Helical Wormlike Research Articles

Dynamic Depolarized Light Scattering by Oligo- and Poly(α-methylstyrene)s in Dilute Solution

The power spectrum JΓ of the excess depolarizedcomponent of scattered light was determined from frequency-domain dynamic depolarized light scattering measurements for 10 samples of atactic oligo- and poly(α-methylstyrene)s (a-PαMS), each with the fraction of racemic diads fr=0.72, in the range of weight-average degree of polymerization xw from 2 to 67.1 in cyclohexane at 30.5 °C (Θ). It is found that, as in the cases of atactic polystyrene (a-PS), atactic poly(methyl methacrylate) (a-PMMA), and isotactic (i-) PMMA, JΓ may be well represented in terms of a single Lorentzian independently of xw, and that the relaxation time τΓ as defined as the reciprocal of the half-width at half-maximum of JΓ evaluated at infinite dilution increases with increasing xw and levels off to its asymptotic value in the limit of xw→∞, being consistent with the theoretical prediction on the basis of the helical wormlike (HW) chain model. A comparison is made of the present data for τΓ with the HW theory, and it is shown that the theory may explain satisfactorily the data in the range of xw ≳ 10, although semiquantitatively. For xw ≲ 10, the rigid sphere model having the radius equal to the apparent root-mean-square radius of gyration of the HW chain may give a good explanation of τΓ. From a comparison of the present results for τΓ for a-PαMS with previous ones for a-PS and a- and i-PMMAs, it is shown that there is good correlation between static and dynamic chain stiffness, the latter being defined as the ratio of the relaxation time associated with the local motion of the long chain to that of its isolated repeat unit (monomer), as predicted by the HW theory.

Nuclear Magnetic Relaxation of Oligo- and Poly(α-methylstyrene)s in Dilute Solution

The spin–lattice relaxation time T1 and nuclear Overhauser enhancement NOE were determined for three samples of atactic oligo- and poly(α-methylstyrene)s (a-PαMS) with weight-average degree of polymerization xw = 2, 3, and 67.1 in cyclohexane at 30 °C. A comparison is made of the present data for T1 and NOE with the helical wormlike (HW) chain theory, and it is shown that the theory may explain well the data for large xw (≳ 10). For smaller xw, the rigid sphere model having the radius equal to the apparent root-mean-square radius of gyration of the HW chain fails to give a quantitative explanation of T1 in contrast to the case of the relaxation time τΓ determined from dynamic depolarized light scattering measurements, indicating that the nuclear magnetic relaxation and dynamic depolarized light scattering cannot be described in terms of a common single relaxation time. However, it is shown that there is an effective (mean) magnetic relaxation time τM approximately equal to 0.6τΓ. From a comparison of the present results for T1 for a-PαMS with previous ones for typical flexible polymers, it is also shown that the behavior of T1 for them, including a-PαMS, may be consistently explained by the use of τM (proportional to τΓ).

Some Comments on the Analysis of the Third Virial Coefficient for Polymer Chains

An analysis is made of experimental data for the third virial coefficient A3 of atactic oligo- and poly(α-methylstyrene)s (a-PαMS) in the Θ solvent cyclohexane at 30.5 °C and in three good solvents, toluene, 4-tert-butyltoluene, and n-butyl chloride, at 25.0 °C on the basis of the helical wormlike (HW) chain model. It is found that A3 at Θ, which is denoted by A3,Θ, becomes a constant of 5.0×10−4 cm6 mol/g3 independent of the weight-average molecular weight Mw for Mw ≳ 104 and deviates from it for Mw ≲ 104 because of effects of chain ends. The observed dependence of A3,Θ on Mw may be well explained by the HW theory that takes account of the effects. For the three good-solvent systems, the behavior of the factor g defined by A3/[A2(HW)]2Mw is examined as a function of the cubed gyration-radius expansion factor αS3, where A2(HW) is the part of the second virial coefficient A2 without the effects of chain ends. It is found that the data points for a-PαMS samples with Mw ≳ 105 in all the three good solvents form nearly a single-composite curve, as predicted by the HW theory, and that they follow the Stockmayer–Casassa theoretical curve for αS3 ≳ 2 but deviate upward from it for αS3 ≲ 2 because of effects of three-segment interactions. Effects of chain stiffness are of minor importance for an explanation of this deviation. Some literature data for polystyrene are also examined.

