slow dynamics in binary liquids : microscopic theory and computer simulation studies of diffusion,...
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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