Biman Bagchi

Biman Bagchi

Indian Institute of Science, Bangalore, India.

Dr. Sarika BhattacharyyaMr. Arnab MukherjeeMr. Dwaipayan ChakrabartiDr. Rajesh Murarka

Adv. Chem. Phys. Vol. 116 (2001)

Plan of the talkPlan of the talk

Introduction (summary of some Introduction (summary of some experimental results).experimental results).

Theoretical description (de Gennes Theoretical description (de Gennes narrowing, self-consistent mode narrowing, self-consistent mode coupling theory), Applications of MCT coupling theory), Applications of MCT (Stokes-Einstein relation, Liquid (Stokes-Einstein relation, Liquid Crystals)Crystals)

Some computer simulation results.Some computer simulation results. More MCT (MCT of glass transition)More MCT (MCT of glass transition)

Basic FeaturesBasic Features

Rapid rise of viscosity with lowering of T in the Rapid rise of viscosity with lowering of T in the deeply supercooled liquid near the glass deeply supercooled liquid near the glass transition temperature. This rapid rise starts transition temperature. This rapid rise starts typically 30-50 deg C above Ttypically 30-50 deg C above Tgg..

This increase in viscosity can be described by This increase in viscosity can be described by Vogel-Fulcher expression. The same set of Vogel-Fulcher expression. The same set of parameters can describe the rise for 4-5 orders of parameters can describe the rise for 4-5 orders of magnitude.magnitude.

/( )GC T TAe

Basic Features Basic Features (continued)(continued)

Relaxation functions (that is, relevant Relaxation functions (that is, relevant time correlation functions) become time correlation functions) become markedly non-exponential in this markedly non-exponential in this temperature range. Stretched temperature range. Stretched exponential (KWW) form provides good exponential (KWW) form provides good fit with a low value of the exponent fit with a low value of the exponent (typically 0.5).(typically 0.5).

Computer simulation studies on binary Computer simulation studies on binary mixtures have shown that at high mixtures have shown that at high pressure and low temperature, pressure and low temperature, hoppinghopping becomes the effective mode of mass becomes the effective mode of mass transport and orientational relaxation. transport and orientational relaxation. The emergence of hopping is rather The emergence of hopping is rather gradual, that is, it coexists with gradual, that is, it coexists with continuous (Brownian) mode of diffusion continuous (Brownian) mode of diffusion until some low temperature where the until some low temperature where the latter ceases to contribute to diffusion.latter ceases to contribute to diffusion.

Stretched-exponential stress relaxation with decreasing temperature P*= at constant 10.0

ln C

s(t)

ln(t)

NPT Simulations of Binary Mixture AM + BB, JCP (2002)

Angel Plot

strong

Anomalous observations on Anomalous observations on dynamicsdynamics

Structural relaxation (diffusion) Structural relaxation (diffusion) decouples from mechanical decouples from mechanical

relaxation(viscosity)- fragility of a liquidrelaxation(viscosity)- fragility of a liquid

[C. A. Angell, Science 1995][C. A. Angell, Science 1995] Orientational relaxation remains coupled to viscosity [Ediger, JCP 1996]

Translational diffusion is decoupled from orientational diffusion [Sillescu, JCP 1996, Ediger]

Theories of Slow RelaxationTheories of Slow Relaxation

Mode Coupling TheoryMode Coupling Theory Adam-Gibbs-DiMarzio Adam-Gibbs-DiMarzio

Entropy Crisis Theory --- Entropy Crisis Theory --- Concept of Cooperatively Concept of Cooperatively Rearranging Region (CRR).Rearranging Region (CRR).

Energy Landscape PictureEnergy Landscape Picture Random first order theory Random first order theory

(RFOT)(RFOT)

How do molecules move in normal How do molecules move in normal liquid?liquid?

Small displacements via structural relaxation and transverse current

For many liquidsD T/

Stokes-Einstein relation

How to describe the continuos diffusion?How to describe the continuos diffusion?

Conventional descriptions:Conventional descriptions:

(A) Kinetic theory extended by Enskog(A) Kinetic theory extended by Enskog

(B) Navier-Stokes Hydrodynamics –Stick/Slip (B) Navier-Stokes Hydrodynamics –Stick/Slip boundary conditionboundary condition

But it fails for small molecules which is due to its failure to describe molecular length scale processes

Extended hydrodynamics and Remormalized kinetic theory

Mode Coupling Theory (MCT)

SingleParticle diffusion

Note that Fick’sLaw is phenomenological--- D is obtained by Green-Kubo relation.

Exponential decay of wave-Number dependent densityfluctuation

( , )sS k

( , )sS k

IS THE INCOHERENTDYNAMIC STRUCTURE FACTOR MEASUREDBY NEUTRON SCATTERING

EXPERIMENTS.

Coupling betweenSingle particleand collectivevariables missingin hydrodynamicdescription.

Collective dynamics

Variables are the conservedQuantities : density, Momentum and energy--- but these are coupledTo each other.

The simplest descriptionOf coupled equations isGiven by Navier-StokesHydrodynamic equations.Linearization of Hydrodynamicequations

Note that stress-stresstcf gives viscosity

The hydrodynamic matrix-- note the decoupling betweenthe transverse and the longitudinal current modes--the determinant (which determines poles and hencethe time constants of relaxation)Becomes a prodcut of two.

The dynamic structure factoris a sum of three peak Lorentzianknown as Rayleigh-Brillouin spectra which can be measuredby Light scattering experiments.

The central peak is due to densityfluctuation and the correspondinghydrodynamic mode is calledthe “heat mode”.

The two branches are due to the sound modes Heat mode

Sound mode

De Gennes’narrowing wasthe first indication that dynamics atsmall length scales cannot be describedby conventional hydrodynamics.

The microscopic derivation ofde- Gennes’ narrowing is simple.It uses what is known as Smoluchowski-Vlasov equation. This is an equationof motion for singlet density with aMean-field force term.The basic physics is that at intermediate to large wavenumbers, both momentum and energy relaxation is very fast. At such Small length scales, momentum conservation is no longer a constrain.

