**Rate Limiting
Step for Crystallization:**

Consider that a monomer (crystallizing material in
solution, here called **A**) must diffuse to the crystal to crystallize, the
monomer at the crystalline interface is here called **B**. The **B**
monomers must then undergo crystallization to form a new crystalline surface
nucleation site, **C**. Our interest is in the overall growth process, **A
=> B => C**, and specifically the rate of crystalline growth, **R**.

R = k_{d}A = k_{g}B
(1)

where k_{d} and k_{g} are exponential
functions in temperature reflecting the rate constants for diffusion and
growth. Both k_{d} and k_{g} have an Arrhenius form, k_{i}
~ exp(-DE_{i}/kT) where
DE_{i} is an energy barrier for
the process. DE_{i} is
positive for transport (diffusion) indicating a drop in transport rate with
temperature and negative for growth indicating an increase in growth rate with
drop in temperature from the equilibrium crystallization temperature. At any
temperature below the equilibrium crystallization temperature one of the two
processes will have a lower rate and the difference between these two rates is
many orders of magnitude near the crystallization temperature and near the Vogel
temperature (glass transition or another kinetic limiting temperature specific
to the system).

From eqn. (1) were have,

B = k_{d}A/k_{g}
(2).

{*Note: The ratio of the diffusion to the growth rate
constants represents a size which reflects a kinetically determined natural size
for the crystal which is* *called the Keith-Padden
d-parameter (size) in polymer
crystallization and reflects the coarseness of spherulitic growth. A similar
parameter is seen in inorganic, organic and metal crystals, reflecting different
features of crystallization in these systems.*}

We also know that the initial composition of **A**
species (away from the crystal, **A _{i}**) plus the initial
concentration of

A_{i} + B_{i} = C_{f}
(3)

we substitute for **B _{i}** using eqn. (2) at
the initial condition which we assume is at equilibrium prior the initiation of
crystallization,

A_{i} (1 + k_{d}/k_{g}) = C_{f}
(4)

Equation (4) can be solved for **A _{i}**. Then,
by substituting in eqn. (1) the initial rate,

R_{i} = k_{d}A_{i} = k_{d}C_{f}/(1
+ k_{d}/k_{g}) = C_{f}k_{d}k_{g}/(k_{d}
+ k_{g}) = C_{f} (1/k_{d} + 1/k_{g})^{-1}
(5)

The latter equality shows that the overall rate of
crystallization is determined by the rate limiting step, that is, the inverse
sum ensures that the lowest rate constant dominates the process (you should
verify this by adding different values of k_{d} and k_{g} (try 1
and 100, 10 and 10, 1000 and 1 etc.). This kind of series mean value differs
from a parallel average where the two values are summed and divided by 2. The
latter is dominated by the largest value.

Equation (5) guarantees that at least the initial growth
rate will display a maximum. For an infinite source of monomer **A** = **A _{i}**,