Polymer Physics

wc¼

1

2

ð

1

ffiffiffiffi

r 1

p þ

1

ffiffiffiffi

r 2

p Þ

(^2) (9.9)
For a symmetric polymer blend, two components share the same molecular
weights, i.e.,r 1 ¼r 2 ¼r, and the critical point becomes
f2c¼

1

2

(9.10)

wc¼

2

r

(9.11)

The extreme case of asymmetric blends is polymer solutions withr 1 ¼1 and
r 2 ¼r. The critical point becomes

’ 2 c¼

1

r^1 =^2 þ 1

(9.12)

wc¼

1

2

þ

1

ffiffi

r

p þ

1

2 r

(9.13)

Whenr!1,

f 2 c! 0 (9.14)

wc!

1

2

(9.15)

Atw¼1/2, the second Virial coefficient in (8.43) becomes zero. Therefore, it is
clear that the theta point of dilute polymer solutions locates in the vicinity of the
critical point of the phase separation. Similarly, polymers in the bulk phase are also
near the critical point. Owning to the strong thermal fluctuations near the critical
point, the long-range correlation could occur along the polymer chains. Therefore,
polymers in both melt and solution phases exhibit typical chain-length scaling laws
with regard to their conformations and motions.
According to (9.11) and (9.13), an increase in molecular weights of polymers
gives rise to the decrease ofwC. In a concentrated solution of polydisperse
polymers, the interaction parameter can be gradually raised by either decreasing
temperatures or by adding droplets of a precipitant agent (a poor solvent to increase
the mixing interactionB). Accordingly, the high molecular weight fraction will
meet the critical condition of phase separation first, and precipitate from the
solution, as illustrated in Fig.9.2. This is the principle of precipitation fractionation
of polydisperse polymers.
The critical point obtained from the Flory-Huggins equation can well explain the
critical condition for phase separation upon temperature drop. This critical point is

9.1 Thermodynamics of Phase Separation 169