Random-coil dimensions and dipole moments of propylene–vinyl chloride copolymers

Mean-square unperturbed dimensions 〈r²〉₀ and dipole moments 〈μ²〉 have been calculated for propylene–vinyl chloride copolymers by means of rotational isomeric state theory. The calculations indicate that for these chain molecules 〈mu;²〉 is much more sensitive to chemical sequence distribution than is 〈r²〉₀, a conclusion in agreement with results of previous studies of ethylene–propylene copolymers and styrene-substituted styrene copolymers. In the case of propylene–vinyl chloride chains, both 〈r²〉₀ and 〈μ²〉 are most strongly dependent on chemical sequence distribution in the case of copolymers which are significantly syndiotactic in stereochemical structure. At equimolar chemical composition, increase in average chemical sequence length generally increases 〈r²〉₀ but decreases 〈μ²〉. Under some conditions, values of these statistical properties go through a minimum with increase in the reactivity ratio product r₁r₂, thus complicating the use of experimental values of these properties in the characterization of chemical sequence distributions in these copolymers.     

MEAN-SQUARE RADIUS OF GYRATION AND DIPOLE MOMENT OF POLY(METHYLPHENYLSILOXANE) CHAINS

The mean-square radius of gyration ,the mean-square dipole moment ,the mean-square end-to-end distance and their temperature coefficients of unsymmetrical disubstituted poly(methylphenylsiloxane) (PMPS) chains, as a function of stereochemical structure,confomational energies and length of polymers,were studied by using an improved configurational-confomational statistical method based on the rotational-isomeric-state theory.It is found that the increase in isotacticity of PMPS chains caus...     

Configurational properties of aromatic polyesters: Dipole moments of poly(diethylene terephthalate) chains

Dielectric constants have been determined for a fraction of poly(diethylene terephthalate) in benzene at several temperatures. The data indicate that the dipole moment ratio 〈μ²〉/Nm² is somewhat higher than that of poly(ethylene oxide), and its temperature coefficient is in the vicinity of zero. Both the dipole ratio and its temperature coefficient are in very good agreement with those predicted by the rotational isomeric state theory. Using this theory, the unperturbed dimensions of poly(diethylene terephthalate) were calculated and it was found that (〈r²〉/M)_∞ = 0.80 Ų (g mol wt)^?1, a value intermediate between those of poly(ethylene oxide) (0.57) and poly(ethylene terephthalate) (1.05).     

Random-coil configurations of the polyformals. VI. Dipole moments of the stereochemically variable polymer prepared from 4-methyl-1,3-dioxolane

Mean-square dipole moments 〈μ²〉₀ for the (atactic) poly(4-methyl-1,3-dioxolane) chain [CH₂OCH(CH₃)CH₂O? ] were determined from dielectric-constant measurements carried out on two fractions dissolved in benzene. Simple examination of the chain structure leads to the predictions that this polymer should have a significantly larger value of 〈μ²〉₀ than poly(1,3-dioxolane) itself, but that 〈μ²〉₀ should be nearly independent of stereochemical structure. The first expectation is confirmed by the experimental results obtained, and the second by calculated results based on rotational isomeric state theory.     

Persistence vectors for polypropylene,polystyrene, and poly(methyl methacrylate) chains

Persistence vectors a ≡ 〈 r 〉 are calculated for polypropylene, polystyrene, and poly(methyl methacrylate) chains as functions of chain length and stereochemical constitution. Differences between the progressions of a with chain length for these vinyl polymers are related to their conformational characteristics. The preferences of the syndiotactic chains for the tt dyad conformation are manifested most strikingly in the behavior of this vector.     

