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Dynamical heterogeneity and relaxation time very close to dynamical arrest

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.

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Dynamical heterogeneity and relaxation time very close to dynamical arrest

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  1. 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

  2. Soft glassy/jammed materials E. Weeks

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

  4. 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

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

  6. j dependence of the relaxation time MCT : ta~ (jc-j)-g jc= 0.59, g = 2.6 VFT: ta~ exp[(j0-j)-d] j0= 0.614, d = 1 j0= 0.637, d = 2 Brambilla et al., PRL 2009

  7. Glass/jamming transition in colloidal HS Fluid Glass (out of equilibrium) MCT jc ~ .57-.59 jrcp ~ .64 Fluid Jamming jc ~ .57-.59 jj = jrcp ~ .64 Fluid Glass Thermodynamic jc ~ .57-.59 j0 jrcp ~ .64

  8. Dynamical heterogeneity x Weeks et al. Science 2000

  9. Dynamical heterogeneity x Weeks et al. Science 2000 var[f(t)] c4(t) dynamical susceptibility

  10. Dynamical susceptibility in glassy systems Supercooled liquid (Lennard-Jones) Lacevic et al., Phys. Rev. E 2002 c4=N var[Q(t)]

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

  12. Dynamical heterogeneity: the theoreticians’ trick Goal: calculate 4-point dynamical susceptibility c4 ~ size of rearranged region For colloidal HS at high j Berthier et al., J. Phys. Chem. 2007

  13. DH in colloidal HS A useful relationship ~(x/a)3 • equilibrium • OK at high j j ~ 0.20 – 0.58 t (sec) dynamical susceptibility c4(t) Berthier et al., J. Phys. Chem.2007 Berthier et al., Science 2005

  14. j dependence of the max of c4 MCT ~ (jc-j)-2 * High j : deviation from MCT Brambilla et al., PRL 2009

  15. 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

  16. lag t time t Time Resolved Correlation (TRC) degree of correlationcI(t,t) = - 1 < Ip(t) Ip(t+t)>p < Ip(t)>p<Ip(t+t)>p Cipelletti et al. J. Phys:Condens. Matter 2003, Duri et al. Phys. Rev. E 2006

  17. Average over t intensity correlation function g2(t) - 1 degree of correlationcI(t,t) = - 1 < Ip(t) Ip(t+t)>p < Ip(t)>p<Ip(t+t)>p Average dynamics g2(t) - 1 ~ f(t)2

  18. Average over t intensity correlation function g2(t) - 1 degree of correlationcI(t,t) = - 1 < Ip(t) Ip(t+t)>p < Ip(t)>p<Ip(t+t)>p fixed t, vs.t fluctuations of the dynamics Average dynamics g2(t) - 1 ~ f(t)2 var[cI(t)] ~ c(t)

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

  20. Non-monotonic behavior of c*(j) Fit: g2(t)-1 ~ exp[-(t/t0)b] relaxation time stretching exponent peak of dynamical susceptibility non monotonic!

  21. Non-monotonic behavior of c*(j) • Competition between: • increasing x (c* as j ) • increasingly restrained displacement • jump size decreases • (many events over t0, c* as j ) • Model: • random events of size x • Poissonian statistics • Quasi-ballistic motion • ( <dr2> ~np, p = 1.65)

  22. Simulating the model cell thickness! inverse jump size volume fraction

  23. 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

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

  25. Local, instantaneous dynamics: cI( t, t,r) < Ip(t) Ip(t+t)>p(r) cI(t, t , r) = - 1 < Ip(t)>p<Ip(t+t)>p(r) Note: <<cI(t, t , r)>t>r = g2(t)-1 [g2(t)-1] ~ f(t)2 2.3 mm

  26. Dynamic Activity Maps Brownian particles g2(t)-1~ exp[-t/tr], tr = 40 s Colloidal gel g2(t)-1~ exp[-(t/tr)1.5], tr = 5000 s cI (t0,tr/200 , r) Movie accelerated 10x 2 mm cI (t0,tr/10 , r) Movie accelerated 500x 2 mm

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

  28. When does x infinity? • G' > G'' : strain propagation in a predominantly solid-like material • Magnitude of G'unimportant (G'onions/G'colloidal gel ~ 6x104 !!) • Origin of elasticity: • entropic(e.g. hard spheres) x very small • enthalpic (attractive gels, squeezed particles...) xsystem-size

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

  30. Conclusions • DH a general feature of glassy/jammed dynamics • Supercooled hard spheres: • equilibrium dynamics above the (apparent) MCT divergence • xlimitedto 5-10a • Jammed materials: • x and cmay decouple! • x ~system size • role of the origin of elasticity

  31. 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

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

  33. Determining the volume fraction j dependence of the short-time diffusing coefficient t = 1/Dssq2 j = 0.2 Absoluteuncertainty: ~4% Relative uncertainty: ~ 10-4

  34. PS gel: time-averaged dynamics g2(q,t) - 1 ~ [f(q,t)]2 • Fast dynamics: overdamped vibrations • (~ 500 nm) Krall & Weitz PRL 1998 • Slow dynamics: rearrangements

  35. PS gels: q dependence of tf and p « ballistic » motion « compressed » exponential

  36. PS gel: temporally heterogeneous dynamics cI(tw,t) @ fixed t : temporal fluctuations <cI(tw,t)>tw:average dynamics

  37. Length scale dependence of c = var(cI) PS gel Increasing q 1.1 +/- 0.1 Duri & LC, EPL 76, 972 (2006)

  38. Scaling of c* c* ~ var(n)/<n> ~ 1/<n> <n> ~ tf ~ 1/q c* ~ q

  39. Dynamical susceptibility in glassy systems Supercooled liquid (Lennard-Jones) Lacevic et al., Phys. Rev. E 2002 c4=N var[Q(t)]

  40. Dynamical susceptibility in glassy systems N regions c4=N var[Q(t)] c4 dynamics spatially correlated

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

  42. The smart trick applied to colloidal HS cf(t) F(t) Df 10-3 t Define

  43. Dynamical heterogeneity: the theoreticians’ trick Goal: calculate 4-point dynamical susceptibility c4 ~ size of rearranged region For colloidal HS at high j Berthier et al., J. Phys. Chem. 2007

  44. Evidence of a growing dynamic length scale j f ~ 0.20 – 0.58 Berthier et al., Science 2005

  45. j dependence of the max of c4 * High j : deviation from MCT MCT Brambilla et al., PRL 2009

  46. Photon Correlation Imaging (PCIm) Dq object plane focal plane image plane q CCD q sample lens annular slit 2.3 mm q = 6.4° q = 1 mm-1. Duri et al., Phys. Rev. Lett. 2009

  47. Spatial correlation of the dynamics

  48. Other PCIm geometries object plane sample beam splitter object plane sample Dq Dq lens focal plane diaphragm Dq Dq image plane lens CCD focal plane diaphragm image plane CCD a) b)

  49. Additional stuff on diverging x

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