Instead, local environment controls density relaxation. Note that numberdensity is conserved at all length scales.

****

Molecular hydrodynamicEquations of motion

Density functionalFree energy

The eigen-valuesOf hydrodynamic Equations undergoSharp changes atMolecular length scalesDue to the presence ofIntermolecular correlations.

Heat mode. NoteThe near zero value

2k ~

What is MCTWhat is MCT??

It is a self-consistent scheme which getsIt is a self-consistent scheme which gets the short time behaviour (nearly) exactly the short time behaviour (nearly) exactly

correct (for two point correlation functions) correct (for two point correlation functions) because the lower order moments and because the lower order moments and satisfied.satisfied.

The long time behaviour is described by a The long time behaviour is described by a correlator which is expanded in terms of the correlator which is expanded in terms of the set of hydrodynamic eigenfunctions. The set of hydrodynamic eigenfunctions. The resulting equations are solved self-consistently.resulting equations are solved self-consistently.

Thus, the long time diffusive limit is described Thus, the long time diffusive limit is described fairly accurately.fairly accurately.

The mode coupling theory The mode coupling theory expressions can be derived in several expressions can be derived in several different ways – they all lead to the different ways – they all lead to the similar (if not the same) expression.similar (if not the same) expression.

One of the early elegant applications One of the early elegant applications of the mode coupling theory of the mode coupling theory expression for liquids was made by expression for liquids was made by Gaskel and Miller who derived an Gaskel and Miller who derived an expression for the velocity time expression for the velocity time correlation function of a tagged correlation function of a tagged particle. They argued that a particle particle. They argued that a particle moves by coupling to the current moves by coupling to the current mode. So, they projected the mode. So, they projected the propagator on propagator on

0 ( ) ( )k j k

The resulting expression involved The resulting expression involved wave vector integration over the wave vector integration over the transverse and longitudinal current transverse and longitudinal current correlation functions.correlation functions.

The longitudinal function decreases The longitudinal function decreases much faster than the transverse time much faster than the transverse time correlation function. The latter correlation function. The latter decay asdecay as

2exp( / )k t m

When you combine all the factors you recover Stokes-Einstein Relation, with 4. This calculation constitutes the first concrete demonstration of MCT.

However, this early success of MCT was based on assumptions which turn out to be untenable.

The reason is that in dense liquids The reason is that in dense liquids (and certainly in the supercooled (and certainly in the supercooled state) it is the density relaxation of state) it is the density relaxation of the surrounding solvent that is the the surrounding solvent that is the slowest relaxation mode. The slowest relaxation mode. The dramatic slowing down of density dramatic slowing down of density relaxation at wave numbers relaxation at wave numbers comparable to comparable to

2 /

This is of course de Gennes narrowing.In principle, all the slow modes should be included.

Power laws in the orientational Power laws in the orientational relaxation near Isotropic-Nematic relaxation near Isotropic-Nematic

phase-transition (INPT)phase-transition (INPT)

The molecular dynamics simulation is run on a system of 576 Gay-Berne ellipsoids in a Micro-Canonical ensemble. The simulations were run at temperatures T*= 1.0, 1.1, 1.2.

The variation of order parameter at different temperatures along the density axis is shown here.

Phase diagram of Gay-Berne ellipsoids with aspect ratio 3.

New Experimental results (New Experimental results (Fayer et Fayer et al.2002)al.2002)

Temperature dependent 5-OCB data sets displayed on a log plot.

optical Kerr effect data displaying the time dependence of orientational dynamics of the liquid crystal, 5-OCB at 347 K on a log plot.

M. Fayer (1996-2004)

Time Scales involvedTime Scales involved

Initial exponential decay occurs with a Initial exponential decay occurs with a time constant in 1-5 ps range.time constant in 1-5 ps range.

The long time Landau-de Gennes The long time Landau-de Gennes exponential decay sets in after 100 ns or exponential decay sets in after 100 ns or so, with a time constant few hundred ns.so, with a time constant few hundred ns.

There is a big window between 10 ps to There is a big window between 10 ps to few hundred ns when decay is very slowfew hundred ns when decay is very slow..

Temperature dependent 3-CHBT data sets displayed on a log plot.

The short time portion of the 5-OCB data at 347 K with the exponential contributions removed on a log plot.

Exponent ≈2/3

The temperature dependence of the The temperature dependence of the time time derivative of collective orientational time derivative of collective orientational time correlationcorrelation function is shown here (Cang et function is shown here (Cang et al. al. J. Chem. Phys., 118, 9303 (2003J. Chem. Phys., 118, 9303 (2003)).)).

Comparison with relaxation in glassy liquids

Mode coupling theory of orientational Mode coupling theory of orientational relaxation near INPTrelaxation near INPT

20 ( 0)

6

LdGR

S k

D

Origin of the slow down in relaxation can be understood from a mean-field theory which gives the following expression for LdG

where S20(k) is the wave number dependent orientational structure factor in the intermolecular frame. DR is the rotational diffusion coefficient. Kerr experiments measure the k= 0 limit of the time derivative of the collective orientational correlation function C2m(k,t).

,

2

1sl

l

R

C

zz z

Zwanzig - Mori Continued fraction

2 1 Bl

l l k T

I

0

0ztR z dte N N t

Single Particle Rotation

Generalized Langevin equation

Mode coupling theory Mode coupling theory calculation of rotational frictioncalculation of rotational friction

The baisc idea is that the torque tcf The baisc idea is that the torque tcf on a tagged ellipsoid slows down on a tagged ellipsoid slows down due to the marked slow down in due to the marked slow down in orientational density relaxation.orientational density relaxation.

Unlike in supercooled liquid, this Unlike in supercooled liquid, this happens at small k.happens at small k.

Expression for the torque can be Expression for the torque can be obtained from the DFT free energy obtained from the DFT free energy functional.functional.

