Solid state reactions nucleation

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. 

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). 

Metal oxidation is a heterogeneous solid state reaction and starts in the same way as other heterogeneous reactions with nucleation and initial growth. This was discussed in Chapter 6. A time-dependent nucleation rate may dominate the overall growth kinetics of thin Films. Even under an optical microscope (i.e., in macroscopic dimensions), preferential sites of growth can still be discerned [J. Benard (1971)). This indicates that lateral transport on the surface (e.g., at sites where screw dislocations emerge) can possibly be more important for the initial reactive growth than transport across thin oxide layers. 

A similar effect has been reported in the crystallization of non-chiral molecules, where the presence of small amounts of chiral additive forces the entire system to crystallize in an enantiomorphous crystal, which upon further solid-state reaction can be converted into polymers of a single handedness [184,185]. Chiral auxiliaries, which affect crystal nucleation enantios-electively, have been successfully used for the large-scale optical resolution of enantiomers [186-188]. 

For a solid-state reaction, one of the solutions of Equation 3.1 is the Avrami-Erofeev equation [3], The phase transition model that derives this equation supposes that the germ nuclei of the new phase are distributed randomly within the solid following a nucleation event, grains grow throughout the old phase until the transformation is complete. Then, the Avrami-Erofeev equation is [3]... 

In the solid-state reaction, nucleation and growth have a fundamental role, because, in essence, the solid-state reaction is a phase transformation. In this type of reactions, nucleation and growth follow similar principles as those previously analyzed in Section 3.1 the principal difference being the increased role of diffusion in solid-state reactions [30],... 

In the last decade, progress has also been made with using superlattices as templates, or structure-directing agents, to kinetically control solid-state reactions. This is accomplished by allowing interdiffusion to reach completion before the occurrence of heterogeneous nucleation, thus trapping the system in the... 

Reactivity in the solid-state is always connected with specific motions which allow the necessary contact between the reacting groups. In most cases solid-state reactions proceed by diffusion of reactions to centers of reactivity or by nucleation of the product phase at certain centers of disorder. This leads to the total destruction of the parent lattice. If the product is able to crystallize it is highly probable that nucleation of the crystalline product phase at the surface of the parent lattice will lead to oriented growth under the influence of surface tension. In such topotactic reactions certain crystallographic directions of parent and daughter phases will coincide. Typical examples for this behaviour are the solid-state polymerizations of oxacyclic compounds such as trioxane, tetroxane or 3-propiolactone... 

These authors observed that binaphthyl crystallize in two polymorphs. The one is stable at lower temperature, is centrosymmetric and is not optically active. This polymorph melts at 145 °C. The second polymorph is stable at higher temperature but is metastable at room temperature. It is optically active and melts at 158 °C. Wilson and Pincock show that as one cycles in temperature between room temperature and 150 °C a sample which is initially the optically inactive low temperature polymoiph transforms to an optically active solid. After three or four cycles one achieves the maximum optical resolution which corresponds to 56% ee. The crux of the Wilson and Pincock experiment is that at 150 °C, the reaction physically resembles a solid state reaction in which the low temperature form is melting in the near presence of high temperature polymorph crystals. These chiral crystals are therefore nucleating sites for further chiral crystal growth. As at room temperature binaphthyl retains its chirality, the resultant samples can then be dissolved with retention of stereochemistry. 

The geometric models of solid state reactions are based upon the processes of nucleation and growth of product nuclei by interface advance. These processes are discussed individually in the next section, followed by a description of the ways in which these contributions are combined to give rate equations for the overall progress of reaction. 

The kinetics of many solid state reactions have been reported as being satisfactorily represented by the first-order rate equation [70] (which is also one form of the Avrami-Erofeev equation (n = 1)). Such kinetic behaviour may be expected in decompositions of fine powders if particle nucleation occurs on a random basis and growth does not advance beyond the individual crystallite nucleated. 

The factors governing the slow thermal decompositions of inorganic azides have been discussed by Fox and Hutchinson [18]. They draw attention to the interest shown in early studies for fitting of kinetic results to rate equations based on nucleation and growth models. Support of kinetic interpretations by microscopic observations (e g., [21]), contributed significantly towards establishment of the role of the active, advancing interface in solid state reactions. The kinetic characteristics of some of the metal azides are summarized in Table 11.1. 

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