Simple Model of glass-formation Itamar Procaccia Institute of Theoretical Physics Chinese University of Hong Kong Weizmann Institute: Einat Aharonov, Eran.

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Simple Model of glass-formation Itamar Procaccia Institute of Theoretical Physics Chinese 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 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.

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

Rigorous Results (J.P. Eckmann and I.P., PRE, 78, (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!

First result :

Specific heat anomalies

The anomalies are due to micro-melting (micro-freezing of crystalline clusters) We have an equation of state !!!

Summary The ‘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)