potential energy landscape description of supercooled liquids and glasses
TRANSCRIPT
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Potential Energy Landscape Description of Supercooled Liquids and Glasses
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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
Riferimenti
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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
Sommario
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Structural Glasses: Self-generated disorder
Nomenclature
Routes to Vitrification:
•Quench•Crunch•Chemical Vitrification•Vapor Deposition•Ion bombardment•Crystal Amorphization
Long Range Order MissingShort Range Order Present
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Local Order IndicatorsRadial Distribution Function - Structure Factor
Conditional probability of finding a particle center at distance r (in a spherical shell of volume 4 r2 dr) given that there is another one at the origin
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Static Structure Factor
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Generalization of S(q) to dynamics
How a density fluctuation decays…..
How a particle decorrelate over a distance of the order of q-1
S(q,t)
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Two well known models for Sself(q,t)
(if xi is a gaussian random process - Kubo)
Free Diffusion
Motion in an harmonic potential,
Two models for Sself
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fq
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Strong-Fragile
P.G. Debenedetti, and F.H. Stillinger, Nature 410, 259 (2001).
A slowing down that cover more than 15 order of magnitudes
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Excess Entropy
A vanishing of the entropy difference at a finite T ?
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van Megen and S.M. Underwood
Phys. Rev. Lett. 70, 2766 (1993)
(t)
(t)
log(t)
Separation of time scales
Supercooled Liquid
Glass
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IS
Pe
IS
Statistical description of the number, depth and shapeof the PEL basins
Potential Energy Landscape, a 3N dimensional surface
The PEL does not depend on TThe exploration of the PEL depends on T
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De Broglie wavelength
1/kBT
Pair-wise additive spherical potentials System of identical particles
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all basins iQ(T,V)= Qi(T,V)
Non-crystalline
‘Formalismo di Stillinger-Weber
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Thermodynamics in the IS formalism Stillinger-Weber
F(T,V)=-kBT ln[(<eIS>)]+fbasin(<eIS>,T,V)
with
fbasin(eIS,T,V)= eIS+fvib(eIS,T,V)
and
Sconf(T,V)=kBln[(<eIS>)]
Basin depth and shape
Number of explored basins
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1-d Cos(x) Landscape
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rN
Distribution of local minima (eIS)
Vibrations (evib)
+
eIS
e vib
Configuration Space
ek
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F(T,V)=-kBT ln[(<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)
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minimization
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BKS Silica Si02
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High TSlow Dyn.
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Time-Dependent Specific Heat in the IS formalism
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BMLJ
V
TA
Liquid Entropy (in B)
CPB
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diagonalization
Basin Shape
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Harmonic Basin free energy
Very often approximated with……
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Vibrational Free Energy
SPC/E LW-OTP
ln[i(eIS)]=a+b eIS +c eIS2
kBTjln [hj(eIS)/kBT]
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Pitfalls
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f anharmonic
eIS independent anharmonicity
Weak eIS dependentanharmonicity
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Differences of 0.1-0.2can arise from different handling of the anharmonicentropy
Example wih soft sphere
V= (/r)n
n=12
D(eIS)
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Thermodynamic Integration
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Frenkel-Ladd (Einstein Crystal)
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n-2n
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BMLJ Configurational Entropy
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T-dependence of Sconf (SPC/E)(SPC/E)
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Excess Entropy
A vanishing of the entropy difference at a finite T ?
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Fine Seconda Parte
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The Random Energy Model for eIS
Hypothesis:eIS)deIS=eN -----------------deIS
e-(eIS
-E0)2/22
22
Sconf(eIS)/N=- (eIS-E0)2/22
Gaussian Landscape
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Partitin function
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Predictions of Gaussian LandscapePrediction 1
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Predictions of Gaussian Landscape II
Eis vs T, Scon vs TEk Tk
Prediction grafics
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eIS=eiIS
E0=<eNIS>=Ne1
IS
2= 2N=N 2
1
Gaussian Distribution ?
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T-dependence of <eIS>
SPC/E LW-OTP
T-1 dependence observed in the studied T-rangeSupport for the Gaussian Approximation
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P(eIS,T)
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BMLJ Configurational Entropy
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T-dependence of Sconf (SPC/E)(SPC/E)
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Come misuriamo
Sigma2, alpha, E0, b
Come misuriamo
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The V-dependence of , 2, E0
eIS)deIS=eN -----------------deISe-(e
IS -E
0)2/22
22
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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)/TPconst(V)= - d/dV [E0-b2]PT(V) =R d/dV [-a-bE0+b22/2]P1/T(V) = d/dV [2/2R]
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Developing an EOS based on PES properties
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SPC/E P(T,V)=Pconst(V)+PT(V) T + P1/T(V)/T
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Non-Gaussian behavior in BKS Silica
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Eis e S conf for silica…
Esempio di forte
Non-Gaussian Behavior in SiO2
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Landscape of Strong Liquid
SW if # of bonded particles <= Nmax
HS if # of bonded particles > Nmax
V(r)
r
Maximum Valency
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Viscosity and Diffusivity: Arrhenius
• =1
• Cv small
• Stokes-Einstein Relation
Other strong properties:
percolating
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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
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It is possible to calculate exactly the vibrational entropy of one single bonding pattern
(basin free energy)
(Ladd andFrenkel)
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sconf
Non zero ground state entropy
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Landscape of strong and fragile liquids
Realistic ModelNetwork
Primitive Model for Network
Fragile Liquid
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Dinamics !
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Correlating Thermodynamics and Dynamics: Adam-Gibbs Relation
BKS Silica
Ivan Saika-Voivod et al, Nature 412, 514 (2001).
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SPC/Ewater
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V ~ (/r)-n
Soft Spheres with different softness
De Michele et al
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SummaryThe 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 !!
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End of Thirth Lecture
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Simple (numerical) Aging Experiment
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Aging in the PEL-IS framework
Starting Configuration (Ti)
Short after the T-change
(Ti->Tf)
Long timeT
i
Tf
Tf
Same Basins as eq.!
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Evolution of eIS in aging (BMLJ)
One can hardly do better than equilibrium !!
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The “TAP” free energies……
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F(T, Tf )=-Tf Sconf (eIS)+fbasin(eIS,T)
S. Franz and M. A. Virasoro, J. Phys. A 33 (2000) 891,
Which T in aging ?
Equivalent form:
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If basins have identical shape …..
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bmlj
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A look to the meaning of Teff
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Heat flows…..(case of basins of identical shape )
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Fluctuation Dissipation Relation (Cugliandolo, Kurcian, Peliti, ….)
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Support from the Soft Sphere Model
F(V, T, Tf)=-TfSconf (eIS)+fbasin(eIS,T)
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From Equilibrium to OOE….
If we know which equilibrium basin the system is exploring…
eIS acts as a fictive T !
eIS, V, T
.. We can correlate the state of the aging system with an equilibrium state and predict the pressure
(OOE-EOS)
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Numerical TestsLiquid-to-Liquid
T-jump at constant V
P-jump at constant T
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Numerical TestsHeating a glass at constant P
TP
time
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Numerical TestsCompressing at constant T
Pf
T
time
Pi
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Breakdowns !
(things to be understood)
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Breaking of the out-of-equilibrium theory….Kovacs (cross-over) effect
S. Mossa and FS, PRL (2004)
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Breakdown - eis-dos From Kovacs
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P(eIS,tw)
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BMLJ
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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
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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
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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. StarrN. Giovambattista, A. Moreno, G. Foffiwith whom I had the pleasure to work on PEL ideas.