Lateral Surface Free Energy Research Articles

An evaluation of the theories of regimes of nucleation controlled crystal growth as applied to polymers

The currently accepted idea that the nucleation controlled growth of polymer crystals and spherulites proceeds in different regimes is evaluated. We start from the concepts of mono- and polynucleation regimes of nucleation controlled growth of 2D crystal ( Frank's theory). We recall and illustrate by a morphological observation a clear-cut distinction between mono- (poly) nucleation regimes and Hoffman's regimes of polymer crystal growth, namely regimes I (and II) respectively. In fact, the concepts of regimes I, II (and III) rely upon additional strong postulates. The currently accepted methods of analysis of the kinetics of polymer crystallization (in the framework of the regime I, II and III theories) are also reviewed. Three criteria are proposed and applied, which show that the breaks observed in the curve of G( T c) giving the thermal dependence of the crystal or spherulitic growth rate are not the signature of regime I–II or II–III transitions. This is very likely because the postulate of a quantified “persistence length” ( Lauritzen) and of the thermal insensitivity of the rate g of spreading of surface nuclei are not satisfied. In addition, the concept of the Hoffman regime III has neither theoretical, computational nor experimental sound basis. In a prospective view, the method of estimation of g is discussed and, in a few cases, alternative explanations of the breaks observed in the G( T c) curves are illustrated. Note that it appears that the values of important physical parameters such as the product of the so-called fold and lateral surface free energies have until now been widely overestimated. Our analysis of the experimental data is not exhaustive and is open to discussion but, on the other hand, we cannot find in the literature any valuable support for the theory of regimes I, II and III.

Is there a connection between the lateral surface free energy of a growing polymer crystal and the mean-square dimensions of the polymer chains in the molten phase?

Is there a connection between the lateral surface free energy of a growing polymer crystal and the mean-square dimensions of the polymer chains in the molten phase?

Relationship between the lateral surface free energy .sigma. and the chain structure of melt-crystallized polymers

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTRelationship between the lateral surface free energy .sigma. and the chain structure of melt-crystallized polymersJohn D. Hoffman, Robert L. Miller, Herve Marand, and Daniel B. RoitmanCite this: Macromolecules 1992, 25, 8, 2221–2229Publication Date (Print):April 1, 1992Publication History Published online1 May 2002Published inissue 1 April 1992https://doi.org/10.1021/ma00034a025RIGHTS & PERMISSIONSArticle Views828Altmetric-Citations175LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (1 MB) Get e-Alerts Get e-Alerts

The relationship of C∞ to the lateral surface free energy σ: estimation of C∞ for the melt from rate of crystallization data

The expression σ = constant × C ∞ −1 is proposed for the lateral surface free energy that appears in polymer crystallization theory; C ∞ is the characteristic ratio of the polymer molecules in the melt and the constant consists of known parameters. Given C ∞, the theory predicts an accurate value of σ for polyethylene. It is shown that C ∞ for polyethylene in the melt state can be obtained directly from the nucleation rate constant K g and the fold surface free energy σ e ( C ∞(theor.) = 6.53 for melt, C ∞(expt) = 6.7 from chain dimensions in dilute θ solvents). The corresponding results for i-polystyrene are C ∞(theor.) = 10.45 for the melt and C ∞(expt) = 10.5 from chain dimensions. The treatment clarifies the meaning of the empirical α parameter and shows prospects for being applicable to other polymers.

Transition from extended-chain to once-folded behaviour in pure n-paraffins crystallized from the melt

