5th Liquid Matter Conference

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5th Liquid Matter Conference
Konstanz, 14-18 September 2002
Potential Energy Landscape Equation of State
Emilia La Nave, Francesco Sciortino, Piero
Tartaglia (Roma)
Stefano Mossa (Boston/Paris)
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Potential Energy Landscape
Statistical description of the number, depth and
shape
of the PEL basins
e
IS
P
IS
w
3
Z(T) S Zi(T)
all basins i
fbasin i(T) -kBT lnZi(T)
fbasin(eIS,T) eIS kBTS ln
hwj(eIS)/kBT
fanharmonic(T)
normal modes j
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Thermodynamics in the IS formalism
Stillinger-Weber
F(T)-kBT lnW(lteISgt)fbasin(lteISgt,T)
with
Basin depth and shape
fbasin(eIS,T) eISfvib(eIS,T)
and
Number of explored basins
Sconf(T)kBlnW(lteISgt)
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Real Space
Distribution of local minima (eIS)

Vibrations (evib)
rN
evib
eIS
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From simulations..
F(T)-kBT lnW(lteISgt)fbasin(lteISgt,T)

• lteISgt(T) (steepest descent minimization)
• fbasin(eIS,T) (harmonic and anharmonic
  contributions)
• F(T) (thermodynamic integration from ideal gas)

In this talk..
Data for two rigid-molecule models LW-OTP,
SPC/E-H20
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Basin Free Energy
LW-OTP
SPC/E

• S lnwi(eIS)ab eIS

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The Random Energy Model for eIS
Hypothesis
e-(eIS -E0)2/2s 2
W(eIS)deISeaN -----------------deIS
2ps2
S lnwi(eIS)ab eIS
Predictions
lteIS(T)gtE0-bs 2 - s 2/kT
Sconf(T)aN- (lteIS (T)gt-E0)2/2s 2
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Gaussian Distribution ?
eISSeiIS
E0lteNISgtNe1IS s2 s2NN s21
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T-dependence of lteISgt (SPC/E)
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T-dependence of lteISgt (LW-OTP)
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T-dependence of Sconf (SPC/E)
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The V-dependence of a, s2, E0
e-(eIS -E0)2/2s 2
W(eIS)deISeaN -----------------deIS
2ps2
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Landscape Equation of State
P-?F/?VT
F(V,T)-TSconf(T,V)lteIS(T,V)gtfvib(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-bE0b2s2/2 P1/T(V) d/dV s2/2R
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Developing an EOS based on PES properties
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SPC/E Water P(T,V)Pconst(V)PT(V) T
P1/T(V)/T
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Conclusion I
The V-dependence of the statistical properties of
the PEL has been quantified for two models of
molecular liquids Accurate EOS can be constructed
from these information Interesting features of
the liquid state (TMD line) can be correlated to
features of the PEL statistical properties
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Aging in the PEL-IS framework
Ti
Tf
Tf
Starting Configuration (Ti)
Short after the T-change (Ti-gtTf)
Long time
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Reconstructing P(T,V)
P-?F/?V
F(V,T)-TSconf(T,V)lteIS(T,V)gtfvib(T,V)
P(T,V) Pconf(T,V)
Pvib(T,V)
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From Equilibrium to OOE.
P(T,V) Pconf(T,V) Pvib(T,V)
If we know which equilibrium basin the system is
exploring

• eIS(V,Tf),V,T
• log(w)
• Pvib

• eIS(V,Tf).V
• Pconf

eIS acts as a fictive T !
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Numerical TestsHeating a glass at constant P
T
P
time
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Numerical TestsCompressing at constant T
Pf
Pi
T
time
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Liquid-to-Liquid
T-jump at constant V
P-jump at constant T
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Conclusion II

• ?The hypothesis that the system samples in aging
  the same basins explored in equilibrium allows to
  develop an EOS for OOE-liquids depending on one
  additional parameter
• Small aging times, small perturbations are
  consistent with such hipothesis. Work is
  requested to evaluate the limit of validity.
• ? This parameter can be chosen as fictive T,
  fictive P or depth of the explored basin eIS

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Perspectives
?An improved description of the statistical
properties of the potential energy surface. ?
Role of the statistical properties of the PEL in
liquid phenomena ? A deeper understanding of the
concept of Pconf and of EOS of a glass. ? An
estimate of the limit of validity of the
assumption that a glass is a frozen liquid
(number of parameters) ? Connections between PEL
properties and Dynamics
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References and Acknowledgements
We acknowledge important discussions, comments,
criticisms from P. Debenedetti, S. Sastry, R.
Speedy, A. Angell, T. Keyes, G. Ruocco and
collaborators
Francesco Sciortino and Piero Tartaglia Extension
of the Fluctuation-Dissipation theorem to the
physical aging of a model glass-forming
liquid Phys. Rev. Lett. 86 107 (2001). Emilia La
Nave, Stefano Mossa and Francesco Sciortino
Potential Energy Landscape Equation of
State Phys. Rev. Lett., 88, 225701
(2002). Stefano Mossa, Emilia La Nave, Francesco
Sciortino and Piero Tartaglia, Aging and Energy
Landscape Application to Liquids and Glasses.,
cond-mat/0205071
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Entering the supercooled region
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Same basins in Equilibrium and Aging ?