Physical Processes - Isoconversional Kinetics of Thermally Stimulated Processes - page 110

Fig. 3.41 Schematic depiction of a cylindrical nucleus of the radius r formed inside a lamellar
crystal of the thickness l . ˃ and ˃ e represent the free energy of the lateral and the fold surface,
respectively

The melt nucleation model is derived as follows [ 127 ]. Similar to Eq. 3.33, the
free energy barrier to the formation of a melt nucleus consists of the surface and
volume components:

∆∆ ∆ GGSGV = + , (3.61)

S V

where Δ G S and Δ G V are the free energy per unit area and unit volume, respec-
tively, and S and V are the nucleus area and volume. The melt nucleus is assumed
to have the shape of a cylinder and form inside a lamellar crystal whose thickness is
l (Fig. 3.41 ). The assumption of the cylindrical shape is quite common for crystal-
line polymers because they crystallize by chain folding so that a crystalline nucleus
presents itself as several chain folds of about the same height. Under this assump-
tion, Eq. 3.61 can be written as:

2 2

∆ Grl = 2 πσ πσπ

e

− 2 r + rl G ∆

V ,

(3.62)

where the first and second terms represent respectively the lateral (side) and fold-
ing (top and bottom) surface free energy of a cylinder. The second term is negative
because during melting, the folding surface disappears, merging with the surround-
ing melt. The third term represents the volume free energy. Its value depends on
temperature as follows:

0

∆ GH TT
T

−

(3.63)

m

= .

V f

0

m

Equation 3.63 is obtained the same way as Eq. 3.38, the only difference being that
for melting Δ H V = Δ H f . When lamellar crystal is sufficiently thin ( l is very small), it
can melt at temperature T m below reaching the equilibrium melting temperature T m 0 .
The temperature difference is determined by the Gibbs-Thomson equation: [ 128 ]

0

TT T
lH

−= σ
∆

2

(3.64)

0 em

.

mm

f