calculating diffusion

8/4/2019 Calculating Diffusion

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Calculation of Transport Coefficients from MD

Simulations in Grant Smiths group.

1)Self-diffusion coefficient (D) (equilibrium MD)2)Shear viscosity () (equilibrium MD)3)Thermal conductivity () (non-equilibrium MD)

References:

1) J.M. Haile,Molecular Dynamics Simulation, (Wiley, NY, 1992).2) M.P. Allen, D.T. Tildesley, Computer Simulation of Liquids, (Oxford, NY, 1987).3) P.J. Daivis and D.J.Evans,J. Chem. Phys. 103, 4261 (1996).4) M. Mondello and G.S. Grest,J. Chem. Phys. 106, 9327 (1997).5) D.K. Dysthe, A.H. Fuchs and B. Rousseau,J. Chem. Phys. 110, 4047 (1999).6) D. Bedrov, G.D. Smith and T.D. Sewell,J. Chem. Phys. submitted.7) F. Muller-Plathe, J. Chem. Phys. 106, 6082 (1997).8) D. Bedrov and G.D. Smith,J. Chem. Phys. submitted.


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Any transport coefficient (K) can be calculated using

generalized Einstein and Green-Kubo Formulas:

where

K A(t) )(tA

Self-diffusion ri(t) vi(t)

Shear viscosity miri(t)vi(t)

Thermal conductivity ri(t)Ei(t) Jq

whereri(t) atom position at time t, vi(t) velocity of atom, m-

mass, -component of the stress tensor andJq-heat flux.

[ ]

==

0

2)0()(2/)0()(lim AAdtAtAK

t


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Self-diffusion coefficient

t

RtRD cmcm

t 6

)]0()([lim

2

= (1)

where Rcm(t)-Rcm(0) is the time dependent center-of-mass

displacement of a given molecule. means averaging overall possible time origins. (see refs. (1) and (2) for more

information).

During our simulation, usually every 1 ps, we output the

positions of all atoms in fort.77 (binary file) which allows

us to calculate D by using analysis subroutine (msd.f ask

Oleg). (Note that before running the long trajectory it

make sense to check that during 1ps (output frequency)

none of the atoms moves more than simulation box size!).

DME diffusion for SCF

time (ps)

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

R2

/6t

0.0

0.1

0.2

0.3

0.4

0.5

0.6

XDME

=0.004

XDME

=0.04

XDME

=0.18

XDME

=0.42

XDME

=0.72


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Shear Viscosity

It was shown by Daivis and Evans that, for an isotropicsystem, the convergence of viscosity calculations can be improved

by including equilibrium fluctuations of diagonal components of

the stress tensor. In this case the generalized Green-Kubo formula

is applied to the symmetrized traceless portion (P) of the stress

tensor with appropriate weight factors for diagonal and off-

diagonal elements:

dtPtPqTk

V

B

=0

)0()(10

,

where Vand Tare volume and temperature of the system, kB is the

Boltzmann constant, q is a weight factor (q=1 if ,q = 4/3 if=), and P is defined as

+=

3

2/)(P

where is the Kronecker delta.


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Haile has shown that in a system with periodic boundary

conditions the viscosity cannot be calculated using the

conventional Einstein formula that involves atomic coordinatesand velocities. However, it can be employed after slight

modificationsyielding[ ]

[ ]2

2

)(20

lim

)0()(20

lim

tAqTtk

V

AtAqTtk

V

Bt

Bt

=

=

where

=

t

dttPtA

0

')'()(

It was shown by Mondello and Grest that this approach gives the

same results as the Green-Kubo formulation for MD simulations

of short-chain alkanes.

In our calculation we employ this Einstein formulation.


