Crystals weight distribution

Once the growth rate at any given retention time is known and if the plot of In n versus L is a straight line, then the crystal weight distribution may be computed by the following equations. 

Figure 11 shows that the molecular weight distribution in the melt (presence of short chains) can account for the coexistence of two types of crystals in the absence of molecular orientation or at a slight stretching of the melt. However, there is a purely thermodynamic reason for the appearance of this main structural feature of samples crystallized under conditions of molecular orientation, even at high degrees of orientation, when virtually the whole distribution function is displaced into the region of /S > /3cr. 

The indirect method described here returns the weight-average crystal size [121], irrespective of the model shape chosen. On the other hand, the direct Fourier inversion according to Warren-Averbach returns the number average of the crystal size distribution. 

Free-radical polyolefin reactions form polymers with many mistakes in addition to the ideal long-chain alkanes because of chain-branching and chain-termination steps, as discussed. This produces a fairly heterogeneous set of polymer molecules with a broad molecular-weight distribution, and these molecules do not crystallize when cooled but rather form amorphous polymers, which are called low-density polyethylene. 

Now we compare the above osmotic pressure data with the scaled particle theory. The relevant equation is Eq. (27) for polydisperse polymers. In the isotropic state, it can be shown that Eq. (27) takes the same form as Eq. (20) for the monodisperse system though the parameters (B, C, v, and c ) have to be calculated from the number-average molecular weight M and the total polymer mass concentration c of a polydisperse system pSI in the parameters B and C is unity in the isotropic state. No information is needed for the molecular weight distribution of the sample. On the other hand, in the liquid crystal state2, Eq. (27) does not necessarily take the same form as Eq. (20), because p5I depends on the molecular weight distribution. 

Macromolecules are very much like the crystalline powder just described. A few polymers, usually biologically-active natural products like enzymes or proteins, have very specific structure, mass, repeat-unit sequence, and conformational architecture. These biopolymers are the exceptions in polymer chemistry, however. Most synthetic polymers or storage biopolymers are collections of molecules with different numbers of repeat units in the molecule. The individual molecules of a polymer sample thus differ in chain length, mass, and size. The molecular weight of a polymer sample is thus a distributed quantity. This variation in molecular weight amongst molecules in a sample has important implications, since, just as in the crystal dimension example, physical and chemical properties of the polymer sample depend on different measures of the molecular weight distribution. 

Batch crystallizers often are seeded with small crystals of a known range of sizes. The resulting crystal size distribution for a given overall weight gain can be estimated by an approximate... 

Nogales, A., Hsiao, B.S., Somani, R.H., Srinivas, S., Tsou, A.H., Balta-Callja, F.J. and Eqzuerra, T.A. (2001) Shear-induced crystallization of isotactic polypropylene with different molecular weight distributions in situ small- and wide-angle X-ray scattering studies, Polymer 42(12), 5247-5256... 

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