Dynamical heterogeneity and relaxation time very close to dynamical arrest L. Cipelletti LCVN, Université Montpellier 2 and CNRS Collaborators: L. Berthier (LCVN), V. Trappe (Fribourg) Students: P. Ballesta, G. Brambilla, A. Duri, D. El Masri Postdocs: S. Maccarrone, M. Pierno

Soft glassy/jammed materials E. Weeks

Dynamical heterogeneity is ubiquitous! Granular matterColloidal Hard Spheres Repulsive disks Weeks et al. Science 2000Keys et al. Nat. Phys. 2007A. Widmer-Cooper Nat. Phys. 2008

Outline Average dynamics and dynamical hetrogeneity of supercooled colloidal HS Temporal fluctuations of the dynamics in jammed/glassy materials Real-space measurements of correlations: ultra-long range spatial correlations and elasticity

Equilibrium dynamics of supercooled HS Brambilla et al., PRL 2009 Multispeckle DLS PMMA in decalin/tetralin a ~ 110 nm polidispertsity 12.2% (TEM)  (sec) f s (qa=2.75,  )

 dependence of the relaxation time Brambilla et al., PRL 2009 MCT :   ~ (  c -  ) -   c = 0.59,  = 2.6 VFT:   ~ exp[(  0 -  ) -  ]  0 = 0.614,  = 1  0 = 0.637,  = 2

Glass/jamming transition in colloidal HS  c ~ FluidGlass (out of equilibrium)  rcp ~.64  c ~ Fluid  j =  rcp ~.64  c ~ FluidGlass  rcp ~.64 00 MCT Jamming Thermodynamic

Dynamical heterogeneity  Weeks et al. Science 2000

Dynamical heterogeneity  Weeks et al. Science 2000 var[f(  )]  4 (  ) dynamical susceptibility

Dynamical susceptibility in glassy systems Supercooled liquid (Lennard-Jones) Lacevic et al., Phys. Rev. E 2002  4  N var[Q(t)]

Dynamical susceptibility in glassy systems Spatial correlation of the dynamics  Weeks et al. Science 2000

Dynamical heterogeneity: the theoreticians’ trick For colloidal HS at high  Berthier et al., J. Phys. Chem Goal: calculate 4-point dynamical susceptibility  4 ~ size of rearranged region

DH in colloidal HS Berthier et al., Science 2005 ~ (  a)  3 dynamical susceptibility  4 (  ) A useful relationship equilibrium OK at high  Berthier et al., J. Phys. Chem.2007  ~ 0.20 – 0.58  (sec)

 dependence of the max of  4 * High  : deviation from MCT Brambilla et al., PRL 2009 MCT ~ (  c -  ) -2

Outline Average dynamics and dynamical hetrogeneity of supercooled colloidal HS Temporal fluctuations of the dynamics in jammed/glassy materials Real-space measurements of correlations: ultra-long range spatial correlations and elasticity

Time Resolved Correlation (TRC) time t lag  degree of correlation c I (t,  ) = - 1 p p p Cipelletti et al. J. Phys:Condens. Matter 2003, Duri et al. Phys. Rev. E 2006

intensity correlation function g 2 (  1 Average over t degree of correlation c I (t,  ) = - 1 p p p Average dynamics g 2 (  1 ~ f(  ) 2

intensity correlation function g 2 (  1 Average over t degree of correlation c I (t,  ) = - 1 p p p fixed , vs. t fluctuations of the dynamics Average dynamics g 2 (  1 ~ f(  ) 2 var[c I (  )] ~  (  )

(Polydisperse) colloids near rcp Ballesta et al., Nature Physics 2008 PVC in DOP a ~ 5 µm polidispertsity ~33% slightly soft  close to rcp multiple scattering (DWS)  ~ 10 nm << a

Non-monotonic behavior of  *(  ) Fit: g 2 (  )-1 ~ exp[-(  /   )  ] relaxation time stretching exponent peak of dynamical susceptibility non monotonic!

