Lifshitz line

For strongly structured microemulsions, g is negative, and the structure functions show a peak at nonzero wavevector q. As long as g < 2 /ca, inverse Fourier transform of S q) still reveals that the water-water correlation functions oscillate rather than decay monotonically. The lines in phase space where this oscillating behavior sets in are usually referred to as disorder lines, and those where the maximum of S q) moves away from zero as Lifshitz lines. ... 

Schubert, K.-V., Strey, R., Kline, S. and Kaler, E.W. (1994) Small-angle neutron scattering near the Lifshitz lines Transition from weakly structured mixtures to microemulsions. /. Chem. 

Morkved, T.L., Stepanek, P., Krishnan, K., Bates, F.S. and Lodge, T.P. (2001) Static and dynamic scattering from ternary polymer blends Bicontinuous microemulsions, Lifshitz lines, and amphiphilicity. /. Chem. Phys., 114, 7247-7259. 

Hence the = 0 point is a fluctuation cross-over point. The locus of such points (a > 0, Cj = 0) in a phase diagram, say the T-X diagram, gives the Lifshitz line, and the point where both a and are zero is the Lifshitz point. 

We can immediately conclude from Eqs. (31) and (32) that there are two lines within the region of stability of a microemulsion phase where the behavior of the scattering intensity or of the correlation function changes qualitatively. The first of these lines is the Lifshitz line... 

Figure 6 The position of the scattering peak as a function of the surfactant volume fraction 0 at constant / = 4.4 A, [idow = 0.1 A -, and a = 2 A. The Lifshitz line is at = 0.545. The data points are for the water-octane-C4E1 systems of Ref 94. (Redrawn from Ref 36.)...

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]. 

As this article is devoted to critical crossover phenomena in polymer blends, we limit our considerations to homopolymer blends of critical composition mixed with diblock concentrations below the Lifshitz line. 

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...

Near the Lifshitz line thermal composition fluctuations are expected to become strong over a larger temperature range because of the reduced smTace energy (c2 a I2 = 0), leading to a lower threshold force for thermal fluctuations. On a more abstract level this effect can also be interpreted in terms of a larger upper critical dimension Du = 8 beyond which thermal fluctuations become irrelevant, and Gi is twice as large as for ordinary binary polymer blends [86]. 

SANS Results from below the Lifshitz Line... 

Figure 25 displays the effects of thermal composition fluctuations on the inverse susceptibility S(0) for a (PEE PDMS) mixture (sample 10 in Table 2) versus 1/T for different diblock concentrations below the Lifshitz line [48]. The critical temperatures determined from S Ho) = 0 decrease with increasing diblock content in a similar way as shown for the (PB PS) blend (Fig. 23). The = 4.3% sample behaves as a pure blend At high temperatures S (0)...

Figure 27 Temperature - diblock concentration plane of the PB/PS blend phase diagram. The composition of the PB/PS homopolymer blend was the critical one of the binary blend. The right figure shows the Lifshitz part. Meaning of symbols (,) line of critical points with (-) the double critical point (() and (v) Lifshitz line between disordered and microemulsion phases (DpE and BpE droplet and bicontinuous microemulsion), respectively (B) transition from disordered to micro emulsion phase, ix) Lifshitz transition point LLT, (A) Ordering transition to lamellar phase. Erom Pipich, V. Schwahn, D. Willner, L. Phys. Rev. Lett. 2005, 94,117801 J. Chem. Phys. 2005, 123,124904-124916 ...

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