Order, local

Al the basis of the polyclustcr model, there lies the assumption of the presence of one or several types of atom LO in a solid. At this point, the polycluster model is close to the so-called stereochemical models [6.8]. Besides, it includes the definition of a cluster as a set of locally ordered atoms and the definition of the boundaries as the closing of this set. Finally, the assumption that clusters conjoin along common boundaries completes the definition of a polyclustcr structure. The polycluster model includes a rather wide set of structures. This model was suggested in [6.26, 27] and developed and applied while describing various properties of metallic glasses [6.28 33]. 

The formulation of the polyclustcr model requires a sufficiently complete definition of LO. For simplification, we will only discuss briefly the presence of atoms of different kinds and the compositional (chemical) order, concentrating instead on the topology and geometry of the structure, though the consideration scheme may be easily generalized. 

The CP orientation is determined by three angles, the values of which determine the orientation of the coordinate axes system rigidly connected with the polyhedron. Assume that a method is given to choose the coordinate system connected with the polyhedron [6.29] and with this the orientation of each of them is determined. Denote by the set of angles giving the orientation of the polyhedron P . 

In the description of LO, it is important to account for elastic deformations. It is natural to assume that if in a solid LO is realized with CP from P, then they differ from the basic polyhedron P° only through elastic deformations. The deformation of the polyhedron P is considered to be elastic if the atom binding energy changes monotonously in the process of deformation transfering P° to P . The deformation magnitude is described by a set of deformation vectors. Let bj, be the polyhedron basis of the i-th atom P (Q ) with the orientation and P°(i2,) the basic polyhedron of the same orientation. Then the set of vectors 

From the definitions of elastic deformation and the topological equivalence, there follows the definition of the admissible deformation region of coordination polyhedrons that we denote by 3). The boundary of the set of elastic deformations 3 possesses the obvious property that on it the atom binding energy achieves the minimum. 

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Typical results for a semiconducting liquid are illustrated in figure Al.3.29 where the experunental pair correlation and structure factors for silicon are presented. The radial distribution function shows a sharp first peak followed by oscillations. The structure in the radial distribution fiinction reflects some local ordering. The nature and degree of this order depends on the chemical nature of the liquid state. For example, semiconductor liquids are especially interesting in this sense as they are believed to retain covalent bonding characteristics even in the melt. 

P is the critical exponent and t denotes the reduced distance from the critical temperature. In the vicinity of the critical point, the free energy can be expanded in tenns of powers and gradients of the local order parameter m (r) = AW - I bW ... 

Atomistically detailed models account for all atoms. The force field contains additive contributions specified in tenns of bond lengtlis, bond angles, torsional angles and possible crosstenns. It also includes non-bonded contributions as tire sum of van der Waals interactions, often described by Lennard-Jones potentials, and Coulomb interactions. Atomistic simulations are successfully used to predict tire transport properties of small molecules in glassy polymers, to calculate elastic moduli and to study plastic defonnation and local motion in quasi-static simulations [fy7, ( ]. The atomistic models are also useful to interiDret scattering data [fyl] and NMR measurements [70] in tenns of local order. 

Stillinger F H and T A Weber 1985. Computer Simulation of Local Order in Condensed Phases of Silicon. Physical Review 831 5262-5271. 

It may occasion surprise that an amorphous material has well-defined energy bands when it has no lattice planes, but as Street s book points out, the silicon atoms have the same tetrahedral local order as crystalline silicon, with a bond angle variation of (only) about 10% and a much smaller bond length disorder . Recent research indicates that if enough hydrogen is incorporated in a-silicon, it transforms from amorphous to microcrystalline, and that the best properties are achieved just as the material teeters on the edge of this transition. It quite often happens in MSE that materials are at their best when they are close to a state of instability. 

In principle, the Knn could be deduced from electronic structure calculations, but the smallest ones amount only to fractions of meV, whereas the calculations deal with binding energies, i.e. some eV it can be understood why the calculation techniques are not yet sufficiently accurate to compute detailed interactions, and why we find it better, until now, to extract them from local order parameters ... 

The thermodynamical properties of the APB will be investigated through the behavior of some local order parameter definec for each fee cube having co-ordinates x,y,z = R in a given MonteCarlo (MC) configuration C as ... 

R. W. Cahn, Correlation of local order with mechanical properties, in "Local Atomic Arrangements... 

Figures 8.8-a through 8.8-d show a few snapshots illustrating intermittent behavior. We have used the logistic-equation driven CML (equation 8.34) and set D = 0.25 and a = 3.83. Each figure shows 16 times overlapped after a certain number of iterations have elapsed (100 iterations in figure 8.7-a, 200 in figure 8,7-b, 300 in 8.7-c, and 500 in 8.8-d). Notice how, depending on the time of the snapshot, some regions are periodic and others are turbulent. Islands cf order tend to come and go as time progresses i.e. the local order is intermittent.

