John D. Hoffman

Extension of theory of growth of chain‐folded polymer crystals to large undercoolings

The kinetic theory of the rate of growth G and the initial lamellar thickness lg* of chain‐folded crystals is extended so that it is applicable at high undercoolings. Attention is centered on the details of how the first step element and the first fold are put down on the substrate. A parameter φ that varies between zero and unity, which apportions the free energy of attachment of the step element between the forward and backward reactions, is used to denote variations in this process. Expressions for G are derived from flux equations for two limiting cases: regime I, single surface nucleation act with rapid substrate completion and regime II, numerous surface nucleation acts with very slow substrate completion. Data from the literature on G for isotactic polystyrene (regime II) and polyethylene single crystals (regime I) are analyzed to obtain surface free energies, and these are used with the revised theory for lg* to predict the lamellar thickness of these polymers. Good agreement between lg* and publi...

Regime III crystallization in melt-crystallized polymers: The variable cluster model of chain folding

The kinetic nucleation theory of chain folding, including the effects of reptation, is extended to predict the increase in crystal growth rate G that is implied by measurements on PE and POM at moderately large undercoolings. The increased growth rate denotes the rather abrupt transition from Regime II where G∝i12 to Regime III where G∝ i (i = surface nucleation rate). The distance between the niches on the growth front in Regime II diminishes rapidly with falling crystallization temperature Tx, and approaches the molecular width at a specified undercooling where Regime III begins. In PE, the Regime I → Regime II transition occurs at ΔT ⋍ 16°C, and the Regime II → Regime III transition is predicted to occur at ΔT ∼ 23°C (ΔT based on T°m(∞) = 145°C). Growth rate data on PE and POM crystallized from the melt suggest conformity with the theoretical predictions. The implications of Regime III crystallization to chain morphology are discussed. The kinetic theory, which predicts narrowly spaced niches on the growth front, taken together with the restrictions on the degree of non-adjacent re-entry imposed by the ‘Gamblers Ruin’ treatment, leads directly to the ‘variable cluster’ model as the relevant morphology in Regime III. Here runs of adjacently chain-folded stems of varying size (averaging about three or so stems) are laid down interspersed with non-adjacent re-entries, leading to a lamellar surface that is about two-thirds ‘regular’ or ‘tight’ folds, most or all of these representing strictly adjacent re-entries. The steady-state reptation process operative in Regimes I and II in PE is impaired at temperatures just below the inception of Regime III, and it is suggested that at lower temperatures the ‘slack’ portions of the chains engage in forming the small clusters of adjacent stems. The variable cluster model leads in a natural way to the amorphous component found in quench-crystallized polymers, and is consistent with neutron scattering data on quench-crystallized PEH-PED.

The Rate of Crystallization of Linear Polymers with Chain Folding

Under a variety of circumstances commonly encountered in practice linear macromolecules crystallize into the form of thin platelets whose large upper and lower surfaces consist of an array of molecular folds. We refer to these as “chain-folded crystals” or “chain-folded lamellae,” the latter term usually being reserved for folded structures in polymers crystallized from the melt. The theory of the rate of formation of these platelets will be outlined, and the prediction and origin of the thin dimension given. The thin dimension of the crystal platelets is determined by kinetic factors, and the elucidation of the kinetics of growth is therefore of importance in polymer morphology on both a molecular and a macroscopic scale.

Role of reptation in the rate of crystallization of polyethylene fractions from the melt

