γλώσσες
Σελίδες
Νομικός
I.Fragility and its Relation to Other Glass Properties
II.Networks
Reinhard Conradt
April 6 & 8, 2010
structural aspects
cristobalite
A fibre glass
float glass
glassy basalt rock
) (... 154.0
,)2sin(2 2
1
α
θ
KCunmX
Xd
=
⋅⋅⋅=
( )180··2·2·
minmax21 π
θθ −=B
Bbroadeningpeak
2·θmin
2·θ in °
2·θmax
silica glass
nmbB
Xb 2)422 ; (
)·2··cos(21
=⇒°±=θ
) ( 154.0 ; )·2··sin(2 2
1α
θKCunmX
Xd ==
( ) 0698.0)422; (180··2·2· minmax2
1 =⇒°±−= BBπ
θθ
Bragg‘s formula relates lattice spacing d and diffraction angle θ:
broadening B of diffraction peaks:
interpretation: peak broadening B ⇔ range b of translational symmetry
interpretation: peak broadening B ⇔ fluctuationε = δd/d of lattice spacing
( ) %9)422; (2tan4 2
1=⇒°±
⋅⋅== ε
θδ
εB
dd
SiO
silica glass
crystalline silica
random packingof length-correlatedclusters1.
2.
W. Zachariasen (1934)
ε = ±0.00
ε = ±0.09
ε = 0.09; d ≤ ε·√2 is accepted as bond
ε = 0.09; d ≤ ε·√2 is accepted as bond
bond percolation diffusion path percolation
simulated 3D structure ofNa2Si2O5
Vessal et al. 1992
„Greaves channels“ of Na+
transport in Na2O-SiO2 glasses
Greaves & Ngai 1995
fractured silica surface, investigated by UHV-AFMFrischat, Poggemann, Heide 2003
evaluate all distance correlations in the window 0.2 - 0.3 nm (O-Si-O distances)
evaluate all distance correlations in the window 0.2 - 0.3 nm (O-Si-O distances)
evaluate all distance correlations in the window 0.2 - 0.3 nm (O-Si-O distances)
viscosity
frozen-in stateloss of degree of freedom"change of lane"
independentmotion
beginning ofcooperative effects
strictly correlated cooperative motion;loss of degree of freedom "velocity"
"glass transition" on the highway
traffic direction
illustration of the concept of configurational entropy
)(·1
logTST c
∝η
Adam & Gibbs 1965
• no translational symmetry
• no flow,• fixed mean position of all elements,• solid state-like degrees of freedom
state „traffic jam“: state „parking lot“:
• translational symmetry
Tammann‘s explanation of sudden transitions due to a continuously tightened constraint
cP = cP(conf) + cP(rot) + cP(vib) cP(rot) + cP(vib) cP(vib)
d
a0
0a3d ⋅> 00 a?a3d ⋅>⋅> 0a?d ⋅<
2.9312p
cos22? ≈⋅⋅=
2 3 40
2
4
6
degr
ees
of fr
eedo
m
d/a0
500 1000 1500
200
250
300
350
400
0 100 200 3000
20
40
60
80
100
120
140
160
180
c P in
J/(
mol
·K)
T in K
c P in
J/(
mol
·K)
T in K
glass
crystal
Tg =981 K
Tm =1665 K
Sm = 85.7 J/(mol·K)
glass crystal
Richet, Robie, Heminway (1986)Robie et al. (1978); Martens et. al. (1987)
Richet & Bottinga (1995)
maximum deviation fromT3 law at 20 KDuval (1990)Chamgagnon (1998)
example: CaO· MgO· 2SiO2 (diopside)
glass
crystal
Tg =981 K
Tm =1665 K
Sm = 85.7 J/(mol·K)
Robie et al. (1978); Martens et. al. (1987)Richet & Bottinga (1995)
example: CaO· MgO· 2SiO2 (diopside)
500 1000 1500
200
250
300
350
400
c P in
J/(
mol
·K)
T in K
short range order S.R.O.essentially the same as in the isochemicalcrystal, hence cP(crystal) ≈ cP(glass)
molperR
atomperkcP
3
3
=
=
0 500 1000 15000.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
(cgla
ss -
c X) /
T in
J/(m
ol·K
2 )
T in K
excessgT
0
Xglass SdTT
cc=
−∫
∫−
=gT
Xglassexcess dTT
TcTcS
0
)()(
low-T vibrational excess entropy:example: CaO· MgO· 2SiO2 (diopside)
nmTk
uhLexcess 52
max
≈⋅
⋅=⋅=λ
max. deviation from Debye‘s T3 lawat 20 K
medium range order M.R.O.