Thermodynamic and conformational properties of partially butyralized poly(vinyl alcohol) in aqueous solution

Phase separation behavior has been studied in aqueous solutions of partially butyralized poly(vinyl alcohol) (BuPVA) with various degrees of butyralization x_Bu and various molecular weights. It is found that these systems exhibit both upper and lower critical solution temperatures. The theta temperatures θ_LCST, evaluated by means of Shultz–Flory plots, are found to be 25.1, 23.3, and 14.4°C for BuPVAs with x_Bu of 7.5, 9.9, and 12.7 mol %, respectively. The unperturbed dimension 〈R²〉₀/M is evaluated as ca. 1.2 × 10^?16 Ų in the above range of x_Bu from viscosity measurement at θ_LCST. Properties of the BuPVA solutions are compared with those of other PVA copolymers.     

Distribution function for the radius of gyration for theta chains attached at one end to an impenetrable surface

Chains that are unperturbed by the intramolecular excluded volume effect are examined in the free state and under conditions where one of the ends is attached to an impenetrable surface. The density of the attachment of the chains to the surface is so low that interchain interactions can be ignored. The shapes of the distribution functions for the squared end-to-end distance, r², and squared radius of gyrations, s², are evaluated in both situations. As is well-known from previous work, attachment of the chains to the impenetrable surface produces an increase in 〈r²〉 and (to a lesser extent) 〈s²〉. It also narrows the distribution function for r². There is, however, no detectable effect on the shape of the distribution function for s². The narrowing of the distribution function for r², coupled with the absence of any effect on the shape of the distribution function for s², is rationalized with a very simple model.     

Stress-optical coefficient of cis-1,4-polybutadiene and cis-1,4-polyisoprene networks. Measurements on cis-1,4-polybutadiene networks and theoretical interpretation

Stress, strain, and birefringence measurements have been carried out on swollen and unswollen networks of ′cis-1,4-polybutadiene polymers. Neither stress-strain nor birefringence-strain relations of unswollen specimens obey the Gaussian network theory, but both can be fitted by the Mooney-Rivlin equation. On the contrary, data on specimens swollen in tetralin, decalin, benzene, and carbon tetrachloride strictly obey the Gaussian network theory. Existing methods for evaluating the temperature coefficient of the unperturbed dimensions, d In 〈r²〉/dT, from the stress-temperature relation are applied to the present data and discussed in some detail. It is concluded that reliable values of d In 〈r²〉/dT are not obtainable from data on unswollen samples because of the pronounced non-Gaussian effect. The value 7.5 ų for the optical anisotropy ų (an alternative to the stress-optical coefficient) for unswollen specimens is markedly larger than values (5.8 ų on the average) for swollen specimens. This is interpreted as due to the shortrange orientational order among polymer segments. The quantities 〈r²〉, ΔΓ, and their temperature coefficients are calculated for both cis-1,4-polybutadiene and cis-1,4-polyisoprene chains, on the basis of the rotational isomeric state approximation for bond rotations. Values of ΔΓ for cis-1,4-polybutadiene calculated using Clément and Bothorel's set of anisotropic bond polarizabilities are in good agreement with observed values for swollen specimens. Those for cis-1,4-polyisoprene obtained using the same set of anisotropic bond polarizabilities are somewhat smaller than observed values for unswollen specimens. This departure is in the direction expected from the behavior of ΔΓ upon swelling (i.e., a decrease in ΔΓ upon swelling).     

Distributions of the unperturbed dimensions of Markov-chain copolymers

This investigation shows that Markov-chain copolymers can be regarded as random copolymers the segment lengths of which depend on the copolymerization parameters. It was possible to derive simple analytical formulae for the mean-square end-to-end distance (〈r²〉), the Kuhn length, and the distribution of r² under theta-conditions. The results of these equations are in excellent agreement with data from simulations. It is shown that 〈r²〉 as well as the non-uniformity of r² increase strongly with increasing probabilities of homopropagation, i.e., with increasing mean homosequence lengths. Furthermore it is demonstrated by simulation that even chains of identical length and composition show a distribution of r² because of different arrangements of the sequences inside the chains. For chains or chain segments shorter than the average homosequences, a double-peak distribution of r² is found. The equations derived in this paper can be applied to real copolymers as well as to chains the curvature of which is altered locally by the association of ligands.     

