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

Dipartimento di Chimica, Universita di Sassari, May, 2008

2

Indian Institute of Science

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.

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

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.

Diffusion in dense fluids

Pradip Ghorai, S. Yashonath, J. Phys. Chem. B, (5th March, 2005).

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

Introduction

Structure of zeolite Y

2

Levitation Effect

Anomalousregime

Linear regime

S. Yashonath and P. Santikary,J. Phys. Chem. 98, 6368 (1994).

Two distinct regimes :

Linear regime (LR) :

D 1/gg2 (for gg << void )

Anomalous regime (AR) :

where D exhibits a maximum (for gg void )

Levitation Effect (LE) : sorbate in zeolites/other crystalline porous solids

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

A larger sorbate diffuses faster than the smaller sorbate. Why ?

7 / 6

/ /

2opt gh

w void w void

These arguments are originally due to Kemball. More recently, Derouane and coworkershave discussed these. The latter passed way recently due to ill-health.

LR is Characterized by :

1) High activation energy2) High friction and high force3) Associated with an a highly undulating potential energy landscape,

along the diffusion path (large amplitude undulations) etc

AR is Characterized by :

1) Low activation energy2) Low friction and lower force.3) Associated with a flat potential energy landscape along the

diffusion path, etc

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.

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 ?

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.

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) )

The parameters for the four sets

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, .

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)

MD Simulation Details

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.

Number of host particles Nh= 500 guest particles Ng=50

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

Snapshots of the host structure

Disordered f.c.c. solid with defects Amorphous

solid

Liquid Liquid with faster dynamics of the host

Host radial distribution function

Void and neck distribution

Stillinger et al, Phys. Rev. E,55, 5530 (1997).

Blue curve (set I) is that of a solid and the two void distributionscorresponding to octahedral and tetrahedral void distributions

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

Diffusivity as a function of sorbate size

Diffusivity vs , the Levitation parameter

Fgh for different sizes of the guest

Velocity auto correlation function

Negative correlation for gg=0.7Å for all sets of parameters.

p.e. landscape is flat for gg=0.9 Å

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 Einstein’s expression

D = lim u2(t)/2dt twhere 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 andVerlet (Phys. Rev. A, 2, 2514 (1970)) have computed this quantity for argon liquid for

1) High density, low temperature fluid : * = 0.8442, T* = 0.7222) Low density, high temperature fluid : * = 0.65, T* = 1.872

k-dependent self diffusion coefficient, D(k)

* = 0.8442, T* = 0.722Liquid argon

Nijboer and Rahman’s result : on a high dense, low temperature fluid

Levesque and Verlet’s result : a low density, high temperature fluid

* = 0.65, T* = 1.872

/2Dk2 as a function of k

oscillating

Smooth decay

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, shouldlead 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.

Interpretation

Decay of Fs(k,t) for linear regime particle

t(ps) t(ps)

F (

k,t)

s

Decay of Fs(k,t) for anomalous regime particleF

(k,

t)s

t(ps) t(ps)

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

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

Stokes relation :

Frictional force f on a spherical solute is given by

f = 6a

where a is the solute radius and is the solvent viscosity.

Einstein relation :Bk T

Df

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 :

6Bk T

Da

where a = /2, is the solute diameter.

Stokes-Einstein relation

Breakdown in Stokes-Einstein relation

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).

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 theliterature in physical chemistry experimental as well as theoretical and computationalstudies of motion of ions/solutes etc in solvents. In materials science motion of animpurity within close packed solids is important in corrosion and alloys. Here also theseresults have implications.

In biology ion motion within biomembranes or even ion motion in water can exhibitanomalous behaviour.

We take the last as an example to show what the present results imply.

Size dependence of Ionic Conductivity in solvents

It is well known (breakdown of Walden’s 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).

Ion in water : dependence of D on ion size

positive ion negative ion

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.

Pedal-like motion in stilbene(Molecular pedals)

Temperature dependence of the difference Fourier map indicated two residual peakswhich 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.

Experimental results

Harada, J.; Ogawa. K.; J. Am. Chem. Soc., 123, 10884(2001)

! INDICATION OF DYNAMICAL DISORDER

Disorder at site 2 and at still higher temperatures even at site 1

Earlier structural studies report thedisorder only at site 2.

Recent studies by Ogawa and HaradaReport the disorder at site 1.

Are site 1 molecules too disordered?

What are the two transitions observed from Ramanspectroscopic 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?

Objectives of our calculations

Snapshot taken from actual simulation

Structure

This potential model is able to predict the structural quantities

well but for b.where the deviation

is 6-10%.

Validity of the potential model

Dynamical disorder?

Pedal-like motion seems to occur at temperatures higher than200K (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.

Nature of transition:

At 170K corresponding to the onset of disorderat site 2

At 250K correspondingto the onset of disorderat site 1

Thermal expansion coefficient

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).

Explaining the two transitions in the range 113K-375K

Thank you