0 500 1000 15000.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
(cgla
ss -
c X) /
T in
J/(m
ol·K
2 )
T in K
vitm
mT
gT
Xliq
SS
dTT
cc
−=
−∫ ( )
( )
∆
+⋅−=∂
∂
⋅∆
+
⋅−+=
≡⋅
+=
⋅∆+≈
−+=
∞
=
∞
∞
∞
∫
vitg
TTg
gvit
gg
C
g
vit
T
T
XliqvitC
Sc
LLTT
L
TT
Sc
TT
LLL
LTST
CLL
TT
cS
dTT
TcTcS(T)S
g
g
1)/(
ln1
log; )(
ln
)()(
η
configurational entropy:
Adam-Gibbs equation:
viscosity slope for T → Tg:
with increasing temperatureincreasingly more possibilitiesto arrange the molecular units
0 500 1000 15000.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
(cgla
ss -
c X) /
T in
J/(m
ol·K
2 )
T in K
vitm
mT
gT
Xliq
SS
dTT
cc
−=
−∫ ( )
( )
∆
+⋅−=∂
∂
⋅∆
+
⋅−+=
≡⋅
+=
⋅∆+≈
−+=
∞
=
∞
∞
∞
∫
vitg
TTg
gvit
gg
C
g
vit
T
T
XliqvitC
Sc
LLTT
L
TT
Sc
TT
LLL
LTST
CLL
TT
cS
dTT
TcTcS(T)S
g
g
1)/(
ln1
log; )(
ln
)()(
η
configurational entropy:
see, e.g., the Adam-Gibbs equation:
viscosity slope for T → Tg:
diopside CaMg[Si2O6]
0.0 0.2 0.4 0.6 0.8 1.0-2-10123456789
10111213
log
η, η
in d
Pa·s
Tg / T
0.663.6slope m ±=
0
logTT
BAL
−+== η
mTT
BT
TTg
TTg g
=−
⋅=
∂∂
=2
0)(/logη
A = -1.899 Tg = 708 °CB = 2493.4T0 = 540.6±δ logη = 0.022
( )
∆+⋅−=
)0(1 ,
0G
LGg S
cLLm
cG,L(Tg) = 254.6 J/(mol·K)cX(Tg) = 246.5 J/(mol·K)3·N·R = 249.0 J/(mol·K)∆cG,L = 80.0 J/(mol·K)SG(0) = 24.8 ± 3.0 J/(mol·K)
⇒ m = 63.7 ± 5.9
⇒ m = 63.3 ± 0.6
14.9
exp. data:Scarfe et. al. (1986)
S L(Tg)
- SX(
T g) /
SL(
T) -
S X(T)
Tg / T
C.A. Angell (2008)
0.0 0.2 0.4 0.6 0.8 1.0-4
-2
0
2
4
6
8
10
12
Tg/T
log
η, η
in d
Pa·
s
SiO2
nepheline NaAl2Si2O8
7BeF2·3NaF
anorthite CaAl2Si2O8
ZnCl2 2BiCl3·KCl
What is „fragility“?
400 600 800 1000 1200 1400 16000
2
4
6
8
10
12
14
16
"length"
working range
log η = A + B/(T-T0)
fixpoints T(13.0), T(7.6), T(4.22) fixpoints T(2.0), T(4.0), T(6.0) fibrization fixpoint T(1.5)
float glass melt
basalt melt
log
η, η
in d
Pa·
s
T in °C
• Technologist call it the „length“ of a glass; it determines the working range.
