Mathematical Description of Primary Nucleation

The thermodynamic description of a nucleus as a small crystal is given in Fig. 3.59. It shows a drawing of a prismatic nucleus with a quadratic cross-section, a. Its free enthalpy, G, can be expressed as a function of dimensions a and (, where the length (is taken always along the molecular chain axis. The term - a CAgf represents the bulk 

A = surface area 8 = specific surface free energy = free enthalpy of fusion (per unit volume) 

The rate of nucleation, I, has been estimated by Turnbull and Fisher [25] from the shape of AG and the influence of local viscosity, governed by a free enthalpy AG,. In Fig. 3.61 this equation is listed. The rate I applies to the case that nucleation is unhindered. The first exponent of the equation expresses retardation of nucleation due to viscosity effects with the given parameters (see Sect. 5.6). It stops nucleation as the glass transition temperature is approached. The nucleation described is homogeneous nucleation and creates a continuous stream of new crystals in the remaining melt or solution. For polymers it takes a supercooling of about 50 K to overcome the 

The right figure gives more quantitative kinetics, gained from isothermal experiments. The fraction of droplets crystallized, n/n, can then be estimated from  

AG (Y Ye 11x10 J cm ). Other polymers may need much larger supercooling for homogeneous nucleation than the just analyzed polyethylene. Typical examples are 105 K for polypropylene, 100 K for nylon 6, 70 K for poly(oxymethylene), and 68 K for poly(oxyethylene). 

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