1. General Aspects of Modeling-Part2

8/9/2019 1. General Aspects of Modeling-Part2

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ENERGY OF THE SYSTEM, FORCES AND STRESSES

Let us consider a system composed of N particles the positions of which are

determined by vectors ri(i = 1,...., N); the Cartesian components of these vectors will

be denoted as ri

! where ! = 1, 2, 3. If several distinct species of particles are

involved an indicator of the species has to be assigned to each particle.The potential energy of this system, E p , is a function of the positions of the particles,

ri and here we assume we know how to calculate it. Hence

Ep = Ep(r1,r2,.....,rN) (G1)

Commonly the system is translationally invariant1and therefore

Ep(r

1,r

2,.....,r

N) =E

p(r

1+ t, r

2+ t,....., r

N+ t)

,

where t is a translation vector. Hence, E p may depend only on the relative positions

of the particles forming the system, i. e. on the vectors ri!rj(i ! j = 1,2,...,N) that

we denote rij

2and their components rij

! v i!

, where ! = 1, 2, 3. In a given structure

there are relations between the vectors rij since when N of these vectors have been

chosen geometrical relations determine all the other vectors.

At finite temperatures particles are moving and their motion is characterized by their

velocities v i(i=1,2, ...., N); their components are marked v i!

, where != 1, 2, 3. The

Hamiltonian of such system of particles is

H =1

2 mivi

2

i=1

N

! + Ep(r1, r2,.....,rN) (G2)

where mi is the mass of the particle i. Assuming that the block of particles does not

move as a rigid body, so that vi

i=1

N

! = 0 , the kinetic energy determines the temperature

of the system in accordance with the equipartition theorem. In the three-dimensional

case 12 mi vi2

i! =NkBT and in the two-dimensional case 12 mi vi

2

i! = 22NkBT ; kB

is the Boltzmann constant.

1If there is an external force field interacting with the system studied such that the corresponding interaction energy

depends on the location of individual atoms and not only on their relative positions, this assumption is not valid.2 It should be emphasized that this does not mean that E p is only a function of the separations of the atoms i and j

rij = ri ! rj . Of course, in the case of pair potentials, such as the Lennard-Jones potential, it is.


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EQUATIONS OF MOTION

Classical equations of motion for the ith particle are

mid2r

i

!

dt2=Fi

!

(! =1,2,3) (G3)

where Fi =!grad

ri

(Ep) is the force acting on the particle i and its components are

Fi!="

#E p

#ri!="

#E p

#rik!

k$ i

% (G4a)

This can be re-written as

Fi!= Fik

!

k" i

# (G4b)

where

Fik!="

#E p

#rik!

(G4c)

can be interpreted as the ! component of the force exerted on the particle i by the

particle k.

MECHANICAL EQUILIBRIUM

The necessary condition for the mechanical equilibrium is that the total force acting onany particle of the system vanishes3, i. e.

Fi!

=0 for i = 1,...,N; ! = 1,2,3 (G5)

The condition which guarantees that the equilibrium corresponds to a minimum of the

potential energy with respect to atomic positions, requires that the matrix of the second

derivatives of the potential energy

Fij!"

=

#2E p

#ri!#rj

" =

#2Ep

#rik!#rj!

"k, !

k$ i, !$j

% , (G6)

called theforce constants matrix, be positively definite4.

3We, of course, assume that the moments of the forces also vanish. 4This matrix determines the phonon spectrum and its positive definiteness also guarantees that thecorresponding eigenvalue problem for the vibrational frequencies has only real solutions. It follows

from the requirement of translational invariance of the system that Fi,j!"

i,j

# =0 .


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The quantities entering equations (G5) and (G6) must be taken as time averages in the

case of molecular dynamics, and ensemble averages in the case of Monte Carlo

simulations.

STRESS TENSOR IN THE SYSTEM OF PARTICLESEquilibrium system at 0K

It is well known from the continuum mechanics that a medium can be in mechanical

equilibrium while it is subject to stresses. These stresses arise owing to application of

forces or displacements at the boundaries or arise locally as internal stresses due to

defects, thermal gradients and generally incompatibilities.

In a continuum the stress field (tensor), !"# , is determined by equations of

equilibrium

!"#$

!x$$=1

3

% +F# =0 (# = 1,2,3)

where F!

is the ! component of the body force (such as weight) per unit volume.

