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Theory of Amorphousness

Category:Science

Amorphous and crystalline materials have been known for long time, for example natural glass (obsidian) and single crystals were known and appreciated for their beauty since antiquity. Theory of amorphousness is a science concerned with the description and analysis of the atomic structure (arrangement) of ideal (model) amorphous solids, in the same way as theory of crystallography describes the atomic structure of perfect (model) crystalline solids.

The theory is based on two axioms:

1. In an amorphous solid the inter-atomic distances viewed along any arbitrarily chosen straight line passing through it will be irregular, in contrast to crystalline solids where inter-atomic distances are regular.
2. In any amorphous solid the statistics of the distribution of distances between atoms nearest to and projected onto an arbitrarily chosen straight line will be the same regardless of the chosen direction of the line (normal distribution in accordance with the Central Limit Theorem).

It follows from the two statements that an ideal amorphous solid will be isotropic. A clear and fundamental distinction between ideal crystalline and amorphous solids is illustrated in Figures 1a and 1b. The graph shown in Fig 1a, represented by a single line peak at the packing fraction of pf = 0.68, signifies that the probability to have this packing fraction in each and every Weiner-Seitz cells in a bcc crystal is exactly one, and is precisely zero for any other packing fraction. The graph shown in Fig 1b is that for an ideal amorphous solid with an average packing fraction, pf ≈ 0.61. By the nature of this arrangement the amorphous structure has a distribution of volumes of the Voronoi cells, and each cell has a unique non-repeatable configuration. This random pattern is repeated ad infinitum, from which one can infer that there can be neither long-, mid-, nor short-range order in such a structure.

Models of Amorphous Solids

The arrangement of atoms in a solid can be considered as a geometrical problem of packing of spheres (also done in crystallograhy). This is referred to as a geometric model or hard sphere model. The elemental unit of structure is a random cluster of touching spheres. An example is shown in Figure 2, comprising one inner sphere and 7 outer spheres. Random packing of clusters results in a distribution of coordination numbers. For the monoatomic ideal amorphous solid the predicted distribution is shown in Figure 3 .

However, atoms are more accurately represented as centres of interatomic forces incorporating both repulsion and attraction. Then, an equilibrium packing of such a system is achieved by allowing an ergodic process to take place with the ultimate goal of minimization of free energy. The result is referred to as a Gibbsian model or soft sphere model.

Classes of real amorphous solids

The large variety of naturally occurring or synthetically produced amorphous materials can be segregated into 4 cognate classes:
1. Inorganic glasses - for example, additions of soda and lime to silica disrupt the pattern of covalent bonding, preventing the natural tendency of silica to crystallise. Inorganic glasses have covalent network of bonds that varies in connectivity between liquid and solid states, forming so-called fragile glasses, and resulting in some variability in coordination numbers in the glass~\cite{Zach-1932}. At high temperature large proportions of the bonds dissociate and the 3-dimensional network dissolves into a mixture of low molecular weight fragments, resulting in a relatively low viscosity liquid. On cooling the network re-forms. The liquid - glass transition is reversible and repeatable. Annealing above the glass transition temperature can lead to some crystallisation.

2. Organic glasses - within which there are two subgroups: (i) random 3-dimensional networks of copolymers, and (ii) polymeric glasses based on atactic macromolecules. Random networks form on copolymerisation of molecules with functionalities greater than 2, for example bisphenol-A epoxy and diaminodiphenyl sulphone~\cite{MinHodgStac-1993}. The reaction between the components takes place in the liquid state (sol) which ensures spatially random bonding, resulting in a random, permanent, non-reversible network. The initial liquid turns into glass through the phenomenon of vitrification caused by increased mass of the molecular network. Annealing does not lead to crystallisation. These glasses do not become free flowing liquids on heating above glass transition temperature, although for low density of crosslinking they become extensible elastomers. At sufficiently high temperatures the network decomposes in a non-reversible way.

The second subgroup includes linear and branched polymers with atactic architecture of the macromolecular chain. These substances are destined to form glasses on cooling from the molten state due to the irregular chain structure. The polymeric materials have covalent bonds unchanging between liquid and solid state, giving ``strong glass formers~\cite{Ange-1995}. Annealing above glass transition temperature does not lead to any crystallisation. The liquid - glass transformation is therefore reversible and repeatable.

3. Metallic glasses - a relatively new group of solids that form amorphous structures. The term metallic symbolises non-directional centro-sym\-metric bonding, in contrast to that found in the two previous groups. Crystallisation on cooling from the liquid phase is suppressed by alloying a number of elements with selectively chosen atomic radii. The liquid - glass solid transition is repeatable, although crystallisation usually will occur on annealing above the glass transition temperature. Metallic glasses have far greater freedom in forming random packing arrangements than organic or inorganic glasses because their bonding is to a large degree non-directional which allows for variation in coordination between neighbouring atoms~\cite{Shen-2006,Shen-2007}. Such metallic glasses are heterogeneous in composition on atomic scale~\cite{Torq-2002}, in contrast to the monoatomic {\ias} described above.

4. Amorphous thin films - here, too, we distinguish two subgroups: (i) materials produced by methods akin to molecular vapour deposition techniques, and (ii) grain boundaries. Usually, not much is known about the positions of the atoms in these films; their amorphous nature is declared on the basis of absence of any crystalline x-ray diffraction peaks. It is not generally known if they exhibit glass transition temperature. On heating, the amorphous structure rearranges and the transformation is non-reversible.

Thin layers between crystalline grains may also exist in amorphous state. In fact, the the original concept in physical metallurgy of amorphous cement at the grain boundaries was proposed nearly a century ago by Rosenhein, as recounted in the book by Professor Cahn.

We should discount solids which are essentially crystalline materials but disordered to such an extent that no measurable long-range atomic order remains. This corresponds to the case when atomic arrangement is disordered, say, by heavy particle bombardment leading to so-called ``amorphous structure in a sense that long range order as detected by x-ray scattering is no longer observed.

End: ZHSANU (talk) 08:38, 27 December 2007 (UTC)