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.     

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.     

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.     

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.     

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     

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.     

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.     

Elastic behavior of single polymer chains adsorbed on rough surfaces

In this paper, elastic behaviors of single polymer chains adsorbed on the rough surfaces with a substrate and some periodically tactic pillars are investigated by the pruned-enriched-Rosenbluth method (PERM). In our simulation, a single polymer chain is firstly adsorbed on the substrate and then pulled along the z-axis direction, which is vertical to the substrate. We investigate the chain size and shape of polymer chains, such as mean-square radii of gyration per bond 〈S²〉_xy/N, 〈S²〉_z/N and shape factor 〈δ^∗〉 in order to show how the size and shape of adsorbed polymer chains change during the desorption process. Due to the occurrences of separation of the chains from the substrate, farther adsorption on the upper surfaces of pillars and complete separation from the whole rough surfaces in the elastic process, the changes of 〈S²〉_xy/N, 〈S²〉_z/N and 〈δ^∗〉 during the process are complicated. On the other hand, some thermodynamic properties such as average energy per bond, average Helmholtz free energy per bond, elastic force f are investigated, and our aim is to study the elastic behaviors of polymer chains adsorbed on the rough surface during the elasticity process. Elastic force f has some plateaus during the desorption process for strong adsorption interaction. If there is no adsorption interaction, the chains can get away from the rough surfaces spontaneously. These investigations can provide some insights into the elastic behaviors of polymer chains adsorbed on the rough surface.     

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.     

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.     

On the use of the gaussian chain as a monte carlo simulation model for the equilibrium properties of polymer solutions

We have explored the performance of a simulation model for Gaussian chains at different concentrations in a good solvent. The Gaussian statistics for the distances between contiguous beads in the model is directly implemented in the individual moves of a Monte Carlo algorithm. When the results of conformational properties for the Gaussian model are compared with those provided by a freely jointed model in the same conditions, significant differences arise at finite concentrations. The modeled Gaussian chain yields incorrect results for the quadratic average dimensions 〈R²〉 and 〈S²〉 at high concentrations, but correctly reproduces the results for the scaled end-to-end distance distribution function at any concentration, showing the effects of the screening of excluded volume when concentration increases. The reason for the wrong behavior of the simulated Gaussian model comes from a strong distortion of the “bond distance” distribution as a result of the concentration increase. We conclude that this model can only be safely applied to infinitely dilute solutions.     

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.     

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.     

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.     

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).     

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.     

Properties of cellulose acetate in solution. I. Light scattering,osmometry, and viscometry on dilute solutions

Light-scattering, osmotic pressure, and viscometric studies on fractions of cellulose acetate (degree of substitution 2.45) in three solvents are described. The data yield the dependence of the mean-square radius of gyration 〈s²〉, the second virial coefficient Γ₂, and the intrinsic viscosity [η] on molecular weight M and temperature. The results are interpreted to show that excluded volume effects on 〈s²〉 are negligible, even though Γ₂ is large and dΓ₂/dT is positive. The large experimental value of d In [η]/d In M is interpreted in terms of partial draining effects. Data on 〈s₂〉 and [η] for other cellulose esters in the literature are similarly interpreted. Significant aggregation found in solutions of cellulose acetate in many solvents is discussed.     

On the behaviour of the wave function for the 23S state of the two-electron atom in the region where both electron–nuclear distances are small

The expansion of the wave function for the 2³S state of the two-electron atom in the neighbourhood of the singularity at r₁ = r₂ = 0 is considered. The restrictions imposed on the variational functions by this expansion are discussed. For the 2³S state of He, Li⁺, N⁵⁺ the behaviour of the variational function based on the Fock expansion in the neighbourhood of this singularity is investigated. The agreement of the variational coefficients with the theoretical coefficients is satisfactory. The calculated values of E and 〈δ(r₂)〉 for He, Li⁺, N⁵⁺ are given.