Nucleation solid state

The Solid State Mechanisms of Nucleation, Solid State Diffusion, Growth of Particles and Measurement of Solid State Reactions... 

Dislocation theory as a portion of the subject of solid-state physics is somewhat beyond the scope of this book, but it is desirable to examine the subject briefly in terms of its implications in surface chemistry. Perhaps the most elementary type of defect is that of an extra or interstitial atom—Frenkel defect [110]—or a missing atom or vacancy—Schottky defect [111]. Such point defects play an important role in the treatment of diffusion and electrical conductivities in solids and the solubility of a salt in the host lattice of another or different valence type [112]. Point defects have a thermodynamic basis for their existence in terms of the energy and entropy of their formation, the situation is similar to the formation of isolated holes and erratic atoms on a surface. Dislocations, on the other hand, may be viewed as an organized concentration of point defects they are lattice defects and play an important role in the mechanism of the plastic deformation of solids. Lattice defects or dislocations are not thermodynamic in the sense of the point defects their formation is intimately connected with the mechanism of nucleation and crystal growth (see Section IX-4), and they constitute an important source of surface imperfection. 

Nucleation in solids is very similar to nucleation in liquids. Because solids usually contain high-energy defects (like dislocations, grain boundaries and surfaces) new phases usually nucleate heterogeneously homogeneous nucleation, which occurs in defect-free regions, is rare. Figure 7.5 summarises the various ways in which nucleation can take place in a typical polycrystalline solid and Problems 7.2 and 7.3 illustrate how nucleation theory can be applied to a solid-state situation. 

In solid state reactions, the rate of nucleation may be given by either of the expressions dN/dt = const, or dN/dt = t° const. For both expressions, the probability (pdf) is proportional to the total volume of the spherical layers at the instant t at the peripheries of nuclei which originated at time r. The radii of the spheres at the inner and outer boundaries of these layers are... 

Solar energy, 6, 488 surface modified electrodes, 6, 30 Sol-Gel process fast reactor fuel, 6, 924 Solid state reactions, 1, 463-471 fraction of reaction, 1, 464 geometric, 1, 464 growth, 1, 464 nucleation, 1, 464 rate laws, 1,464 Solochrome black T metallochromic indicators, 1,555 Solubility... 

Note also that we have just introduced the concepts of nuclei and nucleation in our study of solid state reaction processes. Our next step will be to examine some of the mathematics used to define rate processes in solid state reactions. We will not delve into the precise equations here but present them in Appendices at the end of this chapter. But first, we need to examine reaction rate equations as adapted for the solid state. 

Three types of rate equations are shown here. These rate equations ean be used for quite complieated reactions, but a specific method or measurement approach is needed. How we do this is critical to determining accurate estimation of the progress of a solid state reaction. We will discuss suitable methods in another chapter. We now return to the subject of nucleation so that we can apply the rate equations given above to specific cases. First, we examine heterogeneous processes. 

If we are interested in the nucleation of apeu-ticle prior to completing the solid state reaction, we need to distinguish between surface and volume nucleation of the particle, since these are the major methods of which we can perceive. Several cases are shown in the following diagram. 

Solid state reactions are also very common in producing oxide materials and are based on thermal treatment of solid oxides, hydroxides and metal salts (carbonates, oxalates, nitrates, sulphates, acetates, etc.) which decompose and react forming target products and evolving gaseous products. Solid-state chemistry states that, like in the case of precipitation, powder characteristics depend on the speed of the nucleation of particles and their growth however, these processes in solids are much slower than in liquids. 

Confinement effects may also be employed to characterize the nucleation and growth of porous materials [211]. The underlying mechanisms of self-assembly and crystallization of these complex heterogeneous systems may be traced by solid state NMR methods well before their detection by diffraction methods. 

The rate of nucleation is dependent on the degree of supersaturation as described in section 2.4.1, and because this will always be larger for Form 1 it may be incorrectly assumed that Form I will always precipitate first. The true situation is somewhat more complicated because the critical size, activation energy and nucleation rate also depend on the solid state that is being formed [6]. It is quite feasible and a regular occurrence, that a less stable polymorph will have a higher rate of nucleation than a more stable form, as illustrated in figure 6. 

Experimental determination of Ay for a reaction requires the rate constant k to be determined at different pressures, k is obtained as a fit parameter by the reproduction of the experimental kinetic data with a suitable model. The data are the concentration of the reactants or of the products, or any other coordinate representing their concentration, as a function of time. The choice of a kinetic model for a solid-state chemical reaction is not trivial because many steps, having comparable rates, may be involved in making the kinetic law the superposition of the kinetics of all the different, and often unknown, processes. The evolution of the reaction should be analyzed considering all the fundamental aspects of condensed phase reactions and, in particular, beside the strictly chemical transformations, also the diffusion (transport of matter to and from the reaction center) and the nucleation processes. 

As has already been noted, polymerization is a common output of high-pressure reactions. The kinetics of solid-state pressure-induced polymerizations have been treated within the nuclei growth [see eq. (17)] model. These reactions, as we will discuss in Section IV, are a typical example of how the crystal structure plays a fundamental role in sohd-state chemistry. Kinetic data of polymerizations are usually analyzed according to Eq. (17) by inserting an additional parameter fo accounting for the nucleation step ... 

A kinetic model for single-phase polymerizations— that is, reactions where because of the similarity of structure the polymer grows as a solid-state solution in the monomer crystal without phase separation—has been proposed by Baughman [294] to explain the experimental behavior observed in the temperature- or light-induced polymerization of substimted diacetylenes R—C=C—C=C—R. The basic feature of the model is that the rate constant for nucleation is assumed to depend on the fraction of converted monomer x(f) and is not constant like it is assumed in the Avrami model discussed above. The rate of the solid-state polymerization is given by... 

Aspartame is relatively unstable in solution, undergoing cyclisation by intramolecular self-aminolysis at pH values in excess of 2.0 [91]. This follows nucleophilic attack of the free base N-terminal amino group on the phenylalanine carboxyl group resulting in the formation of 3-methylenecarboxyl-6-benzyl-2, 5-diketopiperazine (DKP). The DKP further hydrolyses to L-aspartyl-L-phenyl-alanine and to L-phenylalanine-L-aspartate [92]. Grant and co-workers [93] have extensively investigated the solid-state stability of aspartame. At elevated temperatures, dehydration followed by loss of methanol and the resultant cyclisation to DKP were observed. The solid-state reaction mechanism was described as Prout-Tompkins kinetics (via nucleation control mechanism). 

The higher the relative supersaturation, the more likely nucleation becomes, and the faster crystal growth proceeds. Molecules or ions will remain dissolved provided that the conditions are energetically favorable. However, all molecules above a certain threshold solution activity will remain in, or become part of, the solid phase. Molecules are in equilibrium between the solid state and the dissolved state. The extent to which the equilibrium balance favors the dissolved phase indicates the degree of solubility. 

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