The Torque Balance Equation

We can now calculate the various torques acting on the director. Because of the two-dimensional geometry, only the y- component of the torque is non-zero. The dielectric and elastic torques are obtained most easily by taking the functional derivative of the free energy with respect to 6,  

Inserting the values for n, and keeping only linear terms, 

This is a restoring torque (r/0 0), i.e., it tends to decrease 6o. The dielectric free energy is 

Inserting Eqs. [15] and [20] into Eq. [27] and averaging over time, one obtains 

Since Ki is larger than K2 and positive, Fvisc/ 0 and this can be seen to be the driving torque. 

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Putting it all together, we obtain the torque balance equation... 

Consider the same bend distortions caused by field and shown in Fig. 11.28 and assume that distortions are small. What happens if we switched the field off In the torque balance equation for Frederiks distortion (a), we shall have two contributions, elastic and viscous ... 

The period of the stripes and the threshold voltage for their appearance have been found [35] by minimising the free energy of the nematic in an electric field, taking into account the flexoelectric (P/E) and dielectric Ea(En) /47i terms. The solution of the torque balance equations for angles

[Pg.332]

We present a detailed theoretical calculation, with experimental verification, of the nonlocal molecular reorientation of the nematic-liquid-crystal director axis induced by a cw Gaussian laser beam. The natures of the torque balance equations and the solutions are significantly different for normally and nonnormally incident laser beams. The nonlocal effects resulting from molecular correlation effects are particularly important for laser spot sizes that are different (smaller or larger) from the sample thickness. Experimental measurements for the transverse dependence of the molecules and the dependence of the Freedericksz threshold as a function of the laser beam sizes are in excellent agreement with theoretical results. We also comment on the effect of these nonlocal effects on transverse optical bistability. 

In a recent study we presented a calculation for the case of a linearly polarized laser incident obliquely upon a nematic film (i.e., where the laser propagation wave vector makes a finite angle with the director axis). We showed that, under physically reasonable assumptions (e.g., all the angles involved are small), the torque balance equations lend themselves to analytical solutions. Some transverse dependences of the reorientation as a function of cell geometry and optical director-axis configurations were discussed. 

In the high speed limit the response is viscous limited. Therefore, ignoring the elasticity gives the torque balance equation. 

In the steady state (dejdt = 0), the relation e = co(n) is obtained from (19). EquatiOTi (21a) with this relation leads to Tq = 0 in the steady state. Therefore, the torque balance equation for the director in the steady state is given from (20) as ... 

The velocity v, in turn, supports the director fluctuation 6(x), in accordance with the torque balance equation ... 

Obtaining the full solution of Maxwell s equations together with the torque balance equations for the liquid crystal remains one of the most challenging problems in the nonlinear optics of liquid crystals [51]. 

In a recent study, the coupled electromagnetic waves and director axis reorientation equations, cf. Chapter 11, for the stimulated orientation scattering process are solved by treating all the molecular and optical parameters involved as spatial-temporal variables. For example, the director axis reorientation angle 0 is represented as 0(z,t)=2 (r)sin( 7tz/Z). In that case, the torque balance equation becomes... 

This is a contribution to be equalized by the surface torque SF /Scpl. The same is valid for the opposite boundary at Z2, see Eqs. (10.21). Thus, two expressions (10.21) are indeed torque balance equations for the director angle cp at the two boundaries. 

Here we ignore the surface energy (Ik s = 0) the director is free to deflect at both botmdaries perpendicular to z. According to Euler equation (8.22), the minimisation over 39/3z results in the torque balance ... 

The solutiOTi of Eq. (13.48) depends on further simplifications. If we assume that the director in the odd layers with i = 0 is unaffected by an external field and only the azimuth in the even layers is changed from Jt to 0, then, for an infinitely thick sample (x —> oo), the free energy is independent of both x and Oi. The corresponding torque balance equation reduces to the form with index (/ -P 1) omitted ... 

When the shear is within the smectic layers, we have a nematic-like behavior, and the torque balance simplifies to the following equation ... 

The solids forwarding angle q> is related to rate by Eq. 5.7, and the force and torque balances provide the relationship between and the pressure gradient and q> using Eq. 5.23. Here, the equation is written in differential form so that integration can be performed numerically in the z direction using variable physical property data. 

When the apparatus begins to rotate, the fluid experiences an initial acceleration, but a stationary state is rapidly attained in which forces balance and acceleration is zero. Equation (5) gives a generalized expression for torque under stationary-state conditions it must be independent of r. If this were not the case, forces would be different in different parts of the fluid, and acceleration would occur. Accordingly, we set the torque on this volume element equal to a constant ... 

Force balances such as Eq. 4.7.1, with the further assumption that the channel is flat and the torque induced by couples of forces can be neglected, lead to the following equations. 

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