Breakdown of Stokes-Einstein Relationship

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Breakdown of Stokes-Einstein Relationship
and
pedal-like motion in stilbene
Yashonath Subramanian1 , 1,2Solid State
Structural Chemistry Unit, Indian Institute of
Science, Bangalore-560012 3Theoretical Science
Unit, Jawaharlal Nehru Centre for Advanced
Scientific Research
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Dipartimento di Chimica, Universita di Sassari,
May, 2008
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Indian Institute of Science
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Acknowledgements
? Thankful to my hosts Prof. Suffritti, Prof.
Demontis and Dr. Marco Masaia Universita di
Sassari for Visiting Professorship. ? Many past
and present students Dr. P. Santikary,
USA Dr. Sanjoy Badyopadhyay, India Dr. R.
Chitra, India Dr. A.V. Anil Kumar, Australia
Dr. C.R. Kamala, USA Dr. S.Y. Bhide, USA
Dr. P. Padmanabhan, USA Ms. Manju Sharma, Mr.
Bhaskar Borah, Mr. Srinivas Rao.
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Our interest

• Diffusion of hydrocarbons in zeolites and carbon
  nanotubes.
• Diffusion in liquids, dense solids (crystalline
  and amorphous)
• Ionic conductivity in polar solvents
• Simulation of Phase transitions in organic
  molecular solids

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Diffusion in dense and porous medium

• It is very well known that diffusion proceeds by
  different mechanisms in different medium. For
  example, diffusion in porous solids, has a
  Knudsen regime that is absent in dense liquids.
• Here we ask if there is any underlying common
  principles governing diffusion in these widely
  differing systems.

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Diffusion in dense fluids
1. Introduction LE (Levitation Effect)
relation 2. Similarity between porous and
dense medium at a conceptual level 3. Binary
liquid mixture (dense liquid) Computational
Details 4. Results and Discussion
? Four sets corresponding to different degrees
of host-guest dynamics ? Diffusion
maximum in a dense liquid and solid ?
Activation energies and friction size
dependence ? k dependence of the fwhm
of the self part of the dynamic
structure factor ? Decay of Fs(k,t)
? Model of Singwi and Sjolander
Pradip Ghorai, S. Yashonath, J. Phys. Chem. B,
(5th March, 2005).
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Introduction
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Structure of zeolite Y
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Levitation Effect (LE)
sorbate in zeolites/other crystalline porous
solids
Two distinct regimes ? Linear regime
(LR) D ? 1/?gg2 (for ?gg ltlt ?void )
? Anomalous regime (AR) where D
exhibits a maximum (for ?gg ? ?void )
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A larger sorbate diffuses faster than the
smaller sorbate.
Why ?
The force on the sorbate due to the zeolite
essentially tethers it to the zeolite wall
thereby reducing the diffusion coefficient.
However, when the sorbate size is similar to that
of the void then the force on it from one side of
the wall or zeolite cancels with the other side
of the wall or zeolite. This mutual cancellation
essentially ensures that the sorbate, although
confined, is effectively free (or more precisely,
nearly free). This leads to an increase in the
self diffusivity. The condition for mutual
cancellation of forces (and for the maximum in D)
can be stated more precisely in terms of the
levitation parameter, ?. It is defined as the
These arguments are originally due to Kemball.
More recently, Derouane and coworkers have
discussed these. The latter passed way recently
due to ill-health.
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• LR is Characterized by
• High activation energy
• High friction and high force
• Associated with an a highly undulating potential
  energy landscape, along the diffusion path (large
  amplitude undulations) etc
• AR is Characterized by
• Low activation energy
• 2) Low friction and lower force.
• 3) Associated with a flat potential energy
  landscape along the diffusion path, etc

