Glass transition in short ranged attractive colloids:

theory, application and perspective

Giuseppe Foffi

Universita’ “La Sapienza”, RomaINFM, Roma

Utrecht University

31/10/2002

Overview of the problem

• Model colloidal particles characterized by an hard core interaction present a kinetic slowing down at high volume fraction.

• This phenomenon has been interpreted within Mode Coupling Theory (MCT) for glass transition.

• As for the phase diagram, once an attraction is considered the scenario becames much richer and new unpredictable features emerge. In particular the range of the attraction plays a key role.

• Some evidence of these peculiar behaviours has been find in simulation and experimental studies.

The ideal glass transition

• In a normal liquid the correlation function decays exponantially. The systems loses memory of the initial configuration after a characteristic time

• Aproacching the glass transition the time scale of the relaxation grows. The typical “two steps” decaying emerges.

• At the ideal glass transition:

The system is frozen

Mode Coupling Theory

If the system is liquid

If the system is a glass

Sq can be calcualted in different ways:• Via integral equation theory• Via simulation• Using experimental data

W.Gotze in Liquids, Freezing and Glass Transition edited by J.P. Hansen , D. Levesque D, and J. Zinn-Justin (Amsterdan: North-Holland) p~287, 1991.

Hard Spheres System (HSS)

• Hard spheres present a a fluid–solid phase separation due to entropi effects

•Experimentally, at =0.58% the system freezes forming disordered agregates. (1)

• MCT understimates the location of the glass transition of about 6% in packing fraction. (2)

• If a scaling for the density is applied, MCT gives very good results for the dynamical properties (3)

MCT transition=51.6%

1. W. van Megen and P.N. Pusey Phys. Rev. A 43, 5429 (1991)

2. U. Bengtzelius et al. J. Phys. C 17, 5915 (1984)

3. W. van Megen and S.M. Underwood Phys. Rev. Lett. 70, 2766 (1993)

Experimental Results for HHS

Once the location of the glass

line has been rescaled. MCT

accounts for experimental data

within 15% accuracy level.

1. W. van Megen and S.M. Underwood Phys. Rev. Lett. 70, 2766 (1993)

Why a glass?At high denisity the system gets so packed that each particle is blocked in a cage formed by its neighbours.

This is is the cage effect. The two time scales of the relaxation reflect this phenomenon.

Adding attractions

• The presence of attraction modifies the behaviour of the system: New phases and their coexistence emerge.

• With narrow interactions the appeareance of metastable liquid-liquid critical point is typical for colloids.

V.J. Anderson and H.N.W. Lekkerkerker Nature 416, 811 (2002)

Gel Line

1) H. Verduin and J. K. G. Dhont, J. Colloid Interface Sci. 172, 425 (1995).2) J. Bergenholtz and M. Fuchs, Phys. Rev. E 59, 5706 (1999).3) L. Fabbian et al. Phys. Rev. E 59, 1347 (1999)

This results (1) suggested the

possibility to use glass

theories to understand gel

transition taking into account

attraction (2-3).

Square Well System (SWS)

The system model presents a larger number of control parameters then hard spheres:

• density

• =1/KbT Temperature (Non Adsorbing-Polymer concentration)

• =/(d+) Range of the attraction

Technical details

• The structure factor Sq has been calculated within PercusYevick approximation (PYA). We have tested results with simulations and the agreement is satisfactory.

• MCT equation has been numerical integrated to obtain the non ergodicity parameter fq.

• The ideal glass line is the locus where fq becomes non-vanishing.

• When a great accuracy was necessary, the eigenvalues of the stability matrix of the equation has been used as guide line for the glass transition

The ideal glass transition line

• It goes to the right Hard spehere limit

• It present a low density branch which passes very clossed to the spinodal line.

Reentrant glass line

• The SWS ideal glass line shows a strong dependece on the range of the potenial.

• A reentrant behaviour emerges: increasing the range this effect diminuishes and eventually disappears

• The system presents a liquid pocket between two glasses.

What is the effect of attraction?

Long Range Attraction

Short Range

Attraction

Glass-Glass line

Mean Square Displacement

(0.1)2

2

t

m.s

.d.

Mechanical Properties

Dynamical Behaviour• In the proximity of an A3 singularity, MCT predicts a new dynamical feature: the logarimtic decay of the correlators.

• The effect of this decaying disappears in the proximity of the liquid-glass transition (A2 singularity).

• The logaritmic decay extendes over a much longer range of time in the proximity of an A4 singularity

Summarizing

In the short range SWS we encounteredthree main features:

1. A reentrant glass line, i.e. a system that, increasing the attraction, passes from a glass to a liquid and then to a new glass

2. A special dynamical behaviour , i.e. a logaritmic decay of the correlation function

3. A glass-glass transition line between to different glassy states due to the competition between attraction and repulsion on short length scale

4. A low density branch which could be seen as gel structure

WHAT CAN WE EXPECT TO BE TRUE IN THIS?

