David M. Sadler

Roughness of growth faces of polymer crystals: Evidence from morphology and implications for growth mechanisms and types of folding

Abstract It is proposed that the growth faces of lamellar polymer crystals can have an equilibrium roughness (or crenellation). This can explain why some polymer crystals show no evidence of faceting. Support for this idea comes from the extensive theoretical developments on the nature of crystal surfaces. The characteristic habits of polyethylene are analysed in terms of a roughness which, on the {110} faces, increases progressively over a temperature range of about 100°C. At a temperature near 110°C the roughness becomes sufficient for there to be no free energy penalty for arbitrary crystal shapes (e.g. rounded) compared with one bounded by {110} faces. Above this temperature of crystallization most of the habits which are observed are leaf-shaped, with an apex along 〈010〉. Below 110°C{110} facets (or microfacets) are normally seen. There is no positive evidence that faces approximately parallel to (100) planes, observed for crystallization temperatures in the range 80°–110°C, are ever other than rough. The relative rates of growth on the {110} compared with the (100) increase with temperature, since {110} faces predominate at low but not high temperatures of crystallization. These changes are attributed to the increase in roughness with temperature on the {110} faces. The existence of surface disorder (roughness) requires that the binding energy between units in the crystal is comparable with KT . Hence this unit is probably several monomer units of polyethylene (rather than, for example, a complete stem which contains a hundred or more monomers). There is therefore a surface lattice on the growth faces with twenty or more units in the direction perpendicular to the lamellae. Monte Carlo calculations are cited for lattices of 20 by 50 units. These show that a cooperative increase in surface roughness with temperature and a transition between faceted and non-faceted growth can be expected for lattices of such limited extents. No explicit allowance has been made as yet for the consequences of the units being linked into chains and, for that reason, not being able to arrive or leave the surface independently. It is noted that changes in the alkane lattice with temperature indicate a possible evolution in binding energy, and in mobility, and hence may influence surface roughness. Theories of crystallization in polymers have normally assumed a growth surface which is molecularly smooth in equilibrium, and have emphasised nucleation events. Since this paper shows that the equilibrium structure may often be rough, it may be necessary to re-examine the basis of these theories. A brief review is included of the experimental evidence for surface nucleation events: nucleation may be a more important barrier at low temperatures than at high. The type of folding will be influenced by equilibrium roughness just as it will be by kinetic roughness, and some comparisons are made with neutron scattering results on this topic. The degree of adjacent folding is higher in the faceted regime as expected. Brief comments are made on the applicability of this idea to polymers other than polyethylene.

On the growth of two dimensional crystals: 2. Assessment of kinetic theories of crystallization of polymers

Abstract Results of analyses of growth processes on two dimensional (2d) crystals are applied to kinetic theories for crystallization of polymers. The purpose is firstly to assess the rate equation methods of calculation which have been used for nucleation (LH) theories, and secondly to estimate free energies of steps σn on the growth faces of the crystal lamellae in a manner independent of detailed models. A systematic critique is then given of the LH theories, on the basis both of the simulation results and previous work. The range of validity of the calculations used in nucleation theories, using rate equation approaches, is assessed in the light of the effects of fluctuations in step positions and of simulation results (Part 1, ref. 1). The simulation results are given in terms of σn and the free energy δf for adding stems at niche sites. It is pointed out that, for those nucleation theories which predict l values in line with experiment, δf kT ≲1 . For this range of δf values the density of steps on the surface which is found from simulation results is fairly near the equilibrium values. The predicted step densities in Regime II of (the nucleation theory) is only 2 −1 2 times the equilibrium density (for low step densities). This difference is discussed in terms of a near approach to equilibrium which is mediated by the growth process. Part 1 emphasized the close connection between growth rates and step densities. The dependence of both on σn is very similar for the simulation results and for the Regime II result (for average spacings between steps greater than about four lattice spacings). For rougher surfaces than this, the growth rate saturates and reaches a plateau at σn = 0. This is readily interpreted since the step density also saturates. This result is contrasted with that of Regime III (for rough surfaces) which predicts an even stronger dependence of growth on σn than in Regime II. The discrepancy between the simulation results and Regime III is discussed in detail, and it is pointed out that the prediction of growth in the absence of a thermodynamic driving force is associated with violation of the principle of microscopic reversibility. The applicability of kinetic models is also discussed. Morphological observations, as discussed in more detail elsewhere, seem incompatible with LH theories. The growth rate plots (rate on a log scale against (TΔT)−1 or against σn) are not fully explained by them: only completely straight growth plots can be explained, or smoothly curved convex ones (in terms of a Regime I/II transition, but even for this explanation to be valid the relevant morphological evidence must be disregarded). Concave or sharply kinked growth plots cannot be directly explained in this way. Alternative kinetic models, based on rough growth surfaces and pinning (as a consequence of molecular connectivity) are also discussed briefly in view of the range of experimental evidence. A review is included of other comparisons of experiment with LH theory, including the lack of crystal size effect on the growth rate and the analysis of growth rates of poly(ethylene oxide). It is concluded that the basic hypothesis of LH theories, that σn is generally large compared with kT and that σn increases linearly in a steep manner with the lamellar thickness, are not consistent with the experimental evidence.