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Dense fluids and close packed solids
? A f.c.c solid has a packing fraction of around
0.74. In other words even in a close packed
solid, a reasonably large fraction of void space
of around 0.26. ? Typically in an
f.c.c. solid of N atoms, there exist N octahedral
voids and 2N tetrahedral voids. They have a
diameter of 0.45R and 0.828R where R is the
radius of the spheres which make up the solid.
Can a particle move through this void space
? The answer is clearly yes since we know that
this is how diffusion within solids occur. (see
Azaroff, L.V, Introduction to solids, TMH,
New Delhi, 1990) ? What is the size of the
particle that can move through such a solid ?
Clearly it will be much smaller than R. We
refer to sphere which make up the solid (of
radius R) as host and the smaller sphere which
diffuse (more easily) as the guests. The
neck dimension (defined as the narrowest part of
the void between two voids) which
interconnect two (tetrahedral or octahedral)
voids is 0.155R.
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? Clearly in a liquid the voids are of
relatively larger size (except probably in
water known for its anomalous expansion on
freezing). ? The question we ask is
Does LE exist in close packed or dense
solids and liquids ?
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Details of simulation
Four sets of calculations have been performed.
The parameters of these four sets have been
selected so as to correspond to different ratios
of the dynamics of host to the guest.
Dg/Dhvaries from 3035 (for set I) to just 4 for
(set IV). This is just to check if the maximum
exists when the host liquid has a relaxation time
as fast as the guest.
The parameters for the four sets
Set ?hh(kJ/mol) mh(amu)
I 1.84 85
II 0.99 85
III 0.25 85
IV 0.25 40
?hh 4.1A mg 40amu ?gg 0.99 kJ/mol ?gh 1.5
kJ/mol and ?gh ?gg 0.7A (non
Lorentz-Berthelot rule (see, for example, M.
Parrinello, A. Rahman, P. Vashistha, Phys.
Rev. Lett., Phys. Rev. Lett.,50, 1073 (1983) )
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Voronoi polyhedral analysis
? We calculate the void and neck distribution
that exists amidst the host (not taking into
account the guest) through the construction of
the Voronoi and Delaunay tesselation as
done previously by several groups. see D. S.
Corti, P. G. Debenedetti, S. Sastry and F. H.
Stillinger,Phys. Rev. E,55,5522 (1997) S.
Sastry, D. S. Corti, P. G. Debenedetti and F. H.
Stillinger, Phys. Rev. E,56,5524 (1997). ?
These have been carried out using the algorithm
of Tanemura et al. see M.Tanemura, T. Ogawa,
N. Ogita, J. Comput. Phys. 51, 191, (1983). ?
This is required to obtain an estimate of the
guest size that can pass through the voids
and necks. Further, it will also indicate the
size at which the diffusion maximum will be
observed based on the value of the levitation
parameter, ?.
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Two dimensional illustration of Voronoi-Delaunay
construction

Taken from D. S. Corti, P. G. Debenedetti, S.
Sastry and F. H. Stillinger, Phys. Rev. E,55,5522
(1997)
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MD Simulation Details

• ? Number of host particles Nh 500
• guest particles Ng50
• ? Simulation cell length, L 33.3A
• ? Time step ?t 5.0 fs
• ? Cut off radius 16.5 A
• ? Positions and velocities stored
• every 0.25ps (once in 50 MD steps)
• Equilibration 1.0ns Properties
• accumulated over 1.0ns

• all simulations in the
• microcanonical ensemble (NVE)
• with better than 1 in 104
• conservation.
• ? reduced density, ? 0.933
• ? reduced temperature, T
• 0.226 (set I), 0.420 (set II),
• 1.663 (set III and IV) all at 50K.

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?Snapshots of the host structure
Disordered f.c.c. solid with defects
Amorphous solid
Liquid with faster dynamics of the host
Liquid
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Host radial distribution function
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Void and neck distribution
Blue curve (set I) is that of a solid and the
two void distributions corresponding to
octahedral and tetrahedral void distributions
Stillinger et al, Phys. Rev. E, 55, 5530 (1997).
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Self-Diffusivity Values for All
Sets set I set II set III
set IV ?gg (Å) D (10-8m2/s) 0.3
1.41 1.70 1.72 1.73
0.4 1.07 1.20 1.26
1.44 0.5 0.93 0.98 1.01
1.32 0.7 0.74 0.73 0.85
1.21 0.8 ---- ---- 1.11
1.43 0.9 1.02 1.10 1.23
1.49 1.0 ---- ---- 1.02
1.34 1.1 1.31 1.17 0.92
1.06 1.3 0.87 0.88
0.56 0.68 1.5 0.45 0.37
0.20 0.46
The maximum shifts to smaller values with
increase in disorder
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Diffusivity as a function of sorbate size
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Diffusivity vs ?, the Levitation parameter
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Fgh for different sizes of the guest
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Velocity auto correlation function