• Fragility is a measure of deviation from an exponential T dependence of relaxation times and transport properties (deviation from Arrhenius behavior).
• Fragility may be quantified, e.g., by the slope for Tg / T → 1 in the so-called Angell plot.
• The non-exponential T dependence is commonly represented by the KWW eq. τ ∝ exp B·(Tg/T)n, by the VFT eq. τ ∝ exp B/(T - T0), etc.
• While the high-T branch reflects a universal feature of liquids, the fragility is likely to reflect the thermo- dynamics of the liquids.
0.0 0.2 0.4 0.6 0.8 1.0-4
-2
0
2
4
6
8
10
12
14
M = 4,000M = 37,000
poly-styrene, M = 390,000
oligo-styrene, M = 500
log
η, η
in d
Pa·s
0.0 0.2 0.4 0.6 0.8 1.0-4
-2
0
2
4
6
8
10
12
14
polystyrene
1,3,5-tri-a-naphthyl benzene
o-terphenyl
log
η, η
in d
Pa·s
Tg / T
B2O3
„It appears that at the same point in T at which the non-Arrhenius regime is entered from aboveglass-forming systems develop heterogeneities in their dynamics.
The heterogeneities are such that, for a limited period of time, one set of particles will locktogether while an adjecent set will become loose will support string-like motions of the patticles,perhaps along the boundaries between nano-regions.
Within each nano-region, relaxation appears to be exponential, and it is therefore thatthe distribution of nano-reagions (which determines the deviation from exponentiality)is seen in the macroscopic relaxation function.
The string-like particles might lie on the Eigenvectors of the low-frequency (or Boson peak)modes of the system, by means of which relaxation occurs.“
C.A. Angell (2008), using findings by Richert (2002), Kob et al. (1998),Weeks et al. (2000), Brito et al. (2007)
non-exponentiality ⇔ development of heterogeneities⇔ nano-regimes⇔ Boson peak, low cP excess of the glass
���
�� ����� �
�����
�����
�TICKETS
����
CINEMA
�diffusion
viscous flowcollective motion
cooperativelyrearrangingcluster
absence of translational symmetry andoccurrence of M.R.O. may by caused by• frustration (entaglement) of polymeric structures or• by frustration of spherical packing
extremely fragile very strong our example: diopside
large ∆cPcP derceases above Tg
tiny ∆cPcP steadily inrceases above Tgand even beyond Tm
C.A. Angell (2008)
500 1000 15000.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
(cG
,L -
c X) /
T in
J/(m
ol·K
2 )
T in K
peak at Tg
( )α
+=
⋅−⋅−+=
−+
−⋅+=
−+=
+=
∞
∞∞
∞
∞
T
TLL
TTa
TTLLLL
TTC
TTCLL
TTB
LL
TbLL
g
g
gg
g
gg
)/ln(1
/
10ln
/
0
2
1
0
SUMMARY OF VISCOSITY MODELS:
Arrhenius
Vogel-Fulcher-Tamman (VFT)
Williams-Landel-Ferry (WLF)
Adam-Gibbs
Avramov
Mauro et. al
atomic mobility
free volume
free volume
configurational entropy
Kohlrausch relaxation
configurational entropy( )
−⋅
−
−⋅⋅−+=
∞∞∞ 11exp
T
T
LLm
T
TLLLL g
g
gg
SUMMARY OF LINEARIZED EQUATIONS:
Adam-Gibbs
modified
VFT; WLF
Avramov
Mauro
xax
Lln1
10 ⋅−⋅+= ββ
xxax
Lln1
10 ⋅⋅−⋅+= ββ
axL ⋅+= 10 ββ
( )xaxa
L⋅−⋅−
⋅+=11
10 ββ
( ))1(exp 10 −⋅⋅⋅+= xaxL ββCONCLUSION:• All established viscosity-temperature functions L = log η(T) are 3-parameter plots of x = Tg / T.• All functions L contain a kind of „structural parameter“ a allowing to generate a linear plot.• The high-T limits L∞ = β0 for x = Tg / T → 0 differ significantly.
