1. June 19 2007 Universite Montpellier II Francesco Sciortino Universita’ di Roma La Sapienza Aggregation, phase separation and arrest in low valence patchy particles Introduzione

2. Motivations The fate of the liquid state (assuming crystallization can be prevented)…. Equilibrium Aggregation, 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….) A four-coordinated patchy particle model - DNA functionalized colloids

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

4. Phase diagram of spherical potentials* * “Hard-Core” plus attraction 0.13<  c <0.27 [if the attractive range is very small ( <10%)] (Foffi et al PRL 94, 078301, 2005)

5. For this class of potentials arrest at low  (gelation) is the result of a phase separation process 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) How to suppress phase separation ? - Zaccarelli et al PRL 94, 218301, 2005 - Sastry et al JSTAT 2006

7. Patchy particles Hard-Core (gray spheres) Short-range Square-Well (gold patchy sites) No dispersion forces The essence of bonding !!! maximum number of “bonds”, (different from fraction of bonding surface)

8. Pine Pine’s particles 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

9. Wertheim TPT for associated liquids (particles with M identical sticky sites ) At low densities and low T (for SW)….. VbVb

10. Wertheim (in a nut-shell) (ideal gas of equilibrium loop-less clusters of independent bonds (Sear, PRL DHS)

11. Steric incompatibilities satisfied if SW width  <0.11 No double bonding Single bond per bond site Steric Incompatibilities No ring configurations !

12. M=2 FS et al J. Chem.Phys.126, 194903, 2007

13. M=2 (Chains) Symbols = Simulation Lines = Wertheim Theory FS et al J. Chem.Phys.126, 194903, 2007 Average chain length Chain length distributions Energy per particle

14. Branching points in the self-assembly process

15. Binary Mixture of M=2 and 3 E. Bianche et al (submitted) X 3 =0.055 =2.055 N 3 =330 N 2 =5670 Each color labels a different cluster

16. =2.055 Wertheim theory predicts p b extremely well (in this model) ! (ground state accessed in equilibrium)

17. “Time” dependence of the potential energy (~p b ) around the predicted Wertheim value ground-state

18. Connectivity properties and cluster size distributions: Flory and Wertheim

19. No bond-loops in finite clusters !

20. Generic features of the phase diagram C v max line Percolation line unstable

21. Wertheim Theory (TPT): predictions Wertheim E. Bianchi et al, PRL 97, 168301, 2006

22. Mixtures of particles with 2 and 3 bonds Wertheim Empty liquids ! Cooling the liquids without phase separating!

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

24. Patchy particles - Critical Parameters

25. A snapshot of =2.025 T=0.05,  =0.01 Ground State (almost) reached ! Bond Lifetime ~ e  u

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

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

28. Noro-Frenkel Scaling for Kern-Frenkel particles

29. 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

30. =2.05 Slow Dynamics at low  Mean squared displacement  =0.1 

31. =2.05  =0.1 Slow Dynamics at low  Collective density fluctuations

32. Connecting colloidal particles with network forming liquids

33. The Primitive Model for Water (PMW) J. Kolafa and I. Nezbeda, Mol. Phys. 161 87 (1987) The Primitive Model for Silica (PMS) Ford, Auerbach, Monson, J.Chem.Phys, 8415,121 (2004) H Lone Pair Silicon Four Sites (tetrahedral) Oxygen Two sites 145.8 o

34. S(q) in the network region (PMW) C. De Michele et al, J. Phys. Chem. B 110, 8064-8079, 2006

35. Structure (q-space) C. De Michele et al J. Chem. Phys. 125, 204710, 2006

36. Approaching the ground state (PMW) Progressive increase in packing prevents approach to the GS PMW energy

37. E vs n Phase- separation Approaching the ground state (PMS)

38. T-dependence of the Diffusion Coefficient Cross-over to strong behavior ! Strong Liquids !!!

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

40. Schematic Summary Optimal Network Region - Arrhenius Approach to Ground State Region of phase separation Packing Region Phase Separation Region Packing Region Spherical Interactions Patchy/ directioal Interactions

41. Summary 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. Directional interactions (suppressing phase-separation) appear to be essential for strong liquids Gels and strong liquids: two faces of the same medal.

42. One last four-coordinated model !

43. Limited Coordination (4) Bond Selectivity Steric Incompatibilities DNA gel model (F. Starr and FS, JPCM, 2006 J. Largo et al Langmuir 2007 ) Limited Coordination (4) Bond Selectivity Steric Incompatibilities

44. Optimal density Bonding equilibrium involves a significant change in entropy (zip-model) Percolation close (in T) to dynamic arrest ! DNA-PMW “Bond” is now a cooperative free-energy concept

45. DNA-Tetramers phase diagram

46. Building an effective potential for particles with valence (cond- mat/0703383)

47. Two “macromolecules” in a (periodic) box Histogram of center-to-center distances Histogram of the bonding angle

49. A two-state model for the bonding process

51. A spherical and a non-spherical potential

52. T-dependence of the effective potential shape Non Spherical CaseSpherical Case

54. Breakdown of pair-wise additivity due to the selective lock-and-key character of the bonding A valence-constrained model

55. Final Message: Universality Class of valence controlled particles

56. Graphic Summary Two distinct arrest lines ? Strong liquids - Patchy colloids: Gels arrest line Fragile Liquids - Colloidal Glasses: Glass arrest line Fluid

57. Coworkers: Emanuela Bianchi (Patchy Colloids) Cristiano De Michele (PMW, PMS) Julio Largo (DNA, Patchy Colloids) Emilia La Nave (M=2.005) Flavio Romano (PMW) Francis Starr (DNA) Jack Douglas (NIST) (M=2) Piero Tartaglia Emanuela Zaccarelli

58. http://www.socobim.de/

59. Unifying aspects of Dynamics (in the new network region)

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

61. N max =4 phase diagram - Isodiffusivity lines Zaccarelli et al JCP 2006 T=0 !

62. Isodiffusivities …. Isodiffusivities (PMW) ….

64. 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 ?

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

66. Angoli modelli Tetrahedral Angle Distribution

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

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

69. 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

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

71. 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

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

73. 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

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

75. 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

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

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

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

79. D along isotherms Diffusion Anomalies

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

81. Mohwald

82. Geometric Constraint: Maximum Valency N max Model (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 Speedy-Debenedetti

83. Equilibrium Phase Diagram PSM

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

85. Equilibrium phase diagram (PMW)

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

87. PMS Structure (r-space)