lecture 9 : topological engineering - constraint theory
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matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
LECTURE 9 : TOPOLOGICAL ENGINEERING- Constraint theory- Naumis models- Temperature-dependent constraints
« Think topology»An early advise from J.C. Phillips
« If you are not creating controversy, then you’re not doing truly good science»
A recent comment from the same JCP
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
D.R. Neuville et al. , JNCS 353 (2007) 180
• Motivation :understanding compositonal
trends in glasses
•Cumbersome study along comp. joins
•Small compositional changes can
dramatically alter system properties.
• Such small compositional changes cannot be
described with brute-force methods such as
Molecular Dynamics (MD) simulations.
•Can all the unnecessary details be filtered
out ? All those which do not influence
ultimately the overall properties.
•There is much to learn from structure and
from approaches which use as a central tool
topology or network connectivity.
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
A) Constraint theory
Quantifying topology in glasses
Bhatia-Thornton number-number functions gNN(r)
gNN(r) -> Measure of the network mean coordination number <r>GexSe1-x glasses follow the 8-N rule, i.e. <r>=2+2x
P.S. Salmon, JNCS 353, 2959 (2007)
GexSe1-x
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
A) Constraint theory
Molecular network (contraint counting)
• Atoms•Covalent bonds
•Stretching and bending interactions
A) Constraint theory
Mechanical structure• Nodes• Bars
•Tension
Basic idea:An analogy with mechanical structures
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
j
Stretching constraints αij
r/2
Bending constraints βijk
2r-3
Enumeration of mechanical constraintsConsider a r-coordinated atom
If r=2, there is only one angle. Each time, one adds a bond, one needs to define 2 new angles
We cpnsider a system with N species of concentration nr. The number of constraints per atom is :
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
=∑ (
2
+ (2 − 3)
∑
A) Constraint theory
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
We introduce the network mean coordination number
e.g. accessed from the Bhatia-Thornton pair distribution function gNN(r)
Then nc can be simply rewritten as :
Invoking the Maxwell stability criterion for isostatic structures nc=D=3we find a stability criterion for:
or :
Networks with nc<3 are underconstrained (flexible). With nc>3, they are overconstrained
Important quantity: number of floppy (deformation) modes : f=3-nc
=∑ (
()
∑
=(
+ 2 − 3 )
=∑
∑
= (
+ 2 − 3 )=3
=
= . !
Phillips, JNCS 1979
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
A) Constraint theoryExamples of application:
GexSe1-x glasses: Ge is 4-fold and Se is 2-fold.
Ge has 2r-3=5 BB and r/2=2 BS constraints Se has 1 BB and 1 BS constraint
nc=2(1-x)+7x=2+5x Stability criterion for nc=3 i.e. for x=0.2
Mean coordination number at 20% Ge
Ge20Se80=GeSe4 glasses are isostatic
= "#$ + %# 1 − $ = 4$ + 2 1 − $
= 2.4
Varshneya et al. JNCS 1991
Ge-SeGe-Sb-Se
B) Naumis models
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
Constraints and dynamics:
Starting point is the classical equation relating the msd to the vibrational density of states
Below Tg, <u2(T)> is linear in T.
At Tg, the Lindemann criteria applied to glasses establish that
<u2(Tg)>= <u2(Tm)>=0.01a2
where Tm is the melting temperature and a the crystalline cell length
10%
U. Buchenau, Zorn, EPL (1992).
Assume that for a given fraction of floppy modes, the vibrational density of states is given by
with gR the density of states for f=0 and ωf the frequency of a floppy mode (4meV).
Remembering that one has :
one can compute the msd:
where:
B) Naumis models
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
Constraints and dynamics:
B) Naumis models
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
Constraints and dynamics:
We apply the Lindemann criterion for an overconstrained glass (f=0). It follows from :
that:
and for f non-zero: with:
B) Naumis models
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
Constraints and dynamics:
Since we have the number of floppy modes f :
We finally have:
with
The glass transition is a function of the mean coordination number <r> of the glass network.
