Kinetics of nucleated assembly

A tell-tale sign of nucleated assembly is the existence of a critical concentration below which assemblies do not form in measurable quantities. Another is the observation of hysteresis in assembly and disassembly experiments 

In the phenomenological theory of phase transitions, it is customary to attribute order parameters to the relevant thermodynamic quantities associated with the macroscopic transition of the state of the material. In our case, 

Let the order parameter Si be associated with the fraction active material, and be defined as S =f. For the second order parameter we choose S2 = Na (Na — 1 )Ka/X as a measure of the degree of polymerization of the active material and presume that the polymerization is highly co-operative, so K.a — 0. The macroscopic Landau free-energy density F that describes the nucleated assembly now reads 

The obvious ( model A ) set of relaxational kinetic equations are (Flohenberg and Flalperin, 1977) 

If the order parameter Si relaxes instantaneously, implying fast conversion of the inactive species into the active one and slow elongation, we find for the time dependence of the fraction active material 

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