AMORPHOUS SOLIDS
a- amorphous
GLASSES
v- vitreous
c- crystalline
- subset of amorphous solids
Definitions
1. A glass is an inorganic product of fusion which has cooled to a rigid
condition without crystallisation
2. A glass is a material which exhibits the glass transition phenomenon
1. Ignores organic and metallic glasses and other formation routes such as
solution synthesis, vapour sources, metamict.
2. Unhelpful
Better: A glass is a solid that possesses no long-range atomic order and which
undergoes the glass transformation from solid to supercooled liquid on
heating.
(Not all amorphous solids exhibit the glass transition – these are not glasses)
•
•
Solid-like
Liquid-like
- rigid, hard, fractures (rapid processes)
- flows (high T or long t) (slow processes)
- structure
1
Natural Glasses:
Meteorite impact
Volcanic/magma Obsidian; viscous melts
Pumice; gassy, low viscosity melts
Used as Arrow tips, scrapers, etc. from 75,000 BC (Paleolithic)
Metamict
Manufactured Glasses:
Egyptian glasses 9000 years ago;
Phoenecian sailors cooking on blocks of Natron noticed glass melts formed in
beach sands around the cooking fires.
Sand (SiO2) + Natron (Na2O) + Sea Shells (CaO)
(Same three components as modern window and container glasses)
Glass windows
Optical glass
1400s
1500s
Thermometers
1800s
Laboratory Glass 1800s
→ microscopes (Huygens) revolutionised biology
→ telescopes (Galileo) revolutionised astronomy
→ underpinned thermodynamics.
→ chemical revolution (Michael Faraday)
Optical fibre; lasers; non-linear optics; display panels; device packaging
2
Stability
- there are NO conditions under
which glasses are
thermodynamically stable
-glasses exist over geological
timescales
-thermodynamically metastable
Gravitational analogy
Stable – c.o.g. at
lowest point
Unstable – c.o.g. already
at highest point – any
change will move it to
stable position
Metastable – c.o.g.
has to rise before it
falls
G
Ea
glass
G
crystal
Glass at high G wrt crystal but needs
to acquire activation energy Ea before
it can convert by rearranging structure.
3
Change in V or H with T
Cooling liquid
TL - liquidus/crystallisation
temperature
Most liquids crystallise below TL
with change in vol
Some continue as supercooled
liquids
Depends on structure and cooling
rate
dV/dT of supercooled liquid same
as liquid until Tg
gradual inc in  until material can
no longer rearrange its structure
on a reasonable timescale
Tg – glass transition temperature
- below Tg, supercooled liquid
behaves like solid - GLASS
V
liquid
supercooled
liquid
faster
cooling
glass
slower
cooling
crystal
Tg Tg
T
TL
Value of Tg depends on cooling rate
If glass held at a temperature near Tg, V
will decrease towards equilibrium value
which supercooled liquid would have
had (arrow) – v.slow.
4
Tg glass-transition temperature
• At Tg,  so high that freedom of
liquid lost and glassy state is
formed
supercooled
liquid
• dV/dT now as for crystal.
glass
• Value of Tg depends on cooling
rate (therefore not a true
thermodynamic function)
• Faster cooling  higher Tg energy required for structural
reorganisation being removed
faster
Tg
• Tg actually a T range
5
Supercooling
• possible because of v. high Ea for phase transformation from
liquid/glass to crystal
• Tg important parameter in technology
- lower limit of use of rubber
- upper limit of use of thermoplastics
- annealing – stress relief – just below Tg
• ALSO liquid structure closer to glass than crystal  Ea for
transformation from liquidglass less than for liquidcrystal
• Ea depends on strength and directionality of bonds in glass 
electron-pair>ionic>metallic
metallic glasses - need cooling rates of > 106 K s-1
soda-lime-silica - “
“
“
“ ~ 10 K s-1
SiO2
- “
“
“
“ ~ 0.1 K s-1
B2O3
- will not crystallise at normal pressures
6
Crystallisation
If the glass is given enough thermal energy, bonds can break and
rearrange to a crystal phase but this may not be the expected phase
Structure of glass often closer to that of
a high temperature polymorph of
crystal and will convert to that phase if
heated to T which is high enough for
structural rearrangement but which
may be below stability Ttr of polymorph.
Ea(total)
Ea(1)
G
G(1)
Ea(2)
glass
G(total)
G(2)
Metastable
Crys 1
T < Ttr
(stable > Ttr)
Stable
Crys 2
(stable < Ttr)
7
ROLE OF DIFFERENT OXIDES IN GLASS
STRUCTURE
network formers
alone
- can form a glass network
O
O
- e.g. SiO2, B2O3
-O-M-O-M-O-
strong, directional e-pair bonds
network modifiers
glass
- e.g. Na2O, CaO
O
O
- break the M-O-M links in the
-O-M-O- Na+
Na+ -O-M-O
ionic bonds
network intermediates
- repair the breaks
- e.g. Al2O3
-O-M-O-Al-O-M-O
8
Na+
BO
Si
NBO
Na
Network former SiO2
all bridging oxygens (BO)
Add network modifier Na2O
form non-bridging oxygens (NBO)
9
form [AlO4]- Na+
groups with all BO
If Na2O = Al2O3
all NBO removed
Add intermediate oxide Al2O3
remove non-bridging oxygens (NBO)
10
GLASS STRUCTURE
•range of M-O-M
bond angles
•range of ring
sizes
crystal
long-range order
glass
short-range order
same structural units
11
Theories of glass structure
Tend to be devised for one glass type only
e.g. Random Network Model – oxide glasses
Bernal (sphere packing) Model – amorphous metals
Random Network Theory (Zachariasen 1932)
Empirical rules derived from observation of oxide glasses AnOy
1.
2.
3.
4.
O not coordinated to more than two atoms of A
Coordination of A small (Si 4; B 3 or 4)
coordination polyhedra of A must share corners not edges or faces
at least 3 corners of each polyhedron must be shared to give a 3D
network and at least 2 corners must be shared for a glass to form
Implies 1st coordination sphere about A is well-defined and nearly
identical to crystal (SHORT RANGE ORDER - SRO)
Most common coordination polyhedron found in oxide glasses is the
tetrahedron (rule 2) e.g. [SiO4], [GeO4], [PO4], [AlO4], [BO4],
Octahedron – edge and face sharing breaks rules 1, 2 and 3.
12
Origin of disorder
Consider two tetrahedra from network
C3
• Rotation of tetrahedron A about C3
axis of B can occur in melt by bond
breaking at other oxygens.
A

