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Orleans July 2004. Networks in physics and biology . Potential Energy Landscape in Models for Liquids. 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

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Orleans July 2004

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Orleans July 2004

Networks in physics and biology

Potential Energy Landscape

in

Models for Liquids

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

Outline


Strong and Fragile liquids Dynamics

A slowing down that cover more than 15 order of magnitudes

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

1


A Decrease in Configurational Entropy: Thermodynamics

Why do we care Thermodyanmics

Is the excess entropy vanishing at a finite T ?

1


The basic idea: Separation of time scales

f(t)

glass

liquid

f(t)

van Megen and S.M. Underwood

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

Glass

Supercooled Liquid

log(t)


Potential Energy Landscape, a 3N dimensional surface

Statistical description of the number, depth and shape

of the PEL basins

e

IS

P

IS

w

The PEL does not depend on TThe exploration of the PEL depends on T


Z(T)= S Zi(T)

allbasins i

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

fbasin(eIS,T)= eIS+ kBTSln [hwj(eIS)/kBT] + fanharmonic(eIS, T)

normal modes j


Thermodynamics in the IS formalism

Stillinger-Weber

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

with

Basin depth and shape

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

and

Number of explored basins

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


Distribution of local minima (eIS)

Configuration Space

+

Real Space

Vibrations (evib)

rN

evib

eIS


<eIS>(T) (steepest descent minimization)

fbasin(eIS,T) (harmonic and anharmonic contributions)

F(T) (thermodynamic integration from ideal gas)

From simulations…..

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

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:

e-(eIS -E0)2/2s 2

W(eIS)deIS=eaN -----------------deIS

2ps2

S ln[wi(eIS)]=a+b eIS

Predictions:

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

Sconf(T)=aN-(<eIS (T)>-E0)2/2s 2


T-dependence of <eIS>

SPC/E

LW-OTP

T-1 dependence observed in the studied T-range

Support for the Gaussian Approximation


BMLJ Sconf

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-bs2]

PT(V) =R d/dV [a-a-bE0+b2s2/2]

P1/T(V) = d/dV [s2/2R]


Non Gaussian Behaviour in BKS silica


Sconf Silica

Non-Gaussian Behavior in SiO2

Eis e S conf for silica…

Esempio di forte

Non gaussian silica


Density minimum and CV maximum in ST2 water

Density Minima

P.Poole

inflection in energy

inflection = CV max


ST2 2650 punti

Isochores of liquid ST2 water

  • HDL

LDL

?


Maximum Valency

Maximum Valency Model (Speedy-Debenedetti)

V(r

)

SW if # of bonded particles <= Nmax

HS if # of bonded particles > Nmax

r

A minimal model for network forming liquids

The IS configurations coincide with the bonding pattern !!!


It is possible to calculate exactly the basin free energy !

Basin Free energy


Ground State Energy Known !

It is possible to equilibrate

at low T !

Energy per Particle


Viscosity and Diffusivity: Arrhenius


Configurational Entropy


Fragile Liquids

Gaussian 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 !

Suggestions for further studies…..


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


Gaussian Distribution ?

eIS=SeiIS

E0=<eNIS>=Ne1IS

s2= s2N=N s21


Diffusivity


Phase Diagram

NMAX-modifiedPhase Diagram


The V-dependence of a, s2, E0

e-(eIS -E0)2/2s 2

W(eIS)deIS=eaN -----------------deIS

2ps2


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)


D ST2

Isobars of diffusion coefficient for ST2 water


Adam-Gibbs Plot


Basin Free Energy

kBTSln [hwj(eIS)/kBT]

LW-OTP

SPC/E

S ln[wi(eIS)]=a+b eIS

…if b=0 …..


BKS Silica

Eis nel tempo


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