Lamellar thickness

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. 

Copolymers of vinyUdene chloride and methyl acrylate have been studied by x-ray techniques (75). For example, the long period (lamellar thickness) for an 8.5 wt % methyl acrylate copolymer was found to be 9.2 nm by smaH-angle x-ray scattering. The unit cell is monoclinic, with a = 0.686 and c = 1.247 nm by wide-angle x-ray scattering. 

Lamellar thickness Minimum stable thickness Thickness deviation l — lmin Surface area of the fold surface Width of a stem Thickness of a stem Fold surface free energy Lateral surface free energy... 

Net forward rate for folding at lamellar thickness / e Pairwise nearest neighbour attractive energy... 

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. 

At this point a third intermediate approach deserves mentioning. It is due to Allegra [43] who proposed that polymer crystallization is controlled by a metastable equilibrium distribution of intramolecular clusters, the so-called bundles , forming in the liquid phase. These subsequently aggregate to the side surfaces of the crystals, driven by van der Waals interactions. The lamellar thickness is determined by the average contour length of the loops within the bundles. Although the model can... 

Fig. 3.16a-c. a A series of fluctuations which are allowed in Frank s model, that is, attachments and detachments are both allowed but the chain is not permitted to fold until it has reached the lamellar thickness, b and c show a series of events which may also be expected to occur... 

A further paper [167] explains the lamellar thickness selection in the row model. The minimum thickness lmin is derived from the similation and found to be consistent with equilibrium results. The thickness deviation 81 = l — lmin is approximately constant with /. It is established that the model fulfills the criteria of a kinetic theory Firstly, a driving force term (proportional to 81) and a barrier term (proportional to /) are indentified. Secondly, the competition between the two terms leads to a maximum in growth rate (see Fig. 2.4) which is located at the average thickness l obtained by simulation. Further, the role of fluctuations becomes apparent when the dependence on the interaction energy e is investigated. Whereas downwards (i.e. decreasing l) fluctuations are approximately independent... 

A very recent application of the two-dimensional model has been to the crystallization of a random copolymer [171]. The units trying to attach to the growth face are either crystallizable A s or non-crystallizable B s with a Poisson probability based on the comonomer concentration in the melt. This means that the on rate becomes thickness dependent with the effect of a depletion of crystallizable material with increasing thickness. This leads to a maximum lamellar thickness and further to a melting point depression much larger than that obtained by the Flory [172] equilibrium treatment. 

Both of the numerical approaches explained above have been successful in producing realistic behaviour for lamellar thickness and growth rate as a function of supercooling. The nature of rough surface growth prevents an analytical solution as many of the growth processes are taking place simultaneously, and any approach which is not stochastic, as the Monte Carlo in Sect. 4.2.1, necessarily involves approximations, as the rate equations detailed in Sect. 4.2.2. At the expense of... 

Table 2. Lamellar thickness 1, as function of annealing temperature at 5.3 Kbar for PE cold-drawn to a natural draw ratio X 8.

The fact that crystalline polymers are multiphase materials has prompted a new approach in characterizing their internal structure (lamellar thickness, perfection, etc.) and relating it to the hardness concept (volume of material locally deformed under a point indenter). In lamellar PE microhardness is grossly a given increasing function of lamellar thickness. In using the composite concept care must be exercised to emphasize and properly account for the non-crystalline phase and its various... 

The thickness of the ordered crystalline regions, termed crystallite or lamellar thickness (Lc), is an important parameter for correlations with thermodynamic and physical properties. Lc and the distribution of lamellar thicknesses can be determined by different experimental methods, including thin-section TEM mentioned earlier, atomic force microscopy, small-angle X-ray scattering and analysis of the LAM in Raman spectroscopy. 

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