The physical meaning of these equations is that the total force is zero at any point of

the medium. These equations, together with the relevant boundary conditions and

constitutive laws relating stresses, strains and displacements (e. g. the Hooks law),

determine fully the stress field !"# . This field is generally varying locally within the

body and is a function of the position. The total stress per unit volume is

!"#total

=

1

V!"# dV

Volume

$ ,

where V is the volume of the body and the integration extends over its whole volume

of the body. If no external loads are applied and no displacements (and/or strains) are

imposed at the boundary then in equilibrium !"#total is equal to zero.

The stress tensor, !"# , which varies through the body, can be gauged by applying an

infinitesimally small strain defined by a strain tensor !"#

o

. This strain tensor is not

related to the stress !"# by the Hooks law. The stress associated with the gaugestrain is infinitesimally small and negligible with respect to the internal stress

#

. In this case the change of the potential energy of the body is, to the first order

in !"#

o

,


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!Ep = "

#$%#$o dV

V

,$=1

3

' , (G7)

where the integration again extends over the whole volume of the body.

Analogously as in the continuum, in a discrete structure the forces acting on

individual atoms vanish in equilibrium but there may be stresses due to applied loads

or sources of internal stresses. In order to measure the stress we again apply as a

gauge an infinitesimally small strain defined by a strain tensor !"#

o

. Any vector rij

changes with the application of the strain !"#

o

such that

rij

!" r

ij

!+ #

!$

orij

$

$=1

3

% . (G8)

In addition, the volume changes to the first order in !"#

o

by

!V = V "##

o

#=1

3

$ = V "#%o &

#%#,%=1

3

$ , (G9)

The corresponding change of the potential energy of the discrete system is then

obtained by inserting (G8) into (G1). Expanding E p into the Taylor series with

respect to !"#

o

we obtain to the first order in !"#

o

!Ep =

"Ep

"rik

#rik

$

k

k%i

&i&'

(

))

*

+

,,#,$=1

3

& -#$o , (G10)

Comparing (G10) with (G7) and realizing that summation over all particles in the

discrete system is equivalent to integration over the volume of the continuous body,

the total stress per unit volume is in the discrete system

!"#total

=

1

V

!Ep

!rik" rik

#

kk"i

$i

$ . (G11a)

Using relation (G4c), which determines the force exerted on the particle i by the

particle k,

!"#total

= $1

V F

ik

"rik

#

k

k!i

%i

% . (G11b)


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Equilibrium system at finite temperature

When the temperature of a system of particles is fixed, the total kinetic energy of the

particles must not change when applying the gauge strain tensor !"#

o

. However, since

the velocity of the particle i is vi= dr

idt , then using equation (G8) velocities of the

particles in the deformed state will change such that

vi

! " vi

!+ #!$

ovi

$

$=1

3

% (G12)

Expanding the kinetic energy Ekin

=1 2 mi[v

i

! ]2

i

"!=1

3

" into the Taylor series with

respect to !"#

o

gives to the first order in !"#o

!Ekin

= mivi

"vi

#

i$%&' (

)*

",#=1

3

$ +"#o (G13).

However, if the temperature is fixed and thus the kinetic energy is to remain the

same, this term must be compensated by a term that arises due to a thermally induced

stress, !"#T

, which invokes the energy change !ET = V "#$

T %#$

o

# ,$=1

3

& that has the same

magnitude but opposite sign than !E kin . Hence

!"#T=

$

1

Vm

ivi

"vi

#

i% (G14)

This is a generalization of the virial theorem. The total stress in the assembly of

interacting moving particles is then at any given time

!"#total

=

1

V

!Ep$r

ik" rik

# %mivi"v

i#

kk& i

'(

)

***

+

,

---i

' (G15a)

or using (G11b)

!"#total

=$ 1V

Fik

"rik

#+m

ivi

"vi

#

kk% i

&'

(

)))

*

+

,,,i

& (G15b)

The stress which would be measured is the time average of the stress given by

(G15a, b), !"#total . In the case of molecular dynamics it is determined by time


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averaging and in the case of Monte Carlo simulations as an ensemble average. (For

more details of evaluation of averages see sections on MD and MC methods).