Response of the mean-square optical anistropy of chain molecules to the imposition of excluded volume

The influence of the chain expansion produced by excluded volume on the mean-square optical anisotropy has been studied in six types of polymers. The mean-square optical anistropy for a specified configuration is calculated using the valence optical scheme. Realistic rotational isomeric state models are used for the configurational statistics of the unperturbed chains. Excluded volume is introduced by hard sphere interactions. Results obtained with chains of 100, 200, 300, and 400 bonds permit extrapolation to the behavior expected for much longer chains. The mean-square optical anisotropy of polyethylene is insensitive to excluded volume. A similar conclusion was obtained several years ago in a study of chains confined to a tetrahedral lattice and weighted in a manner appropriate for the short-range interactions in polyethylene.² Different behavior is seen in poly(vinyl chloride), poly(vinyl bromide), polystyrene, poly(p-chlorostyrene), and poly(p-bromostyrene). The mean-square optical anisotropy of these five vinyl polymers is sensitive to the imposition of excluded volume if the stereochemical composition is exclusively racemic. Much smaller effects are seen in meso chains and in chains with Bernoullian statistics and an equal probability for meso and racemic dyads.     

Conformational characteristics of poly(ethylene phthalate)s

The conformational characteristics, as embodied in the unperturbed mean‐square end‐to‐end distances (〈r²〉_o) and the characteristic ratios of the dimensions [C_n = 〈r²〉_o/(n〈l²〉] are calculated for the para, meta, and ortho isomers of poly(ethylene phthalate)s: poly(ethylene terephthalate) (PET), poly(ethylene isophthalate) (PEI), and poly(ethylene phthalate) (PEP), respectively. Although each of these isomeric and partially aromatic polyesters has identical permissible conformations available to their ethylene glycol fragments, their connections through the ester bonds to the phenyl rings are quite distinct. In addition, for the ortho isomer (PEP), the close spatial proximity of the ester groups bonded to the same phenyl ring results in an interdependence of their orientations with respect to each other and the phenyl ring to which they are attached, unlike the independent orientations of ester groups in the para and meta isomers (PET and PEI). Conformational energy calculations, dependent on the orientation of the ester groups in PEP, are used to characterize their rotational interdependence to modify the rotational isomeric state (RIS) conformational models for PET and PEI and thereby obtain an RIS model appropriate for PEP. This leads to calculated relative dimensions (〈r²〉_o) of 1.0:0.70:0.37 PET:PEI:PEP and characteristic ratios [C_n = 〈r²〉_o/(n〈l²〉)] of 4.13:4.67:2.49 PET:PEI:PEP. These results are discussed in an effort to obtain some understanding of the inherent static (or equilibrium) and dynamic flexibilities of the isomeric poly(ethylene phthalate)s. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 1254–1260, 2002     

Intrinsic viscosity of polymers of low molecular weight

Experimental evidence concerning the dependence of the intrinsic viscosity [η] on molecular weight M in the low molecular weight range (from oligomers to M = 5 × 10⁴) has been collected in a variety of solvents for about ten polymers, i.e., polyethylene, poly(ethylene oxide), poly(propylene oxide), polydimethylsiloxane, polyisobutylene, poly(vinylacetate), poly(methyl methacrylate), polystyrene, poly-α-methylstyrene, and some cellulose derivatives. In theta solvents, the constancy of the ratio [η]Θ/M^0.5 extends down to values of M much lower than those predicted by current hydrodynamic theories. In good solvents, and on decreasing M, the polymers examined, with the exception of polyethylene and some cellulose derivatives, show a decrease in the exponent a of the Mark-Houwink equation [η] = KM^a. This upward curvature gives rise to the existence of a more or less extended linear region where the equation [η] = K₀M^0.5 is obeyed. Below the linear range, i.e., for even shorter chains, the exponent a can increase, i.e., polydimethylsiloxane, or decrease below 0.5, i.e., poly(ethylene oxide), depending on the particular chain properties. These different dependences have been discussed in terms of: (a) variations of thermodynamic interactions with molecular weight; (b) variations of conformational characteristics (as for instance the ratio) 〈r0²/nl²〉, where 〈r0²〉 is the unperturbed mean square end-to-end distance and n is the number of bonds each of length l; (c) hydrodynamic properties of short chains.     