( , , ) ' ' ( ', , ') ( ', ', )N r t dr d C r r r t

MCT expression for rotational frictionMCT expression for rotational friction

2 220 20

0 0

( ) exp( ) ( ) ( , )z C dt zt dkk C k F k t

C20(k) = (20,20) component of the direct correlation functionF20(k,t) = (20,20) component of the dynamic structure factor

Isotropic Isotropic-Nematic coexistence

Nematic* *0.34, 1.0, 576T N

* *0.315, 1.0, 576T N

* *0.2, 1.0, 576T N

The molecular dynamics simulation is run on a system of 576 Gay-Berne ellipsoids in a Micro-Canonical ensemble. The simulations were run at temperatures T*= 1.0, 1.1, 1.2.

The variation of order parameter at different temperatures along the density axis is shown here.

Phase diagram of Gay-Berne ellipsoids with aspect ratio 3.

The coefficients of the spherical harmonic expansion of pair correlation function tend to diverge when isotropic nematic phase transitions approached along the density axis.

1, 2 1 2

2 ', , ,16 , , ( ) ( )l l m l m l mg r g r Y Y

These coefficients of expansion of angular pair correlation functions can be calculated from simulation using the expression

..

20 20( , ) ( )exp( / )LdGF k t S k t

220( ) ~ /S k B k

R

Bz

z

1slC z

z z

2

20 ( )a tC e erfc a t

2

2 2 2 2 3

1 1.3 1.3.5( ) .....

2 (2 ) (2 )(1 )

xeerfc x

x x xx

(JCP (2002))

Collective orientationCollective orientation

20

1,

1

,

cl

B

R

C k zl l f k k T

zI z k z

f20(k) = 1/S20(k)

The time derivative of the theoretical correlation function, C20(t). Also shown is a t -0.63 power law (5-OCB). At short time, thederivative of the theoretical correlation function decays essentially as a power law.

Thus, the leading term in the expansion varies as t-1/2. The above analysis is valid only after the initial short time decay, very close to the INPT.

Fayer et al JCP (2002,2003)

Collective orientational correlation Collective orientational correlation functionfunction

ˆ ˆ0 .

ˆ ˆ0 . 0

l i jic

l

l i ji

P e e t

C t

P e e

Slow down in the relaxation of collective orientational time correlation function. The regions where power law relaxation is dominant are fitted to the function

at density=3.1

0.58 1.1-0.014ty

The Log-log plot of derivative of the collective orientational correlation clearly shows the power law relaxation.

Experimental data in the power law region ; Top 4 curves are of liquid crystals and bottom 5 are of supercooled liquids.

Data shown on left is for liquid crystal and right is for supercooled liquid (Cang et al. J. Chem. Phys., 118, 9303 (2003)).

However, one should add that MCT is However, one should add that MCT is quantitatively accurate at normal liquids quantitatively accurate at normal liquids – far far superior to the Brownian – far far superior to the Brownian oscillator model (BSO). (AD+DR, JLS, oscillator model (BSO). (AD+DR, JLS, Rabani,Egorof ….Rabani,Egorof ….

Please note that BSO has no diffusive Please note that BSO has no diffusive behavior in the proper sense.behavior in the proper sense.

Basic MCT equations for friction

( ) ( ) ( )B Rz z z

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

R BL

BL LL TT

z R z z R z

R z z z R z R z z

( ) ( ) ( ) ( )( )

1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

B BL

B B BL LL L TT

z R z z R zz

R z z R z z R z z R Z R z

B. Bagchi and S. Bhattacharyya Adv Chem Phys, 116, 67 (2001)

1 1 ( )( ) ( ) ( ) TTB

R zz z R z

22 2

3

0 0

( . ) ( )ˆ ˆ2

( , ) ( , ) ( , ) ( , )s s

dqR qq q c qm

F q t F q t F q t F q t

12

4012 0

1 ( )22

3

0

1 ( . )ˆ ˆ2

( , ) ( , ) ( , ) ( , )

td

TT TT

qTT

s

dqR qq

F q t C q t F q t C q t

Relationship with Stokes-Relationship with Stokes-EinsteinEinstein

It is important to realize that the It is important to realize that the Stokes-Einstein expression follows Stokes-Einstein expression follows strictly from the current term, first strictly from the current term, first derived by Gaskell & Miller.derived by Gaskell & Miller.

Self-consistent schemeSelf-consistent scheme

FFss(k,t)=exp(- q(k,t)=exp(- q22<<rr22(t)>/6)(t)>/6)

<r2(t)> = 2 d Cv()(t- )

Cv(z) = kB T / [m(z + (z))]

Power law and mass dependence of diffusion

D1 and D2 are self-diffusion coefficients of solvent and solute with masses m and M

Straight line is the fitting

The slope of the line = 0.099.

MD simulation studies predict the slope to be 0.1

S.Bhattacharyya and B. Bagchi, PRE, 61, 3850 (2000)

Microscopic analysis of Stokes-Einstein relation

Bhattacharyya and Bagchi,JCP(2001)

Projection Operator Projection Operator FormalismFormalism

Basic ingredients of Basic ingredients of mode coupling theorymode coupling theory

Derivation of the Relaxation Equations Derivation of the Relaxation Equations for a set of dynamical variable {A} by for a set of dynamical variable {A} by using projection operator techniqueusing projection operator technique

The resulting equation of motion for The resulting equation of motion for the correlation function is essentially a the correlation function is essentially a GLEGLE

0

0

( , )( ) ( , ) ( , ) ( , )

tdC q tq C q t d M q t C q

dt

This equation is commonly written in the frequency plane In the following form

where M(q,z) is the memory function where M(q,z) is the memory function which is defined in terms of force-force which is defined in terms of force-force time correlation function. The force F(q,t) time correlation function. The force F(q,t) must be orthogonal to the dynamical must be orthogonal to the dynamical variable.variable.

In the next step, one usually writes down a In the next step, one usually writes down a continued fraction representation continued fraction representation (equation) for M(q,z) because one often (equation) for M(q,z) because one often knows the t=0 value of M(q,t=0). knows the t=0 value of M(q,t=0).

