N-SmA-SmC point

Finally, we notice that the N-SmA-SmC point is an example of Lifschitz point defined as the place where some of the coefficients of the gradient terms (C here) vanish [48, 49]. 

Complications arise from the vanishing of the N-SmC latent heat at the N-SmA-SmC point and from the difficulties connected to the smectic state (Lan-dau-Peierls instability) the N-SmA transition (lack of gauge invariance) and the SmA-SmC transition (proximity of a tricrit-ical point). 

The description of the Chen-Lubensky model is reasonably well borne out by experiment [50,51] the universal topology of the phase diagram and the existence of a Cx=0 line in particular are well established. An interesting possibility pointed out by Grinstein and Toner [52] with a model of dislocation unbinding is the existence of a biaxial nematic phase if the N-SmA and SmA-SmC lines are second order with XT critical exponents, the N-SmA-SmC point should be tetracritical [53] and a mixed phase (i.e. a biaxial nematic) should show up. It has however not been observed so far. 

K diverges in the vicinity of the N-SmA-SmC point where Cx vanishes. The type II condition is thus expected to be fulfilled close to the N-SmA-SmC point. In presence of chirality, this is precisely the place where a liquid crystal analog of the Abrikosov flux phase should show up. The structure of this new liquid crystalline state will be described in subsequent section. 

In some mixtures of liquid crystals exhibiting N, SmA and SmC phases, the N-SmA, SmC-SmC, and N-SmC transition lines meet at the N-SmA-SmC multicritical point. Fig. 18 gives the phase diagram with ai N-SmA-SmC point for mixtures of 508 + 608 (4-n-alkyloxyphenyl-4 -n-alkyloxybenzoate) compounds [73]. The N-SmA-SmC point has been the subject of extensive theoretical and experimental studies during the past decade. The nature of the point is, however, still not clearly established. 

Figure 19. Temperature dependence of the enthalpy for several mixtures of 50.S+60.8 with direct N-SmC transitions. For clarity reasons a large linear background C =aR has been subtracted from the direct experimental data. The curve with the smallest steplike increase is closest to the N-SmA-SmC point, and has a latent heat of (1.7 0.3) J/mol [73].

Legrandet al. [102] report an electroclinic effect in the cholesteric phase near a N -SmA-SmC multicritical point of the n = 8 homolog of a biphenyl benzoate series. This chiral analogon to the N-SmA-SmC point is found at 15 MPa and 144 C. 

A point where three fluctuation dominated phase transition lines meet in a 2-dimen-sional parameter space is also expected to exhibit universal features. An extensively studied liquid crystal candidate was the N-SmA-SmC point in mixtures [83], in a pure compound under pressure [84] and at the re-entrant N-SmA-SmC multicritical point [85]. The situation may be summarized as follows. The systems studied showed qualitative and quantitative similarities. However, the exponents exhibited were not in the expected universality class for three second order phase transition lines meeting at a point. This is likely because, in the N-SmA-SmC case, the N-SmC transition line is first order [86] as is the N-SmA transition line, [26] leaving only the SmA-SmC second-order phase transition line. 

On further investigation, Shashidar [20] explains, a remarkable situation was found. In every case, one of the materials had an inherent SmC phase, while the other did not. As there was no observed miscibility gap, in every case there had to be a N-SmA-SMC point at very low temperatures. They also knew from their studies at high pressure [54] as well as their detailed studies on mixtures exhibiting the N-SmA-SmC and the Nre-SmC-SmA point (see Figs. 4 and 12) [16], that the phase diagram has the universal spiral topology where phase boundaries curve as the N-SmA-SmC point is approached. Shashidhar [20] concludes that, as re-entrance in nonpolar compounds results from the universal curvature of the phase boundaries as the N-SmA-SMC point is approached, its origin is the fluctuations associated with the N-SmA-SmC multicritical point. 

Twist grain boundary (TGB) phases [l]-[4] usually appear in the temperature range between a cholesteric N phase with short pitch and a smectic phase, typically SmA or SmC. In particular, they are expected to appear close to a N /SmA/SmC triple point [5]. One of their remarkable properties is the selective reflection of circularly polarized light [2], [3]. This feature shows that the director field has a helical structure similar to the cholesteric phase. On the other hand. X-ray investigations of TGB phases indicate a layer structure as occurring in smectic phases [6]. Chirality of the system is an essential precondition for the occurrence of TGB phases. In mixtures of... 

