Engineering of Small-Molecule Gels Based on the Thermodynamics and Kinetics of Fiber Formation - Soft Fibrillar Materials: Fabrication and Applications - page 86

crystals at high supersaturations owing to the supersaturation-driven mismatch
nucleation and growth. Similarly to normal 3D nucleation, the mismatched do-
main should first nucleate on the growing crystal surface. If the energy cost to
create a mismatch domain per area on the parent crystals is defined as the specific
mismatch free energy γ mis , the barrier of mismatch nucleation is determined

by the surface supersaturation and

γ mis (cf. Equation 2.16). Obviously, the crys-
tallographic mismatch nucleation is a special case of heterogeneous nucleation,
where γ mis = γ sc . If the mismatch growth does not deviate from much from the

orientation of the parent crystal, we can in principle have

γ cf ∼ γ sf . It follows then

that

m = 1 − γ mis / γ cf (2.19)

Likewise, the nucleation barrier of crystallographic mismatch nucleation is given
by Equations 2.15-2.16. As indicated by these equations,

G * mis decreases as the
surface supersaturation increases. Notice that the only difference between normal
heterogeneous nucleation and crystallographic mismatch nucleation is that in
crystallographic mismatch nucleation the substrate is the growing crystal surface,
whereas in normal heterogeneous nucleation, substrates are foreign bodies. In
the following paragraphs, we will discuss briefly some key factors controlling
crystallographic mismatch nucleation and growth:

1) Supersaturation . Similarly to normal nucleation and growth, the kinetics of
crystallographic mismatch nucleation and growth also depends on supersat-
uration. At low supersaturations, the crystallographic mismatch nucleation
is difficult to occur due to the high G * mis (Equations 2.15 and 2.16). As
supersaturation increases, G * mis will drop rapidly (Equation 2.16). It follows
that interfacial mismatch nucleation can occur much more easily in this case.
2) Impurities . Adsorbed impurities may disturb the interfacial structural match
between nucleating layers and the parent crystal surfaces. This gives rise to the
lowering of m , which promotes crystallographic mismatch nucleation.
3) Slow surface integration . Since the crystallographic mismatch nucleation occurs
at the surface of growing crystals, it is governed by surface supersaturation. The
orientations with slow surface integration kinetics will therefore lead to higher
surface supersaturations (much closer to the bulk supersaturation). It follows
that at low supersaturations, the crystal faces with slow surface integration
kinetics can take advantage of the highest possible supersaturation - the
bulk supersaturation of the system, and will induce much more easily the
crystallographic mismatch nucleation.
4) Low specific mismatch free energy . According to Equation 2.16, a low specific
mismatch free energy γ mis corresponds to a low G * mis . This implies that the
crystallographic mismatch nucleation can occur much more easily in crystal
surfaces with low γ mis . Normally crystal surfaces with low γ mis often coincide
with those with slow surface integration kinetics. Therefore, criteria 3 and 4
may be very likely applied to the same crystallographic orientation for a given
crystalline material.