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A MOLECULAR DYNAMICS STUDY OF THE THERMAL DISPLACIVE PROPERTIES OF THE Σ=13 (θ=22.6°) [001] TWIST BOUNDARY IN NOBLE
METALS
G. Evangelakis, M. Hou, C. Maunier, V. Pontikis
To cite this version:
G. Evangelakis, M. Hou, C. Maunier, V. Pontikis. A MOLECULAR DYNAMICS STUDY OF THE THERMAL DISPLACIVE PROPERTIES OF THE Σ=13 (θ=22.6°) [001] TWIST BOUND- ARY IN NOBLE METALS. Journal de Physique Colloques, 1990, 51 (C1), pp.C1-127-C1-132.
�10.1051/jphyscol:1990118�. �jpa-00230276�
A MOLECULAR DYNAMICS STUDY OF THE THERMAL DISPLACIVE PROPERTIES OF THE 2=13
( 8 = 2 2 . 6 ' ) [OOl] TWIST BOUNDARY IN NOBLE METALS
G.A. EVANGEEAKIS, M. HOU*, C. MAUNIER and V. P O N T I K I S
Section d e Recherches de Metallurgie Physique, Centre d'Etudes Fucl6aires de Saclay, F-91191 Gif sur Yvette Cedex, France
Universit6 Libre de Bruxelles CP234, Bd. du Triomphe, B-1050 Brussels, Belgium
RdsumC - Nous avons calculC par simulation, k l'aide de la technique de dynamique moleculaire k temperature constante, le deplacement quadratique moyen des atomes dans le joint de grains de torsion Z=13 (9=22.6') L001
1
pour l e cuivre,l ' a r g e n t e t 1 'or.Les interactions interatomiques ddrivent d'un pseudopotentiel resonnant adapte aux metaux nobles. A proximite du joint la distance interplanaire suivant l'axe de torsion est modifiCe au maximum de +6.5%. La modification des constantes de force locales qui en resulte, induit des amplitudes de vibration plus klevkes qu'en volume. Nous avons trouvC que le deplacement quadratique moyen des atomes du joint B cceur est de 1.5 h 2.5 fois plus Clev6 qu'en volume. Ces rksultats sont en accord qualitatif avec les donnies experimentales pour l'or.Abstraa - We present the results of a molecular dynamics study at constant temperature of the mean square displacements of atoms in the core of the E 1 3 (8=22.6') [OOl] twist grain boundary in copper, gold and silver. The atomic forces derive from a resonant model pseudopotential adapted to noble metals. The relaxations of atomic planes along the twist axis near the boundary, amount a maximum value +6.5%. Consequently, the local force constants are modified with respect to the bulk and enhanced vibration amplitudes are observed. Depending on the temperature, we found that the atomic mean square displacements in the boundary core are 1.5 to 2.5 times larger than in the bulk.
These results are in qualitative agreement with availabIe experimental data for the same boundary in gold.
1- INTRODUCTION
Similarly to crystal surfaces, grain boundaries induce relaxations of the atomic lattice in their vicinity which can, depending of the boundary type and misorientation, reach values as high as 20% /l/. These, in conjunction with the modification of the local environment of atoms in the grain boundary, modify the shape of the local interatomic potential and thus the force constants are different from those of the buIk. Enhanced amplitudes of atomic vibrations and mean square displacements (MSD) with respect to the bulk are therefore expected in the vicinity of grain boundaries, by analogy with surfaces for which such a behavior is well established 12-5/.
Among the various motivations to study of the thermal displacive properties of grain boundary or surface atoms, the possibility they offer to test and improve interatomic potentials belongs to the most important. However, unlike surfaces, such experimental investigations of grain boundaries encounter serious difficulties thus explaining the lack of experimental data. A unique experimental study is available, performed by Fitzsimmons et al. /6/ which succeeded in determining the Debye-Waller factor and the thermal expansion coefficient near a Z=13 (8=22.6') [OOl] twist grain boundary in gold, using quantitative synchrotron X-ray measurements /7/.
In this paper we present the results of a molecular dynamics (MD) study of the temperature dependence of the atomic MSD in the bulk and near a X=13 (8=22.6') [001] twist grain boundary, using a resonant model pseudopotential /8,9/ adapted to the noble metals Ag, Au and Cu. The choice of this particular grain-boundary allows to compare our computations with the above mentioned experimental data. The results we obtained show that the potential we used
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990118
Cl-128 COLLOQUE DE PHYSIQUE
leads to an excellent agreement between MD and experimental data for the bulk systems.
Moreover, good agreement exists also between MD and experimental interfacial MSD values for gold. Depending on the temperature, these are 1.5 to 2.5 times larger than in the bulk.
In section 2 we give the details of the models employed and the computations. Our results and the comparison with the experimental data are presented in section 3, whereas section 4 is devoted to a brief conclusion.