Mean-Square Optical Anisotropy of Oligo- and Poly(α-methylstyrene)s in Dilute Solution

The mean-square optical anisotropy 〈Γ2〉 was determined from anisotropic light scattering measurements for 10 samples of atactic oligo- and poly(α-methylstyrene)s (a-PαMS) with the fraction of racemic diads fr = 0.72 in the range of weight-average degree of polymerization xw from 2 to 67.1 in cyclohexane at 30.5 °C (ϑ). A comparison is made of the present data with the helical wormlike (HW) chain theory with the values of the model parameters previously determined from the mean-square radius of gyration and with those of the local polarizability tensor α0 per repeat unit properly assigned on the basis of the bond and group polarizabilities. The HW theory value of 〈Γ2〉/xw in the limit of xw → ∞ is found to be somewhat larger than the experimental one. It is then shown that if α0 is replaced by 0.90α0, the HW theory may reproduce quantitatively the experimental results for all the samples except those with xw ≤ 4 for which effects of chain ends become appreciable. A comparison is also made of the present results for a-PαMS with previous ones for atactic polystyrene (fr = 0.59), atactic poly(methyl methacrylate) (a-PMMA) (fr = 0.79), isotactic PMMA (fr ≃ 0.01), and poly(n-hexyl isocyanate), and the behavior of 〈Γ2〉 is shown to depend remarkably on chain stiffness and local chain conformation.

Scattering Function of Oligo- and Poly(α-methylstyrene)s in Dilute Solution

The scattering function was determined for four samples of atactic oligo- and poly(α-methylstyrene)s (a-PαMS), each with the fraction of racemic diads fr = 0.72, in the range of weight-average molecular weight Mw from 1.27 × 103 to 6.46 × 104 in cyclohexane at 30.5 °C (ϑ) in the range of the magnitude k of the scattering vector smaller than 1 Å-1 by the use of a point-focusing small-angle X-ray scattering (SAXS) camera. The Kratky function Fs as a function of k increases monotonically with increasing k for Mw ≲ 5 × 103 and exhibits a plateau in the range of 0.15 Å-1 ≲ k ≲ 0.3 Å-1 for Mw ≳ 5 × 103. For k ≲ 0.5 Å-1, all the experimental data may be well explained by the corresponding helical wormlike (HW) chain theory. For k ≳ 0.5 Å-1, however, agreement between theory and experiment is only qualitative. A comparison is made of the present SAXS data for a-PαMS with the previous data for atactic polystyrene with fr = 0.59, atactic poly(methyl methacrylate) (a-PMMA) with fr = 0.79, isotactic PMMA with fr ≃ 0.01, and syndiotactic (s-) PMMA with fr = 0.92. It is then found that the HW theory may give a good explanation of the difference in the behavior of Fs for k ≲ 0.3 Å-1 between the four polymers except s-PMMA, for which the HW model parameters have not been determined.

Transport Coefficients of Oligo- and Poly(α-methylstyrene)s in Dilute Solution

The intrinsic viscosity [η] was determined for 25 samples of atactic oligo- and poly(α-methylstyrene)s (a-PαMS), each with the fraction of racemic diads fr = 0.72, in the range of weight-average molecular weight Mw from 2.94 × 102 (dimer) to 3.22 × 106 in cyclohexane at 30.5 °C (ϑ). The translational diffusion coefficient D was also determined from dynamic light-scattering measurements for 21 of them in the range of Mw from 5.30 × 102 (tetramer) to 3.22 × 106 under the same solvent condition. It is found that the double-logarithmic plots of [η] and MwD against Mw follow their respective asymptotic straight lines of slope 1/2 for Mw ≳ 2 × 105, but deviate upward and downward, respectively, from them with decreasing Mw for smaller Mw. From an analysis of these transport coefficients on the basis of the helical wormlike (HW) chain model, it is shown that the above Mw dependence of [η] and D may be well explained by the corresponding HW theories with the values of the model parameters consistent with those pr...

