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Density functional theory for systems with mesoscopic inhomogeneities

We study effects of fluctuations on the mesoscopic length-scale on systems with mesoscopic inhomogeneities. Equations for the correlation function and for the average volume fraction are derived in the self-consistent Gaussian approximation. The equations are further simplified by postulating the expression for the structure factor consistent with scattering experiments for self-assembling systems. Predictions of the approximate theory are verified by a comparison with the exact results obtained earlier for the one-dimensional lattice model with first-neighbour attraction and third-neighbour repulsion. We find qualitative agreement for the correlation function, the equation of state and the dependence of the chemical potential $μ$ on the volume fraction $ζ$. Our results confirm also that strong inhomogeneities in the disordered phase are found only in the case of strong repulsion. The inhomogeneities are reflected in an oscillatory decay of the correlation function with a very large correlation length, three inflection points in the $μ(ζ)$ curve and a compressibility that for increasing $ζ$ takes very large, very small and again very large values.