Polymer Physics

The traditional dynamic theories treat the molecular motion as a relaxation process
reflected as a characteristic relaxation timet. The decrease in temperature causes the
increase of the relaxation time. For the Arrhenius-type liquids, one obtains

texpð

C

kT

Þ (6.66)

whereCis the potential energy barrier for monomer motions (Goldstein 1969 ). For
fragile liquids,Cchanges with temperature, while for strong liquids,Cremains rather
constant. If the cooling rateqis constant, the time interval for each step of the
temperature scan is |q|^1 .When|dt/dT|<|q|^1 , the fluctuations of the system can be
relaxed in time, and local regions can reach equilibrium. With the decrease of
temperatures, |dt/dT|increases.When|dt/dT|>|q|^1 , the fluctuations of the system
are hardly relaxed in time, and the local regions stay in non-equilibrium states. The
transition from equilibrium to non-equilibrium states can be regarded as the glass
transition. Staying in the non-equilibrium states, the molecular motions cannot be
relaxed, as in the frozen states. This rationale can explain the heating/cooling rate
dependence ofTg.When|qA|>|qB|, one often observesTgA>TgB.Thisisbecause
the decrease of critical relaxation time corresponds to higherTg, empirically one has,

dlnjjq

d

1

Tg

¼

Dh

R

(6.67)

whereRis the gas constant,Dhis the relaxation enthalpy. This implies that when
q!0, one should getTg!0, i.e. no thermodynamic transition at the low-
temperature region.
Adams and Gibbs regardedCin (6.66) as the ratio of the free energy barrier for the
cooperative motion of each molecule to the total configuration entropy of cooperative
rearrangement (Adams and Gibbs 1965 ). This treatment facilitates the interpretation
to the chemical-structure dependence of the glass transition property, for instance,
evidencing the definition above in the Gibbs-DiMarzio theory

T 2 ¼Tg 55  10 % (6.68)

According to WLF equation, whenT¼Tg51.6 K,!1. DiMarzio and
Yang further proposed thatCincludes the configuration entropy of Helmholtz free
energy (DiMarzio and Yang 1997 ).
In the equilibrium liquid phase, the states of molecules can switch into each
other along any thermodynamic route in the phase space. Such a property is called
the ergodicity. The frozen glass state can be regarded as the situation of non-
ergodicity. Such a symmetry break can be theoretically treated by employing the
mode-coupling theory (Go ̈tze and Sjo ̈gren 1992 ), which derives the critical transi-
tion point close to

Tc 1 : 2 Tg (6.69)

6.3 Glass Transition and Fluid Transition 115