Constant potential model

Quantum Free-Electron Theory Constant-Potential Model, The simple quantum free-electron theory (1) is based on the electron-in-a-box model, where the box is the size of the crystal. This model assumes that (1) the positively charged ions and all other electrons (nonvalence electrons) are smeared out to give a constant background potential (a potential box having a constant interior potential), and (2) the electron cannot escape from the box boundary conditions are such that the wavefunction if/ is... 

When two surfaces in an electrolyte environment approach one another, their double layers overlap and several situations may arise. In the case of oxide surfaces, the interaction may itself influence the degree of dissociation of surface groups, such that neither the surface potential nor the surface charge remains constant. A charge regulation model may then be more appropriate [23]. The constant charge and the constant potential model allow, however, both upper and lower limits for the strength of the interaction to be estimated. For most of the model calculations performed, it is assumed that... 

Finite difference techniques are used to generate molecular dynamics trajectories with continuous potential models, which we will assume to be pairwise additive. The essential idea is that the integration is broken down into many small stages, each separated in time by a fixed time 6t. The total force on each particle in the configuration at a time t is calculated as the vector sum of its interactions with other particles. From the force we can determine the accelerations of the particles, which are then combined with the positions and velocities at a time t to calculate the positions and velocities at a time t + 6t. The force is assumed to be constant during the time step. The forces on the particles in their new positions are then determined, leading to new positions and velocities at time t - - 2St, and so on. 

Figure 24-5 contains the general data for the two-conductor model [12]. The conductor phase II has a locally constant potential based on a very high conductiv-... 

Whereas the quasi-chemical theory has been eminently successful in describing the broad outlines, and even some of the details, of the order-disorder phenomenon in metallic solid solutions, several of its assumptions have been shown to be invalid. The manner of its failure, as well as the failure of the average-potential model to describe metallic solutions, indicates that metal atom interactions change radically in going from the pure state to the solution state. It is clear that little further progress may be expected in the formulation of statistical models for metallic solutions until the electronic interactions between solute and solvent species are better understood. In the area of solvent-solute interactions, the elastic model is unfruitful. Better understanding also is needed of the vibrational characteristics of metallic solutions, with respect to the changes in harmonic force constants and those in the anharmonicity of the vibrations. 

Zhu and Nakamura treated the two-state time-independent linear potential model (two linear diabatic potentials coupled by a constant diabatic coupling)... 

The formulas derived in the time-independent framework can be easily transferred into the corresponding time-dependent solutions. The formulas in the time-independent linear potential model, for example, provide the formulas in the time-dependent quadratic potential model in which the two time-dependent diabatic quadratic potentials are coupled by a constant diabatic coupling [1, 13, 147]. The classically forbidden transitions in the time-independent framework correspond to the diabatically avoided crossing case in the time-dependent framework. One more thing to note is that the nonadiabatic tunneling (NT) type of transition does not show up and only the LZ type appears in the time-dependent problems, since time is unidirectional. 

The presence of an (applied) potential at the aqueous/metal interface can, in addition, result in significant differences in the reaction thermodynamics, mechanisms, and structural topologies compared with those found in the absence of a potential. Modeling the potential has been a challenge, since most of today s ab initio methods treat chemical systems in a canonical form whereby the number of electrons are held constant, rather than in the grand canonical form whereby the potential is held constant. Recent advances have been made by mimicking the electrochemical model... 

In initial ET rate measurements, both the NB and aqueous phases contained 0.1 M TEAP, enabling measurements to be made with a constant Galvani potential difference across the liquid junction. In these early studies, the concentration of Fc used in the organic phase (phase 2) was at least 50 times the concentration of the electroactive mediator in the aqueous phase which contained the probe UME (phase 1). This ensured that the interfacial process was not limited by mass transport in the organic phase, and that the simple constant-composition model, described briefly in Section IV, could be used. 

Figure 4. A conjoining/disjoining pressure isotherm for the constant- potential and weak overlap electrostatic model.

According to the Sommerfeld model electrons in a metal electrode are free to move through the bulk of the metal at a constant potential, but not to escape at the edge. Within the metal electrons have to penetrate the potential barriers that exist between atoms, as shown schematically below. 

The assumption of ZDO introduces periodicity into an otherwise constant potential free electron scheme, in the same way that the Kronig-Penney potential modifies the simple Sommerfeld model. 

Alternatively, in the literature, the constant capacitance model and the Stern model were used to describe the dependence of the surface charge density on the surface potential. In the constant capacitance model, the surface charge is defined as ... 

Physical model for colloid stability. Net energy of interaction for spheres of constant potential surface for various ionic strengths (1 1 electrolyte) (cf. Verwey and Overbeck). 

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