Handbook of Solvents - George Wypych - ChemTech - Ventech!

254 Nobuyuki Tanaka
5.2.2 GLASS TRANSITION IN POLYMERS
The glass transition in polymers is the same kind of physical phenomenon as observed generally
for amorphous materials. 12 At Tg in the cooling process, polymers are frozen glasses
and the molecular motions are restricted strictly. However, the actual states of glasses are
dependent on the cooling rate; if the cooling rate is rapid, the glasses formed should be imperfect,
such as liquid glasses or glassy liquids. 13,14 The annealing for imperfect glasses results
in the enthalpy relaxation from imperfect glasses to perfect glasses. At Tg in the
heating process, the strong restriction of molecular motions by intermolecular interactions
is removed and then the broad jump of heat capacity, Cp, is observed. 15 Annealing the
glasses, the Cp jump curve becomes to show a peak. 15,16
5.2.2.1 Glass transition enthalpy
For polymer liquids, the partition function, Ω, normalized per unit volume is given
by: 10,14,17,18
N
2
3Nx / 2
Nx
int
( Z / N! )( 2 mkT / ) ( q / vf) exp { Nxh /( RT)
}
Ω= π h −
[5.2.5]
with vf = qv exp{-h int /(RT)}
where:
h int
the intermolecular cohesive enthalpy per molar structural unit for a polymer
h Planck’s constant
k Boltzmann’s constant
m the mass of a structural unit for a polymer
N the number of chains
q the packing factor of structural units for a polymer
R the gas constant
vf the free volume per molar structural unit for a polymer
x the degree of polymerization
Z the conformational partition function per a chain
From equation [5.2.5], the enthalpy and the entropy per molar chain for polymer liquids,
Hl and Sl, are derived: 10
2 2
H = RT dln Z / dT + ( 3/ 2)
RxT − RxT dln v / dT + xh
1
( ) ( 3 2)
( )
f
int
[5.2.6]
S = R ln Z + RTd ln Z / dT + / Rx − x R lnv + RTd ln v / dT + xS [5.2.7]
1
f f d
with S d = (3R/2)ln(2πmkT/h 2 ) - (1/x)(R/N)lnN! + Rlnq
The first terms on the right hand side of equations [5.2.6] and [5.2.7] are the
conformational enthalpy and entropy per molar chain, xh conf and xs conf , respectively. 19 Assuming
that chains at T g are in quasi-equilibrium state, the criterions on T g are obtained:
f ( = h −T s ) ≈0 [5.2.8]
flow flow g flow
and s flow ≈ 0 (hence h flow ≈ 0) [5.2.9]
with h flow =H l/x - 3RT g/2 and s flow =S l/x - 3R/2
From equations [5.2.8] and [5.2.9], which show the conditions of thermodynamic
quasi-equilibrium and freezing for polymer liquids, the conformational enthalpy and entropy
per molar structural unit at T g,h g conf and sg conf , are derived, respectively: