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!)

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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;

DNA functionalized particles

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

FS et al J. Chem.Phys.126, 194903, 2007

EquilibriumPolymerization(no bond rings)

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)

What happens with (rear) branching ?

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!

Phase Diagram - Theory and Simulations

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

Chemical and physical gelation (in the absence of loops)

t <---->T

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 !

<M>=2.05

Slow Dynamics at low Mean squared displacement

=0.1

<M>=2.05 =0.1

Slow Dynamics at low Collective density fluctuations

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

Tetrahedral Angle Distribution

Energie Modelli

Low T isotherms…..

Coupling between bonding (local geometry) and density

PMSStructure (r-space)

Further check of the absence of loops in finite clusters

S(q) in the network region (PMW)

C. De Michele et al, J. Phys. Chem. B 110, 8064-8079, 2006

Structure (q-space)

C. De Michele et alJ. Chem. Phys. 125, 204710, 2006

E vs n

Phase-separation

Approaching the ground state (PMS)

DNA-Tetramers phase diagram

Largo, J.; Starr, F. W.; FS,. Self-Assembling DNA Dendrimers: A Numerical Study Langmuir, 23, 5896, 2007

Isodiffusivities ….Isodiffusivities (PMW) ….

Wertheim Theory (TPT): predictions

E. Bianchi et al, PRL 97, 168301, 2006

Noro-Frenkel Scaling for Kern-Frenkel particles

G.Foffi and FS, JPCB 2007

Constant B2 lines Constant bond-distance line

“Time” dependence of the potential energy (~pb) around the predicted Wertheim value

ground-state

T-dependence of the diffusion

coefficient

Cross-over tostrong behavior in the network region !

Strong Liquids !!!

Dipolar Hard Spheres…

Tlusty-Safram, Science (2000)

Camp et al PRL (2000)

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.

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Wertheim (in a nut-shell)(ideal gas of equilibrium loop-less clusters of independent bonds

Equilibration in the presence of branching (but no loops !)

(P. van Dongen and M. Ernst, J. Stat Phys 37, 301 (1984).)

At low T (irreversible aggregation)