Landau Tricritical Point

In the mean-field picture one expects the phase transition to be genetically second order. However, as one adjust materials parameters to reduce the temperature width of the nematic phase, parametrized in K-M theory by the dimensionless ratio the coupling drives the transition first order, with the point in material-parameter space where this happens being termed the Landau Tricritical Point (LTP). 

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. )...

Fig. 70. Experimental phase diagram for H adsorbed on Pdf 100), left part, as extracted from the temperature variation of LEED intensities at various coverages 9 right part). Crosses denote the points T /i where the LEED intensities have dropped to one half of their low temperature values (denoted by full dots in the right part). Dashed curve is a theoretical phase diagram obtained by Binder and Landau (1981) for R = ipt/cimn = 1/2 (only the regime of second-order transition ending in tricritical points [dots] are shown). Experimental data are taken from Behm et al. (1980). From Binder and Landau (1981).

Fig. 2. (a) Schematic of Landau phase diagram as a function of the value of parameter b in the development of the critical free energy F as a function of the order parameter p up to sixth order. When b>0, the phase transition is second order. For b< 0, the phase transition is first order. Transition lines are continuous, and for b < 0 the dotted lines show the coexistence region, b — 0 corresponds to a tricritical point. First-order phase transitions may also occur for symmetry reasons when third-order invariant is allowed in the free energy expansion, (b) Schematic representation of the microscopic modification of a variable u(t) = u + p + up(t) in the parent (p — 0) and descendant phases (p/0). Both the mean value < u(t)) — u — p and time fluctuations Sup(t) depend on the phase. 

Holyst and Schick [339] study the phase diagram and scattering of AB symmetric diblock copolymers diluted with A and B homopolymers (in equal concentrations) having the same chain length NA = NB = N as the copolymers. Constructing a Landau expansion, they show that the wave vector q vanishes at a critical copolymer concentration ordering transition there as that of a Lifshitz tricritical point, where the disordered phase, lamellar phase, A-rich and B-rich separated phases can coexist. The critical behavior near this point is expected to deviate strongly from mean field theory [339]. 

Griffiths has given a phenomenological (Landau) treatment of tricritical points which expresses the free energy as a sixth-order polynomial in an order parameter (which is some suitable linear combination of the physical densities , e.g. the mole fractions). The scaling properties of the singular part of the polynomial lead to four numbers = 5/6, 2 = 4/6 = 2/3, 3 = 3/6 = 1/2, = 2/6 = 1/3, in terms of which various critical exponents are expressed. Because this is an analytic (mean field) formulation, these exponents are classical , but it is believed that for experimental tricritical points in three dimensions they should be. ( Nonclassical logarithmic factors may exist, but these do not alter the exponents.)... 

While analyzing the Landau equation of the form (1) or (40), its coeflicients are presumed to have no analytical singularities therefore, the heat cap2 ity has a finite value at the tricritical point (according to Equation 61) rather than diverges m for the common critical point. 

Figure 1., 56. State diagram of a mixture witli the tricritical point O T and x denote the temperature and concentration of component 1, respectively (Landau, 1937a)...

The Landau formalism possesses a universal meaning and is applicable to a wide range of problems. The chief restriction of this version of the mean field theory is in the lack of proper account for the correlation of order parameter fluctuations, which particularly affect the system s properties near the critical point. In the same paragraph, the coiice >t of the tricritical point is introduced, which seems reasonable in connection with the great popularity of this term in polymer theory since de Gennes showed the B point in the P-l LMWL. system to be an analogue of the tricritical point in the field theory formalism. 

In the McMillan model, the smectic A-nematic transition can be continuous or discontinuous. If a is less than 0.7, then o decreases to zero continuously and S is continuous at the smectic A- nematic transition. If a is between 0.7 and 0.98, then a jumps to zero discontinuously and S has a small discontinuity at the smectic A-nematic transition. When a is greater than 0.98, the smectic phase transforms directly into the isotropic phase with discontinuities in both order parameters. So just as in the extended Landau-de Geimes theory for the smectic A phase, a tricritical point is predicted at a=0.7, which corresponds to a ratio in the smectic A—nematic transition temperature to the nematic-isotropic transition temperature of 0.87. A great deal of experimental work has been done on the smectic A-nematic transition, and the results seem to indicate that the tricritical point occurs when the ratio of the two transition temperatures is significantly larger than 0.87. 

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]. 

Am/ T —> oo, (x/T finite an occupation of nearest neighbor sites becomes strictly forbidden, and a hard-square exclusion results. Thus this transition is the end-point of the phase diagram shown in fig. 28a. But at the same time, it is the end-point of a line of tricritical transitions obtained in the lattice gas model when one adds an attractive next-nearest neighbor interaction pnnn and considers the limit R = Ainn/Am - 0 (Binder and Landau, 1980, 1981 fig. 32). 

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