Potential Energy Landscape Description of Supercooled Liquids and Glasses

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

• 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

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

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

Static Structure Factor

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)

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

fq

Strong-Fragile

P.G. Debenedetti, and F.H. Stillinger, Nature 410, 259 (2001).

A slowing down that cover more than 15 order of magnitudes

Excess Entropy

A vanishing of the entropy difference at a finite T ?

van Megen and S.M. Underwood

Phys. Rev. Lett. 70, 2766 (1993)

(t)

(t)

log(t)

Separation of time scales

Supercooled Liquid

Glass

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

De Broglie wavelength

1/kBT

Pair-wise additive spherical potentials System of identical particles

all basins iQ(T,V)= Qi(T,V)

Non-crystalline

‘Formalismo di Stillinger-Weber

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

1-d Cos(x) Landscape

rN

Distribution of local minima (eIS)

Vibrations (evib)

+

eIS

e vib

Configuration Space

ek

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)

minimization

BKS Silica Si02

High TSlow Dyn.

Time-Dependent Specific Heat in the IS formalism

BMLJ

V

TA

Liquid Entropy (in B)

CPB

diagonalization

Basin Shape

Harmonic Basin free energy

Very often approximated with……

Vibrational Free Energy

SPC/E LW-OTP

ln[i(eIS)]=a+b eIS +c eIS2

kBTjln [hj(eIS)/kBT]

Pitfalls

f anharmonic

eIS independent anharmonicity

Weak eIS dependentanharmonicity

Differences of 0.1-0.2can arise from different handling of the anharmonicentropy

Example wih soft sphere

V= (/r)n

n=12

D(eIS)

Thermodynamic Integration

Frenkel-Ladd (Einstein Crystal)

n-2n

BMLJ Configurational Entropy

T-dependence of Sconf (SPC/E)(SPC/E)

Excess Entropy

A vanishing of the entropy difference at a finite T ?

Fine Seconda Parte

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

Partitin function

Predictions of Gaussian LandscapePrediction 1

Predictions of Gaussian Landscape II

Eis vs T, Scon vs TEk Tk

Prediction grafics

eIS=eiIS

E0=<eNIS>=Ne1

IS

2= 2N=N 2

1

Gaussian Distribution ?

T-dependence of <eIS>

SPC/E LW-OTP

T-1 dependence observed in the studied T-rangeSupport for the Gaussian Approximation

P(eIS,T)

BMLJ Configurational Entropy

T-dependence of Sconf (SPC/E)(SPC/E)

Come misuriamo

Sigma2, alpha, E0, b

Come misuriamo

The V-dependence of , 2, E0

eIS)deIS=eN -----------------deISe-(e

IS -E

0)2/22

22

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]

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

Eis e S conf for silica…

Esempio di forte

Non-Gaussian Behavior in SiO2

Landscape of Strong Liquid

SW if # of bonded particles <= Nmax

HS if # of bonded particles > Nmax

V(r)

r

Maximum Valency

Viscosity and Diffusivity: Arrhenius

• =1

• Cv small

• Stokes-Einstein Relation

Other strong properties:

percolating

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 andFrenkel)

sconf

Non zero ground state entropy

Landscape of strong and fragile liquids

Realistic ModelNetwork

Primitive Model for Network

Fragile Liquid

Dinamics !

Correlating Thermodynamics and Dynamics: Adam-Gibbs Relation

BKS Silica

Ivan Saika-Voivod et al, Nature 412, 514 (2001).

SPC/Ewater

V ~ (/r)-n

Soft Spheres with different softness

De Michele et al

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 !!

End of Thirth Lecture

Simple (numerical) Aging Experiment

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.!

Evolution of eIS in aging (BMLJ)

One can hardly do better than equilibrium !!

The “TAP” free energies……

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:

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, ….)

Support from the Soft Sphere Model

F(V, T, Tf)=-TfSconf (eIS)+fbasin(eIS,T)

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)

Numerical TestsLiquid-to-Liquid

T-jump at constant V

P-jump at constant T

Numerical TestsHeating a glass at constant P

TP

time

Numerical TestsCompressing at constant T

Pf

T

time

Pi

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. StarrN. Giovambattista, A. Moreno, G. Foffiwith whom I had the pleasure to work on PEL ideas.