Electric contribution

Lekkerkerker, H. N. W. (1990). The electric contribution to the curvature elastic moduli of charged fluid interfaces, Physica A, 167, 384—394. 

The direct access to the electrical-energetic properties of an ion-in-solution which polarography and related electro-analytical techniques seem to offer, has invited many attempts to interpret the results in terms of fundamental energetic quantities, such as ionization potentials and solvation enthalpies. An early and seminal analysis by Case etal., [16] was followed up by an extension of the theory to various aromatic cations by Kothe et al. [17]. They attempted the absolute calculation of the solvation enthalpies of cations, molecules, and anions of the triphenylmethyl series, and our Equations (4) and (6) are derived by implicit arguments closely related to theirs, but we have preferred not to follow their attempts at absolute calculations. Such calculations are inevitably beset by a lack of data (in this instance especially the ionization energies of the radicals) and by the need for approximations of various kinds. For example, Kothe et al., attempted to calculate the electrical contribution to the solvation enthalpy by Born s equation, applicable to an isolated spherical ion, uninhibited by the fact that they then combined it with half-wave potentials obtained for planar ions at high ionic strength. 

We return to the complex formation equilibria described in Chapter 2 (Eqs. 2.1 -2.10). The equilibrium constants as given in these equations are essentially intrinsic constants valid for a (hypothetically) uncharged surface. In many cases we can use these constants as apparent constants (in a similar way as non-activity corrected constants are being used) to illustrate some of the principal features of the interdependent variables that affect adsorption. Although it is impossible to separate the chemical and electrical contribution to the total energy of interaction with a surface without making non-thermodynamic assumptions, it is useful to operationally break down the interaction energy into a chemical and a Coulombic part ... 

The first integral in Equation 17 is identified as the electrical contribution to the change in free energy in forming the charged interface (3) and may be evaluated using Equation 12... 

In concluding this section, we point out that the effect of any electrical filter composed of purely linear elements, whether they be passive like resistors, capacitors, and inductors or active like linear amplifiers, can be represented as a convolution. The various other spreading phenomena that are described by convolution in the same domain may therefore be lumped together with the electrical contribution and comprehensively called the spectrometer response function. Even inherent line broadening may be included, provided that the convolution does not appear in an exponent, as in the case of absorption spectra. 

Physical properties of liquid crystals are generally anisotropic (see, for example, du Jeu, 1980). The anisotropic physical properties that are relevant to display devices are refractive index, dielectric permittivity and orientational elasticity (Raynes, 1983). A nematic LC has two principal refractive indices, Un and measured parallel and perpendicular to the nematic director respectively. The birefringence An = ny — rij is positive, typically around 0.25. The anisotropy in the dielectric permittivity which is given by As = II — Sj is the driving force for most electrooptic effects in LCs. The electric contribution to the free energy contains a term that depends on the angle between the director n and the electric field E and is given by... 

It is convenient to think of the diffuse part of the double layer as an ionic atmosphere surrounding the particle. Any movement of the particle affects the particle s ionic atmosphere, which can be thought of as being dragged along through bulk motion and diffusional motion of the ions. The resulting electrical contribution to the resistance to particle motion manifests itself as an additional viscous effect, known as the electroviscous effect. Further,... 

The electrochemical potential can be separated into chemical and electrical contributions... 

The standard electrochemical potentials of reactants and products may be written in terms of the chemical and electric contributions according to eqn. (36). 

In order to evaluate the potential dependence of the free energy of activation, without knowledge of the structure of the activated complex, it is assumed that the electrical contribution to the standard free energy of the transition state lies between that to the standard free energy of P and that to the standard free energy of R in the rds. The symmetry factor or transfer coefficient, a, for the rds has the same properties as outlined in Sect. 3.1. 