B
• On cooling, bonds reform at random
and resulting orientations are frozen
in. Ordering to give crystal would
require breaking of bonds.
• Disorder results in range of dihedral
angles, , from 120o – 180o with max
probability at ~140o.
SHORT RANGE ORDER (SRO)
Tetrahedron
But no LONG RANGE ORDER (LRO)
13
Categories of oxides
1.
Glass (network) former – can form a glass as the pure substance
e.g. SiO2, B2O3, GeO2, P2O5
2.
Intermediates – can substitute for network formers but cannot
form glasses independently e.g. Al2O3, TiO2
3.
Network modifiers – disrupt network by breaking interpolyhedral
linkages e.g. alkali metal oxides, alkaline earth oxides
If enough linkages are broken, network cannot be maintained and melt
will crystallise rather than form glass on cooling
Si-O-Si
+ Na2O

Si-O- Na+ Na+ -O-Si
Bridging oxygen (BO)
non-bridging
oxygens (NBO)
Na+ closely associated with NBO for charge balance
14
Na2O-SiO2 system
SiO2
4BO/Si
3D network
33.3Na2O 66.7SiO2
(3BO+1NBO)/Si
2D network – sheets
50Na2O 50SiO2
(2BO+2NBO)/Si
1D network – chains and rings
Q4
Q3
Q2
66.7Na2O 33.3SiO2
(4NBO)/Si
ions SiO44-
60Na2O 40SiO2
(1BO+3NBO)/Si
dimers Si2O76-
Q0
Q1
Qn - tetrahedral (quaternary) species with n bridging oxygens (BO)
15
Q species
Q4 – 3D network
Q3 – 2D network
sheets
Q1 – dimeric
ions
Q2 – 1D network
chains and rings
Q0 – monomeric
ions
NB – in reality, these are 3D entities.
As x increases, number of NBO increases, connectivity of network decreases
and glass becomes less stable.
Zachariasen – need at least two BO per Si to form glass
Question – what combination of species will occur for a given composition 16
(x)?
Distribution of species – xM2O (1-x)SiO2
100
Binary Model
90
80
70
% species
NBO repel
- no more than two species
present at any composition
4
60
50
40
30
xM2O (1  x )SiO2
20
10
total Si  1  x
total O  2  x
NBO  2 x
BO  2  3 x
e.g. 0.33  x  0.5
0
0.0
total Si  1  x  Q  Q
3
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
x - mole fraction M2O
0  x  0.33
Q3  Q2
2
NBO  2 x  Q 3  2Q 2
Q2  3x  1
Q
3
Q
2
Q
1
Q
0
Q
0.33  x  0.5
0.5  x  0.6
Q3  2  4x
0.6  x  0.66
1  3 x 
%Q 4 =100  

