Tricritical phase transitions

A. L. Kholodenko, A. L. Beyerlein. Critical versus tricritical phase transitions in symmetric electrolytes. Phys Lett A 775 366-369, 1993. 

The p tical consequences of these delicate logarithmic effects have been anidyzed in detail for the related problem of tricritical phase transitions. 

The temperature dependence of the spontaneous polarization (Fig. 9.39) of the monomer crystal can be described in the framework of Landau s phenomenological theory assuming a tricritical phase transition [97]. The maximum experimental value of the electric polar-... 

Domb C and Lebowitz J (eds) 1984 Phase Transitions and Critical Phenomena vol 9 (London, New York Academic) oh 1. Lawrie I D and Sarbach S Theory of tricritical points oh 2. Knobler C M and Scott R L Multicritical points in fluid mixtures. Experimental studies. 

It is easily shown that a first-order phase transition is obtained for cases were d < 0, whereas behaviour at the borderline between first- and second-order transitions, tricritical behaviour, is obtained for d = 0. In the latter case the transitional Gibbs energy is... 

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. )...

We note that even short-range interactions may, however, allow a mean-field scenario, if the system has a tricritical point, where three phases are in equilibrium. A well-known example is the 3He-4He system, where a line of critical points of the fluid-superfluid transition meets the coexistence curve of the 3He-4He liquid-liquid transition at its critical point [33]. In D = 3, tricriticality implies that mean-field theory is exact [11], independently from the range of interactions. Such a mechanism is quite natural in ternary systems. For one or two components it would require a further line of hidden phase transitions that meets the coexistence curve at or near its critical point. 

The renormalisation theory (concerning magnetic critical and tricritical phenomena) of Wilson and Kogut (1971,1975) created at that time a new insight in phase transitions. The essence of this theory is that it allows for the effects of fluctuations on different scales. In the neighbourhood of critical transitions (in temperature and concentration) these fluctuations are causing important corrections of the classical theory. 

Table 6. Predicted variations for ehstic constants of a material snbject to a phase transition involving the symmetry change Pm 3 m 14/mcm when the transition is tricritical in character (6 = 0) and the volume strain is small X2 = 0).

Salje E.H, Gallardo MC, Jimenez J, Romero FJ, del Cerro J (1998) The cubic-tetragonal phase transition in strontimn titanate excess specific heat measurments and evidence for a near-tricritical, mean field type transition mechanism. J Phys Cond Matter 10 5535-5543 Schlenker IL, Gibbs GV, Boisen Jr MB (1978) Strain-tensor components expressed in terms of lattice parameters. Acta Crystallogr A34 52-54... 

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 similarity in behaviors of thermally and compositionally induced transitions suggests that the dependence of spontaneous strain on Li+Al content may parallel the dependence of strain on temperature. Xu et al. (2000) determine elastic strains (ei =02 = aloQ - 1 3 = c/co - 1) and volume strains (Vs = V/Fo - 1) by referencing the paraelastic cell dimensions to the P-quartz-like phases within the Lii fAli. fSii+ (04 series. They find that the data are consistent with the relation ei (or or Vg) = A (X-Xff, and with ei oc 3 oc Fs oc the morphotropic transition appears to be tricritical. This transition character conforms closely to that observed by Carpenter et al. (1998), who argued that the thermally induced a-P quartz transition is first-order but very close to tricritical. In... 

One kind of a multicritical point is a point over a critical line where more than two different states coalesce. The common multicritical points in statistical mechanics theory of phase transition are tricritical points (the point that separates a first order and a continuous line) or bicritical points (two continuous lines merge in a first order line) (see, for example, Ref. 166). These multicritical points were observed in quantum few-body systems only in the large dimension limit approximation for small molecules [10,32]. For three-dimensional systems, this kind of multicritical points was not reported yet. 

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. 

The liquid-solid phase transition study has been extended to higher temperatures [17-21]. There is still a debate to know whether the first order transition becomes continuous above a tricritical tenqreratuie located between 125 and 150K. 

Near the bulk isotropic-nematic phase transition temperature, Tjv/ — T = 0.1 K, and in a hybrid film of a typical LC material (such as 5CB) the nematic order is distorted if the film is thicker than dt 47 nm, and it is biaxial otherwise. The metastable biaxial structm e ceases to exist if the film thickness is larger than ds 71 nm. As the temperature is decreased both values decrease and so does the difference between them. The same structural transition can be realized if the film thickness is held constant and the temperature is varied. The (dis)continuity of the structrual transition can be changed if the temperature and film thickness are low enough [10,54,55]. The two different regimes of the transition are separated with a tricritical point below which the transition is continuous. Our estimation for its upper limit is Tjvi — Trp = 0.28 K and drp = 34 nm for 5CB [10]. 

Therefore we again obtain the first order transition for jAi — Ci >0 and second order for IB jA2 — Ci < 0 and a tricritical point for IB /Ai — C =0. The tricritical point (TCP) is located in the continuous phase transition line separating the nematic and smectic A phases [12], see a phase diagram schematically shown in Fig. 6.12. Such a point should not be confused with the triple point common for the isotropic, nematic and SmA phases. In Fig. 6.12, for homologues with alkyl chains shorter than l , the N-SmA transition is second order and shown by the dashed curve. With increasing chain length the nematic temperature range becomes narrower (like in Fig. 6.1) and, at TCP, the N-SmA transition becomes first order (solid curve). 

Consequently, the lines of first-order and continuous phase transitions abut on with each other at the tricritical point (Pu, Tic), which, therefore, defines it. 

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