Slow Dynamics in Binary Liquids : Microscopic Theory and Computer Simulation Studies of Diffusion, Density Relaxation, Solvation and

Composition Fluctuation

Slow Dynamics in Binary Liquids : Microscopic Theory and Computer Simulation Studies of Diffusion, Density Relaxation, Solvation and

Composition Fluctuation

Biman Bagchi

SSCU, IISc, Bangalore.

Decemmber 2003

Outline Outline

Introduction

Local composition fluctuations in strongly nonideal binary mixtures

Diffusion of small light particles in a solvent of large massive molecules

Pair dynamics in a glass-forming binary mixture

Diffusion and viscosity in a highly supercooled polydisperse system

Conclusion

Solvation Dynamics in Binary Mixture.

Polarization Relaxation in Binary Polarization Relaxation in Binary Dipolar MixtureDipolar Mixture

Molecular Hydrodynamic Theory of Chandra and Bagchi. (1990,1991)

The theory uses density functional theory to describe the equilibrium aspect of solvation in a binary mixture.

Definition of Non-idealityDefinition of Non-ideality

Raoult’s Law1 1 2 2P x P x P

Dynamics of Solvation

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

parameters AA BB AB AA BB AB

Equal size model

1.0 1.0 1.0 1.0 0.5 2.0

Kob-Andersen model

1.0 0.88 0.80 1.0 0.5 1.5

xA= 0.8

xB= 0.2

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

Equal size model

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

tcf

AA 0.19 0.47

BB 4.30 0.48

AB 3.00 0.41

tcf

AA 0.20 0.42

BB 8.07 0.58

AB 6.50 0.60

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

Diffusion of small light particles in a solvent of large massive molecules

Diffusion of small light particles in a solvent of large massive molecules

Isolated small light particles in a solvent of large heavy particles can mimic concentrated solution of polysaccharide in water,

motion of water in clay

The coexistence of both hopping and continuous diffusive motion

—MR=5

—MR=25

—MR=50

—MR=250

Relaxation of Solute and Solvent : The Self-intermediate Scattering Function Fs(k,t)

Relaxation of Solute and Solvent : The Self-intermediate Scattering Function Fs(k,t)

Solute

Solvent

k*=k11~2

Fs(k,t) begins to stretch at long time for higher solvent mass !

Sum of two stretched exponential function

No stretching at long times !

Exponential decay

Solute probes progressively more local heterogeneous environment

4

2 22

3 ( )( ) 1

5 ( )

r tt

r t

Non-Gaussian Parameter Non-Gaussian Parameter

—MR=5

—MR=25

—MR=50

—MR=250

Solute

2 2 2

0( ) 4 ( , )n n

sr t r G r t dr

The peak height increases heterogeneity probed by the solute increases with solvent mass

No such increase for the solvent

5 0.082 0.70

25 0.080 0.96 0.49 0.67

50 0.083 0.94 0.59 0.64

250 0.085 0.91 1.01 0.63

1

2R

mM

m

11 2

2

The Self-intermediate Scattering Function of the Solute The Self-intermediate Scattering Function of the Solute

k*~212

Two stretched exponential separated by a power law type plateau, often observed in deeply supercooled liquids

Separation of time scale between binary interaction and solvent density mode — increases with solvent mass

The Velocity Autocorrelation Function of the Solute ParticlesThe Velocity Autocorrelation Function of the Solute Particles

—MR=25

—MR=50

—MR=250

Development of an increasingly negative dip followed by pronounced oscillations at longer times

“dynamic cage” formation in which the solute particle executes a damped oscillatory motion : observed in supercooled liquid

20

( ) 6 ( )( )t

vr t d C t

2 2( )

6( , ) exps

k r tF k t

2( )

( ( ))B

vk T

C zm z z

GeneralizedGeneralized self-consistent schemeself-consistent scheme GeneralizedGeneralized self-consistent schemeself-consistent scheme

21R

5 23.65 13.95

25 25.40 33.80

50 25.45 50.20

250 25.58 99.80

2B1

2R

mM

m

Self-consistent scheme : overestimates diffusion (faster decay of Fs(k,t))

The relative contribution of the binary term decreases with solvent mass

Contribution of the density mode increases !

Gaussian approximation is poor

2simD

2MCTD

Comparison of MCT prediction with simulationsComparison of MCT prediction with simulations

1

2R

mM

m

5 0.135 0.1065

25 0.108 0.0675

50 0.101 0.0530

250 0.0805 0.0320

For larger mass ratio, MCT breaks down more severely !

