Hydrothermal Synthesis and Growth of Zeolites - Hydrothermal Technology - page 307

having the critical size to become viable, i.e., nuclei on which crystal growth
occurs spontaneously.
The nucleation theory with reference to the zeolite synthesis can be approached
from different angles. Subotic and Bronic [81] have developed a theory guided by
their experimental observation. In this connection, the theory proposed by Zhdanov
[64] and Zhdanov and Samulevich [80] infers that zeolite nuclei must be hidden in
the amorphous gel and released during gel dissolution. The basis for their theory is
the following empirical equation:

ð Þ

mz t

f ð t Þ 5

Þ 5 Kf

mz ð tq

where f is the percent of zeolite relative to the final equilibrium value, t is the crys-
tallization time, q is the constant power, and K is the constant coefficient.
Warzywoda and Thompson [112] proposed the autocatalytic nucleation that can
be incorporated in the population balance formalism, which allows the inclusion of
gel and solution material balances. Using the population balance formalism, one
can simulate crystal size distribution behavior and extend these simple analyses to
other crystallizer types. However, it has been shown that the predicted nucleation
rate profile for such a process is that it is unlikely, as observed experimentally.
Nuclei released from the gel are nucleated much later in the crystallization process
than observed experimentally [112] .
The net free energy of the formation of a spherical nucleus can be expressed in
terms of degree of supersaturation (S), the density (

ρ ), and surface energy ( σ )of

the nucleus and is as follows [112] :

2

πσ

3

ð Þ

16 MW

Δ g 5

2

3 ð RT ρ ln S Þ

It is obvious from the above equation that from a saturated solution (S 1,

5

ln S 0), no spontaneous nucleation can occur as the value for Δ g equals infinity.

5

For a supersaturated solution (S 1), Δ g has a finite negative value and hence

.

spontaneous nucleation is possible [112] .
The nucleation rate, J (number of viable nuclei formed per unit of time) is com-
monly described as an Arrhenius rate equation [113] :

J 5 A exp 2 Δ

g

RT

The above relation predicts an exponential increase in the nucleation rate for an
increasing supersaturation degree once a critical degree of supersaturation has been
reached.
Experimentally, the rate of nucleation can be derived from crystal size distribu-
tion measurements of

the final crystallization product and size increase