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Important parameters in the simulation

1)output frequency of the stress tensor

time [fs]

0 100 200 300 400

stresstensor[atm]

-6000

-4000

-2000

0

2000

4000

6000

1fs output

10fs output

20fs output


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Therefore we output the stress tensor every 10fs. (Note that it

is a system dependent parameter, find appropriate frequency for

your system!)

time [ps]

0 200 400 600 800 1000

(asr.h.s.ofeq

.3)[Pa*s]

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

1fs output

10fs output

20fs output

40fs output


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2)length of the trajectory

Criteria for convergence: with additional statistics the average

viscosity fluctuates (+/- 10%) around some mean value.

tsim

[ns]

0 5 10 15 20 25

[Pa*s]

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.050

0.055

T=700K

T=650K


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Thermal conductivity

Calculation of thermal conductivity using equilibrium

MD simulations suffers two major problems (from our

point of view):

1)As in case for shear viscosity the Einstein relationcannot be applied in system with periodic boundary

conditions and therefore the Green-Kubo formula

which uses theheat flux fluctuations should be

employed.

2)In systems with many-body interactions (Ewaldtreatment of the long range electrostatic interactions)

theheat flux is not rigorously defined(at least nobody

derives this in the literature and we were not able to

derive this as well)

Therefore we employed recently developed non-

equilibrium MD method developed byMuller-Plathe.


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General idea of the method is to employ exactly known

heat flux in the system and measure the established

temperature gradient.

Cold slab Hot slab

The heat flux is created by exchanging velocity vectors of

particles in the cold and hot slabs. One of the slabs is located

in the middle of the simulation box and the other is adjacent to

one of the simulation box boundaries. The cold slab donates the

hottest particles (particles with the highest kinetic energy) in the

slab for the exchanging procedure while the hot slab donates its

coolest particles (particles with the lowest kinetic energy).This

process is reaching a steady state with constant temperature

gradient due to thermal conductivity through slabs, which arelocated between cold and hot slabs.


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The imposed heat flux will be known exactly

)(2

)(22

hc

transfers

z vvmtJ >=<

where vc and vh are velocities of the identical mass particles which

participate in the exchanging procedure in the cold and hot slabsrespectively. The temperature gradient can be easily measured by

calculating local temperature in each slab and therefore thermal

conductivity can be determined as

> is total heat flux imposed during simulation on the

system along z-direction, is established temperature

gradient in this direction, A is a surface area perpendicular to z-

axis, and tis total simulation time.


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In order to impose geometric constraints (bonds, bends etc.) in the

simulation using velocity Verlet algorithm one should use wellknown iterative schemes RATTLE or slightly modified from

original version SHAKE In the first scheme in addition to

constraint

= dij2 - (ri-rj)

2 = 0 (2)

where dij is a bond length between atoms i andj the coordinates of

which are ri and rj, another constraint on the velocities of atoms i

andj

(ri-rj)(vi-vj) = 0 (3)

is also required. In the SHAKE version of velocity Verlet

algorithm the condition (2) is satisfied automatically after

convergence of iterative procedure. It is clear that by exchangingthe velocity vectors of atoms participating in different constraints

the equality (3) will be compromised.


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In order to overcome this problem we suggest to exchange

the translational velocities of two molecules rather than to

exchange the velocity vectors of two atoms.

The velocity vector of any atom in the system can be expressed as

vi = vcm + vi

*

and it is trivial to show that total kinetic energy and linear

momentum is conserved after this procedure.

+++++

=+++=

N

jjj

N

jjjcm

N

jjcm

N

iii

N

iiicm

N

iicm

N

jjcmj

N

iicmi

vmvmvmvvmvmvmv

vvmvvmK

2222

22

**2**2

*)(*)(

+++=N

jjjcm

N

iiicm

vmMvvmMvK 2222 **

The proposed technique should work (it will be demonstrated

below) for small molecules, the radius of gyration of which is

comparable with the width of the slab. For systems with the long

polymers this algorithm still would suffersome troubles.


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Important parameters in the simulation

1)frequency of the velocity exchange procedure2)length of the simulation

z A

0 5 10 15 20 25 30 35 40

TK

130

135

140

145

150

155

160

165

170

W=0.0005 fs-1

W=0.001 fs-1

W=0.002 fs-1


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calculating diffusion

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