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Relaxation and Transport in Glass-Forming Liquids

• Motivation (longish)• Democratic motion• Conclusions

G. Appignanesi, J.A. Rodríguez Fries, R.A. Montani

Laboratorio de Fisicoquímica, Bahía Blanca

W. Kob

Laboratoire des Colloïdes, Verres et Nanomatériaux Université Montpellier 2

http://www.lcvn.univ-montp2.fr/kob

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The problem of the glass-transition• Most liquids crystallize if they are cooled below their melting temperature Tm

• But some liquids stay in a (metastable) liquid phase even below Tm

one can study their properties in the supercooled state

Use the viscosity to define a glass transition temp. Tg: (Tg) = 1013 Poise• make a reduced Arrhenius plot log() vs Tg/T

Strong increase of with decreasing T

Questions:

• What is the mechanism for the slowing down?

•What is the difference between strong and fragile systems?

•What is the motion of the particles in this glassy regime?

•...

Angell-plot (Uhlmann)

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Model and details of the simulationAvoid crystallization binary mixture of Lennard-Jones particles; particles of type A (80%) and of type B (20%)

parameters: AA= 1.0 AB= 1.5 BB= 0.5 AA= 1.0 AB= 0.8 BB= 0.85

Simulation:• Integration of Newton’s equations of motion in NVE ensemble (velocity

Verlet algorithm)• 150 – 8000 particles• in the following: use reduced units• length in AA

• energy in AA

• time in (m AA2/48 AA)1/2

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Dynamics: The mean squared displacement• Mean squared displacement is defined as

r2(t) = |rk(t) - rk(0)|2

• short times: ballistic regime r2(t) t2 • long times: diffusive regime r2(t) t• intermediate times at low T: cage effect

• with decreasing T the dynamics slows down quickly since the length of the plateau increases

What is the nature of the motion of the particles when they start to become diffusive (=-process)?

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Time dependent correlation functions• At every time there are equilibrium fluctuations in the density distribution; how do these fluctuations relax?

• consider the incoherent intermediate scattering function Fs(q,t)

Fs(q,t) = N-1 (-q,t) (q,0) with (q,t) = exp(i qrk(t))

• high T: after the microscopic regime the correlation decays exponentially

• low T: existence of a plateau at intermediate time (reason: cage effect); at long times the correlatoris not an exponential (can be fitted well by Kohlrausch-law)Fs(q,t) = A exp( - (t/ ))

Why is the relaxation of the particles in the -process non-exponential? Possible explanation: Dynamical heterogeneities, i.e. there are “fast” and “slow” regions in the sample and thus the average relaxation is no longer an exponential

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Dynamical heterogeneities: I•One possibility to characterize the dynamical heterogeneity (DH) of a system is the non-gaussian parameter

2(t) = 3r4(t) / 5(r2(t))2 –1

with the mean particle displacement r(t) ( = self part of the van Hove

correlation function Gs(r,t) = 1/N i (r-|ri(t) – ri(0)|) )

N.B.: For a gaussian process we have 2(t) = 0.

2(t) is large in the caging regime

•maximum of 2(t) increases with decreasing T evidence for the presence of DH at low T

•define t* as the time at which the maximum occurs

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Dynamical heterogeneities: II

•Define the “mobile particles” as the 5% particles that have the

largest displacement at the time t*

•Visual inspection shows that these particles are not distributed

uniformly in the simulation box, but instead form clusters

•Size of clusters increases with decreasing T

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Dynamical heterogeneities: III

•The mobile particles do not only form clusters, but their motion is also very cooperative:

Similar result from simulations of polymers and experiments of colloids (Weeks et al.; Kegel et al.)

ARE THESE STRINGS THE -PROCESS?

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Existence of meta-basins•Define the “distance matrix” (Ohmine 1995)

2(t’,t’’) = 1/N i |ri(t’) – ri(t’’)|2

T=0.5

•We see meta-basins (MB)

•With decreasing T the residence time within one MB increases

•NB: Need to use small systems (150 particles) in order to avoid that the MB are washed out

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Dynamics: I•Look at the averaged squared displacement

in a time (ASD) of the particles in the

same time window: 2(t,) := 2(t- /2, t+ /2)

= 1/N i |ri(t+/2) – ri(t-/2)|2

ASD changes strongly when system leaves MB

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Dynamics: II

•Look at Gs(r,t’,t’+ ) = 1/N i (r-|ri(t’) – ri(t’+ )|)

for times t’ that are inside a meta-basin

•Gs(r,t’,t’+ ) is very similar to the mean curve ( = Gs(r, ) , the self part of the van Hove function)

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Dynamics: III

•Look at Gs(r,t’,t’+ ) = 1/N i (r-|ri(t’) – ri(t’+ )|) for

times t’ that are at the end of a meta-basin, i.e. the

system is crossing over to a new meta-basin

•Gs(r,t’,t’+ ) is shifted to the right of the mean curve ( = Gs(r, ) )

•NB: This is not the signature of strings!

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Democracy

•Define “mobile particles” as particles that move, within time ,

more than 0.3

•What is the fraction m(t,) of such

mobile particles?

•Fraction of mobile particles in the MB-MB transition is quite substantial ( 20-30 %) ! (cf. strings: 5%)

•Strong correlation between m(t,) and 2(t,)

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Nature of the motion within a MB

•Few particles move collectively; signature of strings (?)

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Nature of the democratic motion in MB-MB transition

•Many particles move collectively; no signature of strings

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Summary•For this system the -relaxation process does not correspond to the fast dynamics of a few particles (string-like motion with amplitude O() ) but to a cooperative movement of 20-50 particles that form a compact cluster

candidate for the cooperatively rearranging regions of Adam and Gibbs

•Slowing down of the system is due to increasing cooperativity of the relaxing entities (clusters)

•Qualitatively similar results for a small system embedded in a larger system

Reference:

PRL 96, 057801 (2006) (= cond-mat/0506577)