Atomic scale modelling of edge dislocation movement in the α-fe Cu system

Transcription

1 Modelling Simul. Mater. Sci. Eng. 8 (2000) Printed in the UK PII: S (00) Atomic scale modelling of edge dislocation movement in the α-fe Cu system S Nedelcu, P Kizler, S Schmauder and N Moldovan Staatliche Materialprüfungsanstalt (MPA), University of Stuttgart, Pfaffenwaldring 32, D Stuttgart, Germany Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439, USA peter.kizler@mpa.uni-stuttgart.de Received 26 September 1999, accepted for publication 26 January 2000 Abstract. The aim of the present work is to investigate by molecular dynamics (MD) calculations the interaction between a moving edge dislocation in an α-fe crystal and a copper precipitate. In the absence of external stresses, two edge dislocations with the same slip plane and opposite Burgers vectors within a perfect α-fe crystal lattice are investigated. In agreement with Frank s rule, the movement of the dislocations under mutual attraction is found and attention is focused on the interaction between one of the dislocations and the Cu precipitate. The critical resolved shear stress of the Fe was calculated and the influence of different sizes of Cu precipitates on the dislocation mobility was studied. The pinning of the dislocation line at the Cu inclusion as derived from the atomistic modelling agrees with previously published continuum theoretical behaviour of pinned dislocations. Therefore, nanosimulation as a way to model precipitation hardening could be established as a useful scientific tool. 1. Introduction A detailed understanding of the behaviour of dislocations is essential in determining the mechanical properties of metals and alloys. In recent years a number of molecular dynamic and static simulations [1] were performed on α-fe single crystals under tensile loading, to account for yield stress, work hardening as well as defect nucleation and growth in this material. Taking into account the presence of precipitates, as for example copper, and the already present dislocations, the flow-stress depends upon the interaction between the dislocation and the obstacle [2, 3]. Ultimately, the strength of the material depends upon the crystallographic structure and dimensions of the precipitates in it [4 6]. While continuum theory can describe the long-range strain fields of cracks and dislocations, atomistic simulations are required to characterize dislocation core structure and dislocation precipitate interactions. In continuum theory, the dislocations are considered to be smooth flexible strings with a line tension [7]. When large dislocation curvatures are encountered, for instance in the presence of precipitates, the results provided by the line tension approximation can be inaccurate. In the present paper, an atomistic simulation model is presented that allows for a smooth movement of two edge dislocations in the absence of applied stresses and thus permits to observe the elastic behaviour of a dislocation line during On leave from National Institute for Research and Development in Microtechnologies (IMT), Bucharest, Romania. Current address: Department of Physics and Astronomy, University of Southampton, UK. Corresponding author /00/ $ IOP Publishing Ltd 181

