Isotropic Lifshitz point

Fig. 2.42 Spinodal lines for a random multiblock copolymer melt of variable X (Fredrickson el al. 1992). On cooling a melt with X > AL —0.268, the first instability is predicted to be phase separation into two homogeneous liquid phases (x = %m)- On further cooling to % = the two liquid phases become unstable with respect to formation of a microphase. In contrast, a melt with X < XL first becomes absolutely unstable to the formation of microphases (x = fom)- At the critical composition of /= j, the point (AL, Xi) is an isotropic Lifshitz point.

The phase behaviour of blends of homopolymers containing block copolymers is governed by a competition between macrophase separation of the homopolymer and microphase separation of the block copolymers. The former occurs at a wavenumber q = 0, whereas the latter is characterized by q + 0. The locus of critical transitions at q, the so-called X line, is divided into q = 0 and q + 0 branches by the (isotropic) Lifshitz point. The Lifshitz point can be described using a simple Landau-Ginzburg free-energy functional for a scalar order parameter rp(r), which for ternary blends containing block copolymers is the total volume fraction of, say, A monomers. The free energy density can be written (Selke 1992)... 

As discussed in Sect. 4.1, one prominent example of a situation where the SCF theory fails on a qualitative level is the microemulsion channel in ternary mixtures of A and B homopolymers and AB diblock copolymers. Figure 5 shows an example of a mean-field phase diagram for such a system. Four different phases are found A disordered phase, an ordered (lamellar) phase (see Fig. 4, right snapshot), an A-rich and a B-rich phase. The SCF theory predicts the existence of a point where all three phases meet and the distance of the lamellar sheets approaches infinity, an isotropic Lifshitz point [100,101]. 

It seems plausible that fluctuations affect the Lifshitz point. If the lamellar distance is large enough that the interfaces between A and B sheets can bend around, the lamellae may rupture and form a globally disordered structure. A Ginzburg analysis reveals that the upper critical dimension of isotropic Lifshitz points is as high as 8 (see also Sect. 4.1). Unfortunately, the lower critical dimension of isotropic Lifshitz points is not known [ 102],... 

Thermodynamically stable, bicontinuous microemulsions have recently been shown to be obtainable in symmetric ternary blends of two homopolymers and a diblock copolymer by formulating alloys with compositions near mean-field isotropic Lifshitz points. In the present paper, it is argued that practical apphcation of this design criterion could require use of homopolymers of unequal molec.wts. and block copolymers of different structure. The existence of, and explicit location of, mean-field isotropic Lifshitz points in ternary blends with homopolymer molec.wt. asymmetry and either AB diblock or ABA triblock copolymer structures were demonstrated. These calculations significantly expanded the parameter space for observing bicontinuous miCToemulsions and allowed for more flexibility in tailoring melt rheological properties and solid-state mechanical properties. 29 refs. 

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. ... 

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]. 

Fig. 23 Phase diagram in the temperature-diblock copolymer plane for the (dPB PS) mixture below the Lifshitz line separating blend like from diblock-like phase behavior. The full dots and the solid line represent the critical points of a two-phase region. The hatched area indicates a crossover from Ising to isotropic Lifshitz critical behavior, and a double critical point DCP is at 7% diblock concentration. The Lifshitz line separates at high and low temperatures the disordered phases and droplet and bicontinuous microemulsion phases ( xE). Its non-monotonic shape near the DCP is caused by the strong thermal fluctuations...

Double critical points in general and the observed one in the (PB PS) system are characterized by a strong increase of y if one applies the scaling law of S"Ho) = C T with the conventional reduced temperature field. If one, however, applies a modified reduced temperature field according to f = (1 - ruc/r)(l - Tlc/T) then the inverse susceptibility delivers a critical exponent y of the corresponding universality class, which here is of the isotropic Lifshitz class [88]. At the DCP one has Tc = Tuc = Tlc and therefore S Ho) = The solid lines in the lower Fig. 26 describe such a behav-... 

A first systematic study of such system was performed on the relatively large-molar-mass symmetric polyolefins PE and PEP and the corresponding diblock copolymer PE-PEP PE being polyethylene and PEP being poly(ethylene propylene). A mean-field Lifshitz like behavior was observed near the predicted isotropic Lifshitz critical point with the critical exponents y=l and v=0.25 of the susceptibility and correlation length, and the stmcture factor following the characteristic mean-field Lifshitz behavior according to S(Q)ocQ". Thermal composition fluctuations were apparently not so relevant as indicated by the observation of mean-field critical exponents. On the other hand, no Lifshitz critical point was observed and instead a one-phase channel of a polymeric bicontinuous miaoemulsion phase appeared. Equivalent one-phase channels were also observed in other systems. 

Let us consider a homogeneous, isotropic solid body. Under the action of applied forces, the solid body exhibit deformation. A point of initial position vector r (with components (x, y, z)) has, after the deformation, a new position r = r + u where u(x, y, z) is the displacement field. The strain tensor Uik is defined as (Landau and Lifshitz, 1986)... 

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