3 Dicembre 2007Firenze

Francesco Sciortino Universita’ di Roma La Sapienza

“Patchy Colloidal Particles:The role of the valence

in gel formation

Introduzione

Main Messages

• Strongly interacting particles (u<<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

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

Phase diagram of spherical potentials*0.13<c<0.27

*One component, “Hard-Core” *One component, “Hard-Core” plus attractionplus attraction

(From van der Waals to Baxter)

Phase diagram of spherical potentials*0.13<c<0.27 [if the attractive range

is very small ( <10%)]

*One component, “Hard-Core” *One component, “Hard-Core” plus attractionplus attraction

(From van der Waals to Baxter)

For this class of potentials arrest at low (gelation) is the result of a phase

separation process interrupted by the glass transition

T T

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(in preparation)

How to go to low T at low (in metastable equilibrium)

reducing “valence”

How to suppress phase separation ?

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)

It enforces the one bond per patch condition

Pine’s particles

Self-Organization of Bidisperse Colloids in Water DropletsYoung-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;

DNA functionalized particles

Wertheim TPT for associated liquids(particles with M identical sticky sites )

At low densities and low T (for SW)…..Vb

M=2

FS et al J. Chem.Phys.126, 194903, 2007

M=2 (Chains)

Symbols = Simulation

Lines = Wertheim Theory

<L>

FS et al J. Chem.Phys.126, 194903, 2007

Average chain length

Chain length distributions

What happens with branching ?

A snapshot of

<M>=2.025

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N3=330

N2=5670

T=0.05, =0.01

<M>=2.055

Wertheim theory predicts pb extremely well (in this model) !

(ground state accessed in equilibrium)

Connectivity properties and cluster size distributions: Flory and Wertheim

Connectivity properties and cluster size distributions: Flory and Wertheim

Connectivity properties and cluster size distributions: Flory and Wertheim

No bond-loops in finite clusters !

Generic features of the phase diagram

Cvmax line

Percolation line

unstable

Wertheim Theory (TPT): predictions

E. Bianchi et al, PRL 97, 168301, 2006

Mixtures of particles with 2 and 3 bonds

Empty liquids !Cooling the liquids without phase separating!

Phase Diagram - Theory and Simulations

theorysimulation

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

• In the newly available density region, at low T the system forms a “equilibrium” gel. Arrest driven by bonding (not by caging).

Functionality 4

One Component(water-like)

Binary mixture

(silica-like)

DNA gel model (F. Starr and FS, JPCM, 2006J. Largo et al Langmuir 2007 )

BondSelectivity

StericIncompatibilities

Isodiffusivities ….Isodiffusivities (PMW) ….

DNA-Tetramers phase diagram

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

A collection of phase diagramsof four-coordinated liquids

Physical Gels <===> Network forming liquids

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 ?

Irreversible aggregation in the absence of bond loops

(Smoluchowski)

Irreversible aggregation in the absence of loops

Smoluchowski coagulation works !

Equilibrium dynamics:

The Flory-Stokmayer distributions are also the equilibrium one !!!

Chemical and physical gelation (in the absence of loops)

t <---->T

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

• 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)

Coworkers:

Emanuela Bianchi (Patchy Colloids)Cristiano De Michele (PMW, PMS)Julio Largo (DNA, Patchy Colloids)Francis Starr (DNA)Jack Douglas (NIST) (M=2)

Piero TartagliaEmanuela Zaccarelli