Configurational statistics of polyamide chains

The statistical mechanical treatment of polymeric chains in terms of the largest eigenvalue of the product of statistical weight matrices for the rotational interactions of skeletal bonds of the repeat unit becomes excessively complicated if the repeat unit spans more than three or four skeletal bonds. Moreover, such treatment is necessarily limited to chains in which the number of repeat units is indefinitely large. Newer methods are readily applicable to chains of any degree of polymerization comprising repeat units of any realizable length. If interdependence of neighboring bond rotations is confined to bond pairs within a given unit, rotations about a pair of bonds belonging to neighboring units being mutually independent, further simplifications may be introduced without sacrifice of rigor. Polyamides, in which rotation about bonds on opposite sides of the amide group are independent, are polymers of this type. Adherence of the amide group to the planar trans conformation favors a more extended configuration of the chain, but this effect is dominated by the smaller steric repulsions affecting rotations about bonds which are first, second, and third neighbors of the amide group. It is for this reason that the characteristic ratio 〈r²〉₀〉/nl² for poly(hexamethylene adipamide), ca. 6.0 according to experimental results of Saunders, is less than the value, 8.0, for polymethylene at 25°C. The characteristic ratios and molecular dipole moments are computed as functions of the degree of polymerization. The poly(εaminocaproamide) chain also is treated.     

Experimental anomalies and the conformational theory of polymers

The influence of the size and shape of the torsional potential upon the theoretical temperature coefficient (T? ln 〈r₀²〉/?T) of a polymeric chain was studied. The uncorrelated end-to-end distance equations for isotactic, syndiotactic, and completely atactic chains were differentiated with respect to temperature and the integrals in the resulting equations were evaluated by the method of Gaussian quadrature. The calculated coefficients were found to be nonlinear functions of the energy ratio U_max/KT, where U_max is the maximum potential barrier and possess real roots which critically depend upon the size and shape of the potential. Qualitative anomalies between experiment and theory disappeared when the entire torsional potential was used in the conformational theory. It appears that quantitative agreement between theory and experiment can be established for polymeric materials if the entire potential is used, rotations within bulky side groups are included, and the potential is determined by the method of minimum conformational energy.     

Dilute solution properties of branched polymers

For calculating the ratio of the intrinsic viscosities of branched and linear polymers of the same molecular weight, [η]_B/[η]_L, a new theory taking into account the excluded volume effect is presented. By using the modified Flory equation, the excluded volume effect of branched polymers is predicted with the aid of the first-order perturbation theory. The linear expansion factor α_s is converted to the hydrodynamic expansion factor α_η by using the Kurata-Yamakawa theory. Our calculated results, i.e., [η]_B/[η]_L and 〈s²〉_B/〈s²〉_L, agree well with experiment for various type branched polymers, i.e., randomly branched and comb-shaped polymers of poly(vinyl acetate).     