Overestimates the friction contribution from the density mode

Solute probes almost quenched system

breakdown of MCT can be connected to its similar breakdown near the glass transition temperature

hopping mode plays the dominant role in the diffusion process

4 : Pair dynamics in a glass-forming binary mixture4 : Pair dynamics in a glass-forming binary mixture

Dynamics in supercooled liquids has been investigated solely in terms of single particle dynamics

The relative motion of the atoms that involve higher-order (two-body) correlations can provide much broader insight into the anomalous dynamics of supercooled liquids

Radial Part of the Time Dependent Pair Distribution Function (TDPDF)

Radial Part of the Time Dependent Pair Distribution Function (TDPDF)

The TDPDF, g2(ro,r;t), is the conditional probability that two particles are separated by r at time t if that pair were separated by ro at time t = 0, thus measures the relative motion of a pair of atoms

Nearest neighbor AA pair

t=500

Jump motions are the dominant diffusive mode by which the separation between pairs of atoms evolves in time

Angular Part of the Time Dependent Pair Distribution Function (TDPDF)

Angular Part of the Time Dependent Pair Distribution Function (TDPDF)

AA pair AB pair

BB pair

Nearest neighbor pair

Compared to AA pair, the approach to the uniform value is faster in case of AB pair

Relaxation of BB pair is relatively slower at short times as compared to the AB pair

Relative Diffusion : Mean-Square Relative Displacement (MSRD)

Relative Diffusion : Mean-Square Relative Displacement (MSRD)

Nearest neighbor pair

Faster approach of the diffusive limit of BB pair separation

Time scale needed to reach the diffusive limit is shorter for the AB pair than that for the AA pair0.

0.006

0

4

.00

4

32

00 8

AAR

BBR

ABR

D

D

D

Relative diffusion coefficients

2( ) (0)

o

ij ijr

MSRD t r r

4

2 22

3 ( ) (0)( ) 1

5 ( ) (0)

ij o

o

ij ijP r

ij ijr

tt

t

r r

r r

The Non-Gaussian Parameter for the Relative MotionThe Non-Gaussian Parameter for the Relative Motion

Single particle dynamicsPair dynamics for nearest neighbor pair

B particles probe a much more heterogeneous environment than the A particles

The dynamics explored by the BB pair is less heterogeneous than the AA and AB pairs

Theoretical Analysis Theoretical Analysis

22 2

( , ; )[ ( , ; ) ( , ; ) ( )]

ijij ijo

o o ijij

g tg t g t W r

r r

r r r r

( ) ln ( )ij B ijW r k T g r

21( ) (0)

6 o

ij ij ijr

t r r

221( ) (0) ( ) (0)

6 i i j jt t r r r r

Mean-field Smoluchowski equation

Potential of mean force

Nonlinear time

Comparison Between Theory and Simulation Comparison Between Theory and Simulation

AA pair AB pair

BB pair

Nearest neighbor pair

Mean-field model successfully describes the dynamics of the AA and AB pairs

Relative diffusion considered as over-damped motion in an effective potential, occurs mainly via hopping

The agreement for the BB pair is less satisfactory !

Nearest neighbor BB pair executes large scale anharmonic motions in a weak effective potential

The fluctuations about the mean-force field experienced by the BB pair are large and important !Next nearest neighbor BB pair

Better agreement compared to nearest neighbor pair

Angell´s ‘strong’ and ‘fragile’ classification

5 : Diffusion and viscosity in a highly supercooled polydisperse system

5 : Diffusion and viscosity in a highly supercooled polydisperse system

exp exp o

o o

E DTT A A

T T T T

exp , 1

T

tt T

T

Fragile liquid: Super-Arrhenius follows VFT equation

Accompanied by Stretched exponential relaxation

Small D more fragile

Progressive decoupling between DT and (DT -, < 1), in contrast to the high T behavior ( = 1 ; SE relation) :

Temperature Dependence of ViscosityTemperature Dependence of Viscosity

Super-Arrhenius behavior of viscosity

VFT fit

Critical temperature for viscosity To

= 0.57

Within the temperature range investigated, Angell´s fragility index, D 1.42

A very fragile liquid

More fragile than Kob-Andersen

Binary mixture, D 2.45

Arrhenius plot

Temperature Dependence of Diffusion CoefficientsTemperature Dependence of Diffusion Coefficients

Arrhenius plots Diffusion shows a super-Arrhenius T dependence

Particles are categorized into

different subsets of width 0.05

0.1, 0.75 0.8 , 1.2 1.25s ls

( ) exp i

i i

Di D D oD T A E T T VFT law

0.46

0.50

0.478

s

l

Do

Do

Do

T

T

T

Critical glass transition temperature for diffusion

Critical temperature depends on the size of the particles !

