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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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Part I

Part II

Free energy

Thermodynamics in the IS formalism

Free energy


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


Basin depth and shape

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


Number of explored basins


The random energy model for e is
The Random Energy Model for eIS

Gaussian Landscape


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

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


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

T dependence of e is
T-dependence of <eIS>



T-1 dependence observed in the studied T-range

Support for the Gaussian Approximation

Bmlj sconf
BMLJ Sconf

BMLJ Configurational Entropy

Density minima
Density Minima


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




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!)


It is possible to equilibrate at low T !

Energy per Particle


Specific Heat (Cv) Maxima

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


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


Excess Entropy

A vanishing of the entropy difference at a finite T ?