0 2 4 6 8 10 12 14 16 18-2
0
2
4
6
8
10
12
14
16
model glass FAL9
model glass FAL7
glassyanorthite
float glass DGG-1
TV panel glass
low α glassE fibre glasslo
g η,
η in
dP
a·s
Tg/(T·SC) in (g·K)/J
CaO·Al2O2· 2SiO2
0 2 4 6 8 10 12 14-2
0
2
4
6
8
10
12
14
16
fibre 1
fibre 2fibre 3
basalt wool from quarry "Westerwald"
2 3
1
2
log
η, η
in d
Pa·
s
Tg/(T·SC) in (g·K)/J
log η(Tg) is not usedfor the linear regression;the lines derived fromthe high-T data hitlog η(Tg)
Adam-Gibbs plot, basalt melts
linear plots of log η(Tg) allowone to assess viscosities in the„crystallization gap“
no direct measure-ment possiblebecause ofcrystallization
networks again
rigid and „floppy“ isostatically rigid
rigid and stressed
under-constrained:one floppy motionin addition to 3 rigidbody motions
perfectly constrained:all bonds maintain theirnatural length; no floppymodes
over-constrained:all bonds are deformed;one bond is redundant
(a floppy motion is amotion that costs noenergy)
different statesof rigidity
(2D example)
J.C. Phillips et alP. Boolchand et alM.F. Thorpe et al.R. Keding et al.
∑=r
rNN
Consider a system of N atoms, classified into groupsof Nr atoms with coordination number r:
The mean coordination number rm is
∑∑ ⋅=r
rr
rm NNrr
An unconstrained atom in 3D space has 3 positional degreesof freedom; an unconstrained system of N atoms thus has
positional degrees of freedom. Let NC be the number ofpositional constraints. Then
denotes the number of zero frequency („floppy“) modesallowing particles to change position at zero energy.
NN F ⋅= 3
CF NNN −=0
∑∑ ∆⋅+∆⋅=ijk
ijij
ijLV 22
22θ
βα
The positional constraints of distance Lij between particlesi and j, and of angles θijk between bonds i-j and j-k, stem fromthe deviations ∆Lij and ∆θijk from the equilibrium positions inthe interparticle potential V:
It is true, V is a simple harmonic potential (V ∝ ∆Lij2, ∝ ∆θijk
2)with simple force constants α and β. But as a constraint fixesa position against very small virtual displacements δLij or δθijk,the actual shape of V is irrelevant.
We have two types of constraints, namely related to inter-particle distance variations ∆L, and to bond-to-bond anglevariations ∆θ. In a system with N atoms, the correspondingnumbers of constraints are NC,∆L and NC,∆θ, respectively:
θ∆∆ += ,, CLCC NNN
constraints related to distance variations ∆L:
one constraint associated with one bond=1/2 constraints per bond for each end member;
for an end member with coordination number r,
nr,∆L = r/2
∑∑ ⋅=⋅= ∆∆r
rr
rLrLC Nr
NnN2,,
r = 2 ⇒ r = 3 ⇒ only one angular constraint: two additional constraints becausenr,∆θ = 1 two angles relative to two existing bonds have to be specified: nr = 3
θijk
i
j
k
αβ