If there is no directional flow and the movement of all the particles is on average

random, then in the three-dimensional case vi

!v

i

"=

13#!" vi

2 . The kinetic energy of

the randomly moving particles determines the temperature of the system according tothe equipartition theorem and in the three-dimensional case 12 m i v i2

i

! = 3 2NkBT ,

where kB is the Boltzmann constant. The average thermally induced stress is then,

following equation (G14),

!"#T

=$1

Vm

i v

i

"vi

#

i

% =$&"#

3Vm

i v

i

2

i

% =$'kBT&"# (G16)

where ! = N/Vis the density of the material, measured as the number of particles per

unit volume. The average stress is in this case

!"#total

= 1V

$Ep$rik

" rik#kk% i

&i& ' (kBT)"# (G17a)

or

!"#total

=$1

V F

ik

"rik

#

k

k%i

&i

& $ 'kBT("# (G17b)

Obviously, the thermally induced stress does not contribute to the shear stresses but

only to the hydrostatic component of the stress tensor.

The hydrostatic pressure in the system is

p =!1

3 "##

#=1

3

$ =1

3V Fik

#rik#

kk% i

$i$

#=1

3

$ +&kBT , (G18)

since !"""=1

3

# = 3 .

In the case of non-interacting particles (ideal gas), when E p =0 , we obtain

p =!k BT " pV = Nk BT = RT .

This is, of course, the well-known isotherm for the ideal gas. When the particles are

interacting the first two terms in equation (G18) contribute. The corresponding

isotherm of the gas of interacting particles is then

pV =RT + f(V),


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where f(V) arises from the interaction term in equation (G18). This is the so-called

van der Waals isotherm.

When some external loads are applied, or boundaries of the system studied are held at

some fixed positions, the stress tensor !"#total is the stress induced by these loads

and/or constraints. However, if there are no loads or constraints, then in equilibrium!"#

total=0 , (G19)

For example, when considering the hydrostatic pressure, the requirement of p = 0

determines the equilibrium volume (density) and is equivalent to the conditiondH

dV=0.

However, the condition (G19) does not imply that there are no local internal stresses

since it only states that the average of all possible internal stresses is zero if external

loads or other constraints are not applied.

When determining the stress (at zero K) the potential energy was expanded to the first

order in !"#o . Expansion of the potential energy to the second order in !

"#o

determines the tensor of elastic moduli (see Appendix). The requirement that the

potential energy attains a minimum for zero strain leads to the condition that these are

positively definite. In fact, the condition that the force constants matrix (G6) is

positively definite already guarantees, together with the condition (G19), that the

elastic moduli are positively definite.

EXAMPLE: INTERACTIONS DESCRIBED BY A PAIR POTENTIAL

SUPPLEMENTED BY A VOLUME (DENSITY) DEPENDENT TERM

The potential energy of a system can be written as

Ep =U(V)+1

2 !(rij)

i, ji"j

# , (G20)

U(V) is the term dependent only on the total volume V or, in other words, only on

the average density, ", of the system. !(rij) is a pair potential depending on the

separation, rij, of the atoms i and j and, possibly, the density ". The pre-factor 1/2

arises because the interaction between atoms i and j is counted twice, once when

evaluating the energy associated with the atom i and the second time when evaluating

the energy associated with the atom j.


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When evaluating the forces on particles

!Ep!rik

" =1

2

d#(r)dr

r=rik

rik"

rik

$d#(r)dr

r=rki

rki"

rki

%

&''

(

)** =

d#(rik )dr

ik

rik"

rik

. (G21)

Since rki!

= "rik!

the force exerted on the particle i by the particle k is

Fik!= "

d#(rik)

drik

rik

!

rik

. (G22)

Following equations (G15a, b), the total stress tensor is

!"#total

=1V

d$(rik)

drik

rik"r

ik#

riki,k

k%i

& ' mivi"v

i#

i&

(

)

***

+

,

---+dUdV

."# ; (G23a)

the termdU

dV!"#

arises due to the explicit dependence of the potential energy as given

by (G20). Similarly as in (G17a)

!"#

total=

1

V

d$(rik)

drik

rik

"rik

#

rik

i,kk%i

& +dU

dV'"#( )k

BT'

"#. (G23b)

The total hydrostatic pressure is, following equations (G18) and (G16),

p =! 1

3V

d"(rik )

drik

(rik# )2

rik#=1

3

$i,kk% i

$ ! dU

dV +&kBT (G24)

In the case of the cubic symmetry,when all quantities are invariant with respect to

the rotation by #/2 around the coordinate axes parallel to the cube axes, !"# $ 0 for

! " # and

!11total =!22total =!33total = 13V d"(rik )

drik

rik#

$%%

&

'((i,k

k)i*

+ dUdV+ 13V mivi

2

i* (G25)

because f(rik)(rik!