Excellent agreement in chalcogenides
G.G. Naumis, PRB 73 (2006)
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
B) Naumis models
Constraints and thermo dynamics:
Hamiltonian of a system containing f floppy modes with zero frequencyenergy:
Out of which can be calculated a partition function:
Floppy modes are cyclic variables of H
Provides a channel in the potentialenergy landscape (PES) since the energy does not depend upon a change in a floppy mode coordinate
B) Naumis models
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
Constraints and thermo dynamics:
For a given inherent structure (local minimum of the PES), the number of channels is givenby f.
Entropy due to floppy modes (available phase space to visit).
At fixed volume, Ω(E,V,N) is proportional to the area defined by the surface f constant E. S=kBlnΩ
)/ln(3 0VVNkfS B≈
f=0
f non zero
Naumis, Phys. Rev. E71, 026114 (2005).
Basics Gupta & Mauro (2009) generalization of the Phillips approach by
inclusion of temperature-dependent constraints:
Required parameters: Ni(x): mole fraction of each network-forming species i wi,α: number of α-type constraints for each species i qα(T): temperature-dependent rigidity of constraint α
Gupta & Mauro, J. Chem. Phys. 130, 094503 (2009)
Mauro, Gupta, Loucks, J. Chem. Phys. 130, 234503 (2009)
C) Temperature dependent constraints
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 200 400 600 800
Temperature (K)
Con
stra
int R
igid
ity
Continuous Form:
Discrete Form:
Conversion between discrete and continuous forms:
qα(T): temperature-dependent rigidity of constraint α
C) Temperature dependent constraints
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
Gjersing, Sen, Youngman, Phys. Rev. B (2010)
x = 10 (Tg = 361 K)
Narrowing of the Se-Se-Se resonance withincreasing T implies rapid rotation of theselenium chain segments
However, the Ge-Se-Se/Ge seleniumenvironments remain relatively rigid in thetemperature range around Tg
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
C) Temperature dependent constraints
Experimental evidence for temperature-dependent constraints
High-temperature 77Se NMR study of GexSe100-x systemTemperature-dependent dynamics of Se atoms around Tg
Theoretical evidence for temperature-dependentconstraints (lecture 10 )
Steps1. Identify and count the number of network-forming species as a function
of composition
2. Identify and count the number of constraints associated with each of those species
3. Rank the constraints in terms of their relative strength (onset temperature)
4. Connect the change in degrees of freedom (f = d – n) with change in specific property of interest
C) Temperature dependent constraints
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
Applied to borate glasses Na2O-B2O3
Addition of modifier oxide to B2O3 can cause boron coordination change formation of NBO
Remember of simple bond models for alkali borates (lecture 3 ) for x>0.33
B
O
O
O + ½ M2O
B
O
OO
O
M
B
O
O
O M
i
ii
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
Step 1: Model the local structure as a function of co mposition
C) Temperature dependent constraints
) $ =(*+),(+*)
) $ =+*
+*)- $ =
-* ,(+*)
Complete statisticsx>0.33
x<0.33
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
Step 1: Model the local structure as a function of co mposition
C) Temperature dependent constraints
) $ =(*+),(+*)
) $ =+*
+*)- $ =
-* ,(+*)
) $ = 1 − . = 1 −*
+*=
+*
+*
)- $ = . =$
1 − $
Can sometimes be re-expressed in terms of bonding oxygens (those participating to the network connectivity, i.e. NB=4 on a B4).
Mauro et al. JCP 2009
α: B-O and MNB-O linear (BS) constraints
Two α constraints at each oxygen
β: O-B-O angular constraints Five β constraints at each Q4 unit. Three at each Q3 unit.
