Polymer Physics

The latter is also calledthe kinetic crystallinitythat can be measured by tracing
the crystallization process with dilatometer, depolarized light intensity, dynamic
X-ray diffraction and DSC. Polymer crystallization is a volume-contraction process.
When the dilatometer is used to measure the change of the sample volume with time
evolution at a constant crystallization temperature, one obtains the relative crystal-
linity as

Xc¼

Vð 0 ÞVðtÞ

Vð 0 ÞVð1Þ

(10.33)

The results are summarized in Fig.10.29. The characteristic timet1/2, the time
scale for the crystallinity to reach its half value, can be used as its reciprocal to
represent the total crystallization rate.
By use of the Poison distribution, Avrami derived the famous Avrami phenome-
nological equation to treat a kinetic process (Avrami 1939 , 1940 , 1941 ).
Kolmogorov first discussed the formulation of this equation (Kolmogorov 1937 ).
Johnson and Mehl also made similar derivation independently (Johnson and Mehl
1939 ). Evans proposed a very concise derivation as introduced below (Evans 1945 ).
The isokinetic condition was assumed in the derivation. Under isothermal crystal-
lization, the heterogeneous nucleation generates a fixed number of centers for
spherulite growth with a constant linear growth rate till the impingement. At the
early stage before any impingement happens, as illustrated in Fig.10.30, the proba-
bility for any point locating outside of theith spherulite in the space volumeVcan be

Pi¼ 1 

Vi

V

(10.34)

Here,Viis the volume of thei-th spherulite. Then, the probability for any point
locating simultaneously outside of all themnumber of spherulites is

P¼P 1 P 2 ...Pm¼

Y

ð 1 

Vi

V

Þ (10.35)

Fig. 10.29 Illustration of the
time evolution of the relative
crystallinity

10.4 Kinetics of Polymer Crystallization 215