2 182 S Nedelcu et al the interaction with an obstacle. The use of atomistic modelling also may offer the chance to simulate dislocation phenomena relying on basic atom-scale data, without empirical data on the mesoscopic scale. The case of a spherical coherent Cu precipitate with a body-centred cubic (bcc) lattice identical to that of the α-fe matrix is studied in the context of the present paper. In the following section the computational model is described. In section 3, the critical resolved shear stress of Fe is calculated and a comparison with related values found in literature is provided. Some discussions upon the elastic behaviour of the dislocation line, pinned to the centre or trapped in the Cu precipitate as the size of the obstacle diameter changes follow in section The computational model In bcc metals slip occurs in close-packed 110 directions [8]. The Burgers vector of the perfect slip dislocation is of the type The motion in the glide plane is that which 2 constitutes the macroscopic phenomenon of slip in crystals [9]. This kind of motion is very easy along the glide direction 111 in the α-fe crystal and assures a smooth movement of the dislocations. Starting from this observation and from Frank s rule [8] for determining whether or not it is energetically feasible for two dislocations to react and combine to form another one, the present simulation model was constructed. The movement of edge dislocations in the absence of externally applied shear stress will be investigated. Edge dislocations in bcc metals, unless they are locked by impurities, are much more mobile than screw dislocations [10]. As a result, in high-purity specimens, yielding takes place first by motion of edge dislocations at a low stress [10]. Only at a later stage of deformation, which is beyond the scope of the present work, the specimens are exhaustion hardened because multiplication cannot occur without the motion of screw dislocations. Figure 1. Schematic representation of a section through the sample, showing the initial position of the edge dislocations and the Cu atoms (grey). In the present model, the atoms were initially placed on perfect bcc crystal lattice sites of α- Fe. The coordinate axes were chosen parallel to the sides of the simulation cell with the x-axis along 1 10, y-axis along 111 and z-axis along, 112. The dimensions of the simulation cell along x-, y- and z-axes are chosen to be 75.3 Å, Å and 76.3 Å, respectively. The outer surfaces perpendicular to x- and z-axes were free and no periodic boundary conditions were applied. Two initial straight edge dislocations, ending at free surfaces, with the same slip plane (1 10) and opposite Burgers vectors b = 111 were introduced along the z-direction by removal of two half-planes of atoms. The initial points where the line of the first and the second dislocation intersected the y-axis were chosen at the origin of the coordinate system and at 115 Å along the y-axis, respectively. In order to preserve the invariance to free translations and rotations, at the left and at the right ends of the sample, see figure 1, six atom layers perpendicular to the y-axis were kept fixed at their initial positions. A schematic representation of a section through the sample, showing the initial position of the edge dislocations and the Cu

3 Modelling of edge dislocation movement 183 atoms is presented in figure 1. In the present molecular dynamics (MD) simulation a number of atoms was considered, and the MD program FEAt developed by Kohlhoff and Schmauder [11] was employed. As input to FEAt, a data file with the undisturbed coordinates of the atoms together with the initial displacement field of the two unlike edge dislocations was created. To calculate the initial displacement field of one single dislocation the theory of a moving edge dislocation described by Stroh [12] was used. For an edge dislocation parallel to the z-axis and with the glide plane z y, the only displacement components are u i = (u x, u y, 0). A general expression for the displacements, according to anisotropic elasticity, may then be written as ( u i = Re 1 2πI ) 3 A i (p α )D α log(x + p α y) α=1 (i = x,y) (1) where A α and D α are tensors of material properties expressed in the chosen coordinate system, depending on the components of the Burgers vector and the elastic constants. The numerical values used [12] are given in table 1. Table 1. Numerical values of the coefficients which define the displacement field (1) of an edge dislocation along the 112 direction in the α-fe crystal. A x (p 1 ) D I A x (p 2 ) D 2 0 A x (p 3 ) D I A y (p 1 ) I p I A y (p 2 ) 0 p 2 0 A y (p 3 ) I p I The displacement field of the second dislocation was simply considered as having the form (1), with the proviso that the initial distance between the dislocations is large enough such that the displacement field of it matches well with that of the first dislocation and interactions do not appear. The introduced Cu precipitate, with a bcc structure and with the same lattice parameter for Cu as for α-fe, is placed close to one of the edge dislocations (figure 1). The MD simulation is carried out for two values of the diameter of the Cu precipitate: 13.2 and 30.4 Å. These precipitates consist of 121 and of 1254 Cu atoms, respectively. The interaction potential for the α-fe Cu system taken into consideration was recently constructed [13] using the embedded atom method (EAM). It had been shown previously that EAM potentials allow us to follow metallic systems where fracture, surfaces, impurities and alloying additions (additives) are included [14]. In addition to pairwise interactions, using the EAM method the total energy includes an embedding energy as function of the local atomic density. The actual parameters used by the EAM in the case of iron and copper are described elsewhere [15, 16]. The model set-up containing the two initial straight edge dislocations and the Cu precipitate was then equilibrated for time steps equivalent to 40 ps at a temperature of T 0 = 10 K. 3. The movement of an edge dislocation hitting a Cu precipitate A detailed view of the core of one dislocation, which has already penetrated the obstacle, is presented in figure 2 for the regions inside and outside the precipitate. For each slice, the intersection between the dislocation line and the lattice plane can be recognized visually. During the interaction between the edge dislocation and the obstacle, the dislocation line does not remain straight. The dislocations do not move as rigid entities, but via the kink pair

