lecture2 martin
TRANSCRIPT
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Advanced Vitreous State: The Structure of Glass
Optimizinginternal for glass formation Structural approach to glass formation
Understand and be able to apply the relationships between atomic levelstructure and ease at which a system will form glass
Understand and be able to apply Zacharaisens Rules for glass formation
Be able to apply understanding of the three different types of additives,modifiers, intermediates, and glass formers, to multi-component systems topredict whether a particular composition will be glass forming or not.
Estimatingexternal for glass formation Kinetic approach to glass formation
Understand and be able to use nucleation and growth theory
Understand and be able to use TTT curves
Understand and be able to calculate critical cooling rates
Section 1: Lecture 2 Fundamentals of Glass Formation: Structural and
Kinetic Approaches
Glass formation results when the internal structural timescale of the liquid
becomes or is forced to become significantly longer than the external time
scale of the surroundings near the melting or liquidus temperature of the
liquid
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Structural Approach to Glass Formation
Crystalline materials exhibit aperiodic array of atoms and/or ions
Each atom/ion in the material has aspecific location that is periodic inthe crystalline structure
Each location can be exactlyspecified once the crystallinestructure is defined
Defects in the structure occur whenthe position and atom/ion type donot agree with that prescribed by thecrystal structure
A2O3 (B2O3)
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Structural Approach to Glass Formation
Amorphous materials lack this long range order
There is no prescription for which atoms/ions
are located at which locations However, the energetics of bond formation are
very strong
Atoms will align themselves chemically to:
Balance charge in ionic materials
Minimize bond energies by fillingappropriate bonding orbitals
Hence, local structure is disordered, but thereare still many similarities to the crystalline phase
Coordination numbers are ~ same
Bond lengths are ~ same
Bond Angles are ~ same
A2O3 (B2O3)
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Structural Approach to Glass Formation
Glass Formation results when
Liquids are cooled to below TM
(TL
) sufficiently fast to avoidcrystallization
Nucleation of crystalline seeds are avoided
Growth of Nuclei into crystallites (crystals) is avoided
Liquid is frustrated by internal structure that hinders both events
Structural Approach to Glass Formation What internal structures promote glass formation?
How can structures be developed that increase the viscosity andfrustrate crystallization processes?
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Structural Approach to Glass Formation
Using structure to promote glass formation
Develop atomic bonding structures in the system that produce largeviscosity near the melting point
Silicate liquids and glasses SiO2, Na2O + CaO + SiO2
Develop large molecular structures that due to their size prevent and/orfrustrate the organization into the crystalline structure
Polymeric liquids with large polymer chains -(CH2)n-
Develop complex local and variable structures in the liquid that oncooling have a large number of possible structural motifs to followand as a result no one structure is favored over another
Molten salt liquids with a number of components
Ca(NO3)2+ KNO3
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Zacharaisens Rules for Glass Formation
Glass formation requires long range continuous bonding in the liquid to
produce:
High viscosity
3 - Dimensional bonding
Strong individual bond strength
Open structure that is not efficiently packed
Corners of polyhedra are shared to increase connectivity Bonds for bridges between corner sharing polyhedra
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Zacharaisens Rules for Glass Formation
Oxygen atoms are linked (bonded) to no more than two atoms
Oxygen coordination around glass forming cations is small, 3, 4 Cation polyhedra share corners and not edges or faces
At least three corners are shared
William H. Zachariasen, Journal of the American Chemical Society
54 (1932) 3841-3851
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Zacharaisens Rules for Glass Formation
Apply these rules to the following:
SiO4/2
B2O3 or BO3/2
Apply these rules to the following:
CaO
Na2O
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Zacharaisens Rules for Glass Formation
SiO4/2
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Zacharaisens Rules for Glass Formation
B2O3 or BO3/2
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Zacharaisens Rules for Modifiers
Ca1O1 (CaO) Closed-packed cubic Ca occupying all octahedral
sites
Octahedral sites = Ca = O
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Zacharaisens Rules for Modifiers M2O
Na2O1 (Na2O) Closed-packed cubic Na occupying tetrahedral sites
Tetrahedral sites = 2 x O = Na
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Suns Bond Strength Model
Glass formation is brought about by both:
Connectivity of bridge bonds Strong Bonds between atoms (ions)
Sun classified oxide according to their bond strengths
Glass formers form strong bonds to oxygen rigid network,
high viscosity
Modifiers from weak bonds to oxygen Disrupt, modify,
network
Intermediates form intermediate bonds to oxygencant form
glasses on their own, but aid with other oxides to form glasses
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Suns Bond Strength Model
Glass formers
Greater than 80 kcal/mole bond strength with oxygen B2O3, SiO2, Geo2, P2O5, Al2O5.