Mean-Square Radius of Gyration of Oligo- and Poly(α-methylstyrene)s in Dilute Solution

The mean-square radius of gyration 〈S2〉 was determined from small-angle X-ray and light scattering measurements for 20 samples of atactic oligo- and poly(α-methylstyrene)s (a-PαMS), each with the fraction of racemic diads fr = 0.72, in the range of weight-average molecular weight from 6.48 × 102 (pentamer) to 3.22 × 106 in cyclohexane at 30.5 °C (ϑ). The ratio 〈S2〉/xw as a function of the weight-average degree of polymerization xw first increases steeply for xw ≲ 20, then passes through a flat hump at xw ≃ 40, and finally approaches its asymptotic value for xw ≳ 1 ×103. On the basis of the helical wormlike (HW) chain model the HW model parameters are λ-1κ0 = 3.0, λ-1τ0 = 0.9, λ-1 = 46.8 A, and ML = 39.8 A-1, where κ0 and τ0 are the differential-geometrical curvature and tortion, respectively, of the characteristic helix of the HW chain taken at the minimum zero of its elastic energy, λ-1 is the stiffness parameter, and ML is the shift factor defined as the molecular weight per unit contour length of the c...

Dynamic Structure Factor of Polystyrene and Poly(methyl methacrylate) in ϑ Solvents

The dynamic structure factor and its first cumulant Ω(k) as functions of the magnitude k of the scattering vector were determined from dynamic light scattering (DLS) measurements for two atactic polystyrene samples having the weight-average molecular weights Mw = 6.40 × 106 and 8.04 × 106 in cyclohexane at 34.5 °C (ϑ) and for two atactic poly(methyl methacrylate) samples having Mw = 4.82 × 106 and 1.31 × 107 in acetonitrile at 44.0 °C (ϑ). The translational diffusion coefficient and the mean-square radius of gyration 〈S2〉 were also determined from DLS and static light scattering measurements, respectively, for all the samples except for the a-PS sample with the smaller Mw. It is found that even in such a range of large Mw, the “universal” behavior cannot be realized; i.e., the plot of η0Ω(k)/kBTk3 against 〈S2〉1/2k depends on the kind of polymer in the k3 -region, where η0 is the solvent viscosity, kB the Boltzmann constant, and T the absolute temperature. This result is consistent with the theoretical prediction on the basis of the helical wormlike (HW) chain model, although the HW theory cannot quantitatively explain the experimental data for the individual polymer.

Excluded-Volume Effects on the Mean-Square Radius of Gyration of Oligo- and Polyisobutylenes in Dilute Solution

The mean-square radius of gyration 〈S2〉 was determined from small-angle X-ray scattering and light scattering (LS) measurements for oligo- and polyisobutylenes (PIB) in n-heptane at 25.0 °C in the range of the weight-average degree of polymerization xw from 1.14 × 10 to 2.48 × 103. The model parameters of the helical wormlike (HW) chain for PIB are then determined by analyzing simultaneously the present data for 〈S2〉/xw and the previous data obtained from LS measurements for samples with large xw in the same solvent condition and also in isoamyl isovalerate at 25.0 °C (ϑ), by the use of the HW theories with and without the excluded-volume effect. With the parameter values so determined for PIB, the plots of the viscosity- and hydrodynamic-radius expansion factors αη and αH against the scaled excluded-volume parameter z̃ for PIB along with those for atactic polystyrene and atactic and isotactic poly(methyl methacrylate)s are shown to form a single-composite curve better than before in each case, confirming the validity of the quasi-two-parameter scheme. The improvement of the consistency of the results for PIB with those for the other polymers arises from the re-evaluation of z̃ with the present values of the model parameters directly determined from 〈S2〉.

Dynamic Depolarized Light Scattering and Nuclear Magnetic Relaxation Studies of Isotactic Oligo- and Poly(methyl methacrylate)s in Dilute Solution

The (excess) power spectrum JΓ of the depolarized component of scattered light intensity was measured for eight samples of isotactic oligo- and poly(methyl methacrylate)s (i-PMMA), each with the fraction of racemic diads fr ≃ 0.01, in the range of weight-average degree of polymerization xw from 4 to 70.1 in acetonitrile at 28.0 °C (ϑ). The spin−lattice relaxation time T1 was also determined for the three samples with xw = 4, 5, and 70.1, and the nuclear Overhauser enhancement NOE, for the two samples with xw = 4 and 70.1, all in acetonitrile at 35 °C. As in the cases of atactic (a-) polystyrene (a-PS) and a-PMMA previously studied, it is found that JΓ may be well represented in terms of a single Lorentzian independently of xw and that the relaxation time τΓ defined from JΓ at infinite dilution increases with increasing xw and levels off to its asymptotic value in the limit of xw → ∞, being consistent with the recent theoretical prediction on the basis of the helical wormlike (HW) chain model. A comparison...

