SimBioMa, Konstanz 2008 Francesco Sciortino Universita’ di Roma La Sapienza “Models for colloidal gelation” Introduzione

Coworkers: Emanuela Bianchi Cristiano De Michele Jack Douglas (NIST) (M=2) Piero Tartaglia Emanuela Zaccarelli

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 A parameter free description of self-assembly (both equilibrium and equilibration !) can be formulated joining Wertheim and Flory-Stockmayer theories for a class of patchy particles systems. Connections to chemical gels.

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 in controlling the width of the gas- liquid instability Physical and chemical gels

Phase diagram of spherical potentials* 0.13<  c <0.27 [if the attractive range is very small ( <10%)] *One component, “Hard-Core” plus attraction (Foffi et al PRL 94, , 2005)

Nature, in press For this class of potentials arrest at low  (gelation) is the result of a phase separation process interrupted by the glass transition CONFOCAL IMAGES (THE REAL STUFF!)

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 Energy= Number of bonds = bond probability

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

Mohwald

DNA functionalized particles

Wertheim TPT for associated liquids (particles with M identical sticky sites ) At low densities and low T (for SW)….. VbVb Wertheim in a nut-shell Appendix A: Bianchi et al JCP (in press)

M=2 FS et al J. Chem.Phys.126, , 2007 Equilibrium Polymerization (no bond rings)

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

M=2 EQUILIBRATION (Growth of the Chains) Low T limit: FS, C. De Michele and J. Douglas Growth of equilibrium polymers under non-equilibrium conditions J. Phys. Condensed Matter 20, (2008)

What happens with (rear) branching ?

A snapshot of =2.025 N 3 =330 N 2 =5670 T=0.05,  =0.01

=2.055 Wertheim theory predicts p b extremely well (in this model) ! (ground state accessed in equilibrium !!!!!) Emanuela Bianchi, Piero Tartaglia, Emilia La Nave and FS, Fully Solvable Equilibrium Self-Assembly Process: Fine- Tuning the Clusters Size and the Connectivity in Patchy Particle Systems, J. Phys. Chem. B 111, (2007).

Generic features of the phase diagram Branching introduces percolation and phase-separation! C v max line Percolation line unstable

Connectivity properties and cluster size distributions: Flory and Wertheim Flory-Stockmayer cluster size distributions observed

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

Phase Diagram - Theory and Simulations

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. ARREST DRIVEN BY BONDING INSTEAD OF PACKING (equilibrium gels !) Message THE WIDTH OF THE GAS-LIQUID UNSTABLE REGION IS STRONGLY CONTROLLED BY THE VALENCE (empty liquids)

Equilibration (to a finite T) in the presence of branching (but no loops !) ( P. van Dongen and M. Ernst, J. Stat Phys 37, 301 (1984). ) At low T (irreversible coagulation) At all times, the cluster size distribution is the same as the equilibrium one, but with p(t) instead of p eq The resulting equation for p(t) CAN be solved analytically !!!

Comparing simulation and theory Evolution of the number of bonds following a T-jump, starting from high-T Quench protocol

Irreversible aggregation in the absence of bond loops Smoluchowski coagulation works ! Chemical Gels….. Quench protocol

Chemical and physical gelation (in the absence of loops) t T

Message Final Message: In the absence of bond-loops, chemical gelation proceeds via a sequence of “quasi- ”equilibrium steps (longer t --> smaller T) The phase-diagram information (gas-liquid instability) are thus of relevance to the process of chemical gelation. Syneresis as a “echo” of the equilibrium phase separation ?

Message Final Message: In the absence of bond-loops, chemical gelation proceeds via a sequence of “quasi- ”equilibrium steps (longer t --> smaller T) The phase-diagram information (gas-liquid instability) are thus of relevance to the process of chemical gelation. Syneresis as a “echo” of the equilibrium phase separation ? Thank you for your attention !

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

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

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

Angoli modelli Tetrahedral Angle Distribution

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

PMS Structure (r-space)

Further check of the absence of loops in finite clusters

S(q) in the network region (PMW) C. De Michele et al, J. Phys. Chem. B 110, , 2006

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

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

DNA-Tetramers phase diagram Largo, J.; Starr, F. W.; FS,. Self-Assembling DNA Dendrimers: A Numerical Study Langmuir, 23, 5896, 2007

Isodiffusivities …. Isodiffusivities (PMW) ….

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

Noro-Frenkel Scaling for Kern-Frenkel particles G.Foffi and FS, JPCB 2007 Constant B 2 lines Constant bond-distance line

“Time” dependence of the potential energy (~p b ) around the predicted Wertheim value ground-state

T-dependence of the diffusion coefficient Cross-over to strong behavior in the network region ! Strong Liquids !!!

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

Functionality 4 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

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

Conclusions (II) 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.

Wertheim (in a nut-shell) (ideal gas of equilibrium loop-less clusters of independent bonds

Equilibration in the presence of branching (but no loops !) ( P. van Dongen and M. Ernst, J. Stat Phys 37, 301 (1984). ) At low T (irreversible aggregation)