Dislocations 2008 IOP Conf. Series: Materials Science and Engineering 3 (2009) 012001
IOP Publishing doi:10.1088/1757-899X/3/1/012001
Annealing of Dislocation Loops in Dislocation Dynamics Simulations Dan Mordehai∗1,3 , Emmanuel Clouet1,4 , Marc Fivel2 and Marc Verdier2 1 2
SRMP, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, France CNRS/SIMAP, INPG, BP 75, 38402 St Martin d’H`eres, France
E-mail: ∗
[email protected] Abstract. We report of 3-dimensional discrete dislocation dynamics (DDD) simulations of dislocation loops coarsening by vacancy bulk diffusion. The calculation is based upon a model which couples the diffusion theory of vacancies to the DDD in order to obtain the climb rate of the dislocation segments. Calculation of isolated loops agrees with experimental observations, i.e. loops shrink or expand, depending on their type and vacancy supersaturation. When an array of dislocation loops of various sizes is considered, and the total number of vacancies in the simulation is maintained constant, the largest dislocations are found to increase in size at the expense of small ones, which disappear in a process known as Ostwald ripening.
1. Introduction One of the main computational tools to study the dynamic collective evolution of dislocations in a solid under an external loading at the mesoscopic scale is Dislocation Dynamics (DD). In this simulation technique each dislocation is represented as an elastic entity, which obeys certain rules for motion and interaction with other dislocations, such as drag force upon gliding, cross-slip, junction formation etc. However, the vast majority of DD calculations either prohibit climb or treat it as the conservative glide motion by considering a typical drag coefficient for the climb motion (eg. [1, 2, 3]). Thus, DD simulations are used to study phenomena as strain hardening [4, 5], nanoindentation [6], low-strain fatigue [7] and crack-tip plasticity [8], but in order to perform analysis such as creep and dislocation annealing in a DD simulation, one should first define reliable rules for dislocation climb. In [9] we introduced a climb model in a three-dimensional Discrete Dislocation Dynamics simulation (DDD), by coupling between the DDD and the diffusion field of vacancies in the bulk. In this work, we present a calculation of dislocation loops annealing using this model. First, we detail briefly in Sec. 2 the climb model we introduced in the DDD. Then, we discuss in Sec. 3 the shrinkage and expansion rates of dislocation loops in constant vacancy supersaturation conditions. Finally, we present and analyze a simulation of the annealing of dislocation loops in bulk conditions in Sec. 4. 3 4
Present address: Department of Materials Engineering, Technion, 32000 Haifa, Israel Present address: LMPGM, Universit´e Lille, 59655 Villeneuve d’Ascq Cedex, France
c 2009 IOP Publishing Ltd
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Dislocations 2008 IOP Conf. Series: Materials Science and Engineering 3 (2009) 012001
IOP Publishing doi:10.1088/1757-899X/3/1/012001
2. The Climb Model In [9] we showed by using atomistic simulations that a climbing edge dislocation segment in the DDD simulation creates or resizes edge segments with the same Burgers vector as the climbing segment, which glide in the same direction, but differ from it in their line vectors (see example in fig. 1a). We emphasize that due to the space discretization in the DDD, we consider the dislocation segments climb only after absorbing or emitting a certain amount of vacancies, i.e. dislocation climb form superjogs, rather than of a single monolayer jog. The climb rate of the segments is obtained analytically under the following assumptions: • • • •
We restrict our discussion to climb due to vacancy bulk diffusion and we omit pipe diffusion. At each time step the vacancy flux reaches a steady-state. The elastic interaction energy between dislocations and vacancies is neglected. A high concentration of jogs exists along the dislocation line, i.e. each dislocation segment acts as a source or sink of vacancies. In addition, we assume that vacancies are absorbed or emitted immediately by the climbing dislocation segment.
We concluded in [9] that under these assumptions the climb model captures the main properties of dislocation climb at high temperatures, where the climb by bulk diffusion is the rate-controlling process, and at low climb rates, where the dominant climb process is by jog migration rather than jog formation.