Dynamics of coarse-grained helical wormlike chains. II. Eigenvalue problems

The eigenvalue problem for the matrix representation of the diffusion operator appearing in the diffusion equation previously derived for the coarse-grained helical wormlike (HW) chain model is considered. Following the procedure already established in the study of the original diffusion equation for the discrete HW chain, the problem is solved in the subspace and block-diagonal approximations in the subspace L(1), where L is the ‘‘total angular momentum quantum number’’ and (1) indicates the one-body excitation basis set. The 1(1) eigenvalues λ1,kj thus obtained form three branches of the eigenvalue spectrum specified by the index j as before, and it is shown that λ1,k0 in the j=0 (lowest) branch at small wave number k is approximately proportional to the corresponding Rouse–Zimm eigenvalue in the Hearst version. The behavior of the eigenvalue spectra is also numerically examined taking atactic polystyrene with the fraction of racemic diads fr=0.59 and atactic poly(methyl methacrylate) with fr=0.79 as typical examples of flexible polymers. It is found that the ratio λ1,k0/λ1,10 at small k depends somewhat (appreciably for large k) on chain stiffness and local chain conformation.

Dynamics of coarse-grained helical wormlike chains. I. Diffusion equation

An alternative representation of the diffusion equation is derived for the discrete helical wormlike (HW) chain model by treating the constraints in a manner different from that previously adopted. In contrast to the previous diffusion equation for the same HW model, which can describe local motions but not global to quasi-global motions of flexible polymers in dilute solution because of the defect arising from the preaveraging of the constraining matrix, the present equation may be considered to be suitable for the description of the latter since it can give explicitly the Rouse–Zimm eigenvalues in the ground (global) state. It is shown that the present and previous equations are exactly the same before the constraining matrix common to them is preaveraged, as is natural, and that the above difference between their characteristics arises from the difference in the effect of the preaveraging approximation. A comparison is also made of the present model (or diffusion equation) with other models for polymer chain dynamics. In particular, the present theory is shown to be rather similar to the Fixman–Kovac theory for the conventional bond chain in the physical meaning as well as the formulation.

Excluded-Volume Effects on the Hydrodynamic Radius of Oligo- and Polyisobutylenes in Dilute Solution

The translational diffusion coefficient D was determined from dynamic light scattering measurements for oligo- and polyisobutylenes in isoamyl isovalerate (IAIV) at 25.0 °C (ϑ) and in n-heptane at 25.0 °C in the range of weight-average molecular weight Mw from 1.01 × 103 to 1.76 × 106. The values of the unperturbed and perturbed hydrodynamic radii RH,ϑ and RH defined from D in IAIV and in n-heptane, respectively, were found to agree with each other in the oligomer region, indicating that the values of RH,ϑ may be adopted as those of the unperturbed hydrodynamic radius RH,0 in n-heptane at 25.0 °C. The values of the hydrodynamic-radius expansion factor αH in n-heptane at 25.0 °C are then obtained as the ratio RH/RH,ϑ from those values thus determined. The data for RH,ϑ are analyzed as usual by the use of the corresponding (unperturbed) helical wormlike (HW) chain theory. The results for αH as a function of the scaled excluded-volume parameter z defined in the Yamakawa−Stockmayer−Shimada theory for the HW ...

Local Chain Motion of Isotactic and Syndiotactic Poly(methyl methacrylate)s Studied by the Fluorescence Depolarization Method

The local chain motion of syndiotactic and isotactic poly(methyl methacrylate)s (s-PMMA and i-PMMA) was examined by the time-resolved fluorescence depolarization method. The reduced relaxation time, T m /η, and its activation energy, E * , were measured in various solvents. Both the T m /η and E * for s- and i-PMMAs decreased with a decrease in the local segment density. As for the effect of the stereoregularity, the activation energy of both PMMAs showed similar values, but the relaxation time of s-PMMA was ca. 1.5-2.0 times larger than that of i-PMMA. Such a difference in relaxation time is closely related to the static chain stiffness predicted by the helical wormlike (HW) chain model.

