SINTESI DI UN VETRO DAL
CORRISPONDENTE CRISTALLO
VETRO
CRISTALLO
 = C
Fusione con successivo
super-raffreddamento e
conseguente
risolidificazione
0 < C
DIAGRAMMA DI FASE DI UN LIQUIDO
GLASS-FORMING
Poiché il volume del vetro è maggiore di quello del corrispettivo cristallo gli atomi
costituenti il vetro avranno un maggior numero di gradi di libertà da cui l’eccesso di
stati vibrazionali di bassa energia.
Fragilità dei Liquidi Glass-forming
(C. A. Angell, JNCS 1985, Science 1995; De Benedetti and
Stillinger, Nature 2001)
La fragilità di un liquido glass-forming misura la degradazione termica della struttura vetrosa
nella regione di transizione vetrosa, considerando lo scostamento da un comportamento
Arrheniano della viscosità. Andamento della viscosità di shear o del tempo di rilassamento
strutturale ts (η=G∞ts ) al variare della temperatura nella regione T ≥ Tg:
d log
m
d T T 
g
d log t 
m
d T T 
s
T Tg
g
T Tg
Differentemente dai sistemi “fragile”, i liquidi “strong” preservano l’ordine strutturale a
medio range nel passaggio liquido ↔ solido.
Elevati valori di m (100) individuano i deboli liquidi molecolari semplici (CKN, OTP),
mentre piccoli valori di m (20-30) corrispondono ai liquidi caratterizzati da forti legami
covalenti (SiO2, GeO2, BeF2).
Stretta relazione tra le proprietà
di
trasporto
e
le
proprietà
termodinamiche dei liquidi glassforming:
I liquidi “fragile” esibiscono un elevato
salto, DCp (=Cp,l-Cp,g), nella capacità
termica alla Tg, contrapposto ad una
ridotta variazione DCp esibita dai liquidi
“strong”.
Correlation between fragility and the ratio of longitudinal and transversal
sound velocities in glassy state (Novikov and Sokolov, Nature 2004)
K 2

1G 3

2 K 1
G 3
Correlation between fragility and anharmonicity in glassy state (Carini et al
JPC 2000, PRB 2005, JPC 2006)
d (ln  )
 
d (ln V )
Nc of NFIs ranges between 2 and 4
i
100
G ,i
Se
C
 
C
80

i
G ,i
fragility, m
G ,th
60
i
3 B V
 
C
Li and Na borates
S
40
th
B2O3
20
0.0
m
G ,th
p
SiO2, GeO2
0.5
1.0
th
1.5
Avogadro, Carini et al, Phil. Mag. B 1987
Se amorphous
Se polycrystal
20
Cv/T
3
-1
-4
(mJmol K )
30
10
0
0
10
20
T (K)
30
40
Se86,6 Te13.4
2.0
crystal
Conducibilità termica a
basse T in un vetro Se-Te
1.5
-1
Thermal conductivity,Wm K
-1
e nel corrispondente
cristallo (Rosenberg and
Carini, 1990)
1.0
0.5
glass
0.0
0
50
T, K
100
ATTENUAZIONE ACUSTICA
NEI VETRI
a) Attenuazione acustica in quarzo
(SiO2) vetroso a 930 MHz
b) Attenuazione acustica in quarzo
vetroso a 507 MHz
c) Attenuazione acustica in quarzo
cristallino a 1 GHz
Dynamics of Strong and Fragile Glass-Formers: Correlation between
Fragility and Low Temperature Properties (Sokolov et al PRL 1993)
4
internal friction, 10
(a)
M=Cs
M=K
M=Li
20
ULTRASONIC ATTENUATION AND SOUND VELOCITY IN BORATE
GLASSES: DEPENDENCE ON THE CATIONIC FIELD STRENGTH
10
(M2O)0.14(B2O3)0.86
0
1
10
100
T (K)
1.02
30
(a)
20
vl(T)/vl(300 K)
M=Cs
M=K
M=Li
4
internal friction, 10 Q
-1
Pure B2O3
M=Cs
M=K
M=Li
10
1
10
100
0
T (K)
Q 
1
i
Pure B2O3
M=Cs
M=Kdb
M=Li
0.23

