Geometric approach to the dynamic glass transition

4 Pages·0.14 MB·English

by  Tomas S. Grigera

Geometric approach to the dynamic glass transition
Toma´s S. Grigera‡, Andrea Cavagna⋆, Irene Giardina†, and Giorgio Parisi‡
‡ Dipartimento di Fisica, Unit`a INFM and Sezione INFN Universit`a di Roma “La Sapienza”, 00185 Roma, Italy
⋆ Department of Physics and Astronomy, The University, Manchester, M13 9PL, United Kingdom
† Service de Physique Th`eorique, CEA Saclay, 91191 Gif-sur-Yvette, France
(October 26, 2001)
2
0 We numerically study the potential energy landscape of a fragile glassy system and find that
0 the dynamic crossover corresponding to the glass transition is actually the effect of an underlying
2 geometrictransitioncausedbythevanishingoftheinstabilityindexofsaddlepointsofthepotential

n energy. Furthermore, we show that the potential energy barriers connecting local glassy minima
a increase with decreasing energy of the minima, and we relate this behaviour to the fragility of
J the system. Finally, we analyze the real space structure of activated processes by studying the

1 distribution of particle displacements for local minima connected by simple saddles.
2

Despite a large number of investigations, there is still a change in the dominant mechanism of diffusion. If en-
]
n much to understand about the dynamic glass transition ergy barriers are large, this change in the mechanism of
n insupercooledliquids. Thebasicproblemisthat,strictly diffusion causes the fast increase of the relaxation time.
- speaking, there is no dynamic transition at all. In sys- This study is part of a more general program aimed
s
i temsknownasfragileliquids[1],experimentfindsasharp at explaining glassy dynamics in terms of properties of
d
rise of the viscosity in a very narrow interval of temper- the potential energy landscape. Our method generalizes
.
t ature upon cooling. The shear relaxation time increases the ideas of Goldstein [2], and Stillinger-Weber [3], by
a
m byseveralordersofmagnitudeinafewdegrees,anditbe- extending to unstable stationary points the analysis for-
comes impossible to perform an equilibrium experiment. merly restricted to minima of the potential energy. The
-
d Nevertheless, sharp as this behaviour may be, it is not a first steps in this direction have been done in [4], build-
n genuinedynamicsingularity. Attheotherextremeofthe ing on the ideas of [5,6], and more concrete results have
o experimental spectrum we find strong liquids [1], which beenrecentlyobtainedin[7,8]. Furtherinspirationcame
c experience a gentle increase of the relaxationtime, often from the approach of Keyes and coworkers [9], which
[
according to the Arrhenius law. Even in such systems related diffusion to the stability properties of instanta-

2 though, when the viscosity becomes too large, equilib- neous configurations. In the present work we firmly es-
v rium can no longer be achieved in experimental times. tablish the connection between topological properties of
8 The glass transition temperature T is conventionally the landscapeandfragileglassydynamics. Furthermore,
g
9 defined as that where the value of the viscosity is 1013 westudy the roleofpotentialenergybarriersandwe an-
1
poise. Below T equilibrium experiments become really alyze the real space structure of activated processes.
