The shape of linear and star-branched nonreversal random walks embedded in a tetrahedral lattice

The shape-asymmetry of linear and star-branched nonreversal random walk polymers on a tetrahedral lattice is studied by means of a Monte Carlo simulation. Properties characteristic of the instantaneous shape based on the mean-square radius of gyration and its principal components as well as based on the end-to-end distance are discussed. Short-chain effects are characterized by polynomial expressions (power series of N⁻¹), simultaneously by extrapolation yielding a result in the limit of infinitely large configurations.     

Off-lattice Monte Carlo studies on size and shape of isolated polymer chains with isotactic and syndiotactic attached side groups

Chains containing side groups were generated with up to 102 bonds in the backbone for both iso- and syndiotactic arrangements. The mean-square end-to-end distance <h ²>, the mean-square radius of gyration <r ²> and the ratios <l _i ² >/<l ₁ ² > of the axes of inertia were investigated. It could be shown that the mean size and shape of such chains are not influenced by the kind of stereoregularity.Dedicated to Prof. Dr. K. Ueberreiter on the occasion of his 70th birthday.     

Mean-square radius of gyration of polymer chains

The calculations of the mean-square radius of gyration for more than thirty sorts of polymer chains are reviewed on the basis of a unified approach. A general expression of the mean-square radius of gyration was developed for polymer chains with side groups and/or heteroatoms. It consists of two parts. The first part is the mean-square radius of gyration of a model chain, in which every side group, R, was considered to be located in the centroid of the substituent flanking the related skeletal atom, and the second one is the total contribution of the square radius of gyration of every substituent around its centroid. Numerical calculations showed that the logarithmic relationship between the mean-square radius of gyration and the degree of polymerization becomes linear when x is greater than 100, and the dependence of the mean-square radius of gyration on the molecular weight can be expressed by the general formula 〈S²〉 = aM^b, which was supported by a number of experimental measurements. A comparison of our expression for the mean-square radius of gyration with that reported by Flory was made. The difference is obvious in the range of lower molecular weight, and gradually declines with increasing degree of polymerization.     

Polymer deformation in various flows

Using a Langevin-like approach, the deformation of a polymer, modelled as a bead-spring chain, is calculated in simple shear, elongational and Kramers potential flow. Analytic expressions for the mean-square end-to-end distance, radius of gyration, segment-segment distance, static structure factor up to O(q⁴) and the intrinsic elongational viscosity are given. Near equilibrium, preaveraged hydrodynamic interaction is taken into account.     

Solution properties and chain conformation characteristics of cellulose tricarbanilate

Fractions of two cellulose tricarbanilate samples were characterized by light-scattering (weight-average molecular weight, second virial coefficient, mean-square radius of gyration), gel permeation chromatography (polydispersity index), and viscometry (intrinsic viscosity) in tetrahydrofuran and acetone. The intrinsic viscosity data were analyzed in terms of the theory developed for the continuous wormlike cylinder model, and the chain parameters (Kuhn statistical segment length λ^?1, chain diameter d, and shift factor M_L) were evaluated. The molecular-weight dependence of the mean-square radius of gyration in tetrahydrofuran was calculated for the Kratky—Porod chain model and compared with the experimental results. Data on the intrinsic viscosity and radii of gyration for other solvents at temperatures from 0 to 100°C were analyzed in the same way, and the effects of solvent and temperature on the statistical segment length were evaluated. Polymer—solvent interaction parameters were estimated from the second virial coefficients.     

Perturbation expansion of polymers in shear flow

The effect of shear flow on the excluded volume properties of bead spring chains is calculated via the perturbation expansion of the mean-square end-to-end distance with respect to the excluded volume parameter z. The coefficient of the series 〈R²〉/〈R²〉₀ = 1 + C₁ · z − … vanishes for large flow rates but shows a strong dependence on the shear rate in the intermediate regime.     