Another interesting behavior under elevated pressures was observed [127] in binary mixtures of 4-(2 -methylbutyl) phenyl 4 - -octyl biphenyl-4-carboxylate (CE8) and 4- -dodecyloxy biphenyl-4 -(2 -methylbutyl) benzoate (C12) which show TGB phases close to a virtual N -SmA-SmC triple point. Krishna Prasad et al. observed, in a mixture which shows a direct SmA-N transition, that the appearance of the TGBa phase between SmA and N can be induced by pressure. From the topology close to the SmA-TGB-N point, it was concluded that this point is a critical end point rather than a bicritical point as predicted by Renn and Lubensky [5]. In a mixture with different concentrations which shows a SmC -TGBA transition, the... 

The existence of a N-SmA-SmC multicrit-ical point (i.e. a point where the N-SmA, SmA-SmC and N-SmC lines meet) was demonstrated in the late seventies [40, 41]. Various theories have been proposed to describe the N-SmA-SmC diagram [42-45]. The phenomenological model of Chen and Lubensky [46] (referred to as the N-SmA-SmC model) captures most of the experimental features. The starting point is the observation that the X-ray scattering in the nematic phase in the vicinity of the N-SmA transition shows strong peaks at wavenumber qA= o - Near the N-SmC transition, these two peaks spread out into two rings at qc = ( g, q cos(p, q sinq>). 

Figure 2. N-SmA-SmC phase diagram from Chen and Lubensky [46]. All transitions are second order in mean field but fluctuations lead to a first order N-SmC transition [47]. The line Cx=0 separates two regions in the nematic phase with diffuse X-ray scattering centered about g i=0 (Cx>0) and g 0 (Cx<0). The point Cx=0, r=0 is a Lifschitz point.

The re-entrant N-SmC-SmA multicritical point and the N-SmA-SmC multicritical point are, by definition, points in the temperature-concentration or pressure-temperature plane at which three second order phase boundaries meet. At this point all in-... 

At a second-order SmA-SmC phase transition, the symmetries are different but the layer spacing is the same. Fluctuations can drive a line of second-order SmA-SmC phase transitions to an N-SmA-SmC mul-ticritical point (see Fig. 4) [53]. Competing N-SmA and N-SmC fluctuations pull the N phase under the SmA phase in the temperature-concentration phase plane, leading to the Nj-e-SmA-SmC multicritical point [16]. High-resolution studies, as a function of both concentration and pressure, resolve the fluctuation-driven N-SmA/ N-SmC step (see, e.g. Fig. 5) into a universal spiral (Fig. 12) [16] around the N-SmA-SmC and the Nre-SmA-SmC multicritical points. Loosely speaking, the N-SmA transition line is dominated by N-SmA fluctuations, and the N-SmC transition line is dominated by Brazovskii fluctuations [54] that drive the N-SmC transition to lower temperatures compared to the N-SmA transition [18]. 

Figure 12. Detail of the Nre-SmA-SmC multicriti-cal point showing the universal spiral topology [16] of the N-SmA-SmC multicritical point [54],...

Scattering experiments have also been carried out close to the N - SmA - SmC mul-ticritical point in binary mixtures of meso-gens (see reference [143] and references therein). Above the first-order N-SmC transition, smectic fluctuations evolve from being SmA-like to SmC-like upon cooling. 

Some investigations have been devoted to the behaviour of elastic constants in particular regions of the mesophase diagram for example, studies near the N-SmA-SmC tricrit-ical point [86,89,93,94] and studies of elasticity in a re-entrant nematic phase [113]. 

SmA phases, and SmA and SmC phases, meet tlie line of discontinuous transitions between tire N and SmC phase. The latter transition is first order due to fluctuations of SmC order, which are continuously degenerate, being concentrated on two rings in reciprocal space ratlier tlian two points in tire case of tire N-SmA transition [18,19 and 20], Because tire NAC point corresponds to the meeting of lines of continuous and discontinuous transitions it is an example of a Lifshitz point (a precise definition of tliis critical point is provided in [18,19 and 20]). The NAC point and associated transitions between tire tliree phases are described by tire Chen-Lubensky model [97], which is able to account for tire topology of tire experimental phase diagram. In tire vicinity of tire NAC point, universal behaviour is predicted and observed experimentally [20]. 

Fig. 13.10 Comparison of the temperature dependencies of viscosity coefficients yi (nematic), (soft mode) and y (Goldstone mode) of the same chiral mixture within the ranges of the N and SmC phases [15]. Note that Yi and y curves may be bridged through the SmA phase black points) where measurement have not been made...

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