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

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).

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

Phase diagram of spherical potentials* * “Hard-Core” plus attraction 0.13<  c <0.27 [if the attractive range is very small ( <10%)]

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

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

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

N MAX -modified Phase Diagram N max phase diagram

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

Mohwald

Pine 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 ; Pine

Steric incompatibilities satisfied if SW width  <0.11 No double bonding Single bond per bond site Steric Incompatibilities

Wertheim Theory

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

Mixtures of particles with 2 and 3 bonds Wertheim Empty liquids !

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

Patchy particles - Critical Parameters

T=0.07 M=2 (Chains) Symbols = Simulation Lines = Wertheim Theory

=2.055

A snapshot of a =2.025 (low T) case,  =0.033 Ground State (almost) reached ! Bond Lifetime ~ e  u

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

Del Gado ….. Del Gado/Kob EPL 2005 Del Gado

Hansen

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 (MODEL FOR GELS) HOW ABOUT DYNAMICS ? HOW ABOUT MOLECULAR NETWORKS ? IS THE SAME MECHANISM ACTIVE ? Message

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

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

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 (1987) Hard-Sphere + 4 sites (2H, 2LP) Tetrahedral arrangement H-LP interact via a SW Potential, of range  0.15 . V(r) r  (length scale) (energy scale) u0u0 Bonding is properly defined --- Lowest energy state is well defined

Equilibrium phase diagram (PMW)

Pagan-Gunton Pagan and Gunton JCP (2005)

The PMS Model Ford, Auerbach, Monson, J.Chem.Phys, 8415,121 (2004) Silicon Four sites (tetrahedral) Oxygen Two sites o  OO =1.6   SW interaction between Si sites and O sites 

Equilibrium Phase Diagram PSM

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

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

S(q) in the network region

PMS Structure (r-space)

Structure (q-space)

E vs n Phase-separation

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

How About Dynamics (in the new network region) ?

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

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

PMW -- Diffusion Coefficient Cross-over to strong behavior

Isodiffusivities …. Isodiffusivities (PMW) ….

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 Question Compare ?

Phase Diagram Compared Spinodals and isodiffusivity lines: PMW, PMS, N max

Analogies with other network-forming potentials SPC/E ST2 (Poole) BKS silica (Saika-Voivod) Faster on compression Slower on compression

Angoli modelli Tetrahedral Angle Distribution

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

Water Phase Diagram  ~ 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  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 ? Strong liquids - Gels Arrest line Fragile Liquids - Colloidal Glasses

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 S conf (E(T))+E(T)+f basin (E,T) with f basin (E,T) and S conf (E)=k B ln[  (E)] Sampled Space with E bonds Number of configurations with E bonds Stillinger-Weber

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

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

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, Palindroms in complementary space DNA Gels 1

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

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

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

Density Anomalies… (and possible 2’nd CP) Density anomalies

D vs (1-p b )

D vs (1-p b ) --- (MC) D ~ f 0 4 ~(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 Foffi aging

Strong-fragile: Dire Stretched, Delta Cp Hard Sphere Colloids: model for fragile liquids

S(q) in the phase-separation region

Potential Energy (# of bonds) for the PMW Optimal density !

PMS E vs 1/T

Critical Point of PMS GC simulation BOX SIZE=  T C =0.075  C = s=0.45 Critical point PSM

Critical Point of PMW GC simulation BOX SIZE=  T C =  C =0.153 (Flavio Romano Laurea Thesis)

E-E gs vs. 1/T

PMS -Potential Energy

Lattice-gas calculation for reduced valence (Sastry/La Nave/FS J. Stat. Mech 2006)

R2 vs t PMW

D along isotherms Diffusion Anomalies