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.     

Elastic behavior of uniform star polymer chains

Elastic behaviors of uniform star polymer chains with two to seven branches (namely, functionality f = 2-7) are investigated using Monte Carlo simulation and the bond fluctuation model. Here chain dimensions and thermodynamic properties of uniform star polymer chains during the process of tensile elongation are studied, and comparisons with linear chain are also made. Static properties of chains such as chain sizes and asphericities of chains are calculated, and g-value of g = 〈S²〉_star/〈S²〉_linear decreases with elongation ratio increasing for different functionality f. Thermodynamic properties such as average energy 〈U〉, free energy per bond 〈A′〉 and elastic force F are also investigated during the process of tensile elongation. In the meantime, scatting functions P(q) are calculated for star polymer chains with different functionality f. Additionally, we also discuss the influence of elongation ratio on scattering form factor. The impenetrability of the star cores is known to cause a discontinuity in the osmotic pressure showed through a peak in the scattering functions, and some different behaviors in the tensile process for uniform star chain are obtained.     

Molecular geometry and chain entanglement: parameters for the tube model

The tube diameter in the reptation model is the distance between a given chain segment and its nearest segment in adjacent chains. This dimention is thus related to the cross-sectional area of polymer chains and the nearest approach among chains, without effects of thermal fluctuation and steric repulsion. Prior calculated tube diameters are much larger, about 5 times, than the actual chain cross-sectional areas. This is ascribed to the local freedom required for mutual rearrangement among neighboring chain segments. This tube diameter concept seems to us to infer a relationship to the corresponding entanglement spacing. Indeed, we report here that the critical molecular weight, M_c, for the onset of entanglements is found to be M_c = 28 A/(〈R²〉₀/M), where A is the chain cross-sectional area and 〈R²〉₀ the mean-square end-to-end distance of a freely jointed chain of molecular weight M. The new, computed relationship between the critical number of backbone atoms for entanglement and the chain cross-sectional area of polymers, N_c = A^0,44, is concordant with the cross-sectional area of polymer chains being the parameter controlling the critical entanglement number of backbone atoms of flexible polymers.     

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.     

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.     

Solution properties of high molecular weight polystyrene

A series of polystyrenes with weight-average molecular weight M?_w up to 1.3 × 10⁷ was prepared by anionic polymerization in tetrahydrofuran (THF). Each sample was characterized by gel-permeation chromatography, light scattering, and viscometry. It was found that each sample had an almost symmetrical and very narrow molecular weight distribution (M?_w/M?_n < 1.07). The mean-square unperturbed radius of gyration 〈S²〉₀ was determined in trans-decalin at 20.4°C as 〈S²〉₀ = 7.8₆ × 10^?18M?_w (cm²). The particle scattering factor was well represented by the Debye equation irrespective of solvent in the range of M?_w < 4 × 10⁶, and only a small deviation was observed in benzene at higher molecular weights. The penetration function Ψ ≡ A₂M²/4π^3/2N_A〈S〉^23/2 was found to approach a relatively low asymptotic value of 0.21–0.23 at molecular weights above 2 × 10⁶ in benzene at 30°C, where A₂ is the second virial coefficient and N_A is Avogrado's number. It was also found that the theta temperature in trans-decalin was affected by the nature of polymer samples. A difference of about 3°C in the theta temperature was observed between two series of anionic polystyrenes, one prepared in THF and the other in benzene, but there was practically no difference in unperturbed chain dimension.     

