Computer simulation of interaction of an edge dislocation with a carbon interstitial in α-iron and effects on glide

Acta Materialia 55 (2007) 93–104 www.actamat-journals.com

Computer simulation of interaction of an edge dislocation with a carbon interstitial in a-iron and effects on glide K. Tapasa a, Yu.N. Osetsky b, D.J. Bacon b

a,*

a Department of Engineering, The University of Liverpool, Brownlow Hill, Liverpool L69 3GH, UK Computer Sciences and Mathematics Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6138, USA

Received 18 April 2006; accepted 8 August 2006 Available online 3 November 2006

Abstract The atomic-scale behaviour of a carbon (C) interstitial atom in the core of a 1=2½1 1 1ð1  1 0Þ edge dislocation in a-iron has been simulated for the first time. C sites with high binding energy to the dislocation have been investigated and the critical stress, sc, for the dislocation to overcome a row of C atoms determined. The effects of temperature, T, and applied strain rate, e_ , on sc have been studied. sc decreases rapidly as T increases to 400 K and becomes almost constant at higher T. It decreases with decreasing e_ and is e_ -independent at T greater than 300 K. The activation parameters in simulation conditions have been obtained. The activation distance of (0.2–0.3)b is consistent with point-obstacle strengthening. However, the activation energy is only 5kBT, where kB is the Boltzmann constant, and 20kBT smaller than that realized in experimental conditions. This implies that the decline of sc over the range 0 to 400 K would occur over 0 to 80 K in experiment, which is where C-edge dislocation effects would be influential. A few jumps of C occur in the core before dislocation unpinning at T P 800 K and give a small T-dependence of sc. Core diffusion of C occurs by 1=2½1 1  1 jumps at 70.5 to [1 1 1]. The diffusivity in the absence of applied stress is 4 · 109exp(0.2 eV/kBT) m2/s compared with 1.9 · 107exp(0.7 eV/ kBT) m2/s for bulk diffusion of C in the same MD model. Hence, the edge dislocation provides a path for rapid diffusion of C, but net transport along the core can only occur by motion of the dislocation itself.  2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Dislocation mobility; Carbon diffusion; Iron alloys; Molecular dynamics; Activation parameters

1. Introduction The properties of carbon in iron have been investigated extensively because of its relevance to steel technology. It is known from experiment that the interaction of C interstitial solute atoms with lattice defects significantly affects mechanical properties of ferritic steels, e.g. their strength [1], and the apparent properties of point defects such as vacancies [2,3]. Carbon has small solubility (0.002 wt.%) in the a-Fe matrix at room temperature. Above the solubility limit, it precipitates as carbides or cementite (Fe3C) in pearlite, which is useful for improving the strength of steels. In quenched irons, C is retained in martensite. C *

Corresponding author. Tel.: +44 151 794 4662. E-mail address: [email protected] (D.J. Bacon).

atoms that remain in solution in ferrite can also have a significant influence on steel properties by their interaction with lattice defects, e.g. acting as traps for vacancy-type defects and thus decreasing the vacancy diffusion coefficient [2,3]. The migration properties of C atoms in a-iron and their interaction with intrinsic point defects have been investigated experimentally, e.g. by positron annihilation [3], and theoretically, e.g. by molecular dynamics [4] and by means of ab initio calculation [5]. Furthermore, firstprinciples calculation [6] and experiment [7–9] provide evidence that C is a cohesion enhancer at grain boundaries in Fe. In the context of the present paper, the interaction between C impurity and dislocations is particularly important. It can have significant impact on the mechanical properties of steel by its influence within the dislocation core. At

1359-6454/$30.00  2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.08.015

94

K. Tapasa et al. / Acta Materialia 55 (2007) 93–104

high enough temperature, solute C atoms can diffuse to a dislocation and form a C-rich atmosphere (the ‘Cottrell atmosphere’ [1]). Core segregation of C immobilizes dislocations and increases the applied stress required for them to take part in yielding. Even without the ageing process that leads to atmosphere formation, C in solution is believed to offer strong resistance to dislocation glide. Variations on the elasticity model of the interaction energy, EI, between an edge dislocation and a C atom have been proposed by several workers (e.g. [1,10–14]). These describe the far-field interaction, but elasticity provides only an approximation for C sites near the dislocation core and so atomic-level computer simulation should be employed for this region [15–17]. In an early study, de Hosson [15] simulated the interaction of a C atom and a 1=2½1 1 1ð1  1 0Þ edge dislocation using the pairwise interatomic potentials for Fe–Fe derived by Johnson [18] and Fe–C by Johnson et al. [19]. The results showed that C atoms with maximum binding to the dislocation lie between the {1 1 0} atomic planes immediately under the extra half-plane where EI is 0.7 eV, which is consistent with elasticity estimates. Recently, Niu et al. [16] and Simonetti et al. [17] investigated electronic effects of C in the core of Æ1 0 0æ and 1/2Æ1 1 1æ edge dislocations in a-iron by ab initio methods based on density-functional theory. It was found that a C atom can weaken interactions between the adjacent Fe atoms and form strong covalent-like bonds with them, thereby leading to carbon–dislocation complexes. In the present work, the interaction of carbon interstitials with an edge dislocation in dilute Fe–C alloy and their effect on the motion of the dislocation are investigated. Molecular statics (MS) and molecular dynamics (MD) techniques are employed to simulate conditions with temperature T = 0 K and T > 0 K, respectively. These methods require interatomic potentials in order to compute potential energy of atoms and forces on them. The development of empirical interatomic potentials for computer simulation of C in a-Fe has lagged behind that for pure a-Fe and Fe alloys with substitutional solutes, and only a few Fe–C potentials have been proposed. Johnson et al. [19] developed a short-range pair potential set for the Fe–C system by fitting to the experimental value of 0.86 eV for the migration energy of C atoms in a-Fe [20], zero activation volume of migration [21] and vacancy-carbon binding energy of 0.41 eV. C–C interaction was not considered in this model. This potential, which was used by de Hosson [15], reproduces the experimental value of the energy of solution of a C atom in a-Fe relative to that in Fe3C [22] and gives the octahedral site as the most stable one for a C interstitial, consistent with experiment [23] and subsequent ab initio calculation [5]. Although the volume expansion due to a C atom is 0.3X, where X is the Fe atomic volume, compared with 0.8X deduced from the experimental value of lattice parameter change with C composition [24], the relaxation of Fe atoms neighbouring a C atom obtained in Ref. [4] by a combination of the Fe–C

