Orleans July 2004

Potential Energy Landscape

in

Models for Liquids

Networks in physics and biology

In collaboration with E. La Nave, A. Moreno, I. Saika-Voivod, E. Zaccarelli

• A 3-slides preamble: Thermodynamics and Dynamics

• Review of thermodynamic formalism in the PEL approach

• Potential Energy Landscapes in Fragile and Strong (Network-Forming) liquids.

Outline

Strong and Fragile liquids Dynamics

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

A slowing down that cover more than 15 order of magnitudes

1

A Decrease in Configurational Entropy: Thermodynamics

Is the excess entropy vanishing at a finite T ?

1

van Megen and S.M. Underwood

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

(t)

(t)

log(t)

The basic idea: Separation of time

scales

Supercooled Liquid

Glass

glassliquid

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

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

all basins i

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

+

fanharmonic(eIS, T)

normal modes j

Z(T)= Zi(T)

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

rN

Distribution of local minima (eIS)

Vibrations (evib)

+

eIS

e vib

Configuration Space

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).

Fragile Liquids: The 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

T-dependence of <eIS>

SPC/E LW-OTP

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

BMLJ Configurational Entropy

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

Non Gaussian Behaviour in BKS silica

Eis e S conf for silica…

Esempio di forte

Non-Gaussian Behavior in SiO2

Density minimum and CV maximum in ST2 water

inflection = CV max

inflection in energy

P.Poole

Isochores of liquid ST2 water

LDL

HDL

?

Maximum Valency Model (Speedy-Debenedetti)

A minimal model for network forming liquids

SW if # of bonded particles <= NmaxHS if # of bonded particles > Nmax

V(r)

r

The IS configurations coincide with the bonding pattern !!!

It is possible to calculate exactly the basin free energy !

Ground State Energy Known !

It is possible to equilibrate at low T !

Viscosity and Diffusivity: Arrhenius

Configurational Entropy

Suggestions for further studies…..Fragile LiquidsGaussian Energy Landscape

Finite TK, Sconf(TK)=0

Strong Liquids:“Bond Defect” landscape (binomial)A “quantized” bottom of the landscape !Degenerate Ground State

Sconf(T=0) different from zero !

Acknowledgements

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

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

… and their collaborators

Stoke-Einstein Relation

eIS=eiIS

E0=<eNIS>=Ne1

IS

2= 2N=N 2

1

Gaussian Distribution ?

Diffusivity

Phase Diagram

The V-dependence of , 2, E0

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

IS -E

0)2/22

22

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

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

Isobars of diffusion coefficient for ST2 water

Adam-Gibbs Plot

Basin Free Energy

SPC/E LW-OTP

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

kBTln [hj(eIS)/kBT]

…if b=0 …..

BKS Silica