Crystal growth molar mass

Irrstead of the molar flux derrsity ft the crystal growth rate is often described by the displacement rate v of a crystal face (for irrstance v, for the 111 face). Note that every face can have different growth rates v at the same supersaturation. Let us assume spherical (poly)crystals. The growth rate is eqttal to the derivative of the radius with respect to time t (y = dr/dt) or the derivative G = 6L/6t with/, as the decisive length which is the diameter L for spheres. With the volume shape factor a = V /L and the srrrface shape factor P =. 4p/Z,, the following relationship between the mass flux density m (m = n M), the displacement rate v of the crystal surface, and the rate G = 2v of crystalhne particles is given... 

Two final points need to be made about secondary nucleation. First, that screw-dislocation defects, described in more detail in Sect. 5.3, prodnce indesttnctible secondary nuclei for growth on top of the fold surfaces of polymer lamellae. This surface would otherwise be inactive for further growth and restrict polymer crystals to single lamellae (see Chap. 5). An example of a series of screw dislocations is shown in Fig. 3.72 on the example of poly(oxyethylene) of 6,000 molar mass grown... 

This concludes the discussion of the nucleation dynamics of macromolecules. It shows that the usually assumed constant number of heterogeneous nuclei and linearly increasing number of homogeneous nuclei is a simplification, and secondary nucleation as the basis for crystal growth is a doubtful concept. Finally, molecular nucleation is not a well enough understood concept to quantitatively explain facts such as the molar mass dependence of crystallization. The further work needed to understand the basis of nucleation in polymers is a big challenge for new research in solid-state polymer science. 

It is interesting to note that the deviations from the Avrami expression occur at a level that is given by the limit of applicability of the free growth approximation in Fig. 3.55. Similar data for a single molar mass, but at different temperatures, are shown in Fig. 3.101. All crystallizations seem to approach a common limit, but deviate at different temperatures from the Avrami equation. 

Its flexibility and segmental mobility are reduced so much that the crystallization of a quenched, amorphous sample of molar mass 28,000 Da needs an induction time of almost 50 h at 460 K, the temperature of fastest growth, and the half-time of crystallization is only reached after about 7 days [43]. The heat capacity of the solid PC has been analyzed, and the heat capacity of the liquid PC was measured and compared to the other aromatic polyesters. 

A well studied example is given by the poly(oxyethylene-Z locfc-styrene). In case of atactic sequences of polystyrene, only the poly(oxyethylene), POE, can crystallize. A typical morphology of the POE is shown in Fig. 5.55. Single crystals of the copolymer can be grown from a common solvent which keeps both components mobile up to the time of crystallization of the POE-component. Figure 7.53 illustrates a growth spiral out of poly(oxyethylene-fclocfe-styrene), grown at 293 K from a solution of ethylbenzene (AB diblock, 28 wt-% oxyethylene block with a molar mass of about 10,000 Da). The crystal is comparable to the lamellar crystals of Fig. 5.55, i.e., the poly(oxyethylene) crystals are chain-folded with about 2.5 nm amorphous polystyrene layers at the interfaces. 

Figure 5.12 Spherulite growth rate of PLA of different molar mass as a function of the crystallization temperature. The glass transition temperatures and the equilibrium...

More importantly, the crystallization kinetics of all samples of different molar mass displays the characteristic discontinuity due to the different radial growth rates of a - and a-spherulites. Independent of the molar mass, the transition from growth of a -crystals to growth of a-crystals occurs at 100-120 C [28]. 

As mentioned above, PLA should be addressed as a random copolymer rather than as a homopolymer the properties of the former depend on the ratio between L-lactic acid and D-lactic acid units. A few studies describe the influence of the concentration of D-lactic acid co-units in the PLLA macromolecule on the crystallization kinetics [15, 37, 77-79]. The incorporation of D-lactic acid co-units reduces the radial growth rate of spherulites and increases the induction period of spherulite formation, as is typical for random copolymers. In a recent work, the influence of the chain structure on the crystal polymorphism of PL A was detailed [15], with the results summarized in Figure 5.13. It shows the influence of D-lactic acid units on spherulite growth rates and crystal polymorphism of PLA for two selected molar mass ranges. 

At identical molar mass, the PLA grades containing only L-lactide units in the chain exhibit faster crystal growth than the samples containing even small amounts of D-units [15]. Moreover, the bimodal shape of the spherulite-growth-rate data presented in Figure 5.13 permits identification of the a /a crystal polymorphism, as described above. 

The nucleation and growth processes, similar to the situation in low molar mass organic crystals, are dependent on the degree of supereooling of the melt or solution phase. The crystal thickness or alternatively the thiekness of eaeh new crystalline layer in a growing crystal is the one that grows fastest rather than the one that is at equilibrium. There is a wealth of information available on the crystallization of many polymers as well as several theories that aim to prediet the crystallization rates, crystal shapes and lamellar thieknesses. 

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