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Chapter II Solution Growth….
F = -π ρ 2 є + 2π σ (2.7)
Where, є is the free energy gained per unit area when an adsorbed molecule
becomes attached to the nucleus and σ is the free energy per unit length of
the step at boundary of the nucleus. From this expression one finds
interesting behavior, initially, as ρ increases from zero, the F also increases
until it reaches a maximum value of Fo for ρc = ρ; thereafter, it decreases
with increasing ρ.
Burton et al [21] have re-examined the shape of the nucleus. Their
arduous analysis shows that the nucleus is, ipso-facto, not circular but varying
in the shape and hence the free energy per unit length of the edge changes
with the crystallographic directions. In more general manner, the ρc can be
regarded to be the radius of circumscribing circle to the critical nucleus. The
rate of the formation of nuclei is given by,
Z = (S / so) exp (- Fo / k T) (2.8)
Where, Z gives the rate of arrival of fresh molecules at single surface lattice
site, S is the surface area of the crystal face under consideration and so is the
area per molecule in the layer. Usually, Z cannot exceed 10 13 sec -1 even in a
dense environment, however, the pre-exponential factor is about 10 22 . For the
observable growth rate on the time scale of a laboratory experiments, for
example, 10 -3 layers per second or one micron per month, it follows that the
logarithm of saturation ratio, log α, must be at least (Φ / k T) 2 /90, where, Φ is
the binding energy of nearest neighbor, with typical value of Φ / k T requires
a super-saturation of at least 25 to 50%. Therefore, the nucleation rate is very
sensitive function of super-saturation. It is worth noting that above this critical
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