Electrostatic functions

Terms in the energy expression that describe how one motion of the molecule affects another are called cross terms. A cross term commonly used is a stretch-bend term, which describes how equilibrium bond lengths tend to shift as bond angles are changed. Some force fields have no cross terms and may compensate for this by having sophisticated electrostatic functions. The MM4 force field is at the opposite extreme with nine different types of cross terms. 

Switched van der Waals and electrostatic functions, with switching... 

The electrostatic function f must contain the Debye-Hiickel limiting law with the parameter... 

As the Fourier coefficients in Eq. (8.17) contain the factor 1 /H2, the high-order structure factors are of decreasing importance in the potential summation. The emphasis on the low-order structure factors is less pronounced for the higher-order electrostatic functions, such as the electric field and the electric field gradient, as summarized in Table 8.1. 

Like the potential, other electrostatic functions can be expressed as Fourier summations over the structure factors (Stewart 1979). The electric field, being the (negative) gradient of the potential, is a Fourier series in which the power of the magnitude of H increases from —2 to —1, as expected from the reciprocal relationship between direct space and Fourier space. Starting with... 

While the calculation of the electrostatic functions from the multipole parameters parallels that of the calculation of the atomic electrostatic moments, there is an... 

Database search identifies a template to fill site. Appropriate hydrophobic and electrostatic functions added... 

The electrostatic function S(r) is computed from the partial charges. Application of conditions (a) and (b) leads to an evaluation of the parameters d and c. The angular dependence arises from the electrostatic interaction of the partial charges (in the S(r) term). Once d and c are... 

Figure 3.5 Plot of the association constant of some 1 1 metal cation-hydroxy complexes at zero ionic strength (see Chap. 4) versus the electrostatic function luZon/irti + Toh). where the association reaction is written Af + OH"=A/OH " , and z and r are the charge and radius in nanometers (nm) or angstroms (A) (1 nm = 1 A) of cation M and OH ( oh = 1-40 nm). Cation radii are from Shannon and Prewitt (1969), log values from Baes and Mesmer (1981). The slope of the straight line suggests the contribution of electrostatic (ionic) bonding to the stability of the complexes. The extent to which species plot above this line presumably reflects the increased contribution of covalency to their stabilities.

The stability constants of ion pairs (their log /Cassoc values) have been shown to be proportional to the electrostatic function ZMzJd, where z Z/. are the charge of metal cation and ligand, and d rM + ri, the sum of their crystal radii (cf. Fig. 3.5). Mathematical models for predicting ion pair stabilities generally assume this proportionality and include the simple electrostatic model, the Bjerrum model, and the Fuoss model (cf. Langmuir 1979). Such models can predict stabilities in fair agreement with empirical data for monovalent and divalent cation ion pairs. 

The calculation of the averaged (electrostatic) functions is reached in two steps. At the first, the proper flexibility of the polymer is evaluated either from conformational calculation or from suitable models, then the mean value of each property is calculated through the averaging procedure described below. 

Pitzer electrostatic function, equation (4.62) fugacity of vapor species V, equation (3.20) number of grams of i, i=0 for solvent Gibbs free energy... 

Other types of surfactants are the polymeric (steric) stabilisers, such as partially hydrolysed polyvinyl acetate. Also oligomeric species formed in situ, when SO radicals react with some monomer units in the aqueous phase, will have surface active properties, and can even form a colloidally stable latex Electrosteric stabilisers combine steric and electrostatic functionalities for example, inclusion of acrylic acid in a recipe results in chains with blocks comprised largely of poly(acrylic acid) which for in the aqueous phase, then pick up enough hydrophobic monomer to enter the particle and continue polymerisation in the particle interior. The hydrophilic component remains in the aqueous phase and provides colloidal stability both sterically and, imder the appropriate conditions of pH, electrostatically. This mode of stabilisation is very common in surface coatings, because it gives excellent freeze-thaw stability. 

Further on, Kunz et al could show that this failure of the simplified dispersion model is not a consequence of the weakness of the Poisson-Boitzmann equation. More elaborate statistical mechanics, using the so-called hypernetted chain equation (HNC), yielded basically the same result. Obviously the problem comes from the neglect of ion-water interactions and their changes near the surface. To introduce such interactions in primitive model calculations, Bostrom et al [see also Refs. 13(b)-13(d)] used Jungwirth s water profile perpendicular to the surface as a basis to model a distance-dependent electrostatic function, instead of a static dielectric constant. Such ideas were used several times over the years, for instance to model activity coefficients of electrolyte solutions. ... 

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