Smectic Compressibility Modulus

Giovanni Carbone, Bruno appone and Riccardo Barberi 3.2.5 Smectic Compressibility Modulus... 

The smectic compressibility modulus B can be directly estimated from the force plots pre nted in Sect. 3.2.4, fitting them to the (3.8). 

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. 

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... 

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-------------------------... 

Sect. 3.2.2 we present the technique used to perform the measurements and the experimental setup. In Sect. 3.2.3, basic ideas are introduced, describing the force acting between surfaces that confine a smectic liquid crystal. In Sect. 3.2.4 we report the observations of the smectic periodic structure performed on two smectic compounds. In Sect. 3.2.5 the compressibility modulus of 8CB is estimated using a force plot. 

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. 

It has also been used to study the second sound resonance in smectic A liquid crystals and measure the compression modulus. For measuring the flexocoefficients (ei — 63) and (ei + 63), hybrid-aligned nematic cells have been used extensively. AC techniques avoid problems associated with ionic impurities, but require elaborate numerical fitting of the data. Some observations on the pnblished measurements of flexo-coefHcients are made in Section 2.4, which ends with a few concluding remarks. 

In explaining the fiber formation of "banana smectics," we have to recognize the analogy between the cross section of tire banana smectic and the columnar fibers. This means that the compression of the smectic layers will have the same effect as the compression of the columns in colimmar materials. Accordingly, we can use the same arguments and equation that was used to describe the stability of colimmar fibers. Even the typical surface tension and compression modulus data are similar (7=2 x 10 /m and B 1(F J/tn ) so we get D, 0.15 fim. Again, similar to colmnnar fibers, this is an order of magnitude smaller than the experimentally observed smallest... 

The director deformations described by that do not lead to layer compressions, in the continuum range where the wavelengths A of the deformation are much larger than the molecular dimensions (A 10 nm) can be induced by stress K 27t/pf <10T N/m. This is usually smaller than of the layer compression modulus B l(f N/m , For this reason, deformations that do not lead to layer compression (such as splay in SmA) are usually called soft deformations, whereas those that require layer compression (such as bend and twist in SmA) are the so-called hard deformations. In SmC there will be six soft and three hard deformations, so it is basically impossible to take into account all elastic terms while keeping the transparent physics. (In the chiral smectic C materials, additional three terms are needed, as shown by de Gennes. ) Fortunately, however, the larger number of soft deformations enable for the material to avoid the hard deformations, which makes it possible to understand most of the elastic effects, even in SmC materials. 

Lyotropic lamellar systems are very similar to thermotropic smectics, and their elastic free energy is identical to that of SmA given in Eq. (4.34). In lyotropic lamellar systems, the origin of the layer compression modulus B is the steric repulsive interaction energy. This can be visualized as illustrated in Figure 4.20. If a stack of membranes, formed, e.g., by lipid bilayers, is placed between two parallel walls, violent thermal out-of-plane fluctuations of the membranes exert a pressure p on the walls. 

Martinoty P, GaUani J, Collin D (1998) Hydrodynamic and nonhydrodynamic behavior of layer-compression modulus B at the nematic-smectic- A phase transition in 80CB. Phys Rev Lett 81 144... 

With one-dimensional periodicity, the smectic phase cannot exhibit true long-range order due to the Landau-Peierls instability [6, 7]. An anisotropic scaling analysis [18] (see [3], page 521 for a summary) predicts the divergence of the layer compression modulus B oc thus a divergence with exponent <]) = vy -2i j. ... 

S. Shibahara, J. Yamamoto, Y. Takanishi, K. Ishikawa and H. Takezoe, Layer Compression Modulus in Smectic Liquid Crystals, J. Phys. Soc. Japan, 71, 802-807 (2002). 

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 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... 

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