Dendrite crystal

R. Kobayashi. Modeling and numerical simulations of dendritic crystal growth. Physica D (55 410, 1993 R. Kobayashi. A numerical approach to three-dimensional dendritic solidification. Exp Math 5 59, 1994. 

K. X. H. Zhao, H. Power, L. C. Wrobel. Numerical simulation of dendritic crystal growth in a channel. Eng Anal Bound Elem 79 331, 1997. 

The extension of generic CA systems to two dimensions is significant for two reasons first, the extension brings with it the appearance of many new phenomena involving behaviors of the boundaries of, and interfaces between, two-dimensional patterns that have no simple analogs in one-dimension. Secondly, two-dimensional dynamics permits easier (sometimes direct) comparison to real physical systems. As we shall see in later sections, models for dendritic crystal growth, chemical reaction-diffusion systems and a direct simulation of turbulent fluid flow patterns are in fact specific instances of 2D CA rules and lattices. 

Growth theories of surfaces have received considerable attention over the last sixty years as summarized by Laudise et al. [53] and Jackson [54]. The well-known model of the crystal surface incorporating adatoms, ledges and kinks was first introduced by Kossel [55] and Stranski [56]. Becker and Doring [57] calculated the rates of nucleation of new layers of atoms, and Papapetrou [58] investigated dendritic crystallization. 

The crystals obtained at 800°C have a square pillar form and almost all of them are hollowed in the central part of the cross section. At 1100°C the formation of specially shaped deposits is observed along with the growth of dendritic crystals of CoB (about 5 mm in length). 

Snow crystals [4] Their macroscopic structure is different from a bulk three-dimensional ice crystal, but they are formed by homologous pair-pair interaction between water molecules and are static and in thermodynamic equilibrium. It should be noted, however, that dendritic crystal growth is a common phenomenon for metals [5-7] and polymers. The crystals grow under non-equilibrium conditions, but the final crystal is static. 

Another example is dendritic crystal growth under diffusion-limited conditions accompanied by potential or current oscillations. Wang et al. reported that electrodeposition of Cu and Zn in ultra-thin electrolyte showed electrochemical oscillation, giving beautiful nanostmctured filaments of the deposits [27,28]. Saliba et al. found a potential oscillation in the electrodeposition of Au at a liquid/air interface, in which the Au electrodeposition proceeds specifically along the liquid/air interface, producing thin films with concentric-circle patterns at the interface [29, 30]. Although only two-dimensional ordered structures are formed in these examples because of the quasi-two-dimensional field for electrodeposition, very recently, we found that... 

It is usually believed that the growth of dendritic crystals is controlled by a bulk diffusion-controlled process which is defined as a process controlled by a transportation of solute species by diffusion from the bulk of aqueous solution to the growing crystals (e.g., Strickland-Constable, 1968 Liu et al., 1976). The appearances of feather- and star-like dendritic shapes indicate that the concentrations of pertinent species (e.g., Ba +, SO ) in the solution are highest at the corners of crystals. The rectangular (orthorhombic) crystal forms are generated where the concentrations of solute species are approximately the same for all surfaces but it cannot be homogeneous when the consumption rate of solute is faster than the supply rate by diffusion (Nielsen, 1958). 

This boundary between two different precipitation mechanisms for barite determined by the experiments at 150°C (Shikazono, 1994) roughly coincides with that between dendritic crystals and well-formed crystals which has been experimentally determined at temperatures lower than 100°C. 

Figure 2. (a) Schematic description for the growth of dendrite crystals on a Li surface. The film consisting of decomposition products as shown in Scheme 1 prevents the growth of large granular crystals but rather promotes the formation of treelike dendrites, (b) Schematic description for the formation of isolated lithium particles from Li dendrites. The uneven dissolution of the dendrites leaves lithium crystals detached from the lithium substrate. The isolated lithium crystals become electrochemically dead but chemically reactive due to their high surface area. 

Crystallization at 283 K Feed melt of 80.0 mol% of benzene a No crystallization b Dendritic crystal... 

Figure 2.1. Various forms exhibited by crystals, (a) Polyhedral crystals (b) hopper crystal (c) dendritic crystal (snow crystal, photographed by the late T. Kobayashi) (d) step pattern observed on a hematite crystal (0001) face (e) internal texture of a single crystal (diamond-cut stone, X-ray topograph taken by T.Yasuda) (f) synthetic single crystal boule. Si grown by the Czochralski method (g) synthetic corundum grown by the Verneuil method.

For example, the treatment of diffusion that is to follow is solely restricted to semi-infinite linear diffusion, i.e., diffusion that occurs in the region between x = 0 and x —> +oo, to a plane of infinite area. Thus, diffusion to a point sink—called spherical diffusion—is not treated, though it has been shown to be relevant to the particular problem of the electrolytic growth of dendritic crystals from ionic melts. 

Intensive research has continued into the mechanism of snowflake formation [15], This research encompasses the broader question of dendritic crystal growth. New approaches, such as fractal models, and copious use of computer simulation have greatly facilitated these attempts. It is fascinating how dendritic growth penetrates even chemical synthetic work witnessed by the development of dendrimer chemistry of ever increasing complexity, which is an example of nanochemistry par excellence [16], An illustration is given in Figure 2-23. 

Physical Properties of Pure Compact Iron.—The properties of iron are affected to such a remarkable and unique extent by the presence of small quantities of alloying elements, chief amongst which is carbon, that these phenomena are an important study in themselves. It is not intended in this section, therefore, to deal with the physical properties of any commercial iron other than the chemically pure and compact metal. For a discussion of the physical and metallurgical properties of various types of commercial iron and its alloys, the reader is referred to Part III. of this volume. Pure iron is a white metal which can be readily machined in a lathe, and even cut with a knife. It crystallises according to the cubic system,3 but crystals are rare, the metal being usually massive. Dendritic crystals may be obtained artificially with branches parallel to the cubic axes.4 Shock apparently assists or induces crystallisation in iron.5... 

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