STATISTICAL PROPERTIES OF THE LANDSCAPE OF A SIMPLE STRONG LIQUID MODEL …. AND SOMETHING ELSE. E. La Nave, P. Tartaglia, E. Zaccarelli (Roma ) I. Saika-Voivod (Canada) A. Moreno (Spain) S. Bulderyev (N.Y. USA) 5th International Discussion Meeting on Relaxations in Complex Systems New results, Directions and Opportunities Francesco Sciortino

Outline * Peter Harrowell (UCGS Bangalore) Part I -- A (numerically exact) calculation of the statistical properties of the landscape of a strong liquid 1.Thermodynamic in the Stillinger-Weber formalism 2.Gaussian Statistic 3.Deviation from Gaussian 4.The model Dynamics ---- STRONG LIQUID Landscape ---- KNOWN ! Part II -- Dynamic and Static heterogeneities (the central dogma * )

Thermodynamics in the IS formalism Stillinger- Weber F(T)=-T S conf (, T) +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 Free energy [for a recent review see FS JSTAT 5, p (2005)]

The Random Energy Model for e IS Hypothesis:  e IS )de IS = e  N de IS e -(e IS -E 0 ) 2 /2  2 2222 S conf (e IS )/N=  - (e IS -E 0 ) 2 /2  2 Gaussian Landscape Predictions of Gaussian Landscape (for identical basins) S conf (T)/N=  - ( -E 0 ) 2 /2  2 =E 0 -  2 /kT

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

Non Gaussian behaviour in BKS silica (low  ) Saika-Voivod et al Nature 412, , 2001 Heuer works Heuer

Density minimum and C V maximum in ST2 water ( impossible in the gaussian landascape Phys. Rev. Lett. 91, , 2003 ) inflection = C V max inflection in energy Density Minima P.Poole

Eis e S conf for silica… Esempio di forte Non-Gaussian Behavior in SiO 2 Non gaussian silica Sconf Silica Saika-Voivod et al Nature 412, , 2001

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 !!! Zaccarelli et al PRL (2005) Moreno et al Cond Mat (2004)

Square Well 3% width Generic Phase Diagram for Square Well (3%)

Square Well 3% width Generic Phase Diagram for N MAX Square Well (3%)

Energy per Particle Ground State Energy Known ! (Liquid free energy known everywhere!) It is possible to equilibrate at low T ! (Wertheim)

Specific Heat (Cv) Maxima Cv

Viscosity and Diffusivity: Arrhenius

Stoke-Einstein Relation

Dynamics: Bond Lifetime

Pair-wise model (geometric correlation between bonds) (PMW, I. Nezbeda)

Connection between Dynamics and Structure !

An IS is a bonding pattern !!!!! F(T)=-T S conf (, T) +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

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

Entropies… S vib increases linearly with the # of bonds S conf follows a x ln(x) law S conf does NOT extrapolate to zero

Self-consistent calculation ---> S(T) Self consistence

Part 1 - Take home message(s): Network forming liquids tend to reach their (bonding) ground state on cooling (e IS different from 1/T) The bonding ground state can be degenerate. Degeneracy related to the number of possible networks with full bonding. The discretines of the bonding energy (dominant as compared to the other interactions) favors an Arrhenius dynamics and a logarithmic IS entropy. Network liquids are intrinsically different from non-networks, The approach to the ground state is NOT hampered by phase separation

Part II -Dynamic Heterogeneities J. Chem. Phys. B 108,19663,2004 (attempting to avoid any a priori definition) Look at differences between different realizations Dynamic Eterogeneities SPC/E Water 100 realizations nn distance =0.28 nm Follow dyanmics for MSD = (2 x 0.28) 2 nm 2

 2 MSD - vs - MSD

peis Connections with the landscape ?

Connessione eis - D Memory of the landscape location…..

Which D(e IS,T) ? 155 BMLJ

Which D(e IS,T) ?

Conclusions… Part II Clear Connection between Local Dynamics and Local Landscape Deeper basins statistically generate slower dynamics Connection with the NGP More work to do ! See you in ……….

Frenkel-Ladd (Einstein Crystal)