Lifshitz 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[. 

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.

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]. 

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. 

Similar arguments hold for a tricritical lifshitz point, where a Lifshitz point happens to merge with a regular tricritical point. Here, fluctuations are important if... 

Three-dimensional modulation in fluids corresponds to a so-called isotropic Lifshitz-point, which can describe three-phase equilibrium with a middle microemulsion (sponge) phase and which can thus be treated as a special tri-critical point. Both the interfadal tensions between the water-rich and microemulsion phase and between oil-rich and microemulsion phases will vanish at the tricritical Lifshitz point much faster than at a tricritical point in ordinary multicomponent fluid mixtures, since the gradient-term coefficient cq also vanishes. ... 

Weakly non-linear theory predicts two tricritical points (TP) with a subcritical bifurcation in the field range 5 < hx < 26. Note that the upper TP at hx = 26 is slightly above the lower LP. One may expect rather complex non-linear behaviour in that range, which has not been worked out in detail for 5CB so far. The situation should be simpler for MBBA where the TPs (hx = 4.15 and 31) are below and well-separated from the Lifshitz points (hx = 36 and 62). 

Mean field theory predicts that the critical lines of blend like and diblock like behavior meet at the isotropic critical lifshitz point and the lifshitz line (LL) which is defined when Q becomes zero. The isotropic critical Lifshitz point represents a new imiversahty class [54-56]. Under special conditions even a tricritical Lifshitz point is predicted [55]. In this article we will discuss in some detail SANS experiments on a mixture of a critical binary (A/B) polymer blend with different concentrations of a symmetric (A-B) diblock copolymer of roughly five times larger molar volume. Under such conditions an isotropic critical Lifshitz point is predicted [55]. 

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