# Intermediate phases

34 Pages · 2013 · 2.8 MB · English

Atomic modeling of glass – LECTURE 10 RIGIDITY THEORY. LECTURE 10 .. GeSe. 2. Standard deviations for tetrahedral network glasses.

34 Pages · 2013 · 2.8 MB · English

Atomic modeling of glass – LECTURE 10 RIGIDITY THEORY. LECTURE 10 .. GeSe. 2. Standard deviations for tetrahedral network glasses.

[email protected]Atomic modeling of glass – LECTURE 10 RIGIDITY THEORY
LECTURE 10 : RIGIDTY TRANSITIONS AND INTERMEDIATE PHASES Rigidity transitions and compositional trends Intermediate phases Cluster expansions MD based rigidity theory
j
Stretching constraintsa
ij
r/2
Bending constraintsb
ijk
2r3
1 Remember (lecture 9) the enumeration of mechanical constraints
Consider a rcoordinated atom
If r=2, there is only one angle Each time, one adds a bond, one needs to define 2 new a ngles
We consider a system with N species of concentration n
r The number of constraints per atom is :
=
∑
(
2
+ (2 − 3)
∑
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Atomic modeling of glass – LECTURE 10 RIGIDITY THEORY A) CONSTRAINTS AND RIGIDITY TRANSITIONS
[email protected] modeling of glass – LECTURE 9 TOPOLOGY
Stretching Bending
2 Rigidity transition: Amorphous silicon and harmonic potential
Bond depletion ( decreases)
Calculation of the eigenmodes (vibrational frequencies) of the system
Example of simple eigenmode calculation: the linear chain
A) CONSTRAINTS AND RIGIDITY TRANSITIONS
[email protected] modeling of glass – LECTURE 10 RIGIDITY THEORY
Force acting on spring j:
Newton’s law gives: with frequencies
Normal mode solution :
which actually reduces the problem to an eigenvalue ( W
WW
W
) problem
:
More general : normal modes=eigenvalues Wof the dynamical matrix
[email protected] modeling of glass – LECTURE 10 RIGIDITY THEORY
2 Rigidity transition:
Thorpe (1983) found that bond depleted aSi with mean co ordination number < 2385 contain
zero frequency normal (floppy) modes W Their number f(rank of the W=0 block of the dynamical matrix) scales as >
<
=
= r
N
f
c
2 5
6
3
Flexible to rigid transition
Control parameter
Order parameter f
Powerlaw C
ii=(24)
p
(p=1,5) in the stressed
rigid phase Elastic phase transition
Isostatic glass n
c=3 is at the R transition
He and Thorpe, PRL 1985
A) CONSTRAINTS AND RIGIDITY TRANSITIONS
19792000: Anomalies at =24
Chalcogenide network glasses r(Se)=2 , r(Ge)=4 Ge
xSe
1x
GeSbSe, GeAsSe, etc…
Varshneya et al, JNCS 1991
Boehmer and Angell, PRB 1994
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Atomic modeling of glass – LECTURE 10 RIGIDITY THEORY A) CONSTRAINTS AND RIGIDITY TRANSITIONS
[email protected] modeling of glass – LECTURE 10 RIGIDITY THEORY A) CONSTRAINTS AND RIGIDITY TRANSITIONS
Binary oxides : (1x)SiO
2xNa
2O
Depolymerization with addition of Na r(Q
4)=4 , r(Q
3)=3, Silica and germania: rigidity under pressure under pressure
Micoulaut, Am Mineral 2008
Trachenko et al, PRL 2005
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B INTERMEDIATE PHASE Experimentally Picture of rigidity transitions has been changed in d epth during the last decade
Calorimetric measurements (modulated DSC)
Use of modulated differential scanning calorimetry (MDSC) across the glass transition Allows for the definition of 2 heat flows Reversible ( DC
p) Nonreversible Nonreversing enthalpy ( DH
nr) Georgiev et al PRB (2003
)
nonrev
rev
T
H
H
H &
&
&
+
=
Modulated DSC and reversibility windows
[email protected] Atomic modeling of glass – LECTURE 10 RIGIDITY THEORY
Reversibility window
B INTERMEDIATE PHASE
Since the RW is found between the flexible and the stressed rigid phase, it is also often called
the « Intermediate phase » Nonmean field phase otherwise one would
have the single =24 transitions
Central idea: stress avoidance under increasing crosslinking density Network selforganization
Nearly vanishing of DH
nr
at Tg
Stress free character (P
c
minimum, Raman)
Vibrational thresholds (Raman, IR)
Generic (chalcogenides, oxides)
Wang et al, PRB 2005
Remarkable properties for selected glass compositions
[email protected] Atomic modeling of glass – LECTURE 10 RIGIDITY THEORY
[email protected] modeling of glass – LECTURE 10 RIGIDITY THEORY
“Marginality of nonreversible component of
complex heat flow in MDSC experiments does not necessary reflect selforganized
intermediate phase ”
PSS 2011 « However, that (mDSC) measurement
might also be subjected to a large experimental uncertainty ” PRB 2009 “ The observation of the reversibility window
might be an experimental artifact ” PRB 2009
A subject which has not been without controversy…
Micoulaut, Bauchy, PSS 2013
B INTERMEDIATE PHASE
[email protected] modeling of glass – LECTURE 10 RIGIDITY THEORY
B INTERMEDIATE PHASE
Senapati, JNCS 1991
Bhosle et al, IJAGS 2012
Why has the RW been missed before ? And why are challengers missing it…and getting so annoye d ?
Three

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