1. “Patchy Colloidal Particles:
3 Dicembre 2007
Firenze
Francesco Sciortino
Universita’ di Roma La Sapienza
“Patchy Colloidal Particles:
The role of the valence
in gel formation
Introduzione

2. Main Messages
Strongly interacting particles (bu<<1)---with simple spherical potentials -- at small and intermediate densities ALWAYS phase-separate (in a dense and dilute phase)
Strongly interacting particles with LIMITED valence ---patchy particles, highly directional interactions, dipolar, quadrupolar --- form equilibrium open structures (GELS, network forming liquids). Empty liquids
Self-assembly as an equilibrium liquid-state problem

3. Outline
The fate of the liquid state (neglecting crystallization): phase diagram of spherical and patchy attractive potentials
A theory-of-liquid approach to self-assembly in equilibrium polymerization (linear and branched)
The role of valence: Universality classes for the liquid-gas transition (analogies between network forming (strong) liquids and gels.
Physical and chemical gels

4. Phase diagram of spherical potentials*
0.13<fc<0.27
(From van der Waals to Baxter)
*One component, “Hard-Core”
plus attraction
(Foffi et al PRL 94, , 2005)

5. Phase diagram of spherical potentials*
[if the attractive range
is very small ( <10%)]
0.13<fc<0.27
(From van der Waals to Baxter)
*One component, “Hard-Core”
plus attraction
(Foffi et al PRL 94, , 2005)

6. For this class of potentials arrest at low f (gelation) is the result of a phase separation process interrupted by the glass transition T T f f

7. (in preparation)

8. How to go to low T at low f (in metastable equilibrium)
How to suppress phase separation ?
reducing “valence”

9. Patchy particles
maximum number of “bonds”, (different from fraction of bonding surface)
It enforces the one bond per patch condition
Hard-Core (gray spheres) Short-range Square-Well (gold patchy sites)
No dispersion forces
The essence of bonding !!!

10. Pine 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 ;
Pine
Pine

11. Mohwald

12. DNA functionalized particles

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

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

15. M=2 (Chains) Symbols = Simulation Lines = Wertheim Theory <L>
FS et al
J. Chem.Phys.126, , 2007
Symbols = Simulation
Lines = Wertheim Theory
<L>
Chain length distributions
Average chain length

16. What happens with branching ?

17. A snapshot of <M>=2.025
T=0.05, f=0.01

18. Wertheim theory predicts pb extremely well (in this model) !
(ground state accessed in equilibrium)

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

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

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

22. No bond-loops in finite clusters !

23. Generic features of the phase diagram
Cvmax line
Percolation line
unstable

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

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

26. Phase Diagram - Theory and Simulations

27. Conclusions (I)
Directional interaction and limited valency are essential ingredients for offering a DIFFERENT final fate to the liquid state and in particular to arrested states at low f.
In the newly available density region, at low T the system forms a “equilibrium” gel. Arrest driven by bonding (not by caging).

28. Functionality 4 DNA gel model One Component (water-like)
Binary mixture
(silica-like)
DNA gel model
(F. Starr and FS, JPCM, 2006 J. Largo et al Langmuir 2007 )
Bond
Selectivity
Steric
Incompatibilities

29. Isodiffusivities ….
Isodiffusivities (PMW) ….

30. DNA-Tetramers phase diagram

31. 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 ?

32. A collection of phase diagrams
of four-coordinated liquids
Physical Gels <===> Network forming liquids

33. Quanto di questo che abbiamo imparato sulla valenza puo’ servirci a capire la gelazione chimica ?
Fino a che punto la gelazione chimica puo’ essere vista come un quench a U/kT --> oo ?

34. Irreversible aggregation in the absence of bond loops
(Smoluchowski)

35. Irreversible aggregation in the absence of loops
Smoluchowski coagulation works !

36. Equilibrium dynamics:
The Flory-Stokmayer distributions are also the equilibrium one !!!

37. Chemical and physical gelation (in the absence of loops)
t <---->T

38. Conclusions
Directional interaction and limited valency are essential ingredients for offering a DIFFERENT final fate to the liquid state and in particular to arrested states at low f.
In the newly available density region, at low T the system forms a “equilibrium” gel (or a network glass).
Equilibrium Gels and network forming liquids: two faces of the same medal.
In the absence of bond-loops, chemical gelation proceeds via a sequence of quasi-equilibrium states (possibility of using phase-coexistence concepts)

39. Coworkers: Emanuela Bianchi (Patchy Colloids)
Cristiano De Michele (PMW, PMS)
Julio Largo (DNA, Patchy Colloids)
Francis Starr (DNA)
Jack Douglas (NIST) (M=2)
Piero Tartaglia
Emanuela Zaccarelli