Angle representation

In some cases, it is simpler to describe the liquid crystal director n in terms of the polar angle 0 and azimuthal angle The angles may vary in one or two or three dimensions. We first consider a simple case Fr edericksz transition in splay geometry. The liquid crystal director is represented by the tilt angle 6 n = cos 0 z)x + sin 0 z)z, where the z axis is in the cell normal direction. The electric field is applied in the cell normal direction. From Equation (4.17) and (7.8) we have the free energy density 

If initially the system is not in the equilibrium state, 6 z) = Omiz)- It will relax toward the equilibrium state. The dynamic equation governing the relaxation of the system is given by Equation (4.74)  

Using this equation, the angle at time t + At can be calculated from the angle at time t as 

When the system reaches the equilibrium state, the director configuration does not change any more, and d0/dt=-5f/50 = O. Equation (7.30) may not describe the actual dynamic process because the hydrodynamic effect is not considered, but the final director configuration obtained is the actual one for the equilibrium state. 

In the numerical calculation, the liquid crystal cell is discretized into a one-dimensional mesh with lattice sites. The length of the lattice unit is Az = h/N. At step t, the tilt angle at the lattice site i is 0(i) (i = 0, 1,2, 3,., N). The derivatives are calculated by 

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Marcus, R.A. (1988). Semiclassical wavepackets in the angle representation and their role in molecular dynamics, Chem. Phys. Lett. 152, 8-13. 

Another set of early studies came from the work of Judson and coworkers [35, 36], which emphasized using GAs for search problems on small molecules and peptides, especially cyclic peptides. A dihedral angle representation was used for the peptides with values encoded as binary strings, and the energy function used the standard CHARMM force field. Mutations were implemented as bit flips and crossovers were introduced by a cut-and-paste of the strings. The small size of the system enabled a detailed investigation of the various parameters and policies chosen. In Ref. [37], a comparison between a GA and a direct search minimization was performed and showed the advantages and weaknesses of each method. As many concepts are shared between search problems on small peptides and complete proteins, these studies have contributed to subsequent attempts on full proteins. 

A classical mechanical % provides insights into intramolecular dynamics, guided by the powerful diagnostic tools of nonlinear classical dynamics. The action-angle representation of a classical mechanical problem is easily derived from the spectroscopic Heff. In fact, the action-angle picture lies at the core of Heisenberg s version of the Correspondence Principle (Heisenberg, 1925). The prescription is that... 

The angle representation is a valid method when the change of the angle within the cell is less than 90°. Otherwise it must be handled carefully in the case where the liquid crystal director at two neighboring lattice sites are anti-parallel. The numerical calculation may produce a large elastic energy while the actual elastic energy is 0, because n and - n are equivalent. 

Fortunately for us, the problem of assigning rotational angles in unambiguous ways has been mathematically resolved. There are several different sets of rotational angle representations, and we will consider only one of these here—the so-called Euler angles. Other representations can be foimd in the further list Reading Section at the end of this chapter. 

Figure 2-4 Contact angle representation between two solids and a liquid gap.

Figure 13.40 Optical flow birefringence studies of400 kDa polystyrene in Aroclor 1248 at concentrations (a) 9.09, and (b) 105 g/1, based on measurements of Lodge and Schrag(50). Results in the original paper were replotted from magnitude-phase angle representation to the in-phase, out-of-phase representation permitting direct comparison of the results above with the mechanical dynamic moduli. In (b) the vertical axis is a log scale covering 17 orders of magnitude.

Based on these definitions, we can obtain the four dimensional Hamiltonian from the six dimensional one by integrating out the Euler angles. We first specify the Euler angle representation of JMK). This is... 

The final result given in eq 25 is easy to apply. It cai be readily deduced that the states of odd 2S mix into double-valued irreps F only, and correspondingly even 2S states mix into single-valued irreps only. This follows from D (g) = r>[, (g) where the plus/minus refer to single/double valued irreps respectively and fi-om D g) = iD (g) where the plus/minus refer to even/odd values of 2S, respectively. Note that g and g differ by a rotation of 27t about the z-axis (in the Euler angle representation). 

Wigner functions in the Euler angle representation can be calculated as follows (31) ... 

The line-angle representations shown at the bottom of page 1221 might lead you to believe that the carbon atoms in cycloalkanes all lie in the same plane. However, this is not generally the case. As we will soon see, cyclopropane is the only cycloalkane in which the carbon atoms form a planar ring. 

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