The electroneutrality constraint

Since a crystal is neutral, the total electron population must equal the sum of the nuclear charges of the constituent atoms, or 

Several ways to introduce the constraint into the refinement are discussed in the following sections. 

In a convenient method, due to Hamilton (1964), the Lagrangian multipliers representing the constraint are algebraically eliminated from the least-squares expressions. The linear constraints are defined as 

The variance-covariance matrix with the constraint is modified in a similar manner, according to the expression 

Equations (4.40) and (4.41) are easily implemented in an existing least-squares program and give both the constrained and the unconstrained results in a single refinement cycle. However, the method fails if the unconstrained refinement corresponds to a singular matrix, as would be the case, for example, if all population parameters, including those of the core functions, were to be refined in addition to the scale factor k. 

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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. 

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,... 

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. 

Electroneutrality in the bulk plasma If one is not interested in resolving length scales of the order of the Debye length, the electroneutrality constraint in the bulk plasma is applicable. 

This is an excellent assumption for the plasmas of interest since the Debye length is exceedingly small (10s of pm) compared to the reactor dimensions. Of course, the electroneutrality constraint can t be applied in the sheath, where the Poisson equation... 

The boundary values of the carrier concentrations, in this limiting case, may be calculated in straightforward fashion from the corresponding partial pressures of CO2, the reaction equilibria, and the electroneutrality constraint. 

Feldberg SW (2000) On the dilemma of the use of the electroneutrality constraint in electrochemical calculations. Electrochem Commun 2 453 56... 

Here, we imposed the electroneutrality constraint, whereby the summation vanishes. We also... 

The minimum principle for il leads to a scheme in which the constants a , related with the response to the external potential, are not affected by the electroneutrality constraint. Now, since ft has the dimension of an energy and contains the electrostatic energy, X)ij b9i9j one can think that the minimum of the functional... 

Consider the special case of an anionic gel matrix with fixed charges of valence (-1) at a concentration c g in a solution of 1 1 electrolyte. If the cations of the salt are the same as the counterions of the gel, the electroneutrality constraints for the gel and for the solution become ... 

Na ions bound to the polyion per charged group. One could even get carried away and interpret the dependence of Equation (25) on + as a decrease in the number of bound Na" due to the competition of the Mg " ions with the Na" for sites on the polyion. Such an interpretation, however, is completely wrong the effects are due to the diffuse Debye-Hiickel atmosphere. Actually, in this case, the effect persists even if ( = 0, that is, even when all activity coefficients are unity the electroneutrality constraint given by Equation (12) is sufficient to cause an asymmetric distribution of small ions. 

The electroneutrality constraint for the concentration, Cm+p> of the fixed polymer sites, M+, and the concentrations of mobile ions in the polymer phase, C+, A , leads to... 

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