7 g
hardto performandasample canbe consideredto be in Weconsiderasoft-spherebinarymixture[10],afragile
0
1 itsglassphase. However,Tg isjustaconventionalexper- model glass-former. In addition to capturing the essen-
0 imentaltemperature,definedoutoftheneedtomarkthe tial features of fragile glasses [10–13], this model can be
/ onset of glassy dynamics. The attempt to give a theo- thermalized below T with the efficient MC algorithmof
t g
a reticaldescriptionofsuchanill-defined“transition”may [12]. Furthermore, previous investigations of the saddle
m
therefore seem pointless. points have focused on Lennard-Jones systems, so it is
- On the one hand, this conclusion is correct for the useful to look at a broader class of models. Most of our
d
strongest liquids: here nothing peculiar happens close data are obtained for N = 70 particles, but we tested
n
to T , and the glass transition fully displays its purely our key results for N =140 as well. We impose periodic
o g
c conventionalnature. Ontheotherhand,themostfragile boundary conditions in d = 3 dimensions. Particles are
: systems resist such an objection, simply by virtue of the of unit mass and belong to one of two species α = 1,2,
v
extremely steep increase of relaxationtime in a small in- present in equal amounts and interacting via a potential
i
X tervaloftemperature aroundT . This factsuggeststhat
g
r somekindofnewphysicalmechanismisindeedresponsi- N N σ +σ 12
a = V (r r )= α(i) α(j) . (1)
blefortheonsetoftheglassyphaseinfragilesupercooled V ij | i− j| (cid:20) r r (cid:21)
liquids. We share this view, and the aim ofthis Letter is Xi<j Xi<j | i− j|
to shed some light on the nature of this mechanism.
The radii σ are fixed by σ /σ = 1.2 and setting the
The key idea is that the sharp dynamic crossover ob- α 2 1
effectivediameterto unity,thatis(2σ )3+2(σ +σ )3+
served in fragile liquids is a consequence of an underly- 1 1 2
(2σ )3 =4l3, where l is the unit of length. The density
ing topological transition, controlled by energy, rather 2 0 0
is ρ = N/V =1 in units of l−3, and we set Boltzmann’s
than temperature. More precisely, the existence of an 0
constant k = 1. We obtain equilibrium configurations
energylevelwhere the instability index ofthe stationary B
at several temperatures by the swap Monte Carlo algo-
points of the potential energy vanishes is responsible for
rithmof[12]. Along-rangecut-offatr =√3isimposed.
c
1
However, to find the stationary points we need a poten- for u < u < u . Therefore, a topological transition
0 th
tial with a continuous second derivative. Thus, instead takes place at u , where the stability properties of the
th
ofsimplyshiftingthepairpotentialbyaconstantC (so landscape change.
ij
that V (r r ) = 0), we use a smooth cut-off, setting A system confined to the minima-dominated regionof
ij c
≥
V (r) = B (a r)3 for r < r < a and V (r) = 0 for the landscape, u < u , must resort to barrier hopping
ij ij c ij th
−
r a, fixing a, B and C by imposing continuity. to diffuse in phase space. It is therefore essential to find
ij ij
≥
the temperature T below which this confinement takes
th
0.04 place. Tothisendwemustrealizethatasystemtrapped
ensity0.03 NIMA DDLES ieanqvuaiablsriatnotgilotehneaplobtcaeornnettierainlbeuwrtgeiylolnohfpatsrhoaepopbrootttioetonnmtailaoltofentkheergTwy.edlTle,hnpeslriuteys-
d0.02 MI A B
x S fore, it is the bare potential energy of the system which
e u u
nd0.01 0 th we must compare with the threshold energy [8]. We
I can write the bare energy as u (T)=u (T) 3/2k T.
b eq B
0.00 When the bare energy u (T) drops below the−threshold
b
u ,thesystemiseffectivelyconfinedtotheminimadom-
0.45 th
0.10
inatedregionofthe landscape. For N =70 this happens
atT =0.242 0.012(Fig.1,bottom). Notethat,unlike
e0.40 th ±
ur 0.05 previous investigations [7,8], for N = 70 we thermalize
at the system below the threshold, giving an accurate de-
r0.35
e
p terminationof T [15]. Hence, the dynamic effect of the
m th
Te0.30 0.00 2.0 2.5 topologicaltransitionatuthmustbeaqualitativechange
in the mechanism of diffusion at T .