4 184 S Nedelcu et al Z[-1-1 2] X[1-1 0] Y[111] Figure 2. Detailed structure of one of the dislocation cores during dislocation migration in the vicinity of the obstacle. The Fe atoms are yellow, the Cu atoms are grey. The distance along the z-axis between the upper and the bottom plane is Å =10.56 Å. mechanism. In order to identify the position of the dislocation lines also by means of an automatized computing algorithm, the maximum of the Burgers vector density distribution [17] was calculated for a cut along a (1 10) plane for all 65 x y layers of the simulation cell and for both precipitate examples. The calculations were performed for a large number of simulational results with equidistant time steps. The most interesting snapshots are presented in figures 3 and 4. Comparing the results obtained by running the MD simulation of α-fe for different situations with and without precipitates it could be seen that the presence of the obstacles on the glide plane of a moving dislocation reduces the internal shear stress and impedes the movement. The simplest case of a simulation model containing two unlike edge dislocations and without the Cu precipitate was considered first. In this case, the initial straight lines of the dislocations pre-served their shapes until the end of the relaxation process, when the two dislocations were no longer distinguishable. The movement of the dislocations was taking

5 Modelling of edge dislocation movement 185 Figure 3. Projections on the glide plane (1 10) of the nearest atoms above and below the glide plane that constitute the edge dislocation lines. The Cu precipitates (diameter 13.2 Å) are represented by the circle. place under no external shear stress. This result is in good agreement with Frank s rule describing several dislocations that might associate to form a single dislocation. The difference in elastic moduli between copper and iron can account for the observed influence of the copper precipitate on the movement of the edge dislocation. From this point of view the copper precipitate has an attractive influence on the edge dislocation. Case 1. Smaller precipitate, diameter 13.2 Å, figures 3(a) 3(i). Starting from the initial position, the movement of the dislocation line takes place such that it is curved toward the precipitate, see figure 3(b) in comparison to figure 3(a). Further on, the edge dislocation passes through the precipitate and after passing, a backward bowing can be recognized, see

6 186 S Nedelcu et al Figure 4. Projections on the glide plane (1 10) of the nearest atoms above and below the glide plane, that constitute the edge dislocation lines. The Cu precipitates (diameter 30.4 Å) are represented by the circle. figure 3(h), indicating the persisting attractive force between the precipitate and the dislocation line. Altogether, the movement of the dislocation takes place almost without any impedement, see also the discussion of the Peierls stress in the previous section and the appendix. Case 2. Larger precipitate, diameter 30.4 Å, figures 4(a) 4(f). In the case of the 30.4 Å diameter Cu precipitate a very elastical behaviour of the dislocation line hitting the obstacle could be observed. Passing does not happen and the dislocation line is pinned by the precipitate, with the free ends oscillating. The dislocation is not able to cut the obstacle. It can only pass through the precipitate completely as soon as an external shear stress is applied to increase the strain beyond the Peierls stress. During such calculations (figures not included), the angle between the dislocation arms did not become sharper than that in figure 4(f). 4. The Peierls stress of edge dislocations in the presence of the Cu precipitate The difference in the shear moduli µ Fe = 86 GPa and µ Cu = 54.6 GPa [7] of the α-fe matrix and of the Cu particle, respectively, gives rise to an elastic interaction force between particle and dislocations. In the following this force will be calculated for the two cases of Cu precipitate diameters and the resulting Peierls stress the minimum stress required to move a dislocation by one lattice plane distance. The interaction energy E µ between the Cu precipitate and the

