1. Structure of Amorphous Materials
Crystalline vs. amorphous materials
Free volume and the glass transition
Radial distribution function, short range order and structure factor
Voronoi polyhehra
Medium-range order and nanocrystalline materials 1

2. Crystalline vs. amorphous
There is long-range order (LRO)in crystals - a unit repeats itself and fills the space
There is no LRO in amorphous materials 1

3. Free volume and the glass transition
Free volume = specific volume (volume per unit mass) - specific volume of the corresponding crystal
At the glass transition temperature, Tg, the free volume increases leading to atomic mobility and liquid-like behavior. Below the glass transition temperature atoms (ions) are not mobile and the material behaves like solid 1

4. Glass transition
Within the free volume theory it is understood that with large enough free volume mobility is high, viscosity is low. When the temperature is decreased free volume becomes “critically” small and the system “jams up”
The glass transition is not first order transition (such as melting), meaning there is no discontinuity in the thermodynamic functions (energy, entropy, density).
Typically Tg is ~ 50-60% of the melting point
The effective glass transition temperature, is a function of cooling rate - higher rate  higher Tg. It is also called the fictive temperature
Sometimes the glass transition it is a first order transition, most prominently in Si where the structure changes from 4 coordinated amorphous solid to to ~ six coordinated liquid. The same applies to water (amorphous ice) 1

5. Characterizing the structure - radial distribution function, also called pair distribution function
Gas, amorphous/liquid and crystal structures have very different radial distribution function 1

6. Radial distribution function - definitiondr r
Carve a shell of size r and r + dr around a center of an atom. The volume of the shell is dv=4r2dr
Count number of atoms with centers within the shell (dn)
Average over all atoms in the system
Divide by the average atomic density <>

7. Properties of the radial distribution function
For gases, liquids and amorphous solids g(r) becomes unity for large enough r.
The distance over which g(r) becomes unity is called the correlation distance which is a measure of the extent of so-called short range order (SRO)
The first peak corresponds to an average nearest neighbor distance
Features in g(r) for liquids and amorphous solids are due to packing (exclude volume) and possibly bonding characteristics 1

8. Radial Distribution Function - Crystal and Liquid
Liquid/amorphous g(r), for large r exhibit oscillatory exponential decay
Crystal g(r) does not exhibit an exponential decay (  ∞)

9. Radial distribution functions and the structure factor
The structure factor, S(k), which can be measured experimentally (e.g. by X-rays) is given by the Fourier transform of the radial distribution function and vice versa
Radial distribution functions can be obtained from experiment and compared with that from the structural model

10. More detailed structural characterization - Voronoi Polyhedra
Draw lines between a center of an atom and nearby atoms.
Construct planes bisecting the lines perpendicularly
The sets of planes the closest to the central atom forms a convex polyhedron
Perform the statistical analysis of such constructed polyhedrons, most notably evaluate an average number of faces
For no-directional bonding promoting packing number of faces is large ~ (metallic glasses)
For directional bonding (covalent glasses) number of faces is small
Ionic glasses - intermediate
In all cases the number of faces is closely related to the number of nearest neighbors (the coordination number)

11. Medium range order and radial distribution function
Radial distribution functions (and also X-ray) of amorphous silicon and model Si with ~ 2 nm crystalline grains are essentially the same - medium range order difficult to see by standard characterization tools. Such structure is called a paracrystal.

12. Radial Distribution Function
Sensitive enough to see medium range order and crystal size
Behavior of g(r), for large r clearly shows differences for CRN and paracrystal models, and also provide a measure of paracrystal size

13. Radial Distribution Function Nanocrystalline material
Nanocrystalline materials shows clear crystalline peaks with some background coming from the grain boundary