0

( , 0)( , )

( ) ( , )

C q tC q z

z q M q z

The calculation of the memory function The calculation of the memory function for M(q,z) is often done by the mode for M(q,z) is often done by the mode coupling theory. The modes are usually coupling theory. The modes are usually the binary product of slow variables the binary product of slow variables which are chosen to be orthogonal to the which are chosen to be orthogonal to the dynamical variable. In the language of dynamical variable. In the language of projection operator technique, one projection operator technique, one projects the force term on the sub-space projects the force term on the sub-space of binary product of slow variables.of binary product of slow variables.

****In the theory of glass transition, In the theory of glass transition, the primary dynamical variable in the primary dynamical variable in question is the wavevector question is the wavevector dependent density. M(q,z) is then dependent density. M(q,z) is then the longitudinal current tcf and the the longitudinal current tcf and the memory function of longitudinal memory function of longitudinal current is longitudinal viscosity current is longitudinal viscosity (vide hydrodynamics(vide hydrodynamics).).

2

0

0t

q lq t q t d D q t q ( , ) ( , ) ( , ) ( , )

2 2 2

0

4 0 t

q qt t t d t ( ) ( ) ( ) ( ) ( )

2

1( )

( , )q

l

z

zz D q z

Replace the expression of the memory function

Start with the continued fraction representation

(1)

(2)

(3a)

EXACTEXPRESSION

2( ) 4 ( )l qD z t 2L

( )lD z

1 2

1 2

( )a a

zz i z i

4( )

222

lD zzz

z

Rewriting the longitudinal viscosity as

In zeroth order, the longitudinal viscosity is

Use of this approximation in the continued fraction gives two poles in the DSF

(3b)

Use Of this DSF gives, to the first order,

( 0) 2 ( 1/ )lD z

( ) (1 ) ( )v

fz f z

z

2( ) 4 / ( )l vD z f z D z

The zero frequency value of the longitudinal viscosity is

This value is greater than the zeroth order value by 1/

Each iteration increases the value of the viscosity,leading to a divergence, leading to an elastic peak in the. Dynamic structure factor. This is the famous feed-back mechanism(Geszti, 1983). Therefore, we make the ansatz

Use of this expression gives the following expression for the longitudinal viscosity

(4)

(5)

2( ) 8 (1 ) ( ) 4 (1 ) ( )v v vD z f f z f t L 2

1 1 1/ / 2f

With the following value of the vibrational contributing part of the of the viscosity

Use of the above expression in Eq.3a gives, after comparisonWith Eq.4 gives the following expression for strength f

The remaining part of the dynamic structure factor is given by

2

1( )

( )

v

v

z

zz D z

(6)

(7)

Where

2 21 4 f

Eq.6 shows that the ansatz gives acceptable solutionAbove the crtitical value =1 . This means the dynamic structure factor does not decay to zero for >=1. Instead,they decay only to f. Thus, the spectrum of density fluctuationWill show a delta function peak at =0. The value of f is wave-number dependent and is called the order parameter of theTransition.

The transition is purely dynamic in nature.

The vibrational part of the longitudinal viscosity followsthe following relation

2

2

[ ( )] ( ) 2 ( )( 2)( )

[ ( )]v v v

v

z z D z D z z D zz

z z D z

where

2 2( ) 4 (1 ) ( ( ))vz f L t

Therefore, theory of dynamics in the glass phase canbe formulated fully in terms of the vibrational componentOf the dynamic structure factor.

The Critical region: Power Law The Critical region: Power Law Decay Decay

The small frequency behaviour near the critical point can bestudied analytically. At small frequencies, z/Dv(z) <<1. At the glass transition point, one gets

2 1/ 2

2

( ) [ ( ) { ( ) 8 ( ) / } ] / 2,

( ) [ ( )]

v

v

D z z z z z

z L t

At small z, Dv(z) exhibits a square root singularity which means a power law decay for the dynamics structure factor,

1

2 1

( )

( )

vD z z

z z

is a solution of (1-2)=22 (1-). This gives a value of =0.395. That is, the square root dependence gets modified by a small extent by the frequency dependence of (z).

The above equations imply a power law decay of the dynamic structure factor

( )t t

This power law decay describes the decay of the vibrational partto the plateau.

As the dynamic structure factor develops an elastic peak,The longitudinal viscosity diverges because the longitudinal Viscosity is time integral of the square of DSF.

The detailed analysis shows that the longitudinal viscosityDiverges as

( 0) ( 1) ,

( 1) / 2 1.765lD z

Detailed numerical solutions have put the value of theExponent between 1.8 and 2.

Predictions of MCTPredictions of MCT

Divergence of viscosity as a power law with Divergence of viscosity as a power law with exponent ~2. This seems to fit the exponent ~2. This seems to fit the experimental and simulation data well at experimental and simulation data well at temperatures significantly above Ttemperatures significantly above Tgg..

Theory predicts non-exponential relaxation Theory predicts non-exponential relaxation functions. The theory also predicts power law functions. The theory also predicts power law decays at the end of beat relaxation and at the decays at the end of beat relaxation and at the beginning of alpha relaxations which seem to beginning of alpha relaxations which seem to be in good accord with experiments and be in good accord with experiments and simulationssimulations..

The predicted critical point (TThe predicted critical point (Tcc) is substantially ) is substantially above the glass transition temperatureabove the glass transition temperature (by 30- (by 30-50 K).50 K).

