Renormalization of a Landau-Ginzburg-Wilson theory of microemulsion

We study the effects of fluctuations on the mean-field phase diagram of a Landau-Ginzburg-Wilson (LGW) action of the form H=1/2${\mathcal{F}}_{\mathbf{q}}$‖${\mathit{cphi}}_{\mathbf{q}}$ ${\mathrm{\Vert }}^2$(${\mathbf{q}}^4$+b${\mathbf{q}}^2$+c)dRdq +λ/4!${\mathcal{F}}_{\mathbf{q}}$ ${\mathrm{\Phi }}_{{\mathbf{q}}_1}$ ${\mathrm{\Phi }}_{{\mathbf{q}}_2}$ ${\mathrm{\Phi }}_{{\mathbf{q}}_3}$ ${\mathrm{\Phi }}_{{\mathbf{q}}_4}$δ (${\mathbf{q}}_1$+${\mathbf{q}}_2$+${\mathbf{q}}_3$+${\mathbf{q}}_4$), where b may be positive or negative. In the latter case the inclusion of fluctuations may produce large quantitative and qualitative effects in the phases and the transitions between them. The renormalization is accomplished by resummation to all orders of two classes of diagrams and is reminiscent of a calculation earlier described by Brazovskii (Zh. Eksp. Teor. Fiz. 68, 175 (1975) [Sov. Phys.—JETP 41, 85 (1975)]). Since the LGW theory (with b<0) may be extracted from an isotropic frustrated-lattice model of microemulsion it is possible to compare the predictions of the present study with earlier Monte Carlo simulations of the Ising model. It is therefore also possible to describe the correlation functions in the disordered phase, which had earlier been identified as the bicontinuous phase of the microemulsion model. In particular, strong renormalizations of the two length scales d and ξ originally introduced by Teubner and Strey [J. Chem. Phys. 87, 3195 (1987)] to describe the bicontinuous-microemulsion phase are derived. In addition renormalizations of the surface energies are derived and discussed in the context of bicontinuous microemulsion.