th
0.25 Thefactthatbarriercrossingbecomesthemainmech-
Crossover temperature T
th anism of diffusion below T does not necessarily imply
th
0.20 a slowing down of the dynamics: large energy barriers
1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00
at u are also needed, i.e. ∆U(u ) substantially larger
Energy density th th
than k T . The value of ∆U(u ) can be estimated
B th th
as the average difference in energy between simple sad-
FIG. 1. Top: Average instability index density k vs. po- dles (K = 1) and threshold minima (K = 0). This
tential energy density u of the stationary points. Bottom: difference can be extracted from the slope of k(u), as
Temperature vs. equilibrium bare potential energy density, ∆U(u ) 1/[3k′(u )] [5,4]. Note that the slope of
th th
ub =ueq−3/2kBT. Inset: k(u)onthewholerangesampled. k(u) for N≈= 70 and N = 140 is the same. This is con-
Open symbols: N =140, filled symbols: N =70. sistent with the fact that activated processes are local
in space (as we will discuss later) and therefore barri-
ers do not depend on the size of the system. We find
We sample the stationary points of the potential en-
∆U(u ) 2.2 10k T . This is an important result:
th B th
ergy by quenching the equilibrium MC configurations ≈ ≈
at the temperature T where activation becomes dom-
th
onto saddle points. This is done by numerically solv-
inant, potential energy barriers are already very large
ing the 3N nonlinear equations ∂ /∂r = 0 by means
i compared to the available thermal energy. Activation is
V
of a backtracking Newton method with finite-difference
therefore highly inefficient at the temperature where for
approximationto the Jacobian[14]. Oncea saddle point
the first time it is actually needed. This, we believe, is
isfound,wemeasureitspotentialenergyU anditsinsta-
the most striking feature of very fragile liquids and it
bilityindexK,thatisthenumberofnegativeeigenvalues
confirms the conjecture of [4] that the fragility of a sys-
of the Hessian matrix at the saddle.
tem is higher the larger the potential energy barriers at
In Fig. 1 (top) we plot the average index density
T . We thus predict that a sharp slowing down of the
th
k = K/3N as a function of the potential energy den-
dynamics must occur at T . Note that the change in
th
sity u = U/N. As in [8], we find a well defined func-
themechanismofdiffusionatT would notbeaccompa-
th
tion k(u) which vanishes at a threshold value of the en-
nied by a dynamic slowing down if barriers at T were
th
ergy, u . For N = 70, a comparison between the lin-
th small. In this case there would rather be a fragile-to-
ear fit of the data and the last point of the curve gives
strong crossoverat T , as discussed in [4].
th
u =1.75 0.01. For N =140, despite worse statistics,
th Totestourpredictionabouttheslowingdownwemust
±
wefindu =1.73 0.01,consistentlywiththe resultfor
th findthetemperaturemarkingtheonsetofglassiness. For
±
N =70. Notethatthethresholdenergyisnottheground
soft-spheres, this is generally accepted to be T 0.226
c
state of the system, and in fact we found minima down ≈
[11]. This value is affected by the same arbitrariness as
to u = 1.68 (N = 70). The threshold energy marks
0 the experimental T , since an arbitrary time scale (set
g
the border between unstable saddle points, dominating
by the simulation times) is involved [11]. However, as
the landscape above u , and stable minima, dominant
th stressed in the introduction, the slowing down of fragile
2
glasses is so sharp that it makes sense to define a T , as andT : Thefirstdependsonthetimescaleoftheexper-
c th
long as one keeps the above proviso in mind. iment and it can be sensibly defined only if the dynamic
crossoveris sharp. The latter marks the point where ac-
0.15
tivation starts ruling the dynamics, and it is uniquely
)
1)(r0.10 T=0.260 defined. In fact, Tth has the same nature as the critical
(s temperature of mode coupling theory [17].
G
0.05
2 Ourestimateofthebarrierscanbecriticized,sincethe
r
0.00 average distance in energy between minima and simple
T=0.246 saddles neglects the requirement that they must be con-
)0.15
(r nectedtoeachother. Totestourestimateweperformfor
(1)s0.10 N = 70 a direct sampling of the potential energy barri-
G
2 0.05 ers. Startingfromasimple saddlewefollowthe gradient
r
in the two opposite directions along the unstable eigen-
0.00
vector, obtaining two connected minima.
