Physical Processes - Isoconversional Kinetics of Thermally Stimulated Processes - page 103

mers requires an empirical adjustment, when the actual maximum extent of crystal-
linity is taken as ʱ = 1. In addition to the rate constant, the Avrami analysis yields
the Avrami exponent, which according to the theory can take some specific values
from ﾽ to 4. Note that even in this case, the values of m do not allow for singular
mechanistic interpretation [ 93 , 94 ]. It is also not uncommon when the m values vary
markedly with temperature. All in all, the Avrami analysis is rather “a convenient
representation of experimental data” [ 102 ] than a way of obtaining physical insights
in the polymer crystallization kinetics.
A widely accepted kinetic theory of polymer crystallization was developed by
Hoffman and Lauritzen [ 103 , 104 ]. The theory makes use of the Turnbull-Fisher
model (Eq. 3.42) and adjusts it to the chain folding mechanism that drives crystal-
lization of polymers. The basic equation of the theory describes the temperature
dependence of the growth rate of polymer spherulites as follows:

*

K
TTf

−

U
RT T

−

g

(3.52)

ΛΛ =

0 exp

exp ,

( − ) ∆

∞

where Λ 0 is the preexponential factor, U * is the activation energy of the segmental
jump, Δ T = T m − T is the supercooling, f = 2 T /( T m + T ) is the correction factor, and T ∞
is a hypothetical temperature where motion associated with viscous flow ceases that
is usually taken 30 K below the glass transition temperature, T g . The kinetic param-
eter K g has the following form:

nb σσ

∆

T
hk

(3.53)

em

K = ,

g

fB

where b is the surface nucleus thickness, ˃ is the lateral surface free energy, ˃ e is
the fold surface free energy, T m is the equilibrium melting temperature, Δ h f is the
heat of fusion per unit volume of crystal, k B is the Boltzmann constant, and n takes
the value 4 for crystallization regime I and III, and 2 for regime II. The dependence
of the growth rate on temperature passes through a maximum (Fig. 3.36 ) in similar
fashion as the rate of nucleation (Fig. 3.32 ).
The parameter U * is typically assumed to have the universal value 6.3 kJ mol −1
(i.e., 1.5 kcal mol − 1 ) [ 103 ]. This assumption in combination with little algebra af-
fords Eq. 3.52 to be transformed to Eq. 3.54:

*

K
TTf

U
RT T

(3.54)

g

ln Λ + = ln Λ − .

0

( − ) ∆

∞

Then K g can be determined from the linear plot of the left-hand side of Eq. 3.54
against ( T Δ Tf ) −1 . The equation is known [ 103 ] to describe adequately the growth
kinetics in a range of supercoolings as wide as 40-100 ﾰC. This means that at least
potentially both melt and glass crystallization kinetics can be described by a single
set of the constant parameters U * and K g .