1. IV Workshop on Non Equilibrium Phenomena in Supercooled Fluids, Glasses and Amorphous Materials Pisa, September 2006 Francesco Sciortino Gel-forming patchy colloids and network glass formers: Thermodynamic and Dynamic analogies Imtroduzione

2. Motivations The fate of the liquid state (assuming crystallization can be prevented)…. Gels and phase separation: essential features (Sticky colloids - Proteins) Thermodynamic and dynamic behavior of new patchy colloids Revisiting dynamics in network forming liquids (Silica, water….) Essential ingredients of “strong behavior” (A. Angell scheme).

3. Glass line (D->0) Liquid-Gas Spinodal Binary Mixture LJ particles “Equilibrium” “homogeneous” arrested states only for large packing fraction BMLJ (Sastry) (see also Debenedetti/Stillinger)

4. Phase diagram of spherical potentials* * “Hard-Core” plus attraction 0.13<  c <0.27 [if the attractive range is very small ( <10%)]

5. Gelation (arrest at low  ) as a result of phase separation (interrupted by the glass transition) T T  

6. How to go to low T at low  (in metastable equilibrium) ? Is there something else beside Sastry’s scenario for a liquid to end ? -controlling valency (Hard core complemented by attractions) -l.r. repulsion (Hard core complemented by both attraction and repulsions How to suppress phase separation ?

7. Geometric Constraint: Maximum Valency (E. Zaccarelli et al, PRL, 2005) SW if # of bonded particles <= N max HS if # of bonded particles > N max V(r) r Maximum Valency

8. N MAX -modified Phase Diagram N max phase diagram

9. Patchy particles Hard-Core (gray spheres) Short-range Square-Well (gold patchy sites) No dispersion forces The essence of bonding !!!

10. Mohwald

11. Pine

12. Self-Organization of Bidisperse Colloids in Water Droplets Young-Sang Cho, Gi-Ra Yi, Jong-Min Lim, Shin-Hyun Kim, Vinothan N. Manoharan,, David J. Pine, and Seung-Man Yang J. Am. Chem. Soc.; 2005; 127 (45) pp 15968 - 15975; Pine

13. Steric incompatibilities satisfied if SW width  <0.11 No double bonding Single bond per bond site Steric Incompatibilities

14. Wertheim Theory

15. Wertheim Theory (TPT): predictions Wertheim E. Bianchi et al, PRL, in press

16. Mixtures of particles with 2 and 3 bonds Wertheim Empty liquids !

17. Patchy particles (critical fluctuations) E. Bianchi et al, PRL, in press (N.B. Wilding) ~N+sE

18. Patchy particles - Critical Parameters

19. Lattice-gas calculation for reduced valence (Sastry/La Nave) cond-mat

20. A snapshot of a =2.025 (low T) case,  =0.033 Ground State (almost) reached ! Bond Lifetime ~ e  u

21. Dipolar Hard Spheres… Tlusty-Safram, Science (2000) Camp et al PRL (2000) Dipolar Hard Sphere

22. Del Gado ….. Del Gado/Kob EPL 2005 Del Gado

23. MESSAGE (so far…): REDUCTION OF THE MAXIMUM VALENCY OPENS A WINDOW IN DENSITIES WHERE THE LIQUID CAN BE COOLED TO VERY LOW T WITHOUT ENCOUNTERING PHASE SEPARATION THE LIFETIME OF THE BONDS INCREASES ON COOLING THE LIFETIME OF THE STRUCTURE INCREASES ARREST A LOW  CAN BE APPROACHED CONTINUOUSLY ON COOLING (MODEL FOR GELS) HOW ABOUT MOLECULAR NETWORKS ? IS THE SAME MECHANISM ACTIVE ? HOW ABOUT DYNAMICS ? Message

24. The PMW model J. Kolafa and I. Nezbeda, Mol. Phys. 161 87 (1987) Hard-Sphere + 4 sites (2H, 2LP) Tetrahedral arrangement H-LP interact via a SW Potential, of range  0.15 . V(r) r  (length scale) (energy scale) u0u0 Bonding is properly defined --- Lowest energy state is well defined

25. Equilibrium phase diagram (PMW)

26. Critical Point of PMW GC simulation BOX SIZE=  T C =0.1095  C =0.153 (Flavio Romano Laurea Thesis)

27. Pagan and Gunton JCP (2005) Pagan-Gunton

28. The PMS Model Ford, Auerbach, Monson, J.Chem.Phys, 8415,121 (2004) Silicon Four sites (tetrahedral) Oxygen Two sites 145.8 o  OO =1.6   SW interaction between Si sites and O sites 

