Backflow effects

The reason for this is a flow of the nematic, which is lunched by the director rotation. The flow arises in the beginning of the director relaxation process when the elastic torque exerted on the director is very high near both interfaces due to a strong curvature of the director field. However, the curvature at the two interfaces has different sign, see Fig. 11.19a, where 89(z) = (ti/2) — 9. Therefore, the flow of nematic fluid coupled to the director rotation (backflow) at the two interfaces is 

director n = (sin9 9,0, cos9 1), the term with Motion coefficients is the standard Navier-Stokes terms (9.15) of the type pdv/dt = rjd ld B is 

11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals 

It is the same general equation (9.22) for director motion adapted to our simple situation we have the familiar form (9.32) for the elastic and viscous torques and, in addition, the coupling term with the same coefficient A describing the torque exerted on the director by shear dvjdz. 

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In the planar to homeotropic transition (fig. 3.4.1(a)) backflow effects are not usually so pronounced near the threshold. In this geometry, the torque exerted by the director on an elementary volume of the fluid is... 

To complete this Section we have to mention the hydrodynamics of a nematic liquid crystal. In the equations described above the macroscopic flow was neglected. Such an approximation is usually justified since we describe only systems with zero total momenum. However, one should bear in mind that due to the coupling between the hydrodynamic and orientational degrees of freedom there are local flows accompanying the orientational changes, the effect known as backflow [38,39]. In some cases the backflow effects can alter substantially the phenomena in question. 

This ansatz was first proposed by Boys and Handy (BH) [142] and Schmidt and Moskowitz (SM) later arrived at the same form by considering averaged backflow effects [143], The SMBH form includes both the e-n function (via the m, n, 0 = m, 0, 0 terms) and the homogeneous e-e Jastrow function (via the m, n, 0 = 0, 0, 0 terms). The remaining e-e-n terms modulate the e-e correlation function according to the e-n distances. 

The coefficients of friction for the director have the dimensions of viscosity and are particular combinations of Leslie coefficients, ji = aj,- (I2, Ji = 3 + 2-It is significant that only two coefficients of viscosity enter the equation for motion of the director. One (72) describes the director coupling to fluid motion. Eor example, if the director tumes rapidly under the influence of the magnetic held, then, due to friction, this rotation drags the liquid and creates flow. It is the backflow effect that will be described in more details in Section 11.2.5. The other coefficient (Yi) describes rather a slow director motion in an immobile liquid. Therefore, the kinetics of all optical effects caused by pure realignment of the director is determined by the same coefficient yj. However a description of flow demands for all the five viscosity coefficients. 

It is simpler to examine the dynamics of the Frederiks effect for the experimental geometry of Fig. 11.15c, since a pure twist distortion is not accompanied by the backflow effect (see the next Section). For a twist distortion we operate with the azimuthal director angle tp(z) (sintp riy) and the equation for rotation of the director that expresses the balance of elastic, magnetic field and viscous torques is given by... 

Fig. 11.19 Backflow effect. The profile of the director in the field-on regime with steep parts close to interfaces at z = 0 and z = d (a). The direction of the torques is shown by small arrows in the right part of sketch (b) and a profile of the velocity is shown by thin arrows in the left part of the sketch. The strongest gradioit of velocity is in the middle of the cell dash arrows)...

We consider the dynamics of the Freedericksz transition in the splay geometry upon the removal of the applied field [24-27]. Initially the liquid crystal director is aligned vertically by the applied field, as shown in Figure 5.17(a). When the applied field is removed, the liquid crystal relaxes back to the homogeneous state. The rotation of the molecules induces a macroscopic translational motion known as the backflow effect. The velocity of the flow is... 

The second term on the right side of the above equation will make the angle increase in the middle of the cell. The translational motion makes the liquid crystal rotate in the opposite direction, which is known as the backflow effect. The strength of the backflow depends on the initial director configuration, which in turn depends on the initially applied field. If the applied field is high, the effect of the backflow is stronger. 

Although backflow effects are completely ignored in the dynamic equations (3.24a) and (3.24b), these equations are still complicated even in the linearized regime. In the following, we consider only the case of normal polarization in the small-distortion regime which can be described by... 

The three assumptions (listed in Section VI. 1.1) that define the Rouse model are often unacceptable. The assumption of localized responses is not correct because of backflow effects. Whenever we tqiply a force f. to one monomer in a fluid, the result is a distorted velocity field in the whole fluid. This backflow decreases only slowly with distaiKe (like r —. ... 

