Duesseldorf Feb 2004Duesseldorf February 5 2004

Potential Energy Landscape Description of Supercooled Liquids and Glasses

Luca Angelani, Stefano Mossa, Ivan Saika-Voivod, Emilia La Nave, Piero Tartaglia

Duesseldorf Feb 2004

• Why do we care ? Thermodynamics and Dynamics

• Review of thermodynamic formalism in the PEL approach

• Comparison with numerical simulations• Development of a PEL EOS• Generalization to non-equilibrium (one or

more effective parameters ?)

Outline

Duesseldorf Feb 2004Why do we care ? Dynamics

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

A slowing down that cover more than 15 order of magnitudes

1

Duesseldorf Feb 2004Why do we care: Thermodynamics

Is the excess entropy vanishing at a finite T ?

1

Duesseldorf Feb 2004

van Megen and S.M. Underwood

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

(t)

(t)

log(t)

Separation of time scales

Supercooled Liquid

Glass

glassliquid

Duesseldorf Feb 2004

Duesseldorf Feb 2004

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

Duesseldorf Feb 2004

fbasin i(T)= -kBT ln[Zi(T)]

all basins i

fbasin(eIS,T)= eIS+ kBTln [hj(eIS)/kBT]

+

fanharmonic(T)

normal modes j

Z(T)= Zi(T)

Duesseldorf Feb 2004

Thermodynamics in the IS formalism

Stillinger-Weber

F(T)=-kBT ln[(<eIS>)]+fbasin(<eIS>,T)

with

fbasin(eIS,T)= eIS+fvib(eIS,T)

and

Sconf(T)=kBln[(<eIS>)]

Basin depth and shape

Number of explored basins

Duesseldorf Feb 2004

rN

Distribution of local minima (eIS)

Vibrations (evib)

+

eIS

e vib

Configuration Space

Duesseldorf Feb 2004

F(T)=-kBT ln[(<eIS>)]+fbasin(<eIS>,T)

From simulations…..

<eIS>(T) (steepest descent minimization)

fbasin(eIS,T) (harmonic and anharmonic

contributions)

F(T) (thermodynamic integration from ideal gas)

E. La Nave et al., Numerical Evaluation of the Statistical Properties of a Potential Energy Landscape, J. Phys.: Condens. Matter 15, S1085 (2003).

Duesseldorf Feb 2004BKS Silica

Duesseldorf Feb 2004Basin Free Energy

SPC/E LW-OTP

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

kBTln [hj(eIS)/kBT]

…if b=0 …..

Duesseldorf Feb 2004The Random Energy Model for eIS

Hypothesis:

Predictions:

eIS)deIS=eN -----------------deIS

e-(eIS

-E0)2/22

22

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

<eIS(T)>=E0-b2 - 2/kT

Sconf(T)=N- (<eIS (T)>-E0)2/22

Duesseldorf Feb 2004

eIS=eiIS

E0=<eNIS>=Ne1

IS

2= 2N=N 2

1

Gaussian Distribution ?

Duesseldorf Feb 2004

T-dependence of <eIS>

SPC/E LW-OTP

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

Duesseldorf Feb 2004BMLJ Configurational Entropy

Duesseldorf Feb 2004The V-dependence of , 2, E0

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

IS -E

0)2/22

22

Duesseldorf Feb 2004Landscape 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-b2]PT(V) =R d/dV [-a-bE0+b22/2]P1/T(V) = d/dV [2/2R]

Duesseldorf Feb 2004Developing an EOS based on PEL properties

SPC/Ewater

Duesseldorf Feb 2004SPC/E P(T,V)=Pconst(V)+PT(V) T + P1/T(V)/T

FS, E. La Nave, and P. Tartaglia, PRL. 91, 155701 (2003)

Duesseldorf Feb 2004Correlating Thermodynamics and Dynamics: Adam-Gibbs Relation

BKS Silica

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

exp(A/TSconf)

Duesseldorf Feb 2004Conclusion IThe V-dependence of the statistical properties of the PEL can be quantified for models of liquids

Accurate EOS can be constructed

Peculiar features of the liquid state (TMD line) can be connected to features of the PEL statistical properties

Relation between Dynamics and Thermodynamics can be explored

Duesseldorf Feb 2004Simple (numerical) Aging Experiment

Duesseldorf Feb 2004

Aging in the PEL-IS framework

Starting Configuration (Ti)

Short after the T-change

(Ti->Tf)

Long timeT

i

Tf

Tf

Duesseldorf Feb 2004Evolution of eIS in aging (BMLJ), following a T-jump

W. Kob et al Europhys. Letters 49, 590 (2000).

One can hardly do better than equilibrium !!

Duesseldorf Feb 2004

F(T, Tf )=-Tf Sconf (eIS)+fbasin(eIS,T)

Relation first derived by S. Franz and M. A. Virasoro, J. Phys. A 33 (2000) 891, in the context of disordered spin systems

Which T in aging ?

Duesseldorf Feb 2004

Duesseldorf Feb 2004Fluctuation Dissipation Relation

(Cugliandolo, Kurcian, Peliti, ….)

FS and Piero TartagliaExtension of the Fluctuation-Dissipation theorem to the physical aging of a model glass-forming liquidPhys. Rev. Lett. 86, 107 (2001).

Duesseldorf Feb 2004

Support from the Soft Sphere Model

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

Duesseldorf Feb 2004From Equilibrium to OOE….

If we know which equilibrium basin the system is exploring…

eIS acts as effective T !

eIS, V, T

.. We can correlate the state of the aging system with an equilibrium state and predict the pressure

(OOE-EOS)

Duesseldorf Feb 2004Numerical Tests

Liquid-to-Liquid

T-jump at constant V

P-jump at constant T

S. Mossa et al. EUR PHYS J B 30 351 (2002)

Duesseldorf Feb 2004Numerical Tests

Heating a glass at constant P

TP

time

Duesseldorf Feb 2004Numerical Tests

Compressing at constant TPf

T

time

Pi

Duesseldorf Feb 2004Breaking of the out-of-equilibrium theory….Kovacs (cross-over) effect

S. Mossa and FS, PRL (2004)

Duesseldorf Feb 2004Conclusion II

The hypothesis that the system samples in aging the same basins explored in equilibrium allows us to develop an EOS for OOE-liquids (with one additional parameter)

Small aging times, small perturbations are consistent with such hypothesis. Work is ongoing to evaluate the limit of validity.

The additional parameter can be chosen as effective T, P or depth of the explored basin eIS

Duesseldorf Feb 2004Perspectives

An improved description of the statistical properties of the potential energy surface.

A deeper understanding of the concept 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

Duesseldorf Feb 2004Acknowledgements

We acknowledge important discussions, comments, collaborations, criticisms from…

A. Angell, P. Debenedetti, T. Keyes, G. Ruocco , S. Sastry, R. Speedy

… and their collaborators

Duesseldorf Feb 2004Entering the supercooled region

Duesseldorf Feb 2004Same basins in Equilibrium and Aging ?

Duesseldorf Feb 2004T-dependence of Sconf (SPC/E)(SPC/E)