Interfacial energy nucleation process

Heterogeneous nucleation, however, is in many cases the predominant formation process for crystals in natural waters. In a similar way as catalysts reduces the activation energy of chemical reaction, foreign solids may catalyze the nucleation process by reducing the energy barrier. Qualitatively, if the surface of the solid substrate matches well with the crystal, the interfacial energy between the two solids is smaller than the interfacial energy between the crystal and the solution, and nucleation may take place at a lower saturation ratio on a solid substrate surface than in solution. 

The above considerations show that the interfacial energy is of utmost importance in determining the thermodynamics and kinetics of the nucleation process. Unfortunately, however, there are considerable uncertainities on the values of interfacial free energies. Values determined from contact angle measurements are significantly lower than those determined from the dependence of solubility upon molar surface of the crystallites. Furthermore, reliable data on yes are lacking. 

The theory that describes these fluctuations was provided by Einstein and Smoluchowski. The nucleation rate depends on the interfacial energy and the supersaturation, which influence the differences in the chemical potential of both phases. This theory employs the fact that the nucleation rate is the same in the whole volume under consideration, i.e., so-called homogenous nucleation (Kondepudi 2008). The kinetic factor of the process can be expressed by (20.1) the nucleation rate (ki) that affects (20.2) cluster growth rate (kj) (Figure 20.4) ... 

The nucleation process has been discussed above in terms of the so-called classical theories stemming from the thermodynamic approach of Gibbs and Volmer, with the modifications of Becker, Doring and later workers. The main criticism of these theories is their dependence on the interfacial tension (surface energy), 7, e.g. in the Gibbs-Thomson equation, and this term is probably meaningless when applied to clusters of near critical nucleus size. 

It has been indicated above, e.g. equation 5.9, that the interfacial tension, 7, is one of the important factors controlling the nucleation process. Figure 5.7 shows an interfacial energy diagram for three phases in contact in this case, however, the three phases are not the more familiar solid, liquid and gas, but two solids and a liquid. The three interfacial tensions are denoted by 7d (between the solid crystalline phase, c, and the liquid 1), 7si (between another foreign solid surface, s, and the liquid) and 7cs (between the solid crystalline phase and the foreign solid surface). Resolving these forces in a horizontal direction... 

Formation of such droplets must then be an activated process whose rate is proportional to exp [—AF /(A 7 ]. We can estimate this rate using equation (4.2.6) for the interfacial energy y, and the result is that the rate of homogeneous nucleation we should expect for polymer systems is vanishingly small. In practice nucleation is usually aided by the presence of other interfaces, for example impurity particles such as dust or the container walls may well be able to nucleate critical droplets with much lower activation energies (heterogeneous nucleation) or indeed with no activation energy at all. We will return to this subject in section 5.3 when we discuss the effects of surfaces on phase separation. 

The persistence length is determined by the ratio of the growth rate (secondary nucleation rate (coi). The LH theory predicts a change in the growth rate because temperature dependent whereas a> is strongly dependent. Alternative explanations for changes in rate have been put forward and include effects such as the temperature dependence of the interfacial energies, viscosity effects and nucleation processes. 

The first exponential term represents the transport of molecules to the growing nucleus with U the activation energy for this process and (T - To) the temperature difference between the crystallization temperature T and the temperature at which backbone motions substantially cease. For most polymers To is about 50°C below the glass transition temperature. The second exponential term represents the work required to form a critical nucleus where TS, is the melting point of an infinitely thick lamellar crystal, AT is the supercooling and AH the heat of fusion. The term C contains various interfacial energy terms and depends upon the precise mechanism of the nucleation process. For homogeneous nucleation... 

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