A Monte Carlo Study of the Elastic Behavior of Polymethylene Chains

The rubber elastic behaviors of long polymethylene (PM) chains are investigated using Monte Carlo simulations and considering the rotational‐isomeric‐state model. Through the Monte Carlo method we can generate many PM chains in the equilibrium states, and obtain the average Helmholtz free energy 〈A〉 and average energy 〈U〉. Chain dimensions and thermodynamic statistical properties of long PM chains under various elongation ratios λ are also calculated. We find that the elastic force f increases with elongation ratio λ, and that energy contribution f_u to the elastic force is negative and significant. The ratio f_u /f ranges from –0.37 to –0.32 at T = 300 K and from –0.53 to –0.40 at T = 413 K, and decreases with increasing temperature, which agrees with the experimental data. Our calculations may provide some insights into the macroscopic phenomena of rubber elasticity.     

Elastic behavior of non-Gaussian polymethylene chains

In this paper, elastic behaviors of non-Gaussian polymethylene (PM) chains with chain length N=100 are investigated by rotational isomeric state model. Here the tetrahedral lattice of PM chain and the non-local interaction of Sutherland potential are adopted. In the metropolis movement of PM chain, a four-bond movement model is used. The average energy and average Helmholtz free energy with various elongation ratios λ are calculated by Monte Carlo simulation method. The average energy increases with elongation ratio λ and the average Helmholtz free energy decreases with elongation ratio λ. The elastic force f and the energy contribution to elastic force f_u can be obtained from f=∂〈A〉/∂r and f=∂〈U〉/∂r. We find that the elastic force f increases with elongation ratio λ and the energy contribution f_u decreases with elongation ratio λ, and f_u is less than zero. The ratio f_u/f is close to −0.21 for λ?1.25, and −0.04 to −0.35 for λ>1.25 at T=364 K. In our calculation, the rubber elasticity may be discussed in terms of the chemical structure of polymer chains.     

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.     

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.     

A Molecular Model of the Elastic Behavior of Polymethylene Chains

In this paper, the elastic behavior of a polymethylene (PM) chain is investigated by using a realistic rotational‐isomeric‐state (RIS) model. In our calculation, the non‐local interactions between pairs of segments in a polymethylene chain are also considered, and the Lennard‐Jones (L‐J) potential function is adopted. Chain dimensions and thermodynamics statistical properties of PM chains with various elongation ratios λ are calculated. We find, that the elastic force increases slowly with elongation ratio for small λ, and abruptly for large λ. In the meantime, the energy contribution to elastic force is negative and significant, especially for large λ. Our calculations may provide some insight into the macroscopic phenomena of rubber elasticity.     

Elastic behavior of bimodal poly(dimethylsiloxane) networks

The rubberlike elastic behavior of bimodal poly(dimethylsiloxane) (PDMS) networks was investigated by the Monte Carlo simulation method and enumeration calculation method on the basis of the rotational‐isomeric‐state (RIS) model. These bimodal PDMS networks consist of short chains (chain length from 10 to 20) as well as long chains (chain length equal to 150). For long PDMS chains, through generating many PDMS conformations in the equilibrium state using the Monte Carlo simulation method we can obtain the average Helmholtz free energy and the average energy. For short PDMS chains with chain lengths from 10 to 20, as the total number of conformations is only from 6.56 × 10³ to 3.87 × 10⁸, we adopt the enumeration calculation method. The deformation is partitioned nonaffinely between the long and short chains, and this partitioning can be determined by requiring the free energy of the deformed network to be minimized. Chain dimensions and thermodynamic statistical properties of bimodal PDMS networks at various elongation ratios are discussed. We find that elastic force f increases with elongation ratio λ; the energy contribution f_u to elastic force is significant, and the ratio of ranges from 0.15 to 0.36 at T = 343 K. In the meantime, elastic force f increases with the average energy 〈U〉. The energy change in the process of tensile elongation is taken over, which has been ignored in previous theories. Our calculations may provide some insights into the phenomena of rubberlike elasticity of bimodal networks. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 40: 105–114, 2002     

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.     

