Order parameter wave smectic

As will be seen later ( 5.3.1), experiments have confirmed that the density wave in smectic A is, in fact, very well represented by a sinusoidal function, indicating that higher terms in the Fourier expansion can be neglected. The form of the potential (5.2.2) ensures that the energy is a minimum when the molecule is in the smectic layer with its axis along z s and O are order parameters which we shall define presently. 

This density wave is usually considered as a complex order parameter pi = exp (itt) of the smectic A phase in the Landau expansion or fi-ee energy at the SmA-N phase transition. Typically, when there is no distortion, one assumes cpi = 0 at z = 0 and operates only with the wave amplitude pi as the real part of the order parameter. 

The dependencies of the dimensionless free energy on the order parameter at T < Tj a (curve 1), T = T /a (curve 2) and T > T a (curve 3) are presented in Fig. 6.10b. The energy is symmetric about pi = 0. For T >T the higher symmetry nematic state is stable curve 3 at finite pi in the nematic manifests short-range smectic order effect. For T < T, in the smectic A state, the two minima in curve 1 situated exactly at pi= 0.31 reflect the symmetry of the energy with respect to the phase (0 or n) of the density wave. 

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. 

The Landau-de Gennes theory for the nematic isotropic transition can be extended to the smectic A-nematic transition. The order parameter for this transition is rl), the amplitude of the density wave describing the formation of layers in the smectic A phase. Since the difference between a value of rlr and -Irlrl only amonnts to a shift of one half layer spacing in the location of all the layers (and therefore no change in the free energy per nnit volume), the expansion in terms of powers of rlr can only contain even powers. Hence the free energy per unit volume in the smectic A phase can be written as... 

The Maier-Saupe theory can also be extended to describe the smectic A-nematic transition in what is called McMillan s model. Two order parameters are introduced into the mean-field potential energy function, the usual orientational order parameter S and an order parameter a related to the amplitude of the density wave describing the smectic A layers,... 

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 order parameter of the SmC phase appears to be more complex because in this phase the director is not parallel to the wave vector q. In a simple case, it is just possible to use the tilt angle 0 as an order parameter. However, this parameter does not specify the direction of the tilt and thus it is analogous to the scalar nematic order parameter 5. The full tensor order parameter of the SmC phase can be constructed in several different ways. One is to define the pseudovector w [8,9] that describes the rotation of the director with respect to the smectic plane normal ... 

Here, yr is the superconductor gap order parameter. It corresponds to the wave function of the superconducting pair in BCS theory and has the X Y symmetry of the smectic order parameter. The magnetic vector potential A comes analogous to the director n (m and e are the mass and charge of a single electron, fi Planck s constant, c the velocity of light and jU the magnetic permittivity). 

Frost showed that the properties and structures of frustrated smectics can be described by two order parameters [72, 82]. The first p(r) measures mass density modulation familiar in SmA phases [1 ]. The second (r), often referred to as a polarization wave, describes long range head-to-tail correlations of asymmetric molecules along the z axis... 

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]. 

Each term in the expansion for p (z) in Eq. (9) corresponds to a pair of Bragg peaks in the diffraction pattern. For example, for 1=2, the term is a cosine wave of period z=a/2, which gives a pair of Bragg peaks at Q2= 4n/a= 2Qi. The intensity of this pair of Bragg peaks, relative to that which would be observed for a perfectly ordered (crystalline) smectic phase, is proportional to the square of the amplitude coefficient (X, and the same is true for all the higher order peaks. We thus see that the intensities of the various Bragg peaks are direct measures of the smectic order parameters. To a good approximation [7] we may write... 

Molecules within the layers move freely, with no defined packing arrangement. There is no correlation between molecules from layer to layer in this phase. The layer-to-layer ordering of the smectic phase can be approximated to a density wave, and this can be incorporated in a modified form of the order parameter. 

We cannot call this wave a second sound keeping this term for the case of smectics, because the physical mechanism permitting the acoustic anisotropy is completely different in nematics and smectics. In nematics it comes from relaxation effects (probably relaxation of the nematic order parameter), i.e. it is a dynamical effect. 

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