Abstract
Effects of chain stiffness on the translational diffusion coefficient D or (effective) hydrodynamic radius RH (∝ D−1) are examined theoretically for the regular three-arm star polymers on the basis of the Kratky–Porod (KP) wormlike chain model. The ratio gH of RH of the regular KP three-arm star touched-bead model to that of the KP linear one, both having the same (reduced) total contour length L and (reduced) bead diameter db, is numerically evaluated on the basis of the Kirkwood formula and/or the Kirkwood–Riseman (KR) hydrodynamic equation. From an examination of the behavior of the Kirkwood value gH(K) and the KR one gH(KR) of gH as a function of L and db, it is found that both of gH(K) and gH(KR) are insensitive to change in L irrespective of the value of db and that gH(KR) is slightly larger than gH(K) in the ranges of L and db investigated. An empirical interpolation formula is constructed for gH(K), which reproduces the asymptotic values 
(=0.947) in the random-coil limit and 1 in the thin-rod limit.
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Appendix
Appendix
Asymptotic form for D of the regular three-arm star in the rod limit
In this appendix, we derive the asymptotic solutions in the limit of L/db→∞ (thin- or long-rod limit) for D(K) and D(KR) of the KP regular three-arm star in the rod limit.
Kirkwood value
The asymptotic form of D(K) for the regular three-arm star in the thin-rod limit may be directly derived from Equation (4) with Equations (5), (6), (7) and (15).
In the case of the regular three-arm star, the summation in Equation (4) may be rewritten in the form,
Recall that L=(n+1)db and m=n/3. In the limit of L/db→∞, that is, m→∞, we may perform the first and second summations on the right-hand side of Equation (39) as follows:
where γE (=0.5772···) is the Euler constant. In this limit, the third summation on the right-hand side of Equation (39) may be converted to an integral and it may be calculated to be
Then we have
From Equation (4) and Equation (42), we obtain Equation (26).
KR value
In the thin-rod limit, we may convert the summations in Equations (16) and (17) to integrals. In the case of the regular three-arm star, Equation (16) with Equation (17) may then be rewritten in the form,
where ψ(x) is the solution of the integral equation,
In Equation (44), K0(x,t) and K1(x,t) are the continuous versions of the mean reciprocal of the distance between the centers of two beads on the same arm and on the different arm, respectively, and they are explicitly given by
From Equation (44), the function F(x) may be defined by
where φ(x)=ψ(x)/2. We then expand φ(x) and Kk(x,t) (k=0, 1) in terms of the shifted Legendre polynomial 
as follows:
where 
is defined by
with Pl(x) the Legendre polynomial. We note that 
satisfies the following orthogonality relation,
where δll′ is the Kronecker delta. In Equations (48) and (49), the expansion coefficients φi and Kk,ij may be given by
and
respectively. Substituting Equations (52) and (53) into the second line of Equation (47) and carrying out the integrations, F(x) may be rewritten in the form,
It can be shown in the limit of L/db→∞ that
and 
. Then we have
From the first line of Equation (47) and Equation (56) along with the relation 
may be written in the from,
Substituting of Equation (57) into Equation (43) and carrying out the integration over x, we obtain Equation (30).
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Ida, D. Translational diffusion coefficient of wormlike regular three-arm stars. Polym J 47, 679–685 (2015). https://doi.org/10.1038/pj.2015.44
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DOI: https://doi.org/10.1038/pj.2015.44
