Location 353 tricritical point

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[. 

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 an example, consider a uniform parent distribution p(°) (a) = const, which can be written as p (o) = p /2, using the fact that pj, = do p (a). Then pf 1 — pff / 3 and p =p /5 and so a tricritical point occurs if the overall density (i.e., the copolymer volume fraction) is pf = 3/(2r + 3). Figure 13 shows the coexistence curve calculated for this parent (with r = 1), which clearly shows the tricritical point at the predicted value X1 = l/(2/4° ) = r + 3/2 = 2.5. Our numerical implementation manages to locate the tricritical point and follow the three coexisting phases without problems we take that as a signature of its robustness [58]. Note that the tricritical point that we found is closely analogous to that studied by Leibler [57] for a symmetric blend of two homopolymers and a symmetric random copolymer that is, nonetheless, chemically monodisperse (in the sense that o = 0 for all copolymers present). In fact, in our notation, the scenario of Ref. 57 simply corresponds to a parent density of the form p (o) S(o — 1) + S(o +1), with the copolymer (o = 0) now playing the role of the neutral solvent. 

Figure 51. Experimental phase diagram of CO physisorbed on graphite with the phases fluid (F), commensurate (CD) and incommensurate (ID) orientationaliy disordered solids, reentrant fluid (RF), second-layer fluid (2F), vapor (2V), liquid (2L), and orientationaliy disordered solid (2SD) phases. Filled circles and triangles represent phase boundary locations from heat capacity and vapor pressure measurements, respectively. Solid and dashed lines indieate phase boundaries believed to be associated with first-order and continuous transitions, respectively dash-dotted lines correspond to speculated boundaries. The large filled triangle and the large filled circle mark the two-dimensional Potts tricritical and critical points, respectively the tricritical point marked with an open triangle is tentative. Lines I-VII with arrows are experimental paths of the heat capacity scans shown in Fig. 52. Coverage unity corresponds to a coverage of CO forming a complete (Vs x VS) monolayer. (Adapted from Fig. 1 of Ref. 112.)...

Fig. 5.6.5. High resolution temperature-concentration T-X) diagram for binary mixtures of 4-n-octyloxy- and 4-n-decyloxy-phenyl-4 -nitrobenzoyloxybenzoate in the vicinity of the Aj-N -A point. The solid lines denote first order phase boundaries. The critical end point (CEP) for the A -N boundary and the approximate location of the tricritical point (TCP) for the Aj-N boundary are indicated in the diagram. (After reference 127.)...

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). 

The surfaces D and B, located symmetrically in the region C > 0 and C < 0, characterize the regions of first-order phase transitions, bounded by the lines of the critical points (the dashed curves in Figure 1..38). Thus, two curves of critical points and their continuous phase transitions curve counterpart intersect at the tricritical point (Griffiths, 1970, 1973),... 

For a ternary system, the diagram is one-dimensional with a triple criticed (tricritical) point which separates the three-phase region from the two-phzise ones with double criticed points (Figure 3.100a see also Figures 3.93 and 3.94). The location of the tricritical point r2,tc = r (pi) depends on pi, varying from r 15.645 for p = 1 to r 9.899 for pi oo (Tompa, 1949 Sole et al., 1984). 

Moreover, the density difference between the fluid and the commensmate solid narrows with increasing temperature and appears to vanish at a tricritical point at a temperature above lOOK (which is a behavior clearly different from the 3D fluid), confirming the proposal of Putnam et al. [44]. Following Butler et al. [105], the tricritical point is located at approximately 115 K on the fluid-commensurate solid-phase boundary (see Fig. 4). 

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