Mean-Square Optical Anisotropy of Isotactic Oligo- and Poly(methyl methacrylate)s in Dilute Solution

The mean-square optical anisotropy was determined from anisotropic light-scattering measurements for 10 samples of isotactic oligo- and poly(methyl methacrylate)s (i-PMMA), each with the fraction of racemic diads f r ≃ 0.01, in the range of weight-average molecular weight from 3.58 x 10 2 (trimer) to 1.07 x 10 4 , and also for methyl isobutyrate (the monomer of PMMA), in acetonitrile at 28.0 °C (Θ). A comparison is made of the present data with the helical wormlike (HW) chain theory with the values of the model parameters previously determined from the mean-square radius of gyration and with the local polarizability tensor properly assigned to the repeat unit of the chain in the same manner as in the previous study of atactic (a-) PMMA, i.e., evaluated with the same bond polarizabilities. It is shown that the HW theoretical values may reproduce quantitatively the experimental results in the range of weight-average degree of polymerization x w ≥ 6. The disagreement between theory and experiment for x w ≤5 may probably be due to effects of chain ends. A comparison is also made of the present results with the previous ones for a-PMMA, leading to the conclusion that the HW theory may explain the f r dependence of as well as of , the scattering function, and steady-state transport coefficients.

Excluded-Volume Effects on the Transport Coefficients of Oligo- and Poly(dimethylsiloxane)s in Dilute Solution

The translational diffusion coefficient D and intrinsic viscosity [η] of oligo- and poly-(dimethylsiloxane)s (PDMS) were determined in toluene at 25.0 °C in the range of weight-average molecular weight M w from 1.08 x 10 3 to 1.12 x 10 6 for the former and from 5.33 x 10 2 to 1.12 x 10 6 for the latter. The hydrodynamic- and viscosity-radius expansion factors α H and α η were then determined from the values of the hydrodynamic radius R H defined from D and those of [η] corrected for the specific interaction between polymer and solvent molecules, respectively, with the use of the corresponding values of R H,0 and [η] 0 for the unperturbed PDMS chain in toluene at 25.0 °C. Here, the latter values were obtained by multiplying the previous values of R H,Θ and [η] Θ in bromocyclohexane at Θ by respective proper factors that take account of the solvent dependence of the bead diameter and may be calculated by the use of the helical wormlike (HW) chain theory. The results show that both α H and α η become functions only of the scaled excluded-volume parameter z defined in the Yamakawa-Stockmayer-Shimada theory for the HW chain with excluded volume, being consistent with the previous results for atactic polystyrene. This implies that the quasi-two-parameter scheme may be valid for a H and α η irrespective of the differences in polymer species and solvent condition and, moreover, that there is no draining effect on α H and α η , although it is significant for the unperturbed PDMS chain. It is again found that the Barrett equation overestimates α H . This disagreement between theory and experiment may be semiquantitatively explained by the Yamakawa-Yoshizaki theory, which takes account of the possible effect of fluctuating hydrodynamic interaction on α H .

Transport Coefficients of Isotactic Oligo- and Poly(methyl methacrylate)s in Dilute Solution

The intrinsic viscosity [η] was determined for 25 samples of isotactic oligo- and poly(methyl methacrylate)s (i-PMMA), each with the fraction of racemic diads f r ≃ 0.01, in the range of weight-average molecular weight M w from 3.58 x 10 2 (trimer) to 1.71 x 10 6 in acetonitrile at 28.0°C (Θ). The translational diffusion coefficient D was also determined from dynamic light scattering measurements for 12 of them in the range of M w from 6.58 x 10 2 (hexamer) to 9.46 x 10 5 under the same solvent condition. It is found that [η] is proportional to M w 1/2 for M w ≥ 5 x 10 4 and its deviation from this asymptotic behavior is small even for smaller M w , while D is inversely proportional to M w 1/2 except for M w ≤ 2 x 10 3 . Such apparent Gaussian behavior of [η] and D over a wide range of M w is the result expected from that previously obtained for the mean-square radius of gyration (S 2 ). From an analysis of these transport coefficients on the basis of the helical wormlike (HW) chain model, it is shown that the above M w dependences of [η] and D may be well explained by the HW theories with the values of the model parameters consistent with those previously determined from (S 2 ). A comparison is also made of the present results for [η] and D for i-PMMA with the previous ones for atactic (a-) PMMA with f r = 0.79. This leads to the confirmation of the previous conclusion derived from (S 2 ) concerning the f r dependence of the chain stiffness and local chain conformation of PMMA. That is, the i-PMMA chain is of weaker helical nature than the chain that retains rather large and clearly distinguishable helical portions in dilute solution.

