Kinetic factors determining the formation of polymorphic modifications

3 Kinetic factors determining the formation of polymorphic modifications 

The existence of the phase boundary between the solid and liquid phase complicates matters, since a phase boundary is associated with an increase in free energy of the system which must be offset by the overall loss of free energy. For this reason the magnitudes of the activated barriers are dependent on the size (i.e. the surface to volume ratio of the new phase) of the supramolecular assembly (crystal nucleus). This was recognized in 1939 by Volmer in his development of the kinetic theory of nucleation from homogeneous solutions and remains our best model today (Volmer 1939). 

One of the key outcomes of this theory is the concept of critical size which must be achieved by an assembly of molecules in order to be stabilized by further growth. The higher the operating level of supersaturation the smaller is this size (typically a few tens of molecules). Now, in Fig. 2.9 the supersaturation with respect to II is simply Go - Gn and is lower than Go - Gj for structure I. However it can now be seen that if for a particular solution composition the critical size is lower for II than for I then the activation free energy for nucleation is lower and kinetics will favour form II. Ultimately form II will have to transform to form I, a process that we discuss later. Overall we can say that the probability that a particular form i will appear is given by 

The rates of nucleation as expressed by the classical expression of Volmer are related to various thermodynamic and physical properties of the system such as surface free energy (y), temperature (T), degree of supersaturation (a), solubility (hidden in the 

From this analysis it is clear that the trade-off between kinetics and thermodynamics is not at all obvious. Consider a monotropic, dimorphic system (for simplicity) whose solubility diagram is shown schematically in Fig. 2.10. It is quite clear that for the occurrence domain given by solution compositions and temperatures that lie between the form II and I solubility curves only polymorph I can crystallize. However, the outcome of an isothermal crystallization that follows the crystallization pathway indicated by the vector in Fig. 2.10 is not so obvious since the initial solution is now supersaturated with respect to both polymorphic structures, with thermodynamics favouring form I and kinetics (i.e. supersaturation) form II. 

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