Francesco Sciortino Universita’ di Roma La Sapienza

October 4 2007ISMC Aachen

Patchy colloidal particles: the role of the valence in the formation of gels

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 - (Zaccarelli talk))

• 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

• Thermodynamic and dynamic behavior of new patchy colloids. Analogies between network forming liquids (silica, water) and colloidal gels.

Phase diagram of spherical potentials*

* “Hard-Core” plus attraction (e.g. LJ)* “Hard-Core” plus attraction (e.g. LJ)

0.13<c<0.27

[if the attractive rangeis very small ( <10%)]

(Foffi et al PRL 94, 078301, 2005)

For sperical potentials (including the depletion potential) arrest at low

(gelation) is the result of a phase separation process interrupted by the

glass transition

T T

E. Zaccarelli, Talk and JPCM, Topical Review 2007

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

reducing “valence”

How to suppress phase separation ?

Valence-Controlled Patchy particles

Hard-Core (gray spheres) Short-range Square-Well (gold patchy sites)

No dispersion forces The essence of bonding !!!(one bond per patch)

maximum # of “bonds”, (as opposed to # patches, fraction of bonding surface)

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;

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Wertheim TPT for associated liquidsparticles with M identical sticky sites -( one bond per patch )

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

M=2

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

Self-assembly

Equilibrium Polymerization

Symbols = SimulationLines = Wertheim TheoryFS et al JCP 126, 194903, 2007

<L>

Average chain length Chain length distributions

Energy per particleM=2 (Chains)

What happens with branching ?

Binary Mixture of M=2 and 3 E. Bianchi et alJPCB (in press)

X3=0.055<M>=2.055

N3=330

N2=5670

Each colorlabelsa differentcluster

<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

Non percolating state points

Percolating state points

Percolation Line (theory)

Phase-separation

Wertheim Theory works (for small M)

Predictions for larger M

Wertheim Theory (TPT): predictions

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

Mixtures of particles with valence 2 and 3A critical point at vanishing packing

Empty liquids !Cooling the liquids without phase separating!

Patchy particles - Critical Parameters

A snapshot of

<M>=2.025

T=0.05, =0.01

Ground State (almost)reached !

Bond Lifetime

~eu

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Del Gado/Kob EPL 2005 Dipolar Hard Spheres (Camp)

Dipolar Hard Spheres (Blaak, Miller, Hansen)

Dipolar Hard Spheres…

Tlusty-Safram, Science (2000)

Camp et al PRL (2000)

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 EQUILIBRIUM GELS !!!

Is there some kind of universal behavior controlled by valence ?

Noro-Frenkel Scaling for Kern-Frenkel particles

G. Foffi and FS, JPCB 2007

Connecting colloidal particles with

network forming liquids

Colloidal Water and Colloidal Silica !

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)

HLone Pair

SiliconFour Sites(tetrahedral)

OxygenTwo sites

145.8 o

LimitedCoordination(4)

BondSelectivity

StericIncompatibilities

4-coordinate “DNA” dendrimed model (F. Starr and FS, JPCM, 2006J. Largo et al Langmuir 2007 )

LimitedCoordination(4)

BondSelectivity

StericIncompatibilities

An example: the PMW phase diagram

E vs n

Phase-separation

Approaching the ground state (PMS)

A collection of phase diagramsof four-coordinated liquids

Schematic Summary

NetworkRegion

-Approach toGround State

-Bond-Activated

Dynamics

Regionof

phaseseparation

Packing Region

Phase Separation RegionPackingRegion

SphericalInteractions

Patchy/directioalInteractions

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.

Collaborators :Emanuela Bianchi (Patchy Colloids)Cristiano De Michele (PMW, PMS)Julio Largo (DNA, Patchy Colloids)Francis Starr (DNA)Jack Douglas (M=2)Emilia La Nave (Mixture M=2-M=3)Giuseppe Foffi (Kern particles)

Piero TartagliaEmanuela Zaccarelli

Patchy particles (critical fluctuations)

E. Bianchi et al, PRL, 2006

(N.B. Wilding method)

~N+sE

Structure (q-space)

C. De Michele et alJ. Chem. Phys. 125, 204710, 2006

T-dependence of the Diffusion

Coefficient

Cross-over tostrong behavior !

Strong Liquids !!!

One last four-coordinated model !

Approaching the ground state (PMW)

Progressive increase in packing prevents approach to the GS

Optimaldensity

Bonding equilibriuminvolves a significantchange in entropy(zip-model)

Percolation close (in T) to dynamicarrest !

“Bond” is now a cooperative free-energy concept

Final Message: Universality Class ofvalence controlled particles

Tetrahedral Angle Distribution

Energie Modelli

Low T isotherms…..

Coupling between bonding (local geometry) and density

<M>=2.05

Slow Dynamics at low Mean squared displacement

=0.1

<M>=2.05 =0.1

Slow Dynamics at low Collective density fluctuations

DNA-Tetramers phase diagram