P - Technische Universiteit Eindhoven

2. Single crystal growth of sodium sulfate heptahydrate 19
growing crystal
solution
r
0 L(t)
R
Fig. 2.2: Model of crystal growth under the assumption of a cylindrical shape of the crystal in
the finite volume of the solution. The boundary, L(t), of the crystal is moving when
crystallization proceeds.
the following boundary conditions:
∂c(r, t)
∂r ∣ r=R
= 0 , (2.3)
c(r = L(t), t) = c h . (2.4)
The corresponding growing process is illustrated in Fig. 2.2. While the crystal is growing,
the surface of crystal will move and therefore the condition (2.4) depends on the
position of the boundary. An analytical result for diffusion-controlled growth with a moving
boundary can be obtained by solving the so-called Stefan problem [51] in an infinite
volume of solution [52]. In our NMR experiments, the volume of solution is finite, so
a numerical approach is needed to calculate the concentration profile and the radius of
the crystal. For that purpose, the COMSOL c⃝ package is used to solve the 2D Stefan
problem assuming radial symmetry and a constant height of the crystal. An example of
calculated concentration profiles during crystal growth is shown in Fig. 2.3 for a circular
salt solution drop with a radius of 7 mm (comparable to the NMR setup) having an initial
concentration of 2.5 mole/kg and a crystal with an initial size of 1 µm in the center of
the drop. A diffusion coefficient of sodium sulfate in solution D = 0.9 × 10 −9 m 2 /s [53] is
assumed as well as a constant temperature of 2.5 ◦ C (the temperature on the heptahydrate
supersolubility line for a concentration of 2.5 mole/kg).
2.4 Interface-controlled growth
If the crystal growth is interface-controlled and the interface is rough, so that every jump
of an ion across the interface results in attachment, see Fig. 2.1, the growth rate depends
on the supersaturation ratio, β − 1, by [49, 54]:
∂L k
= G k (β k − 1) g , (2.5)
∂t
where (k = x, y, z) are the kinetic growth parameters in the x, y, and z directions, respectively,
and β k are the supersaturations of the solution adjacent to the respective surfaces