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Relaxation dynamics of glassy liquids:Meta-basins and democratic motion

• motivation (long)• strings• 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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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 (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) = |r(t) - r(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(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? Motion of system in rugged landscape? Dynamical heterogeneities?

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Dynamical heterogeneities: I•One possibility to characterize the dynamical homogeneity 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) )

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?

ARE THESE DH THE REASON FOR THE STRETCHTING IN 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 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 interval: 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 (ri(t’) – ri(t’+ ))2 for

times t’ that are inside a meta-basin

•Gs(r,t’,t’+ ) is shifted to the left of the mean curve ( = Gs(r, ) ) and is more peaked

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Dynamics: III

• look at Gs(r,t’,t’+ ) = 1/N i (ri(t’) – ri(t’+ ))2 for times t’

that are at the end of a 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 of such

mobile particles?

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

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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

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

Reference:

cond-mat/0506577

K. Binder and W. Kob Glassy Materials and

Disordered Solids: An Introduction to their

Statistical Mechanics (World Scientific,

Singapore, 2005)