In 3 dimension, eachadditional bond constitutestwo further constraints:
r 2 3 4 5 6
n r 1 3 5 7 9
constraints related to angular variations ∆θ:
0 1 2 3 4 5 6-3-2-10123456789
n r = 2· r - 3nu
mbe
r of
anu
lar
cons
trai
nts
n r
coordination number r
( )∑∑ ⋅−⋅=⋅= ∆∆r
rr
rrC NrNnN 32,, θθ
The total number of constraints thus amounts to
( )∑∑ ⋅−⋅+⋅=r
rr
rC NrNr
N 322
The total number of zero frequency („floppy“) modes is
( )
∑
∑∑∑
⋅
−=
⋅−⋅−⋅−⋅=−=
rr
rr
rr
rrCF
Nr
NrNr
NNNN
25
6
322
3
0
Let us calculate the fraction ƒ0 = N0 / (3·N) of „floppy“ modes:
4.2 065
23
25
6
3ƒ 0
0 =→−=⋅
⋅
−
=⋅
=∑
∑mm
rr
rr
rforrN
Nr
NN
number of 3D network
degrees of freedom NF
constraints NC
floppy modes N0 = NF - NC
N0 → 0 for
( ) Nrm ⋅− 3·25
( ) Nrm ⋅− ·6 25
N⋅ 3
4.2=mr
summary: constraint counting
N = number of nodes in the network(e.g., number of Si atoms)
rm = mean number of constraining bonds per node(e.g., number of bridging oxygens per Si atom)
rm
no. o
f ato
ms
in c
ritica
l nuc
leus
critic
al u
nder
cool
ing
∆T
/ Tm
∆T
in K
BaO·TiO2·2SiO2
Avramov, Keding, Rüssel(2000)
coordination number of the network node
critic
al b
ond
num
ber r
m
the „magic“ 2.4 threshold
here, only rigid clusters percolate
here, only floppy clusters percolate
Avramov 2009
xxx OSiNaSiOxOxNa −−⇒−⋅ 21222 )1(
Consider Na2O-SiO2 glasses:
• Mean coordination number bm per network node (each Si in one node):
for SiO2 as the only network former
• Mean number qm of modified („broken“) bonds per network node:
4=mb
xx
qm −=
12
xxx OSiNa −− 212
• Mean number rm of constrained bonds per network node:
xx
xx
qbr mmm −−
=−
−=−=1
6412
4
The average number of constraining bonds (or: constraints) per node is:
xx
rm −−
=1
64
glass N-C-S (mol %) modifier Nrm =
50.0 N + 50.0 S 2.0044.4 N + 55.6 S 2.4033.3 N + 66.7 S 3.00
isostatic case?
Does a glass composition with 55.6 mol-% SiO2 stand out in any way?
0.6 0.8 1.0400
600
800
1000
1200
1400
1600
1800
SN3S8NS2NS
system Na2O-SiO2 (Na2O = N, SiO2= S)
T(13.0)
T(4.0)
T(3.0)
T(2.0)T(1.0)
T in
°C
wt. fraction of SiO2
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0T = 1400 °C
Na2O Na4SiO4
Na2SiO3
NaSi2O5
SiO2
mol
ar fr
actio
n of
spe
cies
molar fraction of SiO2
„chemical structure“ of the binary system Na2O-SiO2
.
0.0 0.2 0.4 0.6 0.8 1.0
-10
-8
-6
-4
-2
0
1400 °C
log ƒ(Na2O)
log ƒ(SiO2)
molar fraction of SiO2
log
activ
ity c
oeffi
cient
„chemical structure“ and chemical potentials are equivalent conceptsboth quantifying the chemical interaction of components
-10 -8 -6 -4 -2 0
[SiO4]4-
[SiO4]2-
[SiO4]1-
[SiO4]0
Na2O-SiO2 melt at 1400 °Caqueous SiO2 solution at 25 °C
Na2O
log [O2-] ≈ log a(Na2O)
-10 -8 -6 -4 -2 00.0
0.2
0.4
0.6
0.8
1.0
[SiO4]3-
[SiO4]4-
[SiO4]2-
[SiO4]1-
[SiO4]0
rel.
