Introduction

We have made theoretical and/or computational studies of the intrinsic viscosity [η]1, 2 and second virial coefficient A23 of the semiflexible regular three-arm stars. The quantities [η] and A2 are measures of the average chain dimension as well as the mean-square radius of gyration , although A2 is related to the chain dimension only in good solvents or perturbed state. The ratio of gη of [η] of an unperturbed regular three-arm star chain to that of the corresponding unperturbed linear one, both having the same molecular weight and chain stiffness, has been shown to become remarkably smaller than the random-coil limiting value as the chains become stiffer, as in the case of the ratio gS of of the former chain to that of the latter.4 Further, for practical purposes, an empirical interpolation formula for gη has been constructed. However, the ratio of A2 of a perturbed regular three-arm star chain to that of the corresponding perturbed linear one has been shown to be rather insensitive to change in chain stiffness.

The (effective) hydrodynamic radius RH is another measure of the chain dimension and is defined from the translational diffusion coefficient D as follows:

where kB is the Boltzmann constant, T is the absolute temperature and η0 is the viscosity coefficient of the solvent. It is then interesting and necessary to examine the effects of chain stiffness on the ratio gH of RH of the regular three-arm star chain to that of the corresponding linear one. In this paper, we evaluate gH of the semiflexible regular three-arm star polymer on the basis of the Kratky–Porod (KP) chain without excluded volume as in the case of the previous study of gη,2 and then, we construct an interpolation formula for gH for practical purposes.

For an evaluation of D for both of the KP regular three-arm star and linear chains, we adopt the touched-bead hydrodynamic model as in the case of the previous study of [η].2 And also, we use the Kirkwood formula5, 6, 7 as in the case of the linear helical wormlike touched-bead model8, 9 including the KP chain as a special case. On the basis of the Kirkwood formula, for unperturbed linear chains in the random-coil limit, there holds the asymptotic relation:

where is the mean-square end-to-end distance of the chains. On the other hand, if D is evaluated from the Kirkwood–Riseman (KR) hydrodynamic equation in the scheme of preaveraged hydrodynamic interaction (HI),7, 10 we have another asymptotic relation given by

Note that the correct numerical factor 0.192 of Equation (3) was obtained by Kurata and Yamakawa11 instead of the original approximate one 0.196 obtained in the so-called KR approximation for D.7, 10 We also note that the Zimm theory12 on the basis of the unperturbed (dynamic) Gaussian spring-bead model in the scheme of preaveraged HI gives the latter relation. The difference between the prefactor of the right-hand side of Equation (2) and that of Equation (3) is arising from the fact that the Kirkwood and KR (or Zimm) values of D correspond to the translational diffusion of the center of mass of the polymer chain and that of the Zimm center of resistance7, 12 as defined as the point where the translational motion of the entire chain becomes independent of the internal (segmental) motions in the scheme of preaveraged HI, respectively.8, 13, 14 This may be expected to be the case with the star polymers. We then also evaluate D (or RH) and gH for the KP regular three-arm star chain following the KR procedure and examine difference between the Kirkwood and KR values of RH and gH. This is another purpose of this paper.

Materials and methods

The model used in this study is the same as that used in the previous one,2 that is, a regular three-arm star touched-bead model composed of 3m+1 identical spherical beads of (hydrodynamic) diameter db whose centers are located on the KP regular three-arm star chain contour (see Figure 1 in Ref. 2). For convenience, the three arms are designated the first, second and third ones, and the m beads on the ith (i=1, 2, 3) arm are numbered (i−1)m+1, (i−1)m+2, …im from the branch point (center) to the terminal end, with the center bead numbered 0. The angle between each pair of the unit vectors tangent to the KP contours at the branch point is fixed to be 120°, so that the three vectors are on the same plane. The linear touched-bead model, the counterpart of the above star one, is the KP touched-bead model composed of n+1 identical beads of diameter db whose centers are located on the KP linear chain contour. We set n+1 equal to 3m+1, so that n=3m. The n+1 beads are numbered 0, 1, 2, …, n from one end to the other. For both the star and linear touched-bead models, the contour distance between the two adjacent beads is set equal to db. In what follows, all lengths are measured in units of the stiffness parameter λ−1 of the KP chain unless otherwise specified.

