Potential Energy Landscape Description of
Supercooled Liquids and Glasses
Riferimenti
http://mc2tar.phys.uniroma1.it/~fs/didattica/dottorato/
D. Wales Energy Landscapes Cambridge University Press
F. Sciortino Potential energy landscape description of
supercooled liquids and glassesJ. Stat. Mech. 050515, 2005
Articoli Gruppo Roma (molti dei quali sul landscape)
http://glass.phys.uniroma1.it/sciortino/publications.htm
Sommario
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Introduzione ai vetri ed ai liquidi
sottorrafreddati
Formalismo Termodinamico nel PEL
Confronti con dati numerici
Sviluppo di una PEL EOS
Termodinamica di fuori equilibrio
Nomenclature
Structural Glasses: Self-generated disorder
Routes to Vitrification:
•Quench
•Crunch
•Chemical Vitrification
•Vapor Deposition
•Ion bombardment
•Crystal Amorphization
Long Range Order Missing
Short Range Order Present
Local Order Indicators
Radial Distribution Function - Structure Factor
Conditional probability of finding a
particle center at distance r
(in a spherical shell of volume
4p r2 dr) given that there is another
one at the origin
Static Structure Factor
S(q,t)
Generalization of S(q) to dynamics
How a density fluctuation decays…..
How a particle decorrelate over a distance of the order of q-1
Two well known models for Sself(q,t)
Two models for Sself
(if xi is a gaussian random process - Kubo)
Free Diffusion
Motion in an harmonic potential,
fq
Strong-Fragile
A slowing down
that cover more
than 15 order of
magnitudes
P.G. Debenedetti, and F.H.
Stillinger, Nature 410, 259
(2001).
Excess Entropy
A vanishing of
the entropy
difference at a
finite T ?
f(t)
Separation of time
scales
f(t)
van Megen and S.M. Underwood
Phys. Rev. Lett. 70, 2766 (1993)
Glass
Supercooled Liquid
log(t)
Potential Energy Landscape, a 3N dimensional surface
Statistical description of the number, depth and shape
of the PEL basins
eIS
PIS
w
The PEL does not depend on T
The exploration of the PEL depends on T
Pair-wise additive spherical potentials
System of identical particles
De Broglie wavelength
1/kBT
Formalismo di Stillinger-Weber
‘
Q(T,V)= S Qi(T,V)
all basins i
Non-crystalline
Thermodynamics in the IS formalism
StillingerWeber
F(T,V)=-kBT ln[W(<eIS>)]+fbasin(<eIS>,T,V)
with
fbasin(eIS,T,V)= eIS+fvib(eIS,T,V)
and
Sconf(T,V)=kBln[W(<eIS>)]
Basin depth
and shape
Number of
explored
basins
1-d Cos(x) Landscape
N
r
eIS
ek
Vibrations
(evib)
evib
Configuration
Space
Distribution of
local minima +
(eIS)
F(T,V)=-kBT ln[W(<eIS>)]+fbasin(<eIS>,T,V)
From simulations…..
<eIS>(T,V) (steepest descent minimization)
fbasin(eIS,T,V) (harmonic and anharmonic
contributions)
F(T,V) (thermodynamic integration from ideal gas)
minimization
BKS Silica Si02
Slow Dyn.
High T
Time-Dependent Specific Heat in the IS formalism
Liquid Entropy (in B)
T
A
CP
B
V
BMLJ
Basin Shape
diagonalization
Harmonic Basin free energy
Very often approximated with……
Vibrational Free Energy
kBTSj ln [hwj(eIS)/kBT]
SPC/E
LW-OTP
S ln[wi(eIS)]=a+b eIS +c eIS
2
Pitfalls
f anharmonic
eIS independent
anharmonicity
Weak eIS dependent
anharmonicity
Example wih soft sphere
V=e (s/r)n
n=12
Differences of 0.1-0.2
can arise from different
handling of the anharmonic
entropy
D(eIS)
Thermodynamic Integration
Frenkel-Ladd (Einstein Crystal)
n-2n
BMLJ Configurational Entropy
T-dependence of Sconf (SPC/E)
Excess Entropy
A vanishing of
the entropy
difference at a
finite T ?
Fine Seconda Parte
The Random Energy Model for eIS
Gaussian Landscape
Hypothesis:
W(eIS)deIS=e
-(e
aN e
IS
-E )2/2s 2
0
-----------------deIS
2ps2
Sconf(eIS)/N=a- (eIS-E0)2/2s 2
Partitin function
Predictions of Gaussian Landscape
Prediction 1
Predictions of Gaussian Landscape II
Eis vs T, Scon vs T
Ek Tk
Prediction grafics
Gaussian Distribution ?
