Jacob Mazur

Monte Carlo Studies of Configurational and Thermodynamic Properties of Self‐Interacting Linear Polymer Chains

Non‐self‐intersecting walks on the simple cubic and face‐centered cubic lattices are used as a model for the linear polymer chain with excluded volume and nearest‐neighbor interactions between the chain elements. The statistical properties of this model are investigated using the modified Monte Carlo technique for inversely restricted sampling. The following properties are investigated: the limiting distribution function of chain dimensions, the dependence of mean square length of the chain on the number of chain elements, and the thermodynamic properties of the chain. The results of these investigations are presented by a set of parametric representations. Each of these representations includes a parameter which is descriptive of long‐range interactions in the polymer chain. These parameters are investigated for their dependence on the nearest‐neighbor interaction parameter. A particular value for the nearest‐neighbor interaction parameter is found, in which long‐range interaction parameters reduce to th...

Small defects in crystalline polyethylene

Abstract A family of five crystallographic defects of three classes (two dislocations, two dispirations and one disclination) which primarily involve only one polymer chain is described for polyethylene. The extra energy associated with each defect in a perfect polyethylene crystal was computed. The crystal model used consisted of a central chain containing the defect with 18 zig-zag chains in two shells around the central chain. The zig-zag chains each had 60 carbon atoms. The conformation of each defect, placed near the centre of the central chain, was adjusted to minimize the sum of the interatomic interactions. A closely related procedure was used to calculate the energy per chain at boundaries where each chain contains an identical ‘partial’ dislocation or disclination. The characterization of these well defined defects and partial defects greatly simplifies the establishment of connections between atomic and macroscopic scale phenomena in crystalline polyethylene.

Erratum: Energetics of defect motion which transports polyethylene molecules along their axis

Defect energy was calculated as a function of dihedral angles of the bonds in a point dislocation for sequences of conformations that resulted in motion of the dislocation along the polyethylene chain. Paths that presented low barriers to diffusive motion of the defect were found by incrementing, in a particular sequence, selected dihedral angles around two separated bonds near the opposite ends of the defect as the computer searched for the lowest energy conformation of all the other parts of the defect. Thus, the diffusion of a point dislocation provides a plausible mechanism for diffusion of the chain along its axis.

Dispirations, disclinations, dislocations, and chain twist in polyethylene crystals

Abstract It is proposed that the twist in polyethylene chains that can result from crystallization and subsequent deformation aggregates at boundaries and becomes a template for further reorganization that results in the long period observed in polyethylene fibres. The observed lower density at the boundaries requires the transport of free volume to the twist boundaries. Dispirations, disclinations and dislocations are crystallographic defects that are involved in the necessary transport mechanism. Twist and bend, derived from the Eulerian angles which are computed from the sets of chain internal coordinates, relate the orientation of different segments of a chain. Twist and bend are useful for the characterization of both crystallographic defects and arbitrary conformations of polymer chains. Defects, along with folds, chain ends, and ordinary edge and screw dislocations provide a basis for interpretation of structure-property relationships in solid polyethylene.

On the Limiting Shape of the Distribution Function of Lengths of a Single Polymer Molecule with Excluded‐Volume Effects

The distribution of the lengths of the polymer molecule is estimated from the computed reduced moments with the excluded‐volume effect being taken into consideration. It is found that the probability distribution function for chain ends can be fitted reasonably well by an exponentially decaying function, raised to the power of t, where t, the shape factor, exceeds its Gaussian value of 2. As a result of the excluded‐volume effect, the distribution of the lengths of a single polymeric molecule results in the chain assuming a more uniform conformation, as compared with the Gaussian coil model.

Modelling of chain twist boundaries in poly(vinylidene fluoride) as a mechanism for ferroelectric polarization

Abstract It is assumed that the process of ferroelectric polarization of the beta phase of poly(vinylidene fluoride) (PVF2) in response to the action of the external electric field in direction perpendicular to the molecular axis and to the film, involves movement of the chain twist boundaries. These boundaries, at which every chain is twisted by 180 degrees, separate domains of opposite polarization. The energy barriers that are surmounted as the boundary was advanced one repeat unit were calculated and compared with the energy gained by reversing the polarization of an unfavourably oriented repeat unit in an electric field that produces polarization in PVF2. It is suggested that the movement of chain twist boundaries, in contradistinction to previously postulated models in which only one chain is twisted at a time, provides a model for the poling of PVF2 that is feasible energetically and kinetically. Theoretical modelling, analogous to that for Bloch wall that separates domains in magnetic materials, suggest that the process of polarization might be described either as a diffusion process or as the propagation of a soliton along the chains.

