Polymer Physics

which is called RPA equation.SD(h,r)is the Debye function that describes the
structure factor of each ideal chain, as given by

SDðQÞ¼

2 ½expðQÞþQ 1 Š

Q^2

(9.23)

Here,Q¼h^2 Rg^2 is related with the radius of gyration of the polymer coilRg.
Based on the analysis above, the Flory-Huggins interaction parameter of the blend
can be directly derived from the experimentally determined scattering intensity.
Whenh!0,SD(h,r)¼1, and then

1

Sð 0 Þ

¼

1

r 1 f 1

þ

1

r 2 f 2

 2 w¼ 2 ðwswÞ (9.24)

The extrapolation of scattering intensity to the lower limit of scattering angles gives
the Flory-Huggins interaction parameter. The experimental definition of this value is

wSANS¼

1

2

ð

1

r 1 f 1

þ

1

r 2 f 2



1

Sð 0 Þ

Þ (9.25)

According to (9.16), (wsw)~(TTs), the divergence ofS(0)indicates the
temperature approaching the spinodal line. At this moment, the sample quickly
becomes opaque. Therefore, the spinodal temperature at a specific concentration
can be measured by the scattering experiment.
At the very large scattering vectorh,

SD

2

Q

¼

2

h^2 R^2 g

(9.26)

In the normal situation, the simple addition of two extreme cases has been taken
as an approximation,

SD^1 ¼ 1 þ

h^2 R^2 g

2

(9.27)

That means

SD¼

1

1 þ

h^2 R^2 g

2

(9.28)

This equation is known as Ornstein-Zernike approximation (Ornstein and
Zernike 1914 ). InsertingRg^2 ¼rb^2 /6 into the RPA equation (9.22), one obtains

1

SðhÞ

¼

1

r 1 f 1

þ

h^2 b^2

12 f 1

þ

1

r 2 f 2

þ

h^2 b^2

12 f 2

 2 w¼

1

Sð 0 Þ

þ

h^2 b^2

12 f 1 f 2

(9.29)

174 9 Polymer Phase Separation