Simple Model of glass-formationSimple Model of glass-formation

Itamar Procaccia

Institute of Theoretical PhysicsChinese University of Hong Kong

Weizmann Institute: Einat Aharonov, Eran Bouchbinder, Valery Ilyin, Edan Lerner, Ting-Shek Lo, Natalya Makedonska, Ido Regev and Nurith Schupper .

Emory University: George Hentschel

CUHK September 2008

Glass phenomenology

The three accepted ‘facts’: jamming, Vogel-Fulcher, Kauzmann

A very popular model: a 50-50 binary mixture of particles interacting via soft repulsion potential

With ratio of `diameters’ 1.4

Simulations: both Monte Carlo and Molecular Dynamics with 4096 particles enclosed in an area L x L with periodic boundary conditions. We ran

simulations at a chosen temperature, fixed volume and fixed N. The units of mass, length, time and temperature are

Previous work (lots): Deng, Argon and Yip, P. Harrowell et al, etc: for T>0.5 the system is a “fluid”; for T smaller - dynamical

relaxation slows down considerably.

QuickTime™ and a decompressor

are needed to see this picture.

The conclusion was that “defects” do not show any ‘singular’ behaviour , so they were discarded as a diagnostic tool .

The liquid like defects disappear at the glass transition!

For temperature > 0.8

For 0.3 < T < 0.8

Associated with the disappearance of liquid like defects there is an increase of typical scale

QuickTime™ and a decompressor

are needed to see this picture.

Rigorous Results(J.P. Eckmann and I.P., PRE, 78, 011503 (2008))

The system is ergodic at all temperatures

Consequences: there is no Vogel-Fulcher temperature!

There is no Kauzman tempearture!

There is no jamming!

(the three no’s of Khartoum)

Statistical Mechanics

We define the energy of a cell of type i

Similarly we can measure the areas of cells of type i

Denote the number of boxes available for largest cells

Then the number of boxes available for the second largest cells is

The number of possible configurations W is then

Denote

A low temperature phase

Note that here the hexagons have disappeared entirely!

QuickTime™ and aCinepak decompressor

are needed to see this picture.

First result :

Specific heat anomalies

The anomalies are due to micro-melting (micro-freezing of crystalline clusters)

We have an equation of state !!!

SummarySummaryThe ‘glass transition’ is not an abrupt transition, rather a very smeared out

phenomenon in which relaxation times increase at the T decreases .

There is no singularity on the way, no jamming, no Vogel-Fulcher, no Kauzman

We showed how to relate the statistical mechanics and structural information in a quantitative way to the slowing down and to the relaxation functions.

We could also explain in some detail the anomalies of the specific heat

Remaining task: How to use the increased understanding to write a proper theory of the mechanical properties of amorphous solid materials. (work in progress).

Since nothing gets singular, statistical mechanics is useful

Strains, stresses etc.

We are interested in the shear modulus

Dynamics of the stress

Zwanzig-Mountain (1965)