(N P T) simulations in Binary Mixture

Isolated Ellipsoid in a sea of spheres

L-J

LJGB

A. Mukherjee, S. Bhattacharyya, B. Bagchi, JCP (2002)

B. S. Bhattacharyya, B. Bagchi, Phys. Rev. Lett. (2002)

Stretched-exponential stress relaxation with decreasing temperature at constant P*=10.0

Dashed line is fitted to the equation below

Cs(t) = Aexp(-t/1) + Bexp[-(t/ 2)]

Normalized stress correlation ln Cs(t) in binary mixture

ln(t)

Stretched exponential relaxation with increasing pressure at constant temperature T*=1.0

Normalized stress autocorrelation function ln Cs(t)

ln(t)

Dashed line is fitted to the equation below

Cs(t) = Aexp(-t/1) + Bexp[-(t/ 2)]

Non-Arrhenius increase in viscosity () with temperature indicates that the studied model liquid is weakly fragile

ln(

)

1/T*

Pressure dependence of Pressure dependence of ViscosityViscosity

at constant at constant temperature(T*=1.0)temperature(T*=1.0)

Pressure dependence found to be exponential --- though there is a distinct change in slope ---- signature of Landscape dominance?

ln(

)

Pressure(P)

Evidence of a dynamic transition?

Pressure dependence of viscosity in another constant temperature (T*=0.5).

ln(

)

Pressure(P)

Nature of pressure dependence remains the same as that of a higher T*.

Computer Simulation Computer Simulation Studies of Composition Studies of Composition Fluctuation in Binary Fluctuation in Binary

MixtureMixture We need to simulate a binary We need to simulate a binary

mixture where the two components mixture where the two components interact strongly among interact strongly among themselves. Also, the interaction themselves. Also, the interaction between the second (B) component between the second (B) component should not be too strong.should not be too strong.

Thus, AB interaction should be Thus, AB interaction should be stronger than BB interaction, while stronger than BB interaction, while AA intercation can be comparable AA intercation can be comparable to AB interaction, preferably a bit to AB interaction, preferably a bit weaker to avoid phase separation.weaker to avoid phase separation.

Local composition fluctuations in strongly nonideal binary mixtures

Local composition fluctuations in strongly nonideal binary mixtures

Spontaneous local fluctuations rich and complex behavior in many-body system

R

V(R)

What is the probability of finding exactly n particle centers within V(R) ?

In one component liquid local density fluctuations are Gaussian

Binary mixtures that are highly nonideal, play an important role in

industry

N P T simulations of Nonideal Binary Mixtures

Study of Composition Fluctuations

N P T simulations of Nonideal Binary Mixtures

Study of Composition Fluctuations

mA = mB = m

parametersparameters AAAA BBBB ABAB EEAA EEBBBB EEABAB

Equal size Equal size modelmodel

1.01.0 1.01.0 1.01.0 1.01.0 0.50.5 2.02.0

Kob-Andersen Kob-Andersen modelmodel

1.01.0 0.880.88 0.800.80 1.01.0 0.50.5 1.51.5

xA= 0.8

xB= 0.2

Two models binary mixtures : Kob-Andersen model (glass-forming mixture)

Equal size model

tcftcf

AAAA 0.190.19 0.470.47

BBBB 4.304.30 0.480.48

ABAB 3.003.00 0.410.41

tcftcf

AAAA 0.200.20 0.420.42

BBBB 8.078.07 0.580.58

ABAB 6.506.50 0.600.60

R = 2.0AA

P* = 2.0

R = 2.0AA

P* = 4.0

Dynamical Correlations in Composition Fluctuation : Kob-Andersen Model

Dynamical Correlations in Composition Fluctuation : Kob-Andersen Model

Non-exponential decay

Distribution of relaxation times

( / )tAeStretched exponential fit

Slow Dynamics

A. Correlated Orientational and Translational Hopping

Translational motion

Orientational motion

Orientational Correlation function at different time intervals

P* = 10.0

Correlated Orientational and Translational Hopping



Orientational Correlation function at different time intervals

P* = 10.0

The Single Particle Potential Energy during the time of Hopping.

Ellipsoid

Neighbour 1

Neighbour 2

B. Heterogeneous Rotational Dynamics of the Ellipsoids

The orientational correlation function of the 4 ellipsoids in two different runs

System in two different runs are in two glassy minima

Bhattacharyya and Bagchi,Phys. Rev. Lett. (2002)

R Orientational relaxation time DT Translational diffusion

Normal liquids R 10 -11 sec and DT 10 10 sec

P* = 2.0 D R = 0.144

P* = 5.0 D R = 0.23

P* = 6.0 DT R > 1

At higher pressures D R >> 1

Decoupling between translation and rotational motion

D/D

SE

*

Decoupling of diffusion of smaller particles from solvent viscosity: Test of ode coupling theory

DSE= kBT/4rr = 0.5 long dashed line

r =0.75 short dashed line

r = 1.0 is solid line

2 /1 = r

B. B

agchi an

d S

. Bh

attacharyya A

C P

, 116, 67 (2001)

MCT predicts weak decoupling -- at variance with expts.

What went wrong with MCT?What went wrong with MCT?

Relaxation functions are decomposed asRelaxation functions are decomposed as

G(r,r’,t) = GG(r,r’,t) = Gαα (r,r (r,r´́,t) + G,t) + G (r,r (r,r´́,t),t)

MCT includes only two point MCT includes only two point correlations.correlations.

Thus decay due to many body Thus decay due to many body fluctuations is neglected!!fluctuations is neglected!!

However, decay can indeed happen due However, decay can indeed happen due to many body fluctuations!! In fact, in to many body fluctuations!! In fact, in highly viscous liquids, these many body highly viscous liquids, these many body fluctuations are the important ones.fluctuations are the important ones.

Idea of Entropy Crisis – Idea of Entropy Crisis – Adam-Gibbs pictureAdam-Gibbs picture

A cooperatively rearranging region (CRR) is defined asthe region which contains at least two distinct configurational states.

As the entropy of the systems decreases, this z* must growin size because by definition, z* contains at least two states.

It is shown that z* ~ 1/Sc, where Sc is the configurationalentropy of the system.Relaxation time scales as ~ exp(z*), so as entropy of the system goes to zero (as in Gibbs-DiMarzio theory), the relaxation time diverges.

