Introduzione

1
Introduzione
3 Dicembre 2007 Firenze
Francesco Sciortino Universita di Roma La
Sapienza
Patchy Colloidal Particles The role of the
valence in gel formation
2
Main Messages

• Strongly interacting particles (bultlt1)---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
0.13ltfclt0.27
(From van der Waals to Baxter)
One component, Hard-Core plus attraction
(Foffi et al PRL 94, 078301, 2005)
5
Phase diagram of spherical potentials
if the attractive range is very small ( lt10)
0.13ltfclt0.27
(From van der Waals to Baxter)
One component, Hard-Core plus attraction
(Foffi et al PRL 94, 078301, 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
Pines 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
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, 194903, 2007
M2
15
M2 (Chains)
FS et al J. Chem.Phys.126, 194903, 2007
Symbols Simulation Lines Wertheim Theory
ltLgt
Chain length distributions
Average chain length
16
What happens with branching ?
17
A snapshot of ltMgt2.025
N25670
N3330
T0.05, f0.01
18
Wertheim theory predicts pb extremely well (in
this model) !
ltMgt2.055
(ground state accessed in equilibrium)
19
Connectivity properties and cluster size
distributions Flory and Wertheim
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Connectivity properties and cluster size
distributions Flory and Wertheim
21
Connectivity properties and cluster size
distributions Flory and Wertheim
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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, 168301, 2006
25
Wertheim
Mixtures of particles with 2 and 3 bonds
Cooling the liquids without phase separating!
Empty liquids !
26
Phase Diagram - Theory and Simulations
theory
simulation
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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).

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Functionality 4
One Component (water-like)
Binary mixture (silica-like)
DNA gel model (F. Starr and FS, JPCM, 2006J.
Largo et al Langmuir 2007 )
Bond Selectivity Steric Incompatibilities
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Isodiffusivities .
Isodiffusivities (PMW) .
30
DNA-Tetramers phase diagram
31
Question Compare ?
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
32
A collection of phase diagrams of
four-coordinated liquids
Physical Gels ltgt 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 --gt 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 lt----gtT
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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)
(M2) Piero Tartaglia Emanuela Zaccarelli