ATOMISTIC MODELING OF DIFFUSION IN ALUMINUM

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1 ATOMISTIC MODELING OF DIFFUSION IN ALUMINUM S. GRABOWSKI, K. KADAU and P. ENTEL Theoretische Physik, Gerhard-Mercator-Universität Duisburg, Duisburg, Germany (Received...) Abstract We present molecular-dynamics simulations of self-diffusion in Al. In order to facilitate the description of elastic and vibrational properties as well as vacancy migration, an embedded-atom method potential was used in the simulations. This potential was specifically designed to reproduce the T = 0 K equation of state of Al obtained by ab initio total-energy calculations. We show that the temperature dependent self-diffusion coefficient obeys an Arrhenius law and that the resulting dynamical migration energy is slightly larger than the static migration energy obtained by using classical rate theory. Keywords: Self-diffusion, Vacany migration, Molecular-dynamics simulation 1. INTRODUCTION Molecular-dynamics can be a useful tool in the study of solid-state reactions provided one is able to cover a sufficiently large time range. By numerically solving the classical equations of motion, the time evolution of a system containing thousands of atoms can be simulated up to the nano-second range. This, in principle, is sufficient to calculate the temperature dependent diffusion coefficient D(T) and to investigate individual atomic jump processes in systems which have a natural high mobility of atoms. Comprehension of mechanisms underlying atomic diffusion in metals can be gained from direct observation of individual atomic jumps occurring during the simulation. Depending on the accurateness of interaction potentials between atoms specific aspects of diffusion can be studied. Of interest are, for example, deviations of D(T) from an Arrhenius law as observed for several metals. Such deviations can be investigated by analyzing the contributions of different jump mechanisms (Lorenzi and Ercolessi, 1992). Another interesting aspect is the dynamical reduction of the migration energy for self-diffusion due to low lying 1

2 phonon modes in bcc metals (Köhler and Herzig, 1988). This effect can be studied by a close inspection of the trajectories of the jumping atom and the motional behavior of surrounding atoms (Mikhin and Osetsky, 1993). Furthermore the validity and limitations of theoretical concepts like classical rate theory can be investigated without essential simplification of the N-body potential energy surface (Jacucci, 1984; Schott, Fähnle and Seeger, 1997). Here we report first results of molecular-dynamics simulations in which we employed an embedded-atom method potential specially designed for a reliable description of self-diffusion in Al. 2. COMPUTATIONAL METHODS 2.1. Construction of an optimized EAM potential The key ingredient of any molecular-dynamics simulation is the force law describing the interactions between the atoms. Embedded-atom method (EAM) potentials of the form proposed by Daw and Baskes (1984) are appropriate to describe lattice defects and lattice dynamics in metals. In the EAM case, the potential energy of a single atom is considered as the energy required to embed the atom into the local electron density provided by the other atoms plus a core-core repulsion potential which takes the form of a pair potential. The potential energy is written as E = i F(ρ i ) + i<j Φ (r ij ), (1) where the summation is over all atoms and r ij is the distance between the atoms i and j. The function F(ρ i ) represents the embedding energy of atom i depending on the background charge density ρ i = j i ρ at j (r ij ) (2) which is the sum over all atomic contributions ρ at j (r ij) from the neighboring atoms. We use double-ζ wave functions Ψ of Clementi and Roetti (1974) for the calculation of the atomic contributions ρ at (r) = N 3s Ψ 3s (r) 2 + N 3p Ψ 3p (r) 2 ρ c. (3) Here N 3s = 2.0 and N 3p = 1.0 are the numbers of s- and p-symmetry valence electrons of Al. In order to get a finite interaction range, ρ at (r) is truncated at a cutoff radius R c and the constant shift ρ c = ρ at (R c ) is applied to obtain a continuous behavior of the embedding charge density at r = R c. The second term in Eq. 1 represents a screened Coulomb potential, Φ(r) = 2 Z(r)Z(r) r (4) 2

