The Landau Model

To predict the experiments, we want to compute the mathematical form of m T) near the critical temperature T Tc, where m(T) 0. We could get this function from the lattice or van der Waals gas models, but instead we will find the same result from a model that is simpler and more general, called the Landau model. 

The Landau Model Is a General Description of Phase Transitions and Critical Exponents 

We now describe the Landau model, named for the Russian physicist LD Landau (1908-1968), who won the 1962 Nobel prize in Physics for his work on condensed phases of matter. The Landau model is a generic treatment of phase transitions and critical points. It is based on the idea that coexistence curves are mathematical functions that have two minima at low temperatures, merging into a single minimum at higher temperatures. You can capture that mathematically by expressing the free energy f (m) as a polynomial function, 

496 Chapter 26. Cooperativity The Helix-Coil, Ising Landau Models 

To capture the correspondence between and temperature, let t represent the fractional deviation of the temperature T away from the critical temperature Tc, 

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W. Gozdz, R. Hotyst. Triply periodic surfaces and multiply continuous structures from the Landau model of microemulsions. Phys Rev E 54 5012-5027, 1996. 

It was shown by Wilson [131] that the Kadanoff procedure, combined with the Landau model, may be used to identify the critical point, verify the scaling law and determine the critical exponents without obtaining an exact solution, or specifying the nature of fluctuations near the critical point. The Hamiltonian for a set of Ising spins is written in suitable units, as before... 

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. 

Once convection starts in either of the two layers, it drives a fluid motion in the other. Using a certain model (the Landau model), Lienhard and Catton [299] predicted the heat transfer in the Rayleigh number range slightly greater than critical. Use of Eq. 4.78 for both layers, with the Rac the one relevant to thicker layer, is also tentatively recommended. 

Detailed stress-optical measurements have been analyzed to yield further information [4]. In Fig. 10 the birefringence (order parameter) was plotted as a function of reduced temperature for several nominal stresses <7 . These results were combined with the predictions of the Landau model and static stress-strain curves and led to a number of interesting consequences. In Fig. 11 the shift in the phase transition temperature is plotted as a function of nominal stress and shifts of up to 7.5 K were found compared to maximum displacements by electric and magnetic fields of about 5 mK in low molecular weight materials. In Fig. 12 the birefringence An is shown as a function of strain X=L/Lq at constant nominal stress f7n = 2.11xlO Nmm. A strictly... 

The field response of the isotropic phase has also been discussed in terms of the Landau model for the N-I phase transition [227]. Apart from the conventional quadratic and cubic terms the gradient of the order parameter was taken into account. The theory predicted the temperature dependence of the saturation field in the form... 

The revolution in understanding critical phenomena began with Guggenheim s experiments. The critical exponent for those data and others is found to be /I 1/3 (see Figure 26.7), in disagreement with the value /I = 1/2 predicted by the Landau model [4. The discrepancy is evidently not in the neglect of atomic detail, because the coexistence curves for different types of atom superimpose upon each other. Rather, the problem is that very near the critical... 

The Landau model for the critical exponent of Cv(T). In the Landau model, show that C oc T asT increases toward the critical temperature T. ... 

We should note that (8.15) is only an approximation. Because of the helical structure in the chiral smectic-C phase, the divergence of /g is in principle incomplete since a real divergence would be obtained only as a response to a helicoidal electric field [68], [69]. Including the helical structure into the Landau model leads to a modification of (8.15) and a truncation of the divergence with a finite value of /g at 7), similar to the case of a first-order transition. However, whereas the truneation at a first-order smectic-yl-smectic-C transition can be observed experimentally, measurements of Xe around a tricritical point, where the transition changes from first-order to second-order, have shown that the influence of the helix on the divergence of Xg is probably beyond experimental resolution [70]. 

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