Web Course
Physical Properties of Glass
1. Properties of Glass Melts
2. Thermal Properties of Glasses
Richard K. Brow
Missouri University of Science & Technology
Department of
Materials Science & Engineering
FS08
Richard K. Brow
[email protected]
Melt properties-1
Melt and Glass Properties
•
•
•
•
•
FS08
Viscosity- chapter 9
Surface Tension- chapter 9
Thermal Expansion- chapter 10
Heat Capacity- chapter 11
Thermal Conductivity- chapter 12
Richard K. Brow
[email protected]
Melt properties-2
Supplementary References on Viscosity
• Structure, Dynamics and Properties of Silicate
Melts, Reviews in Mineralogy, Vol. 32 (1995),
ed. JF Stebbins, PF McMillan and DB Dingwell
(Mineralogical Society of America)- several
outstanding reviews of viscosity, relaxation, etc.
• CA Angell, Science, 267, (1995), 1924- concepts
of melt fragility, configurational entropy, etc.
• JH Simmons and C Simmons, Cer Bull 68[11]
1949 (1989)- Non-Newtonian behavior
• HE Hagy in Introduction to Glass Science, ed.
LD Pye, et al Plenum Press (1972)- nice review
of viscosity measurements
FS08
Richard K. Brow
[email protected]
Melt properties-3
Why should we care about melt viscosity?
1. Glass Forming Tendency
a. Nucleation, crystallization, phase separation kinetics
IVHO
3
⎤
⎛
⎞
⎡
k
T
K
σ
⋅
−
⋅
0
⎟ exp ⎢
= NV ⎜⎜
3
2⎥
⎟
⎝ 3π ⋅ a0 ⋅η ⎠
⎣ T ⋅ ΔGv ⎦
2. Melt Fining
Stoke' s Law :
d 2 g ( ρb − ρl )
V=
12η
3. Manufacturing Process Control
4. Annealing Schedules/Permanent Stress
5. What else??
FS08
Richard K. Brow
[email protected]
Melt properties-4
F,V
d
Viscosity ≡ η = Fd
Units:
FS08
AV
σ
= 0
Newtonian Liquids:
σ0
Time
•
ε
(dynes·cm)/(cm2·(cm/s))
= dyne·s/cm2 = Poise or
N·s/m2 = Pa·s
1 Pa·s = 10 P
1 P = 1 dPa·s
Strain (ε)
A
Shear stress (σ)
Viscosity Definitions
Richard K. Brow
[email protected]
ε = σ0
•
η
Time
Melt properties-5
Practical Consequences
FS08
Richard K. Brow
[email protected]
Melt properties-6
Elastic Solid
Newtonian Liquid
Viscoelastic transition
glass
forming
melting
η(T) for SLS melt
FS08
Richard K. Brow
[email protected]
Melt properties-7
FS08
Richard K. Brow
[email protected]
Melt properties-8
Important Manufacturing Viscosities
FS08
Name
η (Pa·s)
Melting pt (Tm)
101
Melting range
100.5-101.5
Working pt (Tw)
103
Working range
102-106
Forming
Liquidus temp (Tl)
~104
No crystallization for T>Tl
Flow point
104
Softening point (TLit)
106.6
Littleton, flow under own weight
Crystallization temp (Tx)
~107
No crystallization for T<Tx
Deformation temp (Td)
1010-1011
Dilatometric: expansion
compensated by viscous flow
Glass transition (Tg)
1011-1012
Annealing pt (Tap)
1012
Internal stresses relieved <15 min
Strain pt (Tsp)
1013.5
Internal stresses relieved <15 hrs
Remarks
Melting, fining
Richard K. Brow
[email protected]
Melt properties-9
Defined Viscosities
log Pa·s
log P
Working Pt
103
104
Littleton
Softening Pt
106.6
107.6
Annealing Pt 1012
1013
Strain Pt
Annealing
Range
1013.5
1014.5
Working
range
η(T) for SLS melt
FS08
Richard K. Brow
[email protected]
Melt properties-10
Viscosity Classifications
• Working Range: Temperatures (ΔT) between ‘working
point’ and ‘softening point’
– Long glasses: large ΔT (shallow η(T) curves)
– Short glasses: small ΔT (steep η(T) curves)
– Hard glasses: Working range at greater temperatures
than for S-L-S glass
• Borosilicates, aluminosilicates, oxynitrides, silica, etc.
