Smulación de Ondas de Choque

Transcripción

1 Smulación de Ondas de Choque Gonzalo Gutiér r e z Departament o f Physics Universidad de Chile. gonzalo@fisica.ciencias.uchile.cl

2 Teoría ondas de choque 2 IDEA: Gases: se puede hacer un desarrollo bastante completo sobre las ondas de choque Básicamente se escriben ecuaciones de conservación de masa, momentum y energía para adelante y atrás del frente de ondas y luego se ocupa la ecuación de estado (EOS) del material para obtener las relaciones termodinámicas (ver apuntes L. Moraga). Liquidos y sólidos: acá uno de los problemas es que no se conoce la ecuación de estado en forma general.

3 Simulación ondas de choque 3 IDEA: A) Usar relaciones de Hugoniot: Materia en condiciones extremas: Se necesita conocer EOS - Simulación mediante método de Montecarlo (~1950) - Dinámica molecular (~1960) B) Calcular (o simular) el proceso en toda su complejidad - Ecuaciones del continuo tipo Navier-Stokes, incorporando términos de mayor orden (Burnett, etc) - Simulaciones de Dinámica Molecular

4 Molecular Dynamic: moving the atoms 4 IDEA: atoms are classical particles, they follow Newton laws Interatomic potential V(r) Temperature : equipartition theorem F = V(r1,...,r N ) 2 T = mv / 3k Numerical solution of the coupled ordinary differential equations Results: r 1 (t),...,r N (t); v 1 (t),...,v N (t) Physical properties: average over configurations, like in statistical mechanics

5 Molecular Dynamic program 5 Inicial conditions: {r i (t 0 ),v i (t 0 )} N N~2000 V E K =3/2Nk B T Choice of interatomic potential forces empirical pot ab-initio (R) V ( r) = qq i r j CC i j + Bexp( ar) 6 r Numerical solution to the eqs. of motion Verlet t:10-15 s : time step Physical properties: temporal average over configurations:... t Termodynamics prop. structural properties dynamical prop.

6 Interatomic potentials 6 A general expresion for an interatomic potential should contains one-body, two body, three body,...and N body terms For simplicity, people in general use only two and three body terms. The idea is to write a general functional form and to fit it according some experimental data: elastic constants, vacancy energy, etc.

7 Simple interatomic potential 7 Ex: Lennard-Jones potential: good for noble gases

8 Interatomic potential II 8 A good interatomic potential should have the following properties FLEXIBILITY 2. PRECISION 3. TRANSFERABILITY 4. COMPUTATIONAL EFFICIENCY

9 Time discretization r i (t + ) = r i (t) + v i (t) v i (t + ) = v i (t) + DM algorithm a i (t) 2 a i (t) + a i (t + ) 2 a i = 1 V m i r i ri(0) ri() ri(2) 9 Time stepping: Velocity Verlet algorithm Given ( r v i (t)) i (t), 1. Compute a i (t) as a function of 2. v i (t + ) v 2 i (t) + a 2 i (t) 3. r i (t + ) r i (t) + v i (t + 2 ) 4. Compute a i (t + ) as a function of 5. v i (t + ) v i (t + 2 ) + a i (t + ) 2 { r i (t)} { r i (t + )}

10 Physical Properties 10 Structural properties: pair distribution function g( r) static strucuture factor S(q) Thermodynamics properties: pressure, heat capacity, etc Example: p = kt V 1 3V N r (r ) ij ij i, j r ij Dynamical properties diffusion coeficient D Velocity autorrelation function VACF(t) Vibrational density of states D(w)

11 Simulaciones Argón 11 Densidad: 1,4 g/cm 3 Temperatura 5 K V S ~1,5 km/s N atomos Lx=Ly= 34 A; Lz= 680 A Vp 0,5; 0,7;

12 12 Fractura de argón mediante ondas de choque

13 Simulaciones Argón: animación 13 La figura muestra una simulación computacional de una onda de choque propagándose hacia la derecha a través de un paralelepípedo de argón a la temperatura de 5 K. El frente de ondas viaja a una velocidad de 1,2 Mach presentando una estructura muy definida, cuya densidad es ~1,1 veces la densidad del argón en estado normal. (Simulación de dinámica molecular con átomos, realizada por Claudia Loyola, Junio 2008, Características principales: - al propagarse, se produce un deslizamiento de planos en la dirección (111), que produce un desorden en el cristal - al parar el piston, se produce una onda de rarefacción - esta onda de rarefacción, al pasar por partes donde existen defectos puntuales, funde localmente la zona (presion negativa) y finalmente se fractura le material. (ver gráficos a continuacion).

