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IV Workshop on Non Equilibrium Phenomenain Supercooled Fluids,
Glasses and Amorphous Materials
Pisa, September 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 !!!
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, in press
Mixtures of particles with 2 and 3 bonds
Empty liquids !
Patchy particles (critical fluctuations)
E. Bianchi et al, PRL, in press
(N.B. Wilding)
~N+sE
Patchy particles - Critical Parameters
Lattice-gas calculation forreduced valence (Sastry/La Nave) cond-mat
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
MESSAGE (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 MOLECULAR NETWORKS ? IS THE SAME MECHANISM ACTIVE ? HOW ABOUT DYNAMICS ?
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)
Critical Point of PMW GC simulationBOX SIZE=TC=0.1095C=0.153
(Flavio Romano Laurea Thesis)
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
Critical Point of PMSGC simulationBOX SIZE=TC=0.075C=0.0445 s=0.45
Potential Energy (# of bonds) for the PMW
Optimal density !
Potential Energy -- Approaching the ground state
Progressive increase in packing prevents approach to the GS
E-Egs vs. 1/T
Potential Energy along isotherms
Optimal densityHints of a LL CP
Phase-separation
S(q) in the network region
PMS -Potential Energy
PMS E vs 1/T
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
PMW -- Diffusion Coefficient
Cross-over tostrong behavior
D along isotherms
Diffusion Anomalies
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
Water Phase Diagram
~ 0.34
Do we need do invoke disperison forces for LL ?
Tetrahedral Angle Distribution
Energie Modelli
Low T isotherms…..
Coupling between bonding (local geometry) and density
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
The 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