eIS=SeiIS
N
1
E0=<e IS>=Ne IS
s2= s2N=N s21
T-dependence of <eIS>
SPC/E
LW-OTP
T-1 dependence observed in the studied T-range
Support for the Gaussian Approximation
P(eIS,T)
BMLJ Configurational Entropy
T-dependence of Sconf (SPC/E)
Come misuriamo
Come misuriamo
Sigma2, alpha, E0, b
The V-dependence of a, s2, E02
aN
W(eIS)deIS=e
-(e
-E ) /2s 2
e IS 0
-----------------de
IS
2
2ps
Landscape Equation of State
P=-∂F/∂V|T
F(V,T)=-TSconf(T,V)+<eIS(T,V)>+fvib(T,V)
In Gaussian (and harmonic) approximation
P(T,V)=Pconst(V)+PT(V) T + P1/T(V)/T
Pconst(V)= - d/dV [E0-bs2]
PT(V) =R d/dV [a-a-bE0+b2s2/2]
P1/T(V) = d/dV [s2/2R]
Developing an EOS based on PES properties
SPC/E P(T,V)=Pconst(V)+PT(V) T + P1/T(V)/T
Non-Gaussian behavior in
BKS Silica
Non-Gaussian Behavior in SiO2
Eis e S conf for silica…
Esempio di forte
Landscape of Strong Liquid
V(r )
SW if # of bonded particles <= Nmax
HS if # of bonded particles > Nmax
r
percolating
Viscosity and
Diffusivity:
Arrhenius
Other strong properties:
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b=1
DCv small
Stokes-Einstein
Relation
Ground State Energy Known !
•It is possible to equilibrate at
low T !
•E(T) is known and hence free
energy can be calculated exactly
down to T=0
It is possible to calculate exactly the
vibrational entropy of one single bonding pattern
(basin free energy)
(Ladd and
Frenkel)
sconf
Non zero
ground
state
entropy
Landscape of strong and fragile liquids
Primitive
Model for
Network
Realistic Model
Network
Fragile Liquid
Dinamics !
Correlating Thermodynamics and Dynamics: Adam-Gibbs Relation
BKS
Silica
Ivan SaikaVoivod et al,
Nature 412,
514 (2001).
SPC/E
water
V ~ (s/r)-n
Soft Spheres with different softness
De Michele et al
Summary
The statistical properties of the PEL can be quantified
with a proper analysis of simulation data
Accurate EOS can be constructed from these
information (but we may have to go beyond the
Gaussian approximation)
Interesting features of the liquid state (TMD line) can
be correlated to features of the PEL statistical
properties
Connections between Dynamics and Thermodynamics
need further studies !!
End of Thirth Lecture
Simple (numerical) Aging
Experiment
Aging in the PEL-IS framework
Same Basins as eq.!
Tf
Ti
Tf
Starting
Configuration (Ti)
Short after
the T-change
(Ti->Tf)
Long time
Evolution of eIS in aging (BMLJ)
One can hardly do better than equilibrium !!
The “TAP” free energies……
Which T in aging ?
F(T, Tf )=-Tf Sconf (eIS)+fbasin(eIS,T)
Equivalent form:
S. Franz and M. A. Virasoro,
J. Phys. A 33 (2000) 891,
If basins have identical shape …..
bmlj
A look to the meaning of Teff
Heat flows…..
(case of basins of identical shape )
Fluctuation Dissipation Relation
(Cugliandolo, Kurcian, Peliti, ….)
F(V, T, Tf)=-TfSconf (eIS)+fbasin(eIS,T)
Support from the Soft
Sphere Model
From Equilibrium to OOE….
If we know which equilibrium basin the system is
exploring…
eIS, V, T
.. We can correlate the state of the aging system
with an equilibrium state and predict the pressure
(OOE-EOS)
eIS acts as a fictive T !
Numerical Tests
Liquid-to-Liquid
T-jump
at constant V
P-jump
at constant T
Numerical Tests
Heating a glass at constant P
T
P
time
Numerical Tests
Compressing at constant T
Pi
Pf
T
time
Breakdowns !
(things to be understood)
Breaking of the out-of-equilibrium theory….
Kovacs (cross-over) effect
S. Mossa and FS, PRL (2004)
Breakdown - eis-dos
From Kovacs
P(eIS,tw)
BMLJ
Summary II
The hypothesis that the system samples in aging
the same basins explored in equilibrium allows to
develop an EOS for OOE-liquids depending on one
additional parameter
Small aging times, small perturbations are
consistent with such hypothesis. Work must be
done to evaluate the limit of validity.
 The aditional parameter can be chosen as fictive
T, fictive P or depth of the explored basin eIS
Perspectives
An improved description of the statistical properties
of the potential energy surface.
 Role of the statistical properties of the PEL in liquid
phenomena
 A deeper understanding of the concept of Pconf and
of EOS of a glass.
 An estimate of the limit of validity of the
assumption that a glass is a frozen liquid (number of
parameters)
 Connections between PEL properties and Dynamics
Acknowledgements
I acknowledge important comments, criticisms, discussions with
P. Debenedetti, S. Sastry, R. Speedy, A. Angell, T. Keyes, G.
Ruocco, P. Poole and their collaborators
I thank, among others,
E. La Nave, I. Saika-Voivod, C. Donati,
A. Scala, L. Angelani, C. De Michele, F. Starr
N. Giovambattista, A. Moreno, G. Foffi
with whom I had the pleasure to work on PEL ideas.