Distribution of dipoles

Only at extremely high electric fields are the water molecules fiilly aligned at the electrode surface. For electric fields of the size normally encountered, a distribution of dipole directions is found, whose half-widtli is strongly dependent on whether specific adsorption of ions takes place. In tlie absence of such adsorption the distribution fiinction steadily narrows, but in the presence of adsorption the distribution may show little change from that found at the PZC an example is shown in figure A2.4.10 [30]. 

Fig. 1.2 A distribution of dipoles undergoing the effect of a static electric field.

Two additional feature can be incorporated into Eqs. (7.32)—(7.35) the dipole orientation distribution and the concentration distribution in systems consisting of many dipoles. The orientation of the dipole with respect to the surface, described by angles Q = (8, ), affects E and all the other measurables derived from it.(33) Consider a concentration distribution of dipoles in both orientation and distance from the surface specified by C(0, , z). Since the dipoles all oscillate incoherently with respect to one another, the integrated intensity J due to this distribution is simply ... 

Combining the above equations, we can write a useful expression for the collected fluorescence from a distribution of dipoles in terms of the collection efficiency Q for a single dipole ... 

Theoretical Results for a Distribution of Dipoles Random Orientations... 

For several metal - - H2 systems, it has been found that the experimental heat of adsorption is a linear function of the surface coverage S4, 131). This implies that for these systems the distribution of dipoles approximates to a continuous layer of charge and that Equation (11) correctly describes the change in the heat of adsorption when an electron is moved through the double charge layer. Originally Boudart (108) used the relation... 

This expression is made up so that p tends to unity at the limit of random-coil (fN = 0) and approaches the value of length/diameter at the limit of intact a-helix (fN = 1), with the assumption of 15 A for the diameter of the helical rod (2). It roughly accounts for the effect that the distribution of dipoles in the molecule becomes more spherically symmetric in going from helix to random-coil. Since p appears only in the correction terms for /g, its exact values are not needed. It has been shown (117) that the calculated values of depend little on the type of equation employed for p except in the region close to fN = 0. 

Figure 3.3 Hypothetical distribution of dipole moment magnitudes from a simulation of liquid water. The dashed curve is generated by connecting the tops of histogram bins whose height is dictated by the number of water molecules found to have dipole moments in the range spanned by the bin. Note that although the example is illustrated to be symmetric about a central value (which will thus necessarily be (/z)) this need not be the case...

Figure 9.8 Schematic diagram of how angular correlations occur. Panel (a) shows the angular distribution of dipole radiation for Am = 0 and Am = +1. Panel (b) shows the magnetic substates populated in a y y2 cascade from J = 0 to J = 1 to J = 0. When -y, defines the Z axis, then the mi = 0 state cannot be fed and one has only Ami = + 1 and Am2 = +1, causing y2 to have an anisotropic distribution relative to 71 shown in panel (c). [From Marmier and Sheldon, 1969.] Copyright 1969 Academic Press. Reprinted by permission of Elsevier.

The positive charge of a cesium ion will therefore be attracted owing to the discrete distribution of dipoles brought about by the cesium ions already present. Consequently the heat of adsorption of the cesium ion is increased. 

As we have seen in Sec. IX,4 the dipole layer may not be treated as a double layer with a continuous distribution of charge. Discrete charge distributions have to be assumed. Gomer (335) criticizes Boudart on this point. He evaluates potential curves for a discrete distribution of dipole... 

Debye obtained his result by solving a forced diffusion equation Ci.e., with torque of the applied field included) for the distribution of dipole coordinate p - pcosS, with 6 the polar angle between the dipole axis and tSe field, and the same result for the model follows very simply from equation (3) using the time dependent distribution function in the absence of the field (5). The relaxation time is given by td = 1/2D, which for a molecular sphere of volume v rotating in fluid of viscosity n becomes... 

Figure 1.1. Scheme of dipole-moment densities utilized in the method of Ewald (A) Distribution of point dipoles whose field is calculated at the origin 0. (B) distribution of gaussians centered at each point, point 0 included. (C) distribution of dipoles plus gaussians ... 

The treatment of polarization based on the assumption that water has a uniform dielectric constant involves a fundamental difficulty. Indeed, a uniform, continuous distribution of dipoles on a planar surface does not generate a field in a homogeneous medium and hence is not able to polarize the water. If the dipoles are distributed in the sites of a 2D planar square lattice, the field is... 

