8

Forms and textures of polycrystalline aggregates

Spontaneously nucleated crystals can grow with various orientations. In such processes, surviving crystals are selected simply by their geometrical relation with the substrate surface. As a result, various textures of polycrystalline aggregates appear that are controlled by the form of the substrate surface. Spherulite, sheaf-like, and confeito-like* polycrystalline aggregates or the curved banding patterns seen in agate are all formed by a simple geometrical selection process. This principle has been practically utilized in single crystal synthesis and in epitaxial growth. It also acts in the formation of calculus in the internal organs of human bodies.

8.1Geometrical selection

When nucleation occurs freely on a flat substrate surface under uncontrolled conditions, numerous crystals are formed at random orientations if there is no epitaxial relation between the substrate and the crystal. Crystals inclined to the substrate surface will make contact with crystals growing perpendicularly and stop their growth. If we assume the existence of equi-concentration lines parallel to the substrate surface, the growth of crystals perpendicular to the substrate will be promoted further, since their tips are in an ambient phase of higher driving force. In this way, only the crystals growing perpendicularly to the substrate surface will survive and continue to grow among many crystals formed on the substrate surface, and this is accompanied by a decrease in the number of individuals and appearance of textures consisting of many crystals aligned in a specific orientation. Since the selection of the surviving crystals is determined by geometrical

*Confeito is the name given to small pieces of candy made by crystallizing sugar around poppyseed cores.

8.1 Geometrical selection 151

Figure 8.1. (a) Size s versus population density p. Points a, b, c, and d correspond to I (a,

b), II (b, c), and III (d) in (b), which is a schematic illustration of geometrical selection.

orientation only, this is called geometrical selection. Figure 8.1 shows the concept schematically. Kolmogorov’s theory is a probability theory solving the problem [1]; see ref. [14], Chapter 7. The population density n of surviving crystals by geometrical selection is expressed as follows. For the two-dimensional case,

n(h) 1/ h ,

and, for the three-dimensional case,

n(h) 1 / h,

where h is the separation between neighboring crystals.

The textures of polycrystalline aggregates formed by geometrical selection have been widely observed in natural minerals, but the principle has also been actively used in various synthetic methods.

The most representative example is seen in the comb-like texture of quartz veins formed by the precipitation of quartz crystals on the wall of a vein from a hydrothermal solution that has entered into small fractures in the rocks (Fig. 8.2).

If there is no epitaxial (coherent) relation between the substrate and the growing crystals, and the nuclei formed initially have completely random orientation, and the growth rate is more or less isotropic, geometrical selection operates in one direction only: perpendicular to the substrate surface. When there is an epitaxial and coherent relation between the substrate and the crystals, and the growth rate

152 Forms and textures of polycrystalline aggregates

Figure 8.2. Quartz vein showing a comb-like texture.

is remarkably anisotropic, geometrical selection will be operative in both perpendicular and parallel orientations to the substrate surface, leading to the formation of a texture of polycrystalline aggregate in one direction. Whether a coherent relation with the substrate is present or not is determined not only by the misfit ratio, a factor relating to the interface energy, but also by the driving force (see Section 7.4). Even if a coherent relation is achieved under a low driving force condition, the relation will be violated under higher driving force conditions. Close-packed, densely arranged compact and hard textures will appear if a coherent relation is maintained, but the texture will be coarsely packed and soft if the coherency is violated. This relation is important for crystal growth in living bodies (see Chapter 14).

Geometrical selection is used practically in various methods of single crystal synthesis. In one such method from the melt phase, the Bridgman–Stockbarger method, crystallization is achieved by moving the melt zone, either by moving the heater or the crucible. To obtain large single crystals by this method, a technique is adopted in practice to design the crucible so as to encourage the survival of only a small number of crystals, by geometrical selection, out of the numerous crystals that are formed initially. In crystal growth from the vapor phase by the CVT method using a closed tube, a technique is adopted in practice in which the diameter of the tube is narrowed at the crystallization zone so that only a small number out of many crystals will survive and grow. To achieve an epitaxial relation between the substrate and the crystal with a large misfit ratio, a device that forms a buffered layer in between the substrate and the crystal is used; this may also be regarded as a method utilizing geometrical selection.

8.2Formation of banding

When the substrate surface is curved or spherical, various textures of polycrystalline aggregate appear through geometrical selection. Spherulites will be formed when a sand grain or spherical polycrystalline aggregate formed at the early stage of nucleation acts as a substrate, and wavy banding parallel to the sub-

8.3 Spherulites 153

strate surface will appear when the substrate surface is irregular. After one layer is completed via intermittent growth, and the process is repeated, a banding pattern will appear consisting of a unit layer of polycrystalline aggregate running perpendicularly to the substrate surface through geometrical selection, starting from minute crystallites of random orientation. For spherulites, concentric banding will appear in a similar manner.

