Nucleus, critical

The resistance to nucleation is associated with the surface energy of forming small clusters. Once beyond a critical size, the growth proceeds with the considerable driving force due to the supersaturation or subcooling. It is the definition of this critical nucleus size that has consumed much theoretical and experimental research. We present a brief description of the classic nucleation theory along with some examples of crystal nucleation and growth studies. 

Classic nucleation theory must be modified for nucleation near a critical point. Observed supercooling and superheating far exceeds that predicted by conventional theory and McGraw and Reiss [36] pointed out that if a usually neglected excluded volume term is retained the free energy of the critical nucleus increases considerably. As noted by Derjaguin [37], a similar problem occurs in the theory of cavitation. In binary systems the composition of the nuclei will differ from that of the bulk... 

If we compare eqns (7.11) and (7.3) we see that the expressions for the critical radius are identical for both homogeneous and heterogeneous nucleation. But the expressions for the volume of the critical nucleus are not volume is... 

The relative magnitude of these two activation free energies determines the size and shape of the critical nucleus, and hence of the resulting crystal. If sliding diffusion is easy then extended chain crystals may form if it is hard then the thickness will be determined kinetically and will be close to lmin. The work so far has concentrated on obtaining a measure for this nucleus for different input parameters and on plotting the most likely path for its formation. The SI catastrophe does not occur because there is always a barrier against the formation of thick crystals which increases with /. 

The primary nucleation process is divided into two periods in CNT one is the so called induction period and the other is the steady (or stationary) nucleation period (Fig. 2) [16,17]. It has been proposed by CNT that small (nanometer scale) nuclei will be formed spontaneously by thermal fluctuation after quenching into the supercooled melt, some of the nuclei could grow into a critical nucleus , and some of the critical nuclei will finally survive into macroscopic crystals. The induction period is defined as the period where the nucleation rate (I) increases with time f, whereas the steady period is that where I nearly saturates to a constant rate (fst). It should be noted that I is a function of N and t,I = I(N, t). In Fig. 2, N and N mean the size of a nucleus and that of the critical nucleus, respectively. The size N is defined... 

There are two typical definitions of the induction time (ti) in CNT given by Frisch [16] and by Andres and M. Boudart [17]. x is an increasing function of N, t,(N). In previous studies, the special case N = N was usually focused on. As any critical nucleus can not be directly observed, Tj(N ) has been estimated from r (N) of macroscopic nuclei by optical microscopy by correcting the time necessary for growth from N to N. Therefore, x (N ) is named r (OM) in this work. It should be noted that there is no guarantee that the estimated Xi(N ) = r (OM) is correct, that is also an important unresolved problem. 

PE crystallizes into the orthorhombic form from the melt, which results in folded chain crystals (FCCs) [14,15]. In this case, the critical nucleus should... 

Classical nucleation theory (CNT) shows that I is a product of the probability of diffusion and that of formation of a critical nucleus [1,4],... 

This means that the Mn dependence of I is controlled by the diffusion process of polymer chains and not by the formation process of a critical nucleus. 

The reason why AG does not depend on M is because only part of the molecular length of one chain is included within a critical nucleus in this study. This means that only a partial length of one chain forms a critical nucleus. In other words, chain ends are not significantly included within a critical nucleus. Therefore, the whole length does not assume an important role in the formation of a critical nucleus. This is quite different from the case of n-paraffin or an oligomer system [35,36]. 

It should be noted that the critical nucleation process does not depend on M. This can be explained by our model of surface diffusion (Fig. 27). In the model a nucleus will be formed from the absorbed chains. We can estimate the number of repeating units within a critical nucleus (N ) using parameters a, ae, and Ah given in [14]. N is the order of 102-103 for the range of AT in our experiment, which is much smaller than the number of repeating units within a molecule (103-104). This indicates that a critical nucleus should be formed by a part of a molecular chain. Therefore, the nucleation process of the critical nucleus will not depend on M. Thus, it is a natural result that B does not depend on M in this study. This is consistent with the discussion by Hoffman et al. [28] on FCC. They showed that the nucleation process of an FCC does not depend on Mn in the case of Mn > 104. On the contrary they showed that it depends on Mn for Mn < 104, because ae depends on Mn due to the effect of chain ends on the end surface of the 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. 

There is a point at which these aggregates reach a critical size of minimum stability r and the free energy of formation AG is a maximum. Further addition of material to the critical nucleus decreases the free energy and produces a stable growing nucleus. The nucleation rate is the product of the concentration of critical nuclei N given by... 

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