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

Outline 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 Why do we care Thermodyanmics Is the excess entropy vanishing at a finite T ? 1

van Megen and S.M. Underwood Phys. Rev. Lett. 70, 2766 (1993)  (t) log(t) The basic idea: Separation of time scales Supercooled Liquid Glass glass liquid

IS P e  Statistical description of the number, depth and shape of the PEL basins Potential Energy Landscape, a 3N dimensional surface The PEL does not depend on T The exploration of the PEL depends on T

f basin i (T)= -k B T ln[Z i (T)] all basins i f basin (e IS,T)= e IS + k B T   ln [h  j (e IS )/k B T] + f anharmonic (e IS, T) normal modes j Z(T)=  Z i (T)

Thermodynamics in the IS formalism Stillinger- Weber F(T)=-k B T ln[  ( )]+f basin (,T) with f basin (e IS,T)= e IS +f vib (e IS,T) and S conf (T)=k B ln[  ( )] Basin depth and shape Number of explored basins

Real Space rNrN Distribution of local minima (e IS ) Vibrations (e vib ) + e IS e vib Configuration Space

F(T)=-k B T ln[  ( )]+f basin (,T) From simulations….. (T) (steepest descent minimization) f basin (e IS,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 e IS Hypothesis: Predictions :  e IS )de IS = e  N de IS e -(e IS -E 0 ) 2 /2  2 2222  ln[  i (e IS )]=a+b e IS =E 0 -b  2 -  2 /kT S conf (T)=  N- ( -E 0 ) 2 /2  2

T-dependence of SPC/ELW-OTP T -1 dependence observed in the studied T-range Support for the Gaussian Approximation

BMLJ Configurational Entropy BMLJ Sconf

Landscape Equation of State P=-∂F/∂V |T F(V,T)=-TS conf (T,V)+ +f vib (T,V) In Gaussian (and harmonic) approximation P(T,V)=P const (V)+P T (V) T + P 1/T (V)/T P const (V)= - d/dV [E 0 -b  2 ] P T (V) =R d/dV [  -a-bE 0 +b 2  2 /2] P 1/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 SiO 2 Non gaussian silica Sconf Silica

Density minimum and C V maximum in ST2 water inflection = C V max inflection in energy Density Minima P.Poole

Isochores of liquid ST2 water LDL HDL ? ST punti

Maximum Valency Model (Speedy-Debenedetti) A minimal model for network forming liquids SW if # of bonded particles <= Nmax HS if # of bonded particles > Nmax V(r) r Maximum Valency The IS configurations coincide with the bonding pattern !!!

It is possible to calculate exactly the basin free energy ! Basin Free energy

Energy per Particle Ground State Energy Known ! It is possible to equilibrate at low T !

Viscosity and Diffusivity: Arrhenius

Configurational Entropy

Suggestions for further studies….. Fragile Liquids Gaussian Energy Landscape Finite T K, S conf (T K )=0 Strong Liquids: “Bond Defect” landscape (binomial) A “quantized” bottom of the landscape ! Degenerate Ground State S conf (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

Citazioni goldstein, stillinger

e IS =  e i IS E 0 = =N e 1 IS  2 =  2 N =N  2 1 Gaussian Distribution ?

Diffusivity

N MAX -modified Phase Diagram Phase Diagram

The V-dependence of ,  2, E 0  e IS )de IS =e  N de IS e -(e IS -E 0 ) 2 /2  2 2222

SPC/E P(T,V)=P const (V)+P T (V) T + P 1/T (V)/T FS, E. La Nave, and P. Tartaglia, PRL. 91, (2003)

Isobars of diffusion coefficient for ST2 water D ST2

Adam-Gibbs Plot

Basin Free Energy SPC/E LW-OTP  ln[  i (e IS )]=a+b e IS k B T   ln [h  j (e IS )/k B T] …if b=0 …..

Eis nel tempo BKS Silica