Long-distance order

Normally, solids are crystalline, i.e. they have a three-dimensional periodic order with three-dimensional translational symmetry. However, this is not always so. Aperiodic crystals do have a long-distance order, but no three-dimensional translational symmetry. In a formal (mathematical) way, they can be treated with lattices having translational symmetry in four- or five-dimensional space , the so-called superspace their symmetry corresponds to a four- or five-dimensional superspace group. The additional dimensions are not dimensions in real space, but have to be taken in a similar way to the fourth dimension in space-time. In space-time the position of an object is specified by its spatial coordinates x, y, z the coordinate of the fourth dimension is the time at which the object is located at the site x, y, z. 

In the glassy state the molecular structure is disorderly, and comparable to that of a liquid. This is clearly demonstrated by X-ray diffraction patterns, in which only a diffuse ring is visible, which indicates some short-distance order in contrast to to the sharp reflections found with crystals as a result of long-distance order. 

It is considered that liquid crystals that are soft and have long-distance order may form certain complex hierarchical structures other than blue phases. In fact, recent discoveries of new liquid crystal phases have arisen one after another. While many think of liquid crystals as display materials with the range of applications seemingly exhausted, it is more likely that applications other than displays are only beginning to appear. 

The sol-gel method is also used to make very fine spherical particles of oxides. By structuring the solvent with surface-active solutes, other forms can also be realized during condensation of the monomeric reactant molecules to form a solid particle. Figure 8.16 shows that normal or inverse micelles or liquid crystals (liquids having long-distance order) can be formed in such solutes. Micelles are small domains in a liquid that are bounded by a layer of surface-active molecules. In these domains the solid is condensed and the microstructure of the precipitated solid is affected by the micelle boundaries. Monodisperse colloidal metal particles (as model catalyst) have been made in solvents that have been structured with surfactants. In the concentration domains where liquid crystals obtain highly porous crystalline oxides can be condensed. After calcination such solids can attain specific surface areas up to 1000 m /g. Micro-organisms use structured solutions when they precipitate calcite, hematite and silica particles. 

As previously mentioned, phosphates can be prepared as crystalline samples. P04 unit is known to be a very efficient connector leading to various dimensionahties structures (from 0 to 3D architectures). If the technique of choice for the structural characterisation of crystalline samples is the X-ray or neutron diffraction, NMR can also provide valuable information. When the diffraction methods offer a clear picture of the long distance ordering, NMR can be used to probe the very local order and address special issues like the presence of vacancy or a statistical distribution of elements in a specific crystallographic site. 

Like crystals, glasses must be made of an extended three-dimensional lattice, but the diffuse character of x-rays scattering shows that this lattice is neither symmetric nor periodic, unlike in crystals in other words, there is no long-distance order. 

We must keep in mind that real surfaces are part of a usually disordered solid and in addition act as an inseparable ensemble of sites. Therefore, the analytical approach is restricted to methods that do not require free mobility of the target group or a long distance order of the solid. So the most powerful tools of the modem analytical chemistry, NMR (except MASNMR), and X-ray structure analysis, are ruled out. Others may be used with considerable limitations only. 

When we decrease the temperature, the exchange energies increase and, below a certain critical temperature Tc, the interactions begin to be felt at a long distance and we obtain a long-distance order (see section 2.2.2). 

The concept of long-distance order introduced previously is, practically, encountered only for metal alloys. For this reason, we limit our remarks here to solutions of atoms. 

When we decrease the temperature of a solid solution of metal atoms, we see the emergence of distinct families of sites which form sublattices. Thus, each sublattice is occupied by a specific t) e of atom. When each sublattice is occupied by a single type of atom, the solution is completely ordered this is long-distance order . Ordering is generally accompanied by a drop in S5mimetry. 

We have perfectly defined the two states of order and disorder. However, we can imagine intermediary states - e.g. a state where a certain number of atoms of A are on sites that are normally attributed to A in the perfectly ordered solution. This number would obviously be between the average and the total number of atoms of A. In order to characterize such an intermediary state, Bragg and Williams defined a degree of order or long-distance order parameter s, such that this degree is equal to 1 if the solution is perfectly ordered, and 0 in the case of a completely random distribution solution. 

We suppose that there is no short-distance order in the alloy, and therefore all of the configurational partition function is due to the long-distance order. 

Chapter 2 looks at the modeling and characterization of solid solutions. Following a quaUtative description of the different types of solid solution of substitution and insertion, the short-distance and long-distance order coefficients are introduced. Simple solution models are briefly described and the thermodynamics of the order/disorder transformations in alloys is presented. The chapter ends with the experimental determination of the activity coefficients of the components of a solid solution. 

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