“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

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

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<fc<0.27 (From van der Waals to Baxter) *One component, “Hard-Core” plus attraction (Foffi et al PRL 94, 078301, 2005)

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, 078301, 2005)

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

(in preparation)

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

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 !!!

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 15968 - 15975; Pine Pine

Mohwald

DNA functionalized particles

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

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

M=2 (Chains) Symbols = Simulation Lines = Wertheim Theory <L> FS et al J. Chem.Phys.126, 194903, 2007 Symbols = Simulation Lines = Wertheim Theory <L> Chain length distributions Average chain length

What happens with branching ?

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

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

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

Phase Diagram - Theory and Simulations

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).

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

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 Question Compare ?

A collection of phase diagrams of 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 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)

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