Crystal shapes

We have said nothing so far about the shape of crystals, preferring to concentrate instead on their interior structure. However, the shape of crystals is, to the layman, perhaps their most characteristic property, and nearly everyone is familiar with the beautifully developed flat faces exhibited by natural minerals or crystals artificially grown from a supersaturated salt solution. In fact, it was with a study of these faces and the angles between them that the science of crystallography began. 

Nevertheless, the shape of crystals is really a secondary characteristic, since it depends on, and is a consequence of, the interior arrangement of atoms. Sometimes the external shape of a crystal is rather obviously related to its smallest building block, the unit cell, as in the httle cubical grains of ordinary table salt (NaCl has a cubic lattice) or the six-sided prisms of natural quartz crystals (hexagonal lattice). In many other cases, however, the crystal and its unit cell have quite different shapes gold, for example, has a cubic lattice, but natural gold crystals are octahedral in form, i.e., bounded by eight planes of the form 111.  

An important fact about crystal faces was known long before there was any knowledge of crystal interiors. It is expressed as the law of rational indices, which states that the indices of naturally developed crystal faces are always composed of small whole numbers, rarely exceeding 3 or 4. Thus, faces of the form 100, 111, iTOO, 210, etc., are observed but not such faces as 510, 719, etc. We know today that planes of low indices have the largest density of lattice points, and it is a law of crystal growth that such planes develop at the expense of planes with high indices and few lattice points. 

The firinged-micelle model is today considered uruealistic except for potymers of exceptionally low crystalliriity—for a polymer such as PVC, for example, with oystallinity which is hardty detectable. But the central fact of the model remains valid, which is that the two-phase crystal-amorphous solid is pirmed together by the macromolecules which pass from one region to another and back again mar times. It will be noted incidentalty 

5 Banded spherulities of linear polyethylene within a commercial pellet. Optical micrograph of a thin section viewed between crossed polars. Note that sequences of spherulites have nucleated along lines, here in black contrast, which are probably a legacy of extrusion (after D. C. Bassett.). Scale bar -10 un. 

The determination of the proportion of the solid that is crystalline (its ciystallinity) is of considerable fundamental and often practical significance. The easiest method for measuring the crystallinity is the determination of specific volume (the specific volume is the inverse of the density). The simplest methods are Archimedes method and the density gradient colunm. The measurement, Ity either method, is most convenientty performed at room temperature, say 2(f C. At this temperature, let the specific volumes of the crystal and amorphous fractions be and v. Then if X is the fraction of the mass that is crystalline and (1 -x) the fraction that is amorphous, the specific volume v of the specimen is 

ZB Specific voluine versus temperature for semictystalline linear polyethylene showing the effect of heating a specimen from 20°C to above the melting point T . The specific volume of the spcumen at 2(rc is v. The specific volume of the amorphous fraction v, is obtained by extrapolating the v-T curve for the liquid down to 20 D. The specific volume of the crystalline fraction is obtained from the lattice constants of the unit cell (see Problem 2.1). The crystal fraction can be obtained using these quantities in eqn 2.2. 

There are very few polymers for which the specific volume method fails. One is PTFE. This polymer usually contains voids at the 1% level due to the method of fabrication (sinterii. Fluctuations in void content around 1% complete) mask the smaller changes in specific volume due to changes in aystallinity. The method also fails if the specific volumes of the amorphous and crystalline pofymer are almost equal. 

The material in this section draws from similar material in the excellent book by Ewell and Scheel [28]. The shape of a ciystal (i.e., ciystal habit) can be controlled by either thermodynamics or kinetics. Only for crystals grown imder very, very low supersaturation ratios is a crystal habit established thermodynamic considerations. These crystals tend to be of mineralological origin. For most other crystal growth conditions, the kinetics of the slowest growing crystal frees give rise to a crystal shape. 

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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]. 

C. Relative Surface Tensions from Equilibrium Crystal Shapes... 

Analytical Methods. Fluorite is readily identified by its crystal shape, usually simple cubes or interpenetrating twins, by its prominent octahedral cleavage, its relative softness, and the production of hydrogen fluoride when treated with sulfuric acid, evidenced by etching of glass. The presence of fluorite in ore specimens, or when associated with other fluorine-containing minerals, may be deterrnined by x-ray diffraction. 

Crystalline Structures. Crystal shape of amino acids varies widely, for example, monoclinic prisms in glycine and orthorhombic needles in L-alanine. X-ray crystallographic analyses of 23 amino acids have been described (31). L-Glutamic acid crystallizes in two polymorphic forms (a and P) (32), and the a-form is mote facdely handled in industrial processes. The crystal stmeture has been determined (33) and is shown in Figure 1. 

Particle-Size Distribution. Particle size, crystal shape, and distribution of vanillin ate important and gready affect parameters such as taste. 

An approach has been devised (24) to break out of the surface-to-volume relationship imposed by crystal shapes that are nearly spherical. 

