Fusion of the -alkanes and other oligomers

At the melting temperature equilibrium exists between the liquid and crystalline states. There is no reason in principle why equilibrium between the two phases cannot be maintained at finite levels of crystallinity as well. It is thus possible to uniquely specify the properties of the crystalline phase as long as it is recognized that the stipulation of equilibrium must be met. For crystallization under conditions removed from equilibrium this condition cannot be satisfied. The properties of the crystalline phase will depend on the mode by which crystallinity is developed. The development of higher levels of crystallinity that approach equilibrium requirements is not achieved very easily. It is primarily governed by the nature of the crystallization mechanisms. These problems are not unique to the crystallization of polymers. The same difficulties are encountered by crystallizing low molecular weight systems. 

The molar free energy of a chain of n repeating units at any arbitrary temperature T can be expressed as(l 1) 

In this equation AGu(T) represents the free energy of fusion, at temperature T, of a repeating unit in the limit of infinite chain length. AGe(T ) is the end-group contribution which is assumed to be independent of n. The constant /f In c is incorporated into AGe. This latter term plays a role analogous to that of an interfacial free energy. 

The temperature dependence of AG and AGe can be accounted for by performing a Taylor series expansion around the equilibrium melting temperature. By expanding AGu to second order one obtains(ll) 

At the melting temperatures of an n-mer, AG = 0 and T = Tm. After rearrangement, the melting temperature of a series of homologues of length n is given to an approximation that is usually sufficient, by(l 1) 

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