Convection, crystal growth solution

We have so far assumed that the atoms deposited from the vapor phase or from dilute solution strike randomly and balHstically on the crystal surface. However, the material to be crystallized would normally be transported through another medium. Even if this is achieved by hydrodynamic convection, it must nevertheless overcome the last displacement for incorporation by a random diffusion process. Therefore, diffusion of material (as well as of heat) is the most important transport mechanism during crystal growth. An exception, to some extent, is molecular beam epitaxy (MBE) (see [3,12-14] and [15-19]) where the atoms may arrive non-thermalized at supersonic speeds on the crystal surface. But again, after their deposition, surface diffusion then comes into play. 

J. Holuigue, O. Bertrand, E. Arquis. Solutal convection in crystal growth effect of interface curvature on flow structuration in a three-dimensional cylindrical configuration. J Cryst Growth 180 591, 1997. 

The linear instability theory of the behavior of a system near the bifurcation point can be successfully applied to many self-organization problems, such as thermal convection in hydrodynamics4 and crystal growth in solution.5 In these theories, various initial fluctuations play important roles. Occasionally the fluctuations arise from the thermal motion of atoms or molecules. If a system reaches an unstable mode over... 

The crystallization of the D-enantiomer is therefore considered to be induced by crystal growth on the surface of the seed crystal and at the same time initial breeding may play a role that causes small crystals near the seed crystal. The propagation of nucleation in distance from the seed may be caused by convective flow of the solution due to density difference during the crystal growth. 

The scope of kinetics includes (i) the rates and mechanisms of homogeneous chemical reactions (reactions that occur in one single phase, such as ionic and molecular reactions in aqueous solutions, radioactive decay, many reactions in silicate melts, and cation distribution reactions in minerals), (ii) diffusion (owing to random motion of particles) and convection (both are parts of mass transport diffusion is often referred to as kinetics and convection and other motions are often referred to as dynamics), and (iii) the kinetics of phase transformations and heterogeneous reactions (including nucleation, crystal growth, crystal dissolution, and bubble growth). 

Figure 3.3. Various features of diffusion and convection associated with crystal growth in solution (a) in a beaker and (b) around a crystal. The crystal is denoted by the shaded area. Shown are the diffusion boundary layer (db) the bulk diffusion (D) the convection due to thermal or gravity difference (T) Marangoni convection (M) buoyancy-driven convection (B) laminar flow, turbulent flow (F) Berg effect (be) smooth interface (S) rough interface (R) growth unit (g). The attachment and detachment of the solute (solid line) and the solvent (open line) are illustrated in (b).

One-Dimensional, Diffusion-Controlled Crystal Growth. Neglecting bulk convection leads to an idealized picture of diffusion-controlled solute transport of a dilute binary alloy with the solute composition c0 far from an almost flat melt-crystal interface located at z = 0 (I, 21). 

Examples of dimensionless groups that specify ratios of transport mechanisms are listed next in Table II and depend on the size and shape of the domain. The Peclet numbers for heat (Pet) and solute (Pes) and momentum (Re) transport are ratios of scales for convective to diffusive transport and depend on the magnitudes of the velocity field and the length scale for the diffusion gradient. Boundary layers form at large Peclet numbers (Pet or Pes) or Reynolds numbers (Re). The fonnation of a boundary layer at a large Re is particularly important in crystal growth from the melt, because the low... 

These measures of solute segregation are closely related to the spatial and temporal patterns of the flow in the melt. Most of the theories that will be discussed are appropriate for laminar convection of varying strength and spatial structure. Intense laminar convection is rarely seen in the low-Prandtl-number melts typical of semiconductor materials. Instead, nonlinear flow transitions usually lead to time-periodic and chaotic fluctuations in the velocity and temperature fields and induce melting and accelerated crystal growth on the typically short time scale (order of 1 s) of the fluctuations. 

First, the role of system design on the details of convection and solute segregation in industrial-scale crystal growth systems has not been adequately studied. This deficiency is mostly because numerical simulations of the three-dimensional, weakly turbulent convection present in these systems are at the very limit of what is computationally feasible today. New developments in computational power may lift this limitation. Also, the extensive use of applied magnetic fields to control the intensity of the convection actually makes the calculations much more feasible. 

Nerad, B.A., Shlichta, P.J., Ground-based experiments on the minimization of convection during the growth of crystals from solution. J. Cryst. Growth 1986, 75, 591-608. 

Solvents are never 100% pure and these impurities are well-known culprits in reducing diffusive nutrient flux to the growth interface in convection-free solution growth systems. Even though this is not a crystal growth system, the role played by impurities in retarding diffusive nutrient transport to tne interface is the same. 

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