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1. titolo Cluster Phases, Gels and Yukawa Glasses in charged colloid-polymer mixtures. Francesco Sciortino

2. Motivations Outline Dynamic Arrest in Colloidal Systems: Glasses and Gels Excluded Volume Short Range Attraction (SRA) SRA+ Longer Range Repulsion Investigate the competing effects of short range attraction and longer-range repulsion in colloidal systems Dynamics close to arrested states of matter: Cluster Phases, Glasses and/or Gels

3. HS Hard Spheres Potential (No temperature, only density) V(r) r s • Hard spheres present a a fluid–solid phase separation due to entropic effects • Experimentally, at h=0.58, the system freezes forming disordered aggregates. MCT transition =51.6% • W. van Megen and P.N. Pusey Phys. Rev. A 43, 5429 (1991) • U. Bengtzelius et al. J. Phys. C 17, 5915 (1984) • W. van Megen and S.M. Underwood Phys. Rev. Lett.70, 2766 (1993)

4. Explanation of the cage and analysis of correlation function .The Cage Effect (in HS). Rattling in the cage F(t) Cage changes log(t)

5. Colloids: Possibility to control the Interparticle interactions Design Potenziale Hard Sphere Chemistry (surface) r s Asakura- Oosawa Physic Processes (solvent modulation, polydispersity, Depletions) s Yukawa r - + + + + - - r

6. Depletion Interactions: A (C. Likos) Cartoon Depletion Interactions V(r ) s D r D<<s

7. Adding attraction (phase diagram) Adding attraction (phase diagram) • The presence of attraction modifies the behaviour of the system: New phases and their coexistence emerge. • With narrow interactions the appeareance of metastable liquid-liquid critical point is typical for colloids. V.J. Anderson and H.N.W. Lekkerkerker Nature416, 811 (2002)

8. Nat Mat 2

9. Square Well 3% width Phase Diagram for Square Well (3%) Iso- diffusivity lines Percolation Line Repulsive Glass A3 Spinodal (and Baxter) Attractive Glass Liquid+Gas Coexistence Spinodal AHS (Miller&Frenkel)

10. Gelation as a result of phase separation (interrupted by the glass transition) T T f f

11. Nat Nat

12. The quest The quest for the ideal (thermoreversible) gel….model 1) Long Living reversible bonds 2)No Phase Separation 3) No Crystallization Are 1 and 2 mutually exclusive ? Long Bond Lifetime LowTemperature Condensation The quest

13. Surface Tension How to stay at low T without condensation ? Reasons for condensation (Frank, Hill, Coniglio) Physical Clusters at low T if the infinite cluster is the lowest (free)energy state How to make the surface as stable as the bulk (or more)? The quest

14. Cluster Ground State Energy : Only Attraction

15. Competition between short range attraction and long-range repulsion (this talk) Limited Valency (see E. Zaccarelli et al PRL xxx Routes to Arrest at low packing fractions (in the absence of a “liquid-gas” phase separation)

16. Cluster Ground State: Attraction and Repulsion (Yukawa)

17. Cluster Ground State: Attraction and Repulsion (Yukawa) Vanishing of the “surface tension” !

18. Competition Between Short Range Attraction and Longer Range Repulsion: Role in the clustering Short Range Attraction, --dominant in small clusters Longer Range Repulsion Importance of the short-range attraction: Only nn interactions

19. Typical Shapes in the ground state A=8 x =0.5 s A=0.05 x=2 s

20. Size dependence of the cluster shape “Linear” Growth is an “attractor”

21. From isolated to interacting clusters Role of T and f: On cooling (or on increasing attraction), monomers tend to cluster…. In the region of the phase diagram where the attractive potential would generate a phase separation….repulsion slows down (or stop) aggregation.The range of the attractive interactions plays a role. How do clusters interact ?

22. How do cluster interact How do “spherical” clusters interact ?

23. Yukawa Phase Diagram

24. Description of the flow in the Yukawa model N=1

25. N=2

26. N=4

27. N=8

28. N=16

29. N=32

30. N=64

31. Yukawa Phase Diagram

32. Figure gel yukawa Tc=0.23 n=100 lowering T Increasing packing fraction

33. Brief Intermediate Summary Equilibrium Cluster-phases result from the competition between aggregation and repulsion. Arrest at low packing fraction generated by a glass transition of the clusters. Aggregation progressively cool the system down till the repulsive cages become dominant

34. Interacting cluster linear case Interacting Clusters - Linear case The Bernal Spiral Campbell, Anderson, van Dujneveldt, Bartlett PRL June (2005)

35. Pictures of the clusters at f=0.08 T=0.12 T=0.10 T=0.15 Aggshapec=0.08

36. T=0.07

37. Pictures of the aggregation T=0.10 T=0.12 T=0.15 at f=0.125

38. A gel ! Cluster shapec=0.125 T=0.07

39. Cluster size distribution  n ~ s s = 2.2 (random percolation)

40. Fractal Dimension T=0.1 size

41. Bond Correlation funtions stretched exponential ~0.7 (a.u.)

42. Density fluctuations

45. bartlett

46. Shurtemberger

47. Several morphologies can be generated by the competition of short-range attraction (fixing the T-scale) and the strength and length of the interaction. A new route to gelation. Continuous change from a Wigner-like glass to a gel While equilibrium would probably suggest a first order transition to a lamellar phase, arrested metastable states appear to be kinetically favored Possibility of exporting ideas developed in colloidal systems to protein systems (Schurtenberger, Chen) and, more in general to biological systems in which often one dimensional growth followed by gelation is observed. Conclusions……

48. Yukawa Upper Limit Optimal Size Groenewold and Kegel

49. No density dependence in prepeak No strong density dependence in peak position

50. Mean square displacement