Determination of the Equilibrium Melting Temperature of Polymer Crystals:  Linear and Nonlinear Hoffman−Weeks Extrapolations

Abstract

The applicability of the Hoffman-Weeks (HW) linear extrapolation for the determination of equilibrium melting temperatures of polymers is critically reviewed. In the first paper of this series, it is shown that the linear extrapolation of observed melting temperatures cannot, in general, provide a reliable estimate of the equilibrium melting temperature. A combination of the experimentally observed undercooling dependence of the initial lamellar thickness, l* ) C1/∆T + C2, and the finite lamellar thickness dependent melting temperature depression, as described by the Gibbs-Thomson treatment, provides a venue to the general relationship between the crystallization and observed melting temperatures. It is further shown that, for a constant thickening coefficient, the observed melting temperature must vary nonlinearly with the crystallization temperature. The origin of this nonlinearity lies in the term C2, which is neglected in the classical HW treatment. The principal implications of this study in the context of the Lauritzen-Hoffman theory are the following: (1) the linear extrapolation, when carried out for lamellar crystals exhibiting a constant thickening coefficient, invariably underestimates the equilibrium melting temperature; (2) the extent of the underestimation increases with a decrease in the lamellar thickening coefficient, with an increase in the magnitude of C 2 and with an increase in the range of undercoolings where the crystals are formed; (3) the linear extrapolation always leads to an overestimation of the lamellar thickening coefficient. Finally, a more accurate method is proposed for the determination of equilibrium melting temperatures in cases where the thickening coefficient can be assumed constant.

Equilibrium Melting Temperature of Poly(ferrocenyl dimethylsilane) in Homopolymers and Lamellar Diblock Copolymers with Polystyrene

“…Obviously, the HW presuppositions are sharper and, consequently, more often violated than the GT one. The violation of the HW presuppositions causes a nonlinear dependency of T m on T c , [53] as found particularly here for PFDMS as a homopolymer and as a component of a block copolymer. The HW analysis of the dependency of T m on T c -neglecting the mentioned nonlinearity-confirmed a value of T m,0 ¼ 144 8C as reported in the literature [24] and also determined by HW analysis.…”

Crystallization and melting behavior in syndiotactic polypropylene: Origin of multiple melting phenomenon

“…Although the nonlinearity in the observed T m Ϫ T c data over a wide range of temperatures was explained to some extent by Alamo et al, 79 it is the recent contribution by Marand et al 82 …”

“…It is worth noting that for most cases it is safe to assume that e 1 ϭ e GT . 82 Precautionary remarks regarding the use of the nonlinear Hoffman-Weeks procedure to estimate the T m 0 were addressed in detail in the original publication by Marand et al 82 In order to apply eq. (3) to analyze the experimental T m Ϫ T c data in real polymer systems, it is required that the observed T m data be collected from samples crystallized at different tempera- …”

Equilibrium melting temperature and crystallization kinetics of α- and β′-PBN crystal forms

“…15 The popularity of this approach is a result of its simplicity, as only the experimental melting temperature of the crystallites formed at T c is required. Nevertheless, Marand and co-workers 16,17 recently discussed the validity of the basic premise of the linear Hoffmann-Weeks treatment: that is, the thickening coefficient for lamellae, g, taken as independent of T c and time. [18][19][20] As demonstrated by some results in the literature, [16][17][18][19][20][21] the linear extrapolation, when carried out for lamellar crystals exhibiting a constant g value, invariably underestimates T m 1 and leads to an overestimation of the g value.…”

“…Nevertheless, Marand and co-workers 16,17 recently discussed the validity of the basic premise of the linear Hoffmann-Weeks treatment: that is, the thickening coefficient for lamellae, g, taken as independent of T c and time. [18][19][20] As demonstrated by some results in the literature, [16][17][18][19][20][21] the linear extrapolation, when carried out for lamellar crystals exhibiting a constant g value, invariably underestimates T m 1 and leads to an overestimation of the g value. In fact, the HoffmannWeeks procedure does not account for the significant contribution to the difference between the melting and crystallization temperatures that arises from the temperature dependence of the fold surface free energy and the thickness increment above the minimum (thermodynamic) lamellar thickness.…”