Polymer Physics

The second contribution is from the conformational entropy of deformed polymer
chains due to the separation of two blocks at the two sides of the interfaces.
Accordingly,

DSstrð

R

R 0

Þ^2 ð

d

R 0

Þ^2 (9.46)

Here assumeR/dand the ideal coil sizeR 02 ¼rv2/3. The total free energy
contribution

DF

kT

w^1 =^2 v^1 =^3

r

d

þ

d^2

rv^2 =^3

(9.47)

Taking the minimum free energy with respect tod, one obtains

dr^2 =^3 v^1 =^3 w^1 =^6 (9.48)

The scaling relationship betweendand the chain lengthrhas been well verified
by the experimental observations. The total free energy after the phase separation in
diblock copolymers is scaled as

DF

kT

ðwrÞ^1 =^3 (9.49)

From the Flory-Huggins equation, the free energy of the homogeneous mixture
(denoted as the disordered state) of binary blends in parallel to diblock copolymers
isDFm~wr, while the free energy of microphase-separated state (denoted as the
ordered state) isDFm~(wr)1/3. Herewris normally called thesegregation strength.
With the increase of the segregation strength starting from zero, the disordered state
exhibits a relatively smaller free energy than the ordered state at the beginning, and
is more stable. There exists a critical condition(wr)cfor the phase transition, above
which the ordered state exhibits a smaller free energy and becomes more stable, as
illustrated in Fig.9.13. The critical segregation strength calculated from the self-
consistent-field theory is related only to the chain length (Fredrickson and Helfand
1987 ), as given by

ðwrÞc 10 : 5 þ 41 r^1 =^3 (9.50)

The microphase separation of block copolymers is sometimes calledorder–disorder
transition(ODT). The self-consistent-field theory (SCFT) provides a mean-field
method to calculate various geometric shapes of microdomains. Edwards first
introduced the SCFT into polymer systems on the basis of making path integrals
along chain conformations (Edwards 1965 ). Helfand applied it to the mean-field
description of immiscible polymer blends on the basis of the Gaussian-chain model

9.3 Microphase Separation of Diblock Copolymers 181