Introduzione

Francesco Sciortino Universita’ di Roma La Sapienza

“Self-Assembly of patchy particles and DNA-functionalized

dendrimers”

Introduzione

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Motivations and outline of the talk

Self-Assembly requires interaction energies larger than kT (u<<1). Long lifetime of the assembly

Need to study the phase behavior at low T (crystal

formation, phase separation, dynamic arrest, gelation)

Spherical interacting potentials Limited valence potentials (patchy) DNA-functionalized particles Reversible and Irreversible aggregation

Main Messages (and outline of the talk)• 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 (equilibrium gels, network forming liquids). Models for self-assembly Empty liquids

• For (small valence) patchy particles, a parameter free description of self-assembly (both equilibrium and equilibration !) can be formulated joining Wertheim and Flory-Stockmayer theories. Connections to chemical gels (and supramolecular chemistry)

• Valence controlled universality (DNA-dendrimers)

Phase diagram of spherical potentials*0.13<c<0.27

*One component, “Hard-Core” *One component, “Hard-Core” plus attractionplus attraction

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



Two possibilities, on reducing the range of interaction (depletion interactions, proteins)

Simulations Supported(Foffi et al PRL 94, 078301, 2005)

Contradictory exp resultsMCT (Fuchs, Bergenholtz)







For depletion interactions, arrest at low (gelation) is the result of a phase separation process interrupted by the glass transition

CONFOCAL IMAGESFirst Order Transition But.. Where are we ?









How to estimate the position of the system in the phase diagram ?

Use connectivity information (cluster size distributions)

Which potential ?

Use B2* scaling

(Noro-Frenkel)







Cp <---> B2*

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Spinodal Decomposition Behavior before Arrest (S(q) from confocal)



Spinodal Decomposition Behavior before Arrest !

q 1 =d

q S

(q)

q

Gels resulting from arrested phase separation (interrupted by the glass transition)

arrested dense phase

quench

Non-equilibrium route to gelation

From Zaccarelli, Topical Review JPCM 19, 323101 (2007)

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 )

Wertheim in a nut-shellAppendix A: Bianchi et alJ. Chem. Phys. 128, 144504 (2008)


At low densities and low T (for SW)…..Vb



At low densities and low T (for SW)…..Vb


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)

`




Low T limit:


Same l

Same

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

E. Bianchi, J. Largo, P. Tartaglia,E. Zaccarelli, FSPhase diagram of patchy colloids:towards empty liquidsPhys. Rev. Lett. 97, 168301, 2006

First Summary• 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

• Possibility to reach (in homogeneous conditions) states where u>>1 and the bond lifetime is large

• In the newly available density region (whose with is controlled by the valence), at low T the system forms a “equilibrium” gel

DNA functionalized particles: modulating the interaction

DNA-dendrimers

Four Arm Ologonucleotide Complexes as precursors for the generation of supramolecular periodic assembliesJACS 126, 2050 2004

Our minimal model: Selectivity of the bonding

“DNA” : chain of WAC-LJ (purely repulsive) particles, chained by a FENE potential. Three-body bending potential to model chain rigidity

Bases modeled as labeled sites (A,T,G,C), constrained (FENE) to stay at a fixed distance from the monomer center. Site-site interactions are WAC-LJ (purely repulsive) between non-complementary bases andLJ for (A-T) and (G-C) pairs.

Base-sites

F.W. Starr and FSModel for assembly and gelation of four-armedDNA dendrimersJ. Phys. Cond. Matt. 18, L347-L353, 2006

Selectivity of the bonding (single bond per arm !)

A

T

C

C

C

A

A

A

C

T

T

T

G

G

G

G

Single strand Double Strand

“DNA”-pairing (ss-ds) transition

Small T-range where transition takes place

“DNA”-Tetramers (kinetic)phase diagram

Largo, J.; Starr, F. W and FS.Self-Assembling DNA Dendrimers: A Numerical StudyLangmuir, 23, 5896-5905. 2007

(“homogeneous”gel !!!)

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 models

F. SciortinoGel-forming patchy colloids and network glass formers: thermodynamic and dynamic analogiesEur. Phys. J. B e2008-00034-0 (2008)

A collection of phase diagramsof four-coordinated liquids

Physical Gels <===> Network forming liquids

F. SciortinoGel-forming patchy colloids and network glass formers: thermodynamic and dynamic analogiesEur. Phys. J. B e2008-00034-0 (2008)

Message: Valence “fixes” the phase diagram type

Opening of a region of intermediate densities where the system can be cooled down without the intervention of phase separationArrest driven by “bonding” more than by “packing”

Possibility of interpreting the behavior of functionalized particlesin the same “spirit” as patchy colloids

Analogies between “patchy particles” and “network liquids”Physical gels <---> network forming liquids



Il primo amore non si scorda mai…… (P. H. Poole, FS, U. Essmann, H. E. StanleyPhase behavior of metastable water Nature 360, 324-328, 1992)

Liquid-liquid critical point in one-component systems (water!)







