connected-network theory) from the various crystal bond energies, if they
are known. It follows thatDG* will generally be different for each crystal
face.
Note The parametergx=x(in newtons), wherexis the length of
the growth unit in the x-direction, can be seen as a line tension, i.e.,
the one-dimensional analogue of surface tension.
At small supersaturation lnb,DGwill thus be large, generally much
larger thankBT. This implies that adsorption of a single unit is unlikely and
is soon followed by its desorption. If a greater number of growth units is
adsorbed to form a two-dimensionalembryo, the total edge free energy per
unit will be smaller, because there is no edge between two adjacent units. On
the other hand, the negative adsorption free energy will be proportional to
the number of growth units in the embryo. If the latter has reached a certain
size, further growth will lead to a decrease in total free energy, and two-
dimensional growth will occur. This is analogous to the situation described
in Section 14.2.1. The larger lnb, the smaller the radius of the nucleus, and
the greater the rate of surface nucleation.
All this means that growth will primarily occur at the ‘‘step’’ formed
by the nucleus (see Figure 15.7), since it is less likely that an adsorbed
molecule will desorb if it is adjacent to two molecules on the surface rather
than one. The step will thus move over the surface (from left to right in the
figure). As depicted, a ‘‘kink’’ may also form, where a molecule can form
bonds on three sides, being even more strongly bound. Hence the kink will
also move. Taking these phenomena into account, modeling of the growth
can be done, and an example (for a case wherea¼8) is shown in Figure
15.6, curve N. It is seen that the growth rate would be very much smaller
than in the regime of thermodynamic roughening.
However, some modification will be needed. It has been shown that
surface diffusionof an adsorbed molecule can occur and that it can meet a
step (as illustrated in Figure 15.7) before it desorbs. Overall growth will then
be much faster, as is illustrated in Figure 15.6, curve NþD.
Spiral Growth. When growth is two-dimensional, it will stop after
a full monomolecular layer has formed. A flat surface then results and
nucleation is needed again to induce growth. Especially at low values of lnb,
where the nucleation rate is low, this will greatly slow down crystal growth.
However, steps on a crystal face may also be caused bydislocations,mostof
which are due to inclusion of a foreign molecule or tiny particle in the crystal
lattice. Some of these defects, calledscrew dislocations, tend to remain while
molecules are attached in the step, to form agrowth spiral, illustrated in
Figure 15.8; such spirals can often be observed by microscopy, e.g., on