7 Modelling of edge dislocation movement 187 edge dislocations can be obtained using the following formula [18]: N E µ = (ECu i Ei Fe ) (2) i=1 where N is the total number of the Cu atoms for a given diameter of the precipitate, ECu i is the energy of the ith Cu atom and EFe i is the energy of the ith Fe atom, when only Fe atoms are present. The values ECu i and Ei Fe of the ith atom energy were calculated using the EAM potential and running two MD simulations for the same geometry of the model, firstly containing a Cu precipitate and secondly for a simulation cell consisting only of Fe atoms, without Cu atoms. The numbering rule of the atoms was kept the same in both cases. (a) (b) (c) (d) Figure 5. Interaction energy per Cu atom E µ /N as function of the distance y along the 111 axis of the simulation cell (a and c), and the corresponding atom dislocation interaction force F µ (b and d). The pair plots a and b, c and d correspond to a 13.2 Å and 30.4 Å diameter Cu precipitate, respectively. The shaded area marks the location of the precipitate. Movement of the dislocation is from right to left. The energy plots contain also the figure numbers of the corresponding figure numbers in figures 3 or 4. Considering the position y of the edge dislocation as the point where the line crosses the plane z = 0, one can represent the interaction energy E µ as function of the distance y along the 111 axis, see figures 5(a) and 5(c). The dotted lines denote the boundaries of the Cu precipitate and the symbols and the connecting lines denote the position of the dislocation core along the y-axis. Several of the points correspond to the images of figures 3 and 4, respectively. For the precipitate with a diameter of 13.2 Å, an abrupt change of the energy per Cu atom can be observed after the dislocation has reached the precipitate. The same kind of change takes place later when the dislocation exits the precipitate again. Deriving E µ with respect to y yields the particle dislocation interaction force F µ [18], see figures 5(b) and 5(d): F µ = E µ y. (3)

8 188 S Nedelcu et al The maximum interaction force that the whole 13.2 Å diameter Cu precipitate exerts on the dislocation is attained when the edge dislocation is exiting the precipitate. The maximum value equals the forces as displayed in figure 5(b), multiplied with the total number of 121 Cu atoms of the precipitate and amounts to about 0.80 ev Å 1. However, the maximum negative value of the interaction force when the edge dislocation is first in contact with the precipitate is not attained at the particle matrix interface. This value is reached at the time when the edge dislocation is stopped for a short moment inside the precipitate, before completely shearing the particle and moving further. For the 30.4 Å diameter Cu precipitate, the maximum positive value of the interaction force was calculated as the force per atom (see figure 5(d)) times the total number of 1254 Cu atoms in the precipitate and amounts to 5.75 ev Å 1. At this size of the obstacle, the edge dislocation cannot pass through and its motion will be blocked. Additional external force has to be applied to continue the dislocation movement. The energy calculation was currently not possible within the frame of the calculational arrangment presently used. The upper bound of the Peierls stress τ P was calculated using the basic equation [18] of strengthening by shearable particles: bτ P = F 0 (4) L c where b is the Burgers vector, τ P is the calculated Peierls stress, F 0 is the maximum value of the particle dislocation interaction force, derived as explained above, see (3), and L c is the minimum length of the dislocation line, 75.2 Å for the considered simulation cell. The calculated Peierls stress τ P required to move the dislocation through the Cu precipitate in the case of the 13.2 Å diameter Cu precipitate is times the iron shear modulus, i.e. τ P = 0.72 GPa. A value of times the iron shear modulus (τ P = 5.07 GPa) results in the case of the 30.4 Å diameter Cu precipitate. The core structure and Peierls barrier for an edge dislocation lying in the {110} plane with the Burgers vector along 111 in bcc iron without Cu was also investigated by Chang and Graham [19] using an anharmonic potential. The calculated Peierls barrier was about 0.03 ev and the Peierls stress for dislocation motion at absolute zero temperature was dyn cm 2 or times the shear modulus of Fe (0.567 GPa). In the present results of τ P for the smaller precipitate compare closely with those of Chang and Graham, taking into account the presence of the Cu atoms in our model, and also with experimental values for pure Fe at very low temperatures [20], which range between 0.34 and 0.42 GPa. In other words, for the case of the small precipitate, the Peierls Stress is quite close to the case of pure Fe. The present calculations rely on no other physical assumptions than on the interatomic potentials, which base themselves on basic elastic constants. This suggests that the strengthening of the iron-rich iron copper system derives from the modulus mismatch between particle and matrix and from no other strengthening mechanism such as, for example, lattice constant mismatch strengthening. 5. Derivation of dispersion strengthening from modelling The bent dislocation with the backlash inside the precipitate is to be regarded as a part of a whole dislocation line, typically bowing between two obstacles, such as it is known from TEM images [21]. Considering a dislocation bowing between a pair of such obstacles, together with the assumption of a conventional constant line tension approximation, the angle between the two dislocation line branches on either side of the precipitate, together with the distance between the obstacles, is the key parameter to calculate the increase in matrix strength due to

