Constraint electroneutrality

The final equation to complete the electrostatic constraints on the interface is the electroneutrality equation,... 

Electroneutrality may also be implemented by imposing the requirement that F(000) equal the number of electrons in the unit cell. The equation F(000) = ne can be treated as an observation, with a weight sufficient to keep the crystal practically neutral, but sufficiently small such as not to dominate the least-squares treatment. This slack constraint (Pawley 1972) has been applied in electron density analysis by Hirshfeld (1977). 

In all fitting procedures, electroneutrality and electrostatic-moment constraints may be introduced to provide additional observational equations. 

The key argument is rooted in a simple observation concerning any dissociation KL K + L. The individual subunits K and L are in general not electroneutral while they are part of the original host molecule, whereas the corresponding radicals certainly satisfy electroneutrality. A charge neutralization accompanies the transformations K K and L L. This constraint solves our problem. A few examples (Table 12.1) help us understand why this argument is important. 

I wish to respond to Professor Ubbelohde s question regarding what thinness, per se, of biological membranes could be important. For ion movements across membranes as mediated catalytically by carriers or channels, the thinness permits local deviations from the electroneutrality constraint that, for example, enables neutral molecules to carry cations across the membrane as charged species, leaving their counterions behind in the aqueous solutions. This is not possible when the thickness of the system becomes large. 

It has long been known that defect thermodynamics provides correct answers if the (local) equilibrium conditions between SE and chemical components of the crystal are correctly formulated, that is, if in addition to the conservation of chemical species the balances of sites and charges are properly taken into account. The correct use of these balances, however, is equivalent to the introduction of so-called building elements ( Bauelemente ) [W. Schottky (1958)]. These are properly defined in the next section and are the main content of it. It will be shown that these building units possess real thermodynamic potentials since they can be added to or removed from the crystal without violating structural and electroneutrality constraints, that is, without violating the site or charge balance of the crystal [see, for example, M. Martin et al. (1988)]. 

The results of the discussion on the phenomenological thermodynamics of crystals can be summarized as follows. One can define chemical potentials, /jk, for components k (Eqn. (2.4)), for building units (Eqn. (2.11)), and for structure elements (Eqn, (2.31)). The lattice construction requires the introduction of structural units , which are the vacancies V,. Electroneutrality in a crystal composed of charged SE s requires the introduction of the electrical unit, e. The composition of an n component crystal is fixed by n- 1) independent mole fractions, Nk, of chemical components. (n-1) is also the number of conditions for the definition of the component potentials juk, as seen from Eqn. (2.4). For building units, we have (n — 1) independent composition variables and n-(K- 1) equilibria between sublattices x, so that the number of conditions is n-K-1, as required by the definition of the building element potential uk(Xy For structure elements, the actual number of constraints is larger than the number of constraints required by Eqn. (2.18), which defines nk(x.y This circumstance is responsible for the introduction of the concept of virtual chemical potentials of SE s. 

The basis of defect thermodynamics is the concept of regular and irregular SE s and the constraints which crystallography and electroneutrality (in the case of ionic crystals) impose on the derivation of the thermodynamic functions. Thermodynamic potential functions are of particular interest, since one derives the driving forces for the chemical processes in the solid state from them. 

Activity effects. The exchange of trace ions in solution with others in the polymer film might, simplistically, be expected to lead to a linear uptake/solution concentration relationship. Unfortunately, this is seldom the case. The thermodynamic restraint is that of electrochemical potential. Thus electroneutrality is not the sole constraint on the ion exchange process. A second thermodynamic requirement is that the activity of mobile species in the polymer and solution phases be equal. (Temporal satisfaction of these two constraints is discussed below, with reference to Figure 4.) The rather unusual, high concentration environment in the polymer film can lead to significant - and unanticipated - activity effects (8). 

Equation (7) was combined with appropriate chemical reaction rate expressions to yield a set of coupled differential equations expressing rates of change in the dissolved O3 and S(IV) concentrations. The equations were then solved numerically with the usual constraints of electroneutrality and the appropriate ionic equilibria given in Table I. 

For charged sites, it is the asymptotics of the site-site Coulomb interaction potential, and the constraint (25) ensures the condition of local electroneutrality is obeyed. 

Three examples are given here to demonstrate various capabilities of DDAPLUS. In the first example, DDAPLUS is used to solve a system of ordinary differential equations for the concentrations in an isothermal batch reactor.In the second example, the same state equations are to be integrated to a given time limit, or until one of the state variables reaches a given limit. The last example demonstrates the use of DDAPLUS to solve a differential-algebraic reactor problem with constraints of electroneutrality and ionization equilibria. 

The flux condition given by equation 6.15 together with the constraints imposed by the mass balance, electroneutrality, and no net charge transfer gives a non-linear differential rate equation that is only amenable to computerized numerical solutions. Often sufficiently... 

From equation 6.1 and incorporating the constraints demanded by no net charge transfer, electroneutrality, and material mass balance the analytical rate equations for the infinite volume and finite volume boundary conditions are represented by equations 6.19 and 6.20 respectively ... 

In practice, in describing a binary mixture of charged particles, another set of dynamic variables is widely used, namely, instead of partial densities nk,a or the set (43), the mass density pk and the charge density qk are utilized. However, it should be mentioned that due to the electroneutrality constraint the charge density qk can be simply connected with the mass-concentration density xk, introduced above. In particular, one has,... 

Even when the system is dilute, the diffusing ionic species are coupled to one another in a very interesting manner. This coupling arises out of the constraint imposed by the electroneutrality condition. Equation 2.4.3 can be differentiated to give... 

The important features of mixed ion diffusion are brought out very clearly in the calculations. The rapidly diffusing ions are slowed down by the electrostatic pull being exerted on them by the more slowly diffusing Cl ions (Fig. 2.10). At the same time the Cl ions are accelerated by the ions. The Ba ions, which have a low diffusion coefficient already, diffuse even slower because of the constraint of electroneutrality. ... 

This is the most widely used model for ion-exchange kinetics (Schlogl and Helfferich, 1957 Plesset et al., 1958). Combining the Nernst-Planck equation (Eq. [2]) with the constraints of electroneutrality and zero net charge transfer yields... 

In order to determine the electrical potential, the electroneutrality constraint is also imposed ... 

Equations (9), (20), and (21), and the boundary conditions define a nonlinear and coupled system of partial differential equations, solved by an FVM. The equations were linearized around a guessed value. The guessed values were updated iteratively to convergence before executing the next time step. Since the electroneutrality constraint tightly couples the potential and concentration fields, the discretized sets of algebraic equations at each node point were solved simultaneously. Attempts were made to employ a sequential solver in which the electrical field was assumed for determination of the concentration of each species. In this way, the concentration fields appear decoupled and could be determined easily with a commercial, convection-diffusion solver. A robust method for converging upon the correct electrical field was, however, not found. 

Pincus [212] has studied polyelectrolyte chains the monomers of which are in 0 solvent conditions, i.e., the excluded volume is zero. This assumption is justified by the fact that most polyelectrolytes are rather hydrophobic and their solubility in water is only due the presence of charges along the chains. The author shows that electroneutrality is locally achieved within the brush, providing that the fraction of charged monomers p and the densities of adsorption points l/d2 are not too low. Actually, the high concentrations of charged monomers and counterions lead to a Debye length much lower than the brush thickness. Consequently, the only relevant electrostatic contribution to the forces within the brush is the osmotic pressure of the counterions, which behave as a constraint ideal gas with a pressure equal to pckT, in the absence of added salt. Hence the equilibrium thickness of the brush results from the balance between the elastic force and the counterions ... 

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