Polymer Physics

The Ornstein-Zernike approximation also exists in the relationship betweenS(h)
andS(0),

SðhÞ¼

Sð 0 Þ

1 þðhxÞ^2

(9.30)

wherexis the correlation length of the concentration fluctuation. Comparison
between (9.30) and (9.29), one obtains

x¼ð

b^2 Sð 0 Þ

12 f 1 f 2

Þ^1 =^2 ðTTsÞ^1 =^2 (9.31)

This result implies that when the temperature approaches to the spinodal line, the
correlation length of concentration fluctuations diverges.
In the spinodal decomposition mechanism of phase separation, the modulation
of concentration distributions in the stochastic concentration fluctuations exhibits
multiple correlation wavelengths. The concentration modulation with a large wave-
length requires long-distance diffusion of chains, which is relatively slow. On the
other hand, the concentration modulation with a small wavelength is faster, but it
generates too many interfaces, which is energetically unfavorable. Naturally, an
optimized wavelength exists in the concentration modulation, which results in a
periodic distribution of polymer concentrations at the early stage of the spinodal
decomposition.
The optimized wavelength can be calculated from the Ginzburg-Landau free
energy functional considering concentration fluctuations (Ginzburg and Landau
1950 ). The functional adds the interfacial free energy onto the mean-field free
energy, i.e.

DF¼

ð

fDfmþkðrfÞ^2 Šgd

3

r (9.32)

wherekis called the gradient energy coefficient, reflecting the magnitude of
interfacial free energy density. The square term of concentration gradient can be
traced back to the van der Waals work on the non-ideal gas, which is sometimes
called the Ginzburg term or the Cahn-Hilliard term (Cahn and Hilliard 1958 ).
According to Fick’s first law of diffusion, the diffusion flux

Jm¼Drf (9.33)

According to Fick’s second law of diffusion, the diffusion equation

df

dt

¼rJm¼Dðf^00 r^2 f 2 kr^4 fþ:::Þ (9.34)

Its Fourier analytical solution gives

9.2 Kinetics of Phase Separation 175