Duesseldorf Feb 2004Duesseldorf February 5 2004
Potential Energy Landscape Description of Supercooled Liquids and Glasses
Luca Angelani, Stefano Mossa, Ivan Saika-Voivod, Emilia La Nave, Piero Tartaglia
Duesseldorf Feb 2004
• Why do we care ? Thermodynamics and Dynamics
• Review of thermodynamic formalism in the PEL approach
• Comparison with numerical simulations• Development of a PEL EOS• Generalization to non-equilibrium (one or
more effective parameters ?)
Outline
Duesseldorf Feb 2004Why do we care ? Dynamics
P.G. Debenedetti, and F.H. Stillinger, Nature 410, 259 (2001).
A slowing down that cover more than 15 order of magnitudes
1
Duesseldorf Feb 2004Why do we care: Thermodynamics
Is the excess entropy vanishing at a finite T ?
1
Duesseldorf Feb 2004
van Megen and S.M. Underwood
Phys. Rev. Lett. 70, 2766 (1993)
(t)
(t)
log(t)
Separation of time scales
Supercooled Liquid
Glass
glassliquid
Duesseldorf Feb 2004
Duesseldorf Feb 2004
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
Duesseldorf Feb 2004
fbasin i(T)= -kBT ln[Zi(T)]
all basins i
fbasin(eIS,T)= eIS+ kBTln [hj(eIS)/kBT]
+
fanharmonic(T)
normal modes j
Z(T)= Zi(T)
Duesseldorf Feb 2004
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
Duesseldorf Feb 2004
rN
Distribution of local minima (eIS)
Vibrations (evib)
+
eIS
e vib
Configuration Space
Duesseldorf Feb 2004
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).
Duesseldorf Feb 2004BKS Silica
Duesseldorf Feb 2004Basin Free Energy
SPC/E LW-OTP
ln[i(eIS)]=a+b eIS
kBTln [hj(eIS)/kBT]
…if b=0 …..
Duesseldorf Feb 2004The 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
Duesseldorf Feb 2004
eIS=eiIS
E0=<eNIS>=Ne1
IS
2= 2N=N 2
1
Gaussian Distribution ?
Duesseldorf Feb 2004
T-dependence of <eIS>
SPC/E LW-OTP
T-1 dependence observed in the studied T-rangeSupport for the Gaussian Approximation
Duesseldorf Feb 2004BMLJ Configurational Entropy
Duesseldorf Feb 2004The V-dependence of , 2, E0
eIS)deIS=eN -----------------deISe-(e
IS -E
0)2/22
22
Duesseldorf Feb 2004Landscape 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]
Duesseldorf Feb 2004Developing an EOS based on PEL properties
SPC/Ewater
Duesseldorf Feb 2004SPC/E P(T,V)=Pconst(V)+PT(V) T + P1/T(V)/T
FS, E. La Nave, and P. Tartaglia, PRL. 91, 155701 (2003)
Duesseldorf Feb 2004Correlating Thermodynamics and Dynamics: Adam-Gibbs Relation
BKS Silica
Ivan Saika-Voivod et al, Nature 412, 514 (2001).
exp(A/TSconf)
Duesseldorf Feb 2004Conclusion IThe V-dependence of the statistical properties of the PEL can be quantified for models of liquids
Accurate EOS can be constructed
Peculiar features of the liquid state (TMD line) can be connected to features of the PEL statistical properties
Relation between Dynamics and Thermodynamics can be explored
Duesseldorf Feb 2004Simple (numerical) Aging Experiment
Duesseldorf Feb 2004
Aging in the PEL-IS framework
Starting Configuration (Ti)
Short after the T-change
(Ti->Tf)
Long timeT
i
Tf
Tf
Duesseldorf Feb 2004Evolution of eIS in aging (BMLJ), following a T-jump
W. Kob et al Europhys. Letters 49, 590 (2000).
One can hardly do better than equilibrium !!
Duesseldorf Feb 2004
F(T, Tf )=-Tf Sconf (eIS)+fbasin(eIS,T)
Relation first derived by S. Franz and M. A. Virasoro, J. Phys. A 33 (2000) 891, in the context of disordered spin systems
Which T in aging ?
Duesseldorf Feb 2004
Duesseldorf Feb 2004Fluctuation Dissipation Relation
(Cugliandolo, Kurcian, Peliti, ….)
FS and Piero TartagliaExtension of the Fluctuation-Dissipation theorem to the physical aging of a model glass-forming liquidPhys. Rev. Lett. 86, 107 (2001).
Duesseldorf Feb 2004
Support from the Soft Sphere Model
F(V, T, Tf)=-TfSconf (eIS)+fbasin(eIS,T)
Duesseldorf Feb 2004From Equilibrium to OOE….
If we know which equilibrium basin the system is exploring…
eIS acts as effective T !
eIS, V, T
.. We can correlate the state of the aging system with an equilibrium state and predict the pressure
(OOE-EOS)
Duesseldorf Feb 2004Numerical Tests
Liquid-to-Liquid
T-jump at constant V
P-jump at constant T
S. Mossa et al. EUR PHYS J B 30 351 (2002)
Duesseldorf Feb 2004Numerical Tests
Heating a glass at constant P
TP
time
Duesseldorf Feb 2004Numerical Tests
Compressing at constant TPf
T
time
Pi
Duesseldorf Feb 2004Breaking of the out-of-equilibrium theory….Kovacs (cross-over) effect
S. Mossa and FS, PRL (2004)
Duesseldorf Feb 2004Conclusion II
The hypothesis that the system samples in aging the same basins explored in equilibrium allows us to develop an EOS for OOE-liquids (with one additional parameter)
Small aging times, small perturbations are consistent with such hypothesis. Work is ongoing to evaluate the limit of validity.
The additional parameter can be chosen as effective T, P or depth of the explored basin eIS
Duesseldorf Feb 2004Perspectives
An improved description of the statistical properties of the potential energy surface.
A deeper understanding of the concept 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
Duesseldorf Feb 2004Acknowledgements
We acknowledge important discussions, comments, collaborations, criticisms from…
A. Angell, P. Debenedetti, T. Keyes, G. Ruocco , S. Sastry, R. Speedy
… and their collaborators
Duesseldorf Feb 2004Entering the supercooled region
Duesseldorf Feb 2004Same basins in Equilibrium and Aging ?
Duesseldorf Feb 2004T-dependence of Sconf (SPC/E)(SPC/E)