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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
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Glass phenomenology The three accepted ‘facts’: jamming, Vogel-Fulcher, Kauzmann
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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.
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The conclusion was that “defects” do not show any ‘singular’ behaviour, so they were discarded as a diagnostic tool.
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The liquid like defects disappear at the glass transition!
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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
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Rigorous Results (J.P. Eckmann and I.P., PRE, 78, 011503 (2008)) The system is ergodic at all temperatures
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Consequences: there is no Vogel-Fulcher temperature! There is no Kauzman tempearture! There is no jamming! (the three no’s of Khartoum)
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Statistical Mechanics We define the energy of a cell of type i Similarly we can measure the areas of cells of type i
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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
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A low temperature phase Note that here the hexagons have disappeared entirely!
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First result :
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Specific heat anomalies
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The anomalies are due to micro-melting (micro-freezing of crystalline clusters) We have an equation of state !!!
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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
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Strains, stresses etc. We are interested in the shear modulus Dynamics of the stress
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Zwanzig-Mountain (1965)
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