Surface potential equations

What then, is the expression for gdipole in the charge region between these extreme cases We know that in this in-between region it is more likely that there will be a mixture of flip-up and flop-down water molecules. Not all of the monomers are aligned in the same direction, and their individual contribution to the surface potential equations should be considered. And what about the dimers Are they not also present... 

The energy of the electric double layer is directly dependent on the square of the surface potential (Equation 4) and the observed increase of the potassium oleate alcohol ratio should enhance the stability of the inverse micelle. The stability of the inverse micelle is not the only determining factor. Its solution with a maximal amount of water is in equilibrium with a lamellar liquid crystalline phase (7) and the extent of the solubility region of the inverse micellar structure depends on the stability of the liquid crystalline phase. 

Since the complex formation model can estimate the net surface charge, op, and thus in turn the surface potential, (equation v. Table 14.3), the colloid stability and pHp c can, at least in principle, be predicted (see Figure 14.8b). 

At low surface potentials, Equation 4.5 reduces to Equation 4.2. Second, an inner layer exists because ions are not really point charges and an ion can only approach a surface to the extent allowed by its hydration sphere. The Stern model specifically incorporates a layer of specifically adsorbed ions bounded by a plane termed the Stern plane (see Figure 4.2). In this case, the potential changes from at the surface, to i/r(5) at the Stern plane, to i/r = 0 in bulk solution. 

S. Ohki, Membrane Potential of Squid Axons Comparison between the Goldman-Hodgkin-Katz Equation and the Diffusion/Surface Potential Equation, in Charge and Field Effects in Biosystems (M. J. Allen and P. N. R. Usherwood, eds.), pp. 147-156, Abacus Press, Tunbridge Wells (1984). 

FIGURE 4.19 Qualitative comparison of the intermolecular potential (Equation 4.26) and the molecule-surface potential (Equation 4.27). 

On compression, a gaseous phase may condense to a liquid-expanded, L phase via a first-order transition. This transition is difficult to study experimentally because of the small film pressures involved and the need to avoid any impurities [76,193]. There is ample evidence that the transition is clearly first-order there are discontinuities in v-a plots, a latent heat of vaporization associated with the transition and two coexisting phases can be seen. Also, fluctuations in the surface potential [194] in the two phase region indicate two-phase coexistence. The general situation is reminiscent of three-dimensional vapor-liquid condensation and can be treated by the two-dimensional van der Waals equation (Eq. Ill-104) [195] or statistical mechanical models [191]. 

The equations are transcendental for the general case, and their solution has been discussed in several contexts [32-35]. One important issue is the treatment of the boundary condition at the surface as d is changed. Traditionally, the constant surface potential condition is used where po is constant however, it is equally plausible that ag is constant due to the behavior of charged sites on the surface. 

Equations (5,61) and (5.62) can be used to derive a pressure potential equation applicable to thin-layer flow between curved surfaces using the following procedure. In a thin-layer flow, the following velocity boundary conditions are prescribed ... 

The comparison of flow conductivity coefficients obtained from Equation (5.76) with their counterparts, found assuming flat boundary surfaces in a thin-layer flow, provides a quantitative estimate for the error involved in ignoring the cui"vature of the layer. For highly viscous flows, the derived pressure potential equation should be solved in conjunction with an energy equation, obtained using an asymptotic expansion similar to the outlined procedure. This derivation is routine and to avoid repetition is not given here. 

Electrostatic Interaction. Similarly charged particles repel one another. The charges on a particle surface may be due to hydrolysis of surface groups or adsorption of ions from solution. The surface charge density can be converted to an effective surface potential, /, when the potential is <30 mV, using the foUowing equation, where -Np represents the Faraday constant and Ai the gas law constant. 

Nakagaki1U) has given a theoretical treatment of the electrostatic interactions by using the Gouy-Chapman equation for the relation between the surface charge density oe and surface potential /. The experimental data for (Lys)n agrees very well with the theoretical curve obtained. 

Because there is no depletion layer between the substrate and the conducting channel, the equations of the current-voltage curves are in fact simpler in the TFT than in the MISFET, provided the mobility can still be assumed constant (which is not actually the case in most devices, as will be seen below). Under such circumstances, the charge induced in the channel is given, in the case of an /l-channel, by Eq. (14.23). In the accumulation regime, the surface potential Vs(x) is the sum of two contributions (i) the ohmic drop in the accumulation layer, and (ii) a term V(x) that accounts for the drain bias. The first term can be estimated from Eqs. (14.15), (14.16) and (14.19). In the accumulation regime, and provided Vx>kT/q, the exponential term prevails in Eq. (14.16), so that Eq. (14.15) reduces to... 

Amplitude equations and fluctuations during passivation, 279 Analytical formulae for microwave frequency effects, accuracy of, 464 Andersen on the open circuit scrape method for potential of zero charge, 39 Anisotropic surface potential and the potential of zero charge, (Heusler and Lang), 34... 

Henry [ 157] solved the steady-flow continuity and Navier-Stokes equations in spherical geometry, neglecting inertial terms but including pressure and electrical force terms, coupled with Poisson s equation. The electrical force term in Henry s analysis consisted of the sum of the externally applied electric field and the field due to the double layers. His major assumptions are low surface potential (i.e., potentials less than approximately 25 mV) and undistorted double layers. The additional parameter ku appearing in the Henry... 

The surface potential change, besides the surface pressure, is the most important quantity describing the surface state in the presence of an adsorbed substance. However, the significance in molecular terms of this very useful experimental parameter still remains unclear. It is common in the literature to link A% with the properties of the neutral adsorbate by means of the Helmholtz equation" ... 

Equations (25) to (29) concern the case of neutral adsorbates, where there is no ionic double layer to contribute to the surface potential. In the case of charged (i.e., ionic) adsorbates, the measured potential consists of two terms. The first term is due to dipoles oriented at the interface, which may be described by the above formulas, and the second term presents the potential of the ionic double layer at the interface from the aqueous... 

According to the above equations, the surface potential is the difference between the real and chemical potentials of charged particles dissolved in the liquid phase. 

Equation (9.2) can be used to calculate the metal s surface potential. The value of the electron work function X can be determined experimentally. The chemical potential of the electrons in the metal can be calculated approximately from equations based on the models in modem theories of metals. The accuracy of such calculations is not very high. The surface potential of mercury determined in this way is roughly -F2.2V. 

The form of the kinetic equation depends on the way in which the surface potential X varies with electrode potential E. When the surface potential is practically constant, the first factor in Eq. (14.24) will also be constant, and the potential dependence of the reaction rate is governed by the second factor alone. The slope b of the polarization curve will be RT/ F (i.e., has the same value as that found when the same reaction occurs at a metal electrode). When in another case a change in electrode potential E produces an equally large change in surface potential (i.e., E = x + const), while there is practically no change in interfacial potential. Then Eq. (14.24) changes into... 

It should be recalled that the term surface potential is used quite often in membranology in rather a different sense, i.e. for the potential difference in a diffuse electric layer on the surface of a membrane, see page 443.) It holds that 0 = 0 + X (this equation is the definition of the inner electrical potential 0). Equation (3.1.2) can then be written in the form... 

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