Elementary Distortions

The static continuum theory of elasticity for nematic liquid crystals has been developed by Oseen, Ericksen, Frank and others [4]. It was Oseen who introduced the concept of the vector field of the director into the physics of liquid crystals and found that a nematic is completely described by four moduli of elasticity Kn, K22, K33, and K24 [4,5] that will be discussed below. Ericksen was the first who understood the importance of asymmetry of the stress tensor for the hydrostatics of nematic liquid crystals [6] and developed the theoretical basis for the general continuum theory of liquid crystals based on conservation equations for mass, linear and angular momentum. Later the dynamic approach was further developed by Leslie (Chapter 9) and nowadays the continuum theory of liquid crystal is called Ericksen-Leslie theory. As to Frank, he presented a very clear description of the hydrostatic part of the problem and made a great contribution to the theory of defects. In this Chapter we shall discuss elastic properties of nematics based on the most popular version of Frank [7]. 

As already mentioned, for the fixed direction of the nematic director n the shear modulus is absent because the shear distortion is not coupled to stress due to the material slippage upon a translation. The compressibility modulus B is the same as for the isotropic liquid. New feature in the elastic properties originates from the spatial dependence of the orientational part of the order parameter tensor, i.e. director n(r). It is assumed that the modulus S of the order parameter Qij r) is unchanged. In Fig. 8.4 we can see the difference between the translation and rotation distortion of a nematic. 

Let us assume that a liquid is incompressible, B oo, and discuss orientational (or torsimial) elasticity of a nematic. In a solid, the stress is caused by a change in the distance between neighbor points in a nematic the stress is caused by the curvature of the director field. Now a curvature tensor dnjdxj plays the role of the strain tensor ,y. Here, indices i,j = 1, 2, 3 and Xj correspond to the Cartesian frame axes. The linear relationship between the curvature and the torsional stress (i.e., Hooke s law) is assumed to be valid. The stress can be caused by boundary conditions, electric or magnetic field, shear, mechanical shot, etc. We are going to write the key expression for the distortion fi-ee energy density gji, related to the director field curvature . To discuss a more general case, we assume that gji t depends not only on quadratic combinations of derivatives dnjdxj, but also on their linear combinations  

As we shall see further on, the terms linear in dnjdx, allow us to discuss not only conventional nematics with Dooh symmetry but also some biased nematic phases. For example, we can discuss the phases with a spontaneous twist (cholesterics with broken mirror symmetry) or a spontaneous splay (uniaxial polar nematics with broken head-to-tail symmetry, n -n). For a standard nematic only quadratic terms will remain. 

Consider elementary distortions of a nematic. The undistorted director n = (0,0,1) is aligned along the z-axis. Fig. 8.5a. For instance, at a distance 8x from the origin of the Cartesian frame O the director has been turned through some angle in the zOx plane like in Fig. 8.5b. The relative distortion is then described by the ratio of hn, an absolute change of the x-component of the director, to distance 8x, over which the distortion occurs. In the same sketch, but in the zOy plane we see similar fan-shape or splay distortion 8/iy. Thus for the two elementary splay distortions we write  

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Fig. 8.5 Elementary distortions of the director field for particular geometry (a) with nWz splay (b), bend (c) and twist (d)...

Here, symbol comma between suffixes in tii j means spatial derivatives. In the general case, with three missing elementary distortions ay, ag, and ag the curvature distortion tensor is given by ... 

First let us go back to the same particular case with a constraint llz, and discuss the free energy of a conventional (uniaxial, nonpolar) nematic liquid crystal. We combine elementary distortions corresponding to splay (ai + as), bend ( 3 + ae) and twist ( 2 + ad and present the free energy as a sum of these combinations squared. 

To have the free energy density in a more general form including all the nine elementary distortions tzi, 2... agWQ should add the terms dnjdz, dnjdx and dnj dy and rewrite the Eq. (8.15b) in the vector notations for arbitrary distortion of n with respect to the Cartesian frame. Then we obtain Frank formula for the density of elastic energy in the general vector form ... 

