Order parameter smectic

C and I account for gradients of the smectic order parameter the fifth tenn also allows for director fluctuations, n. The tenn is the elastic free-energy density of the nematic phase, given by equation (02.2.9). In the smectic... 

The smectic order parameter / which describes the spatial variation of density due to the layered structure... 

In our model the moduli of the nematic and smectic order parameters play similar roles, so we will deal with both. Since we consider a situation beyond the phase transition regime, the equilibrium value of the order parameter is non-zero (S s for both nematic and smectic) and only its variations s(n,s can enter the energy density (S = S(q s) + s ) ... 

We write the solution as the vector X = (6,(j),u,vx,vy,i ,P,) consisting of the angular variables of the director, the layer displacement, the velocity field, the pressure, and the modulus of the (nematic or smectic) order parameter. For a spatially homogeneous situation the equations simplify significantly and the desired solution Xo can directly be found (see Sect. 3.1). To determine the region of stability of Xq we perform a linear stability analysis, i.e., we add a small perturbation Xi to... 

Equation (39) shows that nematic degrees of freedom couple to simple shear, but not the smectic degrees of freedom the modulus of the nematic order parameter has a non-vanishing spatially homogeneous correction (see (39)), whereas the smectic order parameter stays unchanged. The reason for this difference lies in the fact that J3 and /3 include h and p, respectively, which coupled differently to the flow field (see (22) and (23)). Equation (38) gives a well defined relation between the shear rate y and the director tilt angle 9o, which we will use to eliminate y from our further calculations. To lowest order 0O depends linearly on y ... 

For a measure of the one-dimensional translational order in a layer perpendicular to the director, the smectic order parameter ij/ is defined as the magnitude of the Fourier component of the normalized density along the director... 

The smectic order parameter provides a quantitative measure of the onedimensional translational order, which is a characteristic of the smectic phase. In Fig. 30, we show the evolution of the average smectic order parameter I/ of the inherent structures with temperature. A steady increase in with the concomitant growth of S for the underlying inherent structures is apparent across the nematic phase. 

On scrutiny of Fig. 29 and 30, we find that the reversal in the temperature behavior of D in the nematic phase occurs when the average smectic order parameter for the underlying inherent structures becomes significant (above 0.3) for the first time. The smectic order parameter is a measure of the translational order that appears in a layer perpendicular to the director. The induction of such... 

Figure 30. Evolution of the smectic order parameter for the inherent structures of the calamitic system GB(3, 5, 2, 1) (N = 256) with temperature at two densities. 

Figure 31. Coupling between the nematic order parameter S and the smectic order parameter 4/ for the calamitic system GB(3, 5, 2, 1) (TV = 256) at three state points along the isochor at density p = 0.32. At the nematic phase (T = 1.194 bottom), at the smectic phase (T = 0.502 top), and at the nematic-smectic transition region (T = 0.785 middle). The order parameters are for instantaneous configurations. (Reproduced from Ref. 161.)...

When the liquid crystal exhibits a direct isotropic to smectic-A phase transition, the capillary condensation of the smectic phase in a gap between the surfaces is possible, since the isotropic to smectic-A phase transition is a first order phase transition due to the coupling between the nematic and smectic order parameters [26]. 

The full lines in Fig. 3.8 are best fits to the presmectic interaction, based on the LdG theory [22,24] with an addition of the van der Waals force [2,14]. Just like in previous nematic cases, the mesoscopic LdG theory superbly describes the structural force between the surfaces. The fitting of the measured presmectic forces to the theory allows for the determination of many important surface parameters, that are difficult to obtain using other methods. For example, the amplitude of the smectic order at the surface and the smectic correlation length can be obtained directly. For 8CB on silanated glass, one obtains a typical surface smectic order parameter iPfi which is of the order of 0.1 and the smectic correlation length y, which is in perfect agreement with X-ray data of Davidov et al. [25]. In addition to that, it has been observed that the smectic order is coupled to the nematic order and this amplifies the presmectic interaction close to the isotropic-nematic phase transition [23]. 

In reality, the N-A transition is, as a rule, weak first order transition. There are, at least, two ways to understand this in framework of Landau approach. We still use the same smectic order parameter pi but include additional factors, either (a) higher harmonics of the density wave, or (b) consider the influence of the positional order on the orientational order of SmA, the so-called interaction of order parameters. 

The smaller the temperature difference Tna T, the smaller is the smectic order parameter, that is the amplitude of the density wave. Consequently, the permeation coefficient in SmA should decrease upon approaching the SmA-N transition. Indeed, in experiment, very close to Tna the PoiseuiUe flow is observed, as in the nematic phase, but already at Tna T > 0.3 K the plug flow occurs with apparent viscosity two orders of magnitude larger than q. 

Figure 6.19 shows an example phase diagram for the athermal solvent x = 0. together with the nematic and smectic order parameters, and the number of mesogenic cores vm as functions of the temperature for different composition. 

Electron density Smectic order parameter Mirror plane Switching time... 

Smectic side chain polymers show an oblate equDibrium conformation of the polymer melt where the polymer backbone is partially confined between the smectic layers independent of the attachment geometry. For Sa polymers the confinement depends not only on the smectic order parameter but also on the t3q)e of Sa phase structure. For the monolayer phase structure Sai the anisotropy of the radii of gyration is R /R 0.3 while for the less densely packed partially bUayer structure Saci the confinement of the backbone is less pronounced (R /R 0.7) [63,64, 69-71]. 

In this equation, y is the interaction strength, c(r) the crosslink concentration, the smectic order parameter, and Vz (r) the relative displacement of the rubber matrix. Witkowski and Terentjev [132] evaluated (15) for (r) = 1, which is valid deep in the smectic phase, i.e., far below the smectic-nematic transition. Using the so-called replica trick, they integrated out the rubbery matrix fluctuations and obtained an effective free-energy density that depends only on the layer displacements M(r). Under the restriction that wave vector components along the layer normal dominate over in-layer components, q q, and considering only long-... 

The suppression of director fluctuations near the nematic-smectic A (N-A) transition because of divergence in the twist and bend elastic constants and the twist viscosity 7e [6.6] are now examined. Above the phase transition at T/vaj there are cybotactic smectic A clusters in the nematic phase, whose dimension is measured by a coherence length In fact, it is the coupling between the nematic director and the smectic order parameter that causes the viscoelastic constants to approach infinity at TnA Hence, A22J A33 oc while 7e oc Suppose there is interest in a frequency far below the high-frequency cutoffs such that A >> 1. In this limit,... 

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