potential of Johnson et al. and many-body Fe–Fe potentials (either the Finnis–Sinclair (F–S) potential of Ackland et al. [25] or the embedded-atom method (EAM) potential of Ackland et al. [26]) is in good agreement with that found in a recent ab initio calculation by Domain et al. [5]. There have been several attempts to obtain a better description of C atoms in iron using empirical potentials. Rosato [27] rescaled the Fe–C potential of Johnson et al. [19] to use with an FS-type many-body potential for Fe [28], which provides a better description of the elastic properties of a-Fe than the simple Fe–Fe pair potential in Ref. [19]. Again, C–C interaction was ignored. Although the octahedral site is the most stable for C in this model, the carbon migration energy is found to be 1.14 eV, which is too high. Ruda et al. [29] developed EAM potentials fitted to ab initio data for metastable Fe–C carbide with B1 structure. The C–C interaction was described using the Tersoff [30] potential with no angular dependency. The equilibrium lattice constant, bulk modulus and cohesive energy of stable and metastable carbides were reproduced, but, in contradiction with experiment and ab initio calculation, the tetrahedral site is the most stable position for a C atom in a-Fe with this potential. Lee [31] has developed very recently a modified EAM potential for Fe–C. It produces a CV binding energy of 0.9 eV, which is close to that predicted by Vehanen et al. [3] (i.e. 0.85 eV), but much higher than the value of 0.41 eV deduced by Arndt and Damask [32], which was used to fit the Fe–C potential of Johnson et al. [19], and the ab initio value of 0.47 eV [5]. It gives a large binding energy (0.68 eV) between C and a self-interstitial atom, as do all the empirical potentials, e.g. 0.58 eV with the model used in the present work, compared with 0.19 eV by ab initio calculation. This is an anomaly that has not yet been resolved. Thus, we are not aware of any potential set that accurately represents the Fe–C system, and so the approach adopted here is to use a ‘model’ system for an octahedral interstitial solute in a bcc metal matrix that has many similarities to the real alloy. Fe–Fe interactions are computed from the FS-type many-body potential of Ackland et al. [25], which has been widely used to simulate point and extended defects. This is combined with the pair potential of Johnson et al. [19] for Fe–C. In the absence of a reliable C–C interatomic potential, we treat the dilute concentration limit of only a single C atom interacting with the dislocation within the simulation cell. Despite the simplicity of this approach, the model system has many properties, such as elastic constants, solute site stability, migration energy, neighbour-atom displacements and some intrinsic point defect–solute interaction energy values, that are close to those of the real alloy. The paper is organized as follows. The parameters and method used are presented in Section 2. EI for C in sites within and near the dislocation core at 0 K is calculated using MS and presented in Section 3. Section 4 describes the results for the critical resolved shear stress, sc, required for the dislocation to overcome a C atom situated in a site

K. Tapasa et al. / Acta Materialia 55 (2007) 93–104

of large jEIj at T = 0 K and T > 0 K using MS and MD, respectively. MD is used extensively to investigate the effects of temperature and applied strain rate on sc. The activation energy and activation volume parameters are presented and their interpretation discussed for slip in MD modelling. MD is employed to study diffusion of a C atom along the dislocation line, i.e. ‘core’ diffusion, in Section 5. The conclusions are drawn in Section 6. 2. Model parameters and methods The atomic model used defines a periodic array of edge dislocations, as described by Osetsky and Bacon [33]. The dislocation line in the computational cell was in the ½1 1 2 direction and had Burgers vector b = 1/2[1 1 1]. Periodic boundary conditions were imposed along both ½1 1 2 and [1 1 1]: the latter allowed the dislocation to move freely without encountering fixed boundaries and the former resulted in a periodic obstacle spacing of length L along the dislocation line. Two crystal sizes were used: crystal C1 had L = 3.51 nm and C2 had L = 11.23 nm. In MS, the period length, Lb, of the model along [1 1 1] and the height, H, along ½1  1 0 for MS simulations were 37.5 nm (=151b) and 20.67 nm, respectively. For sufficiently large values of Lb, critical stress sc is independent of this length [33], and so for MD Lb was taken to be 25.1 nm (=101b) to shorten the simulation time. Models C1 and C2 contained up to 230,280 and 736,896 mobile atoms, respectively (153,780 and 492,097 in MD). These sizes are sufficiently large to avoid the effects of model boundaries and prevent significant interaction between the dislocation and C atom with their images. In order to apply stress or strain, blocks of atoms in rigid coordination were created at the top and bottom of the atomic cell in the ½1  1 0 direction. The critical stress, sc, for the gliding dislocation to overcome the periodic array of C atoms was determined by one of the following methods. In statics simulation (T = 0 K), resolved shear strain, e, was applied by the incremental [1 1 1] displacement of the upper block relative to the lower block until the dislocation was free of the obstacle. At each step, the atomic positions were relaxed to minimize the potential energy and the applied shear stress was calculated from the total force on the upper block from the free atoms. Relaxation was achieved by a combination of conjugate gradients energy minimization and quasi-dynamic quenching. An increment of De = 2 · 105 was found to reproduce the stress–strain dependence with an accuracy in stress of better than 10 MPa [33]. This set of simulations is of interest because dynamic and thermal effects are absent and they are the atomic-level equivalent of the elasticity treatment of dislocations where the configuration is determined by the potential energy. In dynamics (T > 0 K), a constant shear strain rate, e_ , was imposed by moving the upper block along the [1 1 1] glide direction at a particular velocity, with the applied stress computed as above, but averaged over 1000 MD time

95

steps because of the fluctuations that can occur from step to step. Three values of e_ were applied, namely 1 · 106, 5 · 106 and 2 · 107 s1. Temperatures in the range 100– 1200 K were considered. The lattice parameter, a0, was adjusted to maintain zero pressure in the system at each temperature. The MD time step, ts, was kept small (ts  3–6 fs) so that the temperature change, DT, in the model crystal was insignificant: for example, at e_ ¼ 5 106 s1 and T = 100 K, DT was less than 1.2 K for a simulation time of 16 ns, an effect due to work done by external force rather than computational inaccuracy. The strain rate used here is of course much larger than the macroscopic rate applied in laboratory tensile tests, but it is compatible with practical computing times. (For example, for e_ ¼ 1 106 s1 at 100 K and ts = 5 fs, a simulation time of 14 ns required 40 CPU days on a Pentium 4 2400 GHz processor.) Furthermore, since the dislocation velocity, vD, in steady state [33] satisfies the classic Orowan relation e_ ¼ vD bqD ¼ vD b=ðLb H Þ, where qD is the dislocation density, vD = 2.1 and 41.5 m/s when e_ ¼ 1  106 s1 and 2 · 107 s1 for the model size considered, this velocity range is reasonable for a dislocation in free flight and is in the regime of viscous damping due to phonons, well below the velocity range where relativistic effects become apparent. 3. Interaction between carbon atoms and dislocations at T=0K A carbon interstitial in an octahedral site in iron produces tetragonal distortion in one of the three Æ1 0 0æ directions. It repels its first-nearest-neighbour atoms (distance a0/2) p and attracts second-nearest neighbours (distance a0/ 2). In the model orientation used here, C solutes are located in either a ð1 1 0Þ atomic plane parallel to the dislocation glide plane and have tetragonal distortion axis (TDA) in the [0 0 1] direction, or they occupy sites between two ð1 1 0Þ planes and have [1 0 0] or [0 1 0] TDAs symmetric about b. The dislocation–carbon interaction energy, EI, obtained by MS calculation is plotted as a function of distance along [1 1 1] from the dislocation core centre in Fig. 1. Results for the C atom with [0 0 1] TDA are presented in Fig. 1(a) and those for sites with [0 1 0] TDA in Fig. 1(b). The sign of the scale on the abscissa would be reversed in Fig. 1(b) for the [1 0 0] TDA. The individual plots are for C in sites in particular ð1 1 0Þ layers above and below the glide plane, as indicated in the inset sketches. (The glide plane for the positive edge dislocation is defined to be midway between the ð1 1 0Þ atomic plane that contains atoms at the bottom of the extra half-plane and the ð1 1 0Þ atomic plane immediately below it.) The plots in Fig. 1(b) show repulsion (EI > 0) on one side of the dislocation and attraction (EI < 0) on the other due to the asymmetry mentioned above. The absence or clustering of data points for some values of the C–dislocation distance results from attraction or repulsion being sufficient for the dislocation to overcome the resistance of the Peierls stress and glide to a new core