Elastic behavior of poly(ethylene terephthalate) chains

The Monte Carlo (MC) method based on the rotational-isomeric-state (RIS) model is adopted in studying the elastic behavior of poly(ethylene terephthalate) (PET) chains in this paper. The mean-square end-to-end distance 〈R²〉, the mean-square radius of gyration 〈S²〉, and the ratio of 〈R²〉/〈S²〉 all increase with elongation ratio λ. The interior conformations are also investigated through calculating the a priori probability of rotational state in the process of tensile elongation. The radius of gyration tensor S is introduced here in order to measure the shape of PET chains, and increases with elongation ratio λ, however, some different behaviors are obtained for . Here , and are the eigenvalues of the radius of gyration tensor . The average energy per repeat unit 〈U〉 and the average free energy per repeat unit 〈A〉 are also calculated, and we find that the average energy decreases with elongation ratio λ, however, the average free energy per repeat unit increases with elongation ratio λ. Elastic force f, energy contribution to force f_U, and entropy contribution to force f_S are also investigated. Both elastic force f and entropy contribution to force f_S increases with λ, however, energy contribution to force f_U and the ratio f_U/f decreases with λ. The ratio of f_U/f is less than zero and almost independent of chain length. The results of these microscopic calculations may explain some macroscopic phenomena of rubber elasticity.     

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.     

Monte Carlo study of star-branched polymers on the tetrahedral lattice. I. Conformation of the macromolecule

The star-branched polymers on the tetrahedral lattice are studied by means of the Monte Carlo method. The influence of solvent quality on the dimensions of the coil is described for both linear and branched polymer systems of different functionality. It has been observed that the ratios of gyration radii 〈S²〉_b/〈S²〉_l are greater than those predicted theoretically for the random-flight model. The fourth reduced moment of S² distribution and the mean-square separation of the branch ends from the center of gravity have been also computed. The changes in segment arrangement in the coil with increasing number of branches have been observed.     

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.     

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.     

Magnetic and Mössbauer Study of the Novel FeIn2S2Se2 Layered Compound

The magnetic properties of the new polymorphic FeIn₂S₂Se₂ compound are presented. The system crystallizes in the α-FeGa₂S₄ structure at low temperatures, and undergoes a transition to a MgAl₂S₄-type structure at T>850°C. For this high-temperature phase, low-field magnetization data show a peak at T₁=12.5(5) K, below which magnetic irreversibility is observed. High-temperature susceptibility fits indicate the presence of antiferromagnetic interactions with a high degree of frustration. The effective magnetic moment μ_eff=4.54(3) μ_B agrees with the expected 3d⁶ (S=2) configuration for Fe²⁺. Mössbauer spectroscopy showed that Fe²⁺ ions are distributed in tetrahedral (A) and octahedral (B) sites with a B:A≈1 ratio. The ac susceptibility data were analyzed according to conventional power law dynamics, giving a freezing temperature T_g=12.5(2) K and critical exponent zν=6.5±1, in agreement with Monte Carlo simulations for 3D short-range Ising spin-glass systems.     

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.     

Effects of the A-site cation number on the properties of Ln5/8M3/8MnO3 manganites

The properties of manganites can be tuned by changing the doping level x in Ln_1−xM_xMnO₃. A second mechanism allows tuning of magnetic and electronic properties, for fixed x values, by varying the average A-cation radius, 〈r_A〉. Moreover, for fixed x and 〈r_A〉 values, the changes in the A-cation size variance, σ², also modify the ferromagnetic and metal-insulator transition temperatures. Here, we investigate the influence of the number of A-site cations on Ln_5/8M_3/8MnO₃ manganites, where x, 〈r_A〉 and σ² values are kept constant, and in the absence of phase separation phenomena. We have found that the number of cation species at the A site (N_A) has a strong influence on the width of the ferromagnetic and metal-insulator transitions, and a small influence on the average transition temperature. This behavior is opposite to that observed for increasing values of the variance σ² in manganites, with the same x and 〈r_A〉 values, where average transition temperatures are strongly reduced.     

Configurational-conformational statistics of stereoirregular polystyrene

The dependence of the configurational-conformational characteristics, such as the mean-square end-to-end distance, the mean-square radius of gyration, and the temperature coefficient of the stereoirregular polystyrene chains on the fraction of meso dyads, P_m was investigated, using the periodic condition method. The calculation shows that polystyrene has the most compact chain when P_m = 0.7, and the temperature coefficient of the characteristic ratio of the mean-square end-to-end distance reaches the minimum, −0.89 × 10⁻³ K⁻¹, at P_m = 0.5. The theoretical result of the mean-square radius of gyration for atactic polystyrene, 〈S²〉^1/2 = 0.2245 M^0.5106 shows good agreement with the experimental measurement in both bulk amorphous state and θ-solvent.     

