Polymer Physics

When the phase separation occurs, the curve ofDfmversusf 2 exhibits a common
tangent line at two points of A and B, as illustrated in Fig.9.1b. This implies that at
A and B states,
Dm1A¼Dm1B (9.3)

Dm2A¼Dm2B (9.4)

The common tangent rule above is the thermodynamic condition for the equilib-
rium between A and B phases. The temperature dependence of the concentrations at
A and B states outlines the phase coexistence curve, which is calledthe binodal line.
When temperature is high enough, A and B points will merge at thecritical point
of phase separation. The thermodynamic condition for the critical point is that
partial derivatives of both the first and the second orders for the free energy with
respect to the concentration are equal to zero. From

f 1 þf 2 ¼ 1 (9.5)

and
@^3 Dfm
@f 13

¼kTð

1

r 2 f 22



1

r 1 f 12

Þ¼ 0 (9.6)

@^2 Dfm

@f 12

¼kTð

1

r 1 f 1

þ

1

r 2 f 2

 2 wÞ¼ 0 (9.7)

Solve the above three simultaneous equations, we can obtain

f 2 c¼

ffiffiffiffi

r 1

p

ffiffiffiffi

r 1

p

þ

ffiffiffiffi

r 2

p (9.8)

Fig. 9.1 Illustration of mixing free energy as a function of polymer volume fractions. (a) The
mixing state are stable over allf 2 ;(b)f 2 are stable only at pointsAandBand the regions outside
of these two points

168 9 Polymer Phase Separation