96

K. Tapasa et al. / Acta Materialia 55 (2007) 93–104

inside the core region where linear elasticity is not strictly valid, the value it gives for Eb is consistent with the atomic-level treatments. 4. Critical resolved shear stress 4.1. Static and dynamic conditions

Fig. 1. Interaction energy between a 1=2½1 1 1ð1 1 0Þ edge dislocation and C atoms at sites with (a) [0 0 1] TDA in ð1 1 0Þ atomic planes and (b) [0 1 0] TDA between ð1  1 0Þ atomic planes. Positions of C above and below the dislocation glide plane shown as a dashed line in the inset figures are indicated.

position during relaxation. The maximum binding energy, Eb = EI, is 0.68 eV and arises for carbon solute in [0 1 0] or [1 0 0] orientation midway between the ð1  1 0Þ planes of Fe atoms above and below the glide plane, and located at 0.5a0 from the core centre, as shown in Fig. 1(b). This is close to the result of 0.7 eV obtained by de Hosson [15] with a pairwise Fe–Fe potential. A carbon atom with [0 0 1] TDA in the first ð1  1 0Þ atomic plane below the glide plane has the second highest Eb = 0.5 eV (Fig. 1(a)). As noted in Section 1, Eb between C interstitials and dislocations has been estimated by elasticity theory many times. For example, Cottrell and Bilby [1] estimated the maximum Eb to be about 0.5 eV in treating the equilibrium concentration of segregation C atoms round stationary dislocations. Later workers [10–14] analysed the dislocation field and/or tetrahedral distortion more rigorously. Nevertheless, Eb was found to fall in the range 0.5–0.8 eV when the distance from the dislocation axis to the C atom is b. (The value for the screw dislocation is approximately 40% of this [14].) Thus, although the distance b is well

In the absence of thermal vibrations (T = 0 K), a dislocation can overcome an obstacle to glide if the mechanical work done by the applied stress is equal to the maximum Eb between the dislocation and obstacle, i.e. the area under the force–distance curve experienced by the dislocation (e.g. [34]). This determines the critical resolved shear stress, sc, for glide at T = 0 K. The effective obstacle energy barrier is diminished by the contribution of thermal energy at T > 0 K and this results in a reduction in the critical stress. It is dependent on temperature and strain rate. In this section, results for sc from simulations of T = 0 and >0 K are presented for a gliding dislocation encountering a carbon interstitial near its glide plane. The activation parameters that describe the glide are derived. Carbon sites that give high values of Eb in Section 3 are considered, namely the site with [0 1 0] TDA located between the two atomic planes adjacent to the glide plane and the [0 0 1] site in the atomic plane immediately below the glide plane. They have maximum Eb equal to 0.68 and 0.50 eV, respectively, and will be denoted A and B in the following. The C atom was placed about 7b and 11b away from the initial position of the dislocation for the C1 and C2 models described in Section 2, i.e. the crystals with short (3.51 nm) and long (11.23 nm) values of inter-carbon spacing, L, along the ½1 1 2 direction of the dislocation line. An additional factor may arise when the obstacles to dislocation motion are interstitial solute atoms. At sufficiently high temperature and low strain rate, solute atoms may be able to diffuse with the moving dislocation. This phenomenon (the Portevin–le Chatelier effect) occurs when the solute atom is able to achieve a velocity of the same order as the dislocation velocity. Tapasa et al. [4] have used MD with the model of C in Fe employed here to determine the jump frequency, vC, and nature of C atom migration. They find vC is 20 and 52 GHz at 1000 and 1200 K, respectively. As noted in Section 2, the dislocation velocity in the present work is 2.1 and 10.4 m/s at e_ ¼ 1  106 s1 and 5 · 106 s1, respectively. To achieve these speeds via steps of length b, a carbon atom would have to jump with a rate of 8.4 and 41.6 GHz, respectively, which is lower than the jump frequency of free carbon at these temperatures. Thus, the possibility exists that solute drag could occur in the dynamic simulations, so we checked to see if it did. 4.2. Stress–strain response at T = 0 K The dislocation was induced to glide in the MS simulations by the incremental increase De = 2 · 105 of resolved shear strain (Section 2). Fig. 2 displays the corresponding

K. Tapasa et al. / Acta Materialia 55 (2007) 93–104

97

Fig. 2. Stress–strain response for strain increasing by increment De = 2 · 105 at T = 0 K for the C atom in the A (solid line) and B (broken line) positions in crystals C1 (black line) and C2 (grey line).

