Tricritical point phase diagram

This study suggests realizing a future research program including a study of the boundaries of global phase diagram (tricritical points, double critical end points, and etc) for binary mixture with polyamorphic components. 

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 also found in the phase diagrams of some systems fomiing liquid crystals. 

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.

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. )...

Fig. 6.42. For < 2, the system is homogeneous at all compositions (regime (i), not shown in Fig. 6.42). The general (multidimensional) phase diagram for %N > 2 is enriched by the presence of tricritical points and Lifshitz tricritical points under certain conditions. The critical line for homopolymer phase separation is given by 0hjOi, = 2fyN, r]ml = 0 where A + V>b and rj = ipA - ipB (Broseta and Fredrickson 1990) (the so-called Scott line (Scott 1949) ). Here ipA and ipB are the volume fractions of A and B monomers. The point at which the Scott line meets the lines of critical points for phase separation (at q = 0) in the A-AB and B-AB systems is termed a tricritical point. This occurs at (Broseta and Fredrickson 1990)...

Fig. 6.43 Phase diagram for a ternary mixture of equal concentrations of A and B homopolymers and symmetric AB diblock (all with equal degrees of polymerization) computed by Holyst and Schick (1992). The Lifshitz tricritical point is shown at L, the line CL is that of continuous transitions from the disordered phase to coexisting A-rich and B-rich phases, and LG is the line of continuous transitions from the disordered to the lamellar phase. LD is the disorder line.

This is designated as Li = V + L2. An L point arises when a low-density liquid (Li) and a high-density liquid (L2) become critically identical in the presence of a vapor phase. An L point is designated as Li = L2 -f V. Tricritical points, where three phases in equilibrium are also critically identical, are designated as Li = L2 = V. Such critical points, while present in phase diagrams and phase projections, are rarely observed in practice. At low temperatures, solid phases such as asphaltenes and wax can, and frequently do, coexist with the fluid phases noted here and are discussed in later sections. 

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). 

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. 

Figure 12. Generalized T-A phase diagram for fatty acids. A typical n-A isotherm is indicated by the dashed line. Two possible intersections of the line of second-order transitions with the LE-LC coexistence curve are shown. In the case of the dotted line, the intersection is a tricritical point.

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]. 

---