Network equations

Attack by alkali solution, hydrofluoric acid and phosphoric acid A common feature of these corrosive agents is their ability to disrupt the network. Equation 18.1 shows the nature of the attack in alkaline solution where unlimited numbers of OH ions are available. This process is not encumbered by the formation of porous layers and the amount of leached matter is linearly dependent on time. Consequently the extent of attack by strong alkali is usually far greater than either acid or water attack. 

Let the network specifications and parameters be collectively denoted by u and the state variables be denoted by x as before. Then we may represent the steady-state pipeline network equations as... 

For a moderately crossllnked network, equation (13) predicts a declining stress with lamellae formation from the amorphous melt. A stress Increase can be achieved with this model only by reorientation of the chain axis to the directions perpendicular (or nearly so) to the stress direction. If then this model is suitable for lightly crystalline materials, its behavior is in good accord with the observations of Luch and Yeh (6) on stretched natural rubber networks. They reported simultaneous lamellae formation and declining network stress. 

These network equations can be solved for the unknown driving force (across each branch) or the unknown flow rate (in each branch of the net-... 

Equations (3.3) and (3.4) have become known respectively as the valence sum rule and the loop, or equal valence, rule, and are known collectively as the network equations. Equation (3.4) represents the condition that each atom distributes its valence equally among its bonds subject to the constraints of eqn (3.3) as shown in the appendix to Brown (1992a). The two network equations provide sufficient constraints to determine all the bond valences, given a knowledge of the bond graph and the valences of the atoms. The solutions of the network equations are called the theoretical bond valences and are designated by the lower case letter 5. Methods for solving the network equations are described in Appendix 3. ... 

In many compounds, the experimental bond valences, S, and the theoretical bond valences, s, are both found to be equal to the bond fluxes, <1>, within the limits of experimental uncertainty. This is an empirical observation that is not required by any theory. For this reason, and because there are occasions when the differences between them are significant and contain important information about the crystal chemistry, it is convenient to retain a different name for each of these three quantities to indicate the ways in which they have been determined. The bond flux is determined from the calculation of the Madelung field, the theoretical bond valence is calculated from the network equations (3.3) and (3.4), and the experimental bond valence is determined from the observed bond lengths using eqn (3.1) or (3.2). 

The network equations (3.3) and (3.4) invite comparison with the Kirchhoff equations (2.7) and (2.11). By choosing the ionic charge, in eqn (2.7) to be the same as the atomic valence, F,-, in eqn (3.3), and recognizing that in unstrained structures is equal to Sy, it follows from eqn (2.11) that loop ijl Cy = 0 which, when compared with eqn (3.4), means that the capacitances in eqn (2.11) must cancel, i.e. they must all be equal. This greatly simplifies the model, since it means that... 

Tertiary bonds (Section 2.6) are excluded. Where they are real, they have very small capacitances and therefore should not be included in the bond network used to solve the network equations. Tertiary bonds are not found among the traditionally assigned chemical bonds. 

Rules 3.3 and 3.4, through the corresponding network equations (3.3) and (3.4), can be solved (see Appendix 3) to give theoretical bond valences which, for unstrained structures, are equal to the bond fluxes and experimental bond valences. Taken together. Rules 3.3 and 3.4 are equivalent to the statement ... 

However, if the atoms are not related by symmetry, the normal rules break down. The homoionic N-N bond in the hydrazinium ion is an electron pair bond, but one in which N1 contributes 1.25 and N2 0.75 electrons. How can we apply the bond valence model in such cases where no solution to the network equations is possible One approach is to isolate the non-bipartite portion of the graph into a complex pseudo-atom. Thus in the hydrazinium ion the homoionic bond and its two terminating N atoms are treated as a single pseudo-anion which forms six bonds with a valence sum equal to the formal charge of —4. 