)2=

i, k

" 13 f(rik)rik2

i,k

" , where f(r) is a function dependent only on the

magnitude of the vectorr.


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U(V) entering the potential energy is often set equal to zero, which means the energy

of the system is defined by the pair potential only (see for example the Lennard-Jones

potential). In other situation when this term is considered it is often approximated as

p0(V ! V

0) / V

0, where V is the current volume of the studied block of a given

material, V0 the equilibrium volume of the same block and p0 a constant pressure,

called the Cauchy pressure.


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APPENDIX

ELASTIC MODULI

When applying a small homogeneous strain tensor, !"#o

, each vector rij changes as

described by equation (G8), assuming that no additional relaxations of atomic positions

occur owing to the application of the strain. This is the case when all the atoms in the

repeat cell are equivalent as, for example, in centro-symmetric structures but not in

general. However, we shall make this assumption at this stage. Furthermore, we shall

not consider the kinetics of the system and thus, effectively, 0K temperature is assumed.

In a continuum the change of the potential energy upon application of the strain tensor

!"#

o

can be written, to the second order in !"#

o

, as

!Ep = V "#$%#$

o

# ,$=1

3

& + 12 C#$'(%#$o %'(o# ,$,',(=1

3

&)

*+

,

-. , (AG1)

where !"# is the stress tensor and C!"#$ the tensor of second order elastic moduli. The

moduli satisfy the following symmetry relations:

C!"#$ = C"!#$ = C!"$# = C"!$#

The total volume changes such that to the second order in !"#

!V =V "#$

o %#$

# ,$=1

3

& '12

"#$

o "(%

o %#(

%$%(1' %

#$)

# ,$ ,(,%=1

3

& + 12

"#$

o "(%

o %#$

%(%(1' %

#()

# ,$ ,(,%=1

3

&)

*+,

-.(AG2)

For the system composed of N interacting particles the change of the potential energy,

E p , to the second order in !"#o

, is obtained by inserting (G8) into (G1) and expanding

E p into the Taylor series with respect to !"#o

to the second order:

!Ep = "Ep

"rik

#r

ik$%

#$o

#,$=1

3

&i,kk' i

& + 12 "2

Ep"r

ik

#"rj!

(r

ik$r

j!) %

#$o %

()o

#,$,(,)=1

3

&i,j,k,!k' i,!'j

& (AG3)

where the summation indices k and !run over all atoms interacting with the atoms i and

j, respectively. Comparison of equations (AG1) and (AG3) yields for the total stress

tensor equation (G11a) and for the elastic moduli the following relation that satisfies the

above mentioned symmetry requirements


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C!"#$

=1

4V

%2Ep%rik

!%rj!# rik

"rj!$+

%2Ep

%rik"%rj!

# rik

!rj!$+

%2Ep%rik

!%rj!$ rik

"rj!#+

%2Ep

%rik"%rj!

$ rik

!rj!#

&

'((

)

*++i,j,k,!

k,i,!,j

-

(AG4)

When additional relaxations of atomic positions occur, for example when the repeat cell

of the particle system contains non-equivalent atoms and there is no centro-symmetry,

the corresponding inhomogeneous atomic displacements are determined by the auxiliarycondition of zero forces on each atom of the strained medium. It was shown by Martin

(J. Phys. C: Solid State Phys., 8, 2837, 2858, 1975) that stresses are not affected by the

relaxation and are still given by equation (G11). However, the relaxation alters theelastic moduli. They are now given by the relation

C!"#$

*= C

!"#$+

!C!"#$

in which C!"#$is given by equation (AG4) and the relaxation part is

!C!"#$ =% D!"&i

g&'ijD'#$

j

&,' =1

3

(j= 2

N

(i=2

N

(

where D!"#i

=

1

V

$2E pot

$%!"$ri1# and g!"

ij is the inverse of the force constants matrix

E!"ij=

1

V

#2Epot

#ri1!#rj1

" when the origin of the repeat cell is identified with the atom number 15.

In practical calculations the evaluation of the elastic moduli is done by applying suitable

small strains to the block of atoms, relaxing, and evaluating the total energy as a

function of the applied strain. In such calculations the internal relaxations are

automatically included if the energy of the structure is minimized with respect to atompositions after each application of the small strain. The elastic moduli are then obtained

by approximating the numerically calculated energy vs strain dependence by a second

or higher order polynomial and taking the appropriate second derivatives with respect tothe strain.