Mean-square radius of gyration of polysiloxanes

The mean-square radius of gyration for polysiloxanes has been derived according to the exact definition. Taking account of the examples of symmetrically substituted poly(dimethylsiloxane) and unsymmetrically substituted poly(methylphenylsiloxane), we find that the dependence of 〈S²〉 on the molecular weight follows the general formula 〈S²〉 = aM^b with b = 1 ± 0.016, which is analogous to the theoretical outcomes for vinyl or vinylidene polymers even though the skeletal bone of polysiloxanes consists of alternating heteroatoms. A numerical comparison of the rigorous expression of the mean-square radius of gyration given in this paper with that reported by Flory shows that the difference is obvious for low-molecular-weight polymer and it gradually declines with increasing degree of polymerization.     

Improved expression of the mean-square radius of gyration, 2. Poly(1,1-disubstituted ethylene)s

The mean-square radius of gyration of poly(1,1-disubstituted ethylene)s is calculated according to a method already developed for poly(methyl methacrylate), poly(α-methylstyrene) and polyitaconate. During the derivation both the effect of side groups and the masses of skeletal atoms were taken into account. A hypothetical polymer chain was introduced, in which the mass of the substituents on every C^α was considered to be concentrated in their center of mass, and the virtual side bond vector runs from C^α to this center. The mean-square radius of gyration of poly(1,1-disubstituted ethylene)s consists of two parts, one of which is the mean-square radius of gyration of the hypothetical molecule described before and the other is related to the geometrical characteristics of the side groups. Numerical calculations indicated that the dependence of the mean-square radius of gyration of poly(1,1-disubstituted ethylene)s on the molecular weight is analogous to that of vinyl polymers, 〈S²〉 = aM^b, where a and b are constants characteristic of the polymer.     

Temperature dependence of limiting viscosity number and radius of gyration of cellulose diacetate in acetone

Acetone solutions of a cellulose diacetate fraction were studied by viscosity and light scattering methods over the range 12.6–50.3². The temperature dependences of the limiting viscosity number [η], the mean-square radius of gyration 〈s²〉, and the second virial coefficient A₂ were determined. The unperturbed mean-square radius of gyration 〈s²〉_o and the expansion factor α, were estimated by using theoretical relations to the interpenetration function. It was found that dln 〈s²〉_o/dT is ?6.4 × 10^?3 deg^?1, while α, is close to unity over the whole temperature range studied. The viscosity results are interpreted to show that the draining effect is not negligible and the Flory viscosity parameter Φ slightly increases with increasing temperature. It is finally concluded that the value of ?6.9 × 10^?3 deg^?1 for dln [η]/dT can be ascribable to the rapid decrease in 〈s²〉_o.     

Estimation and analysis of the rheological properties of a perfluoropolyether through molecular dynamics simulation

Equilibrium and non-equilibrium molecular dynamics simulations of a perfluoropolyether C₈F₁₈O₄ are reported using an atomistic interaction potential. The bulk rheological properties of the perfluoropolyether are investigated through molecular dynamics simulations as a function of both temperature and shear rate. The effect of molecular structure on viscosity is explored in detail. The rotational relaxation time is reported as a function of temperature. Structural properties, including the mean-square end-to-end chain length, the mean-square radius of gyration of chains, and the distribution functions of bond lengths, bond angles, and bond torsional angles are collected and analyzed as functions of shear rate. After an initial plateau, both mean-square end-to-end chain length and mean-square radius of gyration decrease monotonically with increasing shear rate. The behaviors of the rheological and structural properties are explained through an analysis of the individual contributions due to bond stretching, bond bending, and bond torsion, as well as both intramolecular and intermolecular non-bonded interactions. A further analysis is possible through a meticulous breakdown of each contribution into a specific type of mode; e.g., the total bond stretching is comprised of CC, CO, and CF bond stretching terms. In this way, one can relate the shear viscosity to the specific chemical structure of C₈F₁₈O₄.     

Third-order interaction approximation for linear polymer chains

Third-order interactions imposed by a pair of atoms separated by five bonds are taken into account in computations of the mean-square end-to-end distance and the mean-square radius of gyration for linear polymer chains. The statistical weight matrices are established on the basis of the rotational isomeric state model. The conformational energy of n-hexane is calculated as a function of the C? C bond rotation angles. The third-order interaction energy is obtained by comparison with that of n-pentane. The characteristic ratio of polymethylene is 6.6 in the third-order interaction approximation, which is in agreement with experimental data.     

