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.     

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.     

Random-coil dimensions of poly(methylphenylsiloxane) and their dependence on stereochemical structure

Rotational isomeric state theory was used to study the unperturbed dimensions 〈r²〉₀ of poly(methylphenylsiloxane) (PMPS) chains [Si(CH₃)(C₆H₅)? O? ]_x as a function of their stereochemical structure. The required conformational energies were obtained from semi-empirical, interatomic potential energy functions and from known results on poly(dimethylsiloxane). PMPS chains were found to differ from monosubstituted and disubstituted vinyl chains primarily in the larger distance of separation between groups in conformations giving rise to “pentanetype interactions.” In PMPS, the relatively large distance of separation, 3.8 Å, makes such in teractions attractive, particularly in the case of two phenyl groups; in contrast, such interactions are strongly repulsive at the ～2.5 Å separation characterizing vinyl chains. According to the calculated results, PMPS chains are very different from vinyl chains in that increase in isotacticity should cause a significant decrease in 〈r²〉₀ and increase in d ln 〈r²〉₀/dT. Comparison of theory with experimental results in the literature suggests that PMPS polymers which have been studied in this regard must have been significantly syndiotactic in stereochemical structure.     

Influence of the chemical composition and comonomer distribution on the dipole moments of 3,3-dimethyloxetane/tetrahydrofuran copolymers

Experimental values of the dipole moment ratio 〈μ₂〉/nm² and its temperature coefficient d ln 〈μ₂〉²/dT, where 〈μ₂〉 is the mean-square dipole moment of a chain with n skeletal bonds and m² is the mean square of the skeletal bond dipole moments, are reported for well-characterized random copolymers of 3,3-dimethyloxetane and tetrahydrofuran. The results are interpreted in terms of the rotational isomeric state theory in a manner consistent with that developed for the parent homopolymers. The theory gives a good account of the experimental results corresponding to copolymers in which the mole fraction of tetrahydrofuran lies in the range 0.11–0.89. It is found that whenever the copolymerization obeys Bernouillan statistics, the dipole moments are quite insensitive to the comonomer distribution. The theoretical analysis suggests, however, that the value of the dipole moment ratio of alternating copolymers of 3,3-dimethyloxetane and tetrehydrofuran should be near that of the parent homopolymer of lower polarity.     

Lattice Monte Carlo investigations on copolymer systems, 3. Alternating and random copolymers

Alternating and random copolymers in dilute solution are investigated by means of Monte Carlo simulations on a cubic lattice. Each molecule consists of an equal number of A and B segments, either randomly distributed along the chain or forming an alternating sequence. The energy parameters chosen represent selective solvent conditions (the solvent is a good one for monomers of type A and a θ-solvent for B; between A and B repulsive interactions are operative). Comparison with di- and triblock copolymers of equal overall composition reveals that the behaviour of random or alternating copolymers (subject to the same selective solvent) is quite different. Their properties rather resemble those of homopolymers in a solvent of intermediate quality. The absolute chain dimensions (e.g. the mean square radius of gyration, 〈s²〉, and the mean square end-to-end distance, 〈h²〉) of random and alternating copolymers as well as their scaling exponents are significantly larger than those of block copolymers. The ratio between 〈h²〉 and 〈s²〉 as well as the shape of the polymer (expressed by the asphericity δ) are similar to those of athermal polymers indicating that there is no pronounced selectivity of the solvent. In contrast to block copolymers, these parameters exhibit no significant chain-length dependence. The number of the various types of polymer-polymer contacts (A-A, B-B and A-B) is almost independent of the type of contact at least for the solvent conditions investigated. This is in contrast to block copolymers where A-B contacts are widely suppressed and where the number of B-B contacts is approximately twice as high as that of A-A contacts.     

Influence of excluded volume on the conformational property of tail-like chain on the simple cubic lattice

The influence of excluded volume on the conformational property of linear tail-like chain with one end attached to a flat surface is investigated by means of dynamic Monte Carlo method. Conformational properties such as mean-square end-to-end distance 〈R²〉, mean-square radius of gyration 〈S²〉 and mean asphericity parameter 〈A〉 are calculated for random walking (RW) and self-avoiding walking (SAW) tail-like chains on the simple cubic lattice. We find that the EV has nearly the same effect on 〈R²〉 as on 〈S²〉: (1) 〈R²〉_SAW/〈R²〉_RW≈〈S²〉_SAW/〈S²〉_RW∝n^0.204±0.05, where n is the chain length, and (2) the limiting value of 〈R²〉/〈S²〉≈7.7 for both chains. The distribution P(R) of the SAW tail-like chain can be expressed as a R⁴ correction of that of the RW one. We find that the value 〈A〉 of the SAW tail-like chain is bigger than that of the RW tail-like chain for all chain lengths, and the limiting values are 0.446±0.006 and 0.403±0.005 respectively.     