stress vs. strain data for models C1 and C2 as strain is increased from the initial unstrained state when the straight dislocation does not interact with the C atom. The dislocation overcomes the C obstacle at the maximum stress, sc, after which it glides at the Peierls stress for this model of iron, as indicated in the figure. The small drop in stress well below sc for the C atom in A sites corresponds to spontaneous jump of the dislocation past the C atom to the position of maximum attraction, which is when the solute is behind the dislocation by about 0.5a0 (see the minimum in Fig. 1(b)). sc is dependent on the site and spacing of the C solute. For C in site A, sc = 466 MPa for C1 and 146 MPa for C2. For site B, sc is 196 and 62 MPa for C1 and C2, respectively. Note that the ratio of the C2 and C1 values of sc for both A and B layers is 3.2, which is exactly the reciprocal of the ratio of the two values of L, i.e. sc is proportional to 1/L. The line shape in the glide plane at s = sc is shown in Fig. 3(a) and (b) for crystals C1 and C2, respectively. The bow-out of the line is small, even for C2 (<1 nm), and so for these localized obstacles the sc values should be determined principally by the magnitude of Eb and spacing L as follows. The forward force on the dislocation of length L under resolved shear stress s is (sbL) [34] and so it is appropriate to equate (sc  sP)bL to the maximum value of jdEI/dxj, where sP is the Peierls stress (23 MPa for this model of iron). However, it is clear from Fig. 1 that jdEI/dxj cannot be quantified uniquely because the dislocation can glide spontaneously to the solute in the absence of applied stress. We therefore estimate sc at T = 0 K by equating (sc  sP)bLd, the work done by the external load when the dislocation undergoes displacement d to overcome the obstacle, to the maximum Eb, i.e.   1 Eb sc  sP ¼ : ð1Þ L bd The L1-dependence was demonstrated above. Taking d = b = 0.25 nm, which seems a reasonable choice for the

Fig. 3. Critical line shape (visualized by atoms in the dislocation core) for the C atom in the A site in (a) crystal C1 at T = 0 K, (b) crystal C2 at T = 0 K and (c) crystal C2 under e_ ¼ 5  106 s1 at T = 300 K. The C atom is denoted as a small sphere.

small C atom, sc given by Eq. (1) is found to be 473 and 342 MPa for sites A and B, respectively, in the C1 model (L = 3.51 nm), compared with the simulated values of 466 and 196 MPa. For the A and B sites in model C2 (L = 11.23 nm), Eq. (1) gives 132 and 91 MPa, respectively, compared with 146 and 62 MPa from simulation. Thus, the simple analysis leading to Eq. (1) appears justified for the A site with high Eb, but predicts too large a value for the B site. Visualization of C atom behaviour in the B sites showed that it underwent a small displacement upwards from the ð1 1 0Þ atomic plane at sc, so that Eb was effectively reduced. 4.3. Stress–strain response at T > 0 K The effects of T and e_ on sc are investigated by MD in this section. Three values of e_ are treated and, for convenience of presentation, the temperatures considered are divided into two ranges, namely 100 K 6 T 6 450 K and 600 6 T 6 1200 K. Stress–strain dependences obtained for the C1 crystal (L = 3.52 nm) under applied strain rate = 2 · 107 s1 with a carbon atom in the A site ([0 1 0] TDA just below the glide plane) are presented in Fig. 4(a) and (b) for the low and high temperature ranges, respectively. The plot for T = 0 K from Fig. 2 is included in Fig. 4(a) for comparison. Higher temperature produces greater fluctuation in stress and so the stress displayed in the graphs was averaged over 9–18 ps. The dislocation is attracted towards the C atom and reaches it in shorter time with increasing temperature, as seen by the reduction in stress that precedes the increase when the dislocation becomes pinned. This is because the lattice energy barrier for the gliding

98

K. Tapasa et al. / Acta Materialia 55 (2007) 93–104

Fig. 4. Stress–strain response for crystal C1 under constant strain rate e_ ¼ 20  106 s1 at (a) 100 K 6 T 6 450 K and (b) 600 K 6 T 6 1200 K with C atom in position A. The stress–strain plot at 0 K is shown for comparison in (a).

dislocation is reduced by thermal energy. The highest value of the stress was taken to be sc, with ±10 MPa uncertainty. Although the percentage uncertainty increases with increasing temperature, sc decreases, so an absolute uncertainty of ±10 MPa seems reasonable. Computer visualization was carried out to study the behaviour of the solute atom under all conditions. The temperature dependence of sc in the C1 model, with ± 10 MPa taken as the uncertainty across the range of T, is plotted in Fig. 5(a) for all three strain rates. sc is largely independent of temperature with a value of less than 100 MPa for T > (300–400)K, but it increases strongly with decreasing T below that range to reach 466 MPa when T = 0 K. In this lower temperature range, sc increases with increasing strain rate. This is illustrated in Fig. 6 by the stress vs. time plots at T = 100 K for the three applied strain rates in crystal C1. sc increases by 9% and 27% when e_ is changed from 1 · 106 to 5 · 106 and 2 · 107 s1, respectively. The applied strain when the stress equals sc rises from 0.53% to 0.56% to 0.64%, respectively. When e_ ¼ 2  107 s1 , sc is almost constant for T P 400 K but, surprisingly, exhibits a peak at 1000 K. Checks were made to confirm that this is a real effect. It was found that the C atom migrates within the dislocation core during

Fig. 5. Critical stress as a function of temperature for different strain rates. (a) Crystal C1 with C atom in position A. (b) Crystal C2 with C atom in positions A or B. sc at T = 0 K calculated by MS in Section 4.2 is shown by a cross. The dashed line in A indicates osc/oT in the region of strong temperature dependence of sc.

Fig. 6. Stress as a function of time for crystal C1 under various applied strain rates at 100 K with C atom in position A.

pinning when T P 800 K. In this phenomenon, known as ‘core’ or ‘pipe’ diffusion, the direction of C migration is not in the ½1 1 2 direction of the dislocation but along ½1 1 1 at 70.5 to b. The details of this motion are discussed later in Section 5. At 800 and 1200 K, the direction of C motion is along the ½1 1 1 direction, which has a

K. Tapasa et al. / Acta Materialia 55 (2007) 93–104

component in the forward direction of dislocation glide (see Fig. 9 later). At 1000 K, in contrast, the direction of C migration was found to be in the ½ 1 1 1 direction with a component opposite to the glide motion. This may give rise to the higher value of sc at 1000 K. During core jumps at 1200 K, the C atom was observed to jump between the second plane above the glide plane, which has EI > 0, and the plane of the maximum binding energy: in the absence of applied stress at this temperature, it actually dissociated from the dislocation (see Section 5). Fig. 5(a) shows that, when e_ ¼ 5  106 s1 , sc is temperature independent for T P 300 K. Core jumps of carbon occur for T P 600 K and the displacement of the C atom is greater than when e_ ¼ 2  107 s1 because of the longer time available for each step. At 1000 K, the C atom was found to move in the ½1 1  1 direction during this process, and sc is lower than when e_ ¼ 2  107 s1 at this temperature. This lends support to the explanation of the maximum observed in the high strain-rate case. At the lowest strain rate of 1 · 106 s1, sc again rises sharply when T falls below 300 K. In this case, however, the C atom was observed to jump through several planes perpendicular to the glide plane during pinning at 1000 K, and the motion to sites of low Eb seems to be the cause of even lower sc. The dependence of sc on temperature in model C2 (L = 11.23 nm) is shown in Fig. 5(b). It exhibits similar characteristics to that for C1, i.e. sc decreases strongly with increasing T below about 400 K. The line shape at the critical stress of 50 MPa when e_ ¼ 5  106 s1 in the C2 model at 300 K is presented in Fig. 3(c) for comparison with the critical shape at 146 MPa in the same model at T = 0 K shown in Fig. 3(b). The values when L = 11.23 nm are smaller than for the C1 model, consistent with the ratio of the L1 values, as discussed earlier. Again, sc is higher at T P 1000 K than at 600 and 800 K due to migration of the C solute in the dislocation core. One set of simulations was undertaken for carbon in the B site with lower Eb, as indicated. The effect on sc is only strong at T 6 100 K. The static simulations of Section 4.1 showed that sc at T = 0 K is proportional to L1 for C in an A site, as expected for point-like obstacles, but, due to displacement of the carbon atom, not for the B site. This proportionality has also been tested in dynamics, as shown in Fig. 7(a) and (b) for e_ ¼ 5  106 and 20 · 106 s1, respectively. The carbon atom was situated in site A for both sets of simulations and, additionally, in site B in the C2 model at the higher strain rate (Fig. 7(a)). For both strain rates, the values of scL for the A sites are similar at low temperature, but differences are more apparent as T increases. The values are systematically lower for the C1 model, i.e. sc is slightly lower than expected when L is smaller. The cause is not known. Interestingly, the scL values for the C2 model in Fig. 7(a) show that sc for carbon in the B site are the same as those for the A site when T P 300 K, in contrast to the much lower values below this temperature range. It was found that this effect arose because the carbon solute jumped into the A position when the dislocation approached.

