In-plane order parameter

The surface in-plane order parameter =< cos2p >, which is a measure of the substrate alignment strength for the adjacent LC molecules, is given by... 

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]. 

Fig. 5.7. Relative surface density (circles) and surface in-plane order parameter (squares) of trans-cinnamoyl side groups as a function of the LP UV exposure time,...

Table 5.1 In-plane order parameters and correlations for two dimensional single layers...

Figure 7. In-plane ordering of graphene layers. The turbostratic case gives rise to Moire and pseudosymmetric supcrlattice structures, detectable in STM images. In many cases, however, no correlation exists between the individual layers. The turbostratic ordering has very little effect on the interplanar spacing. Enlarged cj2 parameters arise from defects such as sp1 centers which accompany the turbostratic graphene layers.

As the shear rate increases, the numerical solutions of the Smoluchowski equation (11-3) begin to show deviations from the predictions of the simple Ericksen theory. Tn particular, the scalar order parameter S begins to oscillate during the tumbling motion of the director (for a discussion of tumbling, see Sections 11.4.4 and 10.2.6). The maxima in the order parameter occur when the director is in the first and third quadrants of the deformation plane i.e., 0 < 9 — nn < it j2, where n is an integer. Minima of S occur in the second and fourth quadrants. The amplitude of the oscillations in S increases as y increases, until S is reduced to only 0.25 or so over part of the tumbling cycle. 

In this case, there is a symmetry against the change of sign of 0 (fig. 10 shows that this simply corresponds to an interchange of sublattices) and hence a term 03 cannot occur. The fourth order term, however, now contains two cubic invariants rather than a single term [(02)2 =

rotational invariance in the order parameter space (0i, 02), since all directions in the (01, 02) plane are equivalent. No such rotational symmetry applies to the (2x1) structure, of course. So the expansion eq. (26) results, which defines the universality class of the X Y model with cubic anisotropy . Of course, in this approach not much can he said on the phenomenological coefficients r, u, u, R in eq. (26). 

The three ordered stales of the Potts model correspond to a preferential occupation of one of the three sublattices a,b,c into which the triangular lattice is split in the (-/3x-v/3)R30° structure. In the order parameter plane (0x.0r), the minima of F occur at positions (1, 0)MS, (—1/2, i/3/2)yWs, (—1/2, -yf3/2)Ms, where Ms is the absolute value of the order parameter, i.e. they are rotated by an angle of 120° with respect to each other. The phase transition of the three-state Potts model hence can be interpreted as spontaneous breaking of the (discrete) Zj symmetry. While Landau s theory implies [fig. 13 and eqs. (20), (21)] that this transition must be of first order due to the third-order invariant present in eq. (34), it actually is of second order in d = 2 dimensions (Baxter, 1982, 1973) in agreement with experimental observations on monolayer ( /3x /3)R30o structures (Dash, 1978 Bretz, 1977). The reasons why Landau s theory fails in predicting the order of the transition and the critical behavior that results in this case will be discussed in the next section. 

An orientational order parameter can be defined in tenns of an ensemble average of a suitable orthogonal polynomial. In liquid crystal phases with a mirror plane of symmetry nonnal to the director, orientational ordering is specified. 

Positional Distribution Function and Order Parameter. In addition to orientational order, some Hquid crystals possess positional order in that a snapshot at any time reveals that there are parallel planes which possess a higher density of molecular centers than the spaces between these planes. If the normal to these planes is defined as the -axis, then a positional distribution function, can be defined, where is proportional to the... 

This is achieved by coupling the system to a suitably defined order parameter that is sensitive to the crystal order (the stacking sequence of 111 planes in this case), and doing umbrella sampling with this quantity. The result of the simulation is the free energy difference between both candidate structures—and the winner is fed... 

Figure 2 Time sequence of th< spin configuration on a (100) plane at 50% when the system at T=2.5 (snapshot a) is quenched down to T—1.7 and is subject to an isothermal aging. Snapshots demonstrated in figs, b, c and d correspond to time t=20,000, 43,000 and 50,000. The long range and short range order parameters input from the PPM calculations and resultant ones in the simulated lattice are also demonstrated [22, 24, 28]. ...

The temperature dependence of the electrical double-layer parameters has been determined for real393,398 as well as quasi-perfect Ag planes.382,394 For quasi-perfect Ag electrodes, the value of 3 ffa0/9rhas been found to be higher for Ag(100) than for Ag(lll), and so it was concluded that Ag(lll) is more hydrophilic than Ag(100). For real surfaces,382,385,386 dEff=0/BT increases in the order (110) < (100) <(111). The same order of planes has been observed for Au 446-448 BEa /BT linearly increases as AX (interfacial parameter) decreases, i.e., as the hydrophilicity of Ag and Au electrodes decreases.15 32 393 397 398 446 48... 

Normal incidence transmission IRLD measurements are used to study thin films (typically 100 pm thickness and less, depending on the molar extinction coefficient of the bands) with in-plane uniaxial orientation. Two spectra are recorded sequentially with the radiation polarized parallel (p) and perpendicular (s) to the principal (machine) direction of the sample. The order parameter of the transition moment of the studied vibration is calculated from either the dichroic ratio (R — Ap/As) or the dichroic difference (AA = Ap—As) as ... 

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