There is an alternative way of calculating the bond flux using the Kirchhoff equations ((2.7) and (2.11)) in place of the network equations ((3.3) and (3.4)), the problem in this case being to determine the appropriate bond capacitances which are not now all equal. Where the multipole produces a shorter bond, a larger capacitance is needed, and conversely where the multipole produces a longer bond, a smaller capacitance is needed. Transferable bond capacitances have been successfully used to model the asymmetries in d° transition metal environments as discussed in Section 8.3.2 below. 

Fig. 8.9(b), ZnV20g is able to crystallize with the brannerite structure whose theoretical bond valences, calculated from the network equations ((3.3) and (3.4)) and shown in Fig. 8.9(b), already predict an out-of-centre distortion for the ion. ZnV20g thus adopts a bond graph that supports the electronically induced distortion. In this case the adoption of a lower symmetry bond graph is favoured because it is able to reduce the bond strain. 

As pointed out above, the bond flux depends on the connectivity of the compound, that is, on the bond graph. This means that the length of a bond depends not only on its immediate environment, but also on the structure of the whole crystal or molecule of which the bond is part. Thus anions such as PO, which ideally are perfect tetrahedra, will often be distorted when they appear in crystals. However, this distortion can normally be predicted via the network equations provided the graph of the bond network is known. 

It must be possible to choose parameters for the unit cell and atomic coordinates that reproduce as closely as possible the ideal bond lengths calculated using the network equations (3.3) and (3.4), without bringing any atoms into too close contact. 

The bond valence model may also be used to refine the structure since it is based on the same assumptions as the two-body potential method. The network equations (3.3) and (3.4), can be used to predict the theoretical bond valences as soon as the bond graph is known. From these one can determine the expected bond... 

Lattice-induced strains clearly cause the bonds to violate the network equations and their presence may be indicated by a large value of the bond strain index (BSI) defined in eqn (12.1) (Preiser et al. 1999, <73 in table 1) ... 

The network equations constitute a set of A a 1 valence sum rule equations (eqn (3.3)) and A b Xa+1 loop equations (eqn (3.4)) where the network contains atoms and A b bonds. Alternatively one can use the equivalent Kirchhoff equations (2.7) and (2.11). One can readily write down equations of type 3.3 but one of these is redundant since the sum of all atomic valences in the crystal must be zero. There are many more than Ab — Aa + 1 possible loops in most bond graphs, but only Ab —Aa+ 1 are independent. Equations (3.3) and (3.4) thus constitute a set of Ab equations which is exactly the number needed to solve for the Ab unknown bond valences,. s. 

Alternatively the network equations can be solved by the method of simultaneous equations which is illustrated here for the case of CaCrFs whose bond graph is shown in Fig. A3.1. 

One problem with the network equations is that they can, on occasion, give rise to negative bond valences which have no physical significance (expect to indicate that, from a chemical point of view, the bond should not exist). Rutherford (1998) has explored the resonance bond model as an alternative to the use of the loop equation (Section 14.4) while Rao and Brown (1998) have suggested using the method of maximum entropy (Section 11.2.2.1). 

Consider a general linear circuit whose instantaneous electrical state is described by a set of currents and voltages. If we denote all these variables by vv the network equations have the general form... 

Specifically, from Equation (69) follows the property of exceptionally great flatness of a near the optimum point (e = 1). For example, for the turbulent fluid flow (( = 0.19) a twofold pressure loss in comparison to the optimal value increases transportation cost by 4.6% and a twofold reduction of loss decreases the cost only by 3.8%. For the linear electric networks (Equation (69) is also true for them) the corresponding figures are much higher and account for 8.3 and 25.0%. The revealed property of economic function flatness allows a reasonable simplification of the pressure loss optimization methods. 

In nodal analysis, the voltages between adjacent nodes of the network are chosen as the unknowns. This can commonly be achieved by selecting a reference node from the graph of the network. Equations are then formed if KCL is employed. By equating the sum of the currents flowing through admittances associated with one node to the sum of the currents flowing out of the current sources associated with the same node, a set of equations can be established with the form of [F][V] = [/] ... 

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