P.H. Poole, I. Saika-Voivod and FSDensity minimum and liquid-liquid phase transitionJ. Phys. Cond. Matt.17, L431-L437, 2005

Compressibility

Specific heat

density

Liquid-gas spinodal

Liquid-liquid spinodal

Can tetrahedral DNA-functionalized particles display the phenomenology which has been proposed for supercooled water ?

Can this teach us something on the mechanism behind the existence of a liquid-liquid phase transition in network forming systems ?

Four “tetrahedral” bonds in both cases: but hard and soft cores….

Julio Largo, Piero Tartaglia, and FSEffective nonadditive pair potential for lock-and-key interacting particles: The role of the limited valencePhys. Rev. E 76, 011402 2007

Effective potential for DNA-functionalized nanoparticles



Three Ising-like critical points in the effective potential !!!!



Second Messages (summary)• A challenge for functionalized nanoparticles: bonding

selectivity can provide an effective route to generate new networked materials with polyamorphic behavior.

• The interpenetration of networks provides an alternative simple and generic mechanism to understand the generation of multiple liquid phases, such as expected in water, silicon, silica and several other network forming fluids.

• it is possible to expand the potential tool-box of building blocks so as to open the possibility of a hierarchy of amorphous networked phases. With a proper mixing of particles functionalized with distinct sequences of DNA it should also be possible to control the properties of the individual interpenetrating networks, realizing a nanoscopic canvass woven by differently ”colored” DNA wires.

Self assembly of branched structures…..

Connecting time in chemical (irreversible)aggregation and temperature in physical (reversible) one.

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 all times, the cluster size distribution is the same as the equilibriumone, but with p(t) instead of peq

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(for patchy spheres)

QuenchprotocolEvolution of the number of bonds

following a T-jump, starting fromhigh-T

Same densityDifferent T

Same TDifferent

Irreversible aggregation in the absence of bond loops

Chemical Gels….. Quenchprotocol

Irreversible aggregation in the absence of bond loops

Smoluchowski coagulation works !

Chemical Gels….. Quenchprotocol

Chemical and physical gelation (in the absence of loops)

t <---->TAt p(t) =p(T)

Chemical and physical gelation (in the absence of loops)

t <---->TAt p(t) =p(T)

A zero-th order model inspired by epoxy-resin step polymerization

Comparing Simulation and Theory for patchy ellipsoids: Cluster size distributions following a quench



Comparing Simulation and Theory for patchy ellipsoids: Evolution of the bond probability



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 and swelling as a “echo” of the equilibrium phase separation.

Coworkers:Emanuela Bianchi, Cristiano De MicheleFabio Ciulla, Piero Tartaglia, Emanuela Zaccarelli (Roma !)

Silvia Corezzi (Perugia)

Francis Starr, Chia W. Hsu (Wesleyan) Julio Largo (U. Cantabria, Santander)

Jack Douglas (NIST)

Peter Lu, David Weitz (Harvard)

Summary: routes to gels

arrested phase separation: non-equilibrium route

Equilibrium routes to gelation:with long-range repulsion / with patches

Zaccarelli, JPCM 19, 323101 (2007)

Next: Depletion interactionsadd linear polystyrene in good solvent

Two sets of polymers: 1) Rg=33nm; 2) Rg=10nminvestigated size ratios =0.059 =0.018

Investigated packing fractions: =0.045, 0.13, 0.16

Experimental system

PMMA particles (A. Schofield) radius a=560 nm density-and-index-matched solvent mixture (DHN+CXB) dielectric constant 7added 4mM TBAC: Debye length 12nm

Peter J. Lu and D.A. Weitz

Confocal microscopy: for each fluid state 26 3d-stacks are collected @ 10 frame/s; for the gel states 3d-stacks are collected over time.

<M>=2.05

Slow Dynamics at low Mean squared displacement

=0.1

<M>=2.05 =0.1

Slow Dynamics at low Collective density fluctuations

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

ground-state

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.

Structure: center-center radial distribution function

()

Low T

High T

Structure: center-center structure factor

Dynamics (MSD)

(a.u.) (a.u.)

2 )

Effective-potential based on center-to-center distance r and angular orientation of the two dendrimers

A two-state model: bonded and non-bonded configurations

Julio Largo, Piero Tartaglia, and FSEffective nonadditive pair potential for lock-and-key interacting particles: The role of the limited valencePhys. Rev. E 76, 011402 2007

Two-state model effective potential:

Non-Spherical effective potential:

Spherical effective potential:

A look at the resulting effective potential

Soft

Second Message:

Selectivity in the bonding at microscopic level MUST BEretained in any coarse graining procedure. This may be encoded in a many-body contribution to the potential.

DNA-functionalized particles are particularly suitable to realize limited valence systems.

Julio Largo, Piero Tartaglia, and Francesco SciortinoEffective nonadditive pair potential for lock-and-key interacting particles: The role of the limited valencePhys. Rev. E 76, 011402 2007

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)

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