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Outline. Part I Part II. Thermodynamics in the IS formalism. Free energy. Stillinger-Weber. F(T)=-T S conf (<e IS >, T) +f basin (<e IS >,T). with. Basin depth and shape. f basin (e IS ,T)= e IS +f vib (e IS ,T). and. Number of explored basins. S conf (T)=k B ln[ W (<e IS >)].

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Outline

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Outline

Outline

Part I

Part II


Free energy

Thermodynamics in the IS formalism

Free energy

Stillinger-Weber

F(T)=-T Sconf(<eIS>, T) +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>)]


The random energy model for e is

The Random Energy Model for eIS

Gaussian Landscape

Hypothesis:

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

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

2ps2

Sconf(eIS)/N=a-(eIS-E0)2/2s 2


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Predictions of Gaussian Landscape


T dependence of e is

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 Sconf

BMLJ Configurational Entropy


Non gaussian behaviour in bks silica

Non Gaussian Behaviour in BKS silica


Density minima

Density Minima

P.Poole

Density minimum and CV maximum in ST2 water

inflection in energy

inflection = CV max


Sconf silica

Sconf Silica

Non-Gaussian Behavior in SiO2

Eis e S conf for silica…

Esempio di forte

Non gaussian silica


Maximum valency

Maximum Valency

Maximum Valency Model (Speedy-Debenedetti)

SW if # of bonded particles <= Nmax

HS if # of bonded particles > Nmax

V(r

)

r

A minimal model for network forming liquids

The IS configurations coincide with the bonding pattern !!!


Square well 3 width

Square Well 3% width

Generic Phase Diagram for Square Well (3%)


Square well 3 width1

Square Well 3% width

Generic Phase Diagram for NMAX Square Well (3%)


Ground state energy known liquid free energy known everywhere

Ground State Energy Known !(Liquid free energy known everywhere!)

(Wertheim)

It is possible to equilibrate at low T !

Energy per Particle


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Cv

Specific Heat (Cv) Maxima


Viscosity and diffusivity arrhenius

Viscosity and Diffusivity: Arrhenius


Stoke einstein relation

Stoke-Einstein Relation


Dynamics bond lifetime

Dynamics: Bond Lifetime


Basin free energy

It is possible to calculate exactly the basin free energy !

Basin Free energy


S vib

S vib

Svib increases linearly with

the number of bonds

Sconf follows

a x ln(x) law

Sconfdoes NOT extrepolate to zero


Self consistence

Self consistence

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


Take home message

Take home message:

Network forming liquids tend to reach their (bonding) ground state on cooling (eIS 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

Network liquids are intrinsically different from non-networks, since the approach to the ground state is hampered by phase separation


Frenkel ladd einstein crystal

Frenkel-Ladd (Einstein Crystal)


Thermodyanmics

Thermodyanmics

Excess Entropy

A vanishing of the entropy difference at a finite T ?


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