W. R. Krigbaum

Description of Crystallite Orientation in Polycrystalline Materials Having Fiber Texture

The distribution of plane‐normal orientations in a polycrystalline sample can be investigated by x‐ray diffraction measurements. Our problem is to deduce from such a set of data the maximum amount of information concerning the distribution of crystallite orientations within the sample. We restrict our attention to samples having fiber texture. The plane‐normal distribution function is expanded in a series of Legendre polynomials, the coefficients of which are to be evaluated from the experimental diffraction data. The crystallite distribution function is expanded in a series of spherical harmonics, the coefficients of which may be obtained by the solution of a set of simultaneous linear equations. Certain simplifications arising from symmetry within the unit cell, or in the crystallite distribution, are examined. The effects of series termination errors are investigated, and relations are presented which permit one to estimate the number of terms which will be required to achieve a specified degree of acc...

Crystallite Orientation in Materials Having Fiber Texture. II. A Study of Strained Samples of Crosslinked Polyethylene

Diffraction data obtained for a crosslinked polyethylene sample crystallized at fixed strain are utilized to demonstrate the practicality of the method of analysis for samples having fiber texture which was presented in a previous paper. Plane‐normal distributions qi(ζi) reconstructed using data for twelve reflections reproduce the features of the measured distributions quite satisfactorily. The series termination error is only appreciable for planes having a sharply peaked distribution, and fortunately it then appears mainly as false oscillations of small amplitude in the low‐density region, which do not distort the principal features of the distribution. As a test of the accuracy of plane‐normal distributions constructed for unmeasured reflections, qi(ζi) was recalculated for the (200) plane with omission of the intensity data for this plane. Deviations from the experimental distribution are again trivial. Thus, the procedure permits one to calculate the plane‐normal distributions for the (002) and (401...

A theoretical treatment of the modulus of semi-crystalline polymers

Abstract Previous treatments of the modulus of semi-crystalline polymers were based on the assumption that the amorphous chains are disposed in a manner resembling an elastomeric network. While the crystallites may perform the same function as crosslinks, we believe that the formation of crystallites leaves the remaining amorphous chains under considerable tension. The large modulus increase observed on crystallization is thus due to the fact that the network chains are near full extension, even in the absence of an external force. Theoretical treatment along these lines yields an expression for the initial Youngs modulus E0 in terms of the degree of crystallinity ω. This relation contains one adjustable parameter, and the same parameter appeared in our treatment of the variation of crystallinity with temperature. Comparison with data for polyethylene shows that the theoretical modulus values are of the correct order of magnitude. Furthermore, the temperature dependence of both E0 and ω for a given sample can be represented by the two theoretical relationships using the same value for the adjustable parameter. Thus, these equations can be combined to yield an expression for the modulus which has no adjustable parameter. At temperatures sufficiently below the melting point a linear dependence of E0 upon 1 (1−ω) is predicted, and the experimental data appear to obey this behaviour.

Equilibrium Degrees of Crystallization Predicted for ``Single Pass'' and Folded Chain Crystallite Models

The free energy change for the crystallization of a bulk polymer consists of two parts: the ``single crystal term exhibited by low molecular weight compounds, and that arising from the deformation produced in the intervening amorphous chains as crystallization proceeds. The latter term limits the degree of crystallinity which can be achieved in a polymeric material at a given temperature. Beginning with an initially unstretched, isotropic sample, the entropy change associated with the deformation of the amorphous chains is calculated for two models, in which the chain either passes through a crystallite once and departs, or folds back and forth within the crystallite several times before re‐entering the amorphous region. The folded chain morphology imposes less strain upon the amorphous chains, and hence results in the development of higher degrees of crystallinity from the original isotropic melt. The folded chain model, in which the amorphous portions are represented by flexibly linked chains, gives g...

On the Question of the Correct Values for Rayleigh's Ratio

Rayleighs ratio values for benzene, toluene, and a toluene solution of the Cornell polystyrene were determined by absolute transverse scattering measurements. Use of a diffuser in the determination of the intensity of the primary beam is superior to direct measurement from the point of view of both random and systematic errors. Both these absolute measurements, and relative measurements on two polystyrene fractions of known number‐average molecular weight, confirm the ``high absolute scattering values. There exists a second set of absolute values, some recently determined, which is 40% lower. Likely sources of error in these measurements are pointed out.

