Tricritical transitions

The transitional entropy, enthalpy and heat capacity for a tricritical transition is for T < Ttrs ... 

For a tricritical transition (i.e., a high-order transition crossing boundary), B = 0 and C> 0, and condition 2.53 leads to a first-order phase... 

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. 

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). 

The phase transition of DNP could further be characterized by specific heat measurements for monomer and thermally polymerized single crystals (Fig. 9.46) [98]. These data support the description of the phase transition of the monomer crystals as a tricritical transition. This means it is a borderline case between a first-order and second-order phase transition, with a distribution of transition temperatures. The transition enthalpy was much lower than the corresponding order-disorder transition, in agreement with results obtained by Ber-tault et al. via Raman spectroscopy, which proved the importance of displacive contributions to the DNP phase transition [99]. 

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. 

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... 

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. 

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]. 

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[. 

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

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... 

M molar mass), where I and III are the tricritical or -regions. Here, the chain molecules exhibit an unperturbed random coil confirmation. In contrast, I and II are the critical or good solvent regimes, which are characterized by structural fluctuations in direction of an expanded coil conformation. According to the underlying concept of critical phenomena, the phase boundaries have to be considered as a continuous crossover and not as discontinuous transitions. 

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]...

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... 

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... 

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. )...

Figure 2.8 The order parameter g as a function of TIT for second-order (a) and tricritical (b) transitions, and as a function of T/Ttrans for a first-order transition (c) with = 0.99Ttrans nd gtrans 0-4- From Carpenter (1988). Reprinted by permission of Kluwer Academic Publishers.

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. 6.43 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.

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 transition to the continuum fluid may be mimicked by a discretization of the model choosing > 1. To this end, Panagiotopoulos and Kumar [292] performed simulations for several integer ratios 1 < < 5. For — 2 the tricritical point is shifted to very high density and was not exactly located. The absence of a liquid-vapor transition for = 1 and 2 appears to follow from solidification, before a liquid is formed. For > 3, ordinary liquid-vapor critical points were observed which were consistent with Ising-like behavior. Obviously, for finely discretisized lattice models the behavior approaches that of the continuum RPM. Already at = 4 the critical parameters of the lattice and continuum RPM agree closely. From the computational point of view, the exploitation of these discretization effects may open many possibilities for methodological improvements of simulations [292], From the fundamental point of view these discretization effects need to be explored in detail. 

As the density increases, the validity of KT theory becomes, however, more and more questionable. There are conflicting views about the fate of the KT transition. It was suggested that the KT transition is replaced by some discontinuous first-order transition or by a first-order coexistence curve between an insulating vapor and a conducting fluid-like phase [293]. Minnhagen and Wallin [294,295] found that the KT transition terminates in a critical end point. In contrast, DH theory predicts in D = 2 the KT line to terminate in a tricritical point, after which the insulating vapor phase coexists with a... 

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