Poly(lactic acid) crystallization

Kinetic of crystallization from the melt and chain folding in polyethylene fractions revisited: theory and experiment

In this article we shall present a summary of the various stochastic approaches and applications to chemical reaction kinetics, but before discussing these we first briefly introduce the basic ideas and definitions of classical or deterministic chemical kinetics. One of the basic questions to which chemists address themselves is the rate of chemical reactions, or in other words, how long it takes for a chemical reaction to attain completion, or equilibrium. Apparently the first significant quantitative investigation was made in 1850 by L. Wilhelmy [93]. He studied the inversion of sucrose (cane sugar) in aqueous solutions of acids, whose reaction is He found empirically that the rate of decrease of concentration of sucrose was simply proportional to the concentration remaining unconverted, i.e., if S(t) is the concentration of sucrose, then The constant of proportionality is called the rate constant of the reaction. If S o is the initial concentration of sucrose, then Since then an enormous number of reactions has been studied and the field of chemical kinetics is now one of the largest areas of chemical research. The importance of the field lies in the fact that it yields concise expressions for the time dependence of reactions, predicts yields, optimum economic conditions, and gives one much insight into the actual molecular processes involved. The detailed molecular picture of a reaction process is called the mechanism of the reaction.

Stochastic approach to chemical kinetics

To account for spherulitic crystallization from the melt, one must explain the origins (i) of fibrous crystal habits in the absence of appreciable temperature gradients and (ii) of profuse noncrystallographic branching. Attention is drawn to properties held in common by spherulite‐forming melts of various types and, in particular, to the facts that (a) they are multicomponent systems, (b) they exhibit small coefficients of self‐diffusion, and (c) they crystallize slowly. It is shown that a consequence of these properties is that a plane crystal face cannot grow without suffering an instability of profile. Analagous instabilities lead in metal crystals to a cellular interface but, because of unusual growth kinetics, instability in spherulite‐forming melts gives rise to a drastic modification of crystal habit. Bundles of discrete fibers are formed whose widths are determined by δ = D/G, D being the coefficient of self‐diffusion and G being the growth rate. δ is generally small in these systems and commensurate with the scale of crystalline disorder in the fibers. It is this circumstance that allows noncrystallographic branching to occur.

A Phenomenological Theory of Spherulitic Crystallization

Crystallization behaviour of poly(ether-ether-ketone)

A detailed interpretation of the kinetics of homogeneous nucleation and growth of crystals of a linear homopolymer from dilute solution is given. The probability of forming both nuclei with folded chains, and conventional bundlelike nuclei, from dilute solution is analyzed. It is predicted that at sufficiently high dilution, critical nuclei of length lp* will be formed from single polymer molecules by sharp folding of the chain backbone. The step height of the nucleus is given approximately by lp*=4σe/Δf . Here σe is the free energy required to form a unit area of the loop-containing end surfaces, and Δf is the free energy difference per unit volume of crystal between the crystalline and solution states. The quantity Δf is approximately proportional to the degree of supercooling ΔT. The growth of these nuclei is then analyzed. After growth, the resulting crystal is flat and platelike, the loops formed by the chain folds being on the upper and lower surfaces. Kinetic factors determine that the distance between the flat surfaces in the grown crystal will vary over only a narrow range about a value that is in the vicinity of 1*=4σe /Δf. (Neglecting effects due to edge free energies, the theoretical upper and lower limits are 1*=4σe /Δf and 1*=2σe /Δf, respectively.) In some cases the predicted temperature dependence of the step height of the grown crystal, 1* = const./ΔT, may be modified by the existence of a constant term resulting from the presence of an edge free energy ϵp . A grown loop-type crystal is predicted to be stable in comparison with a bundlelike crystal of the same shape and volume in a sufficiently dilute solution. The logarithm of the nucleation rate is approximately proportional to 1/(ΔT)2 near the melting point. The exponent n in the free growth rate law is predicted under various assumptions. To the extent that comparison is possible, the predictions given agree with the experimental results obtained by Keller and O'Connor and others on single crystals of unbranched polyethylene grown from dilute solution. A survey is given of homogeneous nucleation in bulk polymers, where the conventional bundlelike nucleus containing segments from many different molecules is valid, and the essential results compared with those calculated for the dilute solution case. The theory given for loop nuclei is both general and precise enough at the critical points to suggest that, on crystallization from sufficiently dilute solution, crystals of a definite step height are commonly to be expected for other crystallizable linear polymers than polyethylene, provided loop formation is sterically possible.

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Theory of Formation of Polymer Crystals with Folded Chains in Dilute Solution.

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...