0.25
T=0.239 4
)0.20
r
( u
(1)s0.15 th
G
2 0.10 3
r
0.05 ht
g ∆U
ei slope
0.00 h
0.0 0.5 1.0 1.5 2.0 r 2
e
r ri
r
a
B
FIG.2. VanHoveselfcorrelationfunctionsforparticletype 1
1, at times t=88 (full line), 177 (dotted), 265 (dashed) and
353 (dash-dotted). N =70. k T
B c
0
1.70 1.75 1.80 1.85
Given that we use a non-standard cut-off for the po-
Energy density
tential, we perform an independent determination of T
c
for N = 70. To this end we compute the van Hove self-
correlationfunctionGα(r,t)fromconfigurationssampled FIG.3. Average potential energy barriers as a function of
s
in a molecular dynamics (MD) run which uses equilib- thepotentialenergydensityoftheadjacentminimum. Points
rium MC configurations as starting points. Of course, are an average over 1652 barriers. N =70.
theMDsimulationfallsoutofequilibriumathighertem-
peratures than the MC swapdynamics. Gα is defined as
s
InFig.3weplottheaveragebarriersize∆U asafunc-
1 Nα tion of the energy density u of the adjacent minimum.
G(sα)(r,t)= N hδ[ri(t)−ri(0)−r]i . (2) On the same plot we report the value of uth (N = 70),
α Xi=1 and the estimate of ∆U(u ) obtained from the slope of
th
Theprobabilitythataparticleoftypeαhasmovedadis- k(u): this estimate agrees with the value that can be
tance r in a time t is proportional to r2G(α)(r,t), which read off from the plot. We conclude that the function
s
k(u) provides the threshold energy and the potential en-
is plotted in Fig. 2 for several times and temperatures.
ergy barriers at the threshold.
A reliable dynamic diagnostic for T is to look at the
c
Asecondimportantpieceofinformationiscontainedin
evolution of the first peak of r2G(α)(r,t) [11]. In the
s Fig. 3: the typical barriers grow with decreasing energy.
liquid phase, the peak moves to the right and rapidly
The consequence is that below T the dynamic slowing
becomes Gaussian (top panel of Fig. 2). On the other th
downisenhancednotonlybythedecreaseofthethermal
hand, in the glassy phase it takes a huge time to reach
energy available for activation, but also by the increase
the hydrodynamic limit, and the simulations show an
ofthetypicalbarriers. Thus,belowT ,weexpectsuper-
unmoving peak whose area very slowly decreases as a th
Arrhenius behaviour of the relaxation time [18].
secondarypeakgrows(bottompanel). Themiddle panel
A brief comment on barrier crossing is in order here.
shows an intermediate situation. On this basis, we esti-
Thermal activation must be introduced within a canoni-
mateT 0.24,notfarfromthestandardT . Thisvalue
c ≈ c caldescription,sinceinthemicro-canonicalensemblethe
is consistent with the topologicaltransition temperature
total energy is conserved. In other words, an activated
T 0.242wefoundabove,aresultwhichstronglysup-
th ≈ transition is performed by a sub-system, with the rest
portsourscenario. LetusstressthedifferencebetweenT
c
3
actingasathermalbath. The sub-systemmustbe much placements between minimum and intermediate saddle
smaller than the total system, and indeed activated pro- shows no secondary peak at large distances, consistently
cesses involve a finite number of particles. For this rea- withthe factthatonthe saddlethe exchangingparticles
son, potential energy barriers associated with such pro- have not completed their transition yet.
cesses are finite in the thermodynamic limit, implying In this Letter we argued that glassy slowing down in
that simple saddles and minima have the same potential fragile liquids is caused by the presence of a topological
energy density for N . Yet, the equilibrium poten- transition. Potentialenergybarriersaremuchlargerthan
→ ∞
tial energy density is of order k T above minima. This the available thermal energy at the transition, and they
B
fact does not imply that barriers are easy to overcome, increase with decreasing energy. Activated processes in-
northatthermalactivationisirrelevant,butsimplythat volvesmallnumbers ofparticles, eachmoving a distance
activated processes involve a finite number of particles. of the order of the nearest-neighbor separation.
