The Landau Expansion

Calculation of the physical properties of a spatially uniform, macroscopic system requires information about f[(y,T] only in the immediate vicinity of its minimum at temperature T, due to the sharpness of the peak in the function exp —j8V/[(7,T] (See Appendix A). This implies that for a physical system having a second-order phase transi-ton at T = Tc, with a = 0 for T Tc and (r 7 0 for T Tc (such as the ferromagnetic phase transition), many features of the transition can be deduced if /[ r,T] is known only in the neighborhood of = 0, T = Tc. Landau speculated that the first few derivatives of the function f[(T,T] with respect to r exist, and that they have finite values when evaluated at (t = 0, T = Tc. For a second-order phase transition the value of r varies continuously and can be arbitrarily small near T = Tc therefore, the Landau assumption enables one to express /[cr,T] near T - Tc as  

Various properties of the expansion coefficients X, A, B and C can be obtained from quite general considerations. 

For definiteness we will examine the expansion in relation to two physical systems (1) the CuZn (jS-brass) binary alloy and (2) a ferro-magnet such as iron. It is well known that CuZn has a second-order, order-disorder transition at Tc = 742 K. The crystal structure at 0 K can be described by two interpenetrating simple cubic lattices each with No sites. Let us suppose that lattice 1 is occupied by Cu atoms and lattice 2 by Zn atoms. As the temperature is raised above 

some Zn atoms will be found on lattice 1 and some Cu atoms on lattice 2 but, so long as the temperature is less than Tc, NcuW/Nq where we have denoted the number of copper atoms on lattice 1 by JVcu(l)- For temperatures in excess of Tc, complete randomization occurs and NcuW/Nq = iVzn(l)/iVo = A suitable order parameter for the system can be defined as 

For a second order phase transition, Eq. [16] thus takes the form 

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Here A, C and E are phenomenological coefficients in the Landau expansion in tenns of the smectic ordering ... 

Obviously, if we know experimentally the behavior of the macroscopic ordering parameter with T, we may determine the corresponding coefficients of the Landau expansion (eq. 2.52). However, things are not so easy when different transitions are superimposed (such as, for instance, the displacive and order-disorder transitions in feldspars). In these cases the Landau potential is a summation of terms corresponding to the different reactions plus a couphng factor associated with the common elastic strain. 

Coefficients a, B, and C in equation 5.175 have the usual meanings in the Landau expansion (see section 2.8.1) and for the (second-order) displacive transition of albite assume the values = 1.309 cal/(mole X K) and B, = 1.638 kcal/mole (Salje et al., 1985). is the critical temperature of transition = Bla = 1251 K). The corresponding coefficients of the ordering process are = 9.947 cal/(mole X K), B = -2.233 kcal/mole, = 10.42 kcal/(mole X K), and = 824.1 K. With all three coefficients being present in the Landau expansion relative to substitutional disorder it is obvious that Salje et al. (1985) consider this transition first-order. A is a T-dependent coupling coefficient between displacive and substitutional energy terms (Salje et al., 1985) ... 

Replacing the generalized strain e with strain components X4 and Xg and adding the elastic energy term in the Landau expansion results in equation 5.175. 

The Cp change at the transition temperature resulting from displacive disorder is, according to the Landau expansion above, ACp = 0.66 cal/(mole X K). The resulting entropy change associated with Al-Si substitutional disorder is —4.97 cal/(mole X K), near the maximum configurational effect expected by the random distribution of 3A1 -I- 1 Si atoms on four tetrahedral sites. 

Whether the phase transition is first- or second-order depends on the relative magnitudes of the coefficients in the Landau expansion, Eq. 17.2. For a first-order transition, the free energy has a discontinuity in its first derivative, as at the temperature Tm in Fig. 17.1a, and higher-order derivative quantities, such as heat capacity, are unbounded. In second-order transitions, the discontinuity occurs in the second-order derivatives of the free energy, while first derivatives such as entropy and volume are continuous at the transition. 

For a binary A-B alloy, another independent parameter, Xb (or = 1 — Xg) must be added to the fixed-stoichiometry order parameters in the preceding section. The phenomenological form of the Landau expansion, Eq. 17.2, can be extended to include Xb and has been used to catalog the conditions for many transitions in two-component systems [3]. 

In the case of quartz, the only strains are non-symmetry-breaking and would normally be expected to lead to renormalisation only of the fourth order Landau coefficient. Higher order coupling leads to renormalisation of higher order terms as well, however. A renormalised version of the Landau expansion for the 3 a quartz transition (Eqn. 28) would be... 

The case of bilinear coupling is important in the description of the high temperature behaviour of albite, and is considered in more detail here. For the case of biquadratic coupling the reader is referred to Salje and Devarajan (1986) and Salje (1990). If we consider two order parameters, Qi and Q2, which are coupled together by a single spontaneous strain, e, the Landau expansion is ... 

This assumes that only linear coupling between strain and order parameters exists, and that the strain obeys Hooke s law. The Landau expansion can then be renormalized as ... 

How does one obtain the Landau expansion in particular cases If one wishes to consider specific models, a straightforward approach uses a molecular field approximation (MFA), where one then obtains the free energy explicitly and expands it directly. We illustrate this approach here with the -state Potts model (Potts, 1952). The Hamiltonian is... 

In order to make contact with the Landau expansion, however, we consider now the special case q = 3 and expand F in terms of the two order parameter components 0] = ni — 1/3 and 02 = 2 — 1/3 (note that all rij — l/q in the disordered phase). One recognizes that the model for q = 3 has a two-component order parameter and there is no symmetry between 0,-and —0<. So cubic terms in the expansion of F arc expected and do occur, whereas for a properly defined order parameter, there cannot be any linear term in the expansion ... 

