Lauritzen-Hoffman theory equation

Figure 18.8 Analysis of crystallization rate of SPS according to Lauritzen-Hoffman nucleation theory [Equation (1)] (AT = 7, -7), (a) Data plotted for various Mw O 4.70 X 104 1.26 x 105 tx, 3.78 x 105 A, 6.91 x 105 o, 9.23 x 105 , 2.00 x106. (b) Superposed line by horizontal shift with reference to 7°/TAT...

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. 

The spheralite growth rate was analyzed by using Lauritzen-Hoffman (LH) growth theory. TTiis secondary nucleation theory is described by the following equation, ... 

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 theory developed by Hoffman and Lauritzen yields a simple equation for the growth rate. 

The crystallization of PEO was also unveiled at the level of individual lamellae in ultrathin films (278-280). It was observed that although lamellar growth rates were retarded in films thinner than 200 nm because of interfacial interactions, SFM provided the same qualitative and quantitative information as typically applied bulk characterization techniques on lamellar growth rates, lamellar thicknesses, and melting points. The Hoffman-Weeks extrapolation, the Gihhs-Thompson equation, and the Hoffman-Lauritzen theory were apphed and the results compared favorably to the corresponding hulk data. 

This chapter presents first some fundamental aspects of nucleation, and second the general Avrami equation, which is frequently used to describe overall crystallization. The growth theories of Lauritzen and Hoffman and Sadler and Gilmer are discussed in sections 8.4.2 and 8.4.3. Molecular fractionation and orientation-induced crystallization are dealt with in sections 8.5 and 8.6. 

Most of the kinetic work has dealt with the temperature dependence of the growth rate in accordance with the kinetic theory of Lauritzen and Hoffman (see Hoffman et al. 1975). The experimental data, the linear growth rate (G) of spherulites (axialites), are obtained by hot-stage polarized light microscopy at different constant temperatures and the data are adapted to the equation ... 

The quantitative analysis of the growth rate was made on the basis of the Hoffman-Lauritzen theory expressed by Equation 11.1. The universal values of U = 1500cal/mol and T = Tg- 30K were used in these calculations [4]. It is often most convenient to rearrange Equation 11.1 as... 

---