Polymer Physics

1 Xc¼expðm

<Vi>

V

Þ (10.37)

For athermal nucleation (self-seeding nucleation as a special case of heteroge-
neous nucleation), the number of nuclei is fixed. Thus the number of spherulites is
fixed, with a densitym/V,¼ 4 pr^3 /3,r¼vt, and then

m

<Vi>

V

¼

4 pv^3 m

3 V

t^3 ¼Kt^3 (10.38)

Inserting the equation above into (10.36), one derives the general Avrami
equation as given by

1 Xc¼expðKtnÞ (10.39)

The Avrami equation can be applied to treat the time evolution of crystallinity in
the self-acceleration process right after the incubation periodt 0 for the initiation of
crystallization. Here,Kis the rate constant, which is related to the nucleation rate,
the linear growth rate as well as the number of nuclei;nis called the Avrami index,
which is related to the mechanism of crystal nucleation and the dimensionality of
crystal growth. When thermal nucleation generates new growth centers at a con-
stant rate, n¼4. Linear regression to the relative crystallinity in the self-
acceleration stage measured by experiments gives the Avrami index as well as
the total crystallization rate, as

Fig. 10.30 Illustration of the
i-th spherulite induced to grow
in the space volumeV.The
labelarepresents the
amorphous phase and the label
crepresents the crystalline
phase

216 10 Polymer Crystallization