 1 x 
 2  4x 
%Q3 =100  

 1 x 
 3  5x 
%Q 2 =100  

 1 x 
 4  6x 
%Q1=100  

 1 x 
 2x 
%Q 3 =100  

1  x 
 3 x  1
%Q 2 =100  

 1 x 
 4x  2 
%Q1=100  

 1 x 
 5x  3 
%Q0 =100  
 x 
 1
17
Statistical model
 2  3x 
%Q 4  100  

 2  2x 
4
100
90
80
4
70
% species
NBO do not repel.
Distribution of species depends
only on composition and statistics.
xM2O (1-x)SiO2
Q
3
Q
2
Q
1
Q
0
Q
60
50
40
30
20
 x   2  3x 
%Q 3  100  4  


 2  2x   2  2x 
3
2
 x   2  3x 
%Q 2  100  6  
 

 2  2x   2  2x 
10
2
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
x - mole fraction M2O
3
 x   2  3x 
%Q1  100  4  
 

 2  2x   2  2x 
 x 
%Q  100  

 2  2x 
0
4
 2  3x 
BO probability 

 2  2x 
 x 
NBO probability 

 2  2x 
18
Structural techniques for glasses
Magnetic resonance
Diffraction
- NMR
- ESR
- X-ray diffraction
- Neutron diffraction
Vibrational spectroscopy
- EXAFS
– Infra-red
- Raman
Mössbauer spectroscopy
Modelling
Need techniques which will be sensitive to changes in local
environment (SRO) and which do not depend on LRO
19
Nuclear Magnetic Resonance (NMR)
• Requires nuclear spin I > 0
16O I=0 no NMR; 17O I=5/2 (0.037% abundant!)
I = 1/2
-1/2
E
no
field
+1/2
H0
Zeeman
interaction
0
• Degeneracy of nuclear spin states lifted in
presence of a magnetic field – Zeeman
interaction
• Transitions between states
E = h0
0 - Larmor frequency - rf energies
• E depends on magnitude of applied field H0;
magnetic properties of nucleus; local
environment
E = hH/2
 resonance (Larmor) frequency 0 = H/2
•  is the gyromagnetic ratio of the nucleus and H
is the field at the nucleus
20
Chemical shift
• H differs from the applied field H0 because local fields exist due to
chemical environment (motion of electrons in bonds and on
neighbouring atoms)
• H = H0(1-)
• 0 = H0(1-)/2
 - chemical shielding tensor
frequency depends on local environment
•  is an indicator of the local environment and therefore is useful for
glasses as well as crystals
• Difficult to measure H accurately  in practice measure difference in
resonance frequency of sample and a standard reference
• Chemical shift  = 106  (sample - standard)/ standard
(ppm)
21
Quadrupole Interaction
Nuclei with I>1/2 have a quadrupole moment which interacts with electric
field gradient at nucleus
I = 1/2 (e.g. 29Si)
I = 3/2 (e.g. 11B)
-3/2
E
-1/2
E
no
field
-1/2
E
+1/2
+1/2
E
+3/2
I = 3/2
-3/2
E
E + 2
-1/2
E
E
+1/2
E
E - 2
+3/2
Zeeman
interaction
Zeeman
interaction
1st order quadrupole
interaction
0
0
0
22
(a)
xLi2O (1-x)SiO2 29Si
spectra Q Q Q Q Q
0
1
2
3
fit peaks for each
species and
integrate under
peak for
concentrations
4
(b)
NB glasses made
by roller-quenching
x=
0.375
0.444
-40
-50
-60
-70
-80
-90
-100
-110
-120
ppm
Q3
100
0.5
Q2
Q1
Q0
abundance %
80
0.545
0.583
predicted binary
JNCS 116 (1990) 148
"
"
60
40
Q4
20
0.615
0
0.2
0.643
0.3
0.4
0.5
0.6
0.7
0.8
x - mole fraction Li2O
-40
-60
-80
-100
ppm wrt TMS
chemical shift  (ppm wrt TMS)
-120
Close to binary distribution
n-1
disproportionation 2Qn  Qn+1 + Q
23
29Si
xPbO (1-x)SiO2
NMR
Close to statistical distribution
Pb-O bonds mostly electron-pair whereas Li-O bonds ionic
x=
100
0.333
Q4
0.444
Q0
0.5
80
0.667
0.71
0.75
% species
0.6
60
Q1
Q3
Q2
40
0.78
0.8
20
0.818
0.833
0
-30
-40
-50
-60
-70
-80
-90 -100 -110 -120 -130 -140 -150
chemical shift (ppm wrt TMS)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
x - mole fraction PbO
24
Diffraction
roller quenched Sb2O3
60
c-Sb2O3 senarmontite
50
Conventional
diffraction
*
*
Intensity
40
Diffraction from planes of
atoms
30
20
In glasses there is no LRO
therefore no planes of atoms
10
Consider diffraction from
pairs of atoms instead
0
10
20
30
40
50
60
70
2 (degrees)
25
Radial distribution function
Radial distribution function
RDF
J(r) - the number of atoms lying between r
and r + dr from the centre of a given atom
J(r) = 4r2(r) where the density function (r) is the
atomic pair correlation function
(r) = 0 at distances less than the nearest
neighbour distance
(r) = 0, the average value of density at
very large values of r
Diffraction pattern is sum of patterns for all possible
central atoms
The area under a given peak gives the coordination
number for that shell of atoms.
The width of the peak reflects thermal motion and
topological (static) disorder
2 = T2 + D2
26
SnO – SiO2 neutron diffraction
Diffraction from pairs of atoms - superposed
Si-O
Sn-O
O-O
FT
Diffraction pattern
Total correlation function
Q = 4sin/
27
(related to Radial distribution function)
Obtain
- interatomic distances – bond lengths
- bond angles calculated from distances
- coordination numbers from areas under peaks
distances (Å)
Si-O
1.6
Sn-O
2.1
O-O
2.6
Sn-Sn
3.5
rSi-O