conc
entr
atio
n of
spe
cies
log [O2-] = -28 +2·pH
species coexistence is a feature of liquids in general ⇒ „chemical structure“
0.5 0.6 0.7 0.8 0.9 1.02
3
4
5
6
aver
age
no. q
m o
f bon
ds m
odifi
ed p
er N
a + io
n
x(SiO2)
0.0
0.2
0.4
0.6
0.8
1.0
mol
ar fr
actio
n of
spe
cies
k
k = NS
q = 6 for k = Sq = 2 for k = NS2q = 2 for k = NS
k = S
k = NS2
„chemical structure“ of the binary system Na2O-SiO2
0.5 0.6 0.7 0.8 0.9 1.00
1
2
3
4
5
6
aver
age
no. q
m o
f bon
ds m
odifi
ed p
er N
a + io
n
x(SiO2)
0.0
0.2
0.4
0.6
0.8
1.0
mol
ar fr
actio
n of
spe
cies
k
k = NS
k = S
k = NS2
qm
q = 6 for k = Sq = 2 for k = NS2q = 2 for k = NS
qm = Σ xk· qk
mean number of „modified“ bonds in the binary system Na2O-SiO2
0.5 0.6 0.7 0.8 0.9 1.02
3
4
2.4
rm
k = NS2
k = NS k = S
aver
age
bond
no.
per
Si a
tom
rm
x(SiO2)
0.0
0.2
0.4
0.6
0.8
1.0
mol
ar fr
actio
n of
spe
cies
k
)()(1
42
2
SiOxSiOx
qr mm−⋅−=
under-constrained isostatic over-constrained(rigid and floppy) (rigid and stressed)
rm
ηlog; )(
≡⋅
+= ∞ LTST
CLL
C
mol % SiO2
C /
S C(T
g)
M. Toplis (2001)
QUIZ
1. The fragility of a glass is a measure of• its practical strength.• the activation energy in an Arrhenius plot.• the over-exponential increase of relaxation times when approaching Tg.
2. Clusters formation• and random network formation are strictly antagonistic concepts.• is rather supported than discouraged by random network concepts.• is a structural feature exclusively found in metallic glasses.
3. The heat capacities of a glass and its isochemical crystal• are essentially the same since cP is dominated by short-range order.• are usually quite different because the crystal is highly ordered while the glass is not.• deviate in the low-T regime.• None of these answers is correct.• More than one answer is correct.
4. The cP in the glass transition• is of the order of 3R per g-atom.• is approx. (3/2)R per g-atom.• is very different for different glasses depending on their fragility.• depends on the absolute value of Tg.
QUIZ continued
5. The viscosity of a glass-forming melt:• The viscosity-temperature relation essentially depends on thermodynamic properties.• The viscosity-temperature relation is a kinetic property and as such has no relation to
thermodynamics.• The viscosities of glass-forming melts at liquidus temperature are of the same order
of magnitude.• More than one answer is correct.
6. Floppy modes in a 3D random network• occur if the average bond per network node exceeds a threshold value of approx.
2.4.• decrease the critical undercooling of a glass melt.• Both answers are correct.
7. The “chemical structure” of a glass• reflects true structural entities.• is a merely stoichiometric concept which has no real correspondence to the glass
structure.• is based on a nanocrystalline concept of the glassy state.
8. The different 3-parameter viscosity models• although based on different scientific concepts yield – in practical terms –
the same results of the viscosity-temperature function.• agree well within limited temperature ranges, but disagree significantly at very high
temperatures and at T < Tg.• yield considerably different results within the entire T range.
Homework
Use the simplified form of the Adam-Gibbs relation
with abbreviations
Use the „structural parameter“ a as an adjustable parameter to generate a linearfit to the following experimental viscosity data of diopside (viscosity in dPas):
The experimental value of ∆cP is 80.0 J/(mol·K). Derive an estimate for the conventionalentropy of vitrification Svit. Conclusion?
)ƒ(ln1
)(
)(loglog 10 y
ya
yLLL
TSTC g
C
⋅+=⋅−
⋅−+=
⋅+= ∞
∞∞ ββηη
.; ; log yT
Ta
Sc
L gvit ≡≡
∆≡η
T in °C log η 708 ≈13.0 1375 1.0871400 0.9921450 0.8811500 0.6781550 0.5591600 0.465
Thank youfor your attention
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