Unperturbed dimensions of copolymer molecules

Proceeding from the work of Zimm and Stockmayer and of Benoit, a general formula is derived for the calculation of the radius of gyration of block and graft copolymers. It appears that with linear block copolymers the ratio of the mean-square radius of gyration and the mean-square end-to-end distance is usually not far from 1/6. With a great number of blocks the difference from 1/6 can entirely be neglected. Also the influence of the composition of the molecules on the scattered light intensity has been considered.     

Studies of solution properties of copolymers. II. Copolymer of maleic anhydride and styrene

Styrene and maleic anhydride were copolymerized in benzene. The whole polymer thus obtained was fractionated with acetone and petroleum ether as the solvent and precipitant, respectively. The viscosities and osmotic pressures of the fractions were determined in tetrahydrofuran. The relation between the intrinsic viscosity and the molecular weight, [η] = 5.07 × 10^?5 M?_n^0.81, was obtained in tetrahydrofuran. The unperturbed mean square end-to-end distance was estimated by the Stockmayer-Fixman equation. A theoretical equation for the mean square end-to-end distance for a chain of repeating units of different bond lengths a and b with a fixed valence angle θ and without restriction of internal rotation was presented and applied to this copolymer. In addition, the equation of the mean-square end-to-end distance derived by Wall for trans-polyisoprene without rotational restriction was modified for application to this copolymer. The result evaluated with our equation was about 26% smaller than that from the modified Wall equation. A steric parameter for the present copolymer is defined and discussed in comparison with those of polystyrenesulfone and polystyrene.     

Laser light-scattering investigation of the density of pauci-chain polystyrene microlatices

The average density (〈ρ〉) of the pauci-chain polystyrene microlatices (PCPS), which contains a few linear polystyrene chains, was investigated by laser light scattering (LLS) including both angular dependence of absolute integrated scattered intensity (static LLS) and of the line-width distribution G(Γ) (dynamic LLS). In static LLS, the weight-average particle mass (M_w) and the z-average radius of gyration (R_g) were measured; and simultaneously in dynamic LLS, the hydrodynamic radius distribution was obtained from Laplace inversion of very precisely measured intensity-intensity time correlation function. A combination of both the static and dynamic LLS results leads us to a value of 〈ρ〉. For comparison, we also determined 〈ρ〉 of conventional multichain polystyrene latex (MCPS) by following the same LLS procedure. It was found that 〈ρ〉_MCPS = 〈_bulk〉 = 1.05 g/cm^3, but 〈ρ〉_PCPS = 0.92 g/cm^3. This difference in density suggests that the intersegmental distance in MCPS or bulk polystyrene is smaller than that in PCPS, even the chains in PCPS are confined to a smaller volume. This might attribute to the fact, namely the intersegmental approaching inside PCPS is mainly the intrachain crossing which is more difficult in comparison with the interchain crossing inside MCPS or bulk polystyrene.     

Hydrodynamic interaction effects on intrinsic viscosity of perturbed chains: Experimental results for polystyrene

Experimental measurements of intrinsic viscosity and radii of gyration of monodisperse samples of polystyrene of molecular weights 2.33 X 10⁵, 4.11 X 10⁵, 6.70 X 10⁵, and 2.3 X 10⁶ dissolved in the homologous series of 1-chloroparaffins from butane to undecane are reported. The dependence of the viscosity expansion coefficient α_η on the expansion factor α_s for the radius of gyration is discussed in the light of the results obtained by an expansion of the Fixman theory for perturbed chains to include partial draining. These results give support to the finite chain effect on the hydrodynamics of expanded coils in the usual range of molecular weight. The exponent a in the relation α³_η = α^a_s depends on molecular weight and agrees with recent nondraining theoretical calculation for exceedingly high polymers.     