Figure 1
figure 1

Plots of (L,db) and (L,db) against log L. The open and closed circles represent and , respectively, for db=0.001 (pip up), 0.003 (pip right-up), 0.01 (pip right), 0.03 (pip right-down), 0.1 (pip down), 0.2 (pip left-down), 0.3 (pip left) and 0.4 (pip left-up). The dashed curves connect smoothly the theoretical values at constant db. The lower and upper horizontal lines represent the random-coil limiting values (=0.947) of and 0.964 of , respectively. The solid curves represent the values calculated from the interpolation formula for (see the text).

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Kirkwood formula

The Kirkwood formula for D of the chain composed of n+1 beads may be given by5, 6, 7

where ζ=3πη0db is the translational friction coefficient of bead and is the mean reciprocal of the distance between the centers of the ith and jth beads.

For the [(i−1)m+k]th and [(j−1)m+l]th beads (i, j=1, 2, 3; k, l=1, 2, …, m) of the KP regular three-arm star chain, that is, the kth bead on the ith arm and the lth bead on the jth arm, respectively, may be given by

with denoting the contour distance from the branch point to the contour point on the ith arm where the center of the [(i−1)m+k]th bead locates on, so that

The theoretical expression for has been obtained in the previous paper.2 Here we give only the results with a brief description. For the KP regular three-arm star chain under the consideration, may be given by

where is the mean reciprocal of the end-to-end distance of the (unperturbed) once-broken KP chain of total contour length t1+t2 such that two KP subchains 1 and 2 of contour lengths t1 and t2, respectively, are connected with a bending angle θ=120° (see Figure 2 in Ref. 2). We note that and/or represent the mean reciprocal of the end-to-end distance of the KP linear chain of contour length t. The interpolation formula for may be given by

Figure 2
figure 2

Plots of against L−1/2. All the symbols have the same meaning as those in Figure 1. The dashed curves connect smoothly the theoretical values at constant db and the solid lines indicate the respective initial tangents.

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where

and C(t1,t2) is given by

with

In Equations (8), (9), (10) and are the second and fourth moments, respectively, of the end-to-end distance of the once-broken KP chain that are given by

We note that the expression for given by Equation (13) was first derived by Mansfield–Stockmayer.4 In the rod limit, Equation (8) reduces to

For later convenience, D obtained from Equation (2) is designated D(K) hereafter.

Kirkwood–Riseman equation

On the basis of the KR hydrodynamic equation in the scheme of preaveraged HI,7, 10 D of the chain composed of n+1 beads may be written as follows:

where ψi is the solution of the following linear simultaneous equations:

The expression for has already been given by Equation (5) with Equations (7), (8), (9), (10), (11), (12), (13), (14). We note that if we assume , that is, the average force exerted on the solvent of the ith bead equal to the average total force of the entire chain divided by n+1, for all i, Equations (16) and (17) may reduce to Equation (4). We also note that this assumption is equivalent to the KR approximation for D mentioned in the Introduction section. For later convenience, D obtained from Equations (16) and (17) is designated D(KR) hereafter.

Results and Discussion

We have calculated the Kirkwood value D(K) and the KR one D(KR) of the translational diffusion coefficient D from Equation (4) and from Equation (16) with the numerical solution ψi of the linear simultaneous equations (17), respectively, for both the KP regular three-arm star and linear touched-bead models, in the ranges of the total number n+1 of bonds from 4 to 9001 and of the bead diameter db from 0.001 to 0.4. Note that the total contour length L of the chain is equal to (n+1)db, as already mentioned in the Materials and methods. In Equations (4) and (17), is given by Equation (5) with Equations (7), (8), (9), (10), (11), (12), (13), (14). On the basis of the values of D(K) and D(KR) for the star and linear chains having the same L and db so obtained along with Equation (1), we evaluate the Kirkwood value and the KR one of the ratio gH as functions of L and db defined by

and

respectively.

In the following subsections, we first examine the behavior of and as functions of L and db and compare the theoretical values of the two ratios. Then we construct an interpolation formula for .

Comparison between and

Figure 1 shows plots of gH against the logarithm of L. The open and closed circles represent the theoretical values of and , respectively, for db=0.001 (pip up), 0.003 (pip right-up), 0.01 (pip right), 0.03 (pip right-down), 0.1 (pip down), 0.2 (pip left-down), 0.3 (pip left) and 0.4 (pip left-up); the dashed curves connecting smoothly the respective theoretical values at constant db. The solid curves represent the values calculated from an interpolation formula for , as discussed later.