• Sometimes defined as CTE<6x10-6/°C
– Soft glasses: Working range at lower temperatures
than for S-L-S glass
• Soda-lime silicate, Pb-silicates
• Sometimes defined as CTE>6x10-6/°C
FS08
Richard K. Brow
[email protected]
Melt properties-11
From Seward and
Varshneya (2001)
Soft glasses
Short
glass
FS08
Hard glasses
Long glass
Richard K. Brow
[email protected]
Corning Codes:
8363: High PbO radiation
shield
0010: Pb-silicate tube
7070: Borosilicate
0080: SLS lamp glass
7740: Pyrex
1720: Alkaline-earth boroaluminosilicate
Melt properties-12
Measurement of Viscosity
Range
Melting
Method
Viscosity Values
Falling Sphere/Bubble Rise
Margules Rotating Cylinder
Parallel Plate
Softening Penetration Viscometer
and
Fiber Elongation
Annealing Beam Bending
Disappearance of Stress
FS08
Richard K. Brow
[email protected]
η<104 Pa-s
η<106 Pa-s
105 Pa-s<η< 109 Pa-s
105 Pa-s<η< 109 Pa-s
105 Pa-s<η< 1015.5 Pa-s
107 Pa-s<η< 1012 Pa-s
1011 Pa-s<η< 1014 Pa-s
Melt properties-13
Rotating Spindle: 10-106 Pa·s
(ASTM C965-96)
1 ⎛1
1 ⎞⎛ Τ ⎞
η=
⎜ 2 − 2 ⎟⎜ ⎟
4π ⋅ L ⎝ r
R ⎠⎝ ω ⎠
r
R
FS08
Richard K. Brow
[email protected]
L
Τ = torque
ω = rotational velocity
Melt properties-14
Littleton Softening Point: 106.6 Pa·s
(fiber elongation- ASTM C338-93)
L⋅F
Applied Stress= F/A
η=
3 A ⋅ (dL / dt ) Elongation rate=dL/dt
Balance of gravitational force
(density) and surface tension
FS08
2/25/03
Richard K. Brow
[email protected]
Melt properties-15
Annealing/Strain Points: 1012, 1013.5 Pa·s
(fiber elongation: ASTM-C336-69)
dL/dt = 2.5x10-6 l/d2
at 1012 Pa·s (anneal pt)
Strain pt elongation rate is
0.0316 x annealing pt
elongation rate (1.5 log units)
FS08
2/25/03
Richard K. Brow
[email protected]
Melt properties-16
Beam Bending: 108-1013 Pa-s
g ⋅ L3 ⎛ M + A ⋅ L ⋅ ρ ⎞
η=
⎜
⎟
2.4 I c ⋅ V ⎝
1.6
⎠
V=deflection rate
FS08
2/25/03
Richard K. Brow
[email protected]
Melt properties-17
FS08
Richard K. Brow
[email protected]
Melt properties-18
The temperature dependence of viscosity
FS08
Richard K. Brow
[email protected]
Melt properties-19
σyx
layer
velocity
gradient
A
vAx
B
y
hole
C
vBx
vCx
x
Consider the ‘activated’ motion of a hole under the action of a shearing stress
FS08
Richard K. Brow
[email protected]
Melt properties-20
σyx
Va
2
V
σ yx a
2
Potential energy
σ yx
ΔG0
ΔG0
Applied shear biases potential energy function
Jump frequency (υ0), no shear:
• Va is atom volume
• Same l-r as r-l
• Forward jump frequency (υ+) exceeds
• Depends on barrier energy and
reverse (υ-)
probability of finding suitable hole
σ yxVa
as neighbor (Ph)
) / k BT ] ⋅ Ph
υ + = [k BT / h] exp[−(ΔG0 −
υ0 = [k BT / h] exp[−ΔG0 / k BT ] ⋅ Ph
υ + = υ0 exp[σ yxVa / 2k BT ]
υ − = [k BT / h] exp[−(ΔG0 +