Unperturbed chain dimensions of isotactic polypropylene in theta solvents

The unperturbed mean-square end-to-end distance 〈R₀²〉 and its temperature variation d In 〈R₀²〉/dT for isotactic polypropylene have been estimated from intrinsic viscosity data in three theta solvents, i.e., diphenyl, diphenyl ether, and dibenzyl ether, measured at their θ temperatures as determined by precipitation temperature measurements. The characteristic ratios, 〈R₀²〉/nl², where n is the number of bonds of length l in the main chain, evaluated by assuming Φ = 2.87 × 10²¹, are 5.80 in diphenyl (at θ = 125.1°C.), 5.41 in diphenyl ether (at θ = 142.8°C.), and 4.56 in dibenzyl ether (at θ = 183.2°C.). These values lead to the temperature coefficient d In 〈R₀²〉/dT = ?4.09 × 10^?3 deg.^?1 Results are compared with the data previously reported on polyethylene.     

CONFIGURATION-DEPENDENT PROPERTIES OF POLY(DIMETHYLSILOXANE) CHAIN*

The intermolecular rotational potential energies for poly(dimethylsiloxane) (PDMS) chains aredirectly obtained from a priori probability P_(αβ). Here the differing statistical weight matrices for the Si-Oand O-Si bonds are considered in calculating the configuration partition function. In the Bahar's model, asthe same statistical weight matrices for the Si -O and O- Si bonds are adopted, there exists a large deviationof αpriori probability P_(αβ) between the theory and the molecular dynamics (MD) simulation. Our model givessatisfactory agreement with experiment on the mean-square unperturbed end-to-end distance, the mean-square dipole moment and its temperature dependence, and the molar cyclization equilibrium constants fordimethylsiloxane oligomers. This new rotational isomeric state approach can be widely applied to otherchains, such as -CH_2-C[(CH_2)_mH]_2- and -O-Si[(CH_2)_mH]_2 for arbitrary m.     

The nonlinear change in conformation of polyelectrolyte macromolecules in saltless water-organic solvent

High molecular weight polyelectrolyte: poly(dimethylaminoethyl methacrylate) [PDMAEMA] with molecular weights M_W = 28.0×10⁶, 20.0×10⁶, 15.0×10⁶ was investigated in dilute solution by light scattering, flow birefringence and viscometry (at different rate gradients) in a water-acetone system by varying the weight fraction of acetone r in the mixture. At r=0.76 the polymer undergoes a reversible coil-globule transition accompanied by a drastic decrease in intrinsic viscosity [n], mean-square radius of gyration R²_z^1/2 and second virial coefficient A₂, with no change in molecular weight. The coil asymmetry parameter p (p=2.5 at r=0.50) decreases with increasing r and attains unity (completely symmetrical particle) at the transition point (r=0.76). The anomalous behavior of the viscosity of PDMAEMA-water-acetone solutions, detected near the transition point (r=0.6+0.7), is interpreted by formation of local knots of compactization on the molecular chain under the influence of a hydrodynamic field.     

聚丙烯酸烷基酯侧基差异对链构象性质的影响

针对聚丙烯酸烷基酯侧基形状、极性的不同,基于高分子链的构象-构型统计理论和生成矩阵方法,推出均方电偶极矩和均方回转半径的改进公式,应用于聚丙烯酸甲酯(PMA)链和聚甲基丙烯酸乙酯(PEMA)链等的链构象依赖性质的研究.得到无规PMA4(4态PMA)和PEMA的偶极矩特征比分别为0.63和0.66,温度系数为-0.66×10-3 K-1和1.72×10-3 K-1,无规PMA和PEMA的均方回转半径的特征比分别为1.73和1.43,温度系数为-1.05×10-3 K-1和-1.28×10-3 K-1,均与实验结果符合.而且考虑链规整程度的不同和酯类基团方向等的改变,PMA和PEMA的特征比也呈现较大的差异,尤其对间同构型链PMA4短链,偶极矩及其温度系数均伴有极大值的出现,这些结果表明聚丙烯酸烷基酯侧基对其链构象性质的影响不能忽略.