In the case of , the asymptotic value in the random-coil limit, that is, the limit L→∞ (in units of λ−1) may be given by

which may be calculated from the relation obtained for the Gaussian regular f-arm stars by Kurata and Fukatsu15 and by Stockmayer and Fixman16 (the latter authors using the KR approximation for D). The asymptotic value is represented by the lower horizontal line. As L is decreased, first increases from the random-coil limiting value and then decreases and exhibits a minimum after passing through a maximum, in the range of db investigated except for db=0.4. The behavior of depends also on db. It should be noted that the difference between the maximum and minimum of is rather small (4% at most).

As for , its values are slightly (3% at most) larger than those of and exhibit no appreciable maximum in contrast to the case of , in the ranges of L and db investigated. We have evaluated the random-coil limiting value of (L,db) from the numerical theoretical values with large db. Figure 2 shows plots of against L−1/2 for db=0.1, 0.2, 0.3 and 0.4. All the symbols have the same meaning as those in Figure 1. The dashed curves connect smoothly the theoretical values at constant db and solid lines indicate the respective initial tangents. It is seen that as L−1/2 is decreased to 0 (L→∞), approaches a constant value irrespective of the value of db. On the basis of such numerical results, it may be concluded that

Note that the values of for smaller db have been omitted in Figure 2, since we cannot make L−1/2 (=[(n+1)db]−1/2) small enough to evaluate at L−1/2=0. We also note that the asymptotic value so obtained is consistent with an available theoretical value of 0.96 obtained by Irurzun17 for the Gaussian regular three-arm star chain without excluded volume on the basis of the KR equation. In Figure 1, the upper horizontal line represents the asymptotic value 0.964. This asymptotic value is 1.8% larger than that of given by Equation (20).

For an examination the difference between and , it is useful to derive the asymptotic relations between RH and in the random-coil limit for both of the regular three-arm star and linear chains. From Equations (2) and (3), using the asymptotic relation for the linear chain7 in this limit along with Equation (1), we may obtain

As for the regular three-arm star chain in the random-coil limit, from Equations (18), (19), (20), (21), (22), (23) using the asymptotic value 7/9 of gS,18 we then obtain the relations

and

It is seen that for the regular three-arm star chain is 3.8% larger than , while for the linear chain the former value is 2.1% larger than the latter.

Further we give the asymptotic forms of and in the thin-rod limit, that is, the limit of L→0 (in units of λ−1) and L/db→∞. In this limit, D(K) for the regular three-arm star chain may be written in the form (see Appendix):

As for the linear chain, we have7

We note that Equation (27) may be obtained directly from Equation (4) with Equations (5), (7), and (15) along with the relation L=(n+1)db. In the rod limit, should be a function only of L/db, that is,

From Equation (18) with Equations (26), (27), (28) we have

On the other hand, D(KR) for the regular three-arm star chain (see Appendix) and that for the linear one19 in the thin-rod limit may be written in the forms:

and

respectively. In the rod limit, should also be a function only of , that is,

From Equation (19) with Equations (30), (31), (32) we have

All of these equations for D(KR) and have the same forms as the corresponding ones for D(K) and given by Equations (26),(27) and (29).

Such salient results given by Equations (29) or (33) that the translational diffusion coefficient of the regular three-arm star chain becomes identical with that of the corresponding linear chain in the thin-rod limit may be regarded as indicating the defect of the Kirkwood formula or that of the scheme of preaveraged HI as in the case of rigid rings.7, 8, 20, 21, 22, 23, 24

Interpolation formula for

Now we are in a position to construct an interpolation formula for on the basis of the numerical theoretical values of (L,db) as already shown in Figure 1 along with the asymptotic relations given by Equations (20) and (29) in the random coil and thin-rod limits, respectively.

We have evaluated (L/db) numerically in the same manner as (L,db) mentioned above using the expression given by Equations (5),(7) and (15) in place of that for the KP chain. Figure 3 shows plots of against [ln(L/db)]−1. The open circles represent the values so obtained. As [ln(L/db)]−1 is decreased (L/db is increased), first decreases and then increases to the asymptotic value 1 after passing through a minimum. For later convenience, we have constructed an interpolation formula for (L/db) in the range of L/db≳10, which is given by

Figure 3
figure 3

Plots of (L/db) against [ln(L/db)]−1. The open circles represent the theoretical values. The horizontal line segment represent the asymptotic value 1 in the limit of [ln(L/db)]−1→0 (L/db→∞). The curve represents the values of the interpolation formula, the solid part indicating the range of (see the text).