γ: B-O-B and B-O-M(NB) angular constraints One γ constraint at each bridging oxygen
µ: modifier rigidity (due to clustering) Two µ constraints per NBO-forming Na atom
Each involves an onset temperature at which q(T) be comes activeSimilar procedure for borosilicates
C) Temperature dependent constraints
Step 2: Count constraints on each atom (borates)
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
0 10 20 30 40
0
1
2
3
O-
Na+ Na+
Na+
O
T > Tα
Tµ < T < T
α
Tβ < T < T
µ
Tγ < T < T
β
Ato
mic
deg
rees
of f
reed
om
[Na2O] (mol%)
T < Tγ
Cooling
B-O
MNB-O
OB
B BO
O- O-
• Constraints become rigid as temperature is lowered
– Onset temperatures:
Smedskjaer, Mauro, Sen, Yue, Chem. Mater. 22, 5358 (2010)
T T T Tγ β µ α< < <
C) Temperature dependent constraints
Step 3: Ranking of constraints according to temperature
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
Temperature-DependentConstraint Model
Structural Information
Naumis FloppyMode Analysis
Adam -Gibbs Relation
Fraction of Network-Forming Species, N(x)
Topological Degrees of Freedom, f(T,x)
ConfigurationalEntropy, Sc(T,x)
Utimate goal: Tg(x), m(x), Cp(x)
Viscosity, η(T,x)
C) Temperature dependent constraints
Step 4: Calculating properties…the roadmap
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
C) Temperature dependent constraints
Step 4: Calculating properties
A. Use Adam-Gibbs definition of viscosity
B. Use the fact that Tg is the reference temperature at which η=1012 Pa.s. Since η is constant for any composition, we can write:
C. Remember that Naumis’ model leads to Sc # f (floppy modes).
D. This allows writing:
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
D. Remember the definition of fragility :
E. Using Naumis’ definition, once more, we obtain:
F. Application to sodium borates
C) Temperature dependent constraints
Step 4: Calculating properties
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
• Na sets up a locally rigid environment, whereas Ca does not
• Prediction of fragility with only one fitting parameter (νtobs)
Smedskjaer, Mauro, Sen, Yue, Chem. Mater. 22, 5358 (2010)
0 10 20 30 40500
600
700
800
Model Model without rigid Na Experiment (0% Fe
2O
3)
Experiment (1% Fe2O
3)
Tg (
K)
[Na2O] (mol%)
0 10 20 30 4040
50
60
70
80 Experiment (DSC) Model (fit to DSC) Experiment (viscosity) Model (fit to viscosity)
m (
-)
[Na2O] (mol%)
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
C) Temperature dependent constraints
Results: Fragility and Tg variation of calcium borat e glasses
Tg of a borate glass can be predicted from that of a silicate glass with f(x,y,z,T) as the only scaling parameter
Fragility: onset temperatures Tβ,Si and Tµ are treated as fitting parameters (1425 K)
0.0 0.2 0.4 0.6 0.8 1.020
30
40
50
60
Experiment Model (analyzed) Model (ideal)
m (
-)
[SiO2]/([SiO
2]+[B
2O
3])
0.0 0.2 0.4 0.6 0.8 1.0760
785
810
835
860
Experiment Model (analyzed) Model (ideal)
Tg (
K)
[SiO2]/([SiO
2]+[B
2O
3])
Smedskjaer et al., J. Phys. Chem. B 115, 12930 (2011)
C) Temperature dependent constraints
Results: Fragility and Tg variation of sodium borosi licate glass
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
Idea: critical number of constraints (ncrit) must be present for material to display mechanical resistance n = 2: rigidity in one dimension (Se) n = 3: rigidity in three dimensions (SiO2) n = 2.5: rigid 2D structure (graphene) → ncrit
Proposal: hardness is proportional to the number of 3D network constraints at room temperature
?