Dl

Ds

Critical Glass Transition Temperature for Diffusion : Particle Size Dependence

Critical Glass Transition Temperature for Diffusion : Particle Size Dependence

increases with size of the particles iDoT iR

0.46 0.50 0.57s lD Do o oT T T

Size only

Size + mass

Near the glass transition the diffusion is partly decoupled from the viscosity, and for smaller particles the degree of decoupling is more

The increase of critical temperature with size is not an effect of mass polydispersity related to the dynamical heterogeneity induced by geometrical frustration

Size Dependence of Diffusion Coefficient : Breakdown of Stokes-Einstien Relation

Size Dependence of Diffusion Coefficient : Breakdown of Stokes-Einstien Relation

T* = 0.67

SE relation

A marked deviation from Stokesian behavior at low T

A highly nonlinear size dependence of the diffusion

20.0011 0.00132 1 0.000442 1i i iD R R

For the smallest size particles, Ds -0.5

At low T, the observed nonlinear dependence of diffusion on size may be related to the increase in dynamic heterogeneity in a polydisperse system

Self-part of the van Hove correlation functionSelf-part of the van Hove correlation function

Smallest particle

Largest particle

T* = 0.67

The gradual development of a second peak at r 1.0 indicates single particle hopping

For the larger particles hopping takes place at relatively longer times

The Self-intermediate Scattering FunctionThe Self-intermediate Scattering Function

Smallest

Largest

T* = 0.67, 2k

( , ) expi

is

i

tF k t

The long time decay of Fs(k,t) is well fitted by the Kohlrausch-Williams-Watts (KWW) stretched exponential form :

Particle i i

Smallest 242 0.49

Largest 717 0.64

The enhanced stretching (s l) is due to the greater heterogeneity probed by the smaller size particles

6: Hetergeneous relaxation in supercooled liquids: A density functional theory analysis

6: Hetergeneous relaxation in supercooled liquids: A density functional theory analysis

Recent time domain experiments, het 2-3 nm

Spatially heterogeneous dynamics in highly supercooled liquids

Near Tg , dynamics differ by 1-5 orders of magnitude between the fastest and slowest regions

Why do these heterogeneities arise ?

RI=4.0

RI=2.5RI=1.5

Hard sphere liquid

1exp k k

kPS kS k

Large free energy cost to create larger inhomogeneous region

S(k) is nearly zero for small k, density fluctuation only in intermediate k

Unlikely to sustain inhomogeneity, lf 5

22

( ) ( ) exp( )R

R

tC t d P

2

( ) exp( )R

td P

2 20( )R Rdt C t

( ) expog

C

Rotational Dynamics in Relaxing Inhomogeneous Domains Rotational Dynamics in Relaxing Inhomogeneous Domains

VFT form

Orientational correlation function

Av. Rotational correlation time

RI=2.5

1.04

1.1g

The decay is nonexponential and av. correlation time is increased by a factor 1.8

Increase in , slower regions become slower at a faster rate

7 : Isomerization dynamics in highly viscous liquids 7 : Isomerization dynamics in highly viscous liquids

, 1 0isok

Isomerization reactions involve large amplitude motion of a bulky group

Strongly coupled to the enviornment

For barrier frequency, b 1013 s-1, the situation is not starightforward

reactive motion probes mainly the elastic (high frequency) response of the medium

At high viscosities, experiments and simulations predict

28 3 ( )E Bm mk T g

Enskog friction

Frequency-Dependent Friction from Mode-Coupling Theory

Frequency-Dependent Friction from Mode-Coupling Theory

In the high frequency regime the total(z) is much less than E and is dominated entirely by B(z)

E always overestimate total(z) for continuous potential

*=0.85 T*=0.85

Frequency Dependent Viscosity Frequency Dependent Viscosity

MCT

Maxwell relation

Maxwell viscoelastic model fails to describe higher frequency peak : even poorly describe low-frequency peak

The two-peak structure is a clear indication of the bimodal response of a dense liquid

*=0.85 T*=0.73

1 si

Maxwell relation

0s G

Viscoelastic relaxation time

GHr

TSTb

k

k

Transmission coefficient

Barrier Crossing Rate Barrier Crossing Rate

*=0.6-1.05, T*=0.85

b (s-1)

31012 0.72

51012 0.58

1013 0.22

21013 0.045

strongly depends on b

b 21013 s-1, 0 TST result

The values of the exponent appear to be in very good agreement with many experimental results

ACKNOWLEDGEMENTACKNOWLEDGEMENT

Dr. Rajesh Murarka (Berkeley)Dr. Sarika Bhattacharyya (CalTech)Dr. Goundla Srinivas (UPenn)

DSTCSIR