υ − = υ 0 exp[−σ yxVa / 2k BT ]
FS08
Richard K. Brow
[email protected]
2
σ yxVa
2
) / k BT ] ⋅ Ph
Melt properties-21
σyx
Va
2
V
σ yx a
2
Potential energy
σ yx
ΔG0
The net ‘forward velocity’ is
(vBx − v Ax ) = (υ + − υ − )δx
∂v / ∂y = (υ + − υ − )δx / δy ≈ (υ + − υ − )
∂v / ∂y = 2υ0 sinh(σ yxVa / 2k BT ) ≅ υ 0σ yxVa / k BT
shear strain rate is e&xy = ∂v / ∂y,
η = σ yx / [υ0σ yxVa / k BT ] = [h / Va ][exp(ΔG0 / k BT ](Ph )−1
Consider the energy required to create a hole
(ΔEh), then Ph can be described by
Ph = exp[− ΔEh / k BT ]
substituting Ph into the viscosity equation,
η = [h / Va ][exp(ΔG0 + ΔEh ) / k BT ]
Simplifying as an Arrhenius equation:
η = η 0 exp(ΔHη / RT )
FS08
Richard K. Brow
[email protected]
Melt properties-22
Most glass-forming liquids are non-Arrhenius
2000°C 1500°C
1000°C
500°C
SiO2
NTS2
NS3
NS2
Ab=albite
An=anorthite
Di=diopside
Ab
B
logη = A +
T − T0
An
Di
From P. Richet and Y. Bottinga, in Reviews in Mineralogy, Vol. 32, (1995), p. 67-93
FS08
Richard K. Brow
[email protected]
Melt properties-23
Log (viscosity in poise)
Melt
Fragility
Strong
Fragile
From C. A. Angell, Science, 267, (1995), 1924.
FS08
Richard K. Brow
[email protected]
Tg/T
Melt properties-24
Why the non-Arrhenius
temperature-dependence?
1.
Energy for hole formation (ΔEh) is low at high
temperatures
ƒ ΔHη is greater at lower temperatures
2. Free-volume increases with temperature
3. Configurational entropy increases with temperature
(Adam-Gibbs description)
FS08
Richard K. Brow
[email protected]
Melt properties-25
What accounts for viscous flow in
a silicate melt?
What has to happen for flow to
occur?
FS08
Richard K. Brow
[email protected]
Melt properties-26
What happens at the molecular-level that affects viscosity?
Q2
FS08
At the short timescales, ‘melt structures’ are
similar to ‘glass structures’….
• Raman spectroscopy- probing
structure on timescales 10-12-10-14 s
• Si-O stretching/bending modes remain
dominate from room temperature into the
melt
• Some evidence for some melt
speciation reactions: 2Q3 ↔ Q2 + Q4
Richard K. Brow
[email protected]
Melt properties-27
NMR provides structural information about glasses
Q1
Q2
Q3
Q4
Q4
Q3
Q2
Q1
Q0
ni
xLi2O (1-x)SiO2
FS08
Richard K. Brow
[email protected]
Melt properties-28
Chemical exchange in
melts: silicate species
and viscous flow
• NMR exchange
frequencies (kHz range)
are comparable to the
timescales for viscous flow
in silicate melts
• The ‘lifetimes’ for Si-O
bonds in a melt can be
determined and compared
with timescales associated
with viscous flow
• At high temperatures, the
Q3-Q4 exchange (Si-O
bond rupture) is fast
compared to the
experimental time frame.
