Alignment angle

Fig. 16 STM images (b 11.7x 11.7 nm2, c 11.5 x 11.5 nm2) and structure models (d,e) for the enantiomorphous lamella structures induced by adsorption of ISA (a) on hopg from a 1-heptanol solution [47]. For opposite enantiomers, opposite lamella tilt angles (0) are observed. The large lamella ALi is built up from pure ISA enantiomers, while the smaller AL2 lamella consists of coadsorbed achiral 1-heptanol molecules. The ISA chirality is transferred to the coadsorbed solvent molecules via opposite alignment angles (p (f,g). Reprinted with permission from Wiley...

Fig. 8.1. Schematic representation of the collisionally-induced charge cloud in a P-state atom. The collision plane is determined by the ingoing and outgoing electron momenta ko and k,. The excited atom is characterised by the alignment angle y, its inherent angular momentum (Z, ) = L , and the shape of the charge cloud P/. The direction of emission of the photon and its polarisation vectors are also shown.

X of a flow-aligning nematic with flow-alignment angle 9a in a steady shearing flow, with x the flow direction and... 

Problem 10.7 Derive Eq. (10.4) for the alignment angle 6 of the director in shearing flow of a flow-aligning nematic. 

It is often preferable to evaluate 6q by EMD methods because the director fluctuates around the preferred orientation in a shear flow simulation, which makes it hard to obtain accurate estimates. If one performs such a simulation one must fix the director at several alignment angles and calculate the antisymmetric pressure tensor, which, according to Eq. (4.10e), is a linear function of cos 26. One can fit a straight line to the data points and the zero gives... 

Fig. 4 The antisymmetric pressure function of the alignment angle 6 for a shearing...

These results can be cross checked by performing a simulation where the director is constrained to lie in the vorticity plane but leaving it free to select the alignment angle. The angular distribution of the director is shown in Fig. 5 In these simulations only 256 particles were used. Therefore the distribution is fairly wide. As the system size increases the distribution becomes narrower and it is completely sharp in the thermodynamic limit. The maximum of the distribution appears ai 6 - 20° which is in agreement with the zero of the antisymmetric pressure tensor. A similar value of 6q was also found by using the equilibrium fluctuation relations (4.13) and (4.14). One can consequently conclude that the liquid crystal is flow stable. 

In ordinary nematics p and p are both negative and the flow alignment angle 0 is usually small. This equilibrium orientation of the director is... 

Gahwiller discovered that nematics that undergo a transformation to the smectic phase at lower temperatures exhibit an unusual type of instability as the temperature approaches the transition point. The limiting value of the flow alignment angle 0 defined as... 

Fig. 6.5.2. Flow alignment of the director in nematic liquid crystals, (a) For rodshaped molecules the alignment angle 0 with respect to the flow direction lies between 0° and 45°, while (b) for disc-shaped molecules it lies between —90° and —45° (or equivalently between 90° and 135°).

It has been suggested that in the phase, the disc-like shape of the molecule may have a significant effect on p and 3. The stable orientation of the director under planar shear will now be as shown in fig. 6.5.2(0). Thus it can be argued that both p and p should be positive, and the flow alignment angle 0o should lie between —45 and —90°. It then follows that when p < 0, the director tumbles and the flow becomes... 

In this method, one machine axis is chosen as a reference and the relative displacement and relative alignment angle of the other axis is measured directly by physically contacting the machine parts (such as shafts, couplings) to the measuring device (dial indicator). It is assumed that the relative alignment of the axes is stable during the measurement process. It is thus a relative, contact, manual, static method. 

Having determined relative displacement and alignment angle one can extrapolate back to the foundation to shim the machine into the correct alignment... 

In laser-optic measurement systems (such as Optalign), a combined laser source -sensor unit is placed on the reference shaft and the relative displacement vector and the alignment angle is calculated from the position of the incoming laser beam reflected by a prism unit, which is attached to the other shaft. This is an optical counterpart of the rim-and-face method. It is thus also relative, contact, manual and static in nature. 

When the units are rotated from Position 3 to Position 4 in the horizontal direction, the laser dot moves perpendicularly to the previous direction on the laser sensor (alignment angle detection direction) and the vertical component of the alignment angle is computed, independently from the displacement vector. 

The relative alignment angle (relative azimuth) of the axes is the corresponding azimuth difference ... 

A measure for the alignment can eithCT be defined by means of the vector dot product or by the vector cross product For small alignment angles, i.e. almost parallel shafts, the second is more sensitive. 

For small angles m = 0, approximately, thus Eq.(8) gives the alignment angle directly from coordinate measurements. 

Consider a line f, which contains point B and is parallel to a. The angle between f and b is the alignment angle 0. Its components in the plane q and the plane r perpendicular to q are 0p and 0r, respectively. 

The third material, MES, behaves differently. In this case the adsorbed LC film does show an anisotropy (Fig. 5.4b) the 5CB molecules follow the rubbing direction. The surface alignment is characterized by Qa 0.3 and 0 w 71°. The value of the surface in-plane order parameter is in the lowest range of the usual values for rubbed polymer surfaces Qa 0.3 — 0.5 [29,35,36]. The observed tilt angle is the same as for the other silanes and comparable with alignment angles of cyano-biphenyl monolayers usually found on substrates that induce a planar macroscopic alignment [15,22]. 

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