1 Bulk Simulations in a HCP lattice

Size: px

Start display at page:

Download "1 Bulk Simulations in a HCP lattice"

Berniece McBride
5 years ago
Views:

1 1 Bulk Simulations in a HCP lattice 1.1 Introduction This project is a continuation of two previous projects that studied the mechanism of melting in a FCC and BCC lattice. The current project studies the melting process in a HCP lattice which is different than the previous two in some aspects. The method in which is employed in this study is molecular dynamics, this demands a search for a stable potential which can describe good physical behavior and results compared with experimental results and physical behavior. A potential was found for magnesium which has a HCP structure at no external press, the potential was constructed especially for temperatures close to the melting temperature. The potential is part of a family potential called an embedded atom potential, the potential has two parts, one is a pair potential like a Lennard Jones potential and the second is an embedded atom function. The latter holds the effective potential that an atom feels as a result from its surroundings. The first step in a melting simulation is to test the potential, it was chosen to check the thermal expansion and the elastic constants. It is expected that the thermal expansion 1 dl α = LdT will rise slightly with temperature and will have the same order of magnitude like the experimental thermal expansion. This formula is the thermal expansion along one side of the simulated box. From the elastic constants it is expect the decrease with temperature and of course to have the same order of magnitude as the experimental ones. The simulation is run in the NtT ensemble isothermal isobaric ensemble. In this ensemble the number of atoms is held fix, the external temperature and external pressure. This ensemble is chosen so that with the desired temperature and pressure held fixed the simulated box can expand from ground state (zero temperature) to its relaxed state with respect to the external variables. When the system is relaxed the lattice constants can be calculated, they are needed later for the surface simulations where it requires working in a NVT ensemble, V stands for a constant volume that is taken from the NtT simulation by measuring the lattice constants. 1.2 HCP lattice This project is different from the previous two because of the geometric structure of magnesium and the different symmetry compared with FCC and BCC. BCC and FCC both have a cubic symmetry, they belong to a class of structures called Bravais lattice which means that a crystal can be described by using only 3 vectors and their products with an integer. A HCP lattice is not a Bravais lattice and it does not have a cubic

2 symmetry. HCP stand for Hexagonal Closed Packed which is a Bravais lattice of hexagonal with a basis where the base composed of two atoms. Hexagonal lattice have 6-fold symmetry in the XY plane and a 2-fold symmetry in the Z axis. One other difference is that FCC and BCC lattice are defined by only one lattice constant while HCP needs two lattice constants. In the XY plane the atoms minimal distance is the lattice constant ' a ', the XY planes are separated by a lattice constant ' c 2'. All of those differences complicate the understanding of the system as well as the ability to model it correctly and do simulations. 1.3 NtT ensemble The algorithm that produces the NtT ensemble keeps the system coupled to two external baths, the heat bath and the pressure bath, by introducing to the equation of motion, of a particle, imaginary non physical forces. Each bath has it one imaginary force which after a few iterations is periodic in time, this periodic depends on the bath's mass. The baths mass is an arbitrary constant which should be chosen wisely, in the end it doesn't have any effect on the average values but to get the correct averaged values many periods should be performed which sums at the end in the total running time of one simulation. In this algorithm the simulated box is describes with 3 time dependent vectors abc,, defined from the origin of the coordinates. These 3 vectors form a 3 3 matrix h det h = a b c as a result h components are regarded as dynamical such that ( ) variables describing changes in the volume and shape of the simulated box. This procedure scales the atoms position into a simulated unit box where each atom coordinate r i is scaled down into a scaled coordinate s i via a realationship r i = hs. i The Lagrangian for the TtN ensemble T ( ) ( ) 1 2 T 1 L = mf i s i Gs i ϕ ri rj + WTrace h h 2 i i< j VTr 0 ( σε ) + Qf ( 3N+ 1) kbt0ln f 2 where QW, are the masses of the heat and pressure baths respectively, m is the atom's mass, σ is the external stress tensor, k B is Boltzmann constant, T 0 is the heat T bath's temperature, G = h h V = det h is the initial is called the metric tensor, ( ) T volume of the simulated box, N is the number of atom, ε ( h ) strain tensor. The equations of motion can be derived from the Lagrangian: 1 1 ( Gh 1) 0 0 = 2 is the

