PHYSICAL INTERPRETATION OF ATOMISTIC

SIMULATIONS

Interpretation of the results of an atomistic simulation depends on the physical problem studied. However, methods of interpretation fall broadly into two categories: (i) Analyses of the structures. (ii) Evaluation of physical quantities of the system studied. In the case of MD and

MC the temperature and pressure dependence of such quantities is analyzed using statistical mechanics.

ATOMIC STRUCTURE

Structures obtained as the result of calculations may be interpreted in many different visual ways, most commonly by a display of atomic positions and/or local quantities such as local atomic volume, stress, coordination etc. A quantity which describes the average structural features of any system whether crystalline or amorphous is the radial distribution function, also called pair correlation function.

Radial distribution function (RDF)

Let ni (r) be the number of particles found at distances between r and r+Δr from a particle i. We define the average number of particles found between r and r+Δr from any particle of the system as

n(r) = 1

Nni (r)

i=1

N

! , (G26a)

where N is the total number of particles in the system.

r !

r

i

2

The average number of particles per unit volume found between r and r+Δr from any

particle of the system is

n(r)4!r2"r

in three dimensions and

n(r)2!r"r

in two dimensions.

The RDF, marked g(r), is then defined as

g(r) = 1

!n(r)

4"r2#rin three dimensions (G26b.1)

g(r) = 1

!n(r)

2"r#r in two dimensions (G26b.2)

where ρ = N/V is the average density of the material, defined as the average number of particles per unit volume (or surface in two dimensions). In this definition the RDF is normalized such that when

r! " g(r)! 1 since for large r, n(r)! 4"r2#r $ % for every particle in three dimensions; in two dimensions n(r)! 2"r#r $ %. If we know the RDF we can, of course, calculate the average number of particles at a distance interval R, R+ΔR from any particle of the system. It is simply equal to n(R) and thus in three dimensions it is 4!R2"R #g(r) # $ and in two dimensions 2!R"R #g(r) # $ .

r

Crystalline material

g(r)

1st neighbors 2nd neighbors 3rd neighbors

3

Examples of radial distribution functions Practical evaluation of RDF In practice the RDF is calculated by first counting the numbers of particles at various distances between r and r+Δr from each particle in the block and then taking the average over all the particles. In MD and MC calculations the RDF must be taken as a time average or ensemble average, respectively. If a physical quantity, A, depends only on the separation of the particles, then its average value in the three dimensional system is

A =4!V

A(r)g(r)r 2dr0

"

# , (G27)

Local atomic volume-Voronoi polyhedra

Volume associated with a given atom is never determined uniquely but it is often defined as the volume of the corresponding Voronoi polyhedron associated with this atom. It is constructed as follows:

r

g(r)Liquid

1

4

(i) Connect the given atom by straight lines with its neighbors. (ii) Draw planes bisecting these connecting lines. (iii) Construct the polyhedron formed by these planes. This construction is not completely unique but depends on how the neighbors are defined. Use of more distant neighbors may lead to many small areas on the polyhedron. Thus a cut-off of these areas is always introduced. Atomic level stresses

The stress in the studied block is given by equations (G15 a,b) which can be re-written as

!"#total = 1V $i!"#

i

i% (G28)

where

! i is the volume associated with the atom i so that

V = ! ii" , and

!"#i = 1

$i

%Ep%rik"

rik# &mivi"vi

#

kk' i

()

*

+++

,

-

.

.

.= & 1

$iFik"rik

# +mivi"vi#

kk' i

()

*

+++

,

-

.

.

. (G29)

is defined as the atomic level stress evaluated at the position of the atom i. The atomic level hydrostatic pressure is then

pi = !13" i

#Ep#rik

$ rik$ ! mivi

2

kk % i

&$=1

3

&'

(

) ) )

*

+

, , ,

=13" i

Fik$rik

$ + mivi2

kk % i

&$=1

3

&'

(

) ) )

*

+

, , ,

(G30)

The time averages of atomic level stresses, !"#i , give the average stress evaluated at

the position of the atom i. The average hydrostatic pressure

pi relates to the local volume. When it is large positive the local volume is smaller than average and the region is locally compressed; if it is large negative then the local volume is larger than average and the region is locally expanded. In the case of pair potentials, taking U = 0, the atomic level stress is

!"#i =

1$ i

d%(rik )drik

rik" rik

#

rikkk & i

' ( mivi"vi

#

i'

)

*

+ + +

,

-

.

. . (G31)

5

and the atomic level hydrostatic pressure is

pi = !13" i

d#(rik )drik

(rik$ )2

rik! mivi

2

kk % i

&$=1

3

&'

(

) ) )

*

+

, , ,

(G32a)

or

pi = ! 13"i

d#(rik )drik

rikkk$ i

%&

'

(((

)

*

+++

. (G32b)

In molecular statics calculations, when particles have no velocities, the terms containing

velocities are simply zero. However, in molecular dynamics calculations the terms with

velocities need to be included.