This Brazovskii [327] mechanism for a fluctuation induced first order transition hence means that the strong increase of local fluctuations drives the fourth order coefficient of the Landau expansion negative, and thus a critical divergence of the local fluctuation is prevented. 

The cumbersome procedure of generating the Landau expansion from the underlying mean-field theory has recently been made unnecessary because the problem of obtaining a solution to the mean-field equations with any desired symmetry and without further approximation was solved by Matsen and Schick [77]. All functions of position are expanded in a complete set of states that possess the desired symmetry, so that one is left with the equivalent self-consistent equations expressed in terms of the coefficients in the expansion. These equations can be solved numerically to whatever accuracy is possible with the guarantee that the solution has the desired symmetry. In this way solutions with up to 450 different wave vectors have been obtained, providing accuracy to very large incompatibilities / or, equivalently, to very low temperatures. 

Due to the effect of external fields, the order can vary in space and gradient terms have to be added to the Landau expansion (8.9). Usually, only the terms up to the quadratic order are considered. There are many symmetry allowed invariants related to gradients of the tensorial order parameter [29]. However, in the vicinity of the phase transition, one is not interested in elastic deformations of the nematic director but rather in spatial variations of the degree of nematic order. Therefore, the pretransitional nematic system is described adequately within the usual one-elastic-constant approximation. 

The first two terms give the tilt-dependent free energy for an isolated layer. The first coefficient oq = a T — To) is the only temperatme-dependent coefficient in Eq. (5.2). For simplicity we consider only continuous transitions to the tilted phase, found in systems that can be described by a positive coefficient bo. The generalization to the discontinuous transition to the tilted phase does not add any important new physics to the problem. There exist also systems where the first coefficient does not depend on temperature monotonously around the transition temperatme this is a very rare situation where the transition takes place at the temperature where the Landau expansion for the temperature dependence of the leading coefficient ao has to include higher-order terms. 

The angles 6 and 4> are defined in Fig. 5.2. The approximation cosO 1 and sin0 ss 0 is legitimate as the Landau expansion is valid only close to phase transitions where the order parameters are still small. Inserting Elq. (5.3) into Eq. (5.2) gives... 

Neglecting the mass density change at the transition, the Landau expansion for the free energy density is... 

Fig. 6.4 The forms of the free energy density in the high symmetry (A > 0) and low symmetry (A < 0) phases (a) and the temperature dependence of the first term in the Landau expansion (b)...

Concluding this section I would like to underline the significance of coefficient B in the Landau expansion ... 

This density wave is usually considered as a complex order parameter pi = exp (itt) of the smectic A phase in the Landau expansion or fi-ee energy at the SmA-N phase transition. Typically, when there is no distortion, one assumes cpi = 0 at z = 0 and operates only with the wave amplitude pi as the real part of the order parameter. 

However, if you take into account the interaction of SmA and SmC order parameters, a cross term pjt would result in the appearance of the NAC triple point in the phase diagram [8, 16], see Fig. 6.15. In this case, the phase transition lines might correspond to either second or first order transitions it depends on parameters of the Landau expansion. In experiment, such a phase diagram may be observed when a content of binary mixtures is varied. 

There are also other reasons that truncate the order parameter divergence such as spatial inhomogeneities or external fields. For example, to describe a spatial inhomogeneous system, a term quadratic in the gradient of the order parameter G(Vri) must be added to the density of free energy and all the Landau expansion should be integrated over the system volume ... 

Then, for paraelectric SmA phase both = 0 and = 0, for ferroelectric SmC phase 0 but = 0 as discussed in Section 13.1, for antiferroelectric SmC A phase = 0 but f 0, and for ferrielectric phases SmC F/ both 0 and AP 7 0. Now the Landau expansion of the free energy in the vicinity of transitions between the paraelectric, ferroelectric and antiferroelectric phases will operate with two order parameters and both coefficients at the terms in the free energy are considered to be dependent on temperatiue ... 

To extend the range of applicability of the crossover theory based on the Landau expansion, higher-order terms have been included ( six-term crossover Landau model [4, 60]). However, even such an extended crossover Landau expansion still fails to make a connection with the behavior of fluids very far away from the critical point like the ideal-gas limit at low densities. Ideally, one would like to have a simple closed-form equation which... 

A more phenomenological approach to describe crossover critical phenomena in simple fluids has been developed by Kiselev and coworkers [76-79]. This approach starts from the asymptotic power-law expansion including the leading correction-to-scaling terms which is then multiplied by an empirical crossover functions so that the equation becomes analytic far away from the critical point. A comparison of this approach with the crossover theory based on a Landau expansion has been discussed in earlier publications [13, 78]. One principal difference is that in the application of the results of the RG theory to the Landau expansion the leading correction to asymptotic scaling law is incorporated in the crossover function and recovered upon expanding the crossover function [18]. 

If fe > 0 (positive) and c 0, the Landau expansion describes a second-order transition in which the equilibrium value of Q varies as... 

From the Landau expansion of G the extensive thermodynamic variable entropy (S) can be obtained, such as the Gibbs-Hehriholtz equation ... 

The temperature difference AT between the actual and the phase transition temperature is taken into account via the coefficient of the quadratic term of the order parameter in the Landau expansion a T), which is proportional to AT. Ln denotes the elastic coefficient of a nematic liquid crystal. Both coefficients are related via the nematic correlation length, which is defined as = o/AT,... 

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