Brazovskii theory

In Landau-Brazovskii theory, the density modulation (or composition for block copolymers) is written as... 

The principal requirement for validity of the Landau-Brazovskii theory can be expressed for block copolymers as the inequality (Podneks and Hamley 1996)... 

The effect of harmonics in the composition profile has been considered in Landau Brazovskii theory, as well as mean field theory. Olvera de la Cruz (1991) found a hexagonal perforated layer (HPL) structure to be stable for symmetric or nearly symmetric diblocks in addition to the classical phases. Recent work has... 

The mean-field theory of Leibler agrees very well with experimental observations based on X-ray and neutron scattering when obtained relatively far from the microphase separation temperature (MST). In the vicinity of the MST, however, mean-field treatment is less accurate. Both, Leibler and Fredrickson and Helfand noted that the effective Hamiltonian appropriate for diblock copolymers is in the Brazovskii-universality class [15,16]. Based on the Hartree treatment used in the Brazovskii theory, Fredrickson and HeUand found that the structure factor can still be written as the mean-field expression Eq. (7.101), but with renormalized values % and N [16]. The fluctuation renormalization makes the order parameter nonlinear in x, and thereby nonlinear in T". This is shown in the experimental example given in Figure 7.13. 

Although the gross shape of the phase envelope predicted by the mean-field theory, as well as the regions of lamellar and hexagonal phases, are more or less in agreement with experiments on diblock copolymers (see Fig. 13-4), the predictions of the theory near the critical point at / = 0.5, /(V = 10.5 are incorrect. Fredrickson and Helfand (1987) showed that the second-order transition predicted by the mean-field theory is corrected to SL first-order transition when the effects of fluctuations on the free energy are accounted for using a so-called Brazovskii Hamiltonian (Brazovskii 1975). 

A Landau theory for blue phase was proposed by Brazovskii, Dmitriev, Homreich, and Shtrik-man [7-10]. In this theory, the free energy of the blue phase is expressed in terms of a tensor order parameter which is expanded in Fourier components. The free energy is then minimized with respect to the order parameter with the wave vector in various cubic symmetries. In a narrow temperature region below the isotropic transition temperature, the stmctures with certain cubic symmetries have free energy lower than both the isotroic and cholesteric phases. 

Fredrickson and Helfand developed a weak segregation theory with fluctuations [77], starting by mapping the Leibler free energy functional onto one treated earlier by Brazovskii [78]. They used the UCA and treated the three classical phases. This has since been extended to the G phase [79,80] and to triblocks [81], and modified to include a multihannonic approximation for the density profiles, although not for the G phase [82]. 

A Landau theory for the cubic blue phases was first proposed by Brazovskii and Dmitriev [3], Homreich and Shtrikman [32], [33], [34], [35], [36], [37], and Kleinert and Maki [38]. Detailed reviews of this approach have been given by Seideman [19], Belyakov and Dmitrienko [16], Wright and Mermin [20] and, in particular, by Homreich and Shtrikman [39]. 

Since thread PRISM theory at the R-MMSA or R-MPY/HTA closure level predicts a spinodal instability, its description of the disordered phase can be combined with field-theoretic Landau expansion and Brazovskii... 

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