99

Fig. 7. Comparison of scL for crystals C1 and C2 as a function of temperature for (a) e_ ¼ 20  106 s1 and (b) e_ ¼ 5  106 s1 .

4.4. Activation parameters Analysis of slip in a field of obstacles at T > 0 K is usually based on treatment of dislocations meeting barriers, each of which exerts a resisting force with a profile such that the area under the force–distance curve is the total energy required for a dislocation to overcome that obstacle. This energy is Eb and leads to a critical stress given by Eq. (1) at T = 0 K. At T > 0 K, the energy required is partly provided in the form of mechanical work by the applied load: it is (scbLd*) and can be written (scV*), where d* and V* are known as the activation distance and volume, respectively. (In contrast to the analysis for T = 0 K, sP can be ignored when T P 100 K because dislocation glide occurs even when s is less than 1 MPa.) The difference between d* and d in Eq. (1) depends on the shape of the force–distance profile. The remainder of the energy is thermal and is the free energy of activation DG* = Eb*  scV*, where Eb* is the total energy required between the dislocation states separated by d*. DG* is the Gibbs free energy change at constant T and e_ between those two states. The probability of DG* being provided by thermal fluctuations is exp(DG*/kBT) if DG*  kBT, where kB is the Boltzmann constant. Hence, the macroscopic plastic strain rate is

100

K. Tapasa et al. / Acta Materialia 55 (2007) 93–104

 e_ ¼ qD A exp

DH kBT

 ð2Þ

where qD is the mobile dislocation density, A = bDm, D is the glide distance for each obstacle overcome and m is the vibration (attempt) frequency. G* has been replaced by the enthalpy H* by taking the entropy term in e_ to be unity. This leads to a procedure to determine the activation parameters DH* and V* by a series of tensile tests at constant T or constant e_ . If a change in the strain rate from e_ 1 to e_ 2 causes a change in the flow stress Dsc at constant T, then V* can be obtained from [35]: V ¼

k B T ‘nð_e2 =_e1 Þ Dsc

ð3Þ

The activation energy is found by performing tests at constant strain rate over a range of temperatures and using the equation [35]   osc DH ¼ V T ð4Þ oT e_ where osc/oT is the gradient of the curve of effective critical stress as a function of temperature at constant e_ . Consider the application of these equations to the MD data in Fig. 5(a). At T = 100 K, the decrease in sc, i.e. Dsc, as e_ decreases from 2 · 107 to 5 · 106 s1 is 42 MPa, which gives V* = 2.9b3 from Eq. (3). The values at 200 and 300 K are 2.2b3 and 4.7b3, respectively. The gradient of the dashed sc vs. T line in Fig. 5(a) for e_ ¼ 2  107 s1 is 1.04 MPa/K. When this gradient is used in Eq. (4) with the V* values given, DH* is found to be 0.03, 0.05 and 0.14 eV for T = 100, 200 and 300 K, respectively, i.e. 3–6kBT. The estimates for DH* are increased to the range 0.04–0.17 eV when the higher gradient for the e_ ¼ 1  106 s1 data is used. (We shall take DH* = 5kBT in the discussion below.) 4.5. Discussion The dynamic drag of solute described in Section 4.1 was not observed at any strain rate or temperature considered. The dislocation jumps forward by too large a distance as it unpins for a C atom to move and be recaptured, as can be seen by the averaged position of the dislocation core as a function of time at 300 and 800 K in Fig. 8. It is only within the core itself at T P 800 K that the C atom has sufficient kinetic energy and time to move from one site to another, as described in Section 4.3. Thus, although in principle a C atom can jump at sufficient speed in MD to follow a dislocation in steady state (Section 4.1), the dislocation accelerates as it breaks away and leaves the C atom behind under the conditions applied here. A much lower strain rate would be required to investigate this further. Although the carbon atom did jump within the dislocation core at high T, the number of such jumps when the model crystal was under applied strain was insufficient to gain meaningful statistics for quantitative analysis of this

Fig. 8. Position of dislocation core averaged along its length as a function of time at 300 and 800 K for e_ ¼ 1  106 s1 .

core process. Analysis of core jumps was therefore undertaken in the absence of strain in a separate set of simulations. These are presented in Section 5. Turning to the results of thermal activation analysis in Section 4.4, the V* values are in the range 2–5b3. Since the obstacle spacing along the dislocation line in the C1 crystal is 3.51 nm, which equals 14b, this implies that activation distance d* is approximately 0.2–0.3b. This does not seem unreasonable because the obstacle width, d, in the glide direction was shown to be about b at T = 0 K in Section 4.2, and d* would be expected to be smaller than this. The estimates for DH* fall well below the range DH*  kBT and are considerably smaller than the maximum Eb = 0.68 eV in Fig. 2, suggesting that the thermal energy available under the conditions of MD is restricted. To consider this further in the context of Eq. (2), note that, at the three strain rates applied, namely 1, 5 and 2 · 107 s1, the steady-state dislocation velocity in the absence of solute, vD ¼ e_ =bqD , is 2.1, 10.4 and 41.5 m/s, respectively, and the dislocation makes one complete sweep of the MD cell in 12, 2.4 and 0.6 ns. This sweep time is about one-third of the delay time the dislocation encounters at an obstacle when carbon is present (Fig. 6). Thus, with no more than about 30% error, the velocity across an obstacle field is largely determined by the time taken for activation, i.e.   DH vD ¼ Dt0 exp ð5Þ kBT where D is the distance between the activated events and m0 is an attempt frequency. By using Eq. (5) with m0 = 5 · 1012 s1 and any reasonable choice for D, i.e. between a few b and a few tens of b, DH*/kBT is found to be less than 10 for the three vD values quoted above. This is consistent with the DH* values calculated from the MD simulation data. In other words, strain rates orders of magnitude lower than those currently achievable by MD would be required in atomic-level dislocation dynamics to approach the conditions in experiment. This point was made by