We consider first transport of water and salt across a clay membrane. The salt dissociates into v+ cations of valence z+ and z/ anions of valence 2 We assume that all solutions are ideal, so that the the chemical contributions to the electrochemical potentials within the solution on side i of the membrane are //,vv = piVw + RT In xiw ca p,tVw — RT(xi+ + xt-) and Hi p,V I RT In Xi , where the subscripts (w, ) indicate water and ions, R is the gas constant, T the absolute temperature, p the pressure, xw,x are mole fractions and VW,V are partial molar volumes. There will be an additional electrical contribution Fz , where 0 is the electrical potential and F is the Faraday. Transport through the membrane depends upon the difference in electrochemical potentials across the membrane. If these differences are small, there will be linear transport relations of the form [2]... 

The speed of electron rotation on itself is very weak compared to the light velocity, and so the magnetic contribution to absorption is negligible compared to the electric contribution. Therefore, upon absorption, electronic transitions result from the interaction between electrons and electric field. For this reason, during absorption, a displaced electron preserves the same spin orientation. This is why only the So —Sn transitions are allowed, and the So -> Tn transitions are forbidden. 

The electrical contribution, the electrostatic energy ofthe system, can be expressed as... 

Because of the interactions with neighboring dipoles, which generate the field Ev, the electrical contribution to the free energy contains an additional term when compared to the traditional theory... 

Both theories are based on the calculation of the electrical contribution to the free energy of the region bounded by the macroions. In both theories it is assumed that (a) the motion of the macroions is adiabatically cut off from that of the simple... 

Besides this, however, there is the effect of the excess charge in the interfacial layer, which results in the formation of an electric double layer. Therefore, the thermodynamic treatment of interfacial reactions must be divided into two parts chemical and electric contributions. The chemical reactions are taken into account in the usual way the electric contribution means the electric work done by the particle when transferring through the electric double layer. 

As seen from Equations 1.54-1.56, the intrinsic stability constants of surface reactions are dependent on two factors a chemical and an electric contribution. The chemical contribution is taken into consideration by the mass balance the electric contribution is treated by the charge balance. There are several surface complexation models that mainly differ in the description of the electric double layer that is used to calculate the surface potential, which is done by different double-layer models. These models have been mentioned previously in this chapter. Since, however, the terminology usually used in electrochemistry, colloid chemistry and, especially, in the discussions of surface complexation models is different, they are repeated again ... 

The free energy barrier for the flow of ionic charge across the oxide electrode-electrolyte interface has an electrical contribution and consequently the reaction rate can be formally described by Butler-Volmer-type equations [38]. The cation current density corresponding to the process... 

Thus, the electrical contribution to Afij is nearly 500 times larger than the pressure contribution. Contributions of the pressure term to the chemical potential differences of ions across biological membranes therefore are usually negligible compared with electrical contributions and hence can generally be ignored. 

The electrical work of activation corresponds to a free-energy change. It appears therefore that there is a contribution to the total free energy of activation due to the electrical work done on the ion in making it climb the barrier. This electrical contribution to the free energy of activation is... 

Here a° and yr° are the surface charge (density) and surface potential, respectively. The primes Indicate the variable values when the double layer is reversibly charged from cr° =0 to its final value, cr°. As y/° and a° have the same signs, AG°(el) > 0, so a non-electric contribution is needed to make the overall Gibbs energy change negative, as required for a spontaneous process. For relaxed... 

The standard molar Gibbs energy of adsorption now contains electric and non-electric contributions. Formally, we can write... 

The DLVO theory does not explain either the stability of water-in-oil emulsions or the stability of oil-in-water emulsions stabilized by adsorbed non-ionic surfactants and polymers where the electrical contributions are often of secondary importance. In these, steric and hydrational forces, which arise from the loss of entropy when adsorbed polymer layers or hydrated chains of non-ionic polyether surfactant intermingle on close approach of two similar droplets, are more important (Fig. 4B). In emulsions stabilized by polyether surfactants, these interactions assume importance at very close distances of approach and are influenced markedly by temperature and degree of hydration of the polyoxyethylene chains. With block copolymers of the ethylene oxide-propylene oxide... 

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