Figure 1. The climb model. (a) The climbing edge segment in the [¯ 112] direction and Burgers vector [110] (dashed line) creates two superjogs which are pure edge segments in the direction [1¯ 12] and the same Burgers vector. (b) A control volume about a short dislocation line, in which Fick’s equation is solved. Under these conditions Fick’s equation is solved analytically in a hollow cylindrical control volume about each segment, for which the climb rate is calculated (Fig. 1b). The radius of the outer boundary r∞ is determined so that the vacancy concentration on it can be considered as the average vacancy concentration in the bulk c∞ . In practice, this distance is taken as the average distance between dislocations in the computational cell. On the inner surface of the control volume with the radius rd , the vacancies are assumed to be at equilibrium with the dislocation [10], leading to the concentration Fcl Ω
cd = c0 e bkB T ,
(1)
where c0 is the equilibrium vacancy concentration, Fcl is the climb force, Ω is the atomic volume, b is the Burgers vector and kB T has its usual meaning. Fcl is the component of the Peach-K¨ ohler force in the climb direction, obtained in the DDD from the elastic interactions with the external stress and with all the other segments. From the solution of Fick’s equation we obtain the 2
Dislocations 2008 IOP Conf. Series: Materials Science and Engineering 3 (2009) 012001
IOP Publishing doi:10.1088/1757-899X/3/1/012001
number of vacancies which diffuse into the dislocation core per unit time and length, and we conclude that the climb-rate is µ
¶
Fcl Ω c∞ 2πDs , vcl = e bkB T − f b ln(r∞ /r0 ) c0
(2)
where f is the Bardeen-Herring correlation factor and Ds is the self-diffusion coefficient. Ds depends exponentially on the temperature, with an activation energy for self-diffusion Eact , which is the sum of the formation and migration energies of vacancies. One of the parameters that control the climb rate in eq. 2 is the average vacancy concentration in the bulk c∞ . In reality, the value c∞ is spatially dependent and changes according to the distribution of the vacancy sinks and sources in the system. In this work we assume that far from the dislocations the vacancy concentration is homogenous, and we consider two extreme cases, that either the concentration or the initial number of vacancies is a conservative quantity. We choose to work with a set of parameters for Al, given in [9] 3. Shrinkage and Expansion of Isolated Prismatic Loops In the first stage, we assume that the sinks and sources of vacancies in the system manage to maintain the concentration of free vacancies constant. In this extreme case, one can study the shrinkage and expansion of single isolated dislocation loops and to compare the results with analytical models which are based upon the same assumption. Prismatic dislocation loops are constructed in the DDD computational cell of dislocation segments in the [121] and the [121] direction, all having the same Burgers vectors 21 [101] (see initial configuration in Fig. 2a). Firstly, a diamond-shape vacancy dislocations loop is introduced, with no external stress applied and choosing the average vacancy concentration to be the equilibrium concentration, i.e c∞ = c0 . The loop was found to shrink under its own self stress and to annihilate (Fig. 2a). The loop in the calculation rounds while shrinking, as was observed experimentally by Washburn [11], due to a non-homogenous climb force along the loop contour, which arises from its non-circular shape. When a circular vacancy loop is introduced initially, its surface decreases linearly with time, at a rate increasing with temperature (fig. 2b). Additionally, the annihilation time decreases with temperature and obeys the Arrhenius law with activation energy of 1.25 eV. When the vacancy concentration in the bulk is supersaturated, the loops’ radii increases linearly with time, with an expansion rate that obeys the Arrhenius law with an activation energy of 1.28 eV (fig. 2c). Both activation energy values, for shrinkage and expansion, are comparable with experimental values [12] and correspond to the value for the self-diffusion activation energy in Al, given as an input in the simulation. 4. Population of Loops Distribution in the Bulk In the previous section, the concentration of free vacancies was maintained at constant level at some distance from the segments. However, these conditions are not necessarily fulfilled in all cases, and the concentration of free vacancies may change due to absorption or emission of vacancies from various types of sinks and sources, such as surfaces, grain-boundaries, climbing dislocations, irradiation etc. In bulk material, where the climbing dislocations are far from surfaces and grain-boundaries, the main sources of vacancies are the climbing dislocation. Consequently, in the extreme case of neglecting other vacancy sources but the dislocation themselves, we vary the concentration of free vacancies in the calculation to conserve the initial total number of vacancies in our system, i.e. the sum of the free vacancies and the vacancies that are condensed in loops [9]. Thus, the average concentration of vacancies in this case is time-dependent c∞ (t).
