supersaturation. According to Eq. (15.1) the supersaturation ratio b ¼
a/aeq, but it is usually taken asc/ceq, wherecis concentration. The latter
ratio may significantly differ from the real value ofb, since the solution may
be far from ideal. Common causes of nonidealitycan be (a) that the
properties of solute and solvent molecules are very different, and (b) that the
solution is far from dilute. It is in most cases difficult, if at all possible, to
determine the activities. Anyway, the supersaturation, and thereby the
crystallization, rate will be zero if the concentration ratioc/ceqequals unity.
In principle, one would expect the linear growth rate of a crystal face to be
proportional to the supersaturation, but experiments show this to be the
exception rather than the rule.
It may further be noticed that for b close to unity, lnðc=ceqÞ
&ðcceqÞ=ceq. The latter ratio is also called supersaturation (often
expressed as a percentage) by some authors.
15.2.1 Growth Regimes
During growth, molecules diffuse to a crystal surface, and some of these are
incorporated into the crystal lattice. Assuming (a) thermodynamic
equilibrium at the crystal surface and (b) that each molecule arriving at
the surface becomes incorporated, so-called normal growth will occur. Then
the linear growth rateLCðin m?s^1 Þis proportional tob1, and the
maximum (or Wilson–Frenkel) rate can be calculated; an example is in
Figure 15.6.
However, there is generally no thermodynamic equilibrium at the
crystal surface. Moreover, the chance that a molecule becomes and stays
attached to a plane surface often is very slight, since the free energy gain per
molecule is small. Substantial growth thus involves other processes. In
principle, three regimes can be distinguished:
- Thermodynamic roughening
- Smooth surfaces with a number of subregimes
- Kinetic roughening
In each of these the relations governing the incorporation of molecules (or
ions, etc.) in the crystal are different. The regimes will be briefly discussed,
without giving the full rate theories. (Actually, theoretical results generally
are obtained from Monte Carlo simulations.) It is assumed that there is only
one solute present.