Mesoscopic Non-Equilibrium Thermodynamics of Activated Processes

Mesoscopic non-equilibrium thermodynamics provides a description of activated processes. In the case considered here, when crystallization proceeds by the formation of spherical clusters, the process can be characterized by a coordinate y, which may represent for instance the number of monomers in a cluster, its radius or even a global-order parameter indicating the degree of crystallinity. Polymer crystallization can be viewed as a diffusion process through the free energy barrier that separates the melted phase from the crystalline phase. From mesoscopic non-equilibrium thermodynamics we can analyze the kinetic of the process. Before proceeding to discuss this point, we will illustrate how the theory applies to the study of general activated processes. 

Since activation kinetics is affected by the presence of fluctuations in the metastable phase, we will provide a description of the system in terms of the probability distribution function P(y,t). Our task will be then to derive the expression of the activation current, a quantity accessible to experiments in many instances. To this end, our starting point is the expression for the Gibbs equation accounting for entropy variations due to the underlying diffusion process of the probability density 

is the free energy barrier and k is Boltzmann s constant. The presence of impingement effects can be taken into account by introducing an activity coefficient in the chemical potential as done for non-ideal mixtures. 

The evolution of the probability density is governed by the continuity equation 

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