9 Modelling of edge dislocation movement 189 precipitation strengthening [2, 22] : τ = Gb L ( cos φ 2 ) 3/2 (5) where τ is the matrix strengthening by precipitates, G is the shear modulus of the matrix, b is the burgers vector in the matrix, L is the obstacle spacing in the slip plane and φ is the critical angle between the arms of the dislocation at which the obstacle is cut. A detailed study of the dislocation line (see, e.g. figure 4(f)) permits one to determine the angle between the arms of the dislocation line on the glide plane. This smallest achievable angle between the two dislocation branches corresponding to the maximum of strengthening amounts to 140 in the case of the precipitate with a diameter of 30.4 Å. This value agrees very well with the critical angle as calculated from the mesoscopic continuum theory formalism by Russell and Brown, using as input the precipitate radius together with several empirically determined parameters (formula (2) and figure 2 in [2]), see appendix. For the smaller precipitate, the calculation along the Russell Brown formalism yields a critical angle of φ = 171, which means negligible strengthening, in agreement with the simulational results of figure 3. In a recent study on the relationship between structural information about Cu precipitates in a steel as derived from TEM images and macroscopical mechanical data of the same steel, the approach of Russell Brown has been proven to be reliable [5, 6]. The present investigations have shown that these calculations can also base on results from nanosimulation instead of such from the mesoscopic calculations (Russell Brown). In contrast to the Russell Brown formula (formula (2) in [2]), which uses empirically determined mesoscopic material parameters [2], in the case of nanosimulation the atomistic calculations are based solely on physical interatomic potentials. Figure 6. Scheme of a dislocation cutting a precipitate to explain the definition of the critical angle. 6. Conclusions A MD simulation was performed to understand the detailed mechanism of the complex interaction between a moving edge dislocation and differently sized Cu precipitates in the α-fe crystal. The model set-up contained two edge dislocations, sufficient by itself to permit

10 190 S Nedelcu et al an attractive movement under no external stress, and main attention was paid to the interaction of one of them with the Cu precipitate. Based on Frank s rule, the constructed model revealed the elastic behaviour of the edge dislocations and its strong dependence on the size of the obstacle. For a 32 Å diameter of the precipitate, the pinning process of the dislocation centre and also the trapping of the dislocation line in contrast to an obstacle diameter of 13 Å could be made evident. The calculated Peierls stress from the present MD simulation compares closely with other published values. The precipitates acting as obstacles to dislocation movement induce bowing of the dislocation lines. The present calculations enabled us to derive the critical angles of the dislocation lines at the Fe Cu interface, which are in perfect agreement to that obtained from mesoscopic dislocation theoretical calculations. These angles provide a direct connection to the numerical values of the increase in strength of such a model crystal. This means, that the present modelling of dislocation movement through a precipitate provides, for the first time in materials science, a way to simulate the precipitation hardening from basic principles (atomic properties) to a macroscopically relevant material s property. Appendix The critical angle between two edge dislocation branches can be calculated following the mesoscopic continuum theory formalism by Russell and Brown [22]. The strength of an alloy in the overaged condition can be calculated following the methods of Brown and Ham [2, 22]. The stress, at which a dislocation can move through an array of obstacles is identified with the yield stress τ and is a function of the obstacle spacing L in the slip plane and the critical angle φ at which a dislocation can cut an obstacle. The shear stress is given by τ = Gb L ( cos φ 2 ) 3/2 where G is the shear modulus and b is the Burgers vector of the dislocation. Russell and Brown derived the shear stress from the relationship between the energies of the dislocation per unit length inside (E 1 ) and outside (E 2 ) the precipitate as τ = Gb L ( 1 E2 1 E 2 2 ) 3/4. Therefore, ( cos φ ) 3/2 ( ) 3/4 = 1 E2 1 2 E2 2 where the energies of the dislocation length inside (E 1 ) and outside (E 2 ) the precipitate depend on the precipitate radius as E 1 = E 1 log(r/r 0) E 2 E2 log(r/r 0) + log(r/r) log(r/r 0 ) where E1 and E 2 refer to the energies per unit length of a dislocation in infinite media (Fe or Cu, respectively), r is the precipitate radius and R and r 0 are the outer and inner cut-off radii used to calculate the energy of the dislocation. Russell and Brown verified the validity of the following values for the Fe Cu system: E1 /E 2 = 0.6, r 0 = 2.5b with b = Burgers vector of the dislocation = 2.48 Å, and, finally R = 10 3 r 0. For precipitate radii of 16 Å and 6.5 Å, the formalism results in critical angles of φ = 140 and 171, respectively.

