Crystallization of Polymers. Volume 1, Equilibrium Concepts

66 Fusion of homopolymers
can be neglected. By expanding ln(1 − ζρ/xN) Eq. (2.37) can be written as
G f
ζρ
= G u − 2σ ec
− RT
ζ x
+ RT [ ( )]
x − ζ + 1
ln
(2.38)
ζ x
where ln D ≡−2σ ec /RT. Here σ ec represents the interfacial free energy of the basal
plane associated with the nonequilibrium crystallite of thickness ζ . This interfacial
free energy cannot be identified a priori with the interfacial free energy of
the equilibrium crystallite, σ eq , since the corresponding surface structures are not
necessarily the same. The first term in Eq. (2.38) represents the bulk free energy of
fusion for the ζρ units. The second term represents the excess free energy due to
the interfacial contribution of the chains emerging from the 001 crystal face (the
basal plane). The last two terms result from the finite length of the chain and are
only significant at low molecular weights. The first of these represents the entropy
gain which results from the increased volumes available to the ends of the molecule
after melting. The last term results from the fact that only a portion of the units
of a given chain are included in the crystallite. It represents the entropy gain that
arises from the number of different ways a sequence of ζ units can be located in
a chain x units long with the stipulation that terminal units are excluded from the
lattice.
At the melting temperature Tm ∗ of the nonequilibrium crystallite, G f = 0so
that Eq. (2.38) becomes
1
T ∗ m
− 1
T 0 m
= R
H u
[
2σec
RT ∗ m ζ − 1 ζ
( x − ζ + 1
x
)
+ 1 ]
x
(2.39)
Equation (2.39) represents the relation between the melting temperature and crystallite
thickness ζ for different chain lengths. The crystallite thickness ζ is not
constrained to its equilibrium value and σ ec is characteristic of the particular interface
that is developed in the crystallite under the specific set of crystallization
conditions. The melting temperature depression, Eq. (2.39), is calculated from
the equilibrium melting temperature of the infinite chain, Tm 0 . For high molecular
weights Eq. (2.39) reduces to
or
1
T ∗ m
− 1
T 0 m
= 2σ ec
H u T ∗ m ζ (2.40)
T ∗ m = T 0 m [1 − 2σ ec/H u ζ ] (2.41)
Equation (2.40) is identical to the classical Gibbs–Thomson expression for the melting
of crystals of finite size. Thus, following the Flory theory (10) nonequilibrium
crystallites of high molecular weight chains obey the same melting point relation