100
200
T (K)
1.02
300 K)
1.01
1.00
0
1.01
(b)
V
(b)
300
(ADWP) ASYMMETRIC DOUBLE WELL POTENTIAL MODEL
(Gilroy and Phillips, Phil. Mag 1984; Hunklinger et al, PRB 1995)
The application of an asymmetric double-well potential model allows for
a quite coherent linking between high temperature classical relaxation
processes and low temperature quantum effects observed in the
acoustic behaviours of these borate glasses.
T < 20 K
T > 20 K
V 
 D 
t  t exp
 sec h

k T 
 2k T 
0
B
B
Dynamics of two-level tunneling systems in glasses:
Coherent and incoherent tunneling (T< 20 K)
Classical Thermal Activation in glasses (T > 20 K)
 D  t
0.23


Q 
V
 dDdVf (D) g (V ) sec h 

V k T
 2k T  1   t
2
1
db
2
i
i
2
i
2
B
V 
 D 
 sec h

k T 
 2k T 
t  t exp
0
B
B
2
B
V 

k T 
t  t exp
0
0
B
 V
g (V )  V exp   
 V0 
1
0
2


f

a
a
1
0


Qi    2 at 0   C * at 0 
 v 
k BT
a
V0
CLASSICAL ACTIVATION: COMPARISON BETWEEN THE
EXPERIMENTAL DATA AND THE ADWP THEORETICAL FIT
(K2O)0.14(B2O3)0.86
0.0020
0.0010
Q
-1
0.0015
0.0005
0
50
100
150
T (K)
200
250
300
SPECTRAL DENSITIES OF TLSs AND ASYMMETRIES vs CATION
FIELD STRENGTH
-1
relaxation strength, C*
tunneling strength, C
10
Cs K
Li
-2
10
-3
10
47
-3
f0 (J m )
-1
-1
-3
P (J m )
10
46
1x10
45
10
0.0
0.5
1.0
2
1.5
-2
field strength q/r (A )
At variance with the tunnelling strength C, the relaxation strength C*
48
10
Ag= O, D
decreases with decreasing cation size also exhibiting
values which are more
Li= +
10
than one order of magnitude
larger than those of C.
-3
-1
J m )
-3
-1
(J m )
47
M=Ag
10
-2
10
-3
relaxation strength, C*
tunneling strength, C
SPECTRAL DENSITIES OF TLSs AND ASYMMETRIES vs
INCREASING CONCENTRATION OF METAL OXIDE
O, D M=Ag
X, + M=Li
f0 (J m )
-1
-1
-3
-3
P (J m )
47
10
46
1x10
45
10
0.0
0.1
0.2
0.3
0.4
M2O mol fraction
At variance with the tunnelling strength C, the relaxation strength C*
decreases with increasing metal oxide concentration also exhibiting values
which are more than one order of magnitude larger than those of C.

(kg m-3)
vl
(m s-1)
vt
(m s-1)
vD
(m s-1)
D
(K)
G
(GPa)
B
(GPa)
-6 -1
(10 K )
B2O3 (our data)
1838
3367.4
1871.5
2084
267
6.44
12.26
15.1
0.026
(Li2O)0.14(B2O3)0.86
2071
5060.3
2851
3172
427
16.83
30.59
6.5
0.014
(K2O)0.14(B2O3)0.86
2088
4228
2301
2567
331
11.06
22.58
10.53
0.017
(Cs2O)0.14(B2O3)0.86
2484
3578
1961
2186
270
9.55
19.1
12.68
0.0195
Samples
B = Vl 2 -
G =Vt 2
Samples
Cl
x 104
P l
(107 J m-3)
2
4
G
3
th,298
l
f 0  l2
(eV)
(1045 J-1 m-3)
V0/kB
(K)
Cl*
x 103
t 01
(1013 s-1)
(108 J m-3)
(1046 J-1m-3)
l
P
f0
B2O3
2.4a
0.52a
0.21a
4.5a
725
8.28
1.0
1.73
15.3
(Cs2O)0.14(B2O3)0.86
3.79
1.0
0.47
1.76
728
16.2
2.2
5.17
9.1
(K2O)0.14(B2O3)0.86
3.74
1.74
0.55
2.24
685
15.0
4.1
5.60
7.2
(Li2O)0.14(B2O3)0.86
6.61
3.5
0.63
3.44
650
6.37
1.8
3.38
3.3
The different magnitude of C and C* leads to the conclusion that only a
small fraction of the relaxing particles are involved in tunneling local
motions.
SOUND VELOCITY IN BORATE GLASSES: CLASSICAL
ACTIVATION AND VIBRATIONAL ANHARMONICITY
DVl  DVl
 