Effect of total degree of substitution on molecular parameters of cellulose acetate

An attempt was made to establish for cellulose acetate (CA) the relationship between the molecular properties, the total degree of substitution 〈〈F〉〉 and the degree of solvation. For this purpose, the comprehensive solution data for CA with 〈〈F〉〉 = 0.49-2.92 in previous papers were analyzed. In addition, the strength of the solvation was estimated from the chemical shift of O-acetyl- and hydroxyl-protons in NMR spectra and the number of the solvated solvent molecules at infinite dilution (s_o) was determined from the adiabatic compressibility. The strength of solvation and s_o increased with the dielectric constant ? of the solvent. The close correlation between s_o, the unperturbed chain dimension A and the free draining effect was demonstrated; these three quantities and the exponent a in the Mark-Houwink-Sakurada equation attained their maximum at $\text{〈〈F〉〉 ? 2.5}$, if CA is dissolved in dimethylacetamide. The characteristic variation of the chain rigidity, estimated from A, and of the draining effect with ? of the solvent and 〈〈F〉〉 can be reasonably explained by the solvation. The cellulose molecule dissolved in a hypothetical non-polar (? = 1) solvent was shown to be a freely rotating chain, by duplicated extrapolation A for CA to 〈〈F〉〉 = 0 and to ? = 1.     

Analysis of hyperfine interactions of Fe IN FeF2, MnF2 and ZnF2

Analysis of magnetic hyperfine interactions of Fe²⁺ in FeF₂, Fe²⁺:MnF₂ and Fe²⁺:ZnF₂ yields a core polarization hyperfine field H_c = −514 ± 30 kOe and a value of 〈r⁻³〉_eff = 3.9 ± 0.04 au.     

Coupled anion and cation ordering in Sr3RFe4O10.5 (R=Y, Ho, Dy) anion-deficientperovskites

The Sr₃RFe₄O_10.5 (R=Y, Ho, Dy) anion-deficient perovskites were prepared using a solid-state reaction in evacuated sealed silica tubes. Transmission electron microscopy and ⁵⁷Fe Mössbauer spectroscopy evidenced a complete A-cations and oxygen vacancies ordering. The structure model was further refined by ab initio structure relaxation, based on density functional theory calculations. The compounds crystallize in a tetragonal a≈2√2a_p≈11.3 Å, с≈4с_p≈16 Å unit cell (a_p: parameter of the perovskite subcell) with the P4₂/mnm space group. Oxygen vacancies reside in the (FeO_5/4□_3/4) layers, comprising corner-sharing FeO₄ tetrahedra and FeO₅ tetragonal pyramids, which are sandwiched between the layers of the FeO₆ octahedra. Smaller R atoms occupy the 9-fold coordinated position, whereas the 10-fold coordinated positions are occupied by larger Sr atoms. The Fe sublattice is ordered aniferromagnetically up to at least 500 K, while the rare-earth sublattice remains disordered down to 2 K.     

Anomalous values of 〈Ŝ2〉 before and after annihilation of the first spin contaminant in UHF wave function

We present a number of molecular systems for which the average values 〈?²〉 before and after annihilation of the first spin contaminant in the wave function of the unrestricted Hartree-Fock method are anomalously large (they substantially exceed the expected value S (S + 1)). An example of such systems is N@C₂₀, for the doublet state of which the 〈?²〉 values are equal to 4.2595 before and 13.1390 after annihilation, respectively (calculated by UHF/6-31G* method). We show that four, at the least, spin multiplets (S′ = S, S + 1, S + 2, S + 3) contribute comparatively to the wave function of such systems. The relations are derived allowing one to estimate the contributions of the highest multiplets basing on the average values of 〈?²〉 before and after annihilation of the first spin contaminant.