Factors Determining Crystal Shape

Under conditions of very slow growth, the ultimate crystal shape is determined strictly by thermodynamics. Under such conditions, the faces appearing on the crystal correspond to the smallest convex polyhedron having minimum surface free energy. Gibbs 

Under most conditions, the shape (i.e., the habit) of a crystal is determined by kinetics rather than thermodynamics and the resulting habit is termed the growth, as opposed to the equilibrium, habit. The kinetics of each crystal face can be influenced by such external factors as supersaturation, temperature, and mixing. However, the complexity of crystallization operations arises from the fact that the mass transfer processes at the crystal interface, as opposed to the bulk, are often rate-determining. Thus, without consideration of the role of the interface during crystallization it is impossible to predict the influence of impurities or solvents on crystallization, or explain such diverse processes as secondary nucleation and inclusion formation. 

As the next step in understanding the factors influencing crystal shape, and the effect of solvents and impurities, models 

The earliest advances in structure/morphology relationships came from Hauy, which stated that the crystal faces with the simplest indices will predominate on the crystal surface. Gibbs (1906) related the crystal morphology to energetic considerations 

The lattice energy ) ,(( often referred to as the crystal binding or cohesive energy is obtained by summing all the interactions between a central molecule and all the surrounding molecules. 

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In formulating a population balance, crystals are assumed sufficiently numerous for the population distribution to be treated as a continuous function. One of the key assumptions in the development of a simple population balance is that all crystal properties, including mass (or volume), surface area, and so forth are defined in terms of a single crystal dimension referred to as the characteristic length. For example, Eq. (19) relates the surface area and volume of a single crystal to a characteristic length L. In the simple treatment provided here, shape factors are taken to be constants. These can be determined by simple measurements or estimated if the crystal shape is simple and known—for example, for a cube area = 6 and kY0 = 1. 

If not, incorrect conclusions about magnetic moment reductions and the shapes of form factors may be made. Discussion of the extinction corrections (52) in form factor determinations has recently been given in the case of Tb(OH)s (55) where the intensity of some reflections was reduced by as much as 90%, and K2NaCrFe (22). Because of the small crystal size, extinction has not been observed to be significant in any work with polycrystalline samples, which is one of the principal advantages of the latter technique. Preferred orientation can be a nuisance in powder work (especially with X-rays) but does not appear to have been significant in the experiments discussed below. 

The growth rate, characterized by the partial current density is the factor determining the equilibrium shape of a crystal. The faces with the smallest partial current densities... 

For the acid catalysed conversion of hydrocarbons, the reaction mechanisms in absence of sterical hinderance are rather well understood, so that molecular shape-selective effects exerted by constrained environments can be isolated [8,9]. Shape-selective catalysis is also possible when other than acid functions are confined to the intracrystalline void volumes of zeolite crystals, e.g. metal [10,11], bifunctional [12] and basic functions [13]. Nowadays, catalysis on zeolites with organic substrates containing heteroatoms receives much attention. Molecular shape-selectivity seems to be superimposed on electronic factors determining the selectivities [14,15]. 

Determining the crystal structure of nanoparticles is a challenge, particularly, when the particles are <5 nm. The intensity-maxim observed in the x-ray or electron diffraction patterns of such small particles are broadened due to the crystal shape factor, which greatly reduced the accuracy of structure refinement. The quality of the HRTEM images of the particles is degraded because of the strong effect from the substrate. This difficulty arises in our recent study of CoO nanocrystals whose shape is dominated by tetrahedral of sizes <5 nm. Electron diffraction indicates that the crystal has a NaCl type of structure. To confirm that the synthesized nanocrystals are CoO, EELS is used to measure the valence state of Co. Figure 4.10... 

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