SmA-N transition

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. 11. Distribution of backbone units along the normal to the smectic layers. Continuous line just below the N/SmA transition dashed line room temperature...

Due to the conformation asymmetry in rod-coil diblock copolymer systems, the packing is expected to be totally different from conformationally symmetric coil - coil block copolymers. Semenov and Vasilenko [71] have predicted that a N-SmA transition can be either a first-order transition (in the case of large coil fraction) or a second-order transition (in the case of small coil fraction) and a SmC phase in a rod-coil system is also expected for f <036. 

The narrower a range of the nematic phase in a homological series of different compounds (as an example see Fig. 6.1 la) the stronger are first order features of the N-SmA transition. In some sense, the SmA phase feels the proximity of the isotropic phase. In other words, we may say that, in the isotropic phase, there are traces of both nematic and smectic A short-range order. 

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). 

Figure 6. Phase diagram of a fluid of hard sphero-cylinders in the (axial ratio/order parameter) plane. The circles are the simulation results for the smectic A transition [45]. The N-SmA transition obtained in [45] is denoted by squares N and triangles SmA (after Poniwierski and Sluckin [69]).

After the isotropic to nematic transition, the next step towards more ordered mesophas-es is the condensation of SmA order when the continuous translational symmetry is broken along the director. The theoretical description of the N-SmA transition begins with the identification of an order parameter. Following de Gennes and McMillan [1, 16], we notice that the layered structures of a SmA phase is characterized by a periodic modulation of all the microscopic properties along the direction z perpendicular to the layers. The electron density for instance, commonly detected by X-ray scattering can be expanded in Fourier series ... 

If the nematic susceptibility X is low (i.e. far enough from temperature of the isotropic-nematic transition), u is positive and the N-SmA transition is second order at risf A again. 

Close to the N-SmA transition, the vanishing of the elastic constant of compression of the layers amplifies the fluctuations of the phase whereas eritical fluctuations of the amplitude P(r) are expected to be important too. 

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>). 

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). 

A number of open questions remain the N-SmA transition for instance is almost understood but not quite which suggests more efforts have to be done. 

The SmA liquid crystalline phase results from the development of a one-dimensional density wave in the orientationally ordered nematic phase. The smectic wave vector q is parallel to the nematic director (along the z-axis) and the SmA order parameter i/r= i/r e is introduced by P( ) = Po[1+R6V ]- Thus the order parameter has a magnitude and a phase. This led de Gennes to point out the analogy with superfluid helium and the normal-superconductor transition in metals [7, 59]. This would than place the N-SmA transition in the three-dimensional XY universality class. However, there are two important sources of deviations from isotropic 3D-XY behavior. The first one is crossover from second-order to first-order behavior via a tricritical point due to coupling between the smectic order parameter y/ and the nematic order parameter Q. The second source of deviation from isotropic 3D-XY behavior arises from the coupling between director fluctuations and the smectic order parameter, which is intrinsically anisotropic [60-62]. 

Following McMillan s prediction of the conditions under which the N-SmA transition could be second order [22] and de Gennes analogy between the N-SmA tran-... 

The theoretical picture for the N-SmA transition has been succinctly summarized by Lubensky. [28] He explains that there are several important differences between the normal-metal-superconducting and the N-S mA transitions where AT, the splay elastic constant, emerges as a dangerous irrelevant parameter [28]. 

Dislocation-loop melting theory [30], taking into account entropic effects, introduced two new fixed points on the Ki axis, one stable and the other unstable, where V ==2vj is built-in [28]. As the value of the unstable fixed point is at smaller X, than the stable one, Ki has to be larger than its unstable value for fluctuations to converge to dislocation-loop theory s fixed point [31]. Recent self-consistent one loop calculations by Andereck and Patton [32] and the persistent anisotropy in the critical behavior of have renewed interest in this still mysterious aspect of the N-SmA transition [24]. However, Lubensky emphasizes that the only way theoretically to obtain the larger stable fixed point is by ignoring 1 [28]. If one includes 1, the only stable fixed point is at /sTj = 0 where Vn = y. ... 

Renn and Lubensky proposed a model for the analogue of the vortex lattice [43] for the N -SmA transition [42]. Simultaneously and independently such a phase was discovered by direct observation, supported by X-ray analysis as well as freeze-fracture, by Goodby et al. [44] between the isotropic liquid and a SmC phase. The first TGBA phase found to exist between N and SmA was studied in a dynamic experiment [45]. In the Renn-Lubensky model, uniform sheets of SmA of extent separated by parallel planes of screw dislocations, twist relative to each other [46]. 

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 a microscopic scale, the geometry of this model may be more reminiscent of Type I intermediate states rather than a vortex state [41 ]. The analogue of the vortices are the screw dislocations. A analogue of the intermediate state at the N-SmA transition is a static stripe pattern P. E. Cladis, S. Torza, J. Appl. Phys. 1975, 46, 584. 

Figure 5. Step down to lower temperatures of the N-SmC transition line from the N-SmA transition line [17].

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]. 

Upon cooling a nematic phase towards a second-order N-SmA transition, the susceptibility of smectic fluctuations x di-... 

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