Determination of Nucleation and Growth Kinetics

The inverse problems discussed in Sections 6.1 and 6.2 were addressed in the absence of nucleation and growth processes. In this section we investigate inverse problems for the recovery of the kinetics of nucleation and growth from experimental measurements of the number density. It is assumed, however, that particle break-up and aggregation processes do not occur. Determination of nucleation and growth rates is of considerable practical significance since the control of particle size in crystallization and precipitation processes depends critically on such information. We will dispense with the assumption of self-similar behavior, as it is often not observed in such systems. Also, we provide here only a preliminary analysis of this problem, as it is still in the process of active investigation by Mahoney (2000). 

Nucleation includes primary as well as secondary nucleation. In primary nucleation, there is spontaneous generation of new particles in the absence of existing particles. Secondary nucleation includes the formation of nuclei in the neighborhood of an existing particle, and those microparticles that are formed by breakage by impact with the impeller or the container walls (in other words, first-order processes), or by particle-particle collisions (second-order processes). Denoting the supersaturation at time t by cr(t), we may represent the primary nucleation rate by the particle growth 

The left-hand side represents the total nucleation rate as an explicit function of time alone in view of the number density and the supersaturation being functions of time. The function characterizes first-order nucleation processes while P2 represents second-order nucleation processes. 

The boundary condition in (6.3.3) reflects the assumption that the nuclei are assumed to be of size zero. The problem is to estimate at first, given measurements of the number density t), the functions P[t] and L[/, t]. The particle growth rate kinetics ijj, a) can be obtained from L[/, t] by estimating the supersaturation from measurements of appropriate concentrations. In order to obtain the primary and secondary nucleation rates, it is necessary to calculate the functions j i(/, a) and )- This is a 

20 It is possible that the mechanism for creating or renewing supersaturation may include continuous bubbling of, say a gas. Thus, the batch system considered here refers to the system being closed with respect to the particle slurry. 

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Michaelson C, Dahms M. On the determination of nucleation and growth kinetics by calorimetry. Thermochim Acta 1996 288 9-27. 

Since t is fixed by the choice of experimental conditions, nucle-ation and growth rates can be determined simultaneously by a single measurement of the CSD at steady-state. Example 4.3 shows the use of the population balance in determination of nucleation and growth kinetics. 

This means, that the epitaxy of 2D Me overlayer is reflected in the epitaxy of 3D Me crystallites, as summarized in Table 1 [11,18]. The structural and energetic properties of 2D Me overlayers are found to determine the nucleation and growth kinetics of 3D Me phases [11, 15,18]. 

Direct observation of individual nuclei is now possible using in-situ atomic force microscopy (AFM) techniques [32] that allow the determination of nucleation and growth rates of Pb nuclei on boron-doped diamond electrodes as a function of time and overpotential. In-situ tapping-mode AFM proved to be an efficient tool for imaging the electrodeposition process at the micron scale, and showed an agreement between the kinetic parameters... 

Many industrial crystallizers operate in a weU-mixed or nearly weU-mixed manner, and the equations derived above can be used to describe their performance. Furthermore, the simplicity of the equations describing an MSMPR crystallizer make experimental equipment configured to meet the assumptions lea ding to equation 44 useful in determining nucleation and growth kinetics in systems of interest. 

The kinetics of CO oxidation from HClOi, solutions on the (100), (111) and (311) single crystal planes of platinum has been investigated. Electrochemical oxidation of CO involves a surface reaction between adsorbed CO molecules and a surface oxide of Pt. To determine the rate of this reaction the electrode was first covered by a monolayer of CO and subsequently exposed to anodic potentials at which Pt oxide is formed. Under these conditions the rate of CO oxidation is controlled by the rate of nucleation and growth of the oxide islands in the CO monolayer. By combination of the single and double potential step techniques the rates of the nucleation and the island growth have been determined independently. The results show that the rate of the two processes significantly depend on the crystallography of the Pt surfaces. 

System conditions often allow for the measurement of magma density, and in such cases is should be used as a constraint in evaluating nucleation and growth kinetics from measured population densities. This approach is especially useful in instances of uncertainty in the determination of population densities from sieving or other particle sizing techniques. 

In essence, Eq. (16) describes the formation of a two-phase structure in reactive systems, which takes place according to the mechanism of nucleation and growth under the condition that an increase in concentration of the second phase is determined by the chemical reactions. Crystallization may serve as a physical analogue for such a process. Indeed, in Refs. [124,125] a new model of crystallization kinetics was developed, which is reduced to a self-acceleration equation similar to Eq. (16). 

