π⟨ r_Gνi2⟩)d/2exp(-\fracd2⟨ r_Gνi2⟩s2)for a sufficiently large N, where f denotes the number of arms. It is found that the resultant φstar(s) is, unfortunately, not Gaussian. For dendrimers φdend(s)=∑h=1gωh(\fracd2pi⟨ r_Gh2⟩)d/2exp(-\fracd2⟨ r_Gh2⟩s2)where ωh denotes the weight fraction of masses in the hth generation on a dendrimer constructed from g generations, so that ∑h=1gωh=1. To be specific, ω1=1/N and ωh=(f-1)h-2/N for h≥ 2. These distributions can be described by the same grand sum of each Gaussian function for the end-to-end distance from the center of gravity to each mass point. Note that for a large f and g, the statistical weight of younger generations becomes dominant. As a consequence, the mass distribution of unperturbed dendrimers approaches the Gaussian form in the limit of a large f and g. It is shown that the radii of gyration of dendrimers increase logarithmically with N, which leading to the exponent, ν0=0. An example of randomly branched polymers is also discussed.