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Dislocations 2008 IOP Conf. Series: Materials Science and Engineering 3 (2009) 012001
IOP Publishing doi:10.1088/1757-899X/3/1/012001
Figure 2. Prismatic loop shrinkage and expansion. (a) A diamond shape prismatic loop shrinkage at 500 K. The loop shape is plotted at constant time intervals, starting from the outer contour. (b) Variation of the loop surface, normalized by the initial surface, at various temperatures. (c) Variation of the loop radius at various temperatures under vacancy supersaturation. Conserving the total number of vacancies can be also interpreted as a simple approximation for the interaction between the flux fields of two different climbing segments via a ’vacancy reservoir’ in the bulk. Indeed, Silcox and Whelan [12] observed that the number of dislocation loops in quenched Al specimens decreased after annealing, while the size of the ones that did not annihilate increased. The coarsening of the loops was rationalized by the interaction between the flux fields of the different loops, concluding that larger loops grow at the expense of smaller ones. In accordance, we simulate the evolution of a dislocation loop array under the conditions of annealed bulk. We introduce 64 circular vacancy dislocation loops into the computational cell, considering all have the same Burgers vector in the [101] direction (see Fig. 3a). The loops radii are Gaussian¯ 0 and the initial variance as ∆R0 distributed and we shall denote the initial mean radius as R (initial distribution in Fig. 4a). Their centers are located on a 3-dimensional imaginary cubical grid of length Ld = L/4, where L is the size of the computational cell. We assume that the distance between the loops is large enough so that the number of vacancies in the middle of two adjacent loops is the average concentration of vacancies, i.e. r∞ = Ld /2. The temperature is kept constant at 470 K. At the beginning, the smallest loops in the system shrink by emitting vacancies to the bulk, while the largest loops expands by absorbing vacancies from the bulk (smallest and largest loops in Fig. 4b). In result, the concentration of free vacancies in the bulk increases at the beginning, but since after a certain time the expanding loops absorb more vacancies than the ones that shrink, it reaches a maximum value and decreases afterwards, as shown in Fig. 4c. As the concentration of the free vacancies decreases, the condition for some of the loops, which expanded at the beginning of the calculation, are no longer favorable for expansion, and they shrink and annihilate (medium size loops in Fig. 4b). Eventually, at the time where the calculation is terminated, the supersaturation of free vacancies is being relieved from the bulk and the number of dislocation loops decreases, whereas the remaining ones are bigger (Figs. 3b and 4d), as was observed by Silcox and Whelan [12]. Burton and Speight [13] and Bonafos et al. [14] suggested that the number of loops decreases in time according to the law ¶ µ t −1 , (3) N (t) = N0 1 + τ where N0 is the initial number of loops and τ is a typical time for the coarsening process. By fitting this law to our results, we find that τ = 75.2 hours in our simulations, which is 4
Dislocations 2008 IOP Conf. Series: Materials Science and Engineering 3 (2009) 012001
IOP Publishing doi:10.1088/1757-899X/3/1/012001
Figure 3. The (a) initial and (b) final distribution of the vacancy loops within the computational cell. The loops were annealed via bulk-diffusion climb, and only a few bigger loops remind at the time were the calculation terminated.