Bounds to electronic expectation values for atomic and molecular systems

A generalization of a method to calculate lower bounds to expectation values of non‐negative observables is presented. We consider bounds to three electronic expectation values 〈r²〉, 〈r〉, and 〈r^?1〉 in the helium atom as an example. For both 〈r²〉 and 〈r〉, we are able to calculate improved lower bounds. The lower bound to 〈r^?1〉 does not improve, but we are able to calculate an upper bound which is much closer to the expectation value than the lower bound. Although our generalization allows for improved bounds and/or upper bounds, these bounds to general observables are much less precise than energy bounds and even the expectation values calculated from variational wave functions. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Gaussian vs. Slater representations of d orbitals: An information theoretic appraisal based on both position and momentum space properties

A detailed appraisal of Gaussian-type orbital (GTO) and Slater-type orbital (STO) expansions of 3d orbitals is carried out for the ²S state of copper—a case that should be maximally unfavorable for STOs. The appraisal is based on a wide variety of both position and momentum space properties and utilizes an information theoretic quality assessment technique. It is found that GTO expansions are not as useful as STO expansions for the prediction of 〈p⁸〉, 〈p⁷〉, and 〈r^?6〉 because these properties probe the functional deficiencies of GTOs at small r and large p. On the other hand, GTO expansions can predict accurate values of large r properties like 〈r⁸〉 despite the fact that their position space asymptotic decay is too fast. Unlike the case of s orbitals in helium, there does not seem to be any consistent ordering between accuracy in position space and accuracy in momentum space. The quality measures are found to be very useful for pinpointing the deficiencies of various expansions. This information enables us to construct easily a new GTO and a new STO expansion that are more accurate than any of the others in the literature. It is suggested that one STO is worth no more than two GTOs in the case of d orbitals.     

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.     

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.     

Concentration dependence of the global and anisotropic dimensions of confined macromolecules

The chain dimensions 〈R²〉 of nondilute polymer solutions confined to a slit of the width D were studied using lattice simulations. It was found that the chain compression induced in good solvents by the concentration ϕ is enhanced in a slit relative to the bulk. The global dimensions of chains also change with ϕ in confined and unconfined theta solutions. At intermediate slit widths, a region was noted where coils are squeezed along all three axes. This region is manifested as a channel on the three‐dimensional surface 〈R²〉(D,ϕ) in both good and theta solvents. The coil anisotropy, given by the ratio of the parallel and perpendicular components of the chain dimensions 〈R_y²〉/〈R_x²〉, reaches high values at strong confinements, where coils form quasi‐two‐dimensional pancakes. The concentration‐induced reduction of the global chain dimensions in good solvents is almost fully transmitted to the parallel component 〈R_y²〉. The computed effects of concentration and confinement were compared with the predictions of mean‐field and scaling theories, and implications of the results to ultrathin films and layered nanocomposites were discussed. In addition, the distribution functions of the components of the end‐to‐end distance R perpendicular and parallel to the plates, W (R_x) and W (R_y), were calculated. The function W (R_x) combined with the concentration profile ϕ (x) along the pore provided details of the chain structure close to walls. A marked difference in the pace of the filling up of the depletion layer was noticed between chains in theta and good solvents. From the distribution functions W (R_x) and W (R_y), the highly anisotropic force‐elongation relations imply the deformation of chains in confined solutions and ultrathin bulk films.     

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.     

Dynamic Monte Carlo Simulation on the Polymer Chain with One End Grafted on a Flat Surface

A linear polymer chain in good solvent condition with one end grafted on a infinitely large, impenetrable flat surface is investigated using dynamic Monte Carlo simulation on a simple cubic lattice. Chain shape and dimension, angular correlation between the direction of the end‐to‐end vector and that of the longest principal axis of inertia are studied and discussed. Results reveal that the asphericity of end‐grafted polymer chains is greater than that of free ones, the limit ratio 〈L₁²〉 : 〈L₂²〉 : 〈L₃²〉 is about 1 : 3.0 : 14.9. The limit of mean angle 〈θ〉 of end‐grafted chains is about 22°, smaller than that of free chains, indicating angular correlation between the direction of the end‐to‐end vector and that of the longest principal axis is reinforced.     

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).     

