Landau phase transition

Electroclinic Effect Near the Smectic A—Smectic C Phase Transition. The electric field may induce a tilt of the director in the orthogonal smectic A phase near the smectic A-smectic C phase transition (electroclinic effect [189, 191]). The electroclinic effect may be understood within framework of the Landau phase transition theory [7]. If the dielectric anisotropy is negligible, the free energy density of the smectic A phase may be expanded over the field-induced tilt angle 6 ... 

Binder K and Landau D P 1984 Finite size scaling at Ist-order phase transitions Phys. Rev. B 30 1477-85... 

Undoubtedly the most successful model of the nematic-smectic A phase transition is the Landau-de Gennes model [201. It is applied in the case of a second-order phase transition by combining a Landau expansion for the free energy in tenns of an order parameter for smectic layering with the elastic energy of the nematic phase [20]. It is first convenient to introduce an order parameter for the smectic stmcture, which allows both for the layer periodicity (at the first hannonic level, cf equation (C2.2A)) and the fluctuations of layer position ur [20] ... 

Chen J-FI and Lubensky T C 1976 Landau-Ginzburg mean-fieid theory for the nematio to smeotio C and nematio to smeotio A phase transitions Phys.Rev. A 14 1202-7... 

D. P. Landau, K. Binder. Monte Carlo study of surface phase transitions in the three-dimensional Ising model. Phys Rev B 47 4633-4645, 1980. 

At the mesoscopic level of description the Landau-Ginzburg model of the phase transitions in diblock copolymer system was formulated by Leibler [36]... 

Toledano J-C, Toledano P (1987) The Landau theory of phase transitions. World Scientific, Singapore... 

Kuchanov SI, Panyukov SV (2006) A new look at the Landau theory of phase transitions in polydisperse heteropolymer liquids (paper to be published)... 

Figure 3 Landau free energy at different temperatures. Spontaneous symmetry breaking occurs fort < 0, giving rise to a second-order phase transition at t=0.

The Landau theory predicts the symmetry conditions necessary for a transition to be thermodynamically of second order. The order parameter must in this case vary continuously from 0 to 1. The presence of odd-order coefficients in the expansion gives rise to two values of the transitional Gibbs energy that satisfy the equilibrium conditions. This is not consistent with a continuous change in r and thus corresponds to first-order phase transitions. For this reason all odd-order coefficients must be zero. Furthermore, the sign of b must change from positive to negative at the transition temperature. It is customary to express the temperature dependence of b as a linear function of temperature ... 

Fig. 29. Phase diagram of the model Eq. (22) for coadsorption of two kinds of atoms in the temperature-coverage space. Circles indicate a second-order phase transition, while crosses indicate first-order transitions. Point A is believed to be a tricritical point and point B a bicritical point. The dashed curve shows the boundary from the Blume-Capel model on a square lattice with a nearest-neighbor coupling equal to 7 in the present model (for - 0 Eq. (22) reduces to this model), only the ordered phase I then occurs. From Lee and Landau. )...

Such continuous phase transitions are conveniently described in a phenomenological Landau free-energy expansion of the order parameter. Since we... 

Fig. 29). Using Landau theory, Bak et al. (BMVW) have shown that it is the wall crossing energy A which determines the symmetry of the weakly incommensurate phase and the nature of the phase transition ... 

The excess thermodynamic properties correlated with phase transitions are conveniently described in terms of a macroscopic order parameter Q. Formal relations between Q and the excess thermodynamic properties associated with a transition are conveniently derived by expanding the Gibbs free energy of transition in terms of a Landau potential ... 

Salje (1985) interpreted overlapping (displacive plus Al-Si substitutional) phase transitions in albite in the light of Landau theory (see section 2.8.1), assigning two distinct order parameters Q n and to displacive and substitutional disorder and expanding the excess Gibbs free energy of transition in the appropriate Landau form ... 

According to the well-known Landau theory, the eigenvector of the order parameter in any second order solid-solid phase transition transforms according to an irreducible representation of the space group of the parent phase state. Furthermore, the free energy F=U -TS can be expanded around the transition temperature Tc in terms of the scalar order parameter p, which... 

Every phase transition is associated with a change in symmetry as well as in order. To facilitate quantitative analysis, Landau introduced the concept of an order parameter, and expressed the free energy, 0, as follows ... 

Another important contribution by Landau is related to symmetry changes accompanying phase transitions. In second-order or structural transitions, the symmetry of the crystal changes discontinuously, causing the appearance (or disappearance) of certain symmetry elements, unlike first-order transitions, where there is no relation between the symmetries of the high- and low-temperature phases. If p(x, y, z) describes the probability distribution of atom positions in a crystal, then p would reflect the symmetry group of the crystal. This means that for T> T p must be consistent with... 

A wide class of analytic second-order phase transitions is characterized by their Landau bifurcational mechanism [38]. According to this mechanism, a system characterized by order parameter r], possesses a single stable equilibrium solution (rje = 0) for a range of the external parameter T (T > Tcr see a schematic illustration in Fig. 2.3.4a). This single solution corresponds to an absolute internal minimum of the system s free energy F as a function of the order parameter (Fig. 2.3.4b, Curve 1). As the external parameter T decreases, at a critical value T = Tcr, the solution with r)e = 0 becomes unstable with two more stable solutions with r e 0 (for T < TCI) bifurcating from it (Fig. 2.3.4a). In the (F, rf) plane this corresponds to the appearance of two new local free energy minima that flank the former one, which now turns into a local maximum (Fig. 2.3.4b, Curve 2). 

Fig. 2.3.4a. Schematic dependence of the equilibrium order parameter je, on the external parameter T in the Landau mechanism for a second-order phase transition.

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