K. Tapasa et al. / Acta Materialia 55 (2007) 93–104

Domain and Monnet [36] in comparing the temperature dependence of the critical stress for screw dislocation glide in iron as obtained by MD and experiment. Their Eq. (4) is equivalent to Eq. (5) above rearranged to give:   vD ðsimÞ DH ðsimÞ ¼ DH ðexptÞ  k B T ‘n ð6Þ vD ðexptÞ where ‘sim ‘and ‘expt’ refer to simulation and experimental conditions, respectively. Since vD(sim) is typically 107– 109 times vD(expt) estimated from the macroscopic strain rate of a tensile test, DH*(sim) is approximately 20kBT smaller than DH*(expt). (Estimating vD(expt) from e_ in a test provides a lower limit, since it assumes all dislocations move with the average velocity e_ =qD b. In reality, the maximum vD(expt) is considerably higher, and this would decrease the difference between DH*(sim) and DH*(expt). However, we would still expect the difference to be considerable.) Although the reduction in sc with increasing T up to 300–400 K in Fig. 5 is similar to the experimental temperature-dependence of the critical stress in iron [37–39], the latter is accepted as being due to the temperaturedependence of the critical stress for glide of screw rather than edge dislocations. Edge dislocations are not considered to be a controlling influence because their Peierls stress in much smaller than that of screws, a result of the nature of the screw dislocation core structure. The recent simulations of the screw dislocation in iron and accompanying analysis by Domain and Monnet [36] support this interpretation. However, the critical stress for glide of an edge dislocation when carbon solute is present is seen here to be large, and the influence of edge dislocations can be assessed using the data derived as follows. Domain and Monnet noted that the small value of DH*(sim) results in a smaller value of the gradient josc/oTj than would be found in experiment. The reduction of 20kBT of thermal energy from a total of 25kBT means that DH*  5kBT obtained by MD at temperature T(sim) in Section 4.4 is equivalent to DH*  25kBT obtained experimentally at temperature T(expt) = T(sim)/5. In other words, the decline of sc from the high value at T = 0 K to a plateau by 300–400 K would occur by 60–80 K under experimental test conditions. It is in this low temperature region that edge dislocations might be influential. Another feature of the results for sc as a function of T and e_ is that in the high-temperature regime above 400 K, where sc is largely independent of T, sc is at least several tens of MPa for the carbon spacing considered. It is therefore at least an order of magnitude larger than it would be in the absence of carbon. For example, the stress to maintain edge dislocation glide with vD = 10 m/s in the model of pure iron at 300 K is about 1 MPa. Thus, carbon solute has a significant drag effect on edge dislocation glide, even when thermal activation appears to play no part in motion past the obstacle. The origin of this effect is unclear.

101

5. Core diffusion As mentioned in Section 4.3, carbon atoms were found to migrate in the core region below the extra half-plane of the dislocation under stress at high temperature in MD simulations. The direction of carbon jumps was not that of b, i.e. [1 1 1], but either of the ½1 1 1 directions at 70.5 to b. These directions are shown in a perfect crystal in the illustration of Fig. 9, where atoms in two adjacent ð1 1 0Þ planes are plotted as triangles and the filled circles represent atoms that would form the bottom of the extra half-plane of the edge dislocation in the ½1 1  2 direction. Only a few jumps of this nature were observed before the dislocation broke free of the C atoms in these simulations, however. To remain bound to the dislocation core by ½1 1 1 motion while the dislocation glides with velocity vD in the [1 1 1] direction, the C atom would need a speed of 3vD, and this was not achievable under the conditions of Section 4.3. Hence, in this section we consider a dislocation in a model without applied stress or strain. The C1 crystal (L = 3.51 nm) was used for this purpose. First, the energy barrier, ECore m , for C motion in the core at T = 0 K was calculated as follows. The model crystal was fully relaxed with a C atom in its lowest energy state, i.e. in the A position just below the extra half-plane. The static (potential) energy barrier for motion of the C atom was then estimated by displacing it in small increments (0.03a0) in the ½1 1 2 direction: the potential energy was minimized at each step by allowing full relaxation of all atoms except the C atom, which was not allowed to relax in the ½1 1 2 direction. The minimum energy path of the C atom migration is indeed in the ½1 1 1 direction, as observed in the MD simulations of Section 4.3. However,

Fig. 9. Projection of atoms (triangles) in two adjacent ðl 1 0Þ planes in a perfect crystal. Filled circles represent atoms that would form the bottom of the extra half-plane of the 1/2[1 1 1] edge dislocation in the ½1 1 2 direction.

102

K. Tapasa et al. / Acta Materialia 55 (2007) 93–104

to reach an equivalent site to its starting position, two energy barriers have to be overcome, as seen in the energy profile plotted in Fig. 10(a), where the two migration barriers are labelled P1 and P2. The C atom steps are drawn schematically in ½1  1 0 projection in the inset in Fig. 10(a) and in [1 1 1] projection in Fig. 10(b). The length of each of the P1 and P2 steps projected onto the ½ 1 1 2 direction p of the dislocation is a0/ 6, i.e. the spacing of the {1 1 2} atomic planes. The actual displacement vector for the solute shown in Fig. 10 is 1  1  1  ½1 1 1 þ ½1 1 2 ¼ ½ 1 1 1 12 6 4

ð7Þ

for each of P1 and P2, i.e. the two steps result in a total displacement of 1/2Æ1 1 1æ, the shortest translation vector of the lattice. The larger height of the P2 step arises from repulsion of the solute atom from an Fe atom at the bottom of the extra half-plane, as indicated in Fig. 10(b). Relative to the dislocation core, however, the P2 step appears to be 1=6½ 1 1 2, as implied by the inset sketch of Fig. 10(a), because the dislocation is actually pushed in the ½1 1 1 direction by the moving C atom as a result of the large energy barrier. The same model was used to study C atom behaviour in the core of the edge dislocation by molecular dynamics at 600, 1000 and 1200 K. The simulation time was up to 11 ns in each case. The solute was observed to jump back and forth in the dislocation core, but in the ½1 1  1 direction rather than along the ½1 1  2 direction of the edge dis-

