Francesco Sciortino Dynamical Arrest of Soft Matter and Colloids Lugano April (MRTN-CT ) Tutorial: Recent developments in understanding gelation in colloidal systems

coherent mass consisting of a liquid in which particles are either dispersed or arranged in a fine network throughout the mass. A gel may be notably elastic and jellylike (as gelatin or fruit jelly), or quite solid and rigid (as silica gel). (Britannica) What is a gel ?

Recent reviews Cipelletti, Luca; Ramos, Laurence, Slow dynamics in glassy soft matter Journal of Physics: Condensed Matter, 17, R253-R285 (2005); Slow dynamics in glasses,gels and foams, Current Opinion in Colloid and Interface Science 7 (2002) F. Sciortino and P. Tartaglia Glassy colloidal systems Advances in Physics 54, Future reviews K.A. Dawson The glass paradigm for colloidal glasses, gels, and other arrested states driven by attractive interactions, Coll. Int. Sci 7, V. Trappe, Colloidal gels -- low-density disordered solid-like states lassy colloidal systems Current Opinion in Colloid and Interface Science 8 (2004) 494.

--- Low packing fraction of the dispersed phase --- (visco) elastic properties (ability to sustain stress): The need of a “spanning” network (attraction between particles is requested) Two main conditions

Chemical Gels System with a fixed number of bonds (with infinite lifetime) connecting the dispersed particles rubberepoxy-resins Richard A.L. Jones, Condened Matter. Oxford

Percolation theory! Cluster Size distribution (cluster shape) Infinite Cluster (d.c. conductivity, elasticity) Bond percolation in a two-dimension square lattice

C 1 =z p Bond probability=p C 2 = (z-1) p C 1 C 3 = (z-1) p C 2 C N = [(z-1) p] N-1 C 1 (z-1) p c =1 ……………… Critical Value !!! A Caley-Tree with connectivity z=4 Flory-Stockmayer (mean field solution )

Predictions (close to p c ) : suscettibility magnetization (order parameter) Bethe:  =2.5,  =0.5, d f =4 3d (approx):  2.18,  =0.45, d f =2.53 Stauffer Phys. Rep. 1979

Colloids (Lu’s Talk) Polymers-Biopolymers Proteins (Cardineaux and Zaccarelli’s Talk) Dna-coated colloids (Largo’s Talk) Reversible Bonding ---- Bond Lifetime Persistence of the spanning network Equilibrium Thermodynamics ---- Physical Gels

Gels resulting from irreversible processes (kinetic pathways are important) (phase separation) “Ideal” gels. Gels in which dynamic arrest is progressively approached (and the system is (as close as possible) to thermodynamic equilibrium). [ Experimental timescales < Bond lifetime < equilibration time at geometric percolation]. Routes to Physical Gelation

Phase diagram of spherical potentials * Hard-Core plus attraction 0.13<  c <0.27

Coniglio-Klein Bond Lifetime Static Percolation -- Gelation ???

Two (times 2 !) ways to go to low T (at low  ) Suppress phase separation -valency -l.r. repulsion -DLCA -Glass Arrest

Diffusion Limited Cluster Aggregation (a T->0 phase separation) Particles (and clusters) performing brownian motion and sticking with probability one. Diffusion coefficient of the cluster proportional to M -  (DLVO potential without and with salt).

“monodisperse” fractal objects (d f = 1.9) Gels as space-filling of sticking clusters M~M 1 (R/R 1 ) df The cluster average density decreases with R df-d V occ /V=  (R/R 1 ) d-df R=R 1   df-d) Basic DLCA findings

Structure of DLCA gels

The dynamical rules defining DLCA are the T->0 limit of the lattice gas dynamics. M. Carpineti and M. Giglio, Phys. Rev. Lett (1992) F. Sciortino and P. Tartaglia, Phys. Rev. Lett (1995) P. Pouline, J. Bibette and D. A. Weitz, Eur. Phys. J. B 277 (1999) C. Gimel, T. Nicolai, D. Durand, Phys. Rev. E (2002). MC (Lattice Gas) e -  u Gimel-Nicolai 1 if not bonded 0 if bonded { -  u= ln(1-p b ) DLCA and Spinodal Decomposition

Gels as arrested phase separation F. Sciortino et al, Phys. Rev. E 47, 4615 (1993). D. Sappelt and J. Jackle, Europhys. Lett. 37, 13(1997). M. Solomon and P. Varadan, Phys. Rev. E. 63 (2001) E. Zaccarelli et al, Unifying concepts in Granular Matter and Gels, Elsevier 2004 S. Manley et al, Phys. Rev. Lett. 95 (2005) E. Witman amd Z-G Wang, J.Phys. Chem B

Foffi aging G. Foffi et al, J. Chem. Phys. 122, , 2005 Arrested phase separation in a short-ranged attractive colloidal system: A numerical study

How to suppress phase separation ? Sticky patchy colloids ! Maximum Valency* --- Bond Selectivity * three-body interactions (Del Gado-Kob) E. Bianchi, Poster

Even more.. With mixtures of two and three sticky spots….

Del-Gado Kob Gel

N max =4 phase diagram - Isodiffusivity lines E.Zaccarelli et al, Phys. Rev. Lett. 94, , 2005 ; J. Chem. Phys. 124, (2006). C. De Michele et al, J. Phys. Chem. B, jp056380y (2006).

Analogies with other network-forming potentials SPC/E ST2 (Poole) BKS silica (Saika-Voivod)

Short-Range Attraction and Long-Range Repulsion (Yukawa) Vanishing of the “surface tension” ! Clusters as microphase

Short Range Attraction, --dominant in small clusters Longer Range Repulsion Competition Between Short Range Attraction and Longer Range Repulsion: Role in the clustering Importance of the short-range attraction: Only nn interactions

Gels in charged colloids FS,PT,EZ J. Phys. Chem. B 109, , 2005

proteins

What is the lowest  at which it is possible to form a gel In the absence of gravity Thermal stresses (DLCA, R c =100  m) On the earth Gravitational stresses (DLCA, R c =50  m) Interesting question Manley et al, PRL, (2004).

Viscosity Density autocorrelation functions Chemical Gels….. What is known Martin Libro Kon Physical Gels…. Cipelletti - Others Gel Dynamics

Cipelletti

Krall

Density Fluctuations (chemical gels) Below percolationAbove percolation Tagged particle properties: D finite at the transition and above I. Saika-Voivod et al, Phys. Rev. E 70, , 2004 Kurt Broderix et al, Phys. Rev. Lett. 79, 3688–3691 (1997),

Non ergodicity factors in chemical gelation pbpb

Connection Between Gels and Glasses