Mainz, November 28 2006

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* “Hard-Core” plus attraction

0.13<c<0.27

[if the attractive rangeis 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 <= Nmax

HS if # of bonded particles > Nmax

V(r)

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’sparticle

Self-Organization of Bidisperse Colloids in Water DropletsYoung-Sang Cho, Gi-Ra Yi, Jong-Min Lim, Shin-Hyun Kim, Vinothan N. Manoharan,, David J. Pine, and Seung-Man YangJ. Am. Chem. Soc.; 2005; 127(45) pp 15968 - 15975;

Steric incompatibilities satisfied if SW width <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)

E. Bianchi et al, PRL, 2006

(N.B. Wilding)

~N+sE

Patchy particles - Critical Parameters

T=0.07M=2 (Chains)

Symbols = Simulation

Lines = Wertheim Theory

<M>=2.055

A snapshot of a <M>=2.025 (low T) case, =0.033

Ground State (almost)reached !

Bond Lifetime

~eu

Dipolar Hard Spheres…

Tlusty-Safram, Science (2000)

Camp et al PRL (2000)

Del Gado …..

Del Gado/Kob EPL 2005

Hansen

MESSAGE(S) (so far…):

REDUCTION OF THE MAXIMUM VALENCYOPENS A WINDOW IN DENSITIES WHERE THELIQUID CAN BE COOLED TO VERY LOW T WITHOUTENCOUNTERING PHASE SEPARATION

THE LIFETIME OF THE BONDS INCREASES ON COOLINGTHE LIFETIME OF THE STRUCTURE INCREASESARREST A LOW CAN BE APPROACHED CONTINUOUSLY ON COOLING (MODEL FOR GELS)

HOW ABOUT DYNAMICS ?HOW ABOUT MOLECULAR NETWORKS ? IS THE SAME MECHANISM ACTIVE ?

<M>=2.05

Slow Dynamics at low Mean squared displacement

=0.1

<M>=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 modelJ. Kolafa and I. Nezbeda, Mol. Phys. 161 87 (1987)

Hard-Sphere + 4 sites (2H, 2LP)Tetrahedral arrangement

H-LP interact via a SWPotential, of range 0.15 .

V(r)

r

(length scale)

(energy scale)

u0

Bonding is properly defined --- Lowest energy state is well defined

Equilibrium phase diagram (PMW)

Pagan and GuntonJCP (2005)

The PMS ModelFord, Auerbach, Monson, J.Chem.Phys, 8415,121

(2004)

Silicon

Four sites

(tetrahedral)

OxygenTwo sites

145.8 o

OO=1.6

SW interaction betweenSi sites and O sites

Equilibrium Phase Diagram PSM

Potential Energy -- Approaching the ground state

Progressive increase in packing prevents approach to the GS

Potential Energy along isotherms

Optimal densityHints of a LL CP

Phase-separation

S(q) in the network region

PMSStructure (r-space)

Structure (q-space)

E vs n

Phase-separation

Summary of static data

OptimalNetworkRegion

-Arrhenius

Approach toGround State

Regionof

phaseseparation

Packing Region

Phase Separation RegionPackingRegion

SphericalInteractions

PatchyInteractions

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 tostrong 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

Spinodals and isodiffusivity lines: PMW, PMS, Nmax

Analogies with other network-forming potentials

SPC/E ST2 (Poole)

BKS silica(Saika-Voivod)

Faster on compression

Slower on compression

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 SummaryTwo 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 Sconf(E(T))+E(T)+fbasin(E,T)

with

fbasin (E,T)

and

Sconf(E)=kBln[(E)]

Sampled Space with E bonds

Number of configurations with E bonds

It is possible to calculate exactly the vibrational entropy of one single bonding pattern

(basin free energy)

(Ladd andFrenkel)

•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

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 assembliesJACS 126, 2050 2004

Palindroms in complementary space

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

Optimaldensity

Bonding equilibriuminvolves a significantchange in entropy(zip-model)

Percolation close (in T) to dynamicarrest !

Final Message: Universality Class ofvalence 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 TartagliaEmanuela Zaccarelli

http://www.socobim.de/

Density Anomalies…(and possible 2’nd CP)

D vs (1-pb)

D vs (1-pb) --- (MC)

D ~ f04

~(Stanley-Teixeira)

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

Foffi aging

Hard SphereColloids: 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 PMSGC simulationBOX SIZE=TC=0.075C=0.0445 s=0.45

Critical Point of PMW GC simulationBOX SIZE=TC=0.1095C=0.153

(Flavio Romano Laurea Thesis)

E-Egs vs. 1/T

PMS -Potential Energy

Lattice-gas calculation forreduced valence (Sastry/La Nave/FSJ. Stat. Mech 2006)

PMW

D along isotherms

Diffusion Anomalies