Polymer Physics

polymer motion caused by the difference in the coefficients of thermal expansion
between the polymer and the solvent. This mechanism can be described by the
equation-of-state theories (the Flory-Orwoll-Vrij theory, see Sect.8.3.2) based on
the compressibility of polymer solutions. Further, some LCST phenomena in the
blends of polar polymers can be explained by the lattice-cluster theory considering
the geometrical mismatch between the chain units of the two components
(see Sect.8.3.5).

9.2 Kinetics of Phase Separation

The thermodynamic instability of a mixture does not mean immediate occurrence of
the phase separation (Gibbs 1961 ). The influence of minute concentration
fluctuations on the free energy dictates the dynamic instability of the mixture. The
thermal fluctuations cause local concentrations to deviate from the average concen-
trationf. When the local curvature of the free energy curve opens to the downside,
for instance, the CD segment on the curve shown in Fig.9.1b, the average free
energy caused by local fluctuations appears lower than the initialDfm. The decrease
of average free energy implies the dynamic instability of that state, which triggers an
immediate phase separation, as illustrated in Fig.9.4a. In this case,

@^2 Dfm

@f^2

¼

@m

@f

< 0 (9.17)

which implies that the two components at the interfaces spontaneously diffuse
towards the direction of higher concentrations (uphill diffusion). Any minute con-
centration fluctuation is thus enlarged, leading to large-scale phase separation. Such
a mechanism of phase separation is known asspinodal decomposition(SD) (Cahn
1968 ; Hilliard 1970 ). Therefore, zero second derivative of free energy with respect
to the change of concentration is the boundary condition between the metastable and
unstable states. In Fig.9.4b, concentrations at the inflection points C and D are
changing with temperature, which constitute the critical curve called thespinodal
line, as illustrated in Fig.9.5. The thermodynamic conditions of the spinodal line can
be obtained from the Flory-Huggins-Scott equation of binary blends, as given by

@^2 Dfm

@f 12

¼kTð

1

r 1 f 1

þ

1

r 2 f 2

 2 wsÞ¼ 0 (9.18)

Thus

ws¼

1

2

ð

1

r 1 f 1

þ

1

r 2 f 2

Þ (9.19)

9.2 Kinetics of Phase Separation 171