Nucleation theory is applied to extended-chain crystallization as background to the extended-chain to once-folded transition problem and certain unusual effects found in recent experiments. In the ‘partial stem attachment’ model employed, the activated complex leading to stem addition involves most of the length of the molecule, but only in the form of occasional contacts with the substrate. Here the lateral surface free energy σ is of mostly entropic origin. With a normal σ, the model gives the main features of the striking maximum in the extended-chain growth rate found by Ungar and Keller for n-C 246H 494. Complete register of the chain ends is not attained during nucleation, resulting in a transient layer of cilia on the crystal (‘kinetic ciliation’). This layer leads to an end surface free energy σ′, which is deduced from the growth rate data and used to estimate the initial thickness l a of the unstable ciliated surface layer for n-C 246H 494. Shrinkage of l a through annealing is discussed, with special reference to its effect on increasing the melting point toward its equilibrium value. This leads to an interpretation of the remarkable T′ versus T x plot for extended-chain n-C 192H 386 given by Stack and co-workers. Nucleation theory is employed to predict the temperature T∗ 1 at which once-folding begins in n-C 246H 494. With a normal value of the fold surface free energy (acting in a ‘mean field’ to account for chain end effects) the theory predicts both T∗ 1 and a marked increase in the growth rate of the once-folded species relative to that of the extended-chain at T∗ 1 . This explains the extended-chain to once-folded transition in n-C 246H 494, and also accounts for the previously unexplained minimum in the overall growth rate at T∗ 1 observed by Ungar and Keller. The treatment of the onset of once-folding is supported by data on other systems. The disordered nature of the initially-formed once-folded structure and its fate on annealing are discussed.

Experimental criterion for the crystallization regime in polymer crystals grown from dilute solution: Possible limitation due to fractionation

AbstractBy integration of equations previously derived by Frank, the growth rate of polymer crystals is shown to be dependent on their size, provided that the persistence length Lp or the kinetic length Lk = (2g/i)1/2 are significantly larger than the primary nucleus. A new method of decorating the fold surface (isochronous decoration) allows the measurement of the quasi‐instantaneous growth rate of very small crystals obtained from dilute xylene solution of a sharp polyethylene fraction of moderate molecular weight (Mw = 17,000, Mw/Mn = 1.11). Although the theory predicts that the growth rate increases with the size of the crystals as long as its dimension is smaller than the persistence length and/or the kinetic length, such an increase is not observed experimentally with the sharp PE fraction presently used. Therefore it appears that both the kinetic length and the (hypothetical) persistence length are beyond the resolution limit of electron microscopy and that crystallization occurs in the polynucleation regime. An upper bound is obtained for the rate g at which a locally new layer spreads in two directions on the substrate. The rate is lower than is estimated by the commonly Accepted theories. These theories lead also to an abnormally high value for the lateral surface free energy. The possibility that the observed initial linearity of the growth‐rate curve may results from a balance of opposite effects (an increase with the size of the crystals on the one hand, a decrease with decreasing concentration and possible fractionation on the other) is thoroughly examined and ruled out. In fact, it must be stressed that at the early beginning of crystallization, negligible parts of the sample are crystallized and it is only at the end of crystallization that these effects appear. The fall in the growth rate as crystallization ends is due neither to progressive exhaustion of the solution alone nor to a depletion of the concentration by diffusion for this sharp fraction of low‐molecular‐weight PE. The major effect comes from fractionation. This segregation of the various molecular weights is predicted on the basis of a simple model and is verified by gel permeation chromatography (GPC). The fact that in such a sharp fraction significant fractionation occurs precludes any accurate determination of the supercooling and of the concentration of the polymer actually crystallizing. Subtle differences in the molecular weight distributions may result in significant variation of the growth rate. In conclusion, as the data used in the first part of this work were obtained with only a small percentage of the dissolved polymer sample crystallized, the observed constancy of the growth rate does not result from mutual compensation of opposite effects, and our conclusions about crystallization regime, order of magnitude of the kinetic and persistence lengths, and value of the rate of lateral spreading of a secondary nucleus are well founded.

Crystal growth kinetics and the lateral habits of polyethylene crystals

The lateral growth habits of lamellar crystals of polyethylene grown from solution can be typified in terms of the aspect ratio, r, which is the ratio of the dimension of the crystals in the b-axis direction to that in the a-axis direction. The aspect ratio depends on crystallization temperature, undercooling, solvent, polymer concentration and molecular weight. At steady state growth, r can be expressed in terms of the ratio of growth rates normal to the {110} and (200) faces. Expressing the growth rates in terms of the kinetic theory of polymer crystal growth yields an expression which is used to analyse experimental results on the effect of temperature and concentration on the lateral growth habits of crystals grown from xylene. Using as two adjustable parameters the ratio of end surface free energies for the two growth surfaces and the ratio of the lateral surface free energies to fit the r versus ΔT data permits the determination of these ratios with high sensitivity. The actual values obtained are dependent upon concentration, the assumed growth regime, and, most importantly, ∅, the parameter in the growth rate equations apportioning the bulk free energy change to the forward and backward steps in the stem deposition process.

Effect of molecular weight on the crystallization kinetics of isotactic poly(1‐butene)

AbstractIsothermal crystallization of isotactic poly(1‐butene) fractions by calorimetry and microscopy was studied for a range of molecular weights from 96 000 to 964000. An Avrami exponent of 2 was found indicating a two‐dimensional growth rate of the crystal, following a predetermined nucleation mode, proving that such an exponent is independent of both temperature and molecular weight. The temperature and molecular weight dependence of the growth rate of spherulites and overall crystallization rates was analyzed by means of the theoretical expression given by Hoffman and Lauritzen. The product of the lateral and basal surface free energies of the crystallites, σ·σe, increases with molecular weight and approaches its limiting values when a certain molecular weight is attained.

Effect of molecular weight on the crystallization and morphology of poly (ethylene oxide) fractions

AbstractThe temperature dependence of the isothermal growth rate of spherulites and overall crystallization kinetics of different fractions of poly (ethylene oxide) over the range of molecular weight from 300 to 20,000 were studied by calorimetry, dilatometry, and polarized light microscopy, in order to establish the dependence of the kinetic parameters upon molecular weight. The morphology of crystallites was studied by small‐angle x‐ray scattering. In the course of this study it has been established that the growth rates of spherulites and the overall crystallization rates at any fixed supercooling essentially depend upon molecular weight. The minimum in the crystallization rate which occurs at molecular weight 4000 is a consequence of transition from extended‐chain crystallization to folded‐chain crystallization. Small‐angle x‐ray scattering study of the morphological features confirmed this conclusion. This minimum depends very strongly on the degree of supercooling, increasing rather sharply at small supercooling. The temperature and molecular weight dependence of the growth rates of spherulites and the overall crystallization rates has been analyzed by means of the theoretical expression given by Hoffman and Lauritzen. The product of the lateral and basal surface free energies of the crystallites σσe increases with molecular weight and reaches a limiting value at molecular weight 6000. Such behavior is in agreement with morphological transformations.

Rate of Spherulitic Crystallization with Chain Folds in Polychlorotrifluoroethylene

(I) The requirements for making a critical test of whether two- or three-dimensional surface nucleation controls the radial growth of lamellar spherulites in bulk are discussed. Radial-growth-rate data were obtained on spherulites of polychlorotrifluroethylene (PCTFE) for a wide range of supercooling ΔT, and found to agree with a growth-rate law based on coherent two-dimensional surface nucleation, viz., G=G0 exp(—ΔF*/RT) exp[—Kg/T2(ΔT)]. Approximations for ΔF* are given. (II) Parameters related to the recently proposed ``kinetic'' viewpoint of homogeneous nucleation and growth of lamellar polymer crystals with chain folds are obtained. The nucleation constant Kg is analyzed to obtain σσe=184 erg2/cm4[σ is the lateral surface free energy, σe is the end (chain-folded) surface free energy]. A similar value of σσe is obtained from bulk-crystallization studies. The homogeneous nucleation process was identified at ΔTh=70°C, and a value of σ2σe=950 erg3/cm6 calculated from Kh in I=(NkT/h) exp(—ΔF*)exp[—Kh/T3(ΔT)2]. Combination of σ2σe and σσe gives σ=5.2 erg/cm2 and σe=36 erg/cm2, the latter corresponding to a work on chain folding, q, of 3.8 kcal/mole of folds. A value of σ=5.4 erg/cm2 is obtained independently using an empirical method. Further, an independent estimate of σe∼35 erg/cm2 is obtained for PCTFE from electron-microscope studies of the lamellar thickness, after accounting for the increase of thickness resulting from segmental mobility that occurs subsequent to initial growth using melting-point data. The over-all role of q in homogeneous nucleation and growth in linear polymers is discussed in terms of a reduced variable treatment using PCTFE and polyethylene as examples. The ``kinetic'' chain-fold theory gives a consistent picture of surface free energies, rates of homogeneous nucleation and growth, melting behavior, and the variation of initial step height with temperature. (III) A discussion is given concerning the shape of the bulk-crystallization isotherms associated with spherulitic growth, including stage-1 and stage-2 crystallization, and the presence of amorphous material in the spherulites.