Project02_PointDefects分子动力学实验：点缺陷

MD Project # 2Point Defects in Metals:Vacancies and Interstitials1.VacancyA vacancy is created when an atom within a lattice is removed. Here we discuss how to introduce a vacancy using LAMMPS and how to compute the vacancy formation energy, E v.LAMMPS input file, in.vacancy1) Creates a vacancy in a perfect crystal (FCC Copper)LAMMPS creates a perfect crystal of 4 ×N×N×N Cu atoms with edges along <100> directions. Then an atom is removed. (N is the size of the simulation box)2) RelaxationWhen an atom is removed from a crystal, the surrounding atoms will readjust their atomic positions to lower the potential energy. To obtain such a relaxed vacancy configuration one may use energy minimization techniques. In the sample, the conjugate gradient (CG) or an alternative algorithm (sd) may be used to minimize the energy.The input file in.vacancy:3) Run Lammps4) Calculate the Vacancy Formation EnergyThe vacancy concentration at thermal equilibrium is given by[n ] = exp( − F v / k B T )where F v = E v − TS v is the Helmholtz free energy to form a vacancy. Neglect the entropy contribution S v (~ 2k B for Cu), we have[n ] ~ exp( −E v / k B T )The equation also apply for other point defects such as interstitials (see later).Let E 1 be the energy of the perfect crystal with N atoms, and E 2 the relaxed energy of the system containing the vacancy (N – 1 atoms). The vacancy formation energy E v can be computed from211v N E E E N −≡−or ()21v coh E E N E ≡−−where E coh = E 1 / N , the cohesive energy of the perfect crystal. The result for FCC Cu using the EAM potential with a supercell of 4×(20×20×20) atoms is E v ~1.26 eV, comparing to ~1.28 eV provided in literature.The [n] can then be estimated to be ~ 7.59×10-22 and ~ 2.2×10-5 at T=300 K and 1350 K (T m ), respectively (Note: 1 eV/k B = 1.1604×10−4 K; [n]~2×10-4 for Cu in literature)Fig. 1 The vacant site locates in the center of a FCC 4×(6×6×6) crystal of Cu, with 106c a =,206c a = and 306c a =. Colored according to potential energy per atom.2. InterstitialAn interstitial is created when an atom is inserted into a perfect lattice. If the additional atom is of the same type as lattice atoms, it is called a self-interstitial.Similar to vacancy, we use LAMMPS to calculate the formation energy (E i ) of aself-interstitial in Cu.211i N E E E N+≡− E i (relaxed) might depend on the site the atom was initially introduced, but there should be an optimized interstitialcy associated with a minimum E i .In several FCC metals such as Cu, Ni and Pt, the most stable self-interstitial configuration is a split structure, with two atoms displaced in positive and negative [100] directions, the so-called “dumbbell” interstitialcy (Fig. 2).Fig. 2 A Dumb-bell interstitial configuration. Right, that for Cu obtained using LAMMPS.When an interstitial is introduced, the surrounding atoms will readjust their positions to lower the potential energy. To obtain the relaxed configuration one may use the energy minimization technique. In our sample, the ‘sd’-algorithm will be used for such a purpose.To find a configuration with the minimum energy, we choose to use an energy minimization combined with a thermal equilibrations (via a NVT or NVE ensemble). Temperature effects are included in order to help atoms to find the most stable interstitial configuration. Then we “quench” it to 0 K via energy minimization and calculate the energy.The input file in.interstitial looks likeFor a system of 32000 atoms, we found the interstitial energy E i ~ 3.1 eV for Cu.Similarly, the [n i] for Cu can then be estimated to be ~ 8.4×10-53 and ~ 2.7×10-12 at T=300 K and 1350 K (T m), respectively.DiscussionsVarious interstitial configurations may be observed if atoms are not relaxed enough, see Figs.3 and 4.1.Check how Ev depends on the system size.2.Calculate possible interstitial formation energies for Cu, map out the most stableinterstitial configuration for Cu.(a)(b) (c)Fig. 3 Octahedral (red) and tetrahedral (blue) interstitial symmetry polyhedra in a FCC lattice (a). The most stable interstitial site is supposed to be the center of the octahedron instead of the tetrahedron, as revealed in (b) and (c).Fig. 4 Possible configurations of interstitials。