Coexistence coefficients

In simple cases, the coexistence coefficient matrix (12.75) and phase boundary equation (12.78) reduce to forms that were previously recognized. 

The leading tenn in equation (A2.5.17) is the same kind of parabolic coexistence curve found in section A2.5.3.1 from the van der Waals equation. The similarity between equation (A2.5,5t and equation (A2.5.17) should be obvious the fomi is the same even though the coefficients are different. 

Povodyrev et aJ [30] have applied crossover theory to the Flory equation ( section A2.5.4.1) for polymer solutions for various values of N, the number of monomer units in the polymer chain, obtaining the coexistence curve and values of the coefficient p jj-from the slope of that curve. Figure A2.5.27 shows their comparison between classical and crossover values of p j-j for A = 1, which is of course just the simple mixture. As seen in this figure, the crossover to classical behaviour is not complete until far below the critical temperature. 

Motivated by a puzzling shape of the coexistence line, Kierlik et al. [27] have investigated the model with Lennard-Jones attractive forces between fluid particles as well as matrix particles and have shown that the mean spherical approximation (MSA) for the ROZ equations provides a qualitatively similar behavior to the MFA for adsorption isotherms. It has been shown, however, that the optimized random phase (ORPA) approximation (the MSA represents a particular case of this theory), if supplemented by the contribution of the second and third virial coefficients, yields a peculiar coexistence curve. It exhibits much more similarity to trends observed in... 

The theory of seaweed formation does not only apply to solidification processes but in fact to the completely different phenomenon of a wettingdewetting transition. To be precise, this applies to the so-called partial wetting scenario, where a thin liquid film may coexist with a dry surface on the same substrate. These equations are equivalent to the one-sided model of diffusional growth with an effective diffusion coefficient which depends on the viscosity and on the thermodynamical properties of the thin film. 

When the three coefficients oc12, oc13, and a23 are known, the coexistence curve can be found by simultaneous solution of Eqs. (119) and (120). A numerical iterative technique given by Hennico and Vermeulen (HI) was used by Balder for performing these calculations with a digital electronic computer. 

It should finally be noted that the amount of the neutral and zweitterionic forms of a compound in solution is determined by its tatuomeric equilibrium constant, defined as Kz = cz/cn. Therefore, the neutral species and the zwitterion coexist around the isoelectric pH, and membrane permeation is conditioned by Kx and by the partition coefficient values of both tautomers. 

Taura H, Yurimoto H, Kurita K, Sueno S (1998) Pressure dependence on partition coefficients for trace elements between olivine and the coexisting melts. Phys Chem Min 25 469-484 Taura H, Yurimoto H, Kato T, Sueno S (2001) Trace element partitioning between silicate perovskites and ultracalcic melt. Phys Planet Earth Int 124 25-32... 

The crucial question is at what value of <)> is the attraction high enough to induce phase separation De Hek and Vrij (6) assume that the critical flocculation concentration is equivalent to the phase separation condition defined by the spinodal point. From the pair potential between two hard spheres in a polymer solution they calculate the second virial coefficient B2 for the particles, and derive from the spinodal condition that if B2 = 1/2 (where is the volume fraction of particles in the dispersion) phase separation occurs. For a system in thermodynamic equilibrium, two phases coexist if the chemical potential of the hard spheres is the same in the dispersion and in the floe phase (i.e., the binodal condition). 

By extension one may say that the power laws (5-7) which determine the magnitude of the linear and nonlinear optical coefficients are consequences of this strong electron-lattice coupling. We now make the conjecture that the time response of these coefficients is severely affected by the dynamics of the electron-lattice coupling in conjugated chains when two or more resonant chemical structures can coexist this is the case for many of the organic chains of Figure 2. 

Also attracting growing attention is the phase coexistence curve characteristic of ionic systems it plays a role in some ionic solution phenomena, although examples in aqueous solutions are not known at this time. Other new features are the intense concentration dependence - at low concentration - of certain of the Hamed coefficients that characterize mixed electrolyte solutions and the evidence for a solvent-separated state of the hydrophobic bond, the attractive force between hydrophobic ions, even those of zero charge, in water. 

From the formation reaction of protonic defects in oxides (eq 23), it is evident that protonic defects coexist with oxide ion vacancies, where the ratio of their concentrations is dependent on temperature and water partial pressure. The formation of protonic defects actually requires the uptake of water from the environment and the transport of water within the oxide lattice. Of course, water does not diffuse as such, but rather, as a result of the ambipolar diffusion of protonic defects (OH and oxide ion vacancies (V ). Assuming ideal behavior of the involved defects (an activity coefficient of unity) the chemical (Tick s) diffusion coefficient of water is... 

Oka Y. and Matsumoto T. (1974). Study on the compositional dependence of the apparent partitioning coefficient of iron and magnesium between coexisting garnet and clinopyro-xene solid solutions. Contrib. Mineral. Petrol, 48 115-121. 

Raheim A. and Green D. H. (1974). Experimental determination of the temperature and pressure dependence of the Ee-Mg partition coefficient for coexisting garnet and clinopyro-xene. Contrib. Mineral Petrol, 48 179-203. 

Several models for diffusive transport in and among minerals have been discussed in the literature one is the fast grain boundary (FGB) model of Eiler et al. (1992, 1993). The FGB model considers the effects of diffusion between non-adjacent grains and shows that, when mass balance terms are included, closure temperatures become a strong function of both the modal abundances of constituent minerals and the differences in diffusion coefficients among all coexisting minerals. 

The left-hand side is the forward piezoelectric coefficient, in units of coulombs/newton. The right-hand side is the reverse piezoelectric coefficient, in units of meters/volt. They are equal. The coexistence of forward and reverse piezoelectric effects provides a simple method to test the piezodrive used in STM, which is discussed in Section 9.6. 

The cadmium electrodeposition on the solid cadmium electrode from the sulfate medium was investigated [217]. The following kinetic parameters were obtained cathodic transfer coefficient a = 0.65, exchange current density Iq = 3.41 mA cm , and standard rate constant kg = 8.98 X 10 cm s . The electrochemical deposition of cadmium is a complex process due to the coexistence of the adsorption and nucleation process involving Cd(II) species in the adsorbed state. 

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