Lauritzen and Hoffman theory

Fig. 8 a Spherulitic growth rates for PPDX and the PPDX block within D7732C2310 diblock copolymer. Solid lines are fits to Lauritzen and Hoffman theory, b Lauritzen and Hoffman kinetics theory plot for PPDX (K = 17.2 x 104 K2) and the PPDX block within D7732C2310 diblock copolymer (K = 46 x 104 K2). (From [103]. Reproduced with permission of the Royal Society of Chemistry)... 

In this chapter, we take a practical approach to briefly explain how to experimentally determine both spherulitic growth rates by polarized light optical Microscopy (PLOM) and overall isothermal crystallization kinetics by differential scanning calorimetry (DSC). We give examples on how to fit the data using both the Avrami theory and the Lauritzen and Hoffman theory. Both theories provide useful analytical equations that when properly handled represent valuable tools to understand crystallization kinetics and its relationship with morphology. They also have several shortcomings that are pointed out. 

We then turn to a more recent approach to the determination of i by Point [51, 52] in Sect. 3.7 explaining how it differs from that of Lauritzen and Hoffman (LH). Section 3.8 covers other proposed nucleation models and we conclude with an overview of nucleation theories and their successes and most notable shortcomings. 

This relation is consistent with previous results observed experimentally [49]. Although the kinetic theory of Lauritzen and Hoffman predicts the same law as Eq. 16, it predicts a divergence in L at lower undercoolings. The simulations do not show any evidence for such a catastrophe. 

The isothermal crystallization of PEO in a PEO-PMMA diblock was monitored by observation of the increase in radius of spherulites or the enthalpy of fusion as a function of time by Richardson etal. (1995). Comparative experiments were also made on blends of the two homopolymers. The block copolymer was observed to have a lower melting point and lower spherulitic growth rate compared to the blend with the same composition. The growth rates extracted from optical microscopy were interpreted in terms of the kinetic nucleation theory of Hoffman and co-workers (Hoffman and Miller 1989 Lauritzen and Hoffman 1960) (Section 5.3.3). The fold surface free energy obtained using this model (ere 2.5-3 kJ mol"1) was close to that obtained for PEO/PPO copolymers by Booth and co-workers (Ashman and Booth 1975 Ashman et al. 1975) using the Flory-Vrij theory. 

Polymer crystal growth is predominantly in the lateral direction, because folds and surface entanglements inhibit crystalliza- 4 don in the thickness direction. Neverthe-1 less, there is a considerable increase in the fold period behind the lamellar front during crystallization from the melt and, as we have j seen, polymers annealed above their crys-tallization temperature but below Tm also irreversibly thicken. Nevertheless, in most theories of secondary nucleation, the most i widely used being the theory of Lauritzen and Hoffman,28 it is assumed that once a part of a chain is added to the growing crystal, its. fold period remains unchanged. 

The theory of Lauritzen and Hoffman, perhaps still the most commonly used model for the analysis of polymer crystallization data, then seeks to evaluate c5Z (usually of the order of 40 angstroms) by considering the rates at which stems and folds are successively laid down. We will not go into the details of this derivation, but the expressions that have been derived for the growth rate have the following form at low undercoolings (Equation 10-41) ... 

The major theory [3 7] of polymer crystallization, due primarily to Lauritzen and Hoffman (LH), is a generalization of small-molecule crystallization theory of surface nucleation and growth to incorporate chain folding. In the model of LH theory (Fig. 1.3a), polymer molecules are assumed to attach at the growth front in terms of stems, each of length comparable to the lamellar thickness L. For each polymer molecule, the first step is to place its first stem at the growth surface, whose lateral dimension is taken as Lp. This step is assumed to be associated with a nucleation. The barrier for this step was assumed... 

Regime transition is presented when the data are analyzed with the Lauritzen and Hoffman kinetic theory. Di Lorenzo demonstrated that the discontinuity in the spherulite growth rate is not associated to any change in superstructural morphology. Tsuji et al. and Yuryev et al. also observed this unusual bimodal crystallization behaviour for pure PLLA, while the normal characteristic bell-shaped spherulite growth rate dependence was seen for poly(L/D-lactide) copolymers. 

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