The morphology (including crystal shape or habit), size distribution, and purity of crystalline materials can determine the success in fulfilling the function of a crystallization operation. 

Morphology. A crystal is highly organized, and constituent units, which can be atoms, molecules, or ions, are positioned in a three-dimensional periodic pattern called a space lattice. A characteristic crystal shape results from the regular internal stmcture of the soHd with crystal surfaces forming parallel to planes formed by the constituent units. The surfaces (faces) of a crystal may exhibit varying degrees of development, with a concomitant variation in macroscopic appearance. 

Mathews and Rawlings (1998) successfully applied model-based control using solids hold-up and liquid density measurements to control the filtrability of a photochemical product. Togkalidou etal. (2001) report results of a factorial design approach to investigate relative effects of operating conditions on the filtration resistance of slurry produced in a semi-continuous batch crystallizer using various empirical chemometric methods. This method is proposed as an alternative approach to the development of first principle mathematical models of crystallization for application to non-ideal crystals shapes such as needles found in many pharmaceutical crystals. 

In the particular case cited, a 20 per cent reduction in median crystal size is obtained. The process of attrition and methods for its determination were considered in detail in Chapters 4-6. Similar changes may also occur to crystal shape and form. 

In the analysis of crystal growth, one is mainly interested in macroscopic features like crystal morphology and growth rate. Therefore, the time scale in question is rather slower than the time scale of phonon frequencies, and the deviation of atomic positions from the average crystalline lattice position can be neglected. A lattice model gives a sufiicient description for the crystal shapes and growth [3,34,35]. 

A further result of Sadler s 2D-simulation was a relation between the step density and growth rate on the one hand and the inclination of the surface with respect to the principal axes on the other. From this relation crystal shapes were derived which show considerable curvature. This result of an exact treatment stands in contrast to Frank s mean-field curvature expression which gives essentially flat profiles. We will return to the discussion of curved edges in Sect. 3.6.3. 

Why do we get differences in crystal shape or habit This may be a matter of directional rates of growth. Factors affecting directional rates will then affect the habit. Directional rates of growth can be illustrated with a relatively simple crystal structure, that of sodium chloride. 

Both permeability, K, and specific cake resistance, a, depend on the particle shape and size distribution. However, this relationship has not been recognized conclusively. Therefore, information on the crystal shape and size distribution, which can be obtained from experiments on crystallization, cannot be easily translated into the language of filtration characteristics. 

A polymorph is a solid crystalline phase of a compound resulting from the possibility of at least two different crystal lattice arrangements of that compound in the solid state [42], Polymorphs of a compound are, however, identical in the liquid and vapor states. They usually melt at different temperatures but give melts of identical composition. Two polymorphs of a compound may be as different in structure and properties as crystals of two different compounds [43,44], Apparent solubility, melting point, density, hardness, crystal shape, optical and electrical properties, vapor pressure, etc. may all vary with the polymorphic form. The polymorphs that are produced depend upon factors such as storage temperature, recrystallization solvent, and rate of cooling. Table 2 suggests the importance of polymorphism in the field of pharmaceutics [45],... 

The product of chemical reactions, namely precipitations (their appearance, color, and crystal shape), colors in solution, gases (their appearance, bubbles, and color), and sublimates... 

In conclusion, it can be said that there is always some uncertainty as to the correctness of the model(s) chosen. However, as will be pointed out in the next section, studies on several plausible crystal shapes enable us to draw some conclusions that are independent of the actual crystal shapes chosen. 

A large number of compounds of pharmaceutical interest are capable of being crystallized in either more than one crystal lattice structure (polymorphs), with solvent molecules included in the crystal lattice (solvates), or in crystal lattices that combine the two characteristics (polymorphic solvates) [122,123]. A wide variety of structural explanations can account for the range of observed phenomena, as has been discussed in detail [124,125]. The pharmaceutical implications of polymorphism and solvate formation have been recognized for some time, with solubility, melting point, density, hardness, crystal shape, optical and electrical properties, vapor pressure, and virtually all the thermodynamic properties being known to vary with the differences in physical form [126]. 

The (i-naphthol pigment Pigment Red 1 was the first red azo pigment to be submitted to three dimensional X-ray diffraction analysis [1], Today, three different crystal modifications are known, exhibiting divergent hues and crystal shapes [2] ... 

It is recommended that concentration measurements for this type of modeling work are based on analytical standards of mole or mass fraction, to avoid the conversion error caused by density effects. The excess solid phase should always be characterized by a suitable analytical technique, before and after the equilibrium solubility measurements, to confirm that the polymorphic form is unchanged. It should be noted that the crystal shape (habit) does not always change significantly between different polymorphic forms, and visual assessments can be misleading. 

The first indication for the existence of a roughening transition was obtained by Jackson and Miller who studied the crystal shape of chloroethane and ammonium chloride. Above 370 K and 430 K, respectively, they observed a drastic change in crystal morphology, which might be interpreted as roughening. Similar observations for adamantane have been reported by Pavlovska . ... 

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