rSi-O
rO-O
sin /2 = rO-O/2rSi-O
 = 108.7o i.e. close to
tetrahedral
28
X-ray versus neutron diffraction
7
Sn-Sn
6
T(r) (barns Å-2)
5
Si-O
O-O
4
3
Sn-O
2
HEXRD
ND
1
0
-1
0
2
4
r (Å)
6
8
X-ray scattering depends on atomic number  Sn scatters much
more strongly than O and Si
Neutron scattering changes irregularly between nuclei
29
Pb9Al8O21
- glass (red)
- crystal (black
3.0
LRO
-1
T(r) (Barns Å )
2.5
SRO
2.0
1.5
1.0
0.5
0.0
0
1
2
3
4
5
6
7
8
9
10
r (Å)
SRO of glass and crystal very similar
LRO very different
30
EXAFS
Extended X-ray absorption fine
structure for a given element in
the glass.
Measure absorption coefficient
(), as a function of energy , on
going through an absorption edge
(core-level electron being excited)
I  Io exp   ( ) x 
Fine structure arises from
diffraction of out-going electron by
neighbouring atoms.
The central atom is fixed and
therefore get a single distribution
function.
http://srs.dl.ac.uk/XRS/Theory/theory2.html
31
Raman spectroscopy
The three main scattering processes in Raman spectroscopy. An
incident photon of frequency 0 excites an electron which then either:
(a) relaxes back to its initial state releasing a photon of frequency 0 –
Rayleigh scattering
(b) relaxes to an excited state releasing a photon with a lower
frequency - Stokes
(c) or induces photon emission with a higher frequency – anti-Stokes.
v  v0  v 
E f  Ei
h
32
Absorptions due to vibrations of groups of atoms – therefore get
information about SRO and sometimes about medium range order
(MRO).
B2O3 – unusual in having MRO in form
of ring structures
Boroxol group (B3O3 ring) gives
characteristic absorption at ~ 800 cm-1
due to “breathing mode” of oxygens in
the ring.
O4
B1
O5
O1
B3
B4
O6
O2
O7
33
xLi2O (1-x)SiO2
Q
mol% Li2O
3
37.5
Q
44.4
2
Q
50.0
Q
3
3
Q
54.5
4
58.3
800
900
1000
1100
1200
1300
-1
Raman Shift (cm )
600
800
1000
-1
Raman Shift (cm )
1200
1400
1600
1800
100
3
NMR Q
3
Raman Q
Raman (Umesaki [21])
80
3
60
%Q
400
40
20
0
35
40
45
50
mol% Li2O
55
60
65
34