3 with the effective charge Z(r). Here the factor 2 stems from the atomic units used in this paper. The function F(ρ) and the effective charge Z(r) are represented by cubic spline functions. In order to construct an EAM potential which allows for a reasonable description of defect and lattice vibrational properties, the parameters of these functions have been determined by a weighted least square fit to experimental data of Al. It turned out that a minimum set of experimental data is given by the lattice constant a 0, sublimation energy E s, elastic constants C 11, C 12 and C 44, vacancy formation energy Ev f, and some selected phonon frequencies of Al. These experimental data are listed in Table II. However, EAM potentials which reproduce these experimental data in the simulations, do not automatically guarantee a proper description of locally compressed atomic configurations occurring during the diffusional jumps of atoms. The static vacancy migration energy calculated from an EAM potential which has been fitted to the data in Table II, is about a factor two larger than the corresponding experimental value of (0.61 ± 0.03) ev (Erhart, 1990). The description of mono-vacancy diffusion on EAM basis can be improved if the pair potential Φ(r) is more carefully adjusted for interatomic distances which are smaller than the equilibrium lattice constant, i.e. for 0.85 < r/a 0 < 1.0, because the strong repulsive forces between the jumping atom and its nearest neighbors in the unrelaxed saddle point configuration are mainly due to Φ(r). By additionally fitting the EAM potential to the T = 0 K equation of state (see Fig. 1) calculated on ab initio basis, the resulting static vacancy migration energy is much closer to the experimental value. The potential used in our simulations was fitted to the equation of state for lattice constants in the range 0.9 < r/a 0 < 1.05, whereby the compressed lattice constant of 0.9 a 0 corresponds to an external pressure of 40 GPa. If the potential is fitted to even smaller lattice constants, i.e. higher external pressures, then the static migration energy can be reduced to 0.65 ev, which is very close to the experimental value cited above. But in this case we failed to describe the vibrational and elastic properties with comparable accuracy. In order to exclude such unphysical dynamical behavior, a subtle choice has to be made of which compressional states have to be considered in the fit. The effective pair potential corresponding to the fitted EAM potential in the case of Al is shown in Fig Molecular-dynamics In order to calculate the temperature dependence of the self-diffusion coefficient D(T), the EAM potential defined in Tab. I was used in the moleculardynamics study [for details of the method we refer to Allen and Tildesley (1987)] of fcc elementary cells (2048 lattice sites) with periodic boundary conditions. In order to observe a sufficient rate of elementary diffusion jumps, a rather high concentration of vacancies was used corresponding to 16 vacancies in the simulation box. The vacancies were initially arranged 3

4 on a bcc lattice with lattice constant 4 a (with a the lattice constant of the underlying fcc lattice). For each temperature T the lattice constant a was chosen to correspond to its zero pressure value. The classical equations of motion were integrated with the velocity Verlet algorithm with a time step of 1.5 fs and the temperature was controlled by the Nosé-Hoover thermostat (Nosé, 1984; Hoover, 1985). The box volume was kept fixed throughout each run Diffusion coefficient and migration energy The self-diffusion coefficient D can be calculated from the Einstein relation D = lim t 2 r(t) 6t and the long-time limit of the mean square displacement defined by 2 r(t) = 1 N (5) N [r i (t) r i (0) ] 2, (6) i=1 where r i (t) is the time dependent position of atom i; N is the number of atoms in the simulation box (Rapaport, 1995). In order to calculate D(T) we performed molecular-dynamics runs for temperatures between 770 K and 950 K. For each temperature the mean square displacement was sampled over a time interval of length 3.75 ns and D(T) was calculated from the slope of the linear least mean-square fit to 2 r(t). Examples are plotted in Fig DISCUSSION An Arrhenius plot of the resulting diffusion coefficient D(T) is shown in Fig. 4. From an exponential least mean-square fit to our D(T) data we obtain the dynamical migration energy E m = 0.94 ev as well as the prefactor D 0 = m 2 s 1, i.e. the diffusion coefficient can be described by the Arrhenius law ( D(T) = D 0 exp E ) m. (7) k B T From static calculations using classical rate theory (Vineyard, 1957; Flynn, 1972) we obtain the migration energy E m = 0.75 ev which is 0.19 ev smaller than our calculated dynamical migration energy. In order to estimate the error of the dynamical migration energy E m obtained from our D(T) data, we calculated the 99% confidence limit for E m within random sampling theory of regression (Spiegel, 1976). The resulting error corresponds to ev i.e., the dynamical migration energy calculated from D(T) is by at least 0.1 ev larger than the static value. 4

