3.4 Examples of mass transfer 27

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).

3.4Examples of mass transfer

We will consider in this section how mass transfer proceeds, using crystal growth from a solution phase as a representative example of crystal growth in which heat and mass transfer are coupled. We will use the growth of ionic crystals in aqueous solution in a beaker as an example (Fig. 3.3).

For NaCl aqueous solution, which is a dilute ambient phase, all the solute components of Na , Cl–, or NaCl, (NaCl)n are bonded with the solvent component H2O. Whether the solute component is present in an isolated ionic state, or whether NaCl molecules or clusters of (NaCl)n are present, will depend on the driving force. Although the situation is not fully understood, it is expected that, as the supersaturation increases, the proportion of clusters will increase. Although there can be various bonding states between these clusters and H2O, it is generally considered that there are two regions surrounding the solvent component: one structure forming a region with strong bonding, and the other structure forming a breakable region with much weaker bonding between the solvent molecules.

28 Crystal growth

crystal

solution

Figure 3.4. Schematic illustration explaining the concentration gradient around a growing crystal and the presence of the diffusion boundary layer. Bulk diffusion indicated by diffusion boundary layer (for A or B). The concentration gradient is ( B— S), where B is the bulk supersaturation and S is the surface supersaturation.

0 is the surface equilibrium saturation.

When the temperature of the aqueous solution is lowered, or evaporation occurs, a supersaturated state results. If the driving force of the system exceeds the energy barrier necessary for nucleation to take place (for a discussion, see Section 3.6), nucleation of the solute component occurs in the system. Accompanying the nucleation, a more dilute region will appear surrounding the nucleus, and diffusion from the bulk ambient phase will occur due to the concentration difference. This is called volume or bulk diffusion. The concentration gradient is not linear from the bulk ambient phase, away from the nuclei and the surface of the nuclei or crystal. In the region close to the surface of the nuclei or the crystal, the concentration gradient is sharp, whereas that between the bulk ambient solution away from the surface shows almost no gradient. This was already clear from investigations conducted before the 1930s (see Fig. 3.4). The region with a sharp gradient is called the diffusion boundary layer, and the concentration gradient in this layer plays an essential role in crystal growth. For the growth of ionic crystals in an aqueous solution, the thickness of this layer is of the order of 100 m.

The concentration at the surface of a crystal (a nucleus) is not the same as the

3.4 Examples of mass transfer 29

Figure 3.5. The energy barrier at various steps in solution growth. K kink;

SD surface diffusion; DEADS desolvation; DBL diffusion boundary layer.

equilibrium concentration. In the days before we could measure the surface concentration directly, the driving force of a bulk phase was evaluated for simplicity assuming the equilibrium concentration, but now it is assumed that the surface concentration is slightly higher than this value. Therefore, the driving force for crystal growth is now considered to be the difference and its gradient between the concentration at the outermost region of the diffusion boundary layer and the concentration at the surface (surface supersaturation).

Crystallizing particles arriving at the surface will diffuse onto the surface (surface diffusion). As this occurs, some may return to the ambient phase, while some will be caught at kinks or steps (see Section 3.6) on the surface and will be incorporated into the crystal. When these particles are incorporated into the crystal, the solvent component will be dissociated. This process is called desolvation. In solution growth, this process will determine the growth rate. At certain points in these processes, it is necessary to overcome the energy barriers required to climb the respective steps (Fig. 3.5).

The solvent component detached by the desolvation process returns to the ambient phase. As a result, the concentration in the diffusion boundary layer will be lowered. Through competition between the supply of solute component from the bulk phase and that of the solvent component by the desolvation process, the thickness of the diffusion boundary layer will vary according to the change in concentration, or buoyancy-driven convection will occur when crystals are growing in the gravity field of the Earth. This change is related to the bulk supersaturation. In Fig. 3.6, changes in behavior of the diffusion boundary layer around a growing Ba(NO3)2 crystal in aqueous solution are illustrated in relation to the bulk supersaturation . At 0.5%, the thickness of the diffusion boundary layer simply increases, whereas at 0.5% 3.0%, unstable uplifting convection appears, and at 3.0% a steady buoyancy-driven convection plume rising perpendicularly

3.5 Laminar flow and convection 31

from the center of the top face appears. At 8.0%, rising convection plumes appear, creating a considerable depletion at the root, which greatly affects the perfection of the crystal.

Concentration variation in the bulk phase is isotropic around a crystal. Consider a crystal growing in a concentric, equi-concentration distribution field. In a polyhedral crystal bounded by corners, edges, and faces, we expect the concentration difference at a corner to be larger than that at an edge, which is in turn larger than that at a face, and thus corners and edges are at a higher concentration state than is the center of a face. This creates surface diffusion. The concentration differences at the corners, edges, and face centers of a growing crystal will differ depending on the bulk supersaturation or flow rate along the surface. The difference in concentration at different sites on a face was demonstrated by Berg [3] by means of an interferometric technique, which is now referred to as the Berg effect or Berg phenomenon. This is an important phenomenon in analyzing the morphology of crystals.

3.5Laminar flow and convection

Like diffusion, convection is also an important factor for mass or heat transfer. Convection is a mode of flow in a fluid that creates rotation or circulation of the fluid, and is distinguished from laminar flow and turbulent flow. If there is a temperature difference between the fluid at the top and bottom of the beaker shown in Fig. 3.3, convection occurs due to the concentration difference arising from the temperature difference. Convection will also occur when there is a concentration gradient, or it can arise due to buoyancy. Convection due to the surface tension of a solution is called Marangoni convection. The difference in surface tension may be due to a temperature difference or to the concentration of chemical species on the surface. Convection may be taken to mean thermal convection, solutal convection, or a combination of the two. In contrast to these natural processes of convection, various methods, such as rotating the seed crystals or the crucible itself, are adopted when growing single crystals in order to control convection and to effect the homogenization of the liquid phase. Since it is desirable to achieve a homogeneous liquid phase in order to grow homogeneous single crystals, convection occurring in the liquid phase has been extensively investigated by means of computer simulation or various other visualization techniques.

Laminar flow, which is a directional flow, changes into turbulent flow when the critical Reynolds number (Re)† exceeds 200. When there is a flow of solution around

†The most representative non-dimensional value in fluid dynamics is Re, the ratio of the magnitude of inertia force and viscosity, Re l/ l/v, where is the density, is the viscosity, v is the dynamic viscosity, l is the representative length of matter in the flow, and is the representative flow rate.