Stochastic defect diffusion model for relaxation effects in crystalline polyethylene

Abstract It is suggested that some relaxation processes observed in crystalline polyethylene are consequences of the diffusive motion of a particular defect called a point dislocation or twist dispiration loop along the polyethylene stems in lamellar crystals. The motion of the defect, characterized by a diffusion coefficient and a mobility, is described by solutions of the Smoluchowski diffusion equation with boundary conditions that constrain the defect to move along routes that produce experimentally observable results. The fact that passage of the defect causes both a 180° rotation of the chain and moves an extra CH 2 group in the direction of the chain axis is important to the interpretation of the data according to this model. The diffusion coefficient for a defect is estimated to be around 2 × 10 −9 cm 2 s −1 at 70°C. This value is shown to be reasonable both from the viewpoint of detailed computer modelling of defect motion and contemporary ideas about scaling.

Effects of Excluded Volume on Light Scattered from Flexible Macromolecules

The effect of the non‐Gaussian behavior of the chain‐segment distribution in flexible polymer coils on the angular distribution of scattered light is discussed. The explicit form of a general spherical‐segment distribution function, W(r, N)dr = tΓ(3/t)(Γ(5 / t)Γ(3 / t) 〈RN2〉)3/2exp−(Γ(5 / t)r2Γ(3 / t) 〈RN2〉|t/2 r2dr, with different values of t, is used to evaluate the general scattering equation for the particle scattering factor P(θ). P(θ) = (1 / N2) ∑ i ∑ j (sinksrij / ksrij). In addition, the effects of excluded volume on the mean end‐to‐end chain separation 〈RN2〉 is taken to be of the form 〈RN2〉≈N1+e, where N is the number of chain steps and e is a parameter which measures the excluded volume effect on the 〈RN2〉. In this paper, an expansion of the P(θ) function to the first few terms is carried out and numerically evaluated for the dependence of P(θ) on the variable x, where x = ks〈Rexptl,〉 k is the wavenumber of the incident light, s is related to the scattering angle, and 〈Rexptl〉 is the radius of g...

Asymptotic Behavior of the Light‐Scattering Function of Coiled Molecules

This paper discusses the asymptotic solution of the scattering function P(θ) for large x, where x is proportional to the scattering angle, radius of gyration, and the wavenumber. The theoretical model employed for the calculation of the P(θ)‐vs‐x curves was presented in the preceding paper. The results are discussed with respect to the experimental data, after the results for P(θ) are corrected for various values of polydispersity. A general analysis of the theoretical P(θ)‐vs‐x curves is presented, and the effect of the shape of the chain‐end distribution function on the asymptotic behavior of the light‐scattering function is discussed.

Ordered spans of unrestricted and self‐avoiding random‐walk models of polymer chains. I. Space‐fixed axes

An N‐step random walk on a cubic lattice is adopted as a model of a random polymer chain. The spans, or extents, of each random walk configuration in the principal lattice directions are arranged in order of magnitude, ξ3?ξ2⩾ξ1. In the case of the unrestricted random walk, the average values of the ordered spans 〈ξi,u〉 and 〈ξ2i,u〉, i=1, 2, and 3, are calculated analytically in the limit of large N. The limiting relative values of the first moments, 〈ξi,u〉 are 1.637 : 1.267 : 1; and the limiting values of the second moments 〈ξ2i,u〉 are 2.710 : 1,600 : 1. In the case of the restricted or self‐avoiding walk, the corresponding average spans 〈ξi,r〉 and 〈ξ2i,r〉 are estimated for N?150 by using a Monte Carlo procedure. The same Monte Carlo procedure is used to estimate the values of 〈ξi,u〉 and 〈ξ2i,u〉 for N?1000. On the assumption that the rate of approach of the average ordered spans of the self‐avoiding walks to their asymptotic forms is similar to the rate of approach of the average ordered spans of the unres...