2- MODEL AND COMPUTATIONAL D E T A U
Bulk MSD have been computed using models of N=4000 point particles contained in a cubic box of lOxlOxlO lattice cells. For models of this size it has been recently shown I101 that the MSD, <u2>, which are known to be size-dependent (<u2>
-
UN), converge practically to the thermodynamical limit value. Bicrystalline models consisted in N=2080 particles and have been constructed as follows : two crystals with faces respectively parallel to the (OOl), (510) and (150) crystallographic planes are joined together at a distance equal to the (100) interplanar spacing. Periodic boundary conditions are imposed along these directions and thus, two infinitely extended X=13 (8=22.6") [l001 twist GB's exist in the system. The perturbation of the atomic positions in the bulk, related to the presence of a GB, is known to propagate at a distance roughly equal to the boundary periodicity. Therefore the size of the bicrystal is chosen equal to 20x[001]x2[510]x2[1~] periods in order to preserve a bulk region between the two GB's.The particles interact through a resonant model pseudopotential, @(r), adapted to noble metals 181 which satisfactorily reproduces numerous properties of the real materials such as elastic constants 181, phonon dispersion 11 11, formation and migration energies of point defects 191. The long-range Friedel oscillations of this potential require in principle the use of very large systems and therefore an important. computational effort. Therefore we used instead an effective potential, @,ff(r)=exp(-62r2)@(r) with 6=0.3, according to the prescriptions of Lam et al. 191.
The atomic interactions have been computed using a cutoff radius rc=2.4a, where, a, is the lattice parameter. The geometrical configuration constructed as indicated above has been relaxed using a quasidynamic procedure which consists in a MD calculation modified to include a damping force 1121.
The simulations were performed at constant temperature using the Nose MD technique 1131 with a pseudomass Q=5.10-37 erg.s2 at temperatures ranging from T=100 to 1200 K, close to the experimental melting point temperatures of copper, gold and silver. At each temperature the experimental lattice parameter has been chosen 114-161. The equations of motion are integrated using a fifth order predictor corrector algorithm /17/ with a timestep: 6t = %lO-15 S.
To obtain equilibrium at a given temperature, the system is allowed to evolve during 4000 time steps which were excluded from the calculation of thermodynamical averages. Averages of the atomic density :
were computed along the [l001 direction parallel to the x-axis and to the twist axis for the bicrystals during equilibrium trajectories which lasted 4000 timesteps. In equation (1) brackets indicate time averages and 6 represents the Dirac delta function (figure 1). MSD values are calculated from the density profiles on a local basis by :
to the kfh atomic plane, its mean position and the extremum abscissa values delimiting the correponding peak of the atomic density profile.
Fig. 1
-
Average atomic density profile (in arbitrary units) along the twist axis of a gold bicrystal at T=600 K. Each peak corresponds to a given (100) atomic plane. The change in shape and maximum value of the central peaks, with respect to those correponding to.bulk atomic planes, is related to the presence of the grain boundary and permits to locate its position.3
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RESULTSRelaxations normal to the boundary affect the interlayer spacing in the boundary core which differs up to 6.5% from its bulk value. We may therefore conclude that the E 1 3 (0=22.6') [OOl] twist grain boundary we studied disturbs in a lesser extent the surrounding crystalline lattice than some high-angle tilt boundaries for which the extension of the perturbed region and the variation of the interlayer spacing are significantly larger /l/. The interplanar distances profiles along the twist axis show that for the highest temperature we investigated, T=1200 K, the perturbation introduced by the grain boundary does not affects more than six (100) atomic planes.
Fig. 2 - MSD profiles for copper bicrystal along the twist axis. Curves from bottom to top correpond respectively to temperatures T=100,300,600, 800, 1000 and 1200 K.
Cl-130 COLLOQUE DE PHYSiQUE
Hence, the actual size of-the bicrystal guarantees the presence of a bulk region between the two grain boundaries. This is confirmed by examining the MSD profiles illustrated in the case of copper by figure 2.
The results illustrating the MSD temperature dependence for Ag, Au and Cu are displayed on figures 3a-3c (see also table I) together with the experimental values for the bulk 118,191.
Excellent agreement is obtained between experimental and computed bulk MSD data, which further illustrates the adequacy of the potentials we used to correctly represent the dynamical properties of noble metals in the bulk.
Fig. 3 - Bulk and grain boundary MSD temperature dependence for : (a) Ag, (b) Au and (c) Cu.
Full squares
(m)
: MD data, asterisks ( X ) : experimental values 118,191, full dots (@) : MD data in the grain boundary core.temperature and, over the two boundaries present in the system. Atomic vibrations are enhanced in the grain boundary and lead to MSD values 1.5 to 2.5 times larger than in the bulk. This result shows that grain boundaries behave similarly to free surfaces 13-51. Thanks to the recent experimental work by Fitzsimmons et al. 161 a comparison of computed and experimental MSD in the grain boundary can be made in the case of gold. At T=300 K the experimental value
<u2>=0.012f0.002
A2
l61 fits satisfactorily with the MD result we obtained<u2>=0.0146f 0.0036
A2.
Table I - Computed MSD values for Ag, Au and Cu in the bulk (B) and in the grain boundary (GB) core (see text), expressed in A2.
Moreover, we expect that the melting point of the models we studied is comparable to the experimental one, as a consequence of their realistic MSD dependence on temperature. Indeed, the Lindemann empirical criterion 1201, which states that melting occurs in crystals when the square root of MSD reaches a critical percentage of the nearest neighbors distance, is generally satisfied 1211.
4
-
CONCLI JSIONUsing molecular dynamics at constant temperature, we studied the displacive thermal properties of noble metals Ag, Au and Cu in the bulk and near a Z=13 (8=22.6') [OOl] twist grain boundary. Excellent agreement is found between computed and experimental MSD data for the bulk. Moreover, enhanced atomic MSD are observed in the grain boundary in satisfactory agreement with recent experimental data obtained for the same grain boundary in gold.
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