One-constant approximation

Here, for simplicity, we allow the one-constant approximation and then selKi=Ky = K. 

Analyses are simplified by taking all three constants to be equal (the one-constant approximation) K = K] = K2 = K3. It can then be shown that... 

Thus the one-constant approximation is often used in theoretical analysis. 

Experiments demonstrate that at even higher Er, the rolls become unstable and irregular. Ultimately, defect lines called disclinations form in the flow direction. As the linear analysis concerns the behavior of infinitesimal disturbances, the growth of the instability and further bifurcations are inaccessible to such analyses. This motivated Feng, Tao, and Leal to carry out a direct numerical simulation of a sheared nematic. Using the LE theory, with the one-constant approximation, they predicted a cascade of instabilities illustrated in Fig. 3. Steady state rolls first appear at Er = 2368. The director twists toward the flow (z) direction at the center of the cells. With increasing Er, the secondary flow and the director twisting intensify. 

However, an important parameter that has been ignored in this approach is the surface tension at the interface. The interfadal tension T can be taken into account in an elementary way as is generally done for crystal screw dislocations. The total energy of the disclination in the one-constant approximation, including the energy at the core surface, is... 

The cholesteric pitch is altered around the singular line where N is an integer. The pattern for i = j is shown in fig. 4.2.4. Again, the energies and interactions in the one-constant approximation are the same as for nematic twist disclinations. A somewhat more elaborate treatment of this model has been presented by Scheffer and the effect of elastic anisotropy has been investigated by Caroli and Dubois-Violette. ... 

In the one-constant approximation, the distortion free energy per unit volume is... 

Our task is to find an analytical expression for 9(z) at different fields. The scheme is as follows. First we shall write a proper integral equation for the free energy. Then, following the variational procedure discussed in Section 8.3, we compose the Euler equation corresponding to the free energy minimum and solve this differential equation for 9(z). To simphfy the problem we use the one-constant approximation Kii= K22 = Ks3= K.ln our geometry, and only one derivative, namely the bend term with dnjdz, is essential in the Frank free energy form (8.15) ... 

Here, is an effective elastic modulus for the azimuthal motion of the director in the SmC phase that includes factor sin 9 [11]. Due to this factor, in the one-constant approximation, which will be used below, k 10 dyn is roughly one order of magnitude smaller than for nematics. The third term in the equation for Tch determins the difference in the transition temperatures for a helical and unwound ferroelectric. 

If for simplicity the three elashc constants are set equal to K (called the one constant approximation), then the free energy per unit volume is just... 

The theory, used here involves two simplifications. First, it assumes a n shape instead of a Gaussian. This approximation could be overcome by some numerical calculation it is however doubtful whether this would be worthwhile if at the same time the second simplification, the one constant approximation were maintained. 

Even for the case of an infinite plane optical wave, Eq. (12) is extremely difficult to solve. A more meaningful and physically more insightful approach is to make a so-called one-constant approximation (i.e., Xi X2 X3 X). Second, in the first-order approximation, the dependence of 0 on z is a simple sine wave, i.e., we assume that 0(r, z) is of the form ... 

In order to find the threshold intensity for director reorientation, we must linearize and solve Eq. (6) to first order in the perturbation of the director. In the one-constant approximation, we obtain... 

If the amplitude of the complex order parameter ij/ is constant, (10.8) describes changes of the free energy which are due to compression or dilation of the layers (i.e., deviations of Q from Qo ) or due to phase shifts of ij/, i.e., displacements of the smectic layers. In contrast to the gradient term appearing in the Ginzburg-Landau ansatz [44], two coefficients, C and C , appear because of the anisotropy of the liquid crystal. In a one-constant approximation, C = C = C, (10.8) is reduced to... 

Frequently we want to keep the calculations as simple as possible. In this case the so-called one-constant approximation is introduced, setting all elastic constants Ku in (12.16) equal and dropping the surface terms. In this way a simple expression for the elastic distortion energy is obtained... 

We begin with a not particularly realistic example, a two-dimensional nematic. Here the classification of defects is quite illustrative, and in the one-constant approximation (12.20) a calculation of structures with point defects is simple. The equilibrium condition for a point defect, located at the center of the coordinate system, reads... 

Note that the damping rate [r q )] has been modified by the contribution of translational diflFusion D) from the form of Eq. (6.21) under the one-constant approximation. The reduced spectral density ji j (a ) due to director fiuctuations is obtained by Foirrier transformation of q (t)... 

Brochard [6.6] and Blinc et al. [6.37] extended the one-constant approximation to include the set of anisotropic elastic constants and viscosity coefficients. Suppose that the different elastic constants and viscosities are retained while effects of magnetic field and translational diffusion are not included in Eqs. (6.21)-(6.22). The reduced spectral density [Eq. (6.30)] is modified by using an ellipsoidal volume over the q space with two high-frequency cutoffs Qzc and q c [6.38],... 

As the values of the three elastic constants are comparable and the expression for the molecular field is rather complicated, the one constant approximation... 

Figure 3. Free energy per unit length in term.s of anchoring strength for the pianar polar (PP) and non-planar escaped radial (ER) structures (one-constant approximation, Kfi=K). The critical radius for the structural transition is given by the intersection points of the energy graphs. (Reproduced from Crawford et al. 11911.)...

The introduction of is not as straightforward as that of K24, and causes serious mathematical problems. The free energy contribution corresponding to the 13 term is not bound from below, and the simple application of the variational principle with a nonzero Ki coefficient may lead to discontinuities in the director field at the boundaries. This problem is known as the Oldano-Bar-bero paradox [213, 325]. Consider the simple one-dimensional splay geometry having the director field n=[sin d(z), 0, cos 0(z)] in one constant approximation Kn=K =K. The free energy density is... 

Kralj and Zumer [47], in their derivation, have re-expressed of Eqs. (14)-(16). In addition, they included expressions for the surface-like elastic coefficients. In a one-constant approximation (only Lf O), all constants are equal except for /f]3=0 in a second-order approximation (only L O), there is still a splay-bend degeneracy, - 11 = 33 22 24 13 re ains zero. 

Fig. 11.2 The solution

This expression, and the resulting equations of motion and analysis, can be greatly simplified if one makes a frequently used assumption, namely, the one-constant approximation Kx = Ki = = K). n this case Equation (3.6) becomes... 

This equation is greatly simplified under the one-constant approximation (K = K2 = K-i = K). This gives... 

Sometimes, for example, when the relative values of the Kj are unknown or when the resuhing equilibrium equations are rather complicated, the one constant approximation K = K2 = K3 = K is made. In this case the bulk energy Fn can be simplified further by using identities for unit vectors which results in the more amenable form [5, p.l04]... 

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