Lamellar crystallization theory

Fig. 5.33 Growth of a lamellar crystal according to the Lauritzen-Hoffmann (Lauritzen and Hoffmann 1960) theory. The lateral growth rate is denoted g and the linear growth...

Theories about the details of the fold-length dependence of lamellar crystals have gone through many steps, but are not fully resolved at present. References to some... 

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... 

Figure 4 Schematics depicting the basics of the Lauritzen-Hoffman secondary nucleation theory. G indicates the lateral rate of grovrth accessible in the experiments, while stands for the propagation rate of the nucleus in the row direction until it reaches a dislocation defect or impinges on a neighboring nucleus /.stands for the so-called persistence length of the lamellar crystal while /c is the fold length of the lamella a and b are the lateral dimensions of a crystalline stem and a are the surface energies for the fold and lateral surface, respectively.

Figure 6.33 Crystallization according to the entropic barrier theory, (a) Representation of a lamellar crystal, showing stems (chain direction vertical) and a step in the growth face. The inset provides a description of the step in terms of units that are shorter than the length of the surface nucleation theory (one molecule making up a whole stem). The dotted lines indicate where the row of stems in (b) is imagined to occur, (b) The basic row of stems model, showing mers along the chains as cubes, chain direction vertical, as in (a).

Lauritzen-Hoffman (LH) theory is stUl the most widely used theoretical model in the explanation of the growth kinetics of lamellar crystals. Figure 4.21 schematically shows the basic assumptions of folded stems at the growth front in the LH theory, without considering the thickening at the lateral growth front. 

Approaches used for crystallization in homopolymers may be used to calculate the change in melting temperature due to finite crystal thickness (Thompson-Gibbs equation), lamellar crystal surface energies (Flory-Vrij theory), and growth rates (kinetic nucleation theory). Details can be obtained from [1]. 

The effect of different types of comonomers on varies. VDC—MA copolymers mote closely obey Flory s melting-point depression theory than do copolymers with VC or AN. Studies have shown that, for the copolymers of VDC with MA, Flory s theory needs modification to include both lamella thickness and surface free energy (69). The VDC—VC and VDC—AN copolymers typically have severe composition drift, therefore most of the comonomer units do not belong to crystallizing chains. Hence, they neither enter the crystal as defects nor cause lamellar thickness to decrease, so the depression of the melting temperature is less than expected. 

G. Gompper, S. Zschocke. Elastic properties of interface in a Ginzburg-Landau theory of swollen micells, droplet crystals and lamellar phases. Euro-phys Lett 16 13 -136, 1991. 

Having discussed some equilibrium properties of a crystal, we now outline and contrast the bases of the growth theories which will be dealt with in more detail later. The theories may be broadly split into two categories equilibrium and kinetic. The former [36-42] explain some features of the lamellar thickness, however the intrinsic folding habit is not accounted for. Therefore, at best, the theory must be considered to be incomplete, and today is usually completely ignored. We give a brief summary of the approach and refer the interested reader to the original articles. The kinetic theories will be the topic of the remainder of the review. 

In the classical Lauritzen-Hoffman theory for the mechanism of polymer crystal growth [106], it is assumed that the observed lamellar thickness corresponds to those crystallites that happen to have the largest growth velocity. However, this picture is hard to reconcile with the experimental observation that the thickness of polyethylene single crystals can be modulated by varying the temperature at which they are grown [117,118]. In fact, simulations by Doye et al. [119,120] suggest that the observed lamellar thickness does... 

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