Polymer Physics

G¼GgrowthGmelting¼Ggrowthð 1 

Gmelting

Ggrowth

Þ¼Ggrowthð 1 exp

DF

kT

Þ

GgrowthDFGgrowthðllminÞ

Df

kT

ð 10 : 29 Þ

Here, the rate ratio can be further simplified under the assumption that the growth
and the melting share the opposite barriers of the secondary nucleation step. The net
free energy change is small and the further simplification can assign the free energy
gain to the excess lamellar thickness beyond the minimum thicknesslminnecessary
for the thermodynamic stability of the crystal.lmin¼ 2 se/Df, as derived from the
minimum free energy with respect toaorbin (10.20).Ggrowthis named as the barrier
term, while (llmin)Dfis named as the driving force term. Whenl>lminat low
temperatures, the lamellar crystal grows; whenl<lminat high temperatures, the
lamellar crystal melts (Ren et al. 2010 ). The lateral growth profile appears as a
wedge-shape, with the secondary nucleation barrier at the wedge top and the instant
thickening at the wedge root to harvest the free energy, as demonstrated in Fig.10.27.
Recently, with the advanced high-resolution atomic force microscopy, the wedge-

Fig. 10.27 Illustration of the
lateral growth profile of
chain-folding lamellar
crystals, with the secondary
nucleation barrier at the top
and the excess lamellar
thickness harvested instantly
at the root

Fig. 10.28 Height image of
torsional tapping atomic force
microscopy on a sheared PE
film with the shear direction
from down-right to up-left
(Mullin and Hobbs 2011 ).
Thecircleindicates the
wedge-shaped profile at the
growth front of lamellar
crystals (Courtesy of Jamie
Hobbs)

10.4 Kinetics of Polymer Crystallization 213