5 4. SUMMARY The aim of this work was to specifically design an EAM potential which allows for a reasonable description of vacancy diffusion in Al. It turned out that by an additional fit of the EAM potential to the T = 0 K equation of state, the vacancy migration energy calculated within the framework of classical rate theory could be adjusted to the experimental value. For the potential defined in Table I the static migration energy obtained is 0.75 ev. The simulated temperature dependence of the diffusion coefficient exhibits an Arrhenius law behavior and the dynamical migration energy obtained from the slope of the Arrhenius plot in the temperature range K is given by 0.94 ± ev, which is at least 0.1 ev larger than the static value. We do not have an obvious explanation for the difference between the static and the dynamic value of the vacancy migration energy. It is a well known fact that classical rate theory yields an upper limit for the vacancy jump rate and, therefore, overestimates the diffusion coefficient because return jumps are not taken into account (Hänggi, Talkner and Borkovec, 1990). According to the analytical and numerical investigations of Lorenzi et al. (1987) and Marchese et al. (1987), classical rate theory should be an excellent approximation for fcc model crystals with Lennard-Jones-type of interactions because return jumps do not lower the vacancy jump rate by more than 10%. But this error of classical rate theory is much too small to explain the obseved difference between the static and the dynamic migration energy obtained in the case of Al. Contributions to the diffusion coefficient D due to other jump processes like correlated double jumps cannot be excluded at least for the highest simulation temperatures. Therefore, a slight increase of the activation energy obtained from the Arrhenius plot can be expected because migration energies of correlated double jumps are approximately twice the migration energy of single vacancy jumps (Lorenzi and Ercolessi, 1992). Probably an intricate combination of systematical errors has occurred which could originate from vacancy-vacancy interactions due to the large concentration of vacancies used here, periodic boundary conditions and insufficient supercell size as well as insufficient statistics for single vacancy jumps at low temperatures. Nonetheless this is the first observation of elementary diffusion jumps seen in molecular-dynamics simulations for Al. References Allen, M. P. and D. J. Tildesley (1987). Computer Simulation of Liquids. Clarendon Press, Oxford. Clementi, E. and C. Roetti (1974). Atomic Data and Nuclear Data Tables, volume 14. Academic Press, New York. Daw, M. S. (1990). The embedded atom method: A review. In R. M. Nieminen, 5

6 M. J. Puska and M. J. Manninen, editors, Many-Atom Interactions in Solids, Springer Proceedings in Physics 48. Springer, Berlin. Daw, M. S. and M. I. Baskes (1984). Embedded-atom method: Derivation and application to impurities, surfaces and other defects in metals. Phys. Rev. B, 29, Erhart, P. (1990). Properties and interactions of atomic defects in metals and alloys. In H. Ullmaier, editor, Landolt-Börnstein, volume 25 of New Series Group III. Springer, Berlin. Flynn, C. P. (1972). Point Defects and Diffusion. Clarendon Press, Oxford. Gui, J. (1994). Embedded-atom method study of the effect of the order degree on the lattice parameters of Cu-based shape-memory alloys. J. Phys.: Condens. Matter, 6, Hänggi, P., P. Talkner and M. Borkovec (1990). Reaction-rate theory: Fifty years after Kramers. Rev. Mod. Phys., 62, 251. Hoover, W. G. (1985). Canonical dynamics: Equilibrium phase-space distributions. Phys. Rev. A, 31, Jacucci, G. (1984). Defect calculations beyond the harmonic model. In G. E. Murch and A. S. Nowick, editors, Diffusion in Crystalline Solids, 431. Academic Press. Köhler, U. and C. Herzig (1988). On the correlation between self-diffusion and the low-frequency LA 2/3 111 phonon mode in b.c.c. metals. Phil. Mag. A, 58, 769. Kresse, G. and J. Furthmüller (1996). Efficient iterative schemes for ab initio total-energy calculations using a plane wave basis set. Phys. Rev. B, 54, Lorenzi, G. D. and F. Ercolessi (1992). Multiple jumps and vacancy diffusion in a face-centered-cubic metal. Europhys. Lett., 20, 349. Lorenzi, G. D., G. Jacucci and C. P. Flynn (1987). Jump rate of the fcc vacancy in the short-memory-augmented-rate-theory approximation. I. Difference Monte Carlo sampling for the Vineyard rate. Phys. Rev. B, 36, Marchese, M., G. Jacucci and C. P. Flynn (1987). Jump rate of the fcc vacancy in the short-memory-augmented-rate-theory approximation. I. dynamical conversion coefficient and isotope-effect factor. Phys. Rev. B, 36, Mikhin, A. G. and Y. N. Osetsky (1993). On anomalous self-diffusion in body centered cubic metals: A computer simulation study. J. Phys.: Condens. Matter, 5, Nosé, S. (1984). A molecular dynamics method for simulations in the canonical ensemble. Mol. Phys., 52, 255. Rapaport, D. C. (1995). The Art of Molecular Dynamics Simulation. Cambridge University Press, Cambridge. 6

