Density wave vector, smectics

Smectic" means soap in Greek since it tends to have mechanical properties akin to those of soaps. The simplest smectic phase is the smectic-A phase. This phase has traditionally been described as a system that is a solid in the direction along the director and a fluid normal to the director. A smectic-C phase is similar except that the density-wave vector makes a finite angle with the director. 

One can conveniently describe the layering in smectic liquid crystals by employing a density wave vector a, which following Oseen [43] and de Gennes [8] is subject to the constraint... 

Period d of the lamellar structure can be equal to the molecular length (smectics Ai), to the length doubled (bilayered smectics Aq), or to the intermediate value I < d < 21 (smectics A ). The latter subphases are separated by phase transition lines. Sometimes the smectic A phases were observed even with two incommensurate periods of density waves. However, the coexistence of two collinear density waves along the z-direction is not favorable, and modulated structures are often formed with a component of the wave vector lying in the x, y plane. 

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-... 

Smectic ordering in liquid crystals is usually characterized by the complex order parameter Pa introduced by de Gennes [3]. Here p = (cos (q r)) is the amplitude of the density wave, y/ is the phase and q is the wave vector. This order parameter appears naturally in the Fourier expansion of the one-particle density p(r). 

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]. 

The simple, popular picture of a smectic phase (Sm) consisting of elongated molecules in sharp, distinct layers is rather misleading a more realistic picture is one where the molecules are arranged to provide a single sinusoidal density wave [1] with its wave vector either parallel to the molecular director (orthogonal phases such as SmA, SmB, etc. - see Fig. 1) or at some angle to it (tilted phases such as SmC, SmI, etc.). However, it is often convenient to refer to the layered nature of the smectic phase to help explain phenomena such as conductivity anisotropy. 

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