STATISTICAL PROPERTIES OF THE LANDSCAPE OF A SIMPLE STRONG LIQUID MODEL

1
STATISTICAL PROPERTIES OF THE LANDSCAPE OF A
SIMPLE STRONG LIQUID MODEL . AND SOMETHING ELSE.
5th International Discussion Meeting on
Relaxations in Complex Systems New results,
Directions and Opportunities
Francesco Sciortino

• E. La Nave, P. Tartaglia, E. Zaccarelli (Roma )
• I. Saika-Voivod (Canada)
• A. Moreno (Spain)
• S. Bulderyev (N.Y. USA)

2
Outline

• Part I -- A (numerically exact) calculation
• of the statistical properties of the landscape
• of a strong liquid
• Thermodynamic in the Stillinger-Weber formalism
• Gaussian Statistic
• Deviation from Gaussian
• The model
• Dynamics ---- STRONG LIQUID
• Landscape ---- KNOWN !
• Part II -- Dynamic and Static heterogeneities
• (the central dogma)

Peter Harrowell (UCGS Bangalore)
3
Free energy
Thermodynamics in the IS formalism
Stillinger-Weber
for a recent review see FS JSTAT 5,
p.05015 (2005)
F(T)-T Sconf(lteISgt, T) fbasin(lteISgt,T)
with
Basin depth and shape
fbasin(eIS,T) eISfvib(eIS,T)
and
Number of explored basins
Sconf(T)kBlnW(lteISgt)
4
The Random Energy Model for eIS
Gaussian Landscape
Hypothesis
e-(eIS -E0)2/2s 2
W(eIS)deISeaN -----------------deIS
2ps2
Sconf(eIS)/Na- (eIS-E0)2/2s 2
Predictions of Gaussian Landscape (for identical
basins)
lteIS(T)gtE0 - s 2/kT
Sconf(T)/Na- (lteIS(T)gt -E0)2/2s 2
5
T-dependence of lteISgt
SPC/E
LW-OTP
T-1 dependence observed in the studied T-range
Support for the Gaussian Approximation
6
BMLJ Sconf
BMLJ Configurational Entropy
7
Non Gaussian behaviour in BKS silica (low r)
Saika-Voivod et al Nature 412, 514-517,
2001 Heuer works
Heuer
8
Density Minima
P.Poole
Density minimum and CV maximum in ST2 water
(impossible in the gaussian landascape Phys.
Rev. Lett. 91, 155701, 2003)
inflection CV max
inflection in energy
9
Sconf Silica
Non-Gaussian Behavior in SiO2
Eis e S conf for silica Esempio di forte
Saika-Voivod et al Nature 412, 514-517, 2001
Non gaussian silica
10
Maximum Valency
Maximum Valency Model (Speedy-Debenedetti)
SW if of bonded particles lt Nmax HS if of
bonded particles gt Nmax
V(r
)
r
A minimal model for network forming liquids
The IS configurations coincide with the bonding
pattern !!!
Zaccarelli et al PRL (2005) Moreno et al Cond Mat
(2004)
11
Square Well 3 width
Generic Phase Diagram for Square Well (3)
12
Square Well 3 width
Generic Phase Diagram for NMAX Square Well (3)
13
Ground State Energy Known !(Liquid free energy
known everywhere!)
(Wertheim)
It is possible to equilibrate at low T !
Energy per Particle
14
Cv
Specific Heat (Cv) Maxima
15
Viscosity and Diffusivity Arrhenius
16
Stoke-Einstein Relation
17
Dynamics Bond Lifetime
18
Pair-wise model (geometric correlation between
bonds) (PMW, I. Nezbeda)
19
Connection between Dynamics and Structure !
20
An IS is a bonding pattern !!!!!
F(T)-T Sconf(lteISgt, T) fbasin(lteISgt,T)
with
Basin depth and shape
fbasin(eIS,T) eISfvib(eIS,T)
and
Number of explored basins
Sconf(T)kBlnW(lteISgt)
21
Basin Free energy
It is possible to calculate exactly the basin
free energy !
Frenkel-Ladd
22
Entropies
Svib increases linearly with the of bonds
Sconf follows a x ln(x) law
Sconf does NOT extrapolate to zero
23
Self consistence
Self-consistent calculation ---gt S(T)
24
Part 1 - Take home message(s)

• 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 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

25
Dynamic Eterogeneities
Part II -Dynamic Heterogeneities J. Chem. Phys.
B 108,19663,2004
(attempting to avoid any a priori definition)
Look at differences between different realizations
SPC/E Water 100 realizations nn distance 0.28
nm Follow dyanmics for MSD (2 x 0.28)2 nm2
26
s2MSD - vs - MSD
27
peis
Connections with the landscape ?
28
Connessione eis - D
Memory of the landscape location..
29
Which D(eIS,T) ?
155 BMLJ
30
Which D(eIS,T) ?
31
Which D(eIS,T) ?
32

• 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 .

33
Frenkel-Ladd (Einstein Crystal)