Slow Dynamics in Binary Liquids : Microscopic Theory and Computer Simulation Studies of Diffusion, D

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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
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Outline
Introduction
Solvation Dynamics in Binary Mixture.
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
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Polarization Relaxation in Binary Dipolar 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.

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Definition of Non-ideality

• Raoults Law

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Dynamics of Solvation
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Local composition fluctuations in strongly
nonideal binary mixtures
Spontaneous local fluctuations ? rich and complex
behavior in many-body system
What is the probability of finding exactly n
particle centers within ?V(R) ?
R
?V(R)
In one component liquid ? local density
fluctuations are Gaussian
Binary mixtures that are highly nonideal, play an
important role in industry
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N P T simulations of Nonideal Binary
Mixtures Study of Composition Fluctuations
Two model binary mixtures
Kob-Andersen model (glass-forming mixture) Equal
size model
xA 0.8 xB 0.2
mA mB m
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Probability Distributions of Composition
Fluctuation
Kob-Andersen Model
R 2.0?AA
T 1.0 P 2.0
?NA? 27.3 ?A 1.995
Gaussian distribution
? NB ? 6.74 ?B 1.995
Both A and B fluctuations are large
System is indeed locally heterogeneous
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Joint Probability Distribution Function
Kob-Andersen Model
R 2.0?AA
Nearly Gaussian
Corr?NA , ?NB - 0.203 ? Fluctuations in A and
B are anticorrelated
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Dynamical Correlations in Composition
Fluctuation Kob-Andersen Model
Stretched exponential fit
R 2.0?AA P 2.0
Slow Dynamics
R 2.0?AA P 4.0
Non-exponential decay Distribution of relaxation
times
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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
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Relaxation of Solute and Solvent The
Self-intermediate Scattering Function Fs(k,t)
Solute
Fs(k,t) begins to stretch at long time for higher
solvent mass !
kk?112?
MR5 MR25 MR50 MR250
Sum of two stretched exponential function
Solvent
No stretching at long times !
Exponential decay
Solute probes progressively more local
heterogeneous environment
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Non-Gaussian Parameter
Solute
MR5 MR25 MR50 MR250
The peak height increases ? heterogeneity probed
by the solute increases with solvent mass
No such increase for the solvent
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The Self-intermediate Scattering Function of the
Solute
k2???12
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
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The Velocity Autocorrelation Function of the
Solute Particles
MR25 MR50 MR250
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
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Generalized self-consistent scheme



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Self-consistent scheme overestimates diffusion
(faster decay of Fs(k,t))
Gaussian approximation is poor
The relative contribution of the binary term
decreases with solvent mass
Contribution of the density mode increases !
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Comparison of MCT prediction with simulations
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

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4 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
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Radial Part of the Time Dependent Pair
Distribution Function (TDPDF)
The TDPDF, g2(ro,rt), 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

?t500?
Nearest neighbor AA pair
Jump motions are the dominant diffusive mode by
which the separation between pairs of atoms
evolves in time
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Angular Part of the Time Dependent Pair
Distribution Function (TDPDF)
Nearest neighbor pair
AA pair
AB pair
BB 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
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Relative Diffusion Mean-Square Relative
Displacement (MSRD)
Nearest neighbor pair
Faster approach of the diffusive limit of BB pair
separation
Relative diffusion coefficients
Time scale needed to reach the diffusive limit is
shorter for the AB pair than that for the AA pair
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The Non-Gaussian Parameter for the Relative Motion
Single particle dynamics
Pair 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
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Theoretical Analysis
Mean-field Smoluchowski equation
Potential of mean force
Nonlinear time
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Comparison Between Theory and Simulation
Nearest neighbor pair
AA pair
AB pair
Mean-field model successfully describes the
dynamics of the AA and AB pairs
BB pair
Relative diffusion considered as over-damped
motion in an effective potential, occurs mainly
via hopping
The agreement for the BB pair is less
satisfactory !
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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
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5 Diffusion and viscosity in a highly
supercooled polydisperse system
Fragile liquid Super-Arrhenius ? follows VFT
equation
Small D ? more fragile
Accompanied by Stretched exponential relaxation
Angells strong and fragile classification
Progressive decoupling between DT and ? (DT ?
?-?, ? lt 1), in contrast to the high T behavior
(? 1 SE relation)
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Temperature Dependence of Viscosity
Arrhenius plot
Super-Arrhenius behavior of viscosity
Critical temperature for viscosity To? 0.57
VFT fit
Within the temperature range investigated,
Angells fragility index, D ? 1.42 ? A very
fragile liquid
More fragile than Kob-Andersen Binary mixture, D
? 2.45
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Temperature Dependence of Diffusion Coefficients
Diffusion shows a super-Arrhenius T dependence
Arrhenius plots
Particles are categorized into different subsets
of width
Ds
VFT law
Dl
Critical glass transition temperature for
diffusion
Critical temperature depends on the size of the
particles !
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Critical Glass Transition Temperature for
Diffusion Particle Size Dependence
increases with size of the particles
Size only
? Near the glass transition the diffusion is
partly decoupled from the viscosity, and for
smaller particles the degree of decoupling is more
Size mass
The increase of critical temperature with size is
not an effect of mass polydispersity ? related to
the dynamical heterogeneity induced by
geometrical frustration
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Size Dependence of Diffusion Coefficient
Breakdown of Stokes-Einstien Relation
A marked deviation from Stokesian behavior at low
T
T 0.67
? A highly nonlinear size dependence of the
diffusion
SE relation
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
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Self-part of the van Hove correlation function
T 0.67
Smallest particle
The gradual development of a second peak at r ?
1.0 indicates single particle hopping
Largest particle
For the larger particles hopping takes place at
relatively longer times
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The Self-intermediate Scattering Function
T 0.67,
The long time decay of Fs(k,t) is well fitted by
the Kohlrausch-Williams-Watts (KWW) stretched
exponential form
Largest
Smallest
The enhanced stretching (?s ? ?l) is due to the
greater heterogeneity probed by the smaller size
particles
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6 Hetergeneous relaxation in supercooled
liquids A density functional theory analysis
Spatially heterogeneous dynamics in highly
supercooled liquids
Recent time domain experiments, ?het ? 2-3 nm
Near Tg , dynamics differ by 1-5 orders of
magnitude between the fastest and slowest regions
Why do these heterogeneities arise ?
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Hard sphere liquid
Large free energy cost to create larger
inhomogeneous region
RI4.0?
RI2.5?
RI1.5?
S(k) is nearly zero for small k, density
fluctuation only in intermediate k
Unlikely to sustain inhomogeneity, lf ?5?
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Rotational Dynamics in Relaxing Inhomogeneous
Domains
RI2.5?
VFT form
Orientational correlation function
Av. Rotational correlation time
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
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7 Isomerization dynamics in highly viscous
liquids
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
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Frequency-Dependent Friction from Mode-Coupling
Theory
?0.85 T0.85
Enskog friction
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
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Frequency Dependent Viscosity
?0.85 T0.73
Maxwell relation
Maxwell relation
MCT
Viscoelastic relaxation time
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
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Barrier Crossing Rate
?0.6-1.05, T0.85
Transmission coefficient
? strongly depends on ?b ?b ? 2?1013 s-1, ? ? 0
? TST result

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

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