Difficulties with Adam-Difficulties with Adam-Gibbs ScenarioGibbs Scenario

Experimental and computer Experimental and computer simulations have failed to find to any simulations have failed to find to any diverging length scale. The diverging length scale. The heterogeneous domains are found to heterogeneous domains are found to be typically 2-3 nm long.be typically 2-3 nm long.

No prediction on non-exponentialityNo prediction on non-exponentiality..

The failure may be due to neglect of The failure may be due to neglect of interactions among CRR. It is a mean-interactions among CRR. It is a mean-field theory description.field theory description.

Random First Order Theory Random First Order Theory (RFOT(RFOT) (Wolynes et al.)) (Wolynes et al.)

This theory considers relaxation and This theory considers relaxation and fluctuation to occur via formation of an fluctuation to occur via formation of an entropy droplet. The nucleation free energy entropy droplet. The nucleation free energy is obtained by standard method except that is obtained by standard method except that here surface tension is a function of the size here surface tension is a function of the size of the droplet (nucleus) and decreases as of the droplet (nucleus) and decreases as 1/1/√r , where r is the radius of the droplet.√r , where r is the radius of the droplet.

An interesting prediction of the theory is An interesting prediction of the theory is that the activation free energy is inversely that the activation free energy is inversely proportional to the configuration entropyproportional to the configuration entropy

† / 32 / cF T s

A Mescoscopic Model of Glassy A Mescoscopic Model of Glassy DynamicsDynamics

Jump motion is the dominant mode of diffusion Jump motion is the dominant mode of diffusion near Tnear Tgg..

Hopping is a highly collective phenomenon with Hopping is a highly collective phenomenon with strong nearest neighbor correlationsstrong nearest neighbor correlations. .

Growth of spatially heterogeneous domains that Growth of spatially heterogeneous domains that span 2-3 nm near Tspan 2-3 nm near Tgg..

Motivation

Correlated Orientational and Translational Hopping in a Ring Like Tunnel



Orientational correlation at different time intervals

Mesoscopic Model of glassy Mesoscopic Model of glassy dynamicsdynamics

An An α process is promoted by α process is promoted by coherent excitation of a certain coherent excitation of a certain minimum number of β processes minimum number of β processes within a cooperatively rearranging within a cooperatively rearranging region (CRR).region (CRR).

A β process is envisaged as a A β process is envisaged as a transition in a two-level system transition in a two-level system (TLS).(TLS).

As the temperature is lowered As the temperature is lowered towards the glass transition towards the glass transition temperature, the number of temperature, the number of processes which need to be excited processes which need to be excited to attain to attain αα relaxation increases. relaxation increases.

The modelThe model Each of the two wells, labeled Each of the two wells, labeled

1 and 2, comprises a collection 1 and 2, comprises a collection of Nof Nii (i = 1, 2) identical, non- (i = 1, 2) identical, non-

interacting TLSs of such kind.interacting TLSs of such kind.

For a collection of NFor a collection of Nii TLSs, a TLSs, a

variable variable ζζjjii(t) (j = 1, 2 , …, N(t) (j = 1, 2 , …, Nii) )

is defined, which takes on a is defined, which takes on a value 0 if at the given instant value 0 if at the given instant of time t the level 0 is of time t the level 0 is occupied and 1 if otherwise.occupied and 1 if otherwise.

We defineWe define ( ) ( )1

Ni iQ t t

i jj

•An α process occurs only when

all the β processes in a well are

simultaneously excited, i.e.,

Qi = Ni.

The modelThe model A CRR is characterized by an NA CRR is characterized by an Nβ β number of identical non-number of identical non-

interacting TLSs.interacting TLSs.

A A collective variablecollective variable Q(t) is defined: Q(t) is defined:

ζζjj(t) = 0 or 1 is the occupation variable (j = 1, 2 , …, N(t) = 0 or 1 is the occupation variable (j = 1, 2 , …, Nββ).).

The waiting time before a transition can occur from the level The waiting time before a transition can occur from the level ii (= (= 0, 1) is drawn from a Poissonian probability density function:0, 1) is drawn from a Poissonian probability density function:

where where ττii is the average time of stay in the level is the average time of stay in the level ii..

( ) ( )1

N

Q t tjj

1 exp , 0,1i ii

t i

The modelThe model Detailed balance givesDetailed balance gives

where where ε is the separation between the two levels.ε is the separation between the two levels.

From the TST results:From the TST results:

The rate of The rate of α relaxation α relaxation depends crucially on Q. For simplicity, we assume that depends crucially on Q. For simplicity, we assume that the relaxation occurs with unit probability at the instant Q reaches for the first the relaxation occurs with unit probability at the instant Q reaches for the first time Ntime Ncc, an integer greater than the most probable value of Q., an integer greater than the most probable value of Q.

1 1

0 0

( ) ( )exp[ /( )]

( ) ( )( ) B

T Tk T

T T

pK T p

( ) /( )exp[ /( )], 0,1T h k T k T iB Bi i

Theoretical TreatmentTheoretical Treatment Q(t) is a stochastic variable in the discrete integer space Q(t) is a stochastic variable in the discrete integer space [0, N[0, N] (single flip assumption)] (single flip assumption)

The mean first passage time The mean first passage time τ(n, T), which is the mean time τ(n, T), which is the mean time elapsed before Q starting from its initial value n reaches Nelapsed before Q starting from its initial value n reaches Ncc for for the first time at temperature T, is given bythe first time at temperature T, is given by

where F is the hypergeometric function.where F is the hypergeometric function.

( ; , ) [ 1)/ ( )] ( 1; , )0

[( 1)/ ( )] ( 1; , )1

[( )/ ( )] ( ; , )0

[ / ( )] ( ; , )1

P n T t N n T P n T tt

n T P n T t

N n T P n T t

n T P n T t

( 1, ; 1, 1/ ( ))11( , ) ( ) 1 ( )0

N F N N m N m K TNcn T T K T N mm n

Theoretical TreatmentTheoretical Treatment

We assume that an We assume that an α process corresponds to a large scale α process corresponds to a large scale change in configuration within a CRR.change in configuration within a CRR.

