Nucleation energy cost

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... 

As a final remark it must be mentioned that theoretical and experimental works have been dedicated to investigating the effect of the finite size of the chains [65]. In fact, as grows exponentially, at low temperatures it can become comparable with the distance between two consecutive defects (e.g. impurities and vacancies) which are always present in real systems and hardly separated by more than 103 -104 elementary units. In case of Z < , the nucleation of the DW is energetically favoured if occurring at the boundaries, because the energy cost is halved. However the probability to have a boundary spin is inversely proportional to L thus the pre-exponential factor becomes linearly dependent on L, as experimentally found in doped SCMs. As doping occurs at random positions on the chain, a distribution of lengths is observed in a real system. However, as the relaxation time is only linearly dependent on L, a relatively narrow distribution is expected. 

Rgure 5.2. Scheme of the hole nucleation process and variation of the energy cost with hole radius r. 

At a microscopic scale, a single coalescence event proceeds through the nucleation of a thermally activated hole that reaches a critical size, above which it becomes unstable and grows [29]. We shall term E(r) the energy cost for reaching a hole of size r. A maximum of E occurs at a critical size r, E r ) = Ea being the activation energy of the hole nucleation process (Fig. 5.2). 

A central assertion of homogeneous nucleation theory is that interfacial free energy costs induce a spherical symmetry in the phase embryo. However, these simulation studies indicate that inter molecular interactions may not permit the development of spherical symmetry when these interactions are strong and highly asymmetric. 

For crystals to form from a liquid state, the molecules of the crystallizing species must come together in sufficient number (form a cluster) to overcome the energy cost of forming a surface. Once this energy barrier is overcome, the latent heat associated with crystallization is released, and further nucleation is strongly promoted. Thus, there often is a metastable zone where a supersaturated or subcooled system may not nucleate for a very long time. 

The energy cost to create surfaces/interfaces leads to a nucleation barrier in condensed-matter phase transformations. Therefore, nucleation-based phase transformations can only occur if the energy released by creating the new volume of the second phase sufficiently offsets the energy expended in creating the new interfacial area. This leads to a minimum viable nucleation size and thus helps determine the speed at which nucleation can proceed. These issues will be discussed in the next section ... 

Nucleation Theory. Classical theories for nucleation in small molecules balance the reduction in free energy that occurs because the solid is more stable than the supercooled liquid against a surface term that accounts for the free-energy cost of creating a solid-liquid interface. For a spherical crystalhte this is given by... 

The dislocation loop nucleation energy increases as the dislocation energy increases. Thus, relatively soft materials such as II-VI semiconductors would be more likely to nucleate misfit dislocations by loop formation than would harder materials such as Si. Materials with small misfit strains are less likely to form dislocations by loop expansion, because the loop would have to be much larger before the energy payback from strain relief would compensate for the cost of the loop. Stress concentrators can reduce the loop nucleation barrier, although in many materials such concentrators are rare. If the effective additional stress associated with the concentration point adds to the misfit stress, the local energy barrier to loop nucleation can be reduced or eliminated. 

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