29. Equilibrium Phase Diagram PSM

30. Critical Point of PMS GC simulation BOX SIZE=  T C =0.075  C =0.0445 s=0.45 Critical point PSM

31. Potential Energy (# of bonds) for the PMW Optimal density !

32. Potential Energy -- Approaching the ground state Progressive increase in packing prevents approach to the GS PMW energy

33. E-E gs vs. 1/T

34. Potential Energy along isotherms Optimal density Hints of a LL CP Phase-separation

35. S(q) in the network region

36. PMS -Potential Energy

37. PMS E vs 1/T

38. PMS Structure (r-space)

39. Structure (q-space)

40. E vs n Phase-separation

41. Summary of static data Optimal Network Region - Arrhenius Approach to Ground State Region of phase separation Packing Region Phase Separation Region Packing Region Spherical Interactions Patchy Interactions

42. How About Dynamics (in the new network region) ?

43. Dynamics in the N max =4 model (no angular constraints) Strong Liquid Dynamics !

44. N max =4 phase diagram - Isodiffusivity lines Zaccarelli et al JCP 2006

45. R2 vs t PMW

46. PMW -- Diffusion Coefficient Cross-over to strong behavior

47. D along isotherms Diffusion Anomalies

48. Isodiffusivities …. Isodiffusivities (PMW) ….

49. Diffusion PMS De Michele et al, cond mat

51. How to compare these (and other) models for tetra-coordinated liquids ? Focus on the 4-coordinated particles (other particles are “bond-mediators”) Energy scale ---- Tc Length scale --- nn-distance among 4- coordinated particles Question Compare ?

52. Phase Diagram Compared Spinodals and isodiffusivity lines: PMW, PMS, N max

53. Analogies with other network-forming potentials SPC/E ST2 (Poole) BKS silica (Saika-Voivod) Faster on compression Slower on compression

54. Water Phase Diagram  ~ 0.34 Do we need do invoke disperison forces for LL ?

55. Angoli modelli Tetrahedral Angle Distribution

56. Energie Modelli Low T isotherms….. Coupling between bonding (local geometry) and density

57. Comments Directional interaction and limited valency are essential ingredients for offering a new final fate to the liquid state and in particular to arrested states at low  The resulting low T liquid state is (along isochores) a strong liquid. The bond energy scale: is bonding essential for being strong ?. Gels and strong liquids are two faces of the same medal.

58. Graphic Summary Two glass lines ? Strong liquids - Gels Arrest line Fragile Liquids - Colloidal Glasses

59. Appendix I Possibility to calculate exactly potential energy landscape properties for SW models (spherical and patcky) Moreno et al PRL, 2005

60. Thermodynamics in the Stillinger-Weber formalism F(T)=-T S conf (E(T))+E(T)+f basin (E,T) with f basin (E,T) and S conf (E)=k B ln[  (E)] Sampled Space with E bonds Number of configurations with E bonds Stillinger-Weber

61. It is possible to calculate exactly the vibrational entropy of one single bonding pattern (basin free energy) Basin Free energy (Ladd and Frenkel)

62. Comment: In models for fragile liquids, the number of configurations with energy E has been found to be gaussian distributed Non zero ground state entropy ex

63. Appendix II Percolation and Gelation: How to arrest at (or close to) the percolation line ? F. Starr and FS, JPCM, 2006

64. Colloidal Gels, Molecular Gels, …. and DNA gels Four Arm Ologonucleotide Complexes as precursors for the generation of supramolecular periodic assemblies JACS 126, 2050 2004 Palindroms in complementary space DNA Gels 1

66. The DNA gel model (F. Starr and FS, JPCM, 2006)

67. Optimal density Bonding equilibrium involves a significant change in entropy (zip-model) Percolation close (in T) to dynamic arrest ! DNA-PMW

68. Final Message: Universality Class of valence controlled particles

69. Coworkers: Emanuela Bianchi (Patchy) Cristiano De Michele (PMW,PMS) Simone Gabrielli (PMW) Julio Largo (DNA,Patchy) Emilia La Nave, Srikanth Sastry (Bethe) Angel Moreno (Landscape) Flavio Romano (PMW) Francis Starr (DNA) Piero Tartaglia Emanuela Zaccarelli

70. http://www.socobim.de/

71. Density Anomalies… (and possible 2’nd CP) Density anomalies

74. D vs (1-p b )

75. D vs (1-p b ) --- (MC) D ~ f 0 4 ~(Stanley-Teixeira)

76. G. Foffi, E. Zaccarelli, S. V. Buldyrev, F. Sciortino, P. Tartaglia Aging in short range attractive colloids: A numerical study J. Chem. Phys. 120, 1824, 2004 Foffi aging

77. Strong-fragile: Dire Stretched, Delta Cp Hard Sphere Colloids: model for fragile liquids

78. S(q) in the phase-separation region