Isometric and Isotensional Force‐Length Profiles in Polymethylene Chains

Summary: The force‐length curves and related thermodynamic quantities of single polymethylene (PM) chains in the high‐force region were calculated by the statistical mechanics. Two statistical mechanics ensembles (isometric and isotensional) were used to represent the chain stretching under the conditions, respectively, of the fixed length L or of the fixed force F in single‐chain experiments by AFM and related techniques. The input deformation potentials of highly extended conformations of PM chains were obtained from the molecular‐mechanics calculations. Variations of the energy, entropy, and Helmholtz energy with the end‐to‐end length L of chains were computed under the condition of fixed length. The ensuing isometric profile of the mean force 〈F〉(L) is non‐monotonic, featuring a sawtooth‐like pattern of ascending peaks. In contrast, the mechanical equation under isotensional conditions, given by the variation of the mean length 〈L〉(F), shows a conventional monotonous shape with a distinct plateau region at low temperatures. The considerable difference in the shapes of 〈F〉(L) and 〈L〉(F) curves arises from the large fluctuations of mean values due to the small number of conformers present in molecules at high strains. The computed data should be relevant to the AFM stretching experiments on short polyethylenes or on other soft matter materials involving linear paraffinic chains. It is argued that the dual character of elastic response described by the conjugated force profiles is a universal feature of mechanochemistry of chain molecules whenever the chain length discontinuously increases at a transition.

Force‐length curves of short polymethylene chains at 300 K.     

The mechanical moduli of lamellar semicrystalline polymers

Bounds on the elastic constants are derived for semicrystalline polymers whose local morphology is lamellar. Local response matrices (stiffness and compliance) are formulated in three dimensions that simultaneously incorporate uniform in-plane strain and additive forces from layer to layer of crystalline and amorphous phases and uniform stress and additive displacements normal to the lamellar surfaces. Spatial averaging of the stiffness and compliance matrices under the assumption of axially symmetric orientation gives the upper and lower bounds on the longitudinal and transverse tensile moduli and the axial and transverse shear moduli as functions of the separate phase elastic constants, the volume percent crystallinity, and the moments of the orientation 〈cos²θ〉 and 〈cos⁴θ〉. The bounds are much tighter than the Voight upper and Reuss lower bounds that do not recognize phase geometry. Using the known crystal elastic constants of polyethylene, sample calculations on isotropic unoriented materials show that the divergence of bounds at high crystallinity necessitated by the extreme crystal anisotropy shows up only at very high crystallinity. At low temperature the bounds are tight enough to specify G₁, the amorphous modulus, from the measured G and the known crystal elastic constants. At higher temperatures and lower G, the bounds are not tight enough for this purpose but the shear modulus versus crystallinity and temperature data are well fitted by the lamellar lower bound using a temperature-dependent, crystallinity-independent G₁.     

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.     

Molecular orientation in poly(ethylene terephthalate) by means of laser–raman spectroscopy

The intensity of the Raman scattering from uniaxially oriented amorphous poly(ethylene terephthalate) tapes at wave number shifts of 1732, 1616, 1286, 857, and 632 cm^?1 has been observed for various combinations of incident and scattered light polarization vectors with respect to the draw direction. An attempt has been made to analyze the data to provide values of 〈P₂(ζ)〉 and 〈P₄(ζ)〉, where P_n(ζ) is the nth order Legendre polynomial in ζ,ζ is the cosine of the angle between the draw direction and a typtical chain axis in the polymer, and the angle brackets denote the average value over all repeat units. This attempt was successful for the 1616 and 632 cm^?1 lines but less successful for the other three, although the data for the 1732 and 1286 cm^?1 lines could be analyzed to provide quantities proportional, but not equal, to 〈P₂(ζ)〉. In the analysis and discussion two possible models were considered for the conformation of the terephthaloyl residues in the amorphous polymer but it was not possible to reject either model conclusively. The results suggest, in agreement with previous studies by other methods, that the drawing of PET at 80°C takes place essentially as the extension of a rubberlike network which is frozen on subsequent cooling to room temperature.     

Molecular dimensions,configurational partition functions and entropies of linear aromatic polyesters

The values of the configurational partition function Z and of the entropy S. as well as the unperturbed molecular dimensions 〈r²〉_o/M, were calculated for aromatic polyesters having the repeat unit -OC(C₆H₄)COO(CH₂)_mO-. From a qualitative point of view, the values of Z are useful in interpreting the effect of the number of methylene groups on the unperturbed dimensions. For the lower members of the series (2 ? m ? 10), the melting temperatures show some correlation with the flexibility of the chains as measured by the configurational entropy, calculated at low temperature.     

Determination of orientation of the crystalline and amorphous phases in polyethylene by X-ray diffraction

For polyethylene fibers, orientation parameters of the form 〈cos² ?〉 were calculated from traces obtained with an x-ray diffractometer equipped with an orientation integrator. The angle ? is that which a crystal or molecular axis makes with the fiber axis. Traces over a continuous range of diffraction angle were obtained for (2/π)I_av, and also the weighted averages (2/π)I_av 〈cos²?〉 and (2/π)I_av 〈sin²?〉, where I_av is the intensity averaged over all orientations. The orientation for the crystal a, b, and c axes were determined from area measurements of the {110} and {200} lines, and the orientation parameter for the molecular axis in the amorphous phase was determined from area measurements of the amorphous halos. An undrawn fiber showed a slight a and c axis orientation along the fiber axis, and a slight transverse orientation of the b axis and the molecular axis. For a highly drawn fiber the orientation parameter for the c axis was 0.98 and for the molecular axis 0.65. The degree of crystallinity, measured from the (2/π)I_av versus 2θ traces, were 66.1% and 73.3% for the undrawn and drawn fiber, respectively.     

