Landau model, phase transitions

Lynden-Bell RM (1995) Landau free energy, Landau entropy, phase transitions and limits of metastability in an analytical model with a variable number of degrees of freedom. Mol. Phys. 86 1353-1374... 

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] ... 

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]... 

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. )...

Order parameters may also refer to underlying atomic structure or symmetry. For example, a piezoelectric material cannot have a symmetry that includes an inversion center. To model piezoelectric phase transitions, an order parameter, r], could be associated with the displacement of an atom in a fixed direction away from a crystalline inversion center. Below the transition temperature Tc, the molar Gibbs free energy of a crystal can be modeled as a Landau expansion in even powers of r (because negative and positive displacements, 77, must have the same contribution to molar energy) with coefficients that are functions of fixed temperature and pressure,... 

The evolution Eq. 13 of the order parameter has a similar form to the time-dependent Landau equation [17], which is fundamental in nonequihbrium phase transitions. The asymptotic value of the order parameter 4>i,oo is determined as the zero of the velocity 4>i- The main difference from the standard model of phase transitions lies in the time dependence in the coefficients Ait) and B(t) induced by that of the achiral concentration ait) and the total chiral concentration qft). Because the concentrations a and q are nonnegative, A(t) cannot exceed Bit) Ait) < Bit). 

The application of Landau s (1937) theory of symmetry-changing phase transitions to order-disorder in silicates has been described very clearly by Carpenter (1985 1988) and is further discussed in a number of textbooks and seminal papers (Salje 1990, Putnis 1992). The essential feature behind the model is that the excess Gibbs energy can be described by an expansion of the order parameter of the type ... 

The Landau model for phase transitions is typically applied in a phenomenological manner, with experimental or other data providing a means by which to scale the relative terms in the expansion and fix the parameters a, b, c, etc. The expression given in Equation (9) is usually terminated to the lowest feasible number of terms. Hence both a second-order phase transition and a tricritical transition can be described adequately by a two term expansion, the former as a 2-4 potential and the latter as a 2-6 potential, these figures referring to those exponents in Q present. 

The three ordered stales of the Potts model correspond to a preferential occupation of one of the three sublattices a,b,c into which the triangular lattice is split in the (-/3x-v/3)R30° structure. In the order parameter plane (0x.0r), the minima of F occur at positions (1, 0)MS, (—1/2, i/3/2)yWs, (—1/2, -yf3/2)Ms, where Ms is the absolute value of the order parameter, i.e. they are rotated by an angle of 120° with respect to each other. The phase transition of the three-state Potts model hence can be interpreted as spontaneous breaking of the (discrete) Zj symmetry. While Landau s theory implies [fig. 13 and eqs. (20), (21)] that this transition must be of first order due to the third-order invariant present in eq. (34), it actually is of second order in d = 2 dimensions (Baxter, 1982, 1973) in agreement with experimental observations on monolayer ( /3x /3)R30o structures (Dash, 1978 Bretz, 1977). The reasons why Landau s theory fails in predicting the order of the transition and the critical behavior that results in this case will be discussed in the next section. 

We return here to the simple mean field description of second-order phase transitions in terms of Landau s theory, assuming a scalar order parameter cj)(x) and consider the situation T < Tc for H = 0. Then domains with = + / r/u can coexist in thermal equilibrium with domains with —domain with exists in the halfspace with z < 0 and a domain with 4>(x) = +

0 (fig. 35a), the plane z = 0 hence being the interface between the coexisting phases. While this interface is sharp on an atomic scale at T = 0 for an (sing model, with = -1 for sites with z < 0, cpi = +1 for sites with z > 0 (assuming the plane z = 0 in between two lattice planes), we expect near Tc a smooth variation of the (coarse-grained) order parameter field (z), as sketched in fig. 35a. Within Landau s theory (remember 10(jc) 1, v 00 01 < 1) the interfacial profile is described by... 

Undoubtedly the most successful model of the nematic-smectic A phase transition is the Landau-de Gennes model... 

It is concluded [217] that an interpretation of the ideal herringbone transition within the anisotropic-planar-rotor model (2.5) as a weak first-order transition seems most probable, especially since previous assignments [56, 244] can be rationalized. This phase transition is fluctuation-driven in the sense of the Landau theory because the mean-field theory [141] yields a second-order transition. Assuming that defects of the -v/3 lattice and additional fluctuations due to full rotations and translations in three dimensions are not relevant and only renormalize the nonuniversal quantities, these assignments should be correct for other reasonable models and also for experiment [217]. 

Phase transitions in which the square of the soft-mode frequency or its related microscopic order parameter goes to zero continuously with temperature can be defined as second order within the framework of the Ginzburg-Landau model [110]. The behavior is obviously classical and consistent with mean field... 

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