Fig. 10. (a) Static energy barrier for motion of a C atom along the core at T = 0 K. The two barriers P1 and P2 result from steps shown in the inset of (a) and the [1 1 1] projection in (b). Together, they produce a net displacement 1=2½1 1  1 of the solute and are associated with a 1/6[1 1 1] motion of the dislocation (see text).

location line. Thus, motion in dynamics follows the same route as the static path described above. This process can be examined further, as follows. The interstitial solute atom continued jumping for the whole simulation time at 600 and 1000 K, but at 1200 K it dissociated from the dislocation and migrated in the bulk after 3.5 ns. Nevertheless, the number of 1=4½1 1  1 jumps (P1 or P2) was sufficient, namely 144, 644 and 276 at 600, 1000 and 1200 K, respectively, to provide a reasonable estimate of the jump frequency, mC (=13, 57 and 80/ns, respectively). The process was not strictly core diffusion (see below), but for purposes of comparison with diffusion of C in the bulk, a diffusivity, for one-dimensional migration can be obtained from the equation [40]: DCore ¼ mC a2j =2 C

ð8Þ p

where aj is the jump distance, which is 3a0/4 here. The corresponding DCore at 600 and 1000 K is 1.025 · 1010, C 10 4.65 · 10 and 5.6 · 1010 m2/s, respectively. The values are presented in an Arrhenius plot in Fig. 11. This figure also compares DCore with the diffusion coeffiC cient of carbon in the bulk of a-Fe as obtained by either experiment [41,42] (dashed lines) or MD simulation using the same interatomic potential model as here [4] (open circles). DCore is three orders of magnitude higher than C the diffusion coefficient for three-dimensional bulk diffusion at 600 K and an order of magnitude greater at 1000 K, indicating a higher jump frequency of a carbon atom in the dislocation core as compared with the bulk. The activation energy, Em, and the diffusion constant, D0, can be obtained from the straight-line fit shown using the Arrhenius equation:   Em D ¼ D0 exp  ð9Þ kBT

Fig. 11. Diffusion coefficient obtained via Eq. (8) for comparison of motion of carbon in the edge dislocation core with that in the bulk of Fe, as obtained by either experiment or MD simulation. This coefficient does not represent transport of carbon along the core, however, as explained in the text.

K. Tapasa et al. / Acta Materialia 55 (2007) 93–104

This results in DCore ¼ 4  109 expð0:2eV=k B T Þ m2 =s C compared with diffusivity equal to 1.9 · 107exp(0.7 eV/ kBT) m2/s for bulk diffusion in the same MD model. Experimental values for Em in bulk diffusion are in the range 0.81–0.86 eV. Thus, several key results stand out from this analysis. First, the activation energy of 0.2 eV for migration in the core of the edge dislocation line in dynamic conditions (T > 0 K) is smaller than either of the energy barriers (0.3 and 0.5 eV) for the P1 and P2 steps under static conditions (T = 0 K). In particular, extra resistance to migration implied by the existence of barrier P2 in statics was not present in the MD simulations. This is believed to result from the ease with which segments of the dislocation core can move forward or backward (in the direction of b) as the carbon atom moves when T > 0 K. The atoms at the bottom of the extra half-plane have a Æ1 1 1æ crowdion-like nature, and segments as small as one atom in size can readily move ahead of their neighbours [33]. Second, the activation energy for migration in the core of the edge dislocation line is only about a quarter of the value in the bulk crystal. Although D0 is smaller in the core, DCore is orders of magC nitude higher than diffusivity in the bulk, except at a high enough temperature for thermal activation to overcome the C–dislocation binding energy and allow dissociation. Thus, the edge dislocation core provides an environment for rapid jumping of carbon at moderate temperature. However, the solute jumps in the core are in a Æ1 1 1æ direction, which is neither the direction of b nor the direction of the line, so net transport of carbon along the line can only occur by motion of the line itself. In other words, the process analysed here was not true diffusion of carbon. The carbon atom does jump back and forth with frequency mC in the core, but DCore calculated via Eq. (8) is not a diffusion C coefficient representing carbon transport. Since it scales with mC, however, it does provide a useful parameter for comparison with diffusion in the bulk. It is possible to speculate that dislocations in other orientations, such as the 70.5 Æ1 1 1æ line of mixed character, can provide rapid diffusion paths for interstitial solute. However, their simulation is not as straightforward as that of the edge dislocation: we are engaged in that problem at the moment and will report in due course. 6. Conclusions Molecular statics and dynamics simulations have been employed to study the behaviour of a single carbon interstitial atom in the core of a 1=2½1 1 1ð1  1 0Þ edge dislocation in a-Fe using a combination of the FS-type interatomic potential of Ackland et al. [25] for Fe and an empirical pair potential of Johnson et al. [19] for Fe–C. This is the first investigation of the obstacle strength of interstitial solute atoms in dislocation glide under static (T = 0 K) and dynamic conditions (T > 0 K), and of the motion of such atoms in the core of both stationary and moving dislocations. The following conclusions have been drawn.

103

1. C atoms in octahedral sites half an interplanar spacing below the ð1 1 0Þ atomic plane containing the bottom row of atoms of the extra half-plane of the dislocation have the highest binding energy, Eb = 0.68 eV. Solutes in octahedral sites within the ð1 1 0Þ atomic planes have a maximum Eb = 0.5 eV when in the plane one below the extra half-plane. 2. The critical stress, sc, for an edge dislocation to overcome a row of C atoms of spacing L in the sites of maximum binding at T = 0 K decreases with an L1dependence, consistent with strengthening due to point obstacles with barrier energy Eb. 3. The effects of temperature and applied strain rate, e_ , on sc have been simulated using MD. At T < 400 K, sc decreases rapidly as temperature increases and becomes almost temperature independent at higher temperatures. It decreases with decreasing e_ and becomes strain-rate independent at T > 300 K. 4. Jumps of C in the dislocation core occur before the dislocation unpins at T P 800 K and give a small temperature-dependence of sc in this range. 5. sc decreases with increasing spacing L but does not have exact L1-dependence. sc is slightly lower than expected when L is smaller. 6. The activation volume, V*, and energy, DH*, have been calculated from the change in sc with e_ and T. V* values imply an activation distance of (0.2– 0.3)b, which is consistent with a point defect obstacle. 7. DH* is only 5kBT, which is much smaller than the energy Eb that has to be overcome by the dislocation at T = 0 K. This is shown to result from the high e_ that is unavoidable in MD simulation. This supports the conclusion of Domain and Monnet [36] that DH* found by simulation is 20kBT smaller than that realized in experiment. 8. The reduction of 20kBT of thermal energy from a total of 25kBT means that DH* obtained by MD at temperature T is equivalent to that obtained experimentally at 0.2T. Hence, the decline of sc over the range 0 to 400 K would occur over 0 to 80 K under experimental conditions. It is in this low temperature region that edge dislocations might be influential. 9. The dislocation is unable to drag C atoms under simulation conditions. The C atom jumps noted in point 4 are in the ½1 1 1 direction at 70.5 to the glide direction b. Only a few jumps occur before the dislocation breaks free, however, and so core diffusion of interstitial solute has been investigated without application of stress or strain. 10. Static simulation (T = 0 K) shows that the lowest energy path within the core is indeed in the ½1 1  1 direction. Motion by the shortest lattice translation 1=2½1 1 1 occurs in two partial steps, with barrier energy 0.33 and 0.53 eV. The latter is associated with displacement of the dislocation core by ±1/ 6[1 1 1].

