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 = kdA = kgB                                     (1)


where kd and kg are exponential functions in temperature reflecting the rate constants for diffusion and growth.  Both kd and kg have an Arrhenius form, ki ~ exp(-DEi/kT) where DEi is an energy barrier for the process.  DEi 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 = kdA/kg                                           (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, Ai) plus the initial concentration of B species (next to the crystal, Bi) will convert, for the most part, to C species at the end of crystallization (crystallized species Cf).


Ai + Bi = Cf                                         (3)


we substitute for Bi using eqn. (2) at the initial condition which we assume is at equilibrium prior the initiation of crystallization,


Ai (1 + kd/kg) = Cf                               (4)


Equation (4) can be solved for Ai.  Then, by substituting in eqn. (1) the initial rate, Ri, can be determined,


Ri = kdAi = kdCf/(1 + kd/kg) = Cfkdkg/(kd + kg) = Cf (1/kd + 1/kg)-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 kd and kg (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 = Ai, B = Bi at steady state and eqn. (5) pertains throughout crystallization.  For a depleting system where monomers are consumed the rate will drop with time exponentially, but at any given instant of time the growth rate will follow a similar curve with a maximum below the ideal crystallization temperature.