Site fraction

Rigorous treatment of the model assumed would dictate the use of a surface fraction, or "site fraction, in place of the volume fraction Vi. The site fraction of the solvent may be defined as the number of intermolecular site locations adjacent to solvent molecules divided by the total number of such sites for both solvent and polymer molecules. Thus, if the solvent consists of a single segment, its site fraction is... 

Here the site fractions of the defects in eq. (9.102) are expressed in terms of the oxygen non-stoichiometry parameter 8. 

In these equations gv is the change in Gibbs free energy on taking one atom from a normal lattice site to the surface of the crystal and (gt + gv) the change when an atom is taken from a normal lattice site to an interstitial site, both at constant temperature and pressure. cr denotes a site fraction of species r on its sublattice, and is the chemical potential of a normal lattice ion in the defect-free crystal. 

Definition of site fractions. The multiple sublattice model is an extension of earlier treatments of the two-sublattice models of Hillert and Steffansson (1970), Harvig (1971) and Hillert and Waldenstrom (1977). It allows for the use of many sublattices and concentration dependent interaction terms on these sublattices. To woiic with sublattice models it is first necessary to define what are known as site fractions, y. These are basically the fiactional site occupation of each of the components on the various sublattices where... 

Equation (5.30) holds for the simple case of a phase with the formula A, B)i(C, D)i. But for more complex phases the function for the Gibbs reference energy surface may be generalised by arranging the site fractions in a (f + c) matrix if there are I sublattices and c components. 

Finally some site fraction dependence to these parameters can be added such that... 

In some thermodynamic models there are also potential minima associated with different site occupations, even though the composition may not vary, e.g., a phase with an order/disorder transformation. This must be handled in a somewhat different fashion and the variation in Gibbs energy as a function of site fraction occupation must be examined. Although this is not, perhaps, traditionally recognised as a miscibility gap, there are a number of similarities in dealing with the problem. In this case, however, it is the occupation of sites which govern the local minima and not the overall composition, per se. 

We wish here to obtain the thermodynamic equations defining the liquidus surface of a solid solution, (At BB)2, ). It is assumed that the A and atoms occupy the sites of one sublattice of the structure and the C atoms the sites of a second sublattice. For the specific systems considered here Sb and play the role of C in the general formula above. It is also assumed that the composition variable is confined to values near unity so that the site fractions of atomic point defects is always small compared to unity. This apparently is the case for the solid solutions in the two systems considered. Then it can be shown theoretically (Brebrick, 1979), as well as experimentally for (Hgj CdJ2-yTe)l(s) (Schwartz et al, 1981 Tung et al., 1981b), that the sum of the chemical potentials of A and C and that of and C in the solid are independent of the composition variable y ... 

As a simple example, consider the case of the adsorption of a gas-phase molecule, A, on a surface. The surface is composed of either open sites or adsorbed molecules. In this formalism, there are two surface species one corresponding to the adsorption location, the open site, designated O(s), and the adsorbed molecule, A(s). The site fractions of O(s) and A(s) surface species must sum to unity. There is one surface phase in this case. In this trivial example, such overhead and formal definitions are unnecessarily complicated. However, in complex systems involving many surface phases and dozens of distinct surface species, the discipline imposed by the formalism helps greatly in bookkeeping and in ensuring that the fundamental conservation laws are satisfied. 

The composition of surface phases can be specified in terms of site fractions Zk. This array is of total length Ks. It is composed of Ns (the total number of surface phases) subunits of the site fractions of the species in each surface phase n. The site fractions in each phase are normalized ... 

The site fractions of the Si-containing species are one site occupied by SiH4 out of a total of 32 sites, and two sites out of 32 occupied by Si2H4. The site fraction of open sites is 29/32 = 0.906. As is seen in Eq. 11.8, it is necessary to divide the site fraction of each species by the site occupancy number ok to convert to a molar concentration. The concentration of SiH4 (number per unit area) is equal to that of Si2H4. 

A steady-state analysis of reaction 11.25 yields an expression for the site fraction of adsorbed A(s) ... 

Now consider the form of the BET adsorption isotherm written in Eq. 11.59. If multilayer adsorption were not possible, then Km would be zero. The adsorbed site fraction from Eq. 11.59 becomes... 