The relevant time correlation function is the survival The relevant time correlation function is the survival probability that a CRR refrains from undergoing an α probability that a CRR refrains from undergoing an α process:process:

The average relaxation time τThe average relaxation time τCRRCRR(T) of a CRR, characterized (T) of a CRR, characterized

by a set of values for Nby a set of values for Nββ, N, Ncc, and ε, is , and ε, is

1( , ) ( ; , )

0

NcS t T P n t T

n

1( ) ( ;0, ) ( , )

0

NcT P n T n T

CRR n

The Model (Continued)The Model (Continued)

In a heterogeneous environment within a bulk sample, a fluid-In a heterogeneous environment within a bulk sample, a fluid-like region can be envisaged as having at any instant of time, like region can be envisaged as having at any instant of time, on the average, a larger number of on the average, a larger number of β processes in the excited β processes in the excited state than what is in the solid-like region.state than what is in the solid-like region.

A Gaussian distribution of ε among CRRs => Heterogeneous A Gaussian distribution of ε among CRRs => Heterogeneous domains.domains.

The model assumes NThe model assumes Ncc to grow as the reduced inverse to grow as the reduced inverse

temperature Ttemperature Tgg/T increases until N/T increases until Ncc reaches N reaches Nββ at T at Tgg..

ResultsResults

The present calculation takes the The present calculation takes the following set of values for the following set of values for the relevant parameters:relevant parameters:

# # < < ε > = kε > = kBBTTmm

# σ# σεε = 0.05 k = 0.05 kBBTTmm

## T Tgg = (2 / 3) T = (2 / 3) Tmm

# ε# ε11‡ ‡ = 4k= 4kBBTTmm

Time is scaled by Time is scaled by ττ11 at T at Tgg..

The long time behavior of S(t) fits well to the stretched exponential function.Inset: Monotonic decrease of the Stretching exponent βKWW as Tg isapproached from above.

( ) ( )exp( / )0S t d g tCRR CRR CRR

ResultsResults

γ = Nγ = Nc c / N/ Nββ

O => γ varies linearly from 0.6 to 1.0 O => γ varies linearly from 0.6 to 1.0 as Tas Tgg / T increases from 0.68 to 1.0 / T increases from 0.68 to 1.0

with Nwith Nβ β held fixed at 20. held fixed at 20.

=> γ grows linearly with T� => γ grows linearly with T� gg / T, both / T, both

rising from 0.7 to 1.0, with Nrising from 0.7 to 1.0, with Nc c and and

NNβ β varied together, the latter from varied together, the latter from

10 to 20.10 to 20.

TT00 = 0.689 T = 0.689 Tg g (solid line)(solid line)

TT0 0 = 0.789 T = 0.789 Tg g (dashed line)(dashed line)

Arrhenious plot showing the scaled characteristic relaxation time as a function of the reduced inverse temperature.

Anomalous Temperature Dependence of Anomalous Temperature Dependence of Heat Capacity During a Cooling-Heating Heat Capacity During a Cooling-Heating

Cycle in Glassy systemsCycle in Glassy systems

A sharp rise in the measured heat capacity during heatingA sharp rise in the measured heat capacity during heating

A shift to higher values of the limiting fictive temperature TA shift to higher values of the limiting fictive temperature TffLL

obtained upon cooling and the glass transition temperature Tobtained upon cooling and the glass transition temperature Tg g

for faster rates.for faster rates.

For qFor qcc = q = qhh = q, = q,

An identical dependence of TAn identical dependence of TffLL on q on qcc..

ln * /(1/ )d q h R

d Tg

Motivation

The ModelThe Model

The model, based on the framework of The model, based on the framework of β organized α process, β organized α process, envisages a β process as a transition in a two-level system envisages a β process as a transition in a two-level system (TLS).(TLS).

An An α process is conceived as a cooperative transition from one α process is conceived as a cooperative transition from one well to another in a double-well, subject to the establishment well to another in a double-well, subject to the establishment of a certain condition.of a certain condition.


QQii is a stochastic variable in the discrete integer space is a stochastic variable in the discrete integer space

[0, N[0, Nii].].

where the ‘+’ and ‘-’ signs in the indices of the Kronecker where the ‘+’ and ‘-’ signs in the indices of the Kronecker delta are for i = 1 and 2, respectively. delta are for i = 1 and 2, respectively.

The total energy of the system at time t isThe total energy of the system at time t is

( ; , )[ 1)/ ( )] ( 1; , )

0[( 1)/ ( )] ( 1; , )

1[( )/ ( )] ( ; , )

0[ / ( )] ( ; , ) ( ; , )

,1

( ; , ), , 1

1

P n T ti N n T P n T t

i itn T P n T t

iN n T P n T t

i in T P n T t k P n T t

n Ni ii

k P n T tn N j i i

i

1 2( , ) ( ; , ) ( ) ( ; , )

1 2 1 10 0

N NE T t P n T t N N n P n T t n

n n


The system, when subjected to cooling or heating at a The system, when subjected to cooling or heating at a constant rate, can be envisaged to undergo a series of constant rate, can be envisaged to undergo a series of instantaneous temperature changes, each in discrete step of instantaneous temperature changes, each in discrete step of ΔT, in the limit ΔT ΔT, in the limit ΔT 0, at time intervals of length Δt, whence 0, at time intervals of length Δt, whence q = ΔT / Δt.q = ΔT / Δt.

We calculate the heat capacity from the following equation:We calculate the heat capacity from the following equation:

Here tHere tobsobs = Δt. = Δt.

