Konstanz, September 2002 Potential Energy Landscape Equation of State Emilia La Nave, Francesco Sciortino, Piero Tartaglia (Roma) Stefano Mossa (Boston/Paris) 5th Liquid Matter Conference

IS P e  Statistical description of the number, depth and shape of the PEL basins Potential Energy Landscape

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 (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

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) Data for two rigid-molecule models: LW-OTP, SPC/E-H 2 0 In this talk…..

Basin Free Energy  ln[  i (e IS )]=a+b e IS SPC/E LW-OTP

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

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

T-dependence of (SPC/E)

T-dependence of (LW-OTP)

(SPC/E) T-dependence of S conf (SPC/E)

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

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]

Developing an EOS based on PES properties

SPC/E Water P(T,V)=P const (V)+P T (V) T + P 1/T (V)/T

Conclusion I The V-dependence of the statistical properties of the PEL has been quantified for two models of molecular liquids Accurate EOS can be constructed from these information Interesting features of the liquid state (TMD line) can be correlated to features of the PEL statistical properties

Aging in the PEL-IS framework Starting Configuration (T i ) Short after the T-change (T i ->T f ) Long time TiTi TfTf TfTf

Reconstructing P(T,V) P=-∂F/∂V F(V,T)=-TS conf (T,V)+ +f vib (T,V) P(T,V)= P conf (T,V) + P vib (T,V)

From Equilibrium to OOE…. P(T,V)= P conf (T,V)+ P vib (T,V) If we know which equilibrium basin the system is exploring… e IS acts as a fictive T ! e IS (V,T f ).V  P conf e IS (V,T f ),V,T  log(  )  P vib

Numerical Tests Heating a glass at constant P T P time

Numerical Tests Compressing at constant T PfPf T time PiPi

Liquid-to-Liquid T-jump at constant V P-jump at constant T

Conclusion 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 hipothesis. Work is requested to evaluate the limit of validity.  This parameter can be chosen as fictive T, fictive P or depth of the explored basin e IS

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 P conf 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

References and Acknowledgements We acknowledge important discussions, comments, criticisms from P. Debenedetti, S. Sastry, R. Speedy, A. Angell, T. Keyes, G. Ruocco and collaborators Francesco Sciortino and Piero Tartaglia Extension of the Fluctuation-Dissipation theorem to the physical aging of a model glass-forming liquid Phys. Rev. Lett (2001). Emilia La Nave, Stefano Mossa and Francesco Sciortino Potential Energy Landscape Equation of State Phys. Rev. Lett., 88, (2002). Stefano Mossa, Emilia La Nave, Francesco Sciortino and Piero Tartaglia, Aging and Energy Landscape: Application to Liquids and Glasses., cond-mat/

Entering the supercooled region

Same basins in Equilibrium and Aging ?