11 Acknowledgments Modelling of edge dislocation movement 191 This work was supported by the German Bundesministerium fur Bildung, Wissenschaft, For-schung und Technology (BMBF) under grant No and by the Deutsche Forschungsgemeinschaft (DFG) under grant No SCHM746/21-1 References [1] Hu S Y, Ludwig M, Kizler P and Schmauder S 1998 Modelling Simul. Mater. Sci. Eng and further references therein [2] Russell K C and Brown C M 1972 Acta Metall [3] Pretorius T, Rönnpagel D and Nembach E 1998 Proc. 19th Risø Int. Symp. on Materials Science: Modelling of Structure and Mechanics of Materials from Microscale to Product ed J V Christensen et al (Roskilde, Denmark: Risø National Laboratory) p 443 [4] Kelly A and Nicholson 1971 Strengthening Methods in Crystals (Amsterdam: Elsevier) [5] Uhlmann D, Kizler P and Schmauder S 1998 Proc. 19th Risø Int. Symp. on Materials Science: Modelling of Structure and Mechanics of Materials from Microscale to Product ed J V Christensen et al (Roskilde, Denmark: Risø National Laboratory) p 529 [6] Kizler P, Uhlmann D and Schmauder S 2000 Nucl. Eng. Design at press [7] Hirth J P and Lothe J 1982 Theory of Dislocations (New York: Wiley) [8] Hull D and Bacon D J 1984 Introduction to Dislocations (Oxford: Pergamon) [9] Suzuki T, Takeuchi S and Yoshinaga H 1985 Dislocation Dynamics and Plasticity (Berlin: Springer) [10] Suzuki T, Takeuchi S and Yoshinaga H 1985 Dislocation Dynamics and Plasticity (Berlin: Springer) ch [11] Kohlhoff S and Schmauder S 1989 Atomistic Simulation of Materials ed V Vitek and D J Srolovitz (New York: Plenum) Kohlhoff S, Gumbsch P and Fischmeister H F 1991 Phil. Mag. A [12] Stroh A N 1962 J. Math. Phys [13] Ludwig M, Farkas D, Pedraza D and Schmauder S 1998 Modelling Simul. Mater. Sci. Eng [14] Daw M S and Baskes M I 1984 Phys. Rev. B [15] Voter A F and Chen S P 1987 Mater. Res. Soc. Symp. Proc [16] Voter A F Los Almos Unclassified Technical Report #LA-UR [17] Schroll R, Vitek V and Gumbsch P 1998 Acta Mater [18] Nembach E 1997 Particle Strengthening of Metals and Alloys (New York: Wiley) ch 5.2 [19] Chang R and Graham L J 1966 Phys. Status Solidi [20] Brunner D and Diehl J 1992 Z. Metallkd [21] For example, Othen P J, Jenkins M L and Smith GDW1994 Phil. Mag [22] Brown L M and Ham R K 1971 Strengthening Methods in Crystals ed A Kelly and R B Nicholson (Amsterdam: Elsevier) p 12