Chord model

The AP method uses Eq. (4.41) with W = 0 and corresponds to alignment of the alkyl chain through aligning of individual C-C bonds in an uncorrelated fashion. The chord model (W Wq / 0) provides a physical realization of Marcelja s [4.26] ad hoc choice of the chord segment (joining the midpoints of the pair of adjacent C-C bonds) to construct a potential of mean torque for lipid membranes. Rewriting the P2(x) in terms of the chord vector P (Fig. 4.8)... 

The model has also been applied to study the orientational order profiles of the deuterated chain attached to mesogens nCB and nOCB [4.28]. Figure 4.9 reproduces [4.29] the fit to the orientational order profile [4.29] of 8CB in the nematic phase at Tc — T = 4° C using the chord model (Wo = Wi). In this case, the potential of mean torque C/ext( 5 ) has to... 

As far as side chains are concerned, the local order parameters relative to the various CH2 and CH3 groups give a measure of chain disorder, being dependent on chain flexibility and conformation-orientation correlations. Different theories of orientational ordering in flexible chains, such as the AP and the chord models, have been formulated and critically tested by comparison with experimental data (see Figure 3). 

A Bernal chart for a cylindrical camera of any radius may be constructed graphically by drawing the plan and elevation of this model. Thus, if the height of any reciprocal lattice point above the origin is r and its distance from the axis of rotation is r(, the position of the reflection on the film is obtained in the following way. Draw a circle of radius r (Fig. 85 e), and then a chord NUT at a distance r from the centre (this is the circle of contact seen edgewise) UT is the radius of the circle of contact for this reciprocal lattice point. Join OT and produce to W on the line XC which is parallel to OU. WX is then the ordinate y of the spot on the film. Now draw the plan, that is, draw another circle of radius r (Fig. 85/ ) and in it describe a circle of radius UT. On this circle NT mark off the points Pv P2 which are at a distance r from X, and produce UPX to Yx and UP2 to Y2. The arcs XJ and XY2 are the abscissae x of the two reflections on the film produced by this plane. By doing this for a number of different values of r and rg, the complete chart is obtained. 

The local gas holdup and bubble behavior were measured by a reflective optic fiber probe developed by Wang and co-workers [21,22]. It can be known whether the probe is im-merging in the gas. The rate of the time that probe immerg-ing in the gas and the total sample time is gas holdup. Gas velocity can be got by the time difference that one bubble touch two probes and the distance between two probes. Chord length can be obtained from one bubble velocity and the time that the probe stays in the bubble. Bubble size distribution is got from the probability density of the chord length based on some numerical method. The local liquid velocity in the riser was measured by a backward scattering LDA system (system 9100-8, model TSI). Details have been given by Lin et al. [23]. 

The C 0 model has two different kinds of bar forms AA and AA" (circular arcs of radii Rx and R2) as seen in figure 8. In the unit bar length case, the radii and the chord lengths of the curved bar axes are ... 

In order to test the predictions of the model, plates were made up from selected formulations and tested in air. The thicknesses of the various layers were as stated previously, while the length was one meter and the width was one-third meter. Damping measurements were made by two different methods. In both cases, the plates were suspended in a shock chord and accelerometers placed at different locations on the plate. The first used a reverberation meter and the half-power method (16). In the second, an impact hammer was used to tap the structure and the outputs of the accelerometers fed into a Fourier transform based spectrum analyzer to examine the envelope of vibration (17). The results presented here are based upon the second method. 

Lasentec s Focussed Beam Reflectance Measurement Model (FBRM M400L) operates at a wavelength of 791 nm and the beam rotates 75 s at a peripheral speed of 1.9 ms .to describe a circle of diameter 8 mm. It has been used with a 3D-model of chord length distribution for particles of a general shape as well as particles with the same shape but different sizes [124]. The focal point can be changed in the axial position but it is... 

Model Fit or Direct Evaluation Similar to the ID case we can now fit a model to the projected intensity or to the CLD. What we get in this case is the needle-diameter distribution of the microfibrils. Nevertheless, there are two other possibilities to direetly evaluate the data. We consider polydispersity by allowing for varying hard-disc diameter. If we assume that the shape of each disc is circular, the CLD of an uncorrelated hard-disc fluid is the Mellin convolution of the intrinsic chord distribution, gc ri2), of an ideal disc of diameter 1 and the diameter distribution, ho (D) which characterizes the structure. The definition of theMehin convolution (Titchmarsh [202], S. 53 Marichev [203] [86], S. 304) is... 

These two examples are of interest in their own right, but the authors have also struck a chord in regard to neurological or systems models of Zen and existentialism. 

Further information from x-ray scattering can be extracted if a two-phase media model is applied. In this model, the porous body is assumed to consist of two phases with constant mass (or electron) density. The solid phase has a mass density that occupies a volume fraction 0s, and the voids with Pp = 0 occupy a volume fraction 0p = 1 - 0s. By the application of this model, the specific surface area and the mean chord length of each phase can be derived, provided an absolute intensity measurement is made [38]. 

Radial density profiles perpendicular to the fiber axis can be fitted to a hyperbolic tangent function. Equation (1). For fibers with diameters in the range 5-8 nm, the correlation lengths, are about 0.6 nm, which is close to the value obtained with the models of the free-standing thin films. The end beads are enriched in the surface region, as was also the case with the free-standing thin films. The anisotropy of the chord vectors, as assessed by the order parameter, S, is also similar to the result obtained with the free-standing thin films. 

The anisotropy of segmental motion exhibited in Fig. 19 may arise, as noted above, either from the intramolecular or from the intermoleoilar ccmstraint to the rotational motion. The anisotropy d orioitational condadon decay was indeed noted already by Weber and Helfand [47] in their Brownian dynamics simulation of polyethylene of infinite chain length. Their orioitational time-correlation function of the chord vector ( = 0°) decayed much more slowly than those of either the bisector vector ( = 0°, = 0°) or the out-of-plane vector ( = 0°, = 90°). What they modeled was a phantom chain having no... 

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