Orleans July 2004 Potential Energy Landscape in Models for Liquids Networks in physics and biology...
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Transcript of Orleans July 2004 Potential Energy Landscape in Models for Liquids Networks in physics and biology...
Orleans July 2004
Potential Energy Landscape
in
Models for Liquids
Networks in physics and biology
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
Strong and Fragile liquids Dynamics
P.G. Debenedetti, and F.H. Stillinger, Nature 410, 259 (2001).
A slowing down that cover more than 15 order of magnitudes
1
A Decrease in Configurational Entropy: Thermodynamics
Is the excess entropy vanishing at a finite T ?
1
van Megen and S.M. Underwood
Phys. Rev. Lett. 70, 2766 (1993)
(t)
(t)
log(t)
The basic idea: Separation of time
scales
Supercooled Liquid
Glass
glassliquid
IS
Pe
IS
Statistical description of the number, depth and shapeof the PEL basins
Potential Energy Landscape, a 3N dimensional surface
The PEL does not depend on TThe exploration of the PEL depends on T
fbasin i(T)= -kBT ln[Zi(T)]
all basins i
fbasin(eIS,T)= eIS+ kBTln [hj(eIS)/kBT]
+
fanharmonic(eIS, T)
normal modes j
Z(T)= Zi(T)
Thermodynamics in the IS formalism Stillinger-Weber
F(T)=-kBT ln[(<eIS>)]+fbasin(<eIS>,T)
with
fbasin(eIS,T)= eIS+fvib(eIS,T)
and
Sconf(T)=kBln[(<eIS>)]
Basin depth and shape
Number of explored basins
F(T)=-kBT ln[(<eIS>)]+fbasin(<eIS>,T)
From simulations…..
<eIS>(T) (steepest descent minimization)
fbasin(eIS,T) (harmonic and anharmonic
contributions)
F(T) (thermodynamic integration from ideal gas)
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:
Predictions:
eIS)deIS=eN -----------------deIS
e-(eIS
-E0)2/22
22
ln[i(eIS)]=a+b eIS
<eIS(T)>=E0-b2 - 2/kT
Sconf(T)=N- (<eIS (T)>-E0)2/22
T-dependence of <eIS>
SPC/E LW-OTP
T-1 dependence observed in the studied T-rangeSupport for the Gaussian Approximation
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-b2]PT(V) =R d/dV [-a-bE0+b22/2]P1/T(V) = d/dV [2/2R]
Maximum Valency Model (Speedy-Debenedetti)
A minimal model for network forming liquids
SW if # of bonded particles <= NmaxHS if # of bonded particles > Nmax
V(r)
r
The IS configurations coincide with the bonding pattern !!!
Suggestions for further studies…..Fragile LiquidsGaussian 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 !
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