Structure of drawn fibres: 1. Neutron scattering studies of necking in melt-crystallized polyethylene

Abstract Results of neutron scattering experiments performed on fibres of polyethylene drawn through a neck from melt-crystallized sheets are reported. It is shown, from the appearance of an isotopic segregation signal, that at drawing temperatures above ∼70–90° C (depending on polymer molecular weight) there is some local melting during necking. At lower drawing temperatures there is no evidence for any local melting, and the molecules appear to deform affinely with the sample through the neck. The implications of these results for various models of necking are discussed.

Neutron scattering studies on solution-grown crystals of polyethylene: a statistical preference for adjacent re-entry

Abstract Neutron scattering studies of solution-grown polyethylene crystals demonstrate a strong statistical preference for adjacent re-entry, with 75% of stems occupying adjacent lattice sites. The number of sheets of stems for one molecule increases with molecular weight, with an average molecular weight per sheet of 21 000. These are the main features of computer models which give the best agreement with scattering data obtained over an angular range ( q A −1 ). Scattering in this q range is directly sensitive to typical stem separations. Any preference for alternate re-entry would result in a peak at q=0.7 A −1 , which was not observed. The results are discussed in terms of possible crystallization mechanisms.

A neutron scattering study of the melting behaviour of polyethylene single crystals

Abstract The changes in chain trajectories occurring during the melting of polyethylene single crystal mats have been studied by using neutron scattering techniques. It is shown that molecules change from a regular superfolded sheet structure into a random coil configuration in a very short time, ≈ 2−4 s. Further, it is shown that the rate at which this process takes place is substantially independent of molecular weight.

I.r. study of solution-grown crystals of polyethylene: correlation with the model from neutron scattering

Abstract The crystal stem arrangement in solution-grown polyethylene crystals has been investigated by the mixed-crystal infra-red technique. Parallel neutron scattering measurements are reported separately. The behaviour of the CD 2 bending vibration is shown to be consistent with a model having 75% adjacent re-entry, 50% dilution of a molecule along the fold plane (110 direction) and a degree of superfolding equivalent to an average molecular weight ( M w ) of 21 000 for each sheet. This is in agreement with results from neutron scattering, which have previously been interpreted using this model. Computer calculations of stem positions based on these parameters are used to calculate the distributions of doublet splittings for several molecular weights. The main features of low-temperature FT i.r. spectra as a function of molecular weight are reproduced in this way.

Structure of drawn fibres: 3. Neutron scattering studies of polyethylene fibres drawn beyond the neck

Abstract Results of neutron scattering experiments performed on polyethylene fibres drawn beyond the neck are reported. Data are reported for fibres with total draw ratios up to ∼40 and tensile moduli up to ∼100 GPa. It is shown that on post-neck drawing at temperatures below ∼80°C the molecules deform affinely with the sample. At higher drawing temperatures the molecules deform to a lesser extent than the bulk material. It is further demonstrated that in all cases the final tensile modulus of the drawn fibres is a unique function of the molecular draw ratio.

Trajectory of polyethylene chains in single crystals by low angle neutron scattering

Abstract Measurements of coherent neutron diffraction of oriented single crystals of blends of hydrogenous and deuterated polyethylene have been undertaken in order to study the mutual arrangements of the crystalline ‘stems’ of the same molecule. In the appropriate range of diffraction angle, the scatter is fully consistent with thin lamellae, the planes of which contain the stems. The thickness of these lamellae agrees with that expected for neighbouring stems of one molecule being restricted to the same (110) plane. The density of deuterium atoms in the lamellae is consistent with largely adjacent re-entrant folding; for crystals grown at low supercooling there is a possibility of some segregation according to isotope. Outside the above range of diffraction angle, effects are observed which are attributable to the finite lateral dimensions of the proposed lamellae. At the smallest angles of measurement an artifactual signal attributable to voids can be observed, which can be avoided by suitable sample treatment.

Structure of drawn fibres: 2. Neutron scattering and necking in single-crystal mats of polyethylene

Abstract Results of neutron scattering experiments on polyethylene fibres prepared by drawing oriented mats of isolated single crystals through a neck are reported. It is shown (from the appearance of an isotopic segregation signal) that, as the temperature of drawing is raised above ∼90°C, there is an increasing degree of local melting within the neck. It is further shown that, in contrast to the situation pertaining to the drawing of melt-crystallized material, the molecules do not deform affinely through the neck.

On the growth of two dimensional crystals. I. Fluctuations and the relation of step free energies to morphology

Some fundamental aspects of crystallization of two dimensional (2D) crystals are examined, including an analysis of the fluctuations in step positions, computer simulation results for the step density and growth rates, and analysis for predicting crystal shapes. A discussion is given of whether nucleation can occur on the 1D surface of a 2D crystal. During the course of the paper it is shown that a more appropriate description, for any infinite 1D surface, is of a step density related to the equilibrium value and a growth rate which is proportional to the density and speed of these steps. The speed depends essentially linearly on the net thermodynamic driving force for crystallization. (This description is not appropriate for a 2D surface since the equilibrium step density can then be zero.) Nevertheless, one can define a criterion, depending both on the density of steps and on the extent of the fluctuations in step positions, for a regime of growth with some similarities to nucleation (1D nucleation). Ki...