Bulk modulus

B =Vd 2E

pot

dV2, (AG5)

Considering the strain tensor !"# =1

3

$V

V%"# , which represents a pure dilatation, the

second, third and fourth terms in equation (AG3) give

5 The same type of the relaxation part of average elastic moduli arises in the continuum treatment of composite

materials when different components of the material possess different elastic moduli.


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1

2

!VV

"#$

%&'

21

9

(2Ep

(rik)(rj!

*rik

)rj!*

),*=1

3

+i,j,k,!k,i,l,j

+

and therefore

B =1

9V

!2Ep

!rik"!rj!

#rik

"rj!#

",#=1

3

$i,j,k,!k%i,!%j

$ (AG6)

Comparison with equation (AG4) gives then, as known from elasticity,

B =1

9C!!""

! ,"=1

3

# , (AG7)

Description of the potential energy in terms of pair potentials

The potential energy of the system is now given by equation (G20) and only terms of

the type!2E p

!rik"!rik

# rik$rik%

contribute in equation (AG4). For pair potentials these terms

are

!

2Ep

!rik"!rik

# rik$ rik% = 12d

2

&(r ik )drik2 ' 1

rik

d&(rik )dr

ik

(

)* +

,-rik"

rik$

rik#

rik%

rik2 ,

Equation (AG4) then gives

C!"#$ =1

2V

d2%(rik )drik

2 &

1

rik

d%(rik )dr

ik

'

()*

+,rik

!rik

"rik

#rik

$

rik2

-

.//i,k

i0 k

1

+

1

4rik

d%(rik)

drik $!#r

ik

"

rik$+$"#rik

!

rik$+$!$rik

"

rik#+$"$rik

!

rik#

( )2

344

+dU

dV

14[$!"$#$ (1& $!#) +$"!$#$ (1& $"#) +$!"$$#(1& $!$ ) +$"!$$#(1& $"$ )]{

& 12 [$!#$"$ +$"#$!$ ](1& $!")]}+V

d2U

dV2$!"$#$

(AG8)


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and using equation (G25) which defines the stress (without the kinetic term) weobtain

C!"#$ =1

2V

d2%(rik )drik

2 &

1

rik

d%(rik )dr

ik

'

()*

+,rik

!rik

"rik

#rik

$

rik2

-

.//

0

122i,k

i3 k

4

+1

4$!#5 "$ +$"#5!$ +$!$5"# +$"$5 !#( )

+dU

dV14[$!"$#$ (4& $!#& $"#& $!$& $"$ )& $!#$"$& $"#$!${

& $!$$"#& $"$$!#]& 12[$!#$"$ +$"#$!$](1& $!")]} +V

d 2U

dV2$!"$#$

(AG9)

Cubic symmetry

There are three independent moduli:

c11

=C1111

, c12

=C1122

, c44

=C1212

c11 =1

2V

d2!(r ik )

drik2 "

1

rik

d!(rik )

drik

#

$%%

&

'((i,k

k) i

* (r ik

+=1)4

rik2

+,11"dU

dV+V

d2U

dV2

c12 =1

2V

d 2!(rik )

drik2 "

1

rik

d!(r ik )

drik

#

$%%

&

'((i, j

j) i

* (rik

+=1)2 (rik+=2 )2

rik2 +

dU

dV+V

d2U

dV2

c44 =1

2V

d 2!(rik )

drik2 "

1

rik

d!(r ik )

drik

#

$%%

&

'((i, j

j) i

* (r

ik+=1)2 (r

ik+=2 )2

rik2

+1

2,11"

dU

dV

(AG10)

In equilibrium, when ! 11= 0, we obtain

c12! c 44 = 2dU

dV+V

d2U

dV2

(AG11)

1/2(c12! c44 ) is called the Cauchy pressure and it is in equilibrium independent of

the pair potential. In the approximation in which U =p0(V! V

0) / V

0 the Cauchy

pressure is equal to p0.


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When the energy of a system of particles is determined solely by the pair potential, i.

e. if U = 0, then the Cauchy pressure is zero. This implies that if interaction in a

material is described only by a pair-potential then c12 = c 44, which is the so-called

Cauchy relation. This relation between elastic moduli is not satisfied for metallic and

covalent materials. However, it is almost true in a number of ionic solids, indicating

that in these materials the description of the total energy by pair-potentials is a

reasonable approximation.

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