14 14 Simulaciones Argón V p =0.5 km/s

15 15 Simulaciones Argón Vp=0.1 km/s

16 ab-initio MD: ions + electrons 16 The only interaction there exists is the Coulomb interactions! The idea of ab-initio MD is to calcul everything from the solid state hamiltonian 1. Electronic structure calculation to obtain the total energy e( R) in terms of the positions of the ions. 2. Forces for each t, forces are obtained from the electronic str. calc. F I = R I ({R}) = R I E KS ({ i },{R i }) = R I i H KS i = i H KS i R I i Teo.Hellman-Feynman i

17 Splitting different characteristics time 17 Equation of the system H = E Solid state physics hamiltonian: ( T T + V + V + V ) r,..., r ; R,..., R ) = E( r,... r ; R,..., R ) ion + e ee eion ionion ( 1 N 1 M 1 N 1 M Adiabatic approximation: splitting of the ionics and electronics degree of freedom M ion ~ m e ion >> e Writing (r;r) =(r) (R), the problem split in two parts: electron ( T V + V + V ) r,... r ;{ R}) = ({ R}) ( r,... r ;{ }) e + ee eion ionion ( 1 N 1 N R ion ( + ({ R}) ) ( R) = E' ( R) T ion

18 Electrons and ions 18 Electrons ( T V + V + V ) r,... r ;{ R}) = ({ R}) ( r,... r ;{ }) e + ee eion ionion knowing (R) we can have: -Ground state (T=0) -Crystalline structure and PV diagram -Elastic, optics, magnetics, etc, properties ( 1 N 1 N R -Potential energy surface -Interatomic potential interatomic forces Iones ( + ({ R}) ) ( R) = E' ( R) T ion This is a quantum problem, but classical physics work pretty well! - to look for an effective interatomic potential (R) - to treat the ions as classical particles, i.e. obeying Newton Laws

19 Electrons : solution of the Schrödinger Eq. 19 Hamiltonian of a many electrons system: N 2 2m 2 j + V e e + V ext ( r 1,.., r N ) = E( r 1,.., r N ) j =1 To look for the GS by means of a variational method: ~ ( r,.., r 1 N ) E ~ [ ] = Min ~ 0 0 H 3N dimensions trial function Approximate Methods : one electron selfconsistent equations Hartree: r,.., )... ( 1 r N = 1 Z r n( r') ', con ( ) 2 dr = = ' i ii n r m r r i Hartree-Fock: r,.., r ) det[... ] ( 1 N = 1 N 1 N N i i 2 ~

20 Density Functional Theory 20 The electronic density n(r) as the basic variable Hohenberg-Kohn (1964): 1. The external potential v(r) is totally determined from the electronic density n(r). E [] n = n( r) v( r) dr + T[ n] + V [ n] N.B. a) We had a 3N variables trial function (r 1..r N ), but now only a 3 variables trial function ñ(r) b) DFT gives a strict formulación of the search of the GS from the electronic density ñ(r). For example, Thomas-Fermi can be derived from DFT. ee 2. For the trial density ñ(r) we obtain E 0 E[ñ]

21 Electrons II: Kohn-Sham Eqs. 21 Total energy reads as (Kohn & Sham, 1965) E {R} [ñ]=t s [ñ]+e Hartree [ñ]+e ext [ñ]+ E xc [ñ] Minimization of E {R} [ñ] is equivalent to solve Kohn-Sham Eqs. Selfconsistent solution LDA approximation: E xc [n] with xc = r s 0.44 r s m 2 + v eff ( r ) i ( r ) = i i ( r ) n( r ') v eff = r dr ' +V ext ( r ) + v xc ( r ) r ' n( r N ) = i ( r ) 2 i =1 n( r ) xc (n( r ))dr homogeneous electron gas

22 Electrones III: implementation 22 non-relativistic relativistic 2m 2 + V( r ) + v xc ( r ) k i = k k i i 2 periodic no-periodic spin-polarized non spin-polarized all-electron full-potential all-electron muffin-tin pseudopotenciales jellium LDA GGA I II linearized augmented plane wave: LAPW scattering functions (ex.:. Hankel): LMTO plane wave LCAO I

23 Implementation: selfconsistent cycle 23 n ini (r) q ( r) = n( r) + Z( R ) R 2 V C (r) = 4q(r) v xc (r) = v xc [n(r)] Poisson Eq. Electronics cycle H S kl kl = k H l = k l ( H S) c = 0 { i },{ cij} Matrix elements Find the de eigenvectors and eigenvalues Dynamics j i = c ij j n(r) = i,oc i 2 KS orbitals in terms of base functions E tot [n(r), R ], F = E R Calculation of properties

24 Technical issues 24 Input: crystalline structure, atomic numbers Accuracy in energy, in forces Full potential calculation: FPLAPW (Wien2k), FPLMTO Pseudopotential calculation PPPW method (VASP, CPMD, ABINIT) Other expansion: Gaussian, SIESTA