Disordered Structures Lecture 1, Part 1

Disordered Structures
Lecture 1, Part 1
Outline
• Simple structural models.
random sphere packings
random networks
composites, fractals, rough surfaces
• How to describe a disordered structure
Voronoi polyhedra
distribution functions
local density distributions
Connectivity: percolation models, fractals
Repetition: Crystal structure
14 space lattices in three dimensions
Random sphere packings (RSP)
• Ordered structures of
equal spheres occupy <74
% of space (fcc,hcp)
• Disoredered structures
occupy <64 %
(empirically)
• Model for amorphous
close packed structures
• Physical ”toy” models
• Computer models starting
from seed cluster. Details
depend on seed geometry
and construction rules.
Source: cherrypit.princeton.edu
Computer models
• Relaxation of the
structure to an
energetically favorable
configuration
• Interatomic potential
(Lennard-Jones, metallic
binding…)
• Adjust position until
potential energy is a
minimum
• a) before relaxation
• b) after relaxation
Source: Cusack, TPSDM
Voronoi polyhedra
• Constructed as the
Wigner-Seitz cell of
crystalline lattices
Source: bioinfo.mbb.yale.edu
Statistical distributions:
Cell volume, faces per cell,
edges per face
Source: Zallen, The Physics of Amorphous solids
Example: Metallic glasses
• Amorphous metals
• Produced by very rapid
cooling from the melt
• Some alloy glasses may
be formed at less rapid
cooling rates
• Evaporation on a low
temperature substrate
• Liquid-like structure
• RSP model with
interatomic potential
• Mg-Cu glass simulated
cooling from the melt
Source: dirac.ruc.dk/~nbailey/BMGs/index.html
Random networks
• Model of amorphous materials (”ideal glassy structures”)
with chemical (covalent) binding
• Random network of bonds
• Number of bonds per atom preserved – no unsatisfied
bonds
• Distribution of bond lengths, bond angles, ring sizes.
• Adjacent structural units may be rotated relative to one
another.
• Bond analogue of Voronoi polyhedra – The Simplicial
Graph.
Simplicial Graph
• For the sphere packing,
the simplicial graph is the
dual of the Voronoi
polyhedra description.
• It is the thick lines
connectiong the points
(”atoms”) in the figure.
• Triangles in 2-D and
irregular terahedra in 3-D.
• For covalent materials we
use a simpler graph, a
covalent graph, where the
lines denote the chemical
bonds.
Continuous random network (CRN)
Comparison between crystal and CRN structure of A2O3.
Source: Zachariasen, 1932.
A ”basis” in CRN models
• The connectivity of the
CRN can be illustrated by
a topological network.
• Ex: Triangles in twodimensional case, with
three bonds from each
unit.
• An atom or a group of
atoms (”basis”) can be
placed at the center of
each triangle with bonds
to all the triangles
connected to it.
Source: Cusack
Example
Source: Cusack
• Same topological network with (a) 1 atom and
(b) an AB3 unit in each triangle.
Short-range order
• Chemical bond constraints
lead to rather well-defined
local coordination polyhedra
• Coordination number
• Bond lengths
• Bond angles
• Bond lengths more
constrained than bond
angles
• More random connection of
local polyhedra, rings
• Statistical distributions of
angles, rings
rij
θijk
Bond lengths and
bond angles
Statistics for a model structure
Source: Zallen, The Physics of Amorphous Solids
Medium range order
• Connection of local polyhedra may lead to
structures on a larger length scale (~5–20
Å)
• Corner sharing, edge sharing, face sharing
connections
• Superstructural units: Rings, clusters
• Clustering may lead to development of
nanocrystals.
• How disordered is ”amorphous”?
• Is there a clear demarcation between
amorphous and nanocrystalline structures?
Ex: Amorphous silicon
• Built from tetrahedra, Z=4
• Similar bond lengths, but
significant spread in bond
angles
• In reality dangling bonds
appear, giving possibility
for doping
Source: F. Wootten, Acta Cryst. A 28 (2002) 346
Source: ysolars.com
Ex: Vitreous silica, SiO2.
•Z(Si)=4, Z(O)=2
•Spread in mainly
Si-O-Si bond angles
Local connections
Source: Cusack
Source: Bell and Dean, NPL Report, 1967
Ex: Tungsten oxide
• Many transition metal oxides
are built up of octahedra
• Crystallaine WO3: corner
sharing
• Substoichiometric: also edge
sharing
• Amorphous WO3: Disordered
arrangement of octahedra
• Empty spaces (”tunnels”)
between the octahedra
• Suitable ion (H+, Li+, Na+,…)
intercalation materials
Source: webelements.com
Distribution functions
• Statistical description of atomic positions
• Describes important features of the structure as
an input to theories for physical properties
• Image analysis of optical or electron microscopy
pictures
• Scattering experiments (see next lecture)
• Convenient for amorphous materials and for
comparison with the RSP and CRN models.