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

A systematic study of the problem of spherulitic growth in linear polymers in bulk has been carried out. A calculation of the radial growth of polymer spherulites is given for four models. These concern growth where the surface nuclei that control the rate are (1) bundlelike and coherent, (2) chain folded and coherent, (3) chain folded and noncoherent, and (4) bundlelike and noncoherent. The required modifications of nucleation theory are given. Then the radial growth rate laws are derived for each model, and the type of "spherulite" that would be formed discussed. The model with chain folded and coherent growth nuclei leads to a typical lamellar spherulite. The properties of the individual chain folded lamellae that form the spherulite are predicted, including the change of step height with growth temperature, melting behavior, and the behavior on recrystallization. (Chain folded lamellae may also occur in specimens that are not obviously spherulitic.) Under certain conditions, the noncoherent model with chain folds can lead to a modified lamellar spherulite. None of the bundlelike models will lead to a typical lamellar spherulite, though a spherical microcrystalline object might be formed. It is concluded that lamellar spherulites consist largely of chain folded structures. The factors that could cause chain folded crystals to appear in profusion in bulk polymers are discussed. The case of homogeneous initiation is considered first. Homogeneous initiation of chain folded nuclei in bulk will prevail if the end surface free energy of the bundlelike nucleus exceeds that of the folded. It is shown that the end surface free energy of the bundlelike nucleus, as calculated with a density gradient model, will be larger than had been supposed previously. It is therefore considered to be theoretically possible that the end surface free energy of the bundlelike nucleus may in some cases exceed that of the folded nucleus. Attention is given to the possibility that folded structures appear in large numbers because cumulative strain or large chain ends prevent the growth of bundlelike nuclei to large size, even when the latter type of nucleus is energetically favored when small. Heterogeneous initiation of folded structures is then considered. Other topics mentioned include: (1) Conditions that might lead to nonlamellar or nonspherulitic crystallization in bulk, (2) the origin of the twist that is frequently exhibited by the lamellae in spherulites, (3) the transitions that may sometimes occur in the radial growth rate law, and (4) interlamellar links.

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Crystallization of Bulk Polymers With Chain Folding: Theory of Growth of Lamellar Spherulites.

Th e growth rate C of the cr ys tallin e bodies a ppearing in each of a se t of 35 characte rized po ly. e th ylene fractions ra ngin g from 3600 to 807 ,000 in molec ular we ight has been meas ured as a fun ction of the unde rcooling tJ.T. In isothe rmal c rys tallization, only ax ial ites we re found from M\",= 3600 to 18 ,000. (For th ese runs, tJ.T < 17.5 °C.) From M\".= 18,000 to M\". \"\" 115,000 coa rse·grained non ·banded s phe ru · lit es we re found for tJ.T > 17.5 °C , and axialites for tJ.T < 17.5 °C : a rathe r s ha rp break occurred in th e log .o C ve rsus T data at tJ.T \"\" 17.5 0c. The morphologica l changes we re more gradual. Above Mil' \"\" 115,000, only nea rly s tructureless \" irregular\" spherulit es we re found a t all unde rcoolings corresponding to isothermal growth. Typica l ringed sphe rulites were o btained only on quenching. Wide·ang le x·ray data showed that the usual orthorhombic subce ll predominated in all the morphologies encounte red. Low· angle x·ray data showed tha t the s pecime ns exhibit ed lamellar crystallization irrespective of the particular gross morphology involved. The growth rat e data on each frac tion we re analyzed us ing

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On the Growth Rate of Spherulites and Axialites from the Melt in Polyethylene Fractions: Regime I and Regime II Crystallization.

The polymer crystal lamellar growth rate G is an often‐measured property of semicrystalline polymers. In this paper, we investigate the dependence of G upon the length of the substrate, L, the surface nucleation rate of new growth layers per unit length per unit time, i, the velocity with which the growth layer covers the substrate, g, and the thickness of the growth layer, b. For the purposes of this paper, L is the average length where the polymer segments in the substrate are in crystallographic register. We show that G = biL/n, where n is the average number of nuclei that contribute to each growth layer. We show that n depends only on the dimensionless parameter z = iL2/4g. We have not obtained n explicitly, but we can place upper and lower limits on n that closely define G in regime I, where G ≃ biL and into a transition region between regime I and regime II where G ∞ b(ig)1/2. The experimental results for several polymers are analyzed and some are found to be in or near regime I, and others in ...

John I. Lauritzen

Papers

Theory of Formation of Polymer Crystals with Folded Chains in Dilute Solution.

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

Crystallization of Bulk Polymers With Chain Folding: Theory of Growth of Lamellar Spherulites.

On the Growth Rate of Spherulites and Axialites from the Melt in Polyethylene Fractions: Regime I and Regime II Crystallization.

Effect of a finite substrate length upon polymer crystal lamellar growth rate