LECTURE 7 : CALCULATING PROPERTIES Space Correlation Functions Radial Distribution Function Structure Factor Coordination Numbers, Angles, Etc

Home , Correlation function, Structure factor

We have generated sets of (x,y,z) positions for various times at various given thermodynamic conditions (N,V,T,P). Now, we use them.

LECTURE 7 : CALCULATING PROPERTIES Space correlation functions Radial distribution function Structure factor Coordination numbers, angles, etc.

[email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING A) RADIAL DISTRIBUTION FUNCTION

ρ (1) ρ (2) Structural caracterization: Probability densities N and N

i.e. one can find N particles and N(N-1) pairs of particles in the total volume, respectively.

Note that these are n=1 and n=2 cases of a general correlation function g(n) given by :

ZN the partition function, V the volume, N the nb of particles [email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING So that one can write for e.g. n=2:

2 which defines the radial distribution (rdf) g N (r 1,r 2) function, also given by

ρ (1) ρ For an homogeneous isotropic system, one has N = . Dependence of the rdf only on relative distance between particles:

So that ρg(r) is the conditional probability to find another particle at a distance r away from the origin.

[email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING g(r) is the pair correlation function

In a simulation box with PBC, one cannot obtain the structure beyond r > L/2.

Alternatively: ß Radial distribution function : 1 () = ( + − )

ß Pair correlation function: 1 1 () = ( − − ) 4 Alternative definitions :

ß Total distribution function: = 4() ß Differential distribution function = 4 − 1 [email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING Pair distribution function : examples

Visual inspection allows to distinguish between a crystalline and an amorphous structure

CCP Ni lattice with EAM potential

[email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING Effect of thermodynamic variables : temperature

The integral of g(r) allows to Amorphous Se determine the number of neighbors around a central atom. Remember

The integral to the first minimum gives the coordination number.

r = 4 m Running coordination number N(r) Caprion, Schober, PRB 2000 () = ′ 4′ ′ [email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING Effect of thermodynamic variables : pressure

d-GeO 2 ß Direct comparison with experiments can fail

ß Simple force fields can not account for pressure-induced changes (metallization)

ß Additional structural insight is provided by partial correlation functions : Ge-Ge, Ge-O, O-O

Expt. (neutron) Salmon, JPCM 2011

MD: 256 GeO 2 using Oeffner-Elliot FF Micoulaut, JPCM 2004 [email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING Effect of composition: extending to multicomponent systems This is case for most glasses and materials

SiO 2, GeSe 2, SiO 2-Na 2O,…

Consider a system with n components having

N1, N 2, …Nn particles. We can write

() = ( − − ) with α between [1,n] And for : ≠ () = ( − − )

Out of which can be computed the pair correlation function:

1 (can be also neutron or XRD weighted) = () , [email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING Pair distribution function in multicomponent systems: GeO 2

Evidence for neighbour correlations Ge 3

Ge 2 Ge 1

O1 O3

Ge 1 O2 O3

D. Marrocchelli et al., 2010 O1 O2 [email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING Pair distribution function in multicomponent systems: GeSe 2

ß Evidence for homopolar bonds Ge-Ge and Se-Se

ß Prepeaks in relevant pdfs

[email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING Thermodynamic quantities from the computed g(r): Energy

First, one needs to remember that the energy is related to the partition function β as: Q(N,V, )

where : 1 () , , = = ! !

λ comes from the integration of the momentum p in the phase space given by:

so that:

[email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING and

In order to compute the average energy, one needs to compute the average potential U. Assume the case of a pairwise potential, so that:

i.e. U is a sum of terms depending only between 2 particles, and thus contains N(N-1) terms. Then, we can write:

(i.e. all terms in the first line are the exact same integrals, just with different labels) [email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING ρ(2) which contains a two-body probability distribution function (r 1,r 2) so that can be rewritten as :

where we remind (slide 2) that :

One thus arrives to the result:

[email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING Example : Glass transition in a model A-B glass

Flores-Ruiz et al. PRB 2010

Reproduction of the glass transition phenomenon

[email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING Thermodynamic quantities from the computed g(r): Pressure

We use the Maxwell relation:

The volume dependence can be made explicit by changing the variables:

so that we have from the partition function ZN the desired derivative from V:

And:

which involves the force Fi acting on a particle i [email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING Again, for, pairwise interactions, Fi can be written as : and since one has (interchanging i-j summations):

Using Newton’s third law, we furthermore have :

and the Ensemble average is given by:

[email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING We do the same as for the energy:

and for a pair potential Upair :

