Bonding lattices

In the hydrogen-bonded lattice of ice, the individual water molecules cannot pack together as tightly as they would if there was no hydrogen bonding. Consequently, the density of ice is lower than that of water. Ice cubes float benzene cubes sink. 

Fig. 20. The relationship between the bond lattice and the parent lattice (a) unexcited, (b) with two excitations and local collapse (from Ref. 7>)...

In the simplest version of the transformation, the state of occupancy of a given bond lattice point is independent of the states of occupancy of other bond lattice points this corresponds to neglecting interaction between hydrogen bonds. The calculation of the distribution of possible occupation states of the bond lattice replaces the enumeration of occupancy states of the basic lattice section in the cell model, but in the simplest model the bond lattice occupancy distribution only accounts for a subset of possible basic lattice section occupancies. 

The random network bond lattice transformation very clearly displays some aspects of the relationship between the cell model and broken bond (multistate) models. We have already remarked on the analogies between states of occupancy... 

The basic crystal structure consists of P04 (or As04 ) tetrahedra alternating with the K" " (or NH4 ) ions along the c-axis. The P04 units are connected by 0-H...0 hydrogen bonds in the ab plane, forming a three-dimensional hydrogen-bonded lattice [2]. In the ferroelectric phases, the H atoms are localized such that the two close protons are both on the top of the oxygen ions of the XO4 units, as depicted in Fig. 4b. In the antiferroelectric... 

The predisposition of a material to deform in a particular manner depends on its lattice structure, in particular whether weakly bonded lattice planes are inherently present. In definite terms, most of the materials cannot be classified distinctly into individual categories. Pharmaceuticals exhibit all three characteristics, with one of them being the predominant response, thus making it difficult to clearly demarcate the property favorable for compressibility. 

Nordlander, E. H., Bums, J. H. (1986), Stmcture of a Hydrogen-bonded Lattice Coumpound between Phosphoric Acid and 18-Crown-6 H3PO4.0.5(Ci2H24O3).3H2O, Inorg. Chim. Acta 115, 31-36. 

Using this, the tracer atom is described as if it forms a pair with the vacancy on one of the bonds adjacent to its original site, it walks on the bond lattice, and at the end of the walk (which happens after each move with probability prec) it is released with equal probability at either end of the last visited bond. Results for the probabilities of the different jump lengths (beginning-to-end vectors of these trajectories) are shown in Fig. 9. Note, that the model calculations in Fig. 9 contain no adjustable parameters. 

We use our previous results for the return and recombination probabilities of the vacancy, and consider the random walk of the tracer-vacancy pair on the bond lattice. Let p(r, re) denote the probabihty that the tracer-vacancy pair is at position r at instance n, where re counts the number of moves the tracer-vacancy pair has already made. Since the subsequent moves of the pair are independent, we can write an effective diffusion equation for the evolution of p(r, n) ... 

The first term on the right hand side corresponds to the moves the pair makes on the bond lattice, here Deff denotes the mean square displacement per move of the pair. The second term corresponds to the recombination of the vacancy.1 When this occurs the pair breaks up. In the continuum approximation for space and re, the solution for a Dirac-delta initial condition at the origin is... 

Note that since n counts the number of moves of the vacancy-tracer pair on the bond lattice, the term —ce does not imply that the vacancy can recombine at any lattice site - in fact it recombines at terrace steps, between subsequent returns to the In atom. 

Mix pure molten copper with up to 11% molten tin. When the mixture cools and solidifies, the tin atoms will have replaced some of the copper atoms in the metallic-bonded lattice. This type of alloy is called a substitutional alloy. Some of the atoms of the parent element are replaced in the metals lattice by atoms of the added element. Substitutional alloys occur when the elements in the alloy are about the same size. Copper has an atomic radius of... 

Steel is different. Most forms of steel are made by alloying iron with carbon. High-carbon steels, which contain up to 1.7% carbon, are stronger and harder than either of their constituents, iron and carbon (in the form of coke or charcoal). This change in properties is similar to that produced by adding tin to copper, but the structure of the alloy is entirely different. Iron has an atomic radius of 140 pm, but that of carbon is only 67 pm. So small is the carbon atom in relation to iron, that it cannot replace iron in the metallic-bonded lattice. Instead, the carbon atoms slip into the interstices between the iron atoms. This type of alloy is called—not surprisingly—an interstitial alloy. 

Interstitial alloy An alloy in which the atoms of the alloying agent(s) are so small that they cannot replace the parent metal in a metallic-bonded lattice. Instead, the agent fits into the interstices of the lattice. 

Ion core An atomic nucleus in a metallic-bonded lattice surrounded by all but one or two of its electrons. The ion core s mobile electrons are part of the electron sea found in metals and alloys. 

As mentioned before, the disordered solids will be mostly modelled in this book using randomly diluted site or bond lattice models. A knowledge of percolation cluster statistics will therefore be necessary and widely employed. Although this lattice percolation kind of disorder will not be the only kind of disorder used to model such solids, as can be seen later in this book, the widely established results for percolation statistics have been employed successsfully to understand and formulate analytically various breakdown properties of disordered solids. We therefore give here a very brief introduction to the percolation theory. For details, see the book by Stauffer and Aharony (1992). 

Figure 13.05 Variation of Tg with compositions O and A are experimental values. The variation of average bond energy is shown by the broken line and the corresponding scale is given on the right-side ordinate. The full line represents the Tg variation from bond-lattice calculations. (After Rao and Mohan, 1980).

---