Abstract The theory of polymer crystallization with chain folding is extended to include the effect of reptation in the melt on the rates of crystallization G I and G II in regimes I and II. The result is that the pre-exponential factors for G I and G II contain a factor 1 n , Where n is the number of monomer units in the pendant chain being reeled onto the substrate by the force of crystallization; n is proportional to the molecular weight. The predicted fall in growth rate with increasing molecular weight is found experimentally in nine polyethylene fractions M z =2.65 × 10 4 to M z =2.04 × 10 5 , corresponding to n z =1.90 × 10 3 to 1.45 × 10 4 . The data on these fractions are analysed to find the reptation or ‘reeling’ rate r and the substrate completion rate g . The values g nuc ∼0.5/ n z cm s −1 and r nuc ∼21/ n z cm s −1 at 400K are obtained from the data in conjunction with nucleation theory adapted to account for reptation assuming a substantial degree of regular folding. These results are consistent with a melting point in the range of ∼142° to ∼145°C. (The analysis using T ° m (∞)=145°C gives values of such quantities as σ σ e and α that are quite similar to those deduced in earlier studies.) An estimate of g (denoted g expt ) that is independent of the molecular details of nucleation theory gives g expt ∼0.4/ n z cm s −1 and r ∼17/ n z cm s −1 at 400K. Calculations of the reptation rate from r 1,2 = (force of crystallization ÷ friction coefficient for reptation in melt), where the friction coefficient is determined from diffusion data on polyethylene melts, leads to r 1,2 ∼17/ n z to 34/n z cm s −1 at at 400K, or g 1,2 ∼0.4/ n z to 0.8/n z cm s −1 . The conclusion is that the reptation rate characteristic of the melt is fast enough to allow a significant degree of adjacent re-entry or ‘regular’ folding during substrate completion at the temperature cited, and that the substrate completion process is governed jointly by the activation energy for reptation Q∗ D and the work of chain folding q . The nucleation theory and the friction coefficient theory approaches are compared, and the formulations found to be essentially equivalent; the ‘reeling’ rate r is found to be proportional to ( 1 n )A 0 (Δf)v 0 exp [− (Q∗ D +q) RT ] , where v 0 is a frequency factor, and A 0 ( Δf ) is the force of crystallization on the pendant chain. The data analysis on the fractions confirms the detailed applicability of regime theory. The growth rate theory presented allows the possibility that the growth front may be microfaceted in regime I.

Modelling the amorphous phase and the fold surface of a semicrystalline polymer—the Gambler's Ruin method

Abstract A semicrystalline polymer with lamellar morphology consists of alternating amorphous and crystalline regions. If sufficiently long, each molecule in this system traverses both the crystalline and amorphous zones. The amorphous portion is comprised of portions of a molecule that form loops that re-enter the same lamella at some distance from the point of emergence, and bridges that form connections between two different crystal lamellae. (A tight fold is not considered to be a loop). The statistics of loops and bridges are shown to be identical to the classical Gamblers Ruin problem in mathematical statistics. This is a useful observation because the extensive existing literature on the Gamblers Ruin problem allows us immediately to transcribe results to the polymer system. Using this approach, the ratio of the number of loops to the number of bridges is determined to be M , the thickness of the amorphous zone in unit statistical steps. Also, the average number of steps comprising the amorphous run is determined to be 3 M +1 for a simple cubic lattice in three dimensions. This modelling leads to a calculation of the minimal fraction of crystal stems involved in tight folding in a semicrystalline polymer. For a simple cubic lattice this is found to be 2 3 . The effects of crystal structure and stiffness of the chain in the melt on this bound are discussed.

Theory of Dielectric Relaxation for a Single‐Axis Rotator in a Crystalline Field. II

Previous work on the relaxation times associated with a polar single‐axis rotator where the dipole may occupy three or four orientational sites by turning in a lattice point has been extended to include the more general case where the energies of the sites (as well as the intervening barriers) are all different. The results for these more general models tend to confirm the view that the presence of barriers of different magnitudes between the equilibrium orientations of a dipole can be the source of a set of discrete dielectric relaxation times.The dielectric properties of some specialized three‐ and four‐position models are then considered in detail. A very narrow or an extremely wide set of relaxation times can be obtained depending on the nature of the barrier system, and accordingly, a wide range of dielectric behavior can be accommodated. Despite the wide range of behavior, certain qualitative features (noted in Sec. VII in italics) appear to persist in many of the models. These features may be usefu...

