Structure of Amorphous Materials

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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

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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

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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
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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)

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

• Carve a shell of size r and r dr around a
  center of an atom. The volume of the shell is
  dv4?r2dr
• Count number of atoms with centers within the
  shell (dn)
• Average over all atoms in the system
• Divide by the average atomic density lt?gt

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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
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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 (? ? 8)
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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
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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 13-14 (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)

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