7 Rubini, S. and P. Ballone (1993). Quasiharnonic and molecular-dynamics study of the martensitic transformation in Ni-Al alloys. Phys. Rev. B, 48, 99. Schott, V., M. Fähnle and A. Seeger (1997). Molecular-dynamics study of selfdiffusion in Na: Validity of transition-state theory. Phys. Rev. B, 56, Spiegel, M. R. (1976). Theory and Problems of Statistics. McGraw-Hill. Vineyard, G. H. (1957). Frequency factors and isotope effects in solid state rate processes. J. Phys. Chem. Solids, 3,

8 Table I: Parameters determining the embedding energy function F(ρ) and the effective charge function Z(r). Energies are given in units of Ry and charges in units of e. The equilibrium charge density is given by ρ 0 = a.u., the equilibrium lattice constant is given by a 0 = a.u. and the cut off radius R c is 10.1 a.u.. ρ/ρ 0 F F r/a 0 Z Z

9 Table II: Experimental data used in the fit of the EAM potential and the corresponding calculated values. Phonon frequencies are given in mev. The calculated values correspond to an EAM potential which has also been fitted to the ab initio data shown in Fig.1. Experimental phonon frequencies have been taken from Rubini and Ballone (1993) and the other experimental data have been taken from Gui (1994). Expt. Calc. a 0 (a.u.) E s (ev) E f v (ev) C 11 (10 12 dyn/cm 2 ) C 12 (10 12 dyn/cm 2 ) C 44 (10 12 dyn/cm 2 ) q = 2π a 0 [100] q = 2π a 0 [110] q = 2π a 0 [111]

10 0.21 E pot (V ) (Ry) V (a.u.) Figure 1: Potential energy per atom E pot (V ) as a function of the volume V per atom. Curves for two different EAM potentials fitted to the experimental data listed in Table II are shown. The E pot (V ) curve which corresponds to an EAM potential fitted to the experimental data of Table II, is represented by a dashed line and the full line represents the E pot (V ) curve corresponding to an EAM potential with an additional fit to ab initio data, represented by full circles. Open and full circles represent E pot (V ) calculated by the Vienna ab initio Simulation Package (Kresse and Furthmüller, 1996) using ultrasoft Vanderbilt pseudopotentials. The ab initio data are shifted so that their minimum energy and the corresponding lattice constant coincide with the experimental values of E s and a 0 in Table II. 10

11 F(ρ) (mry) 300 φ (r) (mry) ρ (ρ 0 ) r (a 0 ) F eff (ρ) (mry) 195 φ eff (r) (mry) ρ (ρ 0 ) r (a 0 ) Figure 2: Embedding energy, F(ρ), and pair potential, Φ(r), for the EAM potential defined in Table I. The corresponding effective embedding energy, F eff (ρ), and effective pair potential, Φ eff (r), are also shown. According to Daw (1990) the effective potentials are defined by Φ eff (r) = Φ(r) + 2 c ρ(r) and F eff (ρ) = F(ρ) c ρ with c = F (ρ 0 ), where ρ 0 is the equilibrium embedding charge density from Table I. Both EAM potentials, defined by the pairs (F, Φ) and (F eff, Φ eff ), are equivalent. The second minimum of F eff, approximately at 2.0 ρ 0, results from a fit of the potential to the equation of state for high external pressures, i.e., smaller lattice constants. 11

12 16 < r 2 (t) > (a.u.) K 910 K 870 K 810 K t (ps) Figure 3: Mean square displacement, 2 r(t), for different temperatures. Linear least mean-square fits to these data are represented by dashed lines. The simulation time of 3.75 ns corresponds to time steps (a time step correponds to 1.5 fs). 12

13 10 11 D (T) (m 2 s 1 ) / k B T (ev 1 ) Figure 4: Arrhenius plot for the Diffusion coefficient (full circles) calculated from the mean-square displacements by using the Einstein relation. The exponential least-mean square fit to these data is given by the straight line. Simulation temperatures are between 770 and 950 K. From the result of the exponential least mean-square fit to D(T) we obtain the dynamical migration energy E m = 0.94 ev as well as the prefactor D 0 = m 2 s 1. Error bars correspond to standard deviations calculated according to σ/d = 1/ n with n the number of diffusional jumps in the simulation time. 13