Intermediates
Between 60 to 80 kcal/mol bond strength with oxygen
TiO2, ZnO, PbO.
Modifiers
Less than 60 kcal/mole bond strength with oxygen
Li2O, Na2O, K2O, MgO, CaO.
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Glass Formers (Oxides)form glasses on their own
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Intermediates (Oxides)assist in glass formation
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Modifiers (Oxides)degrade glass formation
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Classifying Oxides
How would each of the following be classified?
SiO2, B2O3, P2O5 TiO2, PbO
Na2O, CaO, ZnO
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Dietzels Field Strength Criteria
Sun classifies Al as both a glass former and an intermediate
Al2O3 does not form glass at normal quenching rates More factors are important than just bond strength
Small cations with high charge glass formers
Large cations with small charge modifiers
Medium sized cations with medium charge - intermediates
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Dietzels Field Strength Model
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Intermediatesassist in glass formation
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Glass forming oxidesform glass on their own
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Glass forming compositions
How would you classify the following compositions? Glassforming or not?
0.15Na2O + 0.35Al
2O
3+ 0.50SiO
2
0.35Na2O + 0.15CaO + 0.25Al2O3 + 0.25SiO2
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Advanced Vitreous State: The Structure of Glass
Kinetic Theory of Glass Formation
Understand and be able to use nucleation andgrowth theory
Understand and be able to use TTT curves Understand and be able to calculate critical
cooling rates
Section 1: Lecture 2 Fundamentals of Glass Formation: Structural and
Kinetic Approaches
Glass formation results when the internal structural timescale of the liquid
becomes oris forced to become significantly longer than the external
time scale of the surroundings near the melting or liquidus temperature of
the liquid
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Kinetic Approach to Glass Formation
Glass formation requires the by-passing of the crystallization eventsat Tm Structural approach is to create high viscosity to frustrate
nucleation and growth processes
SiO4/2 easily supercools due to the high connectivity of the liquidthrough strongO-Si-O- bonding
Kinetic approach to glass formation asserts:
All liquids can be made into the glassy state
The question is how fast must the liquid be cooled?
Fast quenching, >> 100oC/sec, implies marginal glass formingability
Slow cooling,
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Critical Cooling Rates for Various Liquids
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Calculating the Critical Cooling Rate
The kinetic approach to glass formation then becomes:
What is the Rc value for a particular liquid? If Rc >> 100
oC/sec, then the liquid is a poor glass former
If Rc is
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Nucleation and Growth Rates Control Rc
Nucleation, the first step
First process is for microscopic clusters (nuclei) of atoms or ions toform
Nuclei possess the beginnings of the structure of the crystal
Only limited diffusion is necessary
Thermodynamic driving force for crystallization must be present
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Growth of crystals from nuclei
Growth processes then enlarge existing nuclei
Smallest nuclei often redissolve
Larger nuclei can get larger
Thermodynamics favors the formation of larger nuclei
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Nucleation and Growth Control Rc
Poor glass formers:
Liquids which quickly form large numbers of nuclei close toTm
That grow very quickly
Good glass formers Liquids that are sluggish to form nuclei even far below Tm
That grow very slowly
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Nucleation and Growth RatesPoor Glass Formers
Strong overlap of
growthand nucleation rates
Nucleation rate is high
Growth rate is high
Both are high at the
same
temperature
Tm
T
Rate
Growth Rate (m/sec)
Nucleation Rate (#/cm3-sec)
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Nucleation and Growth RatesGood Glass Formers
No overlap of growth
and nucleation rates Nucleation rate is small
Growth rate is small
At any one temperature
one of the two is zero
Tm
T
Rate
Growth Rate (m/sec)
Nucleation Rate (#/cm3-sec)
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Nucleation Rate Theory
Rate at which atoms or ions in the liquid organize into