To truly control crystallization to give the desired crystalline microstructure requires an advanced knowledge of both the equilibrium phase behavior and the kinetics of nucleation and growth. The phase behavior of the particular mixture of TAG in a lipid system controls both the driving force for crystallization and the ultimate phase volume (solid fat content) of the solidified fat. The crystallization kinetics determines the number, size, polymorph, and shape of crystals that are formed as well as the network interactions among the various crystalline elements. There are numerous factors that influence both the phase behavior and the crystallization kinetics, and the effects of these parameters must be understood to control lipid crystallization. 

Early (1930 to 1940) kinetic studies of dehydrations contributed much to the development of the concept of the reaction interface as the important feature of nucleation and growth reactions [2]. Kinetic equations applicable to the decompositions of a vnde range of crystalline substances were developed. Large, well-formed crystals of hydrates could be prepared relatively easily and studies of these were particularly rewarding. The interpretation of kinetic data was supplemented by microscopic evidence concerning the formation and development of product nuclei. Recent work on dehydrations has included more precise determinations of the crystal structures of reactants, products and their interrelationships, including interface textures, in the attempt to resolve unanswered questions. 

However, Ostwald s law of stages is not universally valid because the appearance and evolution of solid phases are determined by the kinetics of nucleation and growth under the specific experimental conditions (Bernstein et ak, 1999 Davey et ak, 1997 Threlfall, 2000) and by the link between molecular assemblies and crystal structure (Davey et ak, 2002 Blagden et ak, 1998 Gu et ak, 2001). 

In the formation of a crystal two steps are required (1) the birth of a new particle and (2) its growth to macroscopic size. The first step is called nucleation. In a crystallizer, the CSD is determined by the interaction of the rates of nucleation and growth, and the overall process is complicated kinetically. The driving potential for both rates is supersaturation, and neither crystal growth nor formation of nuclei from the solution can occur in a saturated or unsaturated solution. Of course, very small crystals can be formed by attrition in a saturated solution, and these may act just like new nuclei as sites for further growth if the solution later becomes supersaturated. 

Upon successful fit of data the nucleation and growth kinetics can be determined. 

The structure and interrelationship of the batch conservation equations (population, mass, and energy balances) and the nucleation and growth kinetic equations are illustrated in an information flow diagram shown in Figure 10.8. To determine the CSD in a batch crystallizer, all of the above equations must be solved simultaneously. The batch conservation equations are difficult to solve even numerically. The population balance, Eq. (10.3), is a nonlinear first-order partial differential equation, and the nucleation and growth kinetic expressions are included in Eq. (10.3) as well as in the boundary conditions. One solution method involves the introduction of moments of the CSD as defined by... 

The phenomenon of polymorphism demonstrates that metastable erystal struetures are observed, and it is not always obvious that sueh crystal structures are metastable. The energy differences between different polymorphs crystallized out of different solvent are small, and those between concomitant polymorphs presumably are very small. Kinetics must play a major role in determining which of the approximately equi-energetic hypothetical crystal structures are actually observed. How do the kinetics of nucleation and growth, and the variations with crystallization conditions, affect which thermodynamically feasible crystal structures are actually seen How can this be incorporated in the crystal structure prediction model to produce a polymorph prediction model ... 

Besides agglomeration and attrition, the rates of nucleation and growth are the main kinetic parameters that determine the size distribution. These parameters, which are reqnired in crystallizer design and simulation, can be determined on the basis of the population balance equations. Section 64.2 discusses the methods for controlling the particle size and particle size distribution. 

Hu WB (2005) Molecular segregation in polymer melt crystallization simulation evidence and unified-scheme interpretation. Macromolecules 38 8712-8718 Hu WB, Cai T (2008) Regime transitions of polymer crystal growth rates molecular simulations and interpretation beyond Lauritzen-Hoffman model. Macromolecules 41 2049-2061 Jeziomy A (1971) Parameters characterizing the kinetics of the non-isothermal crystallization of poly(ethylene terephthalate) determined by DSC. Polymer 12 150-158 Johnson WA, Mehl RT (1939) Reaction kinetics in processes of nucleation and growth. Trans Am Inst Min Pet Eng 135 416-441... 

The perfectly mixed, continuous, steady-state mixed-suspension mixed-product removal (MSMPR) crys-tttllizer is restricrive in the degree to which characteristics of a crystal size distribution can be varied. Indeed, examination of Eqs. (11.2-32) and (11.2-40) shows that once nucleation and growth kinetics are fixed in these systems the ciysral size distribution is determined in its entirely. In addition, such distributions have the following characteristics ... 

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