Tricritical points

In a general way, the coefficients of the Landau equation are functions of pressure. When pressure is varying, b may decrease, in magnitude, and the temperature hystere.sis diminishes according to Equation 35 rind disappears at 6 = 0. 

On the state diagram P vs T, the line corresponding to o = 0 may intersect the line 6 = 0. The point of intersection defined by 

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. 

The tricritical point and its vicinity have their own peculiarities in comparison with the common-type critical point. If the configurative point approaches the tricritical point along the line 6 = 0, Ekjuation 33 gives 

Substituting Equation 38 into Equation 32 allows us to obtain 

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Mixtures of such polymeric substances with solvents show a line of critical points that in theory end at a tricritical point. (See section A2.5.9 for fiirther discussion of tricritical phenomena.)... 

In the absence of special syimnetry, the phase mle requires a minimum of tliree components for a tricritical point to occur. Synnnetrical tricritical points do have such syimnetry, but it is easiest to illustrate such phenomena with a tme ternary system with the necessary syimnetry. A ternary system comprised of a pair of enantiomers (optically active d- and /-isomers) together with a third optically inert substance could satisfy this condition. While liquid-liquid phase separation between enantiomers has not yet been found, ternary phase diagrams like those shown in figure A2.5.30 can be imagined in these diagrams there is a necessary syimnetry around a horizontal axis that represents equal amounts of the two enantiomers. 

Syimnetrical tricritical points are predicted for fluid mixtures of sulfur or living polymers m certain solvents. Scott (1965) in a mean-field treatment [38] of sulfiir solutions found that a second-order transition Ime (the critical... 

Syimnetrical tricritical points are also found in the phase diagrams of some systems fomiing liquid crystals. 

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. ... 

While the phase rule requires tliree components for an unsymmetrical tricritical point, theory can reduce this requirement to two components with a continuous variation of the interaction parameters. Lindli et al (1984) calculated a phase diagram from the van der Waals equation for binary mixtures and found (in accord with figure A2.5.13 that a tricritical point occurred at sufficiently large values of the parameter (a measure of the difference between the two components). 

One can effectively reduce the tliree components to two with quasibinary mixtures in which the second component is a mixture of very similar higher hydrocarbons. Figure A2.5.31 shows a phase diagram [40] calculated from a generalized van der Waals equation for mixtures of ethane n = 2) with nomial hydrocarbons of different carbon number n.2 (treated as continuous). It is evident that, for some values of the parameter n, those to the left of the tricritical point at = 16.48, all that will be observed with increasing... 

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.

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. 

Wheeler J C and Pfeuty P 1981 Critical points and tricritical points in liquid sulfur solutions J. Chem. Phys. 74 6415-30... 

Changing the distance between the critical points requires a new variable (in addition to the three independent fractional concentrations of the four-component system). As illustrated by Figure 5, the addition of a fourth thermodynamic dimension makes it possible for the two critical end points to approach each other, until they occur at the same point. As the distance between the critical end points decreases and the height of the stack of tietriangles becomes smaller and smaller, the tietriangles also shrink. The distance between the critical end points (see Fig. 5) and the size of the tietriangles depend on the distance from the tricritical point. These dependencies also are described scaling theory equations, as are physical properties such as iuterfacial... 

However, as given by group renormalization theory (45), the values of the universal exponents depend on the (thermodynamic) dimensionality of the system. For four dimensions (as required by the phase rule for the existence of tricritical points), the exponents have classical values. This means the values are multiples of 1/2. The dimensions of the volume of tietriangles are (31)... 

As confirmed by experiment (47), the dependence of the optimal tension on the distance from the tricritical point is given (40) by... 

Another interesting version of the MM model considers a variable excluded-volume interaction between same species particles [92]. In the absence of interactions the system is mapped on the standard MM model which has a first-order IPT between A- and B-saturated phases. On increasing the strength of the interaction the first-order transition line, observed for weak interactions, terminates at a tricritical point where two second-order transitions meet. These transitions, which separate the A-saturated, reactive, and B-saturated phases, belong to the same universality class as directed percolation, as follows from the value of critical exponents calculated by means of time-dependent Monte Carlo simulations and series expansions [92]. 

To determine the position of the tricritical point and the structure of the ordered phases stable below the bifurcation we analyze the asymptotic form of Qeff for e 0. At local minima the functional derivative of Qeff with respect to all the OPs vanishes. From this condition and from (45), (58), (47), and (64) we find that at the metastable states... 

From (62) and (70) it follows that the Lifshitz and tricritical points coincide giving the Lifshitz tricritical point [18,66] for 7 = 27/4. 7 = 27/4 can be considered, as a borderline value between the weak (7 <27/4) and the strong (7 >27/4) surfactants. For the weak surfactants the tricritical point is located at the transition between the microemulsion and the coexisting uniform oil- and water-rich phases, whereas for the strong surfactants the tcp is located at the transition between the microemulsion and the liquid-crystal phases. The transition between the microemulsion and the ordered periodic phases is continuous for p < Ps < Ps and first order for p > p[. 

An example for a partially known ternary phase diagram is the sodium octane 1 -sulfonate/ 1-decanol/water system [61]. Figure 34 shows the isotropic areas L, and L2 for the water-rich surfactant phase with solubilized alcohol and for the solvent-rich surfactant phase with solubilized water, respectively. Furthermore, the lamellar neat phase D and the anisotropic hexagonal middle phase E are indicated (for systematics, cf. Ref. 62). For the quaternary sodium octane 1-sulfonate (A)/l-butanol (B)/n-tetradecane (0)/water (W) system, the tricritical point which characterizes the transition of three coexisting phases into one liquid phase is at 40.1°C A, 0.042 (mass parts) B, 0.958 (A + B = 56 wt %) O, 0.54 W, 0.46 [63]. For both the binary phase equilibrium dodecane... 

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. )...

Bourrel M, Chambu C (1983) The Rules for Achieving High Solubilization of Brine and OU by AmphiphUic Molecules. Soc Petrol Eng J 23 327-338 Kunieda H, Shinoda K (1985) Evaluation of the hydrophile-lipophile balance (HLB) of nonionic surfactants I. Multisurfactant systems. J Colloid Interface Sci 107 107-121 Kahlweit M, Strey R, Eirman P (1986) Search for tricritical points in ternary systems Water-oil-nonionic amphiphile. J Phys Chem 90 671... 

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 point for four-dimensional amphiphile—oil—water—electrolyte-temperature...

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. 

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