Let us fix attention on a particular H20 molecule A in the interior of water (if we wish to identify this molecule we can suppose that it contains a nucleus of the oxygen isotope 01S) and let us consider the water molecules which happen to be nearest neighbors of this molecule at the moment. These molecules have been in contact with A for different lengths of time. Since all the molecules in the liquid wander about, there was a time when none of these molecules was in contact with A. Further, if we could now begin to watch these molecules, we should find that, after the lapse of different periods of time, they become separated from A and each is replaced by another molecule. Similar remarks can be made about the molecules which come into contact with any chosen molecule. We can now raise the question—-What is the rate of turnover of this process The rate depends on the degree of local order and disorder, which in turn depends on the strength and character of the forces between adjacent molecules. 

In Chapter 3, in discussing water near its freezing point, we took the point of view that at any moment the liquid contains many groups of molecules that have a local order similar to that of ice. In Fig. 20 the molecules numbered 2, 3, 4, 5 are nearest neighbors of molecule 1, while molecules 6, 7, 8 are neighbors of 5, and consequently arc next-nearest neighbors of 1. To be definite, let us identify the molecules 2, 3, 4, 5 with the outer tetrahedra of Fig. 19, and let us suppose that the protons... 

We may now emphasize the mutual control that is exercised between adjacent water molecules. The orientation of molecule 5, for example, is controlled by the orientations of 1, 6, 7, and 8 and we may say that, in turn, molecule 5 does its share in controlling the orientations of 1, 6, 7, and 8. We may add that, throughout the liquid, near its freezing point, any local ordered arrangement arising from this kind of mutual control is not easily upset by the thermal agitation present in the liquid. 

In the case of amphiphilic molecules, characterized by the coexistence of spatially separated apolar (alkyl chains) and polar moieties, both parts cooperate to drive the intermolecular aggregation. This simple but pivotal peculiarity makes amphiphilic molecules soluble in both polar and apolar solvents and able to realize, in suitable conditions, an impressive variety of molecular aggregates characterized by spatially separated apolar and polar domains, local order at short times and fluidity at long times, and differences in size, shape (linear or branched chains, cyclic or globular aggregates, extended fractal-like molecular networks), and lifetime. 

Though silica supports are amorphous, the surface may exhibit some local order, such as that of the mineral /3-crystoballite (Fig. 5.23). The surfaces of silica support contain OH groups at densities of between 4 and 5.5 OH per nm that of cristobal-lite is 4.55 OH per nm. Silica surfaces contain only terminal OH groups, i.e. bound to a single Si atom. Heating leads to dehydroxylation, and at high temperatures only the isolated OH groups remain. 

However, another obstacle to high bulk conductivity of such a polymer exists. In view of the hnite length and vast number of constitutive polymer chains, bulk conduction requires electrons to jump between chains (or polarons to transfer between chains in the opposite direction). It is obvious, then, that local order, crystallinity, and good contact between the different crystal domains in the polymer are further prerequisites for conduction. 

From this equation and Eqs. (2) and (4) the energy of the system can be obtained. The entropy is more difficult to derive, and we refer to the literature [4,6]. Generally, the quasichemical gives better results than the mean-field approximation, since it allows for local order. We note that for the three-dimensional lattice gas no exact analytical solution exists. 

One important use of X-ray probes is in the study of local order and displacements, but this is not within the scope of the present book. The recent availability in intense synchrotron sources with selectable X-ray energies permits high-precision measurements of chemically specific atomic-pair correlations in solid solution alloys. A recent review of the technique is given by G.E. Ice and C.J. Sparks (Modern Resonant X-ray studies of alloys local order and displacement) in Annual Reviews of Materials Science 1999, 29, 25-52. 

Fig. 2 Illustration of protein structure levels. Shown are primary structure (amino acid sequence), secondary structure (local order of protein chain, a-helix shown as an example), tertiary structure (assembly of secondary structure elements), and quaternary structure (relationship of different protein chain in multisubunit protein). (From Ref. 66.)...

In NSR catalysts, the Ba-Pt interface plays an important role in the storage of NOx, which occurs by the formation of Ba(N03)2. Recent results [95] using scanning transmission microscopy (STM) and a model catalyst formed by deposition of a Ba thin films on Pt(lll) showed that, at room temperature, a film of Ba was formed with few individual Ba atoms, which were locally ordered. Upon annealing, particles are produced, of which atomic resolution is achieved with an atomic spacing consistent with the (111) plane of Ba. 

Polymeric networks held together by noncovalent interactions, mostly disordered, but typically with regions of local order... 

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