Conference on "Nucleation, Aggregation and Growth”, Bangalore January 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)…. Equilibrium Aggregation, 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) Debenedetti,Stillinger, Sastry

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

For this class of potentials arrest at low  (gelation) is the result of a phase separation process 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) How to suppress phase separation ? - Zaccarelli et al PRL 94, , Sastry et al JSTAT 2006

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

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

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

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

M=2 Cond-mat/ GC simulations (particles and chain insertions)

M=2 (Chains) Symbols = Simulation Lines = Wertheim Theory Cond-mat/ Average chain length Chain length distributions Energy per particle

Binary Mixture of M=2 and 3 N 2 =5670 N 3 =330 X 3 =0.055 =2.055 La Nave et al (in preparation)

=2.055 Wertehim theory predicts p b extremely well (in this model) ! (ground state accessed in equilibrium)

Connectivity properties and cluster size distributions: Flory and Wertheim

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

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

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

Patchy particles - Critical Parameters

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

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) Message

Connecting colloidal particles with network forming liquids

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

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

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

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

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

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

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

DNA-Tetramers phase diagram

Final Message: Universality Class of valence controlled particles

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

Verbal Summary 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. Are directional interactions (i.e. suppression of phase- separation) essential for being strong? Gels and strong liquids: two faces of the same medal.

Graphic Summary Two distinct arrest lines ? Strong liquids - Patchy colloids: Gels arrest line Fragile Liquids - Colloidal Glasses: Glass arrest line Fluid

Coworkers: Emanuela Bianchi (Patchy Colloids) Cristiano De Michele (PMW, PMS) Simone Gabrielli (PMW) Julio Largo (DNA, Patchy Colloids) Emilia La Nave, Srikanth Sastry (Bethe) Flavio Romano (PMW) Francis Starr (DNA) Jack Douglas (M=2) Piero Tartaglia Emanuela Zaccarelli

Unifying aspects of 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 T=0 !

Isodiffusivities …. Isodiffusivities (PMW) ….

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 ?

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

One last four-coordinated model !

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

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

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

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

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

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)

D along isotherms Diffusion Anomalies

Hansen

Water Phase Diagram  ~ 0.34 Do we need do invoke dispersion forces for LL ?

Mohwald

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

Geometric Constraint: Maximum Valency N max Model (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 Speedy-Debenedetti

Equilibrium Phase Diagram PSM

Pagan-Gunton Pagan and Gunton JCP (2005)

Equilibrium phase diagram (PMW)

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

PMS Structure (r-space)