Tricritical temperature

While, in principle, a tricritical point is one where three phases simultaneously coalesce into one, that is not what would be observed in the laboratory if the temperature of a closed system is increased along a path that passes exactly tlirough a tricritical point. Although such a difficult experiment is yet to be perfomied, it is clear from theory (Kaufman and Griffiths 1982, Pegg et al 1990) and from experiments in the vicinity of tricritical points that below the tricritical temperature only two phases coexist and that the volume of one slirinks precipitously to zero at T. ... 

Fig. 5 Magnetic phase diagram of [Mn(Cp )2][Pt(tds)2] M(T) (filled diamonds) M(H) (//] (filled triangles), H (filled inverted triangles), x (T) (open circles) x (H) (open squares) Tt is the tricritical temperature I denotes the first-order MM transition II denotes a second-order transition (AF-PM phase houndary) and III denotes a higher order transitions (from a PM to a FM like state). From [45]...

R — —1) plotted versus temperature. The transition is second order for temperatures higher than the tricritical temperature 7", while for T < 7j it is of first order. From Binder and Landau (1981). 

Fig. 32. (a) Phase diagram of the square lattice gas with nearest neighbor repulsion /nn > 0 and next-nearest neighbor attraction 7I)m, < 0, in the plane of variables temperature and coverage, for three choices of R = 7nnn/7nn. Insert shows the variation of the maximum transition temperature (at 9 = 1 /2) and of the tricritical temperature Tt with R. From Binder and Landau (1981). 

The liquid-crystal transition between smectic-A and nematic for some systems is an AT transition. Depending on the value of the MacMillan ratio, the ratio of the temperature of the smectic-A-nematic transition to that of the nematic-isotropic transition (which is Ising), the behaviour of such systems varies continuously from a k-type transition to a tricritical one (see section A2.5.91. Garland and Nounesis [34] reviewed these systems in 1994. 

Figure A2.5.31. Calculated TIT, 0 2 phase diagram in the vicmity of the tricritical point for binary mixtures of ethane n = 2) witii a higher hydrocarbon of contmuous n. The system is in a sealed tube at fixed tricritical density and composition. The tricritical point is at the confluence of the four lines. Because of the fixing of the density and the composition, the system does not pass tiirough critical end points if the critical end-point lines were shown, the three-phase region would be larger. An experiment increasing the temperature in a closed tube would be represented by a vertical line on this diagram. Reproduced from [40], figure 8, by pennission of the American Institute of Physics.

Fig. 5. Lower and upper critical tielines in a quaternary system at different temperatures and a plot of the critical end point salinities vs temperature, illustrating lower critical endline, upper critical endline, optimal line, and tricritical poiat for four-dimensional amphiphile—oil—water—electrolyte-temperature...

The variation of the order parameter with temperature thus distinguishes second-order transitions from tricritical behaviour. In general the variation of the order parameter with temperature for a continuous transition is described as... 

Figure 5. Phase diagram for Nj =2 quark matter in the NCQM. The critical temperature for color superconductivity (2SC phase) can be high enough for this phase to reach close to the tricritical point which shall be explored in future heavy-ion collision experiments.

Fig. 27. Phase diagram of an adsorbed film in- the simple cubic lattice from mean-fleld calculations (full curves - flrst-order transitions, broken curves -second-order transitions) and from a Monte Carlo calculation (dash-dotted curve - only the transition of the first layer is shown). Phases shown are the lattice gas (G), the ordered (2x1) phase in the first layer, lattice fluid in the first layer F(l) and in the bulk F(a>). For the sake of clarity, layering transitions in layers higher than the second layer (which nearly coincide with the layering of the second layer and merge at 7 (2), are not shown. The chemical potential at gas-liquid coexistence is denoted as ttg, and 7 / is the mean-field bulk critical temperature. While the layering transition of the second layer ends in a critical point Tj(2), mean-field theory predicts two tricritical points 7 (1), 7 (1) in the first layer. Parameters of this calculation are R = —0.75, e = 2.5p, 112 = Mi/ = d/2, D = 20, and L varied from 6 to 24. (From Wagner and Binder .)...

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. )...

In a blend of immiscible homopolymers, macrophase separation is favoured on decreasing the temperature in a blend with an upper critical solution temperature (UCST) or on increasing the temperature in a blend with a lower critical solution temperature (LCST). Addition of a block copolymer leads to competition between this macrophase separation and microphase separation of the copolymer. From a practical viewpoint, addition of a block copolymer can be used to suppress phase separation or to compatibilize the homopolymers. Indeed, this is one of the main applications of block copolymers. The compatibilization results from the reduction of interfacial tension that accompanies the segregation of block copolymers to the interface. From a more fundamental viewpoint, the competing effects of macrophase and microphase separation lead to a rich critical phenomenology. In addition to the ordinary critical points of macrophase separation, tricritical points exist where critical lines for the ternary system meet. A Lifshitz point is defined along the line of critical transitions, at the crossover between regimes of macrophase separation and microphase separation. This critical behaviour is discussed in more depth in Chapter 6. 

Fig. 6.42 Schematic of the critical line as a function of ip - 0A -+ n, jj = 0A

Finally, we recall that in high-temperature aqueous solutions of NaCl near the L-G critical line, crossover has also been observed. Again, it has been concluded [152] that the critical locus may be affected by a virtual tricritical 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. 

The four-phase line shows an upper critical endpoint (UCEP) at 313.10 K and 8.232 MPa. At higher temperatures an interesting phenomenum can be oberseved. With increasing temperature the two critical lines (L2=L3)V and L2(L3=V), which bound the three-phase region L2L3V, appproach and finally meet at a common endpoint, a tricritical point where phases L2, L3, and V become critical simultaneously. The procedure to determine the tricritical point has been described previously [4], The tricritical point TCP was determined to 320.75 K and 9.26 MPa. 

What does your best value for the exponent j8 indicate about the nature of the critical behavior underlying the N-I transition (mean-field second-order or tricritical) Note that /3 and P describe the temperature variation of the nematic order parameter S, which is a basic characteristic of the liquid crystal smdied. Thus, the same j8 and P values could have been obtained from measurements of several other physical properties, such as those mentioned in the methods section. 

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