If the surface fraction of Si(s) were unity, then a fraction y of the collisions of SiH2 with the surface result in a reaction. However, for Si(s) coverages less than 1, the reaction rate decreases in proportion with the site fraction of Si(s). Any collisions of SiH2 with another surface species are not addressed by the reaction as written above. 

The time-rate-of-change of surface species k due to heterogeneous reaction is given by Eq. 11.102. As discussed above, the effects of surface chemistry must be accounted for as boundary conditions on gas-phase species through flux-matching conditions such as Eq. 11.123. For a transient simulation, a differential equation for the site fraction Zk of surface species k can be written... 

In a steady-state calculation, it must be the case that the surface species concentrations (or site fractions) are not changing with time, that is,... 

Arrhenius parameters for the rate constants written in the form k = AT exp(-E/RT). The units of A are given in terms of moles, cubic meters, and seconds. E is in kJ/mol. Coverage of surface species (e.g., [O(s)]) specified as a site fraction. 

For a Cd(g) gas-phase mole fraction of 0.2, plot the predicted Open(s) site fraction and the growth rate (in /xm/min) as a function of Te2(g) mole fraction (over the range 0 to 0.5). Write a paragraph discribing the physical origin of the shapes of the theoretical curves. 

The time rate of change of CO(s) coverage (site fraction) was written as d9co... 

From the reaction mechanism above, derive an expression for the adsorbed ammonia site fraction, 6 = [NH3(ad)]/T. Make the simplifying assumption that kip(NH3) + k-1 >> 2[Si(s)] this is equivalent to assuming that reaction 1 is at equilibrium, and that reaction 2 makes a negligible difference in the adsorbed ammonia concentration. Determine the NH3(ad) site fraction for an ammonia pressure of 124 Pa. 

The surface species (denoted by the asterisk) represent site fractions of individual species, which are constrained as A + B + C = 1. 

The surface composition, usually represented by site fractions Z, must adjust itself to be consistent with the local gas-phase composition, temperature, and the heterogeneous reaction mechanism. When the surface composition is represented by site fractions, the definition requires that... 

The site-fraction constraint (Eq. 16.64) means that all the s in Eq. 16.63 are not independent. Therefore only Ks — 1 of Eq. 16.63 are solved. Solving the plug-flow problem requires satisfying the algebraic constraints represented by Eqs. 16.63 and 16.64 at every point along the channel surface. The coupled problem is posed naturally as a system of differential-algebraic equations. 

The net rate of production of gas-phase species by heterogeneous reaction participates in the boundary condition through the surface mass balance (Eq. 17.24). However, since the heterogeneous production rates of gas-phase species depends on the surface composition, the transient state of the surface composition must also be determined. As discussed in Section 11.10 the surface site fractions Z are found from the solution of... 

The baseline reactor conditions in the following reactor analysis are susceptor temperature Ts = 1273 K, inlet temperature Tm = 333 K, reactor pressure p = 400 mTorr, gas velocity through the inlet manifold Vin = 100 cm/s, and the gap between inlet and susceptor L = 1 cm. Incoming gas-mixture mole fractions (e.g., from a gas-cylinder) are TEOS 0.25 and N2 (carrier gas) 0.75. You may use the files teos. gas and teos. surf for the gas-phase and surface reaction mechanisms. (Hint You may need the following initial guesses at the surface species site fractions SiG3(OH) 0.98, SiGsE 0.02, SiG(OH)2E 0.001. More details on the surface reaction mechanism and nomenclature are found in Ref. [69].)... 

The main difficulty with the first mode of oxidation mentioned above is explaining how the cation vacancies that arrive at the metal/oxide interface are accommodated. This problem has already been addressed in Section 7.2. Distinct patterns of dislocations in the metal near the metal/oxide interface and dislocation climb have been invoked to support the continuous motion of the adherent metal/oxide interface in this case [B. Pieraggi, R. A. Rapp (1988)]. If experimental rate constants are moderately larger than those predicted by the Wagner theory, one may assume that internal surfaces such as dislocations (and possibly grain boundaries) in the oxide layer contribute to the cation transport. This can formally be taken into account by defining an effective diffusion coefficient Del( = (1 -/)-DL+/-DNL, where DL is the lattice diffusion coefficient, DNL is the diffusion coefficient of the internal surfaces, and / is the site fraction of cations located on these internal surfaces. 

---