Some Contributions to the Kinetics of Growth of Multicomponent Chains with Application to the Problems of Ciliation and Fractionation in Polymer Crystallization

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Abstract
The problem of the growth rate and purity of crystals formed by the sequential deposition of molecules in a mixed system is solved. The solution is given in terms of the fundamental rates aνij(βν+1ij) for laying down (taking off) species j at the (ν+1)th position of the growing crystal given that species i occupies the (ν)th position. The fundamental rates are thus dependent on nearest‐neighbor interaction but not on neighbors further removed. The method used complements a previously used method and additionally solves the problem for ν‐dependent ανij and βνij. The ν dependence of the α's and β's allows for the stochastic‐like distribution of the various species in the substrate upon which we are growing our linear chain of molecules. It is shown that there exists one unique flux‐determined steady‐state solution for a wide variety of initial conditions. Several diagrams descriptive of various physical processes are displayed. The one describing ciliation (the growth process which results in the ends of polymer chains dangling out of the crystal lamella) is treated in some detail. It is found that there are two classes of α's and β's. One class gives large growth rates and very little ciliation. The second class gives growth rates which are orders of magnitudes smaller and results in much ciliation. It is concluded that when chain‐folded crystallization occurs the amount of material dangling out of the crystal is less than 50% of the total amount of material participating. The problem of fractionation in polymers is treated. Formulas are given for the purity of polymer crystals which are formed by high molecular weight material crystallizing concomitantly with polymeric material of lower molecular weight and/or of different kind. Fractionation can be substantial even when the molecular weights of the mixed system are all large. Simple formulas are derived. More complicated diagrams that allow for more than nearest‐neighbor interaction can be reduced to the nearest‐neighbor case but with a larger number of components.

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