FS08
Experimental Simulated Spectra
697°C
2 kHz
774°C
10 kHz
800°C
25 kHz
847°C
50 kHz
997°C
500 kHz
Richard K. Brow
[email protected]
Melt properties-29
NMR exchange and viscosity timescales coincide
1. Maxwell relationship: η = τshear · G∞
2. Assume that τshear ≈ τex
3. Calculate diffusivity (D) from τex : D =d2/6τ, d is ‘jump distance’
4. Calculate viscosity from η = kBT/(dD)
FS08
Richard K. Brow
[email protected]
Melt properties-30
Stebbins model for viscous flow
Si-O bond-rupture through Q3-Q4
site exchange
• Conversion of one bridging
oxygen to a nonbridging oxygen
• ‘Diffusion’ of modifying cation
from one silicate unit to another
initial
Potential energy
• Creation of an SiO5 transitional
site?
transitional
?
FS08
transitional
initial
final
final
Richard K. Brow
[email protected]
Melt properties-31
NMR evidence for
transitional sites
‘frozen into’ quenched
glass structures
Q3/Q4 peak
SiO5
Na2Si4O9
Fast
quench
Slow
quench
FS08
Richard K. Brow
[email protected]
Cs2Si4O9
Melt properties-32
Effects of composition on viscosity
Viscosity is determined by
• Molecular attractive forces, especially
associated with glass-forming oxides
– Si-O vs. Ge-O
• Number of non-bridging oxygens in
structure
– Alkali oxide additions reduce viscosity
– Water (-OH) and fluorine reduce
viscosity
• Coordination number of the cation
B[3] vs B[4]
FS08
Richard K. Brow
[email protected]
Melt properties-33
FS08
Richard K. Brow
[email protected]
Melt properties-34
Reminder: Effect of Modifier Additions on
Silicate Glass Networks
⏐
O
⏐
⎯
⏐
⏐
O
O
⏐
⏐
⏐
R+
O
⏐
O⎯Si ⎯ O ⎯ Si ⎯O ⎯ +R2O → ⎯ O⎯Si ⎯ O- -O ⎯ Si
⏐
⏐
⏐
⏐
R+
O
O
O
O
⏐
⏐
⏐
⎯O ⎯
⏐
B0 +R2O → 2NBO
2Q4 + R2O → 2Q3
FS08
Richard K. Brow
[email protected]
Melt properties-35
Increasing the modifier
content reduces viscosity
• Water is a particularly effective
flux
Na-K-Zn-Al-silicate
(Wu, JNCS, 41 381, 1980)
FS08
Richard K. Brow
[email protected]
Melt properties-36
The effect of
water on Tg
(η≈1012 Pa·s) of
silicate glasses
Deubener, et al., JNCS 330, 268 (2003).
FS08
Richard K. Brow
[email protected]
Melt properties-37
The effects of modifier
content on melt viscosity
FS08
Richard K. Brow
H. Rawson,
Properties and Applications of Glass, Elsevier,
1980.
Melt properties-38
[email protected]
Isokom temperatures for mixed alkali melts
Log η (Poise)
8
10
12
16
Nemilov (1969)
FS08
Richard K. Brow
[email protected]
Melt properties-39
Reminder: Effect of Alumina Additions on Silicate Glass
Networks
⏐
O
⏐
⎯
⏐
O
R+
⏐
O⎯Si ⎯ O- -O ⎯ Si
⏐
⏐
R+
O
O
⏐
⎯O ⎯
1/2Al2O3 for
⏐
⏐
O
⏐
⎯
⏐
O
⏐
R+
⏐
O
⏐
O⎯Si ⎯ O ⎯ Al ⎯O ⎯ Si ⎯O ⎯
⏐
O
⏐
FS08
1/2R2O
Richard K. Brow
[email protected]
⏐
O
⏐
⏐
O
⏐
Melt properties-40
Viscosity of alkali aluminosilicate melts is greatest when
Al/Na≈1- fully cross-linked
networks….
Toplis et al., Geochim.
Cosmochim. Acta.