It is not unexpected that problems often occur in the fundamental analysis of emergent properties. Maybe the prudent response of the chemist should then be a critical reexamination of those assumptions that underpin the partially successful theory. In any theory, there is a reductionist limit, beyond which there are no data to guide the recognition of more fundamental principles. In the theory of matter, this limit occurs in the vacuum, or sub-ether [2], seen as the primaeval form of matter, continuously spread across the endless void. On deformation of this featureless cosmos, ponderable matter emerges from the void as elementary distortions, which are perpetually dispersed, except in a closed system. We propose such a structure as... 

By adopting a perspective from the philosophy of science I will attempt to cross levels of complexity from the most elementary chemical explanations based on electron shells to those based on ab initio methods. Such a juxtaposition is seldom contemplated in the chemical literature. Textbooks provide elementary explanations which necessarily distort the full details but allow for a more conceptual or qualitative grasp of the main ideas. Meanwhile the research literature focuses on the minute details of particular methods or particular chemical systems and does not typically examine the kind of explanation that is being provided. To give a satisfactory discussion of explanation in the context of the periodic table we need to consider both elementary and deeper explanations within a common framework. 

Figure 4. Evidence of the distortion to the central part of the Ek map in Fig. 1, by transporting an elementary unit cell around the isolated critical point, which is marked by a large dot at the origin. Note the presence of locally smooth lattice vectors, in the direction of the arrows, at every point on the cycle.

Look again at Figure 16-1 If two NO2 molecules can form a bond when they collide, then that bond also can break apart when an N2 O4 molecule distorts. The concept of reversibility is a general principle that applies to all molecular processes. Every elementary reaction that goes in the forward direction can also go In the reverse direction. As a consequence of reversibility, we can write each step in a chemical mechanism using a double arrow to describe what happens at chemical equilibrium. 

Although the application of elementary ligand field theory is adequate to explain many properties of complexes, there are other factors that come into play in some cases. One of those cases involves complexes that have structures that are distorted from regular symmetry. Complexes of copper(II) are among the most common ones that exhibit such a distortion. 

The behaviour of suspensions of fine particles is very considerably influenced by whether the particles flocculate. The overall effect of flocculation is to create large conglomerations of elementary particles with occluded liquid. The floes, which easily become distorted, are effectively enlarged particles of a density intermediate between that of the constituent particles and the liquid. 

Apparently the geometry of the transition state for adsorption is approximately that of a ir-complexed olefin in that its structure seems to be only slightly distorted from that of the isolated alkene. However, this does not necessarily mean that the adsorbed state which is formed in the elementary reaction to which the stereochemistry refers is a tt complex, because the same geometry also represents a stage in the progression of olefin to the eclipsed 1,2-diadsorbed alkane. Hopefully other experi-... 

Figure 5.4 (A) Olivine structure dashed line contours elementary cell (axes c and b). (B) Details of coordination state and distortion of Ml-Ml sites 0(1), 0(2), 0(3) oxygens occupy nonequivalent positions distortion from perfect octahedral symmetry is more marked for Ml site.

This can be seen from elementary considerations. The distortion of the surrounding medium that leads to the intermediate state with energy WH can be described by simple harmonic motion, with a parameter x for the displacement, a potential energy px2 and a wave function of the form r=const xexp(—ax2). The probability P 2 of a configuration with potential energy WH is thus... 

In order to control elementary process (i), an effective scheme based on the concept of quadratic chirping has been proposed [12-17]. It has been demonstrated that this idea can be applied to process (i) and that fast and near-complete selective excitation of a wavepacket can be achieved without significant distortion of its shape through the utilization of specially designed quadratically chirped pulses [18,19]. This method is discussed in the first part... 

Since modern views of catalysis regard active centers as local distortions of the primary crystalline lattice it is important to characterize these elementary crystallites by the following attributes ... 

It turns out, in fact, that the electron distribution and bonding in ethylene can be equally well described by assuming no hybridization at all. The "bent bond" model depicted at the right requires only that the directions of some of the atomic-p orbitals be distorted sufficiently to provide the overlap needed for bonding. So one could well argue that hybrid orbitals are not real they do turn out to be convenient for understanding the bonding of simple molecules at the elementary level, and this is why we use them. 

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