Surface terms

While smooth pair potentials are the rule in the literature, surface terms have traditionally been discontinuous the only potential using smooth surface terms seems to appear in Lund et al. [19], where the surface term is a function of a smooth approximation to the number of neighbors of a Ca atom.)... 

Fig. 1. Aquifers and monitoring wells where denotes the well screen and Hthe water-filled space in the monitoring well. (—) denotes the water table level, (—), the potentiometric surface, and ( ) the ground surface. Terms are discussed in text.

Similar surface terms are commonly used in the description of interfacial systems. They correspond to the idea of a localized interaction with the wall. This has been used in the description of adsorption (see, e.g., [29]), wetting phenomena [30] and interfacial criticality [31],... 

The validity of mean field theory for N —y oo has striking consequences for the initial stages of phase separation. " In a metastable state slightly inside the coexistence curve, the nucleation free energy barrier is due to spherical droplets with a radius R The free energy excess of a droplet is written in terms of bulk and surface terms " "... 

If 5v //v /coex is not small, the simple description Eq. (14) in terms of bulk and surface terms no longer holds. But one can find AF from Eq. (5) by looking for a marginally stable non-uniform spherically symmetric solution v /(p) which leads to an extremum of Eq. (5) and satisfies the boundary condition v /(p oo) = v(/ . Near the spinodal curve i = v /sp = Vcoex /a/3 (at this stability limit of the metastable states both and S(0) diverge) one finds "... 

Ostwald ripening is driven by the positive surface energy, the specific, per gram amount of which can be lessened with particle growth. The change in Gibbs free energy when a particle forms is composed of two terms, a bulk term and a surface term as... 

Here av — 15.5 MeV represents a constant term in B.E. per nucleon, as — 16.8 MeV provides a surface term allowing for a reduced contribution to B.E. from... 

For electrons in a metal the work function is defined as the minimum work required to take an electron from inside the metal to a place just outside (c.f. the preceding definition of the outer potential). In taking the electron across the metal surface, work is done against the surface dipole potential x So the work function contains a surface term, and it may hence be different for different surfaces of a single crystal. The work function is the negative of the Fermi level, provided the reference point for the latter is chosen just outside the metal surface. If the reference point for the Fermi level is taken to be the vacuum level instead, then Ep = —, since an extra work —eoV> is required to take the electron from the vacuum level to the surface of the metal. The relations of the electrochemical potential to the work function and the Fermi level are important because one may want to relate electrochemical and solid-state properties. 

Considering the contribution of the solvent and surface terms to the total free energy change, it is apparent that the enhanced selectivity in the presence of e.g. ethylenediamine corresponds in sign to variations in the solution term and is (in part) due to smaller AG values of hydration of the complex cations. This is exemplified for the Ca-Cu and Ag-Cu cases in the presence and absence of ethylenediamine by the equations ... 

The real potential, a , of electrons in metals, as shown in Eqn. 2-4, comprises the electrostatic surface term, - ex, due to the surface dipole and the chemical potential term, M., determined by the bulk property of metal crystals. In general, the electrostatic surface term is greater the greater the valence electron density in metals whereas, the chemical potential term becomes greater the lower the valence electron density in metals. 

Figure 2-11 compares the observed work function, 4>, with that calculated based on the jeUium model as a function of the electron density, n.,in metals here, n, is represented in terms of the Wigner-Seitz radius which is inversely proportional to the cube root of n.. The chemical potential term (p. = —1.5 to-2.5 eV) predominates in the work function of metals of low valence electron density, while on the contrary the surface term (- e x = -0-1 -5.0 eV) predominates for... 

The unitary real potential, ay., of the surface metal ion consists of the chemical potential, Py, and the electrostatic surface term e x as shown in Eqn. 3-7 ... 

The combined energy is a weighted sum of pair and surface terms ... 

In much of the above analysis, the relative magnitude of the surface and bulk contribution to the nonlinear response has not been addressed in any detail. As noted in Section 3.1, in addition to the surface dipole terms of Eq. (3.9), there are also nonlocal electric-quadrupole-type nonlinearities arising from the bulk medium. The effective polarization is made of a combination of surface nonlinear polarization, PNS (2co) (Eq. (3.9)), and bulk nonlinear polarization (Eq. (3.8)) which contains bulk terms y and . The bulk term y is isotropic with respect to crystal rotation. Since it appears in linear combination with surface terms (e.g. Eq. (3.5)), its separate determination is not possible under most circumstances [83, 129, 130, 131]. It mimics a surface contribution but its magnitude depends only upon the dielectric properties of the bulk phases. For a nonlinear medium with a high index of refraction, this contribution is expected to be small since the ratio of the surface contribution to that from y is always larger than se2(2co)/y. The magnitude of the contribution from depends upon the orientation of the crystal and can be measured separately under conditions where the anisotropic contribution of vanishes. 

A surface term that reflects the opposing tendencies of the molecules to crowd closely together minimizing the unfavorable hydrocarbon-water interactions and to spread apart as a result of the electrostatic (head group) repulsion, hydration and steric hindrance,... 

With increasing the hole concentration the Fermi surface of the t-J model is transformed to a rhombus centered at Q [16], This result is in agreement with the Fermi surface observed in La2-xSrxCuC>4 [19] [however, to reproduce the experimental Fermi surface terms describing the hole transfer to more distant coordination shells have to be taken into account in the kinetic term of Hamiltonian (1)]. For such x another mechanism of the dip formation in the damping comes into effect. The... 

The lamellar habit adopted by crystalline polymers adds surface terms to the specific Gibbs function (chemical potential), most importantly the fold surface free energy, ae, which contributes 2ae/Xg for a lamella of thickness k and crystalline density q. In consequence melting points are lowered from T, for infinite thickness, to Tm according to the Hoffman-Weeks equation... 

Now integrate over the whole xyt domain. Note that all terms except the first and the dz (implicit in V.) are derivatives that give rise to surface terms after integration over x,y,t. Since we consider spatially and temporally localized pulse-like solutions, these terms vanish. The only surviving derivative will be dz from V. ... 

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