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In Figure 3, the curve represents the values calculated from Equation (34) with x=L/db. The error in the value of (L/db) in the range of L/db≳10 (solid part) does not exceed 0.1%.

Next, we consider the ratio . Figure 4 shows plots of against the logarithm of L, where has been evaluated by dividing the values shown in Figure 1 by the values calculated from Equation (34) with x=L/db. All the symbols in Figure 4 have the same meaning as those in Figure 1. It is seen that as L is increased, as a function of L and db, which is represented by f(L,db) hereafter, first increases from unity and then decreases after passing through a maximum in the range of db investigated. Considering the asymptotic conditions limL→0 f(L,db)=1 and , which hold in the limit of L/db→∞, we have constructed an interpolation formula for f(L,db), which may be written in the form,

Figure 4
figure 4

Plots of against log L. All the symbols have the same meaning as those in Figure 1. The solid curves represent the values of the interpolation formula with the corresponding values of db (see the text).

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In Equation (35), the coefficients ai (i=1, 2) and bi (i=0, 1, 2, 3) may be given by

and

respectively, where f′(L/db)=∂f(L/db)/∂L. Note that in Equation (36) the values of f(6,db) and f′(6,db) may be calculated from Equation (35) and Equation (37). In Figure 4, the solid curves represent the values calculated from Equations (35), (36), (37) with the corresponding values of db. It is seen that the interpolation formula may well reproduce the numerical theoretical values in the ranges of L and db so examined, although for the numerical theoretical values seem to deviate downward slightly (up to 0.5%) from the corresponding values of the interpolation formula. Such a slight deviation is within experimental error (1% at least) in D determined by conventional methods and then causes no significant error in a practical use of the present interpolation formula for analysis of experimental data.

The factor (L,db) may therefore be approximately expressed as

where (L/db) and f(L,db) are given by Equation (34) and Equation (35) with Equations (36) and (37), respectively. In Figure 1, the solid curves represent the approximate values calculated from Equation (38) with Equations (34), (35), (36), (37) with the corresponding values of db. It is seen that the interpolation formula for (L,db) so proposed may well reproduce the numerical theoretical values in the ranges of db investigated and of . The error in the value of in those ranges of db and L/db does not exceed 0.4%.

Comparison with experiment

Finally, we make a comparison of the present theoretical result with experimental data in a literature. Figure 5 shows plots of gH against the logarithm of the weight-average molecular weight Mw for the regular three-arm star polystyrene in cyclohexane at 34.5 °C (Θ) obtained by Huber et al.25 The open circles represent the experimental values. The curve represents the KP theory values of , where and is the total contour length and bead diameter, respectively, of the KP regular three-arm star touched-bead model in real length units, calculated from Equation (38) with Equations (34), (35), (36), (37) along with the relation with ML the molecular weight per unit contour length. The solid part of the curve indicates the range of . The necessary KP parameter values used in the calculation of the KP theory values are λ−1=20.0 Å and ML=39.0 Å−1 determined by Norisuye and Fujita26 for (linear) atactic polystyrene in cyclohexane at 34.5 °C (Θ) and =9.9 Å estimated from the relation27 with d the hydrodynamic diameter of the KP cylinder model8 using with d=8.8 Å determined for the same system by Huber et al.28 It is seen that the present theory may explain qualitatively the behavior of the experimental values, which increase with decreasing Mw, although including the range of L/db≲10.

Figure 5
figure 5

Plots of gH against the logarithm of Mw for regular three-arm star polystyrenes in cyclohexane at 34.5 °C (Θ). The open circles represent the experimental data obtained by Huber et al.25 The curve represents the corresponding KP theory values, the solid part indicating the range of L/db≳10 (see text).

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Conclusion

We have evaluated the Kirkwood value and the KR one of the ratio gH of RH of the unperturbed KP regular 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. From an examination of the behavior of and that of as functions of L and db, it is found that both of and are insensitive to change in L irrespective of the value of db and that is 3% at most larger than in the ranges of L and db. The empirical interpolation formula for has been constructed, which reproduces the asymptotic values (=0.947) in the random-coil limit and 1 in the thin-rod limit.