On the problem of crystallization of polymers from the melt with chain folding

It is shown that the “reptation” process proposed by de Gennes allows molecules to be reeled from the melt onto the crystal surface with chain folding by the force associated with crystallization at a rate that is comparable with that demanded by the observed crystallization kinetics for polyethylene fractions n= number of C atoms = 1290–5310. Hence, the rate of transport in the melt is sufficient to permit a considerable amount of chain folding, and an objection due to Flory and Yoon is thereby countered for the range of n noted. The deductions of Yoon and Flory from the neutron scattering data of Schelten and co-workers on PEH + PED mixtures (nped≅ 3750) quench-crystallized from the melt are considered next. It is shown that Yoon and Florys favoured pes= 0.3 model, which gives a probability of adjacent re-entry par close to zero, is deficient despite the good fit of the scattering data, since it exhibits a large density anomaly in the region between the crystal lamellae. This opposes their own view that the material in the interlamellar region has essentially normal amorphous state properties. A “central core” model is proposed that does not possess a density anomaly, and which predicts the scattering curve, characteristic ratio and crystallinity with fair accuracy. This and certain other models give par≈ 0.65, indicating that the adjacent position is by a considerable margin the most probable site for re-entry, in contrast to the analysis of Yoon and Flory. The core model exhibits a mean throw distance of ≈ 22 A for the non-adjacent re-entry loops. This is comparable with the mean “niche” distance calculated from nucleation theory. The number of ties between the lamellae is less than one per chain. Hence the connections of this type between the lamellae are less profuse than have sometimes been depicted.

Calculation of SANS intensity for polyethylene: effect of varying fold planes and fold plane roughening

Abstract The intensity of the small angle neutron scattering (SANS) for polyethylene crystallized in the lamellar habit from the melt at large supercoolings is calculated for μ = 0.01 to μ = 0.14 [ μ = ( 4π λ ) sin ( θ 2 ) ]. Computations are made on models which allow various amounts and types of chain folding and varying degrees of ‘tight’ or ‘regular’ folds. The models that fit the SANS data best have folding along lattice planes in which the stem separation is larger than 0.5 nm (5 A) or which allow for fold plane roughening on a variety of fold planes. the ‘leapfrog’ type folds mentioned by Sadler were also considered, and a possible cause for their existence suggested. As an example, the variable cluster model gives a good account of the SANS data with the surface roughening suggested by nucleation theory with fold planes [110], [200], and [310], or a mixture. Even though the conditions of crystallization used in preparing the SANS specimens (large supercoolings) were conducive to the maximum surface disorder, the probability of ‘tight’ or ‘regular’ folding, ptf, was found to be ∼0.7 for the best models. This corresponds closely to the theoretical lower bound p tf = 2 3 which is rigorous for the case of non-tiled stems. The probability of strictly adjacent re-entry in a single specified fold plane, par, was ∼0.4 to ∼0.7 depending on the particular model chosen. The best models fit not only the SANS data, but also the liquid and crystal density, degree of crystallinity, and characteristic ratio (or radius of gyration). None of the models show the density anomaly inherent in the switchboard or random re-entry models of Yoon and Flory.

Is crystallization from the melt controlled by melt viscosity and entanglement effects

An estimate is made of the time required to reel in a polymer molecule from the melt onto the growing crystal surface. Two models, one a reptation model involving transport within a worm-hole (tube-flow), one a more conventional model, both yield reeling-in times which are orders of magnitude faster than a recent estimate by Flory and Yoon. These times are shown to be consistent with crystal growth rate data. Experimental and theoretical evidence is adduced to show that polymer chains disentangle during crystallization.

Monte Carlo calculation of SANS for various models of semicrystalline polyethylene

Small-angle neutron scattering (SANS) of semicrystalline polyethylene is computed using a Monte Carlo technique similar to that used by Yoon and Flory. Models of polymer chains with substantial amounts of chain folding (with a probability of adjacent re-entry of 0.6–0.8) are shown to give the following: (1) proper density in the crystalline and amorphous regions, (2) the experimentally obtained radius of gyration, (3) scattering close to the experimentally obtained scattering. While SANS alone cannot decisively distinguish between the Yoon and Flory “switchboard” model and models with folds, present indications are that only models with a substantial amount of folding satisfactorily meet all conditions (1)-(3). The “switchboard” model used by Yoon and Flory to explain the SANS of semicrystalline polyethylene is shown not to meet criterion (1) above.