Acrylonitrile–4-vinylpyridine copolymers effect of comonomer sequence distribution on properties

Acrylonitrile–,4-vinylpyridine copolymers were prepared in chloroform solution at 60°C with AIBN as initiator. Copolymer compositions were determined from their 15.01-MHz ¹³C-NMR spectra. Reactivity ratios of r_AN = 0.093 and r_4VP = 0.32 were calculated by the Kelen and Tudos method. The run number, number-average sequence lengths, and monomer sequence distributions were also calculated. The T_g values of the copolymers, their dye uptake, and degree of alkaline hydrolysis were influenced by the overall copolymer composition but particularly by the monomer sequence distribution in the copolymers.     

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.     

Correlation effects to the expectation values of atomic systems

The Hartree-Fock and first natural spin determinants were compared as reference determinants for calculating various one-electron properties such as ρ(0), 〈½?〉, 〈r^?2〉,…, 〈r³〉, and r^?1₁₂〉. Calculations were made on various small atoms and their positive and negative ions. For nearly all the expectation values studied, the first natural spin orbital determinant gave consistently superior results. In particular, the Hartree-Fock functions gave markedly inferior results for some long range properties such as the magnentic susceptibilities of negative ions. The major correlation error in the expectation values is primarily an orbital effect which may be accounted for by including correlation terms in the one-particle Hamiltonian. Such approximate Brueckner or best overlap orbitals should reproduce most one-electron expectation values accurately.     

Optical anisotropy of polymer chains and Markov processes. I. Polyethylene

A study of the average molecular optical anisotropy 〈γ²〉 of tetrahedral lattice polyethylene chains of any length has been carried out by using Markov processes. The results of this treatment include simplified models of polymer chains, as well as more elaborate models, which are in current use.     

Small-angle x-ray scattering from solution of rodlike particles with length distribution

The scattering function of rods with a constant radius, 8 Å, and a length distribution of the Schulz–Zimm type was calculated on a computer as a model of helical synthetic polypeptide. The influence of length and length distribution on the scattered intensity of small-angle x-ray scattering was clarified. As length grows and length distribution broadens it is difficult to obtain reliable values of molecular weight and radius of gyration from a Zimm plot. The influence of length distribution on the mass per unit length, M_q, and the radius of gyration of the cross section, 〈S〉^1/2, from the Guinier plot of the cross-sectional factor decreases as the length increases, and reliable values of M_q and 〈S〉^1/2 can be obtained even for rods with wide distribution for rods more than 600 Å long. In particular, it is pointed out that the value of 〈S〉^1/2 is little influenced by length and length distribution.     

Effects of composition on the solution properties of styrene–acrylonitrile copolymers

Short-range interactions between chain units of random copolymers in solution may be influenced by the composition or precisely by the distribution of sequence lengths of the same monomer units. Steric factors were derived for random copolymers of styrene and acrylonitrile with different compositions from the relation between the limiting viscosity number and the molecular weight. Mark-Houwink relations were obtained in methyl ethyl ketone (MEK) or in N,N′-dimethylformamide (DMF) at 30°C. for random copolymers containing 0.383 (Co-1) and 0.626 (Co-2) mole fraction of acrylonitrile, the expressions are: [η] = 3.6 X 10^?4 M _w^0.62, for Co-1 in MEK; [η] = 5.3 X 10^?4 M _w^0.61, for Co-2 in MEK; [η] = 1.2 × 10^?4M _w^0.77 for Co-2 in DMF. With the Stockmayer-Fixman expression, these correlations become, respectively: [η]/M^1/2 = 1.24 × 10^?3 + 8.0 × 10^?7 M^1/2; and [η]/M^1/2 = 1.70 × 10^?3 + 6.3 × 10^?7 M^1/2; and [η]/M^1/2 = 1.68 × 10^?3 + 31.3 × 10^?7 M^1/2. From the unperturbed mean-square end-to-end distances, 〈L²〉₀, determined from the first terms of the latter expressions, together with 〈L²〉_0f calculated by assuming the completely free rotation, gives the steric factor σ = (〈L²〉₀/〈L²〉_0f)^1/2 as 2.25 ± 0.05 for Co-1, and 2.31 ± 0.10 for Co-2. These values of σ are close to those for polystyrene (σ = 2.22 ± 0.05) and for polyacrylonitrile (σ = 2.20 ± 0.05). Therefore, it is concluded that the dimensions of random copolymers of styrene and acrylonitrile in solution are not significantly influenced by the composition. In other words, the unperturbed dimensions are not affected by a change in the alternation tendency between styrene units with phenyl side groups having a large molar volume and acrylonitrile units with nitrile groups responsible for the electrostatic interactions. On the other hand, the long-range interactions reflect the effect of sequence length. The Huggins constant and the second virial coefficient obtained from the light-scattering measurements have optimum values at about 0.5 mole fraction of acrylonitrile, where the greatest tendency for alternation seems to exist.