Polymer Physics

Although the mixing interaction parameter exhibits the same formulas with the
Flory-Huggins parameter, it contains contributions from both the mixing energy
and the parallel packing energy.
If one takes the approximations of long-chain polymer melt, withr!1,
N 1 !0,N!N 2 r, from (10.9), one obtains

DF

NkT

¼ 1 lnzcþ

ðq 2 Þ^2 Ep

2 qkT

(10.13)

At the equilibrium melting point, the crystallization and melting processes are
balanced with each other, soDF¼0. FromZc 1 þ(q2)exp[Ec/(kT)], one
can calculate the melting pointTm.

1 þðq 2 Þexpð

Ec

kT

Þ¼exp½ 1 þ

ðq 2 Þ^2

2 q



Ep

kT

Š (10.14)

When the coordination numberqis large, one can omit the one at the left-hand
side, and obtain the semi-quantitative result as

Tm

Ecþ

ðq 2 Þ^2

2 q

Ep

k½lnðq 2 Þ 1 Š

(10.15)

One can see that, the largerEcmeans the more rigid polymer chains, resulting in
higher melting points. On the other hand, the largerEpmeans the more regular
sequences, or the smaller substitutes, or the symmetric substitutes, favoring the
compact packing of polymer chains, resulting in higher melting points as well.
Let us compare the equilibrium melting points in association with chemical
structures of some practical polymers. If polymer chains contain larger side groups
that make the internal rotation of the backbone chain more difficult, the chains will
appear more rigid, and their melting points will be higher. Such examples can be
found from the substituted polyolefins –(–CH 2 –CHR–)n–, where the substitute
R¼–H givesTm¼ 146 C, R¼–CH 3 givesTm¼ 200 C, and R¼–CH(CH 3 ) 2
givesTm¼ 304 C. If polymer chains contain rigid groups on the backbone, and the
conjugated rigid groups are longer, the chains appear more rigid, and the melting
points will be higher. Such examples can be found fromTm¼ 146 C for polyeth-
ylene –(–CH 2 –)n–,Tm¼ 375 C for –(–CH 2 – f–CH 2 –)n–, andTm¼ 530 C for
–(–f–)n–. Many conductive polymers even cannot be melted or dissolved, causing
significant difficulty for processing. On the other hand, if polymer chains contain
polar substitutes that cause stronger interactions between molecules, the melting
points of the polymers will be higher. Such examples can be found again from the
polymer –(–CH 2 –CHR–)n–, where the substitute R¼–H givesTm¼ 146 C, R
¼–Cl givesTm¼ 227 C, and R¼–CN givesTm¼ 317 C. Nylon has a high
melting point because of the hydrogen-bonding interactions between chains, while

10.2 Statistical Thermodynamics of Polymer Crystallization 195