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: Radial distribution function : 1 () = ( + − ) Pair correlation function: 1 1 () = ( − − ) 4 Alternative definitions : Total distribution function: = 4() 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 Direct comparison with experiments can fail Simple force fields can not account for pressure-induced changes (metallization) 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 O3 O1 O3 Ge 1 O2 O3 O1 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 Evidence for homopolar bonds Ge-Ge and Se-Se 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,,, ) 7 where : / 1 4&5(6) - ., #, = 1 = 1 3 0 2 0 2 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 <U> 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<L/2, i.e. k< πππ/L 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 + $& ) =$& (>) = eApCD−E. − G 2 out of which can be computed a total structure factor: Neutron weighted : I,( JJ(KK(=( (H) = H = I,( JJ(KK( with the neutron scattering cross section , the concentration of the species bi ci =5.68 fm for Te (tabulated, depends on the isotope) bi X-ray weighted: I,( JJ(L(H)L((H)=( (H) = H = I,( JJ(L(H)L(( H) 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 5 Ag25 4 3 Ag15 ) Q ( 2 S Ag5 1 0 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 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 Bond angle distributions Depending on the system, provides information about the local geometry 1173 K (tetrahedral, octahedral,…). SnSe 2 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 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.