Vl ,0  Vl ,0
 DVl

V
 l ,0

 DV
  l


 rel  Vl ,0



 anh

 i2
D 
1
2
 


d
D
dVf
(
D
)
g
(
V
)
sec
h
2
2 2



Vl k B T
 2k B T  1   t
 rel
 DVl

V
 l ,0





 f 0 2 


 





t

1

C
*
t
1
0
0
2
 V 

 rel  l 
0.000
-0.005

 
  L 
 

 anh  Lo 
1
2

 T 
1


F
  1
l 

  

  T 4 T x 3dx 
T
F   3   x

0 e
 1

   
Dv/v0
 DVl

V
 l ,0
3
2
-0.010
-0.015
-0.020
0
Classical activation also regulates
the sound velocity between 20 and
120 K, whereas the vibrational
anharmonicity results to be the
dominant mechanism for higher
temperatures.
100
200
T (K)
300
DEPENDENCES ON THE CATION FIELD STRENGTH OF
COMPRESSIBILITY, LINEAR THERMAL EXPANSION COEFFICIENT th,
SPECTRAL DENSITY OF ASYMMETRY f0, AND ANHARMONICITY
COEFFICIENT I
20
G =Vt 2
B2O3
-1
K
10
Li
5
B = Vl 2 -
4
G
3
Compressibility = 1/B
5
Cs
B2O3
0.02
K
-3
Li
15
l
46 -1
-6
10
20
f0 (10 J m )
15
Cs
th (10 K )
-11
-1
compressibility (10 Pa )
15
0.01
10
5
0.0
0.5
1.0
2
-2
field strength q/r (A )
0.00
1.5
The anharmonicity of the
glassy network decreases
with increasing field strength
of the modifier ion, in close
correlation with the parallel
reduction of the local
molecular mobility.
Hypersonic and Ultrasonic Attenuation
(M2O)0.14(B2O3)0.86
0.003
M=Li, 50 MHz
M=K, 50 MHz
M=Cs, 50 MHz
Q
-1
0.002
background
0.005
Q -Q
-1
0.001
-1
background by TLS
-1
M=Li, Q back: 0.146 GHz
M=K, Q
-1
back
-1
M=Cs, Q
: 0.049 GHz
back
: 0.049 GHz)
0.000
0
100
200
T (K)
300
(K2O)0.14(B2O3)0.86
Arrhenius plot
10
10 MHz
50 MHz
25.5 GHz
5
3
10 Q
-1
(M2O)0.14(B2O3)0.86
10
11
10
10
0
50
100
150
200
250
300
350
frequency (Hz)
0
T (K)
max
9
Eact/kB=765 K
Eact/kB=587 K
10
8
10
7
0.5
-1
Q /Q
-1
1.0
10
M=Li
M=K
M=Cs
Eact/kB=685 K
0.0
1
10
100
T (K)
t  t 0e
E / KT
t  t 0e
10
6
5
10
15
20
25
-1
E / KT
1000/Tpeak (K )
log( t )  log( t 0 )  E / KT
log( t 0 )   Ea / KTmax
Tpeak,hyper WELL FIT THE ARRHENIUS PLOTS DETERMINED BY Tpeak,ultra
Hypersonic vs Ultrasonic
(M2O)0.14(B2O3)0.86
0.008
THE
HYPERSONIC
ATTENUATION
0.006
exp
0.004
BY
THE
ULTRASONIC
RELAXATION PROCESS IS MARKEDLY
Q
-1
EVALUATED
Rel. contributions
0.002
SMALLER THAN THE EXPERIMENTAL
OBSERVATION.
Anharmonic contribution
anh
0.004
Akhiezer mechanism: “phonon viscosity”
Q
-1
0.006
M=Cs
M=K
M=Li
0.002
0.000
50
100
150
T(K)
200
250
300
CONCLUSIONS
•The fragility of glass-forming liquids scales with the average global anharmonicity of
their glassy state.
•The spectral density of relaxing defects decreases with increasing cation field strength
and is more than one order of magnitude larger than the spectral density of tunnelling
states.
• The anharmonicity of the glassy network decreases with increasing field strength of the
modifier ion, in close correlation with the parallel reduction of the local molecular
mobility.
•Tunnelling TLS’s confirm their universal character, as inherent to the glassy state: they
cause an internal friction which lies in the range between 10-4 and 10-3, independent of
structural changes and of bond strengths characterizing the glassy network.
•The hypersonic attenuation arises from the contributions of classical relaxation
processes and of anharmonic interactions of phonons (Akhiezer mechanism or
“phonon viscosity”).
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