As illustrated in Fig.6.16a, melting transition occurs at the high-temperature
region, and below the melting point, the entropy difference between the
supercooled liquid and the crystal is
DS¼SlSc¼DSmþðTTmCpðlÞCpðcÞ
TdT (6.53)Melting is normally driven by an entropy gain, thenDSm>0. With the decrease
of temperatures fromTm, the integral at the right-hand side of (6.53) decays
gradually from zero toDSm, as demonstrated in Fig.6.16b. However, a linear
extrapolation toDS¼0 reaches a finite temperature rather than zero absolute
temperature, which can be defined asTs. This result implies that belowTs,Sl<SC.
Apparently, the amorphous liquid state could not be more ordered than the crystal-
line solid state, which is against the third law of thermodynamics. Early in 1931,
Simon pointed out this problem (Simon 1931 ). In 1948, Kauzmann gave a detailed
description, and proposed that there should exist a phase transition such as crystal-
lization before extrapolation toTsto avoid this disaster (Kauzmann 1948 ). There-
fore, this scenario is also called theKauzmann paradox.
In 1956, Flory introduced the energy parameterEcof the chain semi-flexibility into
the lattice model of polymer chains, to explain the spontaneous crystallization behav-
ior of polymer chains (Flory 1956 ). He calculated the total numberWwith combina-
torial methods from the conformational statistics of lattice polymers, and found that
when the volume fractionf 2 of polymers approaches one, the disorder parameterf
decreases with temperature, and soon there occurs lnW<0(i.e.W<1) (see the
definition in (8.55)). This result was named as “entropy catastrophe”. Gibbs thought
that the result corresponded exactly to the situation described by the Kauzmann
paradox, and definedT 2 at the condition of lnW¼0. WhenT<T 2 ,W¼1, and
the system will be vitrified with the disordered state atT 2 (Gibbs 1956 ). Gibbs and
DiMarzio further applied Huggins approximation to calculateT 2 , and proved that at
T 2 , the system entropy continues but the heat capacity discontinues, appearing as a
typical second-order thermodynamic phase transition (Gibbs and DiMarzio 1958 ).
Fig. 6.16 Illustration of (a) heat capacity changes and (b) the entropy differencesDSbetween the
liquid and the solid changing with temperatureT
112 6 Polymer Deformation