And finally:

The pressure can be computed from the derivative of the pair potential and from g(r). [email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING Examples: equation of state of liquids SiO 2

GeO 2

Shell et al. PRE 2002

Micoulaut et al. PRE 2006

[email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING B) STATIC STRUCTURE FACTOR

For a simple liquid, the static structure factor is given by :

ρ ρ where k is the Fourier transform of the microscopic density (r). This reads as:

‘

and, remembering that ρ

one has:

If isotropic and uniform medium, remember that ρ(r,r’)= ρ2g( r,r’) [email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING Also, g( r,r’) depends only on | r-r’ |, i.e.:

Or (isotropic fluid, everything depends only on k=| k|):

The calculation of the structure factor S(k) is achieved via a Fourier transform of the pair distribution function g(r).

[email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING Calculating a structure factor S(k) from a MD simulation : 2 options

1. Calculate the pair correlation function g(r) from the MD trajectory, then use :

2. Calculate directly S(k) from the trajectory using:

As 2Se 3

Differences between both methods can arise, and one is limited to r

Effects of the components of the wavevector

[email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING Direct comparison with experiments : neutron or X-ray diffraction

Intensity scattered in the direction kf

Kob et al. 1999 Equal to the computed structur efactor.

[email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING Structure factor in multicomponent systems

SiO 2, GeSe 2, SiO 2-Na 2O,…

Consider a system with n components having N1, N 2, …Nn particles. We can write Faber-Ziman partial structure factors:

(1 + ) () = exp [−. − ] 2

out of which can be computed a total structure factor:

ß Neutron weighted : ∑, () = ∑, with the neutron scattering cross section , the concentration of the species bi ci =5.68 fm for Te (tabulated, depends on the isotope) bi

ß X-ray weighted: ∑, ()() () = ∑, ()( ) With the X-ray form factor (elastic or inelastic XRD) fi(k) [email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING l-GeSe 2 v-SiO 2

Micoulaut et al., PRB 2009 [email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING Various levels of agreement between theory and experiments can be found

Ag25 4

3 Ag15 ) Q

( 2 S Ag5 1

0 2 4 6 8 10 Q (Å-1 )

Micoulaut et al. 2013 Data Petri, Salmon, 1991 [email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING Detailed structural analysis from MD

ß Neighbor distribution

Remember = 4 First minimum of g(r) can be used to define the coordination number. But this is an average.

• Details are provided from the statistical analysis of each atom.

• Allows to characterize the nature of the neighborhood

• Can be extended to partial CN = 4 [email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING Examples-1 Amorphous Ge 1Sb 2Te 4

Raty et al. Solid State Sciences 2010

Information on local geometry - short and long bond distance around Ge Straightforward - CN Te >2 4 neighbours around Ge,Si [email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING Examples-2: Statistics of neighbors with homopolar bonds

Ga 11 Ge 11 Te 78

Voleska et al., PRB 2013

[email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING Detailed structural analysis from MD

ß Bond angle distributions ß Depending on the system, provides information about the local geometry 1173 K (tetrahedral, octahedral,…).

SnSe 2 ß Directional bonding vs non-directional

SiO -Na O 300 K 2 2

Cormack and Du, JNCS 2001 Micoulaut et al. PRB 2008 [email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING Detailed structural analysis from MD

ß Ring statistics: serve to characterize the intermediate range order • Remember the early work of Galeener (lecture 3 ) and the Raman characterization of rings • Simulated positions can serve to define nodes and links. When connected sequentially without overlap, one has a path. A ring is therefore simply a closed path. Each of these rings is characterized by its size and can be classified.

S. Le Roux, P. Jund, Comp. Mater. Sci. 2010 http://rings-code.sourceforge.net/ [email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING Conclusions

ß MD trajectories, once properly generated can lead to various structural informations under various thermodynamic conditions • pair distribution functions g(r) bond distances partial coordination numbers, coordination numbers neighborhood • Structure factor S(k) information at intermediate lengthscales ß Many other quantities of interest • Rings • Bond angle distributions • Topological constraints - > Rigidity transitions (next lectures)

ß With imagination, one can find out much more or invent much more

Next lecture : time correlations and linear response [email protected] Atomic modeling of glass – LECTURE5 MD CALCULATING Final recommendation regarding structural properties

1. Do the best you can. 2. Don’t fool the reader. 3. Be honest with the data 4. Do not attempt to dissimulate 5. Unique S(k) ?

Uncarefully simulated liquid GeTe 4

[email protected] Atomic modeling of glass – LECTURE7 MD CALCULATING