Physical Reasons

It is well known that a nematic liquid crystal is nonpolar as a result of the free or hindered rotation of its constituent molecules around their axes. In the absence of an external field the distribution of the dipoles in an undistorted nematic liquid crystal has a nonpolar cylindrical symmetry. This is shown schematically in Fig. 4.29(a). However, as Meyer [183] has shown, a polar axis can arise in a liquid crystal made up of polar pear-shaped molecules when it is subjected to splay deformations, or in a liquid crystal made up of banana-shaped molecules subjected to bend deformations. In this case, the polar structure corresponds to closer packing of the molecules (Fig. 4.29(b)). Thus, the external mechanical deformation of the nematic liquid crystal results in the occurrence of a charge at electrodes perpendicular to the polar axis, i.e., there is a similarity to the piezoelectric effect in solid crystals. 

This effect has been called the piezoelectric effect in many publications on liquid crystals, but there is good reason for giving it a different name. The piezoelectric effect corresponds to the occurrence of a charge on the surface of the crystal when there is a translational deformation, e.g., with compression or extension. The crystal in this case must be non-centrosymmetric. An effect of this type is also characteristic of polar liquid crystalline phases, e.g., of the chiral smectic C (Chapter 7). The effect, however, in which we are interested here is caused by flexion, a purely orientation deformation in a nematic liquid crystal. Consistent with this argument [1], we will call 

FIGURE 4.29. The flexoelectric effect, (a) The structure of an undeformed nematic hquid crystal with pear- and banana-shaped molecules (b) the same nematic hquid crystal subjected to splay and bend deformations, respectively (c) the experimental configuration atnd (d) the distribution of the director along the z-axis due to the flexoelectric effect. 

This condition holds for two-dimensional deformations (e.g., in the xz surface). However, as shown in Chapter 5 three-dimensional flexoelectric deformations can occur even with strong anchoring of the molecules to the surfaces of the cell (Section 5.1.1). 

For small deformations the components of the director, corresponding to the geometry of the experiment, are 

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The physical reasons for the benefits of EOR on recovery are discussed in Section 8.7, and the following gives a qualitative description of how the techniques may be applied to manage the production decline period of a field. 

Traditional vs regression approach to automatic material characterization The traditional approach to automatic material characterization is based on physical reasoning where a. set of features of the signals that we assume to be the most relevant for solving the characterization problem is. selected. However, in situations with a complicated relation between the measurements and the material property to be characterized, this approach is not always applicable due to limited understanding of the underlying physical relations. 

If the signal features already have been chosen, another important problem is how to optimally combine these features in order to obtain the best estimate of the material property. The physical reasoning will give us ideas of how to combine the features but there will be no guarantee that we are using the chosen features in an optimal way. One reason for this is that we have to take into account the uncertainties that always are present in measurement data. 

Electronic spectra are almost always treated within the framework of the Bom-Oppenlieimer approxunation [8] which states that the total wavefiinction of a molecule can be expressed as a product of electronic, vibrational, and rotational wavefiinctions (plus, of course, the translation of the centre of mass which can always be treated separately from the internal coordinates). The physical reason for the separation is that the nuclei are much heavier than the electrons and move much more slowly, so the electron cloud nonnally follows the instantaneous position of the nuclei quite well. The integral of equation (BE 1.1) is over all internal coordinates, both electronic and nuclear. Integration over the rotational wavefiinctions gives rotational selection rules which detemiine the fine structure and band shapes of electronic transitions in gaseous molecules. Rotational selection rules will be discussed below. For molecules in condensed phases the rotational motion is suppressed and replaced by oscillatory and diflfiisional motions. 

The Langevin model has been employed extensively in the literature for various numerical and physical reasons. For example, the Langevin framework has been used to eliminate explicit representation of water molecules [22], treat droplet surface effects [23, 24], represent hydration shell models in large systems [25, 26, 27], or enhance sampling [28, 29, 30]. See Pastor s comprehensive review [22]. 

Despite the fact Chat there are no analogs of void fraction or pore size in the model, by varying the proportion of dust particles dispersed among the gas molecules it is possible to move from a situation where most momentum transfer occurs in collisions between pairs of gas molecules, Co one where the principal momentum transfer is between gas molecules and the dust. Thus one might hope to obtain at least a physically reasonable form for the flux relations, over the whole range from bulk diffusion to Knudsen streaming. 