3 where p int ( ij ) 1 ϕ r s f T 1 T h = pintv ( h ) V0hh0 σ ( h 0 ) W 1 T kt B 0 f = mi fs i Gs i ( 3N + 1) Q i f ij 1 si = 2 G G+ si mf i j rij rij f is the microscopic internal stress tensor ( r ) ϕ p m f hs hs r r α, β 1 2 α β ij α β int = i i i ij ij V i i< j rij 1.4 Integrator The simulation uses a predictor-corrector integrator to solve the equation of motion. 1.5 Stages in the simulation The simulation start by initializing all of the atoms in their ground state, then it gives them random initial velocities depending on the temperature and Maxwell's distribution. The simulation than creates for each atom an array of 12 for its neighbors, this list is constant throughout the entire simulation, later the list is used for the lattice constants calculations. The simulation every 100 time steps, each time step is femtosecond, updates a neighbor list for each atom with respect to some cutoff radius, it then used in the calculation of the forces acting on that atom. The order of the simulation: 1. Building the neighbor list for each atom. 2. Computation of the forces on each atom. 3. Computation of the value of f. 4. Computation the element matrix of h. 5. Prediction of the next position and velocities of each atom. 6. Prediction of the next values of the f and f. 7. Prediction of the next values of the matrices h and h. 8. Computation of the forces on each atom calculated from the potential. 9. Computation of the value of f. 10. Computation the element matrix of h. 11. Adding to the forces acting on each atom the correction terms (the second and third term in s ). 12. Correction of the next position and velocities of each atom. 13. Correction of the next values of the f and f.

4 14. Correction of the next values of the matrices h and h. 15. Repetition of steps 5 to 12 until the total number of step is executed. Every 100 time steps the simulation calculates and writes the following in files: 1. The diagonal elements of the matrix h. 2. The off-diagonal elements of the matrix h. 3. Kinetic energy, potential energy, current temperature and pressure. 4. Lattice constants and volume of the simulated box. 5. Order parameters, one describing the XY plane and the second the Z axis. 1.6 Calculation of lattice constants and thermal expansion The lattice constants are calculated from the diagonal elements of the averaged h matrix over a long period of time. The spacing between XY planes is c = h 33 avg 2 ( N ) where N Atom is defined at the beginning of the simulation, it is the number of atoms in each direction of the simulation box. Atom The minimal distance between atoms in the XY plane is 1 h11 2 h22 avg avg a = + 2 ( NAtom 2) 3 ( NAtom 2) The factor 2 3 is a result from the vectors that composite the XY plane: a 3 a1 = [ a 0 0 ] ; a2 = a The calculated lattice constants enable the calculation of their thermal expansion: 1 dl 2 ΔL 2 1 N Atom N Atom α = = = a( T +ΔT) a( T) L T dt N a T ΔT N a T ΔT 2 2 ( ) ( ) ( ) Atom 1 1 α = +Δ a T ΔT ( ) Atom ( a( T T) a( T) )

5 2 Results All results are averaged starting from time step 4000 and ending at time step Figure 1: fluctuation of the matrix element h 11 as a function of time steps in 5 different temperatures

6 Figure 2: fluctuation of the matrix element h 22 as a function of time steps in 5 different temperatures Figure 3: fluctuation of the matrix element h 33 as a function of time steps in 5 different temperatures Figure 4: lattice constants as a function of temperature

7 Figure 5: thermal expansion of the lattice constants Figure 6: lattice constants as a function of temperature, each point is an average over 1000 time steps starting from step number The solid line connects between the averages of all the point in the same temperature

8 Figure 7: thermal expansion of the lattice constants from figure 5 vs. thermal expansion as a result of the segment division. Figure 8: fluctuation of the matrix element h 11 as a function of time steps, 5 different runs in the same temperature T = K

Figure 10: fluctuation of the matrix element h 33 as a

9 Figure 9: fluctuation of the matrix element h 22 as a function of time steps, 5 different runs in the same temperature T = K Figure 10: fluctuation of the matrix element h 33 as a function of time steps, 5 different runs in the same temperature T = K

10 Figure 11: average of each lattice constant for every one of the runs, the title contents the average of the averages and its standard error deviation. Figure 12: figure 4 vs. lattice constants as a function of temperature with new value at T = K

11 Figure 13: figure 5 vs. thermal expansion of the lattice constants with new value at T = K a(t) [Angstrom] c(t) [Angstrom] temperature [K] temperature [K] Figure 14: lattice constants as a function of temperature, each point is an average over 1000 time steps starting from step number The solid line connects between the averages of all the point in the same temperature. This figure includes all the runs for T = K

12 2.48 x 104 Volume expansion as a function of temperature Volume(T) [Angstrom 3 ] Volume(T) [Angstrom 3 ] temperature [K] 2.46 x Volume at T=K as a function of run number run # Figure 15:

What is Classical Molecular Dynamics?

What is Classical Molecular Dynamics? Simulation of explicit particles (atoms, ions,... ) Particles interact via relatively simple analytical potential functions Newton s equations of motion are integrated