Dendritic crystal growth

R. O. Gmbel, ed.. Metallurgy of Elemental and Compound Semiconductors, Interscience Pubhshers, New York, 1961. Discusses eady work on semiconductor dendrites and other methods of growing shaped crystals. The special issue of / Cyst. Growth (Sept. 1980) is devoted to shaped crystal growth. 

G. Dziuk. A boundary element method for curvature flow. Application to crystal growth. In J. E. Taylor, ed. Computational Crystal Growers Workshop, AMS Selected Lectures in Mathematics. Providence, Rhode Island American Mathematical Society, 1992, p. 34 A. Schmidt. Computation of three dimensional dendrites with finite elements. J Comput Phys 125 293, 1996. 

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. 

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... 

Experimental results on crystal growth might be affected by dendritic growth and the effect is difficult to evaluate. Hence, experimental data on the melting rate of a pure crystal in its own melt are expected to be a more reliable indication of interface reaction rate than those on crystal growth rate. 

Kassner. K. Pattern Formation in Diffusion-Limited Crystal Growth Beyond the Single Dendrite, World Scientific Publishing Company, Inc., Riveredge, N). 1995. Stuart G.. N. Spruslon and M. Ilausser Dendntes, Oxford University Press, Inc., New York. NY, 2000. 

Several hybrid simulations on crystal growth can be found in recent literature. Examples include dendritic solidification by coupling finite-different discretization of a phase field model to a MC simulation (Plapp and Karma, 2000), coupling a finite difference for the melt with a cellular automata for the solidification (Grujicic et al., 2001), a DSMC model for the fluid phase with a Metropolis-based MC for the surface to address cluster deposition onto substrates (Hongo et al., 2002 Mizuseki et al., 2002), a step model for the surface processes coupled with a CFD simulation of flow (Kwon and Derby, 2001) (two continuum but different feature scale models), an adaptive FEM CVD model coupled with a feature scale model (Merchant et al., 2000), and one-way coupled growth models in plasma systems (Hoekstra et al., 1997). Some specific applications are discussed in more detail below. 

Three types of reactions that have been performed in gels are covered in the following sections (1) standard growth, (2) the complex dilution method, and (3) reduction to form metallic crystals or dendrites. 

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. 

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