Single particle distribution function
• N atoms in the volume V of the material
• Average density n0=N/V
• Single particle distribution function n1(r)
n1 (r ) =
N
∑ δ (r − R )
i =1
i
• Ensemble average over possible configurations –
statistical description
• n1(r) dr is the average number of atom centers in a
volume element dr around r
• 0< n1(r) dr <1, since particles do not coincide
• n1(r) dr can be interpreted as the probability to find
an atom center in dr
Two-particle distribution function
• Two-particle distribution function
n2 (r1 , r2 ) =
N N −1
∑∑ δ (r
1
i =1 j ≠ i
− R i )δ (r2 − R j )
• The probability to find atom centers
simultaneously in dr1 around r1 and
dr2 around r2 is n2(r1,r2) dr1 dr2
∫ n (r , r ) dr
2
1
2
2
= ( N − 1) n1 (r1 )
Atomic correlations
• Probability of having an atom at r2 is correlated
with the probability for an atom at r1, if these
positions are not too far apart.
• This is expected due to interparticle forces and
constraints due to chemical bonds.
• Homogeneous material: n1(rj)=n0
2
• If r1 and r2 are far apart then n2 (r1 , r2 ) = n0
• Generally we can write
n2 (r1 , r2 ) = n 20 g 2 (r1 , r2 )
Pair distribution function (PDF)
• General definition:
• Put r1=0, then define
r=r1-r2
• Homogeneous material
• g2(r) describes the
correlations in the atom
positions
• n0 g2(r)dr is interpreted as
the probability to find an
atom in dr, a vector
distance r from another
atom at the origin.
n2 (r1 , r2 )
g 2 (r1 , r2 ) =
n1 (r1 ) n1 (r2 )
g 2 (r ) = n2 (r ) / n02
n0 ∫ g 2 (r ) dr = N − 1
V
Isotropic materials
• Isotropy: r is the distance between r1 and r2.
• n0g2(r) can be integrated to give the number of
atoms in a spherical shell around the origin
r0 + ∆r
∫n g
0
2
(r ) dr = 4πr n g 2 (r0 ) ∆r = n0 ρ (r )∆r
2
0 0
r0
• Radial distribution function ρ(r)
ρ (r ) = 4πr g 2 (r )
2
Illustration of PDF
• Pair distribution functions
show oscillations due to
nearest neighbor, next
nearest neighbor shells,
etc.
• Oscillations damped out
as r increases
• g2(r) ~1 for large r
• z(r)=n0 g2(r) + δ(r)
is somtimes used. It
includes also the atom at
the origin
Source: Elliott, PoAM
Examples
• Atoms interacting by a
Lennard-Jones potential
• v.d. Waals and repulsive
interactions
• Bond length, bond
angles seen in PDF
g(r)
r
θ
Source: Wikipedia
Radial distribution function
• Coordination number –
number of nearest
neighbors
• Integrate over first peak
of RDF
r2
Z = n0 ∫ ρ (r ) dr
r1
• Second, third neighbors
might be possible to
obtain
Binary systems
•
•
•
•
Atoms A and B, denoted by indices i,j
Homogeneous, isotropic material
Concentrations cA=NA/N and cB=NB/N
Partial pair distribution functions gij(r)
g 2 ,ij (r ) = n2 ,ij (r ) / n0 c j
• Three independent pair distribution functions since
gAB(r)=gBA(r)
•  partial radial distribution functions
•  partial coordination numbers Zij
Order parameters
• Binary systems
• Complete chemical
disorder: Zij=Z*ij
• Consider only the
coordination number ZAB.
• Short-range order
parameter given by, ξοAB.
• Long range order
parameter: For example
like in a binary alloy.
ξ AB = ξ AB / ξ AB
o
max
Z AB
ξ AB = * − 1
Z AB
• Here ξAB describes the
preference for AB
associations. Its
maximum value is ξmaxAB.
Example: Amorphous SiSe2
• Amorphous SiSe2
• Partial pair distribution
functions
• The material exhibits
considerable short range
order – Si centered
tetrahedra.
• Also some intermediate
range order
• Intermediate size
structural units
• Edge-sharing tetrahedra
play a major role
Source: Celino and Massobrio, Comp.
Mater.Sci. 33 (2005) 106-111.
Higher order distribution functions
• nm(r1,r2,….rm) dr1 dr2….drm is the probability of finding
atom centers simultaneously in each of the m volume
elements
• Homogeneous system:
g m (r1 , r2 ,....rm ) = nm (r1 , r2 ,....rm ) / n0m
• Experiments usually provide only g2 (maybe g3), but for a
complete characterization all gm are needed
• Superposition approximation for g3(r1,r2,r3)
g 3 (r1 , r2 , r3 ) = g 3 (r12 , r13 ) ≈ g 2 (r12 ) g 2 (r13 ) g 2 (r23 )