61[13] 2605 (1995)
FS08
Richard K. Brow
[email protected]
Melt properties-41
Reminder: Effect of Alkali Addition on Borate Glass Networks
R+
O
—
B—O —
+1/2R2O
O
[O]/[B]=1.5
—
—
O
B
—
O
O—
[O]/[B]=2.0
O—
+1/2R2O
+1/2R2O
R + -O
R + -O
B — O- R+
B—O—
R + -O
[O]/[B]=3.0
FS08
R + -O
[O]/[B]=2.5
Richard K. Brow
[email protected]
Melt properties-42
2BØ3 + Na2O →
Alkali borate melt viscosity
2(BØ4-Na+)
2(BØ4-·Na+) + Na2O → 2(BØO22-·2Na+)
Note the loss of the ‘borate
anomaly’ effect at high
temperatures (low viscosity)
FS08
Richard K. Brow
[email protected]
Melt properties-43
38.6Li2O·61.4B2O3 glass and melt
Raman spectra indicate that the
BØ2O- triangles replace BØ4tetrahedra in the melt
Cormier et al., JACerS, 89 13 (2006)
FS08
Richard K. Brow
[email protected]
Melt properties-44
Raman spectra indicate that
the BØ2O- triangles replace
BØ4- tetrahedra in the melt
BØ4-Li+ ↔ BØ2O-Li+
Note that B-O bonds are
broken, and that such
configurational changes will
contribute to changes in
heat capacity….
Cormier et al., JACerS, 89 13 (2006)
FS08
Richard K. Brow
[email protected]
Melt properties-45
Effects of composition on viscosity
Component
Alkali oxide
Alkaline earths
PbO
Al2O3
B2O3
OH-/F-
Effect on Viscosity
Low
High
Temp
Temp
+
+
+
+
-
Glass
Softer
Shorter
Softer
Harder
Shorter
Softer
Modifications to soda-lime silicate melt viscosity, after Beerkens, 1997.
FS08
Richard K. Brow
[email protected]
Melt properties-46
Viscosity dependence on
temperature and composition
• Vogel-Fulcher-Tamman (VFT) Equation
B
logη = A +
T − T0
• Lakatos Method*: empirical additivity factors
A = −2.4550 + ∑ ai ⋅ pi
B = 5736.4 + ∑ bi ⋅ pi
T0 = 198.1 + ∑ ti ⋅ pi
For S-L-S melts, T (°C) and
η (Pa·s), pi (mole fraction
oxide per mole SiO2)
*T. Lakatos, et al., Glass Technology, 13 88 (1972)
FS08
Richard K. Brow
[email protected]
Melt properties-47
Lakatos additivity parameters (after Beerkens, 1997)
Na2O
K2O
MgO
CaO
B2O3
Al2O3
PbO
ai
bi
ti
+1.4788
-0.8350
-5.4936
-1.6030
-15.880
+1.5183
+1.3058
-6039.7
-1439.6
+6285.3
-3919.3
+7272.1
+2253.4
-5880.0
-25.07
-321.0
-384.0
+544.3
+521.4
+294.4
-275.5
Valid for the log(viscosity) range 1-12 Pa·s
FS08
Richard K. Brow
[email protected]
Melt properties-48
Not all liquids exhibit Newtonian viscosity behavior
Shear Stress (σ)
Bingham Plastic
Pseudoplastic (decreasing η)
Newtonian (constant η)
Dilatant (increasing η)
Shear Rate
FS08
Richard K. Brow
[email protected]
Melt properties-49
Non-Newtonian Viscosity
Shear thinningdecreasing effective
viscosity with increasing
deformation rates
• fiber drawing
• press-and-blow
Greater problem at
higher temperatures
Yue & Brückner
Glastech. Ber (1996)
FS08
Richard K. Brow
[email protected]
Melt properties-50
Non-Newtonian Viscosity
Simmons and Simmons, Cer Bull
68[11] 1949 (1989)
η / η 0 = 1 / [1 + (e&xy )η 0 / σ ∞ ]
η0 is Newtonian viscosity
σ∞ is the cohesive shear strength
FS08
Richard K. Brow
[email protected]
Melt properties-51
Cohesive strength increases with viscosity
•Viscosity becomes nonlinear at high shearing rates
•Shear stress builds up if stress relaxation rate is sufficiently low
•If shear stress>σ∞, then ‘liquid fracture’ can occur
FS08
Richard K. Brow
[email protected]
Simmons and Simmons, Cer Bull
68[11] 1949 (1989) Melt properties-52