Reentrant behaviour

• Experimentally it has been shown that a glass can be formed either by cooling or heating the same sistem (1-2). So there can be a reentrant behaviour.

• A first numerical approach has shown to agree with this picture (3) in the case of a short range Asakura-Oosawa like potential

• We performed extensive Molecular Dynamics simulation for the SWS, in order to confirm this prediction.

1) K.N. Pham et al. Science 296, 104-106 (2002) 2) T. Eckert and E. Bartsch Phys. Rev. Lett. 89, 125701 (2002)

3) A.M. Puertas et al. Phys. Rev. Lett. 88, 098391 (2002)

Simulation Results

Simulation Details:

• 1237 Particles

• =0.03

• D0 diffusivity of the relative HS system

Extension to Binary Mixtures

NA=350

NB=350

d1/d2=1.2

i=0.03

Logarithmic decay

• Experimentally it has been found in micellar systems (1)

• In simulation has been seen a logartmic decay of the correlation function in Osakura Osawa system(2).

• Again we performed the simulation for the binary Square well system. Confirming this behaviour.

1) F. Mallamace et al. Phys. Rev. Lett. 84, 5431 (2000)

2) A.M. Puertas et al. Phys. Rev. Lett. 88, 098391 (2002)

Glass-Glass transition• No experimental nor numerical results yet.

• Work in progress for the simulation.

• Very difficult situation.

Gel Line• Even if the low density glass line presents analogy with the gel line the theory probably is not so reliable.

• Experimental gel lines show a similar behaviour to MCT ideal glass transition (1), but numerical studies would give a better answer

Conclusions (1)

Phase equilibria and glass transition in Hard-Core Yukawa (HCY)

• We have studied the interplay between phase-equilibria and the ideal glass transition for a system of hard spheres interacting with a short ranged yukawa potential potential.

• The aim is to propose a possible exploration of the protein crystallization problem within glass theories. In this sense the disordered precipitates or aggregates are regarded as glassy structures.

Hard Sphere Yukawa System

b screening parameter

Diameter

energy depth

Technical DetailsFor the fluid phase:HCY model has been solved by a Self Consistent Ornestein-Zernike Approximation (SCOZA)(1):• It gives a good static structure factor Sq (input for the MCT equations) • It takes into account thermodynamical constrains. It is genraly in good

agreement with simulation

:

Where:

• F0 hard spheres free energy (Hall)

• g0(r) hard spheres radial distribution function (Kincaid and Weiss)

For the Solid Phase (assumed to be FCC):We used a high temperature perturbation theory

1) D. Pini, G. Stell and N. B. Wilding , J. Chem.Phys 115, 2702 (2001).

The large b values

The narrow range

Why narrow range attraction?

Long Range Attraction

Short Range

Attraction

And for the SWS?

Conclusions (2)

• Our phase diagrams show all the typical features of the short ranged attractive systems: a meta-stable liquid-liquid phase separation and an isostructural solid-solid coexistence

• The glass line follows the equilibrium phase diagram, suggesting a possible interplay between equilibrium and non equilibrium phenomena

• The location of the phase boundaries and of the ideal glass transition show clear analogies with the crystallization behaviour of some proteins systems. (1)

CollaborationAll the work presented is the results of close collaboration betweenRome and Dublin (University College of Dublin).

Rome:E.Zaccarelli, F. Sciortino, P.Tartaglia

Dublin:K.A. Dawson

Also:

For the Hard Core Yukawa model:G. Stell (State University Of New York)D. Pini (INFM, Milan)A. Lawlor, G. D. McCullagh (Dublin, UCD)

For the SWS model and good advise on MCT:

W.Gotze, M.Sperl,Th. Voitgman and M.Fuchs (Munich)

For the simulations results:

S.V. Buldyrev (Boston University)

References

K. A. Dawson, G. Foffi, M. Fuchs, W. Gotze, F. Sciortino, M. Sperl, P. Tartaglia, Th. Voigtmann and E. Zaccarelli, Phys. Rev. E 63, 011401 (2000). (SWS Model)

E. Zaccarelli, G. Foffi, P. Tartaglia, F. Sciortino and K. A. Dawson, Phys. Rev E, 63, 031501 (2001). (Mechanical properties)

G. Foffi, G. D. McCullagh, A. Lawlor, E.Zaccarelli, K. A. Dawson, F. Sciortino, P. Tartaglia D.Pini and G. Stell, Phys. Rev. E, 65,931407 (2002) (Yukawa Model)

G. Foffi, K. A. Dawson, S.V. Buldyrev, F. Sciortino, E.Zaccarelli and P. Tartaglia, Phys. Rev. E 65, 050802 (2002). (One component Simulation)

E.Zaccarelli, G. Foffi, K. A. Dawson, S.V. Buldyrev, F. Sciortino and P. Tartaglia, Phys. Rev. E 65, 050802 (2002). (Two component Simulation)