Smectic layer compression

The smectic layers may themselves compress or dilate in a full general theory of deformations in smectic liquid crystals. These effects have not been considered in the theory presented above because, in many instances, it may be assumed that layer compression or dilation may be neglected in basic planar layered geometries of SmC liquid crystals. Nevertheless, for the sake of completeness, we record for the interested reader some details from the literature. The additional term for smectic layer compression in SmA liquid crystals, which also serves as a first approximation to the compression term in SmC for the planar geometry pictured in Fig. 6.3, is of the form mentioned by de Gennes [106] and de Gennes and Frost [110, pp.345-346] 

We record here that to obtain a common energy density describing layer rni-dulations for SmA in terms of u(Xjy,z), we may make the approximation a = —du/dxy—du/dy,l) [149] and construct the energy as the sum of WAeiast and comp to give, from (6.40) and (6.148), (for example, see [103]) 

The incorporation of a magnetic energy in terms of u to the above energy for SmC has also been investigated [264]. For example, in the geometry of Fig. 6.3, if we set H = (iif, 0,0) then one possibility for the additional magnetic energy density may be written as 

The wavelength of the undulation u x z) along the x-axis is set to be 27t/A for some wave number A , in which case the boundary condition at z = 0 may be taken 

This result can be applied to wa to find that the equilibrium equation is 

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By adding the function u(x) to the phase factor in (4) one can describe departures from the planar (lamellar, one-dimensional) layer arrangement, which is characteristic for the 2D structures. The first term in (3) is the smectic layer compressibility energy. It is zero when layers are of the equilibrium thickness. If cx(T) > 0, the second term in (3) requires the director to be along the smectic layer normal (the smectic-A phase). If c (T) < 0, this term would prefer the director to lie in the smectic plane. So the last term in (3) is needed to stabilize a finite tilt of the director with respect to the smectic layer normal. In addition this term gives the energy penalty for the spatial variation of the smectic layer normal. 

L /polymer extensibility smectic-layer compressive modulus E E, Finger strain tensor B , Cauchy strain tensor yriso/r, capillary number characteristic ratio, defined by R )q — Ccotib translational diffusivity-------------------------... 

The addition of a layer compression energy Wcomp such as those of the forms introduced at equations (6.147) and (6.148), can also be added to w. Some preliminary theoretical results involving the onset of layer undulations in a Helfrich-Hurault transition in SmC using such an energy obtained, for example, via (6.147), (6.299), (6.301) and (6.302), has been reported by Stewart [263]. Also, the smectic layer compression constant B has been measured for various materials that exhibit SmA and SmC phases see the comments and references on page 284. 

It should also be mentioned that an earlier attempt at SmC d3mamic theory was made by Schiller [245], whose approach differs from that outlined here, and results in fewer viscosity coefficients. Among much more recent theoretical descriptions which include nonlinear hydrodynamics and smectic layer compression effects in SmC and SmC liquid crystals is the work by Pleiner and Brand [224] the reader is referred to their article and its references for further details. 

In thermotropic (solvent-free) smectic-A phases, two types of distortion are permitted, namely, splaying of the director (which corresponds to bending of the layers) and layer compression. Note The material itself is assumed to remain incompressible only the layers compress.) For weak distortions, the free energy cost of these is given by (de Gennes and Frost 1993)... 

For small-molecule thermotropic smectic-A phases, typical values of two elastic constants are K 10 dyn and B 10 dyn/cm (Ostwald and Allain 1985). For lyotropic smectics, such as those made from surfactants in oil or water solvents, the layer compression modulus B can be much lower (see Chapter 12). From B and K, a length scale A. = ( 1 /B) 1 nm is defined it is called the permeation depth and its magnitude... 

When the layers of a lamellar block copolymer are distorted, the free energy density is augmented by a distortional term that can, like the smectic-A phase, be described as the sum of layer compression/dilation and layer-bending energies ... 

The elasticity of multilamellar vesicles can be discussed in reference to that of emulsion droplets. The crystalline lamellar phase constituting the vesicles is characterized by two elastic moduli, one accounting for the compression of the smectic layers, B, and the second for the bending of the layers, K [80]. The combination has the dimension of a surface tension and plays the role of an effective surface tension when the lamellae undergo small deformations [80]. This result is valid for multilamellar vesicles of arbitrary shapes [81, 82]. Like for emulsion droplets, the quantity a/S is the energy scale that determines the cost of small deformations. 

The deformations in the smectic A phase liquid crystals are the bending of the smectic layer (accordingly to the splay of the directors) and the dilation or compression of the layers. The energy is thus... 

The smectic layers are flexible, i.e., it is relatively easy for bending deformation, but the layers are difficult to compress. The bending of the smectic layers can be expressed by two principal curvatures R and R2, i.e.,... 

