Surface nucleus critical radius

Figure 17. Energy for the nucleation of a surface film on metal electrode. M, metal OX, oxide film EL, electrolyte solution. Aj is the activation barrier for the formation of an oxide-film nucleus and rj is its critical radius. 7 a is the interfacial tension of the metal-electrolyte interface, a is the interfacial tension of the film-electrolyte interface. (From N. Sato, J. Electro-chem. Soc. 129, 255, 1982, Fig. 5. Reproduced by permission of The Electrochemical Society, Inc.)...

The critical radius, R is the size where the embryo (nucleus) has a 50 50 chance of either redissolving or growing into a stable nucleus it is determined by the balance between the surface energy required to form the embryo,... 

It is clear from Equations (9.1) to (9.4) that the free energy of formation of a nucleus and the critical radius r, above which the cluster formation grows spontaneously, depend on two main parameters, namely a and (S/S ), both of which are influenced by the presence of surfactants, a is influenced in a direct way by the adsorption of surfactant onto the surface of the nucleus this adsorption lowers y and this in turn reduces r and AG in other words, spontaneous cluster formation will occur at a smaller critical radius. In addition, surfactant adsorption stabilises the nuclei against any flocculation. The presence of micelles in solution also affects the processes ofnucleation and growth, both directly and indirectly. For example, the micelles can act as nuclei on which growth may occur, and may also solubilize the molecules of the material this can affect the relative supersaturation and, in turn, may have an effect on nucleation and growth. 

The discussion above applies to heterogeneous nucleation on flat surfaces. For atmospheric particles the curvature of the surface complicates the situation. Fletcher (1958) showed that the free energy of formation AG of an embryo of critical radius r on a spherical nucleus of radius R is given by... 

They indicate that the wurtzite stmcture is more stable than the rocksalt one for small nuclei. This property was shown to stem from the fact that y (the dominant term for small nuclei) is lower in the wurtzite structure by 10 %, although Ag is larger by 20 %. The authors explained the lower values of y for the wurtzite structure by its stronger ability to solvate its surface, indicated by the higher number of hydrogen bonds formed at the interface. The difference in y turned out to be even more pronounced (70%) in the case of an interface with pure water, instead of the solution. The CNT provides an estimated critical radius of about 5.7 nm, below which the wurtzite nucleus is more stable. The conversion to the rocksalt phase should thereby occur when the nucleus exceeds this size. 

The first term on the right is positive, as energy is expended to form the surface of the crystal, and the second term is negative. An increase in the size of the crystal will decrease the free energy of the system only if d Ag/dr < 0, so the critical radius of the nucleus (i.e., the size beyond which it will grow spontaneously) is found by setting d Ag/dr = 0 in Eq. 4. The result is... 

Nucleation energy barriers depend upon the surface and volume energies associated with the reaction. A critical radius for the nucleus exists beyond which the new phase may grow spontaneously. 

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