Consequences of Non-Newtonian Viscosity
• High-speed glass processing
– Fiber attenuation
– Container processing
• Source for glass inhomogeneities
– Induced phase separation or crystallization in high
shear regions
FS08
Richard K. Brow
[email protected]
Melt properties-53
• Fibers heated below Tx
• Stress in ‘bent’ regions
reduces ‘effective’ viscosity
• Examine differences in
crystallization behavior
FS08
Richard K. Brow
[email protected]
Melt properties-54
FS08
Richard K. Brow
[email protected]
Melt properties-55
Another example
• Melts prepared in micro-gravity have
greater glass-forming tendencies than
melts on earth at comparable quench
rates
– Hypothesis*: gravity-driven fluid-flow
increases overall strain rate within melt
• Reduced ‘local’ viscosity through shear thinning
• Increased ‘local’ crystallization rates
*CS Ray et al, Trans. Indian Inst. Met. 60[2] 143 (2007)
FS08
Richard K. Brow
[email protected]
Melt properties-56
Wall shear stress (Pa)
Maximum strain rate (s-1)
Calculated for LS2 melts at 1400°C and a 5°C temperature gradient
Normalized acceleration due to gravity
Trans. Indian Inst. Met. 60[2] 143 (2007)
FS08
Richard K. Brow
[email protected]
Melt properties-57
Normalized acceleration due to gravity
Shear-thinning
behavior is reduced
when gravitational
effects are reduced.
FS08
Richard K. Brow
[email protected]
Melt properties-58
From J. H. Simmons, in Experimental Techniques
of Glass Science, (1993), p. 383-432.
Phase separation
affects viscosity
FS08
Richard K. Brow
[email protected]
Melt properties-59
Crystallization affects
viscosity
• Example: crystallizable
sealing glass
From H. E. Hagy, in Introduction
to Glass Science, (1972), p. 343371
FS08
Richard K. Brow
[email protected]
Melt properties-60
Viscosity Summary
• Viscosity is the most important melt property
– Critical for processing, from melting through annealing
• Compositional dependence can be understood in terms
of melt/glass structure
– Stronger networks = greater viscosity
• Fraction of NBO’s on silicate tetrahedra
• Aluminosilicate networks
• Borate ‘anomaly’- tetrahedral sites
– Viscosity is sensitive to changes in melt structure
• Temperature dependence is non-Arrhenian
– VFT equation is a useful empirical description
– Fragile/strong classification can be related to configurational
changes
• Shear-thinning has processing consequences
FS08
Richard K. Brow
[email protected]
Melt properties-61
Glass formation is ‘crystallization avoidance’
⎡
⎛ 0.3ΔH
ATm
Tc ≈
⋅ exp(− 0.212 B ) ⋅ ⎢1 − exp⎜⎜ −
η (at 0.77Tm )
⎝ RTm
⎣
•
2
Free energy
Nucleation
driving force
barrier
Reduce critical cooling rate
η(Tm)
Tm(°C)
to improve glass formation
BeF2
540
>106 P
• Lower Tm
B2O3
460
105 P
• Increase η at Tm
GeO2
1150
107 P
If Tg (η=1013 P) is near Tm,
107 P
SiO2
1710
then η(Tm) will be high
⎞⎤
⎟⎟⎥
⎠⎦
3/ 4
Kinetic
barrier
Glass formation expected
when Tg/Tm>2/3
FS08
LiCl
613
0.02 P
Zn
420
0.03
Fe
1535
0.07
Richard K. Brow
[email protected]
Good
glass
formers
Poor
glass
formers
Melt properties-62
Eutectic compositions are good glass-formers
W. Vogel, Chemistry of Glass, 1985
From Dingwell in Rev. Mineral.
32 (1995)
G=U-1
Tg(°C)
ΔTg~30°C
Mole% Na2O
ΔTliq~800°C
Tg/Tliq is a maximum (~0.7) at the eutectic
FS08
Richard K. Brow
[email protected]
Melt properties-63