Retrodiffusion of solvent
On binding to the crystal, the solute (or, more generally, the crystallization unit) takes the place of the solvent that was in contact with the face in question. Let us write that the global volume is unchanged.
N: flow density towards lining (kmol.m-2 .s-1)
О : molar volume (m3 .kmol-1).
Index “s” relates the solvent and index “u” to the crystallization units.
The Maxwell-Stefan equation is written as:
x: molar fractions.
Using equations [2.2] and [2.4], let us replace xs and Ns in [2.3].
Indeed, the Fick law can be written out:
We observe that the retrodiffusion of solvent divides flow density Nu towards the crystal via crystal corrective coefficient K. However, we should note that the coefficient is quite close to 1, and moreover, can be above or below this value.
1) The kinetics referred to as mononuclear and polynuclear, such as those presented by Dirkson et al. [DIR 91], give rates that are far too low and disconnected from experiment data.
Indeed, the population density (frequency) in a continuous homogenous crystallizer is:
L: crystal dimension (m)
G: crystal growth (m.s-1)
т : residence time in the crystallizer (s)
where G is constant and Ln[n(L)] is a decreasing linear function of L (see Figure 2.7). The curve, in absolute value, increases significantly if
G decreases, which is the case of crystals at the beginning of their existence as they are not carrying dislocations and cannot begin to grow except by “formation and spreading”, which is very low. This explains the rounded form of the curve Ln[n(L)] = f(L) for the very low L. The transition
between the rounded form and the rectilinear form occurs for a value of L between 50 and 100 pm. This is the size at which dislocations begin to appear.
When large and small crystals are kept in a continuous phase, the small crystals disappear, benefiting the larger crystals. This phenomenon is significant in metallurgy where the continuous phase, the “matrix”, is in a solid state. For the matter at hand, this phase is liquid and typically of low viscosity, so that a decantation, or more commonly, a separation, intervenes long before maturation occurs. On this question, which we will not address here, there are several works of reference value [RAT 02, MAR 84].
Nonetheless, we should note that the influence of gravity could be neutralized by placing the suspension in a cylindrical recipient that turns slowly on its horizontal axis. A good example of this would be the maturation of sugar seeds.
Figure 2.7. Aspect of logarithmic variations in population density according to crystal dimension
Drawing crystal shapes [DOW 80]
Relative to the center (starting point) of the crystal, we represent each face by the classic plane equation:
R : face growth rate
h, k, l: Miller indices
Triplet faces are resolved in order to give the summits. The required polyhedron is the minimum polyhedron that can be obtained by eliminating the summits that are further from the center than all of the faces. The edges are obtained by connecting the summits that have two faces in common.
Rotation matrices allow for the crystal to be turned in space.