C) Temperature dependent constraints
Results: Calculating the hardness from constraints
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
• Glass hardness can be predicted from the average number of room temperature constraints, with only an unknown proportionality constant (dHV/dn)
2.4 2.6 2.8 3.0 3.2 3.40
2
4
6
8
10
12 Experiment (P = 98 mN) Model (P = 98 mN) Experiment (P = 0.25 N) Model (P = 0.25 N)
HV (
GP
a)
Atomic constraints
(a)
0 10 20 30 404
6
8
10
12
Experiment (P = 98 mN) Model (P = 98 mN) Experiment (P = 0.25 N) Model (P = 0.25 N)
HV (
GP
a)
[Na2O] (mol%)
(b)
dHV/dn
Smedskjaer, Mauro, Yue, Phys. Rev. Lett. 105, 115503 (2010)
C) Temperature dependent constraints
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
Results: Hardness H v in borates
• Correlating the kinetic fragility index m with thermodynamic property change at Tg
– Adam-Gibbs model
conf conf conf conf confp
conf conf
ln ln1( , , , )
ln ln lnP P PP P
H H S H SC x y z T
T S T T S T
∂ ∂ ∂ ∂ ∂ ∆ = = = ∂ ∂ ∂ ∂ ∂
conf
expB
TSη η∞
=
g
conf0
ln ( )1
ln T T
S Tm m
T =
∂ = + ∂
confp,pgplp CCCC ≅−=∆
Smedskjaer et al., J. Phys. Chem. B 115, 12930 (2011)
C) Temperature dependent constraints
Results: Calculating the specific heat from constraints
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
According to temperature-dependent constraint theory, configurational
entropy at Tg is inversely proportional to Tg.
This is a configurationaltemperature of the glass at Tg. For a normal cooling rate (10 K/min),
Tconf = Tg.
g
confp g
g conf 0, ( , , )
1 ( , , )[ , , , ( , , )] 1
( , , ) lnP T T x y z
H m x y zC x y z T x y z
T x y z S m=
∂∆ = − ∂
g
conf g confp g
g conf 0, ( , , )
[ , , , ( , , )] ( , , )[ , , , ( , , )] 1
( , , )P T T x y z
S x y z T x y z H m x y zC x y z T x y z
T x y z S m=
∂∆ = − ∂
R Rp g conf2
g 0 g 0
(( , , ) ) (( , , ) )( , , ) ( , , )[ , , , ( , , )] ( , , ) 1 1
[ ( , , )] ( , , )
A x y z A x y zm x y z m x y zC x y z T x y z T x y z
T x y z m T x y z m
∆ = − = −
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
C) Temperature dependent constraints
Results: Calculating the specific heat from constraints
∆Cp(x,y,z) can be predicted with A is the sole fitting parameter (19 kJ/mol)
Thermodynamic property changes during the glass transition are connected to the kinetic fragility index
100 200 300 400 500 60050
80
110
140
170
Cp (
J m
ol-1 K
-1)
T (oC)
75B 63B-12Si 51B-24Si 37B-37Si 24B-51Si 12B-63Si 6B-69Si 75Si
0.0 0.2 0.4 0.6 0.8 1.010
20
30
40
50
60
Experiment Model (analyzed) Model (ideal)
∆Cp
(J m
ol-1 K
-1)
[SiO2]/([SiO
2]+[B
2O
3])
Smedskjaer et al., J. Phys. Chem. B 115, 12930 (2011)
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
C) Temperature dependent constraints
Results: Results for the specific heat (borosilicates)
Topological modeling: exploring new composition spaces where glasses have not yet been melted
Difference in scaling is due to T-dependence of constraints
1.0
0.5
0.4
0.3
0.2
0.1
0.0
0.9
0.8
0.7
0.6
CaOB 2
O 3
Na2O
500
585
670
755
840
925
Tg (K)
0.0 0.1 0.2 0.3 0.4 0.5
0.5
1.0
0.5
0.4
0.3
0.2
0.1
0.0
0.9
0.8
0.7
0.6
CaOB 2
O 3
Na2O
4.5
5.4
6.3
7.2
8.1
9.0
HV (GPa)
0.0 0.1 0.2 0.3 0.4 0.5
0.5
C) Temperature dependent constraints
Results: Quantitative designe of glasses (borates)
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
Conclusion Topology is a useful tool for the understanding compositional trends in glasses
The scaling of glass properties with composition can be quantitatively predicted from mechanical constraints
Account for the temperature dependence of network constraints leads to the prediction of glass properties
Comparison with other modeling approaches Disadvantages: fewer details; requires a priori knowledge of structure and
constraints Advantages: simple; isolates key physics; analytical
matthieu.micoulaut@upmc.fr Atomic modeling of glass – LECTURE 9 TOPOLOGY
Next lecture: Rigidity transitions and intermediate phases
Home reading: Topological constraint theory of glass, J.C. Mauro, 2012
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