AC was supported by EPSRC-GR/L97698, TSG
7
partlybyCONICET(Argentina). WethankJ.L.Barrat,
101 J.-P.Bouchaud,B. Doliwa, G. Ehrhardt, J.P.Garrahan,
6
A. Heuer, E. Marinari, V. Martin-Mayor, M.A. Moore,
F. Ricci-Tersenghi, A. Stephenson and F. Thalmann.
y5
sit 100
n
e
d4
y
bilit3 1100--11
a
b
o
r
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(1982).
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[5] J. Kurchan and L. Laloux, J. Phys. A 29, 1929 (1996).
Displacement
[6] A. Cavagna et al., Phys. Rev. B 57 11251, (1998); A.
Cavagna et al., Phys.Rev.B 61 3960, (2000).
FIG. 4. Distribution of particle displacements. Full line: [7] L. Angelani et al., Phys. Rev.Lett. 85, 5356 (2000).
distancebetweenthetwominima. Dashed(dotted)line: dis- [8] K. Broderix et al., Phys. Rev.Lett. 85, 5360 (2000).
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Barrat et al., Chem. Phys. 149, 197 (1990).
[11] J.N.Rouxetal.,J.Phys.-Condens.Mat.1,7171(1989).
As we have shown, activated processes are crucial be- [12] T. S. Grigera and G. Parisi, Phys. Rev. E 63, 045102
low Tth and therefore a real-space description of barrier (2001).
crossingisveryimportant. Tothis endwecomputedthe [13] G. Parisi, Phys. Rev.Lett. 79, 3660 (1997).
distribution of the displacement, that is the distance be- [14] W.H.Pressetal,NumericalRecipes,(CambrideUniver-
tweenthepositionofaparticleinaminimumanditspo- sity Press, 1992).
sition in the crossing-connected minimum (Fig. 4) [19]. [15] For N = 140 we could not equilibrate below u , and
th
To interpret this result we need first to fix a reference wehadnotenoughpointstoperformanextrapolationof
distance: Fig. 2 shows that, even in the glassy phase, ub(T). However, it has been shown that standard static
particles can easily travel a distance r 0.5. The pri- properties (as the bare energy) do not show significant
m
marypeakinthedisplacementdistributio≈nindicatesthat finite-size effects for N > 60 in soft spheres [13] and
thelargemajorityofparticlesmoveslessthanrm,whilea Lennard-Jones systems [16]. Thus, u[bN=70] can be com-
smallsecondarypeakcanbeseenatr 1,involvingonly pared with u[N=140], giving T[N=140] ≈0.22.
≈ th th
2 particles [20]. From the radial distribution function [16] S.BuchnerandA.Heuer,Phys.Rev.E60,6507(1999).
≈
(not shown) we know that r 1 is the nearest-neighbor [17] U.Bengtzeliusetal.,J.Chem.Phys.17,5915(1984); E.
distance for type 1 particles.≈These facts thus suggest Leutheusser, Phy.Rev.A 29, 2765 (1984).
that in this system activated processes involve a small [18] The distribution of ∆U at fixed uis fairly broad, with a
number of nearest neighbor particles exchanging posi- smallfractionofbarriersmuchsmallerthankBTth.These
tions,whilemanyparticlesmoveasmallamounttomake rare,butfastbarriersmaybeconnectedtonon-activated
way for them. We also measure the largest displacement off-equilibriumdynamics(ACthanksD.KivelsonandG.
Tarjus for important discussions on this point).
for each pair of minima and find that its distribution
[19] R. DiLeonardo et al., cond-mat/0106214 (2001).
hasnosecondarypeakatshortdistances,confirmingthe
[20] W. Kob et al., Phys. Rev.Lett. 79, 2827 (1997).
aboveinterpretation. Finally,the distributionofthe dis-
4