Using your optimized expression for W, calculate the estimated total energy of each of these atoms and ions. Also calculate the percent error in your estimate for each ion. What physical reason explains the decrease in percentage error as Z increases ... 

It should behave in a physically reasonable manner in those parts of the interaction regions for which no experimental or theoretical data are available. 

It should smoothly connect the asymptotic and interaction regions in a physically reasonable way. 

Of all the topics discussed in this text, mesoscale simulations are probably at the most infantile stage of development. The idea of the mesoscale calculations is very attractive and physically reasonable. However, it is not as simple as one might expect. The choice of bead sizes and parameters is crucial to obtaining physically relevant results. More complex bead shapes are expected to be incorporated in future versions of these techniques. When using one simulation technique to derive parameters for another simulation, very small errors in a low-level calculation could result in large errors in the final stages. 

The rotational isomeric state (RIS) model assumes that conformational angles can take only certain values. It can be used to generate trial conformations, for which energies can be computed using molecular mechanics. This assumption is physically reasonable while allowing statistical averages to be computed easily. This model is used to derive simple analytic equations that predict polymer properties based on a few values, such as the preferred angle... 

It is equivalent to say that entropy of vaporization is a constant value for non-associating Hquids. Associating Hquids, eg, ammonia, water, methanol, and ethanol, do not obey the rule of Pictet and Trouton. Despite its simplicity, the Pictet-Trouton view of Hquid vaporization (19) is an exceUent example of the many rules of thumb that have been useful aids in engineering calculations for decades (5,7,8,9,21). However, proper appHcation requires an understanding of the physical reasoning behind each rule. 

In this case, the shear stress is linear in the shear strain. While more physically reasonable, this is not likely to provide a satisfactory representation for the large deformation shear response of many materials either, since most materials may be expected to stiffen with deformation. Note that the hypoelastic equation of grade zero (5.117) is not invariant to the choice of indifferent stress rate, the predicted response for simple shear depending on the choice which is made. 

In the energy-horizon theory of fragmentation two physically reasonable... 

Evaluating the model in tenns of how well the model fits the data, including the use of posterior predictive simulations to determine whether data predicted from the posterior distribution resemble the data that generated them and look physically reasonable. Overfitting the data will produce unrealistic posterior predictive distributions. 

Two physically reasonable but quite different models have been used to describe the internal motions of lipid molecules observed by neutron scattering. In the first the protons are assumed to undergo diffusion in a sphere [63]. The radius of the sphere is allowed to be different for different protons. Although the results do not seem to be sensitive to the details of the variation in the sphere radii, it is necessary to have a range of sphere volumes, with the largest volume for methylene groups near the ends of the hydrocarbon chains in the middle of the bilayer and the smallest for the methylenes at the tops of the chains, closest to the bilayer surface. This is consistent with the behavior of the carbon-deuterium order parameters,. S cd, measured by deuterium NMR ... 

The physical reason for the third term on the right of equation 2.4-7 is to correct t or a mating the overlap twice as seen in Figure 2.2-1. The technique of Venn diagrams is used in some PRAs to calculate mutually exclusive power states from non-mutually exclusive states... 

Determining which accident sequences lead to which states requires a thorough knowledge of plant and process operations, and previous safety analyses of the plant such as, for nuclear plants, in Chapter 15 of their FSAR. These states do not form a continuum but cluster about specific situations, each with characteristic releases. The maximum number of damage states for a two-branch event trees is 2 where S is the number of systems along the top of the event tree. For example, if there are 10 systems there are 2 = 1,024 end-states. This is true for an "unpruned" event tree, but. in reality, simpler trees result from nodes being bypassed for physical reasons. An additional simplification results... 

Now, release the constraint of having the first layer still oriented at a. That constraint must surely seem quite arbitrary and not at all physically reasonable. Also, we must admit that the second layer probably is not in its proper orientation either. Thus, we will allow the two laminae orientations to float from [01/02 to something else. And we will call that procedure for changing the laminae lamina reorientation. There are two stages of lamina reorientation (1) coarse reorientation and (2) fine reorientation. 

In these circumstances a decision must be made which of two (or more) kinet-ically equivalent rate terms should be included in the rate equation and the kinetic scheme (It will seldom be justified to include both terms, certainly not on kinetic grounds.) A useful procedure is to evaluate the rate constant using both of the kinetically equivalent forms. Now if one of these constants (for a second-order reaction) is greater than about 10 ° M s-, the corresponding rate term can be rejected. This criterion is based on the theoretical estimate of a diffusion-controlled reaction rate (this is described in Chapter 4). It is not physically reasonable that a chemical rate constant can be larger than the diffusion rate limit. 

