Model-Based Analysis: Crystallization Kinetics

The rate of chemical reaction for each crystallization step j can be written as the product of two functions, where the first function fj(ej,pj) depends on the concentrations of reactant (ej) and product (pj). The second function Kj(T)depends on temperature.

(1)

Where

  • ej - concentration of non-crystallized material

  • pj – concentration of product (crystallized material)

The dependence on concentrations f(e,p) can be described by the following crystallization types:

Nk: (2) Nakamura Crystallization contains Avrami nucleation

SbC: (3) Sbirrazzuoli Crystallization, contains Sestak-Berggren equation

Where

  • n is the dimension of nucleation

Function K(T) is non-Arrhenius dependence on temperature.

If K(T) is known then from (2) it for cooling rate β is easy to get Nakamura equation for degree of crystallization α:

(4)

For both Nk and SbC types the analytical dependence K(T) according to the Hoffman-Lauritzen theory is used:

(5)

Where

  • A is the pre-exponential factor

  • U – activation energy of segmental jump in polymers, this parameter has universal value 6.3kJ/mol

  • T∞ =Tg-30 -  temperature at which crystallization transport is finished, this temperature is 30K below the glass transition temperature Tg.

  • KG – kinetic parameter for nucleation

  • ∆T=Tm-T - undercooling from the equilibrium melting point Tm

  • f=2T/(Tm+T) – correction factor.

 

Parameters:

  • Melting temperature Tm and glass transition temperature Tg must be written by the user. Melting temperature can be then a little bit optimized by the optimization of the model.

  • KG, A, and n will be found by the software.

  • U=6.3kJ/mol is the universal value which is fixed and does not changes during optimization

 

Equation (4) has positive values only for the temperature range between T∞ and Tm. Above melting temperature the material is in liquid state and no any crystallization happens. Below T∞ the material is in the glassy state, where any viscous motion is finished, and here is no any crystallization too.

 

Related Literature:

[1] NAKAMURA, K., WATANABE, T., KATAYAMA, K., AMANO, T., Some aspects of non-isothermal crystallization of polymers — Part I: Relationship between crystallization temperature, crystallinity and cooling conditions, Journal of Applied Polymer Science, Vol. 16, pp. 1077-1091, 1972

[2] Patel, R.M., Crystallization kinetics modelling of high density and linear low density polyethylene resins. Journal of Applied Polymer Science 2011, 124(2): 1542-1552.

[3] Vyazovkin S., Sbirrazzuoli N. 2004 Isoconversional Approach to Evaluation the Hoffman-Lauritzen Parameters (U* and Kg) from the Overall Rates of nonisothermal Crystallization, Macromolecular Rapid Communications, 2004, 25. 733-738.

[4] Nathanael Guigo, Jesper van Berkel, Ed de Jong, Nicolas Sbirrazzuoli 2017 Modelling the non-isothermal crystallization of polymers: Application to poly(ethylene 2,5-furandicarboxylate), Thermochimica Acta, 2017, V650. 66-75.