Kinetic shape

Because the parameter depends on temperature, crystal habit will normally change when the growth temperature changes drastically. Crystal defects (dislocations, twins, and inclusions) are also responsible for morphological changes. The flow of solution around a crystal also influences its shape as is discussed in the next section. But the most important factor that can be used to change crystal habit is the addition of impurities to the precipitating system. 

FIGURE 6.19 Comparison of hypothetical growth rates, G, tnd G2, for two F faces as a fimction of the saturation ratio, S. Redrawn, with permission from Elwell and Scheel [28], 

FIGURE 6.20 Diffusion controled growth of a square ciystal. The left-hand side shows a hypothetical concentration gradient around the crystal. The ri t-hand side shows the growing crystel with time. Redrawn, with permission from Nielsen [2]. 

The surface of a screw dislocation will convert an F face to the shape of a pjramid or a cone. The angle, 6, of the cone can be calculated by [51] 

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Many developmental phenomena depend on a sequence of patterns, for example from simple extending tip growth to branching. RD theory provides a means for understanding the kinetic constraints involved in such symmetry-breaking transitions. The development of RD theory for growing domains, in conjunction with experimental tests, illuminates how chemical kinetics shape the plants around us, from ferns to spruce trees. 

The transformatioiis that create a new solid phase present any of the kinetic shapes of curves. For the same reaction, it is even possible to switch from one family of curves to the other by modifying a variable, such as the particle size of a powder. Moreover, one should not exaggerate the difference between both families a displacement of the point of inflection in one direction or the other makes it possible to switch fiom the sigmoid curves to any of the curves shown in Figure 1.8a, and this can occur by modifying the values of certain variables such as the partial pressures, the temperature, or the shape and size of particles in the case of a powder. 

Since the crystal shape, or habit, can be determined by kinetic and other nonequilibrium effects, an actud crystal may have faces that differ from those of the Wulff construction. For example, if a (100) plane is a stable or singular plane but by processing one produces a plane at a small angle to this, describable as an (xOO) plane, where x is a large number, the surface may decompose into a set of (100) steps and (010) risers [39]. 

Qualitative examples abound. Perfect crystals of sodium carbonate, sulfate, or phosphate may be kept for years without efflorescing, although if scratched, they begin to do so immediately. Too strongly heated or burned lime or plaster of Paris takes up the first traces of water only with difficulty. Reactions of this type tend to be autocat-alytic. The initial rate is slow, due to the absence of the necessary linear interface, but the rate accelerates as more and more product is formed. See Refs. 147-153 for other examples. Ruckenstein [154] has discussed a kinetic model based on nucleation theory. There is certainly evidence that patches of product may be present, as in the oxidation of Mo(lOO) surfaces [155], and that surface defects are important [156]. There may be catalysis thus reaction VII-27 is catalyzed by water vapor [157]. A topotactic reaction is one where the product or products retain the external crystalline shape of the reactant crystal [158]. More often, however, there is a complicated morphology with pitting, cracking, and pore formation, as with calcium carbonate [159]. 

If adsorbed electroactive species are present on the electrode surface, the shape of the cyclic voltaimnogram changes, since the species do not need to difflise to the electrode surface. In this case the peaks are syimnetrical with coincident peak potentials provided the kinetics are fast. 

A completely different approach, in particular for fast imimolecular processes, extracts state-resolved kinetic infomiation from molecular spectra without using any fomi of time-dependent observation. This includes conventional line-shape methods, as well as the quantum-dynamical analysis of rovibrational overtone spectra [18, 33, 34 and 35]. 

As the air leaves the blade tips, it contains kinetic energy by virtue of its velocity. The directional component of this velocity is both rotative and radial. When the fan blades ate inclined forward, these components are cumulative. With backward-iaclined blades, the components are ia opposition. The purpose of the fan volute or scroU-shaped casiag is to convert a portion of the kinetic energy of the air leaving the blades iato static pressure. 

Adsorption Kinetics. In zeoHte adsorption processes the adsorbates migrate into the zeoHte crystals. First, transport must occur between crystals contained in a compact or peUet, and second, diffusion must occur within the crystals. Diffusion coefficients are measured by various methods, including the measurement of adsorption rates and the deterniination of jump times as derived from nmr results. Factors affecting kinetics and diffusion include channel geometry and dimensions molecular size, shape, and polarity zeoHte cation distribution and charge temperature adsorbate concentration impurity molecules and crystal-surface defects. 

Volumetric heat generation increases with temperature as a single or multiple S-shaped curves, whereas surface heat removal increases linearly. The shapes of these heat-generation curves and the slopes of the heat-removal lines depend on reaction kinetics, activation energies, reactant concentrations, flow rates, and the initial temperatures of reactants and coolants (70). The intersections of the heat-generation curves and heat-removal lines represent possible steady-state operations called stationary states (Fig. 15). Multiple stationary states are possible. Control is introduced to estabHsh the desired steady-state operation, produce products at targeted rates, and provide safe start-up and shutdown. Control methods can affect overall performance by their way of adjusting temperature and concentration variations and upsets, and by the closeness to which critical variables are operated near their limits. 

This development has been generalized. Results for zero- and second-order irreversible reactions are shown in Figure 10. Results are given elsewhere (48) for more complex kinetics, nonisothermal reactions, and particle shapes other than spheres. For nonspherical particles, the equivalent spherical radius, three times the particle volume/surface area, can be used for R to a good approximation. 

Population balances and crystallization kinetics may be used to relate process variables to the crystal size distribution produced by the crystallizer. Such balances are coupled to the more familiar balances on mass and energy. It is assumed that the population distribution is a continuous function and that crystal size, surface area, and volume can be described by a characteristic dimension T. Area and volume shape factors are assumed to be constant, which is to say that the morphology of the crystal does not change with size. 

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