freezing of liquids into crystalline and amorphous states · functional taylor series around the...
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Freezing of Liquids into Crystalline and Amorphous states
Shankar DasSchool of Physical SciencesJawaharlal Nehru UniversityNew Delhi 110067
UNIVERSITY OF FLORIDA, GAINESVILLE (2009)
Liquid to Crystal : Freezing transitionThermodynamic phase transition
Supercooled Liquid : Metastable.Sharp increase in relaxation time
The Glass transition
: depends on
Tg is fixed by the choice of
variation of Tg is not very sensitive to the expt time scale beyond a certain point.
The Angell Plot
The Kauzmann TemperatureAt extrapolated difference of entropy of the supercooled liquid and crystal goes to zero
Vibrational contribution to entropy same for crystal and the liquid stateConfigurational Entropy goes to zero at T=
Kauzmann paradox The existance of a kinetic spinodal : The deeeplysupercooled liquid will eventually crystallize
3N dimensional potential energy landscapeGoldstein picture (1969)Crystalline minimum and glassy minimaLocal rearrangents involving a small number n of particles Tx : crossover from barrier hopping to continuous liquid type motion.Often compared to the theoretical result of mode coupling theories and Tc
DYNAMIC MODEL obtained from a different theoreticalapproach using generalized hydrodynamics of liquids
Shankar P. Das : Reviews of Modern Physics 76, 785 (2004)
Glass and the CrystalNon-ergodic : Only vibrational motion
Strongly non-equilibrium state, Aging
Different temperatures
Characteristic temperatures for the supercooled liquid
The Stokes-Einstein relation
Single particle dynamics. Tagged particle motion. Einstein relation self-diffusion coefficient D Brownian motion : The macroscopic relaxation time is related to the viscosity of the liquidHolds as well for normal liquids.
Violated in the supercooled state
The Density Functional Theory of Freezing transition
The Density Functional Theory of Freezing Liquid Solid : Tm ( melting point ) Below Tm : The Liquid is less stable thermodynamically than the Crystal
Density is a crucial order parameter Uniform density - Liquid Non-uniform ( Inhomogeneous) Solid : Crystal
Inhomogeneous Solid Equivalent uniform system
FR E E E N E R G Y Functional of inhomogeneous density function n(r) Thermodynamic Variation principle is invoked
FREE ENERGY FUNCTIONAL F [n(r)] Equilibrium density function n(r)
Free energy functional : F = Fideal + Φ
Fideal : Ideal Gas Part, Entropic contribution.
Coarse grained description
Interaction part of the free energy Functional Taylor series around the uniform liquid state.
( Ramakrishnan-Yussouff Theory)
The second derivative of Φ is in the liquid state = Direct correlation function C(r)
Statistical mechanical model for inhomogeneous solidsLow order in a perturbation series
Poor approximation : ∆n(r) is large for the crystalline state
Weighted Density Functional Methods Effective Medium type approach
(Tarazona, Ashcroft et. al. )For computing thermodynamic properties the highly
nonuniform density of the crystal is replaced in terms of an weighted average with an weight function w(r)
Mapping
The non-uniform solid A low density Liquid
The functional f(n) is taken from liquid state theory
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