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Relaxation dynamics of glassy liquids: Meta-basins and democratic motion. G. Appignanesi, J.A. Rodr í guez Fries, R.A. Montani Laboratorio de Fisicoqu í mica, Bah í a Blanca W. Kob. Laboratoire des Collo ïdes, Verres et Nanomatériaux Universit é Montpellier 2

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Relaxation dynamics of glassy liquids:

Meta-basins and democratic motion

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

  • motivation (long)
  • strings
  • democratic motion
  • conclusions

Model and details of the simulation

Avoid 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

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)?

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(qrk(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 correlator
  • is 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?

Dynamical heterogeneities: I

  • 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
  • One possibility to characterize the dynamical homogeneity of a system is the non-gaussian parameter
  • 2(t) = 3r4(t) / 5(r2(t))2 –1
  • with the mean particle displacement r(t) ( = self part of the van Hove correlation function Gs(r,t) )

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

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.)


Existence of meta-basins


  • 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
  • define the “distance matrix” (Ohmine 1995)
  • 2(t’,t’’) = 1/N i|ri(t’) – ri(t’’)|2

ASD changes strongly when system leaves MB

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

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

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!


  • 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%)

Nature of the motion within a MB

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

Nature of the democratic motion in MB-MB transition

  • many particles move collectively; no signature of strings

K. Binder and W. KobGlassy Materials and

Disordered Solids: An Introduction to their

Statistical Mechanics (World Scientific,

Singapore, 2005)


  • 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