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.     

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.     

A comparative study of the magnetic properties and phase separation behavior of the rare earth cobaltates, Ln0.5Sr0.5CoO3 (Ln=rare earth)

A comparative study of the magnetic properties of a few members of the Ln_0.5Sr_0.5CoO₃ family with different radii of the A-site cations, 〈r_A〉, in the range 1.19-1.40 Å has been carried out. The apparent T_c (where the magnetization undergoes an abrupt increase) decreases markedly with 〈r_A〉 as well as the size-disorder arising from the mismatch in the size of the A-site cations. The value of the magnetization at low temperatures decreases markedly with decrease in 〈r_A〉 or increase in size-disorder, suggesting that the relative proportion of the ferromagnetic (FM) species decreases relative to that of the paramagnetic (PM) species. Such a variation of the FM/PM ratio with composition and temperature is evidenced from the Mössbauer spectra of La_0.5Sr_0.5CoO₃ as well. The variation of the FM/PM ratio with 〈r_A〉 and size-disorder, as well as a local-probe study using ⁵⁹Co Nuclear magnetic resonance spectroscopy suggest that electronic phase separation is an inherent feature of the Ln_0.5Sr_0.5CoO₃ type cobaltates, with the nature of the different magnetic species in the phase-separated system varying with 〈r_A〉 and size disorder.     

Crystal structure and radiation-induced solid-state polymerization of 1,3,5,7,9,11-hexoxecane

One of the cyclic oligomers of formaldehyde, 1,3,5,7,9,11-hexoxecane, was found to undergo polymerization in the solid state to form polyoxymethylene both during and subsequent to x- or γ-irradiation. The polymer yield increases with polymerization temperature but decreases drastically near the melting point of the hexoxecane crystal. In order to clarify the specificity of the solid-state polymerization, the crystal structure of hexoxecane was first analyzed; hexoxecane forms a trigonal crystal with cell dimensions of a = b = 7.917 Å, c = 11.345 Å, space group R3 –C_3i², three molecules of 3 symmetry per unit cell. The polyoxymethylene as polymerized from a single crystal of hexoxecane is highly crystalline and the crystallites are definitely oriented with respect to the original crystal. There are three kinds of oriented trigonal polyoxymethylene: i.e., with the polymer chains oriented along the 〈100〉, 〈001〉, and 〈210〉 axes of the hexoxecane crystal. The relative yields of these crystallites depend upon the polymerization temperature. In addition to the ordinary trigonal polyoxymethylene, oriented orthorhombic polyoxymethylene was also found in the case of polymerization during treatment with x-rays.     

Crystallite orientation in stretched polychloroprene networks. II

The orientation of crystallites grown isothermally in several drawn trans-polychloroprene networks is studied as a function of crystallization temperature t_x, degree of crystallinity ω, and elongation ratio α. The orientation distribution is particularly simple for this polymer since the crystallographic c axis (chain axis) orients preferentially along the stretching direction, while a and b are randomly arranged about c. Hence the parameter cos² χ_c adequately characterizes the distribution, where χ_c is the angle between the c axis and the fiber axis, and the average is taken over all crystallites. A treatment due to Krigbaum and Roe is utilized to obtain values of v (the number of statistical segments comprising the crystallization nucleus of critical size) through comparison of the average orientation of crystallites and amorphous statistical segments. The behavior observed falls into two categories. First, if the initial amorphous network is well oriented, 〈cos² χ_c〉 is independent of crystallinity during both crystallization and melting, and v varies with t_z (or the degree of supercooling) as predicted by nucleation theory. If different networks are to have the same crystallite orientation distribution, they must not only be crystallized at the same supercooling, but must also have the same distribution of amorphous segment orientations. Both the relative elongation and the network crosslink density affect the latter distribution. Next, we consider the second category. If the initial amorphous orientation is poor, 〈cos² χ_c〉 decreases linearly during crystallization and increases along approximately the same path during melting. Further, 〈cos² χ_c〉 for a given t_z yields v values which are too large. These two behaviors can be explained if, in the former case, nucleation involves the best oriented statistical segments of all network chains, while in the latter there is a selection according to the chain displacement vector orientation. Thus, if the amorphous orientation is poor, both the orientation and thermodynamic stability of the crystallites decreases with further crystallization. If this decreased stability is reflected in shorter fold lengths, the reversible variation of long period spacing with temperature reported earlier for an oriented polychloroprene network can also be explained as a preferential melting process.     

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.