• Negative correlation
• for ?gg0.7Å for all
• sets of parameters.
• p.e. landscape is flat
• for ?gg0.9 Å

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k-dependent self diffusion coefficient, D(k)

• From neutron scattering measurements, we are
  aware that the width of the self part of the
  dynamic structure factor provides an estimate of
  the self diffusivity. Thus, the k-dependence of
  the full width at half maximum (fwhm) of the
  Ss(k,w), ?? is useful and note that it depends on
  k. In the hydrodynamic limit (k ? 0), ??(k) ?
  2Dk2,or ??(k)/2Dk2 ? 1. Phenomenologically
  speaking, ??(k) gives us a k-dependent D, or
  D(k).
• Unlike, the self diffusivity obtained from
  Einsteins expression
• D lim u2(t)/2dt
• t??
• where u2(t) is the mean squared displacement,
  which is the self diffusivity in
• in the long time limit, the above provides us
  with a more detailed D(k).
• In fact, previously, Nijboer and Rahman (Physica,
  32, 415 (1966) and Levesque and
• Verlet (Phys. Rev. A, 2, 2514 (1970)) have
  computed this quantity for argon liquid for
• High density, low temperature fluid ??
  0.8442, T 0.722
• 2) Low density, high temperature fluid ??
  0.65, T 1.872

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Nijboer and Rahmans result on a high dense,
low temperature fluid
? 0.8442, T 0.722 Liquid argon
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Levesque and Verlets result a low density,
high temperature fluid
? 0.65, T 1.872
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??/2Dk2 as a function of k
oscillating
Smooth decay
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Interpretation

• A lowering of ??(k)/2Dk2, at some wavevector
  suggests lowered D at that k.
• Note that a smaller guest has a lower value of D
  at k 0.9A-1.
• Surprizingly, a bigger guest (0.9A) size, has no
  lowering of D at this k and therefore no
  difficulty at this k.
• This suggests that for the 0.7A particle, the
  difficulty at k 0.9A-1, should
• lead to two time scales, one for motion at small
  distances and another at long distances. These
  should be seen in decay of the density-density
  correlation function, Fs(k,t) at small k (or long
  distance).
• For the larger guest (of 0.9A or larger size), a
  single decay should be seen.

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Decay of Fs(k,t) for linear regime particle
F (k,t)
s
t(ps)
t(ps)
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Decay of Fs(k,t) for anomalous regime particle
F (k,t)
s
t(ps)
t(ps)
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• Values of ?1 for the Particle in the Anomalous
  Regime and ? 1
• and ? 2 for the Particle in the Linear
  Regime for Sets III
• and IV
• ? gg 0.7 Å ?
  gg 0.9 Å
• k 0.57 Å-1 k 0.76 Å-1
  k 0.57 Å-1 k 0.76 Å-1
• Set ? 1 ? 2 ? 1
  ? 2 ? 1 ?
  1(ps)
• III 1.84 9.09 1.01
  4.13 3.71
  2.23
• IV 1.15 4.60 1.32
  2.50 3.17
  2.01

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• Self-Diffusivity Values at Different
  Temperatures for Set III
• for Two Different Sized Particles, One from
  the Linear
• Regime (0.7 Å) and Another from the
  Anomalous Regime (0.9 Å)
• Temperature (K) D
  ( 108 m2/s)
• 
  0.7A 0.9A
• 50
  0.85 1.23
• 70 2.03 1.88
• 100 3.42 3.12
• 150 5.20 3.89

E(0.7A) 1.21 kJ/mol E(0.9A) 0.77 kJ/mol
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Stokes-Einstein relation

• Stokes relation
• Frictional force f on a spherical solute is
  given by
• f
  6??a
• where a is the solute radius and ? is the solvent
  viscosity.
• ? Einstein relation

