South China University of Technology

1. South China University of Technology Molecular Dynamics Phase transition & Quasi-crystals Xiaobao Yang Department of Physics www.compphys.cn

3. Quasi-crystals

4. Many body dynamics

5. Multiscale modeling of materials • Understand the molecular level origins of materials behavior • Predict the behavior of materials from first principles Electrons Atoms Mesoparticles Elements T i m e Macroscale second Mesoscale microsec Molecular dynamics nanosec. Quantum Mechanics picosec. L e n g t h femtosec. nanometer micron mm meters

6. Structural relaxation To obtain the ground state relaxed geometry of the system. the equilibrium lattice constants a given ionic configuration the forces obtained these forces are greater than some minimum tolerance the ions are moved in the direction of the forces

7. Structure of an MD code

8. Key issues in MD • Classical Molecular Dynamics - Potentials to describe the particles in the system formulism + parameterization formulism + parameterization - Algorithm to solve Newton EQ: accuracy + efficiency accuracy + efficiency - Analysis of physical properties • Ab initio Molecular Dynamics - Coupling the motion of electrons and ions.

9. Why classical MD works for Atoms? • Energy View: Typical Kinetic energy < 0.1 eV, while it is about >1eV to remove/excite an elec. • Debroglie wavelength view: Typical distance between atoms > 1 Å, while the DeBroglie wavelength is ~ 10-7Å. • MD vs. MC: MC: be convenient for studying the equilibrium properties. MD: reflects the real process of a system from one microstate to another.

10. EoM: Verlet method Equation of Motion of particle i: Taylor expansion of positions with time Verlet method What is the difference with Euler or Euler-Cromer method?

11. Properties of a dilute gas • Lennard-Jones potential

12. Morse potential

13. Many body potential

14. Properties of a dilute gas • Periodical boundary condition PBC affects the force calculation and position update. • thermodynamic ensembles Temperature: Instantaneous temperature (T*):

15. Structure of an MD code

16. MD.m [ pos, vel, acc] = initialize ( np, nd, box); [ force, potential, kinetic ] = compute ( np, nd, pos, vel, mass ); for step = 1 : step_num [ force, potential, kinetic ] = compute ( np, nd, pos, vel, mass ); [ pos, vel, acc ] = update ( np, nd, pos, vel, force, acc, mass, dt ); end Initialize.m Compute.m Update.m

17. Initialize.m • pos(1:nd,1:np) = rand ( nd, np ); • for i = 1 : nd • pos(i,1:np) = box(i) * pos(i,1:np); • end • vel(1:nd,1:np) = 0.0; • acc(1:nd,1:np) = 0.0; • return • end

18. Compute.m Ri = pos - repmat ( pos( :, i ), 1, np ); % array of vectors to 'i' D = sqrt ( sum ( Ri.^2 ) ); % array of distances Ri = Ri( :, ( D > 0.0 ) ); D = D( D > 0.0 ); % save only pos values

19. Compute.m • pot = pot + 0.5 * sum ( 1./(D.^12)-1./(D.^6) ); % accumulate pot. energy • f( :, i) = Ri * ( (6./(D.^6)-12./(D.^13)) ./ D )'; % force on particle 'i‘ % Compute kinetic energy. kin = 0.5 * mass * sum ( diag ( vel' * vel ) );

20. Update.m • % x(t+dt) = x(t) + v(t) * dt + 0.5 * a(t) * dt * dt • % v(t+dt) = v(t) + 0.5 * ( a(t) + a(t+dt) ) * dt • % a(t+dt) = f(t) / m • pos(1:nd,1:np) = pos(1:nd,1:np) + vel(1:nd,1:np) * dt ... + 0.5 * acc(1:nd,1:np) * dt * dt; • vel(1:nd,1:np) = vel(1:nd,1:np) ... • + 0.5 * dt * ( f(1:nd,1:np) * rmass + acc(1:nd,1:np) ); • acc(1:nd,1:np) = f(1:nd,1:np) * rmass;

21. Properties of a dilute gas Trajectories Speed distribution 20 particles in a 10x10 box with PBC

22. The Melting Transition

23. The Melting Transition

24. MD: isothermal molecular dynamics How can we modify the EoM so that they lead to constant temperature? Nose-Hoover thermostat Berendsen’s thermostat Direct feedback

27. AWK and VASP aa=`awk '$2 == "TOTAL-FORCE" {print NR}' OUTCAR` a=`expr $aa + 1` bb=`expr $aa + $num + 2` awk 'NR > a && NR < bb {print $4,$5,$6}' a=$a bb=$bb OUTCAR >force.1 paste site.1 velocity.1 force.1 >all.1 awk '{print $1+$4*h+$7*h*h/2/m*0.0096,$2+$5*h+$8*h*h/2/m*0.0096, $3+$6*h+$9*h*h/2/m*0.0096}' h=$dt m=$mass all.1>site.2 paste velocity.1 force.1 force.2 >all.2 awk '{print $1+$4*h/2/m*0.0096+$7*h/2/m*0.0096, $2+$5*h/2/m*0.0096+$8*h/2/m*0.0096, $3+$6*h/2/m*0.0096+$9*h/2/m*0.0096}' h=$dt m=$mass all.2>velocity.2

28. How to find the proper parameters?

29. Simulation of Quasi-crystals

30. Annealing simulation Phase Transitions

32. Homework For lecture notes, refer to http://www.compphys.cn/~xbyang/ 主题：学号+姓名+第？次作业 Sending one word/pdf file to 17273799@qq.com when ready