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Mesoscopic theory for systems with competing interactions near a confining wall

Mesoscopic theory for self-assembling systems near a planar confining surface is developed. Euler- Lagrange (EL) equations and the boundary conditions (BC) for the local volume fraction and the correlation function are derived from the DFT expression for the grand thermodynamic potential. Various levels of approximation can be considered for the obtained equations. The lowest-order nontrivial approximation (GM) resembles the Landau-Brazovskii type theory for a semiinfinite system. Unlike in the original phenomenological theory, however, all coefficients in our equations and BC are expressed in terms of the interaction potential and the thermodynamic state. Analytical solutions of the linearized equations in GM are presented and discussed on a general level and for a particular example of the double-Yukawa potential. We show exponentially damped oscillations of the volume fraction and the correlation function in the direction perpendicular to the confining surface. The correlations show oscillatory decay in directions parallel to this surface too, with the decay length increasing significantly when the system boundary is approached. The framework of our theory allows for a systematic improvement of the accuracy of the results.