Small-angle x-ray and light scattering studies of the morphology of blends of poly(ϵ-caprolactone) with poly(vinyl chloride)

Solvent-cast films of blends of poly(?-caprolactone) (PCL) with poly(vinyl chloride) (PVC) were examined by low-angle x-ray scattering and by small-angle light scattering. X-ray scattering from crystalline compositions were analyzed using the Tsvankin–Buchanan technique and led to values of the repeat period of the lamellar structure and the thickness of the crystalline and amorphous layers. With increasing content of PVC, the amorphous layer thickness increased sufficiently to accommodate the PVC, leading to values of the linear crystallinity consistent with macroscopic measurements by density and DSC techniques up to about 50% PVC by weight. Above this concentration, the lamellar structure no longer appeared to be volume filling. At high concentration of PCL, the polymer consisted of volume-filling spherulites containing the lamellar substructure. Spherulite sizes were measured by light scattering and absolute light scattering intensities were consistent with calculations based upon the degree of crystallinity and anisotropy of the spherulites. Compositions containing more than 60% PVC were amorphous. Low-angle x-ray scattering was interpreted in terms of the Debye–Bueche theory which leads to values for a correlation distance l_c and the mean-square electron density fluctuation 〈η²〉 (which was also obtained from the invariant). By the method of Porod, the correlation distances were resolved into persistence lengths within the two phases, which were determined as a function of composition. The fluctuation 〈η²〉 was analyzed in terms of a two-phase model to show that its value was somewhat larger than would be obtained if the phases were composed of the pure components. It was not possible to uniquely determine their compositions. The data were consistent with the existence of a transition zone of the order of 30 Å thick between phases.     

Conformational analysis of poly(2-hydroxyethyl methacrylate)

The poly(2-hydroxyethyl methacrylate) (PHEMA) is a disubstituted vinyl chain in which the substituents CO₂CH₂CH₂OH and CH₃ differ in size and shape. In order to verify the various characteristics of the PHEMA chain, the conformational energy calculations for meso and racemic diads, which are the segments consisting of the stereoregular isotactic and syndiotactic chains, respectively, were carried out using ECEPP/2 potential. From these calculations, the averaged geometry and the statistical weights were obtained in a local minima. The characteristic ratio, C∞ = (〈r²〉_o/nl²)∞, was determined from the statistical weights and geometries. The calculated C∞ for the isotactic and syndiotactic chain are 10.2 and 2.3, respectively. The characteristic ratio for isotactic chain is larger than that for syndiotactic chain. This shows that the syndiotactic chain is more folded than the isotactic chain is, and that the calculated tendency is in reasonably agreement with the experimental tendency of acrylate polymers.     

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.     

Excluded-volume effect for various chain models. I. Application to optical anisotropy and end-to-end distance of linear polymers

A study of the effect of excluded volume on the mean-square optical anisotropy 〈γ²〉 and the mean-square end-to-end distance 〈R²〉 has been carried out for polymer chains of up to 2000 bonds. The calculations have been made for models assuming (a) equiprobability of internal rotations and (b) interdependence of short-range intramolecular interactions. All the results indicate that 〈γ²〉 is practically insensitive to the excluded-volume effect. Concomitantly the behavior of 〈R²〉 has been analyzed.     

Anisotropy in the dimensional and elastic parameters of confined macromolecules

Using lattice simulations the effect of confinement on the size, orientation and elastic properties of athermal chains was investigated. For chains confined in a slit or in a “cylinder” with square profile a minimum was observed in the dependence of the mean‐square end‐to‐end distance 〈R²〉 on the plate distance D. However, the components of the mean chain dimensions perpendicular and parallel to the walls, 〈R²_⟂〉 and 〈R²_∥〉, steadily diverge with reduction of the pore size. In a slit the distribution functions of the chain vector perpendicular and parallel to the plates, W〈R²_⟂ 〉 and W〈R²_∥〉, respectively, were computed. The marked difference between these distribution functions is interpreted as a sign of enhanced alignment of chains of the shape of elongated ellipsoids along the pore walls. A major part of the free energy of confinement ΔA_cf stems from this mechanism of pore‐induced macromolecular orientation. A striking anisotropy was observed in the elastic free energies A^el_⟂ and A^el_∥ of chains deformed in the direction perpendicular and parallel to the walls and in the corresponding force‐displacement functions. Finally, the relation between the elastic free energy A^el and the free energy of confinement ΔA_cf and between the forces f and f_solv derived thereof is analysed.