Cubic Lattice Model Chain

The partition functions for conditions of constant force Zf or fixed displacement length Zr are derived for the three‐dimensional cubic lattice model chain having two possible positions of different energy for each link. The problem can be set up so that it reduces to a one‐dimensional Markov chain, thus Zf can be obtained in closed form. The network treatment of Wang and Guth for non‐Gaussian chains is used to obtain the partition function for a network of cubic lattice chains. Qualitative comparison with experiment indicates that most of the deviations from Gaussian behavior which have been observed for cross‐linked natural rubber at low relative elongations may be ascribed to intramolecular rotational energy effects.

Application of irreversible thermodynamics to the kinetics of polymer crystallization from seeded nuclei

Abstract The kinetics of crystallization of polymers from a melt containing seeded nuclei is treated theoretically by applying irreversible thermodynamics to a model for semi-crystalline polymers. The latter model was previously shown to be useful in predicting the equilibrium degree of crystallinity and the initial Youngs modulus of such materials. Comparison of the theoretical result with experimental crystallization kinetics data obtained with fractionated linear polyethylene, containing a small amount of seed crystallinity due to partial melting, shows that the overall course of the crystallization, and the fractional value of the Avrami constant, can be explained successfully. The slight disagreement between theory and the experimental data arising during the very early stages, and that observed during the last part of the crystallization process, is discussed in terms of the assumptions involved in applying irreversible thermodynamics in the linear approximation to the present system.

Theory of Dilute High Polymer Solutions. II

The second virial coefficient is calculated according to the smoothed distribution method. Our treatment differs from those of Flory‐Krigbaum and Isihara‐Koyama through inclusion of the numerous short‐range intramolecular contacts arising by virtue of the connected nature of the chain. The result is to replace the previous F(X) by F(X, ρ), where the parameter ρ depends upon the ratio of the total number of intramolecular contacts to the number of long‐range contacts counted using a radial segment distribution. The number of intermolecular contacts created when two molecules overlap is estimated by the use of an additional smoothed radial segment distribution which is assumed to be uniformly expanded by intramolecular excluded volume effects.Comparison with experiment reveals that both the temperature and molecular weight dependences of A2 in the vicinity of the theta temperature are described in a reasonably quantitative fashion by the present treatment. In this case there are no adjustable parameters. Fu...

Configuration of Polymer Molecules with Excluded Volume

The displacement distribution for a polymer molecule with excluded volume is derived by rejecting the unacceptable configurations available to random‐flight chains. The fraction of those configurations corresponding to a displacement length L which are acceptable is calculated through use of a model which considers the chain to be composed of a sequence of identical subchains. Interactions between the elements of a given subchain, and between these and elements of the various possible clusters of adjacent subchains, are summed to obtain the total interference for all elements of the chain. The average interference encountered by the elements of one subchain is calculated therefrom. The resulting displacement distribution is considerably sharper than the corresponding Gaussian distribution for positive values of the excluded volume, but somewhat more extended than the corresponding Gaussian for negative excluded volumes. When the present treatment is compared with that of Flory, the former predicts smaller...

Crystallite Orientation Distribution in Biaxially Oriented Polyethylene

The procedure of Roe for describing the crystallite orientation in polycrystalline samples having general texture is tested using two samples of biaxially drawn polyethylene. An automated four‐circle diffractometer was used to obtain 12 pole figures (4332 fixed points) for the film having stretching ratios 3.0u2009×u20092.0, and 13 pole figures (4693 fixed points) for the film having stretching ratios 1.7u2009×u20091.6. The selection of measurement geometry, procedures for data reduction and correction, and methods for computer calculation of the crystallite orientation distribution are described. While the best representation of the crystallite orientation distribution was obtained from the series of generalized associated Legendre polynomials by retaining terms through lu2009=u200912, the characteristic features of the distribution are already evident when the series is truncated at lu2009=u20096. Thus, for materials belonging to the orthorhombic crystal class, as few as four or five pole figures suffice to determine the principal fea...