If we incorporate backflow effects at the Kirkwood level, the friction is modified Ctot 6 irq, R. If the chain is still an ideal phantom chain R = Ro), we get ... 

At finite frequencies o> the viscosity increase due to solute coils 8i) = T) — 1), becomes dependent on Mechanical data on this function are difficult to obtain in the dilute regime, but some results have been achieved. Their classical analysis was in terms of Zimm modes, allowing for backflow effects but ignoring any excluded volume effects. We present here a more general scaling picture. 

On the theoretical side, the exponents can be calculated rather simply at high dimensionalities (between d = 4 and d = 6). In polymer language, this amounts to computing the friction inside each cluster in a Rouse approximation (ignoring all backflow effects). The viscosity is then proportional to the (weight average) square gyration radius of the clusters [eq. (V.9)], and s = 2v — fi.ln dimensions lower than 4, the backflow terms become essential, and how to iiKlude them remains a problem. 

A novel electromechanical effect has been observed in FLCs [130]. A periodic shear flow occurred parallel to the bounding plates and perpendicular to the helical axis (FLC layers were perpendicular to the substrates). The frequency of oscillation of the shear flow was equal to that of the applied field and the amplitude was proportional to the field strength. The electromechanical effect in FLCs seems to have many common features with a backflow effect in nematic liquid crystals, as it is caused by the coupling between the Goldstone mode and flow [131]. 

If long-range backflow effects are assumed negligible, then the friction force v/Aube is essentially proportional to the number of atoms in the chain. 

The dynamics of the splay and bend distortions inevitably involve the flow processes coupled with the director rotation. Such a backflow effect usually renormalizes the viscosity coefficients. Only a pure twist distortion is not accompanied by the flow. In the latter case, and for the infinite anchoring energy, the equation of motion of the director (angle variation) expresses the balance between the torques due to the elastic and viscous forces and the external field (and... 

Backflow effects may accompany the transient process of the director reorientation [64,65]. The process is opposite to the flow orientation of the director known from rheological experiments. Disregarding the back-flow, we can use the same equations for the splay (with A",) and bend (K33) small-angle distortions. The backflow effects renormalize the rotational viscosity of a nematic ... 

In some cases, magnetically induced transient twist distortions have been observed in both thermotropic (MBBA [89]) and lyotropic (PBG [90]) systems. In this case, backflow effects are allowed only in a nonlinear regime, for strong distortions. The physical origin of this phenomenon could be the faster response times of modulated structures, as compared with uniform ones. When the equilibrium director distribution is approached, i.e. a relaxation process is over, the transient structures disappear. The emergence and subsequent evolution of the spatial periodicity of the transient structures have been considered theoretically [89,90]. In addition, the pattern kinetics have been studied in detail experimentally [91] on a mixture of a polymer compound with a low-molecular-mass matrix. The polymer considerably increases the rotational viscosity of the substance and reduces the threshold for pattern formation. This indicates the possibility of recording the pattern using a video camera. A typical transient pattern is shown in Fig. 14 [91]. 

The different modes may interfere with each other, even near their thresholds, and a variety of patterns may be observed. For example, chevrons occur due to interference between the electrolytic and inertial modes, and a transient periodic pattern due to backflow effects accompanies the Frede-riks transition. Well above the threshold, when the induced distortion is nonlinear with the applied field, many other patterns... 

B distortions, external fields 489 B-SmB-SmA transtions 383 backflow effect... 

As the transmission curve of a liquid crystal display during the switching process depends on all the Leslie coefficients due to backflow effects, it is possible to determine the coefficients from the transmission curve. In analogy to light scattering, the coefficients are obtained with different accuracies [38, 39]. The investigation of torsional shear flow in a liquid crystal [31-35] allows the determination of quantities from which some Leslie coefficients can be determined, if one shear viscosity coefficient is known. 

F. Brochard, Backflow Effects In Nematic Liquid Crystals, MoL Cryst and Liq. Cryst, 23, p. 51 (1973). 

The dynamics of the other two types of orientation distortions, bend and splay deformations, are more complicated because such distortions are necessarily accompanied by flow (i.e., physical translational motion of the liquid crystal) this phenomenon is sometimes called the backflow effect, which may be regarded as the reverse effect of the flow-induced reorientation effect discussed in Chapter 3. A quantitative analysis of these processes shows that the half-widths of the spectra associated with pure splay and pure bend deformations are given, respectively, by... 

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