The intermission and resumption of growth are inevitably involved in a system where there is an imbalance between the diffusion rate and the growth rate and a critical value such as the energy barrier is involved. When growth resumes, the particle size is small and the density is high, but the size increases and the density decreases as growth proceeds. Since the impurity concentration will also vary, the color intensity will be different.

Most mineral crystals formed under normal temperatures and pressures on or near the Earth’s surface are minute, and often show a banding texture. Malachite, Cu2(OH)2CO3, and azulite, Cu3(OH)2(CO3)2, are copper carbonate minerals resulting from secondary precipitation from an aqueous solution containing Cu ions and CO2, which result from the dissolution of primary copper sulfide minerals in underground water. Due to the beauty of its banding pattern, consisting of green malachite or alternating layers of green malachite and indigo blue azulite, these minerals have been prized as ornamental materials since ancient times. Rhodochrosite, MnCO3, is a mineral with a beautiful banding pattern, pink in color, formed from a low-temperature hydrothermal solution, and has been used as an ornamental material under the name Inca rose. The familiar banding pattern seen in agate is formed in a similar manner; agate will be considered in more detail in Chapter 10. Several examples of these banding patterns are shown in Fig. 8.3.

The free surface of these polycrystalline aggregates takes various forms depending on the degree of unevenness, and various terms including botryoidal, reniform, stalactite, and colloform have been used to describe them (see Fig. 8.4 and Table 8.1). In addition, the term oolitic is applied to an aggregate of equal sized spherulites. Terms used in mineralogy to express the characteristic forms of polycrystalline aggregates are given in Table 8.1. In most cases, the characteristic morphology is derived from geometrical selection. Other terms, such as bladed, foliated, and plumose, may be used to describe characteristic morphologies. These are, in most cases, forms shown by sub-parallel aggregation of thin platy crystals.

8.3Spherulites

Spherulites are formed if geometrical selection takes place on a spherical substrate particle. Substrate particles may be a completely different material from those materials forming the spherulites, such as a sand grain, or a spherical particle of polycrystalline aggregate of the same species formed under a higher driving

154 Forms and textures of polycrystalline aggregates

(a)

(b)

(c)

Figure 8.3. Examples of banding patterns formed by geometrical selection in

(a) malachite; (b) rhodochrosite; (c) agate.

8.3 Spherulites 155

(a)

(b)

(c)

Figure 8.4. Forms of polycrystalline aggregate: (a) botryoidal; (b) mamillary;

(c) spherulitic; (d) oolitic (see p. 156).

156 Forms and textures of polycrystalline aggregates

(d)

Figure 8.4 (cont.)

force condition, or it may be a polycrystalline aggregate of random orientation formed by agglutination of crystals formed under a low driving force condition through the movement of an ambient phase. Since the formation of spherical polycrystalline aggregates occurs under a high driving force condition in the absence of effect of flow, spherulites are a form of polycrystalline aggregate generally appearing under a high driving force condition. If there is a flow, or turbulence, the situation changes. Crystals formed under small driving force conditions in polyhedral form agglutinate by moving flow, forming polycrystalline aggregates with random orientation, which acts as a substrate for further growth to form spherulites. NaCl spherulites occurring on the shore of the Dead Sea have spherical nuclei of polycrystalline aggregate formed by agglutination due to the movement of seawater.

In the case of sodium uric acid, which is the cause of gout, needle-shaped single crystals are formed in the red corpuscles, transported by blood flow, and deposited at the knuckle of the big toe, where they accumulate, and this acts as a site for the nucleation of spherulites. The formation of kidney stones and gallstones is a similar process; this process will be discussed in Chapter 14. Compared with isotropic crystals, needle, long prismatic, and thin platy crystals are more likely to form spherulites. Among the zeolite group of minerals (silicates with a cage structure), natrolite, Ba2Al2Si3O10 2H2O, whose Habitus is needle-like, always occurs as spherulites, and stilbite, (Ca, Na2, K2)Al2Si7O18 7H2O, with characteristic platy

Habitus occur as sheaf-like or bladed aggregates, which correspond to uncompleted spherulites. In contrast, analcite, NaAlSi2O6 H2O, and chabazite, CaAl2Si4O17 H2O, whose Habitus are isotropic, occur only exceptionally as spherulites.