104

K. Tapasa et al. / Acta Materialia 55 (2007) 93–104

11. Motion of C in the core occurs by the same 1=2½1 1 1 jumps under dynamic conditions (T > 0 K). Diffusivity has been obtained from the jump frequency and is 4 · 109exp(0.2 eV/kBT) m2/s. The lower value of the migration energy compared with the static barriers arises from the ease with which dislocation core segments move when T > 0 K. 12. This diffusivity compares with 1.9 · 107 2 exp(0.7 eV/kBT) m /s for bulk diffusion of C in the same MD model of Fe. Thus, the edge dislocation core provides an environment for high jump frequency of C at moderate temperature. However, the solute jumps are at 70.5 to b, so net transport along the line can only occur by motion of the line itself. It is possible that the 70.5 Æ1 1 1æ mixed dislocation line can provide a rapid diffusion path for interstitial solute. Acknowledgements The authors acknowledge many discussion with Dr. A.V. Barashev (Liverpool) and helpful correspondence with Dr. G. Monnet (EDF). K.T. would like to thank the Science Service Division of the Ministry of Science and Technology, Thailand, for providing a studentship grant. The research was supported by a research grant from the UK Engineering and Physical Sciences Research Council; grant PERFECT (F160-CT-2003-508840) under programme EURATOM FP-6 of the European Commission; and partly by the Division of Materials Sciences and Engineering and the Office of Fusion Energy Sciences, US Department of Energy, under contract DE-AC0500OR22725 with UT-Battelle, LLC. References [1] Cottrell AH, Bilby BA. Proc Phys Soc Lond A 1949;62:49. [2] Takaki S, Fuss J, Kugler H, Dedek U, Schults H. Radiat Eff 1983;79:87. [3] Vehanen A, Hautojarvi P, Johansson J, Yli-Kauppila J, Moser P. Phys Rev B 1982;25:762. [4] Tapasa K, Barashev AV, Bacon DJ, Osetsky YN. Acta Mater, in press. [5] Domain C, Becquart CS, Foct J. Phys Rev B 2004;69:144112. [6] Wu RQ, Freeman AJ, Olson GB. Phys Rev B 1995;53:7504. [7] Olson GB. In: Olson GB, Azrin M, Wright ES, editors. Innovations in ultrahigh-strength steel technology. Sagamore Army Materials Research Conference Proceedings No. 34. US Army Laboratory Command; 1990. p. 3.

[8] Rice JR, Wang J-S. Mater Sci Eng A 1989;107:23. [9] Anderson PM, Wang J-S, Rice JR. In: Olson GB, Azrin M, Wright ES, editors. Innovations in ultrahigh-strength steel technology. Sagamore Army Materials Research Conference Proceedings No. 34. US Army Laboratory Command; 1990. p. 619. [10] Cochardt AW, Schoeck G, Wiedersich H. Acta Metall 1955;3:533. [11] Hirth JP, Cohen M. Scripta Metall 1969;3:107. [12] Hirth JP, Cohen M. Scripta Metall 1969;3:311. [13] Schoeck G. Scripta Metall 1969;3:239. [14] Bacon DJ. Scripta Metall 1969;3:735. [15] de Hosson JThM. Solid State Commun 1975;17:747. [16] Niu Y, Wang S, Zhao D, Wang C. J Phys: Condens Matter 2001;13:4267. [17] Simonetti S, Pronsato ME, Brizuela G, Juan A. Appl Surf Sci 2003;217:56. [18] Johnson RA. Phys Rev 1964;134:A1329. [19] Johnson RA, Dienes GJ, Damask AC. Acta Metall 1964;12: 1215. [20] Wert CA. Phys Rev 1950;79:601; Doremus RH. Trans Amer Inst Min (Metall) Eng 1960;218:596; Stanley JK. J Metals 1949;1:752; Wagenblast H, Damask AC. J Phys Chem Solids 1962;23:221; Fujita FE, Damask AC. Acta Metall 1964;12:341. [21] Bosman AJ, Brommer PE, Rathenau GH. Physica 1957;23:1001; Bosman AJ, Brommer PE, Eijkelenboom LCH, Schinkel CJ, Rathenau GH. Physica 1960;26:533; Bass J, Lazarus DJ. Phys Chem Solids 1962;23:1820. [22] Smith RP. Trans Metall Soc AIME 1962;224:105. [23] Williamson GK, Smallmann RE. Acta Crystallogr 1953;6:361. [24] Pearson WB. Lattice spacings and structures of metals and alloys. London: Pergamon Press; 1958. p. 919. [25] Ackland GJ, Bacon DJ, Calder AF, Harry T. Philos Mag A 1997;75:713. [26] Ackland GJ, Mendelev MI, Srolovitz DJ, Han S, Barashev AV. J Phys: Condens Matter 2004;16:S2629. [27] Rosato V. Acta Metall 1989;37:2759. [28] Marchese M, Jacucci G, Flynn CP. Philos Mag Lett 1988;57:25. [29] Ruda M, Farkas D, Abriata J. Scripta Mater 2002;46:349. [30] Tersoff J. Phys Rev Lett 1988;61:2879. [31] Lee B-J. Acta Mater 2006;54:701. [32] Arndt RA, Damask AC. Acta Metall 1964;12:341. [33] Osetsky YN, Bacon DJ. Modell Simul Mater Sci Eng 2003;11: 427. [34] Hull D, Bacon DJ. Introduction to dislocations. Oxford: Butterworth-Heinemann; 2001. [35] Reed-Hill RE, Abbaschian R. Physical metallurgy principles. Boston: PWS-Kent Publishing Co.; 1992. [36] Domain C, Monnet G. Phys Rev Lett 2005;95:215506. [37] Kuramoto E, Aono Y, Kitajima K. Scripta Metall 1979;13:1039. [38] Spitzig WA, Keh AS. Acta Metall 1970;18:611. [39] Stien DF, Low JR, Seybolt AU. Acta Metall 1963;11:1253. [40] Wert C, Zener C. Phys Rev 1949;76:1169. [41] Da Silva JRG, McLellan RB. Mater Sci Eng 1976;26:83. [42] Brandes EA, editor. Smithells metals reference book. London: Butterworth; 1983.