Francesco Sciortino Universita’ di Roma La Sapienza October ISMC 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) 0.13<  c <0.27 [if the attractive range is very small ( <10%)] (Foffi et al PRL 94, , 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 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

Wertheim TPT for associated liquids particles 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, , 2007 Self-assembly Equilibrium Polymerization

Symbols = Simulation Lines = Wertheim Theory FS et al JCP 126, , 2007 Average chain length Chain length distributions Energy per particle M=2 (Chains)

What happens with branching ?

Binary Mixture of M=2 and 3 E. Bianchi et al JPCB (in press) X 3 =0.055 =2.055 N 3 =330 N 2 =5670 Each color labels a different cluster

=2.055 Wertheim theory predicts p b 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 Wertheim E. Bianchi et al, PRL 97, , 2006

Mixtures of particles with valence 2 and 3 A critical point at vanishing packing Wertheim Empty liquids ! Cooling the liquids without phase separating!

Patchy particles - Critical Parameters

A snapshot of =2.025 T=0.05,  =0.01 Ground State (almost) reached ! Bond Lifetime ~ e  u

Del Gado/Kob EPL 2005 Del Gado Dipolar Hard Spheres (Camp) Dipolar Hard Spheres (Blaak, Miller, Hansen)

Dipolar Hard Spheres… Tlusty-Safram, Science (2000) Camp et al PRL (2000) Dipolar Hard Sphere

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 !!! Message

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 (1987) The Primitive Model for Silica (PMS) Ford, Auerbach, Monson, J.Chem.Phys, 8415,121 (2004) H Lone Pair Silicon Four Sites (tetrahedral) Oxygen Two sites o

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

An example: the PMW phase diagram

E vs n Phase- separation Approaching the ground state (PMS)

A collection of phase diagrams of four-coordinated liquids

Schematic Summary Network Region - Approach to Ground State - Bond-Activated Dynamics Region of phase separation Packing Region Phase Separation Region Packing Region Spherical Interactions Patchy/ directioal Interactions

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 Tartaglia Emanuela Zaccarelli

Patchy particles (critical fluctuations) E. Bianchi et al, PRL, 2006 (N.B. Wilding method) ~N+sE

Structure (q-space) C. De Michele et al J. Chem. Phys. 125, , 2006

T-dependence of the Diffusion Coefficient Cross-over to strong behavior ! Strong Liquids !!!

One last four-coordinated model !

Approaching the ground state (PMW) Progressive increase in packing prevents approach to the GS PMW energy

Optimal density Bonding equilibrium involves a significant change in entropy (zip-model) Percolation close (in T) to dynamic arrest ! DNA-PMW “Bond” is now a cooperative free-energy concept

Final Message: Universality Class of valence controlled particles

Angoli modelli Tetrahedral Angle Distribution

Energie Modelli Low T isotherms….. Coupling between bonding (local geometry) and density

=2.05 Slow Dynamics at low  Mean squared displacement  =0.1 

=2.05  =0.1 Slow Dynamics at low  Collective density fluctuations

DNA-Tetramers phase diagram