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SimBioMa, Konstanz 2008
Francesco Sciortino Universita’ di Roma La Sapienza
“Models for colloidal gelation”
Introduzione
Coworkers:
Emanuela BianchiCristiano De MicheleJack Douglas (NIST) (M=2)
Piero TartagliaEmanuela Zaccarelli
Main Messages• Strongly interacting particles (u<<1)---with simple
spherical potentials -- at small and intermediate densities ---ALWAYS phase-separate (in a dense and dilute phase)
• Strongly interacting particles with LIMITED valence ---patchy particles, highly directional interactions, dipolar, quadrupolar --- form equilibrium open structures (GELS, network forming liquids). Empty liquids
• A parameter free description of self-assembly (both equilibrium and equilibration !) can be formulated joining Wertheim and Flory-Stockmayer theories for a class of patchy particles systems. Connections to chemical gels.
Outline• The fate of the liquid state (neglecting crystallization): phase
diagram of spherical and patchy attractive potentials
• A theory-of-liquid approach to self-assembly in equilibrium polymerization (linear and branched)
• The role of valence in controlling the width of the gas-liquid instability
• Physical and chemical gels
Phase diagram of spherical potentials*0.13<c<0.27 [if the attractive range
is very small ( <10%)]
*One component, “Hard-Core” *One component, “Hard-Core” plus attractionplus attraction
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Nature, in press
For this class of potentials arrest at low (gelation) is the result of a phase separation process interrupted by the glass transition
CONFOCAL IMAGES (THE REAL STUFF!)
How to go to low T at low (in metastable equilibrium)
reducing “valence”
How to suppress phase separation ?
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)
It enforces the one bond per patch condition
Energy= Number of bonds = bond probability
Pine’s particles
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 Yang J. Am. Chem. Soc.; 2005; 127(45) pp 15968 - 15975;
Wertheim TPT for associated liquids(particles with M identical sticky sites )
At low densities and low T (for SW)…..Vb
Wertheim in a nut-shellAppendix A: Bianchi et al
JCP (in press)
M=2 EQUILIBRIUM (Chains)
Symbols = Simulation
Lines = Wertheim Theory
<L>
FS et al J. Chem.Phys.126, 194903, 2007
Average chain length L
Chain length distributions
M=2 EQUILIBRATION (Growth of the Chains)
Low T limit:
FS, C. De Michele and J. DouglasGrowth of equilibrium polymers under non-equilibrium conditionsJ. Phys. Condensed Matter 20, 155101 (2008)
A snapshot of
<M>=2.025
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N3=330
N2=5670
T=0.05, =0.01
<M>=2.055
Wertheim theory predicts pb extremely well (in this model) !(ground state accessed in equilibrium !!!!!)
Emanuela Bianchi, Piero Tartaglia, Emilia La Nave and FS, Fully Solvable Equilibrium Self-Assembly Process: Fine-Tuning the Clusters Size and the Connectivity in Patchy Particle Systems, J. Phys. Chem. B 111, 11765 (2007).
Generic features of the phase diagramBranching introduces percolation and phase-separation!
Cvmax line
Percolation line
unstable
Connectivity properties and cluster size distributions: Flory and Wertheim
Flory-Stockmayercluster size distributionsobserved
Mixtures of particles with 2 and 3 bonds
Empty liquids !Cooling the liquids without phase separating!
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. ARREST DRIVEN BY BONDING INSTEAD OF PACKING (equilibrium gels !)
THE WIDTH OF THE GAS-LIQUID UNSTABLE REGION IS STRONGLY CONTROLLED BY THE VALENCE (empty liquids)
Equilibration (to a finite T) in the presence of branching (but no loops !)
(P. van Dongen and M. Ernst, J. Stat Phys 37, 301 (1984).)
At low T (irreversible coagulation)
At all times, the cluster size distribution is the same as the equilibriumone, but with p(t) instead of peq
The resulting equation for p(t) CAN be solved analytically !!!
Comparing simulation and theory
Evolution of the number of bondsfollowing a T-jump, starting fromhigh-T Quench
protocol
Irreversible aggregation in the absence of bond loops
Smoluchowski coagulation works !
Chemical Gels….. Quenchprotocol
Final Message:
In the absence of bond-loops, chemical gelation proceeds via a sequence of “quasi-”equilibrium steps (longer t --> smaller T)
The phase-diagram information (gas-liquid instability) are thus of relevance to the process of chemical gelation.
Syneresis as a “echo” of the equilibrium phase separation ?
Final Message:
In the absence of bond-loops, chemical gelation proceeds via a sequence of “quasi-”equilibrium steps (longer t --> smaller T)
The phase-diagram information (gas-liquid instability) are thus of relevance to the process of chemical gelation.
Syneresis as a “echo” of the equilibrium phase separation ?
Thank you for your attention !
Conclusions• Directional interaction and limited valency are essential ingredients for offering a DIFFERENT final fate to the liquid state and in particular to arrested states at low
• In the newly available density region, at low T the system forms a “equilibrium” gel (or a network glass).
• Equilibrium Gels and network forming liquids: two faces of the same medal.
• In the absence of bond-loops, chemical gelation proceeds via a sequence of quasi-equilibrium states
DNA-Tetramers phase diagram
Largo, J.; Starr, F. W.; FS,. Self-Assembling DNA Dendrimers: A Numerical Study Langmuir, 23, 5896, 2007
Noro-Frenkel Scaling for Kern-Frenkel particles
G.Foffi and FS, JPCB 2007
Constant B2 lines Constant bond-distance line
T-dependence of the diffusion
coefficient
Cross-over tostrong behavior in the network region !
Strong Liquids !!!
Functionality 4
One Component(water-like)
Binary mixture
(silica-like)
DNA gel model (F. Starr and FS, JPCM, 2006J. Largo et al Langmuir 2007 )
BondSelectivity
StericIncompatibilities
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
A collection of phase diagramsof four-coordinated liquids
Physical Gels <===> Network forming liquids
Conclusions (II)• Directional interaction and limited valency are essential ingredients for offering a DIFFERENT final fate to the liquid state and in particular to arrested states at low
• In the newly available density region, at low T the system forms a “equilibrium” gel (or a network glass).
• Equilibrium Gels and network forming liquids: two faces of the same medal.