Nucleation and growth of many crystals

The total volume growth of crystals in a melt is the sum of the volume growth of all individual crystals. For a crystal nucleated at time x with linear growth rate u (which may depend on time), the radius and volume of crystal i at time t can be written as 

However, the melt volume shrinks as crystallization proceeds. Hence, one can write 

Ignoring the density difference between the melt and the solid, we have V s + Vm=t o, where l o is the initial volume of the melt. Let F=VsIVq we have 

The parameter y (given in Equation 4-64) is zero at time zero and increases with time. In different nucleation and growth regimes, the rate of increase is different. Hence, it is assumed that y is proportional to time raised to some power n, leading to 

The above discussion is for the crystallization of one single mineral (i.e., many crystals of the same mineral). If several minerals are crystallizing and they have different crystallization temperature (and hence different crystallization time), then the relation between the degree of crystallization and time would be much more complicated. 

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