2. 1 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

3. 2 The problem of the glass-transition Most liquids crystallize if they are cooled below their melting temperature T m But some liquids stay in a (metastable) liquid phase even below T m  one can study their properties in the supercooled state Use the viscosity  to define a glass transition temp. T g :  (T g ) = 10 13 Poise make a reduced Arrhenius plot log(  ) vs T g /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 )

4. 3 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 in NVE ensemble (velocity Verlet algorithm) 150 – 8000 particles in the following: use reduced units length in  AA energy in  AA time in (m  AA 2 /48  AA ) 1/2

5. 4 Dynamics: The mean squared displacement Mean squared displacement is defined as  r 2 (t)  =  | r k (t) - r k (0) | 2  short times: ballistic regime  r 2 (t)   t 2 long times: diffusive regime  r 2 (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)?

6. 5 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 F s (q,t) F s (q,t) = N -1   (-q,t)  (q,0)  with  (q,t) = exp(i q  r k (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) F s (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

7. 6 Dynamical heterogeneities: I One possibility to characterize the dynamical heterogeneity (DH) of a system is the non-gaussian parameter  2 (t) = 3  r 4 (t)  / 5(  r 2 (t)  ) 2 –1 with the mean particle displacement r(t) ( = self part of the van Hove correlation function G s (r,t) = 1/N  i  (r-|r i (t) – r i (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

8. 7 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

9. 8 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?

10. 9 Existence of meta-basins Define the “distance matrix” (Ohmine 1995)  2 (t’,t’’) = 1/N  i |r i (t’) – r i (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

11. 10 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 |r i (t+  /2) – r i (t-  /2)| 2 ASD changes strongly when system leaves MB

12. 11 Dynamics: II Look at G s (r,t’,t’+  ) = 1/N  i (r-|r i (t’) – r i (t’+  )|) for times t’ that are inside a meta-basin G s (r,t’,t’+  ) is very similar to the mean curve ( = G s (r,  ), the self part of the van Hove function)

13. 12 Dynamics: III Look at G s (r,t’,t’+  ) = 1/N  i (r-|r i (t’) – r i (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 G s (r,t’,t’+  ) is shifted to the right of the mean curve ( = G s (r,  ) ) NB: This is not the signature of strings!

14. 13 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,  )

15. 14 Nature of the motion within a MB Few particles move collectively; signature of strings (?)

16. 15 Nature of the democratic motion in MB-MB transition Many particles move collectively; no signature of strings

17. 16 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)