Landau free energy expression

Equation (5.3.3) is analogous to the Peierls-Landau free energy expression for a two-dimensional crystal, and leads to a logarithmic divergence of the mean square fluctuation as Writing the free energy in terms of the Fourier components of u... 

The simplification, which enables reducing expression (Eq. 42) into (Eq. 58) remains in force in considering the generating function of the arbitrary chemical correlator. This means that in order to use expression (Eq. 54) for the calculation of the dependence of the vertices of the Landau free energy expansion on wave vectors at region Qs <[Pg.164]

Our purposes require a Ginzburg-Landau free energy for generally inhomogeneous gels [12,19-21]. First, Fmix + Fio is expressed as a functional of space-dependent variable tj>,... 

For Landau free energy (m = 0, - the order parameter), Zt is a partition function of an extended system with additional variable . The integral denoted in square brackets in Eq. (3-12) is simply the configurational partition function of the system with a fixed value of -Z. The statistical definition of the Gibbs free energy function combined with Eq (12) results in the following expression ... 

When concentrations of substitutional atoms are high, the presumption generally has been that transition temperatures will vary in a simple linear fashion with dopant concentration. This supposition can be rationalized by a Landau-Ginzburg excess Gibbs free-energy expansion (Salje et al. 1991). A simple second order phase transition for phase A with the regular free energy expression... 

Crytal chemitry. Solid solutions of lead phosphate compounds with structures like that of palmierite ((K,Na)2Pb(S04)2) exhibit transition sequences that are well modeled by Landau-Ginzburg excess free energy expressions. Systems that have been especially heavily studied include Pb3(Pi (As (04)2 (Toledano et al. 1975, Torres 1975, Bismayer and Salje 1981, Bismayer et al. 1982, 1986 Salje and Wruck 1983) and (Pbi jcA c)3(P04)2, where A = Sr or Ba (Bismayer et al. 1994, 1995). The phase diagram for the phosphate-arsenate join is not completely mapped, but Bismayer and collaborators have identified a series of transitions for the P-rich and the As-rich fields. The structure consists of two sheets of isolated (P,As)04-tetrahedra with vertices pointing towards each other and a sheet of Pb atoms in the plane of the apical oxygen atoms. Layers of Pb atoms lie between these tetrahedral sheets (Fig. 18) (Viswanathan and Miehe 1978). 

The most straightforward way to obtain free energy differences by computer simulation is to express the free energy as a function of the set of order parameters A. Up to a constant this Landau free energy [21] is given by... 

The free-energy expression for conduction electrons in a magnetic field with quantized Landau levels was given by Lifshitz and Kosevich (1956) ... 

Let us now consider the Landau free-energy density expression for liquid crystals. All isotropic-nematic (cholesteric) phase transitions... 

Eq. [133] states the result that in a spatially uniform system, where the partition function can be expressed by Eq. [129], the free energy density at temperature T is equal to the value of the minimum of the Landau free energy density function /[o-,T] at that temperature. 

The Landau type free energy expression can be supplemented by the free volume contribution to produce a model free energy for the composite system. In general, the standard Landau free energy density for SmC -SmA transition can be described by taking the tilt angle 0 as the primary order parameter (Musevic et al. 2002) and... 

The mean atomic mass m = (1 - x)mo + xmi, where mo and mi are a Ughter isotope mass (e.g., mo = 16) and heavier isotope mass, respectively, m/ = mo + 8m, and x is the fraction of the heavier isotope. The Landau expression of the free energy of the displacive FE as a function of temperature is given by using the amplitude of FE mode Xf, which is taken as the order parameter ... 

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 ... 

The free energy F of the body depends on the deformation and is determined (Landau and Lifshitz 1969) by the general expression... 

The RPA can be improved on by the Landau-Ginzburg (LG) formalism [47] appropriate in a quasistatic regime. One introduces a complex order parameter i[f( ) (dimensions of energy) associated with Apld(jc), which can also be related to the amplitude of the lattice distortion [Eqs. (4 and 5)] qt oc e,2fc, vjf(jtj) + e 2kF i (xi). It is complex because the phase of the CDW or BOW at +2kF is independent of the one at -2kF. TTie partition function is expressed as a functional integral weighing all fluctuations in the order parameter Z = J3)i ie-p/w, where the free-energy functional is... 

For the magnetic system in zero external field, the Landau expression for the free energy can be written as an expansion in

[Pg.218]

Unlike ferromagnets, where G((p) = G(-(f), the free energy of nematic liquid crystals exhibits an asymmetry G(S) G(-5), as discussed previously and shown in Fig. 3b. To reproduce this behavior, a term that is an odd power of the order parameter S is needed in the Landau expression for the free energy of a nematic liquid crystal. A term linear in 5 is not allowed since the equilibrium condition dQdS = 0 could not then be satisfied in the disordered isotropic phase where 5=0. The presence of a cubic term will lead to the desired asymmetry in G as a function of 5 and to the emergence on cooling of a second minimum in G at a finite 5 value. 

In some cases, one is interested in the structures of complex fluids only at the continuum level, and the detailed molecular structure is not important. For example, long polymer molecules, especially block copolymers, can form phases whose microstructure has length scales ranging from nanometers almost up to microns. Computer simulations of such structures at the level of atoms is not feasible. However, composition field equations can be written that account for the dynamics of some slow variable such as 0 (x), the concentration of one species in a binary polymer blend, or of one block of a diblock copolymer. If an expression for the free energy / of the mixture exists, then a Ginzburg-Landau type of equation can sometimes be written for the time evolution of the variable 0 with or without flow. An example of such an equation is (Ohta et al. 1990 Tanaka 1994 Kodama and Doi 1996)... 

The structure of the interface formed by coexisting phases is well described by the Cahn-Hilliard approach [53] (developed in a slightly different context by Landau and Lifshitz [54]) extended to incompressible binary polymer mixtures by several authors [4,49,55,56]. The central point of this approach is the free energy functional definition that describes two semi-infinite polymer phases <]), and 2 separated by a planar interface (at depth z=0) and the composition ( )(z) across this interface. The relevant functional Fb for the free energy of mixing per site volume Q (taken as equal to the average segmental volume V of both blend components) and the area A of the interface is expressed by... 

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