Mainz November 28 2006 Gelforming patchy colloids and

Mainz, November 28 2006 Gel-forming patchy colloids and network glass formers: Thermodynamic and dynamic analogies Francesco Sciortino

Motivations • The fate of the liquid state (assuming crystallization can be prevented)…. Gels and phase separation: essential features (Sticky colloids - Proteins) • Thermodynamic and dynamic behavior of new patchy colloids • Revisiting dynamics in network forming liquids (Silica, water…. ) • Essential ingredients of “strong behavior” (A. Angell scheme).

Liquid-Gas Spinodal Glass line (D->0) Binary Mixture LJ particles “Equilibrium” “homogeneous” arrested states only for large packing fraction (see also Debenedetti/Stillinger)

Phase diagram of spherical potentials* 0. 13<fc<0. 27 [if the attractive range is very small ( <10%)] * “Hard-Core” plus attraction

Gelation (arrest at low f) as a result of phase separation (interrupted by the glass transition) T T f f

How to go to low T at low f (in metastable equilibrium) ? Is there something else beside Sastry’s scenario for a liquid to end ? How to suppress phase separation ? -controlling valency (Hard core complemented by attractions) -l. r. repulsion (Hard core complemented by both attraction and repulsions

Geometric Constraint: Maximum Valency V(r ) (E. Zaccarelli et al, PRL, 2005) SW if # of bonded particles <= Nmax HS if # of bonded particles > Nmax r

Nmax phase diagram

Patchy particles Hard-Core (gray spheres) Short-range Square-Well (gold patchy sites) No dispersion forces The essence of bonding !!!

Pine’s particle

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;

Steric incompatibilities satisfied if SW width d<0. 11 No double bonding Single bond per bond site

Wertheim Theory

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

Mixtures of particles with 2 and 3 bonds Empty liquids !

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

Patchy particles - Critical Parameters

M=2 (Chains) Symbols = Simulation Lines = Wertheim Theory T=0. 07

<M>=2. 055

A snapshot of a <M>=2. 025 (low T) case, f=0. 033 Ground State (almost) reached ! Bond Lifetime ~ebu

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

Del Gado …. . Del Gado/Kob EPL 2005

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 f CAN BE APPROACHED CONTINUOUSLY ON COOLING (MODEL FOR GELS) HOW ABOUT DYNAMICS ? HOW ABOUT MOLECULAR NETWORKS ? IS THE SAME MECHANISM ACTIVE ?

Slow Dynamics at low F Mean squared displacement <M>=2. 05 T=0. 05 F=0. 1

Slow Dynamics at low F Collective density fluctuations <M>=2. 05 F=0. 1

Message: Gel dynamics: dynamic arrest due to percolation (in the limit of long-living bonds).

The PMW model J. Kolafa and I. Nezbeda, Mol. Phys. 161 87 (1987) V(r) Hard-Sphere + 4 sites (2 H, 2 LP) Tetrahedral arrangement r u 0 (energy scale) H-LP interact via a SW Potential, of range l=0. 15 s. s (length scale) Bonding is properly defined --- Lowest energy state is well defined

Equilibrium phase diagram (PMW)

Pagan and Gunton JCP (2005)

The PMS Model Ford, Auerbach, Monson, J. Chem. Phys, 8415, 121 (2004) Silicon Four sites (tetrahedral) Oxygen Two sites 145. 8 o SW interaction between Si sites and O sites s. OO=1. 6 s 1/2 l=[1 -3 /2]s

Equilibrium Phase Diagram PSM

Potential Energy -- Approaching the ground state Progressive increase in packing prevents approach to the GS

Potential Energy along isotherms Phase-separation Optimal density Hints of a LL CP

S(q) in the network region

PMS Structure (r-space)

Structure (q-space)

Phase-separation

Summary of static data Spherical Interactions Patchy Interactions Phase Separation Region of phase separation Optimal Network Region Arrhenius Approach to Ground State Packing Region

How About Dynamics (in the new network region) ?

Dynamics in the Nmax=4 model (no angular constraints) Strong Liquid Dynamics !

Nmax=4 phase diagram - Isodiffusivity lines Zaccarelli et al JCP 2006

PMW -- Diffusion Coefficient Cross-over to strong behavior

Isodiffusivities …. (PMW) …. Isodiffusivities

Diffusion PMS De Michele et al, cond mat

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

Spinodals and isodiffusivity lines: PMW, PMS, Nmax

Analogies with other network-forming potentials ST 2 (Poole) SPC/E Faster on compression BKS silica (Saika-Voivod) Slower on compression

Tetrahedral Angle Distribution

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

Water Phase Diagram F ~ 0. 34 Do we need do invoke dispersion forces for LL ?

Comments • Directional interaction and limited valency are essential ingredients for offering a new final fate to the liquid state and in particular to arrested states at low f • The resulting low T liquid state is (along isochores) a strong liquid. The bond energy scale: is bonding essential for being strong ? . • Gels and strong liquids are two faces of the same medal.

Graphic Summary Two glass lines ? Fragile Liquids Colloidal Glasses Strong liquids - Gels Arrest line

Appendix I • Possibility to calculate exactly potential energy landscape properties for SW models (spherical and patcky) Moreno et al PRL, 2005

Thermodynamics in the Stillinger-Weber formalism F(T)=-T Sconf(E(T))+E(T)+fbasin(E, T) with fbasin (E, T) Sampled Space with E bonds and Sconf(E)=k. Bln[W(E)] Number of configurations with E bonds

It is possible to calculate exactly the vibrational entropy of one single bonding pattern (basin free energy) (Ladd and Frenkel)

Non zero ground state entropy ex ex • Comment: In models for fragile liquids, the number of configurations with energy E has been found to be gaussian distributed

Appendix II • Percolation and Gelation: How to arrest at (or close to) the percolation line ? F. Starr and FS, JPCM, 2006

Colloidal Gels, Molecular Gels, …. and DNA gels Four Arm Ologonucleotide Complexes as precursors for the generation of supramolecular periodic assemblies JACS 126, 2050 2004 Palindroms in complementary space

DNA gel model (F. Starr and FS, JPCM, 2006)

Optimal density Percolation close (in T) to dynamic arrest ! Bonding equilibrium involves a significant change in entropy (zip-model)

Final Message: Universality Class of valence controlled particles

Coworkers: Emanuela Bianchi (Patchy) Cristiano De Michele (PMW, PMS) Simone Gabrielli (PMW) Julio Largo (DNA, Patchy) Emilia La Nave, Srikanth Sastry (Bethe) Angel Moreno (Landscape) Flavio Romano (PMW) Francis Starr (DNA) Piero Tartaglia Emanuela Zaccarelli

http: //www. socobim. de/

Density Anomalies… (and possible 2’nd CP)

D vs (1 -pb)

D vs (1 -pb) --- (MC) D ~ f 04 ~(Stanley-Teixeira)

G. Foffi, E. Zaccarelli, S. V. Buldyrev, F. Sciortino, P. Tartaglia Aging in short range attractive colloids: A numerical study J. Chem. Phys. 120, 1824, 2004