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