Neighboring occupied sites

Figure 2.9.3 shows typical maps [31] recorded with proton spin density diffusometry in a model object fabricated based on a computer generated percolation cluster (for descriptions of the so-called percolation theory see Refs. [6, 32, 33]).The pore space model is a two-dimensional site percolation cluster sites on a square lattice were occupied with a probability p (also called porosity ). Neighboring occupied sites are thought to be connected by a pore. With increasing p, clusters of neighboring occupied sites, that is pore networks, begin to form. At a critical probability pc, the so-called percolation threshold, an infinite cluster appears. On a finite system, the infinite cluster connects opposite sides of the lattice, so that transport across the pore network becomes possible. For two-dimensional site percolation clusters on a square lattice, pc was numerically found to be 0.592746 [6]. 

There are commonly void spaces (holes) in a crystal that can sometimes admit foreign particles of a smaller size than the hole. An understanding of the geometry of these holes becomes an important consideration as characteristics of the crystal will be affected when a foreign substance is introduced. In the cubic close-packed structure, the two major types of holes are the tetrahedral and the octahedral holes. In Fig. 10-1(h), tetrahedral holes are in the centers of the indicated minicubes of side a/2. Each tetrahedral hole has four nearest-neighbor occupied sites. The octahedral holes are in the body center and on the centers of the edges of the indicated unit cell. Each octahedral hole has six nearest-neighbor occupied sites. 

Figure 1.4 A 6 X 6 square lattice site model. The dots correspond to multifunctional monomers. (A) The encircled neighboring occupied sites are clusters (branched intermediate polymers). (B) The entire network of the polymer is shown as a cluster that percolates through the lattice from left to right.

The origins of percolation theory are usually attributed to Flory and Stock-mayer [5-8], who published the first studies of polymerization of multifunctional units (monomers). The polymerization process of the multifunctional monomers leads to a continuous formation of bonds between the monomers, and the final ensemble of the branched polymer is a network of chemical bonds. The polymerization reaction is usually considered in terms of a lattice, where each site (square) represents a monomer and the branched intermediate polymers represent clusters (neighboring occupied sites), Figure 1.4 A. When the entire network of the polymer, i.e., the cluster, spans two opposite sides of the lattice, it is called a percolating cluster, Figure 1.4 B. 

Percolation theory is a statistical theory that studies disordered or chaotic systems where the components are randomly distributed in a lattice. A cluster is defined as a group of neighboring occupied sites in the lattice, being considered an infinite or percolating cluster when it extends from one side to the rest of the sides of the lattice, that is, percolates the whole system [38],... 

Following the notation used by Lee and Yang,14 we use 1 for an occupied site and t for a vacant site and we denote by [j ], [tt], and [It] the number of pairs of nearest-neighboring occupied sites, the number of pairs of nearest-neighboring vacant sites, and the number of pairs between nearest-neighboring occupied and vacant sites, respectively. There are two identities ... 

Figure 5.1 A two-dimensional illustration of a portion of an infinite lattice. The open circles represent unoccupied lattice sites and the closed circles represent occupied sites. The segments connecting two nearest neighbor occupied sites represent a local connection, or...

In random bond percolation, which is most widely used to describe gelation, monomers, occupy sites of a periodic lattice. The network formation is simulated by the formation of bonds (with a certain probability, p) between nearest neighbors of lattice sites, Fig. 7b. Since these bonds are randomly placed between the lattice nodes, intramolecular reactions are allowed. Other types of percolation are, for example, random site percolation (sites on a regular lattice are randomly occupied with a probability p) or random random percolation (also known as continuum percolation the sites do not form a periodic lattice but are distributed randomly throughout the percolation space). While the... 

When the random-walk model is expanded to take into account the real structures of solids, it becomes apparent that diffusion in crystals is dependent upon point defect populations. To give a simple example, imagine a crystal such as that of a metal in which all of the atom sites are occupied. Inherently, diffusion from one normally occupied site to another would be impossible in such a crystal and a random walk cannot occur at all. However, diffusion can occur if a population of defects such as vacancies exists. In this case, atoms can jump from a normal site into a neighboring vacancy and so gradually move through the crystal. Movement of a diffusing atom into a vacant site corresponds to movement of the vacancy in the other direction (Fig. 5.7). In practice, it is often very convenient, in problems where vacancy diffusion occurs, to ignore atom movement and to focus attention upon the diffusion of the vacancies as if they were real particles. This process is therefore frequently referred to as vacancy diffusion... 

An alternative mechanism by which interstitial atoms can diffuse involves a jump to a normally occupied site together with simultaneous displacement of the occupant into a neighboring interstitial site. This knock-on process is called interstitialcy diffusion. 

Diffusion of relatively small atoms that normally occupy interstitial sites in the solvent crystal generally occurs by the interstitial mechanism. For example, hydrogen atoms are small and migrate interstitially through most crystalline materials. Carbon is small compared to Fe and occupies the interstitial sites in b.c.c. Fe illustrated in Fig. 8.8 and migrates between neighboring interstitial sites. 

The relationship between jump rate and diffusivity in Eq. 8.3 can be obtained by an alternate method that considers the local concentration gradient and the number of site-pairs that can contribute to flux across a crystal plane. A concentration gradient of C along the y-axis in Fig. 8.86 results in a flux of C atoms from three distinguishable types of interstitial sites in the a plane (labeled 1, 2, and 3 in Fig. 8.8). The sites are assumed to be occupied at random with small relative populations of C atoms that can migrate between nearest-neighbor interstitial sites. If d is the number of C atoms in the a plane per unit area, the carbon concentration on each type of site is c /3. Carbon atoms on the types 1 and 3 sites jump from plane a to plane (3 at the rate (c /3)T. The jump rate from type-2 sites in plane a to plane (3 is zero. The contribution to the flux from all three site types is... 

Self-Diffusion by the Vacancy Mechanism in the F.C.C. Structure. Each site on an f.c.c. lattice has 12 nearest-neighbors, and if vacancies occupy sites randomly and have a jump frequency IV,... 

Lattice Monte Carlo (LMC) simulations are another approach for the study of the principles of liquids thermodynamics. In these simulations it is assumed that molecule-like objects can occupy one or more sites of a predefined regular lattice. Mostly, only nearest-neighbor interactions of the objects are allowed, resulting in a kind of surface interaction, Ey, between objects, i and j, on neighboring lattice sites. Given certain mole fractions xt of the different objects, the configuration space of such a lattice... 

Copper(I) nitride, Cu3N, crystals are also cubic, OPm3m, a0 = 3.814 A, with one molecule in the unit cell. Each N has an octahedral arrangement of six Cu atoms. Cu atoms have linear bonds to two N atoms and eight nearest Cu neighbors (see Figure 5.24). These are two sets of four planar Cu atoms bonded to the N atoms. The N atoms occupy sites for a simple cubic cell. To accommodate Cu and N in the P system, Cu atoms occupy P sites in an ABC sequence with N atoms in O sites. The P sites are three-quarters occupied and O sites are one-quarter occupied to give the notation 3 2P3/4Oi/4. 

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