1. Mainz, November 28 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 Pine’s particle

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, 2006

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

17. Patchy particles (critical fluctuations) E. Bianchi et al, PRL, 2006 (N.B. Wilding) ~N+sE

18. Patchy particles - Critical Parameters

19. T=0.07 M=2 (Chains) Symbols = Simulation Lines = Wertheim Theory

20. =2.055

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

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

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

24. Hansen

25. MESSAGE(S) (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 DYNAMICS ? HOW ABOUT MOLECULAR NETWORKS ? IS THE SAME MECHANISM ACTIVE ? Message

26. =2.05 Slow Dynamics at low  Mean squared displacement  =0.1 

27. =2.05  =0.1 Slow Dynamics at low  Collective density fluctuations

28. Message: Gel dynamics: dynamic arrest due to percolation (in the limit of long-living bonds).

29. 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

30. Equilibrium phase diagram (PMW)

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

32. 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 

33. Equilibrium Phase Diagram PSM

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

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

36. S(q) in the network region

37. PMS Structure (r-space)

38. Structure (q-space)

39. E vs n Phase-separation

40. 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

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

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

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

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

45. Isodiffusivities …. Isodiffusivities (PMW) ….

46. Diffusion PMS De Michele et al, cond mat

48. 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 ?

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

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

51. Angoli modelli Tetrahedral Angle Distribution

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

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

54. 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.

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

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

57. 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

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

59. 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

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

61. 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

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

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

65. Final Message: Universality Class of valence controlled particles

66. 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

67. http://www.socobim.de/

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

71. D vs (1-p b )

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

73. 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

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

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

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

78. PMS E vs 1/T

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

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

81. E-E gs vs. 1/T

82. PMS -Potential Energy

83. Lattice-gas calculation for reduced valence (Sastry/La Nave/FS J. Stat. Mech 2006)

84. R2 vs t PMW

85. D along isotherms Diffusion Anomalies