The proximity effect. This is the simple idea that in an intramolecular reaction the substrate function may be exposed to a larger local concentration of the reagent than in an intermolecular reaction, because the two functions are covalently constrained to occupy adjacent space. This effect has been called the approximation or propinquity effect. The proximity effect certainly seems physically reasonable and is likely to make some contribution to intramolecular reactivity, but it cannot be a major contributor when EM is large, because EM is itself a measure of a presumed local concentration, and the observed large EM values are physically impossible concentrations. The magnitude of rate enhancement achievable by prox-... 

It is difficult to treat the effect of a heteroatom on the localization energies of aromatic systems, but Brown has derived molecular orbital parameters from which he has shown that the rates of attack of the phenyl radical at the three positions of pyridine relatively to benzene agree within 10% with the experimental results. He and his co-workers have shown that the formation of 1-bromoisoquinoline on free-radical bromination of isoquinoline is in agreement with predictions from localization energies for physically reasonable values of the Coulomb parameters, but the observed orientation of the phcnylation of quinoline cannot be correlated with localization ener-... 

In critical applications, if stainless steel is to be used near its limit (in terms of corrosion), and for cases such as welds, where a good finish cannot be otherwise achieved, additional passivation is required. Nitric acid (10-15 per cent by volume) is the best passivator. It also dissolves iron contamination. In circumstances where the use of nitric acid is not possible for safety or physical reasons (such as the underside of vessel roofs) passivation paste is appropriate. Both materials are used at ambient temperature and require a contact time of approximately 30 minutes. They must be removed by thorough rinsing with low chloride-content water. 

Now let us add the possibility of collisions. Before we proceed, we make the following two assumptions (1) only binary collisions occur, i.e. we rule out situations in which three or more hard-spheres simultaneously come together (which is a physically reasonable assumption provided that the gas is sufficiently dilute), and (2) Boltzman s Stosszahlansatz, or his molecular chaos assumption that the motion of the hard-spheres is effectively pairwise uncorrelated i.e. that the pair-distribution function is the product of individual distribution functions ... 

Svozil also suggests a third possibility, whereby a discretized field theory is strictly local in a higher dimensional space d > 4 but appears to be nonlocal in d = 4. While the physical reasons for a such a dimensional reduction remain unclear, such a dimensional shadowing clearly circumvents the no-go theorem by postulating a local dynamics in a higher dimension (see figure 12.9). 

The general idea of using different orbitals for different spins" seems thus to render an important extension of the entire framework of the independent-particle model. There seem to be essential physical reasons for a comparatively large orbital splitting depending on correlation, since electrons with opposite spins try to avoid each other because of their mutual Coulomb repulsion, and, in systems with unbalanced spins, there may further exist an extra exchange polarization of the type emphasized by Slater. 

Of course, any set of experimental data can be described by selecting an appropriate empirical equation with an arbitrary set of constants. However, comparing a vast wealth of the known results of measurements of suspension viscosity, it should be admitted that a universal formula for ther (cp) dependence does not exist, and significant discrepancies may begin already from a linear term, so that physical reasons for exagerated values of the coefficient bt as compared to 2.5 should be looked for. 

Nevertheless, from a practical viewpoint physical reasons for viscosity reduction during the surface treatment of the filler play a minor role first of all the effect of viscosity reduction itself is significant. 

Equation (3-307) also leads to a very important physical interpretation of the function Sx(f) namely, that Sx(f) describes the way in which the total power of X(t) is distributed in frequency. This comes about if we accept the physically reasonable definition that the power in X(t) contained in the frequency band /x < / < f2 is equal to the total power output of the ideal band-pass filter H(f) shown in Fig. 3-11, when X(t) is the input. Equation (3-307) now yields the result,... 

Since E0, the vacuum energy, must for obvious physical reasons, be the lowest energy of the negaton-positon field system, En — E0 s 0, so that Eq. (10-247) indicates that for x0 > x 0, GA contains only positive frequencies. Similarly one verifies that (under the assumption of a stable vacuum state, the state of lowest energy) GA for 0 < x Q contains only negative frequencies. 

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