Non-monotonic behavior of  *(  ) Competition between: increasing  (  as  ) increasingly restrained displacement jump size decreases (many events over  0,  as  ) Model: random events of size  Poissonian statistics Quasi-ballistic motion ( ~n p, p = 1.65)

Simulating the model volume fraction cell thickness! inverse jump size

Outline Average dynamics and dynamical hetrogeneity of supercooled colloidal HS Temporal fluctuations of the dynamics in jammed/glassy materials Real-space measurements of correlations: ultra-long range spatial correlations and elasticity

2.3 mm Measuring  by Photon Correlation Imaging diaphragm lens  object plane image plane focal plane CCD  sample  = 90° 1/q ~ 45 nm Duri et al., Phys. Rev. Lett. 2009

2.3 mm Local, instantaneous dynamics: c I ( t, ,  r) p(r) p p(r) c I (t, , r) = - 1 Note: t > r = g 2 (  )-1 [g 2 (  )-1] ~ f(  ) 2

Dynamic Activity Maps Brownian particles g 2 (  )-1~ exp[-  /  r ],  r = 40 s Colloidal gel g 2 (  )-1~ exp[-(  /  r ) 1.5 ],  r = 5000 s c I (t 0,  r /200, r) c I (t 0,  r /10, r) Movie accelerated 10x Movie accelerated 500x 2 mm

Spatial correlation of the dynamics:  ~ system size in jammed soft matter! Maccarrone et al., Soft Matter 2010

When does  infinity? G' > G'' : strain propagation in a predominantly solid-like material Magnitude of G' unimportant (G' onions /G' colloidal gel ~ 6x10 4 !!) Origin of elasticity: entropic (e.g. hard spheres)  very small enthalpic (attractive gels, squeezed particles...)  system-size

Evidence of internal stress relaxation J. Kaiser, O. Lielig,, G. Brambilla, LC, A. Bausch, submitted Dynamics of actin/fascin networks strain late stages of network formation age average strain fieldmicrocopic dynamics

Conclusions DH a general feature of glassy/jammed dynamics Supercooled hard spheres: equilibrium dynamics above the (apparent) MCT divergence  limited to 5-10a Jammed materials:  and  may decouple!  ~ system size role of the origin of elasticity

Thanks to... L. Berthier (LCVN), V. Trappe (Fribourg) Hard spheres G. Brambilla, M. Pierno, D. El Masri (LCVN), A. Schofield (Edinburgh), G. Petekidis (FORTH) TRC A. Duri, P. Ballesta (LCVN), D. Sessoms, H. Bissig (Firbourg) PCI A. Duri, S. Maccarrone (LCVN), D. Sessoms (Fribourg), E. Pashkowski (Unilever) Funding CNES, ACI, ANR, PICS, Unilever

Open questions... DH and aging? Origin of the dynamics in (undriven) soft materials

Determining the volume fraction  dependence of the short-time diffusing coefficient  = 0.2  = 1/D s s q 2 Absolute uncertainty: ~4% Relative uncertainty: ~ 10 -4

PS gel: time-averaged dynamics g 2 (q,  ) - 1 ~ [f(q,  )] 2 Fast dynamics: overdamped vibrations (~ 500 nm) Krall & Weitz PRL 1998 Slow dynamics: rearrangements

PS gels: q dependence of  f and p « compressed » exponential« ballistic » motion

PS gel: temporally heterogeneous dynamics tw : average dynamics c I (t w,  fixed  temporal fluctuations

Length scale dependence of  var(c I ) Increasing q Duri & LC, EPL 76, 972 (2006) PS gel 1.1 +/- 0.1

Scaling of  *  * ~ var(n)/ ~ 1/ ~  f ~ 1/q  * ~ q

Dynamical susceptibility in glassy systems Supercooled liquid (Lennard-Jones) Lacevic et al., Phys. Rev. E 2002  4  N var[Q(t)]

Dynamical susceptibility in glassy systems  4  N var[Q(t)]  4 dynamics spatially correlated N regions

Dynamical susceptibility in glassy systems Spatial correlation of the dynamics  Weeks et al. Science 2000

The smart trick applied to colloidal HS t   (t)(t) F(t)F(t) Define

Dynamical heterogeneity: the theoreticians’ trick For colloidal HS at high  Berthier et al., J. Phys. Chem Goal: calculate 4-point dynamical susceptibility  4 ~ size of rearranged region

Evidence of a growing dynamic length scale  ~ 0.20 – 0.58 Berthier et al., Science 2005 

 dependence of the max of  4 * MCT High  : deviation from MCT Brambilla et al., PRL 2009

2.3 mm Photon Correlation Imaging (PCIm) sample annular slitlens   object plane image plane  focal plane CCD  = 6.4° q = 1  m -1. Duri et al., Phys. Rev. Lett. 2009

Spatial correlation of the dynamics

Other PCIm geometries diaphragm lens  object plane image plane focal plane CCD  sample diaphragm lens  object plane image plane focal plane CCD beam splitter  a)b)

Additional stuff on diverging 