Free energy for formation of a critical nucleus

Since Jo and C are proportional to the diffusion coefficient (D) and activation free energy for formation of a critical nucleus (AG ), respectively, it is concluded that the Z dependence of J is mainly determined by the diffusion process of the polymer chain and not by the formation process of a critical nucleus. 

The free energy necessary for the formation of a critical nucleus AG in both primary and secondary nucleation processes does not depend on Mn, i.e., AG ps const, while only the diffusion coefficient D depends on Mn, i.e., I ex D(Mn). Therefore, the Mn dependences of I and V are not controlled by the formation process of a critical nucleus but are mainly controlled by the chain sliding diffusion process. 

As the presence of a suitable foreign body or sympathetic surface can induce nucleation at degrees of supercooling lower than those required for spontaneous nucleation, the overall free energy change associated with the formation of a critical nucleus under heterogeneous conditions must... 

For small nuclei the surface term dominates and the free energy increases. Only if this nucleus exceeds a critical size does its free energy decrease and the crystallite can grow spontaneously (see Fig. 1). The probability for the formation of a critical nucleus depends exponentially on its free energy of formation ... 

Polymer crystallization is usually initiated by nucleation. The rate of primary nucleation depends exponentially on the free-energy barrier for the formation of a critical crystal nucleus [ 110]. If we assume that a polymer crystallite is a cylinder with a thickness l and a radius R, then the free-energy cost associated with the formation of such a crystallite in the liquid phase can be expressed as... 

Expanding the free energy of formation of a spherical nucleus of radius r [given by Eq. (1.38)] around that (F ) for a critical nucleus, we get... 

Gj is a pre-exponential constant assitmed to be constant or proportional to 7, AF is the activation energy responsible for the transport of molecnlar chains across liquid-sohd interface. A is the free energy of formation of a snrface nucleus with critical size. R and is gas constant and Boltzman constant, respectively. The quantity G is influenced by the type, molecular weight, composition, and so on of the semicrystalhne polymer. 

Up to this point, the preexponential has been taken to be a constant, independent of the temperature and pressure. An empirical justification for this is that the nucleation rate depends on the exponential of the free energy of formation of the critical nucleus, and that this strong dependence effectively swamps any variation in the preexponential. An illustration of the relative importance of the two terms is shown in Fig. 5. 

For homogeneous nucleation in condensed systems diffusion to the phase boundary becomes an important factor and A/, the energy of activation for this diffusion, must be added to AF, the free energy of formation of a critical size nucleus, AF = 6na l3 AGy. A detailed treatment of this problem has been given by Turnbull and Fisher ( ). 

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