( , , ) ( ,0, )( , , ) lim

0

E T T t q E T qobsC T t q

obs TT

ResultsResults

NN1 1 = 6= 6 NN2 2 = 10= 10 |ΔT| = 0.0015 in reduced |ΔT| = 0.0015 in reduced

units of kunits of kBBT / εT / ε εε11

‡‡ = 8 ε = 8 ε Time is expressed in Time is expressed in

reduced units, being scaled reduced units, being scaled by τby τ11(T(Thh))

kk-1 -1 = 0.50 in reduced time = 0.50 in reduced time unitsunits

For ε / kFor ε / kBB = 600 K, the = 600 K, the cooling and heating rates cooling and heating rates explored here range from explored here range from 0.0085 to 0.35 K s0.0085 to 0.35 K s-1-1..

q = 7.5 x 10-5 in reduced units

ResultsResults

q= 3 X 10q= 3 X 10-4-4 (solid line), 7.5 x 10 (solid line), 7.5 x 10-5-5 (dashed line), both in reduced (dashed line), both in reduced units.units.

Inset: q = 7.5 x 10Inset: q = 7.5 x 10-5-5 The fictive temperature evolves in The fictive temperature evolves in

an identical fashion as the energy.an identical fashion as the energy.

q = 3.0 x 10-4, 7.5 x 10-5, 2.0 x 10-5, 7.5 x 10-6

in reduced units from top to bottom.

Activated dynamics for intra-well transitionTrapped into non-equilibrium glassy state Delayed energy relaxation on subsequent heating

The fictive temperature is calculated in termsof energy.

Mode Coupling Theory of Mode Coupling Theory of frequency dependent specific frequency dependent specific

heatheat Total energy-energy time correlation function Total energy-energy time correlation function

can be expressed by using the mode can be expressed by using the mode coupling theory in terms of dynamic structure coupling theory in terms of dynamic structure factor.factor.

C(t) is given in terms of wave number (q) C(t) is given in terms of wave number (q) integral over [F(q,t)/S(q)]integral over [F(q,t)/S(q)]2, 2, where F(q,t) is the where F(q,t) is the intermediate scattering function and S(q) the intermediate scattering function and S(q) the structure factor of the liquid.structure factor of the liquid.

Thus, MCT predicts a large vibrational peak, Thus, MCT predicts a large vibrational peak, well-separated from the alpha peak, in the well-separated from the alpha peak, in the frequency dependence of the specific heat.frequency dependence of the specific heat.

Conclusions

Non-exponentiality enters naturally into the dynamics.

Even a modest growth in the number of coupled beta transition is sufficient to provide a severe entropic bottle-neck to make relaxation slow.

The fragility of the system is related to the growth of beta states. But this growth is limited. No divergence of length scale is involved.

Acknowledgement

• Sarika Bhattacharyya

• Arnab Mukherjee

• Rajesh Murarka

• Dwaipayan Chakrabarti

VFT fitting signifying the super-Arrhenius nature of the temperature dependence of

Viscosity ()

= exp { - C/(T-T0) }

ln

()

1/(T-T0)

MCTMCT prediction for temperature prediction for temperature dependence of dependence of

viscosity(viscosity())

= (T - T = (T - Tcc))--

ln(

ln(

))

ln(T-Tc)

Correlated Orientational and Translational Hopping in a Ring Like Tunnel



Orientational correlation at different time intervals

The Single Particle Potential Energy during the time of Hopping.

Ellipsoid

Neighbor 1

Neighbor 2

Heterogeneous rotational dynamics and decoupling of orientational and translational motion are also observed.

ResultsResults

Single exponential dependence of the average relaxation time within a CRR on Nβ at fixed γ.Inset: slope m ~ γυ, υ = 3.4

γ = Nc / Nβ

ε = kBT, Nβ = 10, Nc = 6, 7, 8

Single exponential decay of SCRR

Conclusion

Translational motion of the ellipsoid in ring like tunnels is observed. Both orientational and translational hopping are gated process where the free energy barrier is entropic in nature.

One of the possible scenario of decoupling between rotation and translational diffusion is that molecule can jump retaining their orientation

Decoupling between diffusion and viscosity might occur due to the presence of the hopping mode. Hopping found to persist even when stress autocorrelation function ceases to decay

Anomalous observations in Anomalous observations in Supercooled Liquids: Supercooled Liquids:

StaticsStatics

• The overshoot of the heat capacity

during heating is taken to be a

signature of a glass to liquid

transition. Is there any underlying Is there any underlying

thermodynamic phase thermodynamic phase

transition ?transition ?

Potential Energy Potential Energy landscape Picturelandscape Picture

Goldstein originally proposed that Goldstein originally proposed that below certain temperature, activated below certain temperature, activated processes become important in the processes become important in the dynamics of glassy liquids. “Inherent dynamics of glassy liquids. “Inherent structure” concept of Stillinger and structure” concept of Stillinger and Weber substantiated this view.Weber substantiated this view.

Angell has proposed a powerful Angell has proposed a powerful classification of glass forming liquids in classification of glass forming liquids in terms of strong and fragile liquids terms of strong and fragile liquids which can be “explained” in terms of which can be “explained” in terms of topology of the potential energy topology of the potential energy surfacesurface (density of local minima, (density of local minima, barrier heights)barrier heights)

Potential Energy Potential Energy landscape Picture(landscape Picture(continued)continued)

Difficulty is that the role of spatial Difficulty is that the role of spatial correlations is left unclear. Strong correlations is left unclear. Strong liquids are expected to have significant liquids are expected to have significant static pair and higher order correlations static pair and higher order correlations even in less viscous liquids.even in less viscous liquids.

Lack of correlation of fragility with Lack of correlation of fragility with molecular/microscopic properties. molecular/microscopic properties.

Correlation with free energy landscape Correlation with free energy landscape has been proposed.has been proposed.

Probability Distributions of Composition FluctuationProbability Distributions of Composition Fluctuation

Kob-Andersen Model R = 2.0AA

T* = 1.0 P* = 2.0

Gaussian distribution

NA = 27.3 A = 1.995

Both A and B fluctuations are large NB = 6.74 B = 1.995

System is indeed locally heterogeneous

Joint Probability Distribution FunctionJoint Probability Distribution Function

Kob-Andersen Model R = 2.0AA

Nearly Gaussian

Corr[NA , NB] = - 0.203

Fluctuations in A and B are anticorrelated

Biman Bagchi

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