Kinetics, crystal growth description

The development and refinement of population balance techniques for the description of the behavior of laboratory and industrial crystallizers led to the belief that with accurate values for the crystal growth and nucleation kinetics, a simple MSMPR type crystallizer could be accurately modelled in terms of its CSD. Unfortunately, accurate measurement of the CSD with laser light scattering particle size analyzers (especially of the small particles) has revealed that this is not true. In mar cases the CSD data obtained from steady state operation of a MSMPR crystallizer is not a straight line as expected but curves upward (1. 32. 33V This indicates more small particles than predicted... 

All seven steps require time, resulting in a rate of incorporating clusters into the growing crystal surface, which is called crystal growth kinetics. The following two sections consider translation of such a rate into a macroscopic equation for correlation and prediction. It is difficult to say which of the steps control the process, or even if the conceptual picture is valid. However, the first step—species transport to the solid surface—is well established and a brief description is given in Section 3.2.1.2. 

In the description of crystals and crystal structures the two terms/om and habit have very specific and very different meanings. Form refers to the internal crystal structure and etymologically is the descendant of the Greek morph. Hence, polymorph refers to a number of different crystal modifications or different crystal structures, and the naming of different structures as Form F or a Form follows directly from this definition and usage. As we have seen above, the difference in crystal structure is very much, although not exclusively, a function of thermodynamics. Certainly, only the structures which are thermodynamically accessible can ever exist, but there often is a question of thermodynamic vs kinetic control over which particular structure may be obtained under any particular set of crystal growth conditions. 

Avrami Analysis The Avrami equation, a general approach for description of isothermal phase transformation kinetics originally developed for polymers (46), is often used for describing nucleation and crystal growth in fats. The Avrami equation is given as... 

Size-dependent structure and properties of Earth materials impact the geological processes they participate in. This topic has not been fully explored to date. Chapters in this volume contain descriptions of the inorganic and biological processes by which nanoparticles form, information about the distribution of nanoparticles in the atmosphere, aqueous environments, and soils, discussion of the impact of size on nanoparticle structure, thermodynamics, and reaction kinetics, consideration of the nature of the smallest nanoparticles and molecular clusters, pathways for crystal growth and colloid formation, analysis of the size-dependence of phase stability and magnetic properties, and descriptions of methods for the study of nanoparticles. These questions are explored through both theoretical and experimental approaches. 

There have been many attempts to develop theories to explain the important aspects of crystaUizatirMi [3,4]. The most widely accepted approach to the analysis of the linear crystal growth rates is the kinetic description due to Laur-itzen and Hoffman [3]. There are alternative approaches which will not be considered here since this is not meant to be a comprehensive review chapter of theoretical approaches. 

It has been emphasized that the validity of these kinetic models as descriptions of crystal growth has yet to be provided. The advance of a crystal-liquid interface should depend critically on the detailed structure (on a molecular scale) of the interface. In considering crystal growth it is in general important to know the number and distribution of solid-like atoms on an interface plane as well as the population and form of molecular groups in the liquid. Unfortunately such structural details are very difficult to calculate on a statistical basis because the relevant statistics relate to small cluster size and limited numbers of configurations. 

The basic, macroscopic theories of matter are equilibrium thermodynamics, irreversible thermodynamics, and kinetics. Of these, kinetics provides an easy link to the microscopic description via its molecular models. The thermodynamic theories are also connected to a microscopic interpretation through statistical thermodynamics or direct molecular dynamics simulation. Statistical thermodynamics is also outlined in this section when discussing heat capacities, and molecular dynamics simulations are introduced in Sect 1.3.8 and applied to thermal analysis in Sect. 2.1.6. The basics, discussed in this chapter are designed to form the foundation for the later chapters. After the introductory Sect. 2.1, equilibrium thermodynamics is discussed in Sect. 2.2, followed in Sect. 2.3 by a detailed treatment of the most fundamental thermodynamic function, the heat capacity. Section 2.4 contains an introduction into irreversible thermodynamics, and Sect. 2.5 closes this chapter with an initial description of the different phases. The kinetics is closely link to the synthesis of macromolecules, crystal nucleation and growth, as well as melting. These topics are described in the separate Chap. 3. 

Recently the statistical approach was developed [5] for the description of the kinetics of conversion of melt to spherulites and the kinetics of formation of spherulitic pattern during both isothermal and nonisothermal crystallizations. The final spherulitic pattern can also be described. The rates of formation of spherulitic interiors and boundaries (boundary lines, surfaces and points) as well as the their final amounts could be predicted if spherulite growth and nucleation rates are known. Applied to iPP crystallized during cooling with various rates, the approach allowed for the predictions of tendencies in the kinetics of formation of spherulitic structure and its final form. 

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