Surface-chemical potential, analogy

Equation (55) also suggests the introduction of the name surface-chemical potential for [i thus underlining the facts that fi enters into the fundamental equation for a surface phase in the same manner as enters into the corresponding equation for a bulk phase and that generally /t. In these respects is analogous to the electrochemical potential. However, the counterpart of Eq. (55) in the thermodynamics of electrochemical systems is the well-known relation... 

For any surface morphologies to appear, a surface reconstruction mechanism is required that would transform initially planar surface into a structured one. In epitaxially grown solid films such mechanism is surface diffusion in which atoms jump along the surface from one site to another, driven by the gradient of the surface chemical potential. The surface flux of atoms, j, is given by the analog of the Pick s law js = —MVsPs, where ps is the sur-... 

Equation (10.3) states that (given P, T) the boundary state, s, and the composition N) depend on juh the chemical potential of the components in the system, which has already been illustrated in Figure3-7. The 8( Ag) change in Ag2+(5S after the (/ - ) transformation at 176 °C indicates that point defects are adsorbed at the newly formed internal surfaces introduced into the crystal by this transformation, quite analogous to a Gibbs adsorption isotherm. For (isotropic) internal surfaces, the isotherm is... 

One assumption which is usually made in determining the reaction orders with respect to the various carriers in the semiconductor is that the electrons and holes are essentially in translational equilibrium across the space-charge layer. Their concentrations may be very far from equilibrium with respect to recombination, analogous to having water with an ion product (at room temperature) very different from 10 . However, the very high mobility of holes and electrons tends to make the gradients of their chemical potentials (Fermi levels) quite small. Under such conditions, the hole and electron concentrations at the surface (ps and ns) are related to their concentrations (pi and nj) just to the semiconductor side of the space-charge layer by the equations... 

Surface complexation models (SCM s) provide a rational interpretation of the physical and chemical processes of adsorption and are able to simulate adsorption in complex geochemical systems. Chemical reactions at the solid-solution interface are treated as surface complexation reactions analogous to the formation of complexes in solution. Each reaction is defined in terms of a mass action equation and an equilibrium constant. The activities of adsorbing ions are modified by a coulombic term to account for the energy required to penetrate the electrostatic-potential field extending away from the surface. Detailed information on surface complexation theory and the models that have been developed, can be found in (Stumm et al., 1976 ... 

An iteration scheme is used to numerically solve this minimization condition to obtain Peq(r) at the selected temperature, pore width, and chemical potential. For simple geometric pore shapes such as slits or cylinders, the local density is a function of one spatial coordinate only (the coordinate normal to the adsorbent surface) and an efficient solution of Eq. (29) is possible. The adsorption and desorption branches of the isotherm can be constructed in a manner analogous to that used for GCMC simulation. The chemical potential is increased or decreased sequentially, and the solution for the local density profile at previous value of fx is used as the initial guess for the density profile at the next value of /z. The chemical potential at which the equilibrium phase transition occurs is identified as the value of /z for which the liquid and vapor states have the same grand potential. 

Another problem which obscures the analogy between different phase transitions is the fact that one does not always wish to work with the corresponding statistical ensembles. Consider, for example, a first-order transition where from a disordered lattice gas islands of ordered c(2x2) structure form. If we consider a physisorbed layer in full thermal equilibrium with the surrounding gas, then the chemical potential of the gas and the temperature would be the independent control variables. In equilibrium, of course, the chemical potential jx of subsystems is the same, and so the chemical potential of the lattice gas and that of the ordered islands would be the same, while the surface density (or coverage 9) in the islands will differ from that of the lattice gas. The three-dimensional gas acts as a reservoir which supplies adsorbate atoms to maintain the equilibrium value of the coverage in the ordered islands when one cools the adsorbed layer through the order-disorder transition. However, one often considers such a transition at... 

Primarily, this approach was based on the formal analogy between a first order phase transition and the micellisation. When a new phase of a pure substance is formed the chemical potential of this substance and its concentration in the initial phase do not change with the total content of this substance in the system. A similar situation is observed above the CMC, where the adsorption and the surface tension become approximately constant. In reality variations of these properties are relatively small to be observed by conventional experimental methods. The application of the Gibbs adsorption equation shows that the constancy of the surfactant activity above the CMC follows from the constancy of the surfactant adsorption T2 [13]... 

By analogy with solid particle dispersions, adsorption isotherms can be determined as a function of the chemical potential of the solute. An indirect, but rapid and powerful, example is given by surface tension measurements, which can detect very low amounts of surface-active solute in the form of contaminant. This property is used to determine purity in easily hydrolysable surfactants such as SDS or sodium bis(2-ethylhexyOsulfosuccinate (AOT). A minimum in the surface tension near the CMC directly detects the presence of a surface-active contaminant such as dodecanol or octanol, which will desorb from the interface at high surfactant concentrations. This desorption is the origin of the a priori counter-intuitive increase of surface tension with surfactant content (16). 

Equation 22.6 defines surface tension in terms of Gibbs energy. Borrowing an analogy from chemical potential, we submit that surface tension can also be defined in terms of enthalpy, internal energy, or Helmholtz energy. Write partial derivatives for those definitions. 

Sometimes, it is convenient not to express the spreading pressure in terms of the chemical potentials, but of the surface excesses. This may be done in order to draw an analogy between the equations of state used for adsorbates and for bulk phases [9]. Transition from 0(/r ) to 0(ng) is performed by the use of Eq. (52). For the DR potential, we obtain... 

Here we shall review an application of methods of non-equilibrium thermodynamics to diffusion processes description (Murch and Thorn 1979). The following three types of particles are assumed to exist at the surface adatoms beyond clusters (particles of the sort 2 that form a gas with a small gradient of the chemical potential /i2) condensed adatoms (particles of the sort 1 that form a media with a large gradient of /ii) and snbstrate vacancies (particles of sort 3). Diffusion fluxes of particles of the sort j = 1,2,3 can be written by analogy with (10.2.2) ... 

This is analogous to the swelling of polyacrylamide gels when immersed in water (21], where the lower chemical potential of the pore liquid drives flow into Che gel. During drying, the source of liquid is the interior of the body, so flow is toward the surface. 

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