Dynamics of helical wormlike chains. XII. Dynamic depolarized light scattering

Dynamic depolarized light scattering from dilute solutions of flexible chain polymers is studied on the basis of the discrete helical wormlike (HW) chain. A preliminary consideration leads to the conclusion that the spring-bead model with the Kuhn–Grün expression for the spring polarizability tensor is, in principle, inappropriate for the description of the depolarized scattering, so that the Ono–Okano theory is invalid for real chains. In the approximations that the orientational fluctuations of the subbodies composing the HW chain may be decoupled from their density fluctuation at very small magnitude of the scattering vector and that the former relax much faster than the latter, which are good enough under the condition of usual measurements, it is shown that the spectrum JΓ of the excess depolarized component of the scattered light may be written in terms of the same class of basic time-correlation functions with the ‘‘total angular momentum quantum number’’ L=2 and the number of ‘‘excited’’ subbodies n=1, as in the case of nuclear magnetic relaxation and fluorescence depolarization. Numerical evaluation of JΓ is carried out, taking atactic polystyrene (a-PS) and atactic poly(methyl methacrylate) as typical examples of flexible polymers. It is then found that for both polymers JΓ may be represented in terms of a small number of those eigenvalues λj2,k of the diffusion operator for the HW model with very small wave numbers k which belong to two branches specified by the superscript j, one corresponding to the low frequency modes and the other to the high frequency modes. For a-PS, this result is consistent with the experimental result obtained by Bauer, Brauman, and Pecora [Macromolecules 8, 443 (1975)] except that the present low frequency modes do not correspond to the Rouse–Zimm modes of the spring-bead model.

Transport coefficients of helical wormlike chains. 5. Translational diffusion coefficient of the touched-bead model and its application to oligo- and polystyrenes

The translational diffusion coefficient D of the helical wormlike (HW) touched-bead model is evaluated theoretically using the Kirkwood formula in the same spirit as in the evaluation of the intrinsic viscosity. The results are compared with those obtained previousvly on the basis of the HW cylinder model. Experimental results for D determined from dynamic light scattering are also reported for atactic oligo- and polystyrenes in cyclohexane at 34.5°C (θ). The agreement between the experimental and theoretical values is shown fto be fairly good over the whole range of M w down to the trimer

Dynamics of helical worm-like chains. XI. Translational diffusion with fluctuating hydrodynamic interaction

The translational diffusion coefficient D of long enough polymer chains without excluded volume is studied on the basis of the discrete helical worm-like (HW) chain with partially fluctuating (orientation-dependent) hydrodynamic interaction (HI). In order to evaluate D(∞) in the long time limit, which corresponds to diffusion experiments usually done on the long time scales, the time-dependent part of D is explicitly treated by an application of the projection operator method. The result may be written as D(∞)=D(Z)(1−δ0−δ1), where D(Z) is the Zimm value with preaveraged HI, δ0 is the relative decrease at time t=0 due to constraints, and δ1 is the additional relative decrease at t=∞ due to coupling between the translational motion and the Rouse vector (dielectric) modes for the HW chain, especially the long wavelength ones. It is found that D(∞) is decreased as the local conformation of the chain becomes rather compact, and that D(∞)=D(Z) (i.e., δ0=δ1=0) in the stiff-chain limit. Thus the ratio ρ of the root-mean-square radius of gyration to the hydrodynamic radius defined from D is not a universal constant but depends on the chain conformation and stiffness. A comparison of theory with experiment is made with respect to ρ, and it is found that the theory may well explain experimental results. For comparison, also for the Gaussian chain (spring–bead model), which is an unconstrained chain, D(∞) is evaluated, and it is found that δ0=0 but δ1≠0.