A more complete description of smectic A needs to take into account the compressibility of the layers, though, of course, the elastic constant for compression may be expected to be quite large. The basic ideas of this model were put forward by de Gennes. > We consider an idealized structure which has negligible positional correlation within each smectic layer and which is optically uniaxial and non-ferroelectric. For small displacements u of the layers normal to their planes, the free energy density in the presence of a magnetic field along z, the layer normal, takes the form... 

Several techniques are used to study structural properties of smectic phases X ray [29,30], SFA [31,32], ellipsometry [33,34], etc. In Sects. 3.2.4 and 3.2.5, the AFM spectroscopy force is introduced as a simple and straightforward method to measure the smectic periodicity and the compressibility modulus of a stack of smectic layers. 

For separation D comparable to, decreasing or increasing D will not equally compress or stretch the pre-smectic layers as in smectics. The layers that are nearer to the surface are expected to be more smectic , thus stiffer and less -deformed compared to those at the center of the cell. This is confirmed by the theory, developed in [22,54,55]. 

In this part we discuss the use of the AFM Force Spectroscopy mode to study the smectic periodicity and compressibility of smectic layers. In... 

Here D is the separation between the surfaces, R is the sphere radius, B is the compressibility modulus, d is the smectic period and rii is the number of layers contained in the central cell. By measuring the structural force one can therefore determine simultaneously the thickness of the smectic layers and the smectic compressibility modulus B. The model of reference [31] was originally considered for the crossed cylinders of the SFA geometry, that is equivalent to a plane-sphere geometry with a local radius R 2 cm. Some of the approximations used to obtain the (3.8) are however questionable in view of the much smaller size of the AFM tip radius, which is 10 nm. 

It has been shown that the AFM in the force spectroscopy mode is a very simple, accurate and straightforward method to measure the smectic layer thickness with a precision of 0.1 nm using a very small drop of a liquid crystal material. The method is less accurate in measuring the smectic compressibility modulus, which is due to the surface tension on a partially immersed AFM tip and the small, not very well defined radius of the AFM tip. 

N in SI system). Modulus B found for a liquid crystal 80CB at temperature 60°C is B = 8TO erg/cm (or 8TO J/m in the SI system) [18]. In that experiment, the compression-dilatation distortion of smectic layers was induced by an external force from a piezoelectric driver. 

In the smectic C phase the director is free to rotate about the normal z to the smectic layers. In the general case, the smectic layers are considered compressible. The elastic... 

For discussion of dynamics of lamellar smectic phases it is important to include another variable, the layer displacement u (r) [3] or, more generally, the phase of the density wave [4]. This variable is also hydrodynamic for a weak compression or dilatation of a very thick stack of smectic layers (L oo) the relaxation would require infinite time. On the other hand, the director in the smectic A phase is no longer independent variable because it must always be perpendicular to the smectic layers. Therefore, total number of hydrodynamic variables for a SmA is six. For the smectic C phase, the director acquires a degree of freedom for rotation about the normal to the layers and the number of variables again becomes seven. 

Smectic liquid crystals possess partial positional orders besides the orientational order exhibited in nematic and cholesteric liquid crystals. Here we only consider the simplest case smectic-A. The elastic energy of the deformation of the liquid crystal director in smectic-A is the same as in nematic. In addition, the dilatation (compression) of the smectic layer also costs energy, which is given by [23]... 

Both cholesteric and smectic mesophases are layered. In the former case, the periodicity arises from a natural twist to the director field, and in the latter, from a center-of-mass correlation in one dimension. There are many types of smectic phases distinguished by their symmetry and order. The set of field-induced phenomena is quite different for these two materials, owing primarily to the very different layer compressibility. That is, the cholesteric pitch can be unwound by an external field, whereas the smectic layering is typically too strong to be altered significantly. However, because of the common layered structure, there are also strong similarities. 

In the smectic A phase the director is always perpendicular to the plane of the smectic layers. Thus, only the splay distortion leaves the interlayer distance unchanged [7], and only the elastic modulus K i is finite while K22 and Kzz diverge when approaching the smectic A phase from the nematic phase. On the other hand, the compressibility of the layered structure and the corresponding elastic modulus B is taken into account when discussing the elastic properties of smectic phases. The free energy density for the smectic A phase, subjected to the action of an external electric field, is... 

Smectic side-chain polymers prefer locally oblate chain conformations, independent of the spacer length or attachment geometry. Analogous to oblate nematic polydomain elastomers, biaxial mechanical stretching or uniaxial compression can be used to orient Sa polydomain elastomers. This achieves a simultaneous orientaticai of the director and the smectic layer normal in a uniform homeotropic fashion [74],... 

In Sect. 3.2 it was shown that uniaxial stretching or compression of Sc elastomers does not produce macroscopically oriented samples. Due to the Sc symmetry, uniaxial deformation only induces a uniform orientation of the director but leaves the smectic layer normals conically distributed around the stress axis. Usually such elastomers remain opaque. For the preparation of Sc LSCEs a more complex orientation strategy is necessary which can be realized by deploying two successive orientation processes. 

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