• where D is the diffusion coefficient of the
  solute and T is the temperature.
• Combining the two equations we get the well
  known
• Stokes-Einstein relationship

where a ?/2, ? is the solute diameter.
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Breakdown in Stokes-Einstein relation
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Implications of the breakdown

• van der Waals interaction plays an important role
  in enhancing D when solute size is about 1/4th of
  the solvent.
• Experimental proof required. We shall be happy to
  collaborate/assist in any such ventures.
• Breakdown will likely be more easily observable
  in systems dominated by electrostatic
  interactions (e.g., ions in water).

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Implications of the existence of LE or diffusion
maximum in dense liquids and solids
Our understanding of the transport in condensed
media is altered. There exist in the literature
in physical chemistry experimental as well as
theoretical and computational studies of motion
of ions/solutes etc in solvents. In materials
science motion of an impurity within close packed
solids is important in corrosion and alloys. Here
also these results have implications. In biology
ion motion within biomembranes or even ion motion
in water can exhibit anomalous behaviour. We
take the last as an example to show what the
present results imply.
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Size dependence of Ionic Conductivity in solvents
It is well known (breakdown of Waldens rule)
that smaller ions such as Li does not have the
maximum conductivity. Larger ions such as Cs has
a higher ionic conductivity. This is generally
true for any ion in any solvent. Theories such
as Solvent-berg model, continuum theories
(proposed first by MaxBorn, and later developed
by Onsager, Zwanzig, etc) to explain this
observation have suggested that this can be due
to dielectric friction arising from relaxation
of the solvent around an ion. However, this does
not explain all the known experimental
observations. We have recently carried out
studies (Pradip Ghorai, S. Yashonath, R.M. Lynden
Bell, J. Phys. Chem. B, (to appear)) on ion
motion in water (charge and size dependence).
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Ion in water dependence of D on ion size
positive ion
negative ion
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Conclusions

• Widely differing systems such as porous solids,
  close-packed solids, simple liquids, ions
  dissolved in water, ion in solids (such as
  nasicon or AgI) etc exhibit similar size
  dependent maximum in self diffusivity.
• This maximum is therefore ubiquitous, generic and
  universal.

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Pedal-like motion in stilbene (Molecular pedals)
Experimental results
Temperature dependence of the difference Fourier
map indicated two residual peaks which
disappeared at low temperature and increased in
intensity at higher temperatures. The
disorder has been attributed to the
inter-conversion between the conformers through
pedal-like motion.
! INDICATION OF DYNAMICAL DISORDER
Harada, J. Ogawa. K. J. Am. Chem. Soc., 123,
10884(2001)
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Disorder at site 2 and at still higher
temperatures even at site 1
Earlier structural studies report the disorder
only at site 2. Recent studies by Ogawa and
Harada Report the disorder at site 1.
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Objectives of our calculations
Are site 1 molecules too disordered? What are
the two transitions observed from
Raman spectroscopic measurements in the T range
115-375 K Does the anomalous ethylene bond
length variation exist or is it an artifact of
fitting procedure used in the disorder model for
solving structure?
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Snapshot taken from actual simulation
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Structure
Validity of the potential model
This potential model is able to predict the
structural quantities well but for b. where
the deviation is 6-10.
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Dynamical disorder?
Pedal-like motion seems to occur at temperatures
higher than 200K (actually at 180K). The
energies of the minor conformer is not equal to
the major conformer (with 0o dihedral angle)
which is not the case in the gaseous phase.
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Nature of transition
Thermal expansion coefficient
At 170K corresponding to the onset of
disorder at site 2 At 250K corresponding to the
onset of disorder at site 1
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Explaining the two transitions in the range
113K-375K
Raman spectrocopic studies report two
transitions in the temperature range 113K-375K
which is comparable to this observation. The
first transition can be attributed to the
disorder occurring at site 2. The second
transition can be attributed to the disorder at
site 1.
Chakrabarti, S. Misra, T.N. Bull. Chem. Soc.
Jpn., 64, 2454(1991).
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• Thank you

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