microscopic crystals, nuclei I = number of nuclei formed per unit time per unit volume of
liquid
Nucleation Rate (I) number density of atoms x
fastest motion possible xthermodynamic probability of
formation x
diffusion probability
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Nucleation Rate Theory
I = nexp(-NW*/RT)exp(-ED/RT)
n = number density of atoms, molecules, or
formula units per unit volume
= N/Atomic, molecular, formula weight = vibration frequency ~ 1013 sec-1
N = Avogadros number
= 6.023 x 1023 atoms/mole
W* = thermodynamic energy barrier to form
nucleiED = diffusion energy barrier to form nuclei
~ viscosity activation energy
Number density Fastest motion Thermodynamic probability Diffusion probability
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A word about the f(x,T) = exp(-x/T) function
This function is bounded
between 0 and 1 As x >> 0, f >> 1
As x >> , f >> 0
As T >> 0, f >> 0
As T >> , f >> 1
Sketch a series of curves
for T dependence on
linear f and log f
Linear T and 1/T
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Nucleation RateThermodynamic barrier W*
At r*, (W(r)/ r)r=r* = 0
r* = -2/ Gcryst(T)
W(r*) W* = 163/3(Gcryst(T))2
r
WS = 4r2, surface is the surface energy
WB = 4/3r3Gcrsyt(T), bulk
Gcrsyt
(T),
the Gibbs Free-Energy
of Cryst. per unit volume, Vm
Wtot = WS + WB
W*
r*
+
-
0
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Nucleation Rate I(T)
I = nexp(-N 163/3(Gcrsyt(T))2 /RT)exp(-ED/RT)
Gcryst(T) = Hcryst(Tm )(1 T/Tm)/VmHcryst(Tm )(Tm/Tm)
Gcryst(T)
0
+
-
Tm
Approx. for :
~ 1/3Hmelt/N1/3Vm
2/3
note Hmelt= - Hcryst
RT
E
T
T
RT
HnI Dm
crystexp
81
16exp
2
Liquid is Stable
Crystal is Stable
Liquid and Crystal are in equilibrium
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Growth Rates (T)
Crystal growth requires
Diffusion to the nuclei surface Crystallization onto the exposed crystal lattice
Gcryst
ED
lc = exp(-ED/RT)
cl = exp(-(ED- Gcryst) /RT)
net = lc - cl =
exp(-ED/RT) -
exp(-(ED- Gcryst) /RT)
= a net = a exp(-ED/RT) x
(1exp(Gcryst) /RT)
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Growth Rates - (T)
Diffusion coefficient, D
Stokes-Einstein relation between D and
Hence:
m
m
T
T
RT
H
TaN
fRTT exp1
)(3)(
2
)(3exp)(
2
TaN
fRT
RT
E
aTD
D
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Nucleation and Growth Rates
Nucleation and Growth Rates for Water
0
5E+22
1E+23
1.5E+23
2E+23
2.5E+23
3E+23
3.5E+23
4E+23
4.5E+23
100 150 200 250 300
Temperature (K)
Nucleationra
te(sec-1)
0.E+00
2.E-09
4.E-09
6.E-09
8.E-09
1.E-08
1.E-08
1.E-08
Growthrate
m-sec-1
Nucleation
rate(sec-1)
Grow th rate
(m/sec)
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Nucleation and Growth Rates
Nulceation and Growth for Silica
0.E+00
2.E+07
4.E+07
6.E+07
8.E+07
1.E+08
1.E+08
1.E+08
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100
Temperature (K)
NucleationRa
te(sec-1)
0
5E-26
1E-25
1.5E-25
2E-25
2.5E-25
GrowthRate
(m-sec-1)
Nucleation rate (sec-1)
Growth rate (m/sec)
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TimeTemperatureTransformation Curves (TTT)
How much time does it take at any one temperature for a
given fraction of the liquid to transform (nucleate and grow)into a crystal?
X(t,T) ~I(T)(T)3t4/3
where X is the fractional volume of crystals formed, typically
taken to be 10-6, a barely observable crystal volume
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TTT curves and the critical cooling rate, Rc
T
Tm
time
Rc very fast Rc much slower
Poor glass former Better glass former
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Time Transformation Curves for Water
T-T-T Curve for water
100
150
200
250
0 1 2 3 4 5 6 7 8 9 1
Time sec
Temperature(K)
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Time Transformation Curves for Silica
T-T-T Curve for Silica
200
400
600
800
1000
1200
1400
1600
1800
2000
1 1E+13 1E+26 1E+39 1E+52 1E+65 1E+78
Time (sec)
Tempera
ture(K)
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Summary
Glass formation results when the internal structural timescale ofthe liquid becomes or is forced to become significantlylonger than the external time scale of the surroundings nearthe melting or liquidus temperature of the liquid
Create high viscosity of the liquid near the melting point of theliquid that frustrates crystallization
Network bonding favorable for high viscosity
Configurational complexity that frustrates crystallizationpathways
Suppress the melting point through compositionalcomplexity to slow crystallization process
Surpass crystallization processes by limiting available tosystem for them to occur
Exceed critical cooling rate in region near and below thefusion point of the liquid