Relation to Ginzburg-Landau Models

In order to make the connection to the Landau-Ginzburg theory for binary blends, we study the behavior of the structure factor at small wavevectors q for a symmetric mixture. Using g q) 1 -F +. .. obtain  

If we assume composition fluctuations to be Gaussian, we can write down a free energy functional compatible with Eq. 91. 

Note that this free energy functional is Gaussian in the Fomier coefficients of the composition, and, hence, the critical behavior still is of mean-field type. Transforming back from Fourier expansion for the spatial dependence to real space, we obtain for the free energy functional  

Expanding the free energy of the homogeneous system (Flory-Huggins free energy of mixing), we can restore higher order terms and obtain the Landau- 

A field-theory based on this simple expansion will already yield three-dimensional Ising critical behavior. Note that the non-trivial critical behavior is related to fluctuations vaW or a- I b- Fluctuations in the incompressibility field U or the total density a + are not important. 

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