Figure 4. Analysis of the annealing process in the simulation. (a) The evolution in the distribution of the loops size. The time evolution of (b) a few representative loops, (c) the average vacancy concentration in the bulk, (d) the total number of loops, (e) the mean radius, normalized by its initial value, and (f) the variance of the radius size distribution, normalized by its initial value. in agreement with the value which is obtained analytically if using the expressions in [13, 14]. Kirchner [15] has suggested that an initial Gaussian distribution will spread in time, with a time dependent mean radius and variance. The mean average radius satisfies µ
¯2 = R ¯2 1 + t R 0 τ 5
¶
,
(4)
Dislocations 2008 IOP Conf. Series: Materials Science and Engineering 3 (2009) 012001
IOP Publishing doi:10.1088/1757-899X/3/1/012001
having the same typical time τ as in eq. 3. Thus, we consider the distribution to be Gaussian at any time of the simulation, and we calculate the mean radius and the variance at various times along the simulation, as being plotted in Figs 4e and 4f. One can see that the mean radius size evolution in our simulation is comparable with the results of Kirchner, taking the value for τ found by eq. 3. The variance is also found to be time dependent and to increase with time. 5. Conclusion In this paper we presented DDD simulations of the shrinkage and expansion of dislocation loops, as well as annealing of dislocation loops in bulk material, by introducing a climb model based upon the diffusion theory of vacancies. While the model presented in this work does not include the interaction of vacancies with jogs and pipe diffusion, it is shown to capture important physical parameters, such as the activation energies for isolated loops shrinkage and expansion, and time evolution of loops coarsening in annealed bulk. As such, the model should be considered as a first step in obtaining a reliable rigorous treatment of dislocation climb in DDD simulations. In the future, this model should be extended to take into account climb by jogs formation and migrations and pipe diffusion, in order to study climb-related phenomena in a wide range of temperature and stress conditions. This additional step would enhance our results into a wider range of temperatures and climb rates. In addition, while in this work glide is omitted in order to isolate only pure-climb related phenomena, one should incorporate this model with the glide and cross-slip models in the DDD in order to study creep in a complex dislocation microstructure. References [1] N. M. Ghoniem, S. H. Tong, and L. Z. Sun. Parametric dislocation dynamics: A thermodynamics-based approach to investigations of mesoscopic plastic deformation. Phys. Rev. B, 61:913–927, 2000. [2] Y. Xiang, L. T. Cheng, D. J. Srolovitz, and E. Weinan. A level set method for dislocation dynamics. Acta Mater., 51:5499–5518, 2003. [3] Y. Xiang and D. J. Srolovitz. Dislocation climb effects on particle bypass mechanisms. Phil. Mag., 86:3937– 3957, 2006. [4] B. Devincre and L. P. Kubin. Mesoscopic simulations of dislocations and plasticity. Mater. Sci. Eng. A, 234–236:8–14, 1997. [5] H. M. Zbib, T. D. de la Rubia, M. Rhee, and J. P. Hirth. 3d dislocation dynamics: stressstrain behavior and hardening mechanisms in fcc and bcc metals. J. Nucl. Mater., 276:154–165, 2000. [6] M. C. Fivel, T. J. Gosling, and G. R. Canova. Implementing image stresses in a 3d dislocation simulation. Modelling Simul. Mater. Sci. Eng., 4:581–596, 1996. [7] C. D´epr´es, C. F. Robertson, and M. C. Fivel. Low-strain rate fatigue in alsl 316l steel surface grains: a three dimensional discrete dislocation dynamics modelling of the early cycles i. dislocation microstructure and mechanical behavior. Phil. Mag., 84:2257–2275, 2004. [8] E. Van der Giessen, V. S. Deshpande, H. H. M. Cleveringa, and A. Needleman. Discrete dislocation plasticity and crack tip fields in single crystal. J. Mech. Phys. Solids, 49:2133–2153, 2001. [9] D. Mordehai, E. Clouet, M. Fivel, and M. Verdier. Introducing dislocation climb by bulk diffusion in discrete dislocation dynamics. Phil. Mag., 88:899–925, 2008. [10] J. P. Hirth and J. Lothe. Theory of Dislocations. Wiley, New York, 2nd edition, 1982. [11] J. Washburn. Annealing of loops and tetrahedra. In J.W. Corbett and L.C. Ianniello, editors, Proc. Int. Conf. on Radiation-Induced Voids in Metals, pages 647–662, Springfield, Virginia, June 9-11 1971. [12] J. Silcox and M. J. Whelan. Direct observations of the annealing of prismatic dislocation loops and of climb of dislocations in quenched aluminium. Philos. Mag., 5:1–23, 1960. [13] B. Burton and M. V. Speight. The coarsening and annihilation kinetics of dislocation loop. Phil. Mag. A, 53:385–402, 1986. [14] C. Bonafos, D. Mathiot, and A. Claverie. Ostwald ripening of end-of-range defects in silicon. J. Appl. Phys., 83:3008–3017, 1998. [15] H. O. K. Kirchner. Size distribution of dislocation loops. Acta Metall., 21:85–91, 1973.
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