Average dielectric coefficient

Inhomogeneous particles. For particles composed of a matrix and inclusions one approach for calculating optical properties is to assume an average dielectric coefficient (e) for the composed particle. A number of so-called mixing rules have been proposed a frequently used one is the Maxwell-Gamett mixing rule (cf. Bohren Huffman 1983). For a matrix with dielectric coefficient em, and a number of different kinds of inclusions with dielectric coefficients ej and volume fractions f) one uses... 

The surface integral in Eq. [381] has contributions from only those grid positions k j) sharing a common boundary with element /. Denoting < )(ry) by < )y, 1 1 by Tjk, the common surface area between elements / and k by Sjk, and an average dielectric coefficient (to be defined later) by Eq. [381] can be written... 

In Eq. [383] the average dielectric coefficient between two neighboring elements is often taken as the inverse of the arithmetic mean of the inverses " ... 

Here e is the dielectric constant of the gas, F the strength of the applied field, N the number of molecules in unit volumes, n the permanent electric moment of a molecule, and a the coefficient of induced polarization of a molecule cos 9 is the average value of cos 9 for all molecules in the gas, and cos 9 is the time-average of cos 9 for one molecule in a given state of motion, 6 being the angle between the dipole axis and the lines of force of the applied field. 

We shall now discuss the depression of the static permittivity of water by the addition of eiectrolyte solutes, which is a phenomenon of some importance in the understanding of the hydration sheath of the ions. It is essentially a dielectric saturation phenomenon the strong electric fields in the neighbourhood of the ions produce a non-linear polarization, which renders the local water moleodes ineffective as regards orientation in the applied field. It is possible to make estimates of the extent of hydration, or hydration number , of water molecules considered to be bound irrotationally to the average ion these estimates are in reasonable agreement with hydration numbers estimated on the basis of activity coefficients, entropies, mobilities, and viscosities. The hydration number must be distinguished from the number of water molecules actually adjacent to the ion in the first or second layers of hydration (the hydration sheath) it does not follow that all of these molecules can be considered to be attached to the ion as it moves in the solution. 

EXAMPLE 4-1 Calculate the expected electrostatic contribution to the transfer activity coefficient from differences in dielectric constant for ethanol and water (dielectric constants 24.3 and 78.3) at 25°C for a 1 1 electrolyte with an average ionic radius of 1.5 x 10 cm. 

For a number of calculated properties (average dipole moment, dielectric permittivity and its frequency dependence, diffusion coefficient and dielectric and NMR relaxation time) the TIP4Pfq potential performs better than SPCfq, presumably thanks to its more realistic molecular geometry [100]. All calculated properties, except D = 1.9 10 cm /s vs =2.4 10 cm /s at 25 C, are in good agreement with the corresponding measured data. 

Here e is the electron charge, e is the dielectric constant of the medium, a is the distance of the maximum approach of a small ion and a spherical particle, often taken equal to (b-H 2.5A), X is the Debye-Hiickel parameter, 3.57 is the constant coefficient for aqueous systems at 25°C on condition that distances are measured in angstroms), pkjnt is the intrinsic ionization constant, Z = —on is the average value of the total charge of a particle (it is equal numerically to the number of protons having formed on dissociation), n is the number of dissociating groups in a particle. 

For both problems, we optimized both the surface coefficients and the dielectric constant in the energy function (Sect. 7.4). A typical predicted structure is shown in Fig. 7.5, and compared to the known, experimental, Xray structure. Most of the sidechains in the predicted structure have been correctly placed, and overlap nicely with the experimental sidechains. On average, for our test set of 29 proteins (about 3000 sidechains), 80 % of the amino acids have their sidechains in the correct xi rotamer, similar to previous work with similar models. See [46] for a detailed description of our data. 

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