However, these results are highly model dependent (64-66). For example, a model based on a random distribution of dipole-dipole interactions gives an exponential decay in the slow modulation limit (64). As another example, a nonexponential frequency correlation decay gives different intermediate results than the Kubo-Anderson model (67,68). 

Generally, to fit the observed FID, a series of exponential functions (Equation (1)) are used because the distribution of dipole interaction is expressed by Lorentzian function. This is true for the solution, melt and amorphous phases of the polymers. Actually, a PE melt with low MW exhibits a single exponential curve.14-17 The shape of the relaxation curve of amorphous molecular motion still retains the combined exponential types on cooling. On the other hand, Weibullian functions (Equation (2))6 18 are also applicable for the phase with partially restricted motion such as the interfacial phase.19 Therefore, it is reasonable to introduce the exponential and Weibullian functions as the amorphous relaxation ... 

If the electric field is the only orientational force acting on the molecules, the distribution of dipole orientations will be given by Boltzmann s law. Thus the number of dipoles pointing in the directions confined within the element of solid angle dfi will be... 

The above argument does not depend on the assumption that the dipoles lie on a lattice since only the residue of the sum E at its pole at = — 1 comes into the final answer, and this will be the same even if the dipoles are irregularly distributed so long as they have an average density I/a. Indeed, if we consider the correlation energy of a statistical distribution of dipoles we have... 

If the ferrofluid is dilute, so that particle-particle interactions can be neglected, then the distribution of dipole orientations in a field H is given by the usual Boltzmann equation -----------------------------------------------------------------------------------... 

Fig. 12.4. Left The dipole moment in Debye plotted against the time step. Right Distribution of dipole moments (left-hand-side region smaller dipole moments right-hand-side region larger dipole moments). Both data sets were calculated from a Car-Parrinello molecular dynamics simulation of an isolated tetralactam macrocycle [227].

A rather novel scheme for modeling molecular polarizabilities as distributed dipole polarizabilities has recently been reported [141]. In this approach, the overall quadrupole induced in a molecule by an external field, as calculated with ab initio methods, is decomposed into induced dipoles distributed to atomic sites. In turn, this yields the dipole polarizability values at those sites. In effect, this relates the overall dipole quadrupole polarizability to a distribution of dipole polarizabihties. 

A way to analyze different distributions of dipoles at the membrane interface is by studying the behavior of the surface potential as a function of the group esterified to the phosphate (Diaz et al., 2001 Disalvo et al., 2002). 

Refinements in vuv spectroscopy W, aided by the development of synchrotron radiation (7 ) and equivalent-photon electron-impact ( ) tunable light sources, and closely related advances in photoelectron, fluorescence-yield, and electron-ion coincidence spectroscopy measurements of partial cross sections (9), have provided the complete spectral distributions of dipole intensities in many stable diatomic and polyatomic compounds. Of particular importance is the experimental separation of total absorption and ionization cross sections into underlying individual channel contributions over very broad ranges of incident photon energies. 

A charge distribution, however, can be associated directly with a moment expansion. The first such moment of interest is the dipole moment. Using the data presented elsewhere, the error distribution of dipoles for NDDO methods are summarized in Table 4. 

To find the mean value (y/h) in the interval h [0, u of reduced (divided by kBT) energies h, we take into account the Boltzmann distribution of dipoles... 

Fig. 21-14. Distribution of dipoles according to the polarization theory of the trans effect.

When there is no preferential orientation like in a random distribution of dipoles, the contributions of all dipole moments parallel to E cancel each other, because in Eq. (3.99) for a random distribution of permanent dipoles is zero. In this case fij = Hj and no macroscopic polarization of the medium occurs. 

Fig. 5.1. Above The red arrow marks the molecular dipole moment for a static molecule, the dashed arrow is its projection on the layer. Below The T sign marks the average tilt. The thin arrows mark the instantaneous dipole moments of several molecules illustrating the distribution of dipoles. The thick arrow gives the average. The axes of the static coordinate system bound to the layer are marked by x, y and z. For the non tilted molecule the axes with and without stars coincide. For the tilted molecule only the direction along the y axis is the same as the static y direction. The two coordinate systems x, y, z and x, y, z ) are used for the explanation of rotation hindrances only, (a) Three axes of rotation for an average molecule, (b) Above A tilted molecule. Below Rotation around the long axis (z) is hindered resulting in a polarization distribution with a net polarization, (c) Above A molecule can flip around its short axis (y). Below The polarization distribution for the flipped molecule.

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