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Dislocation multiplicationin bcc molybdenum: Adislocation dynamicssimulationMoono Rhee , David H. Lassila , Vasily V.Bulatov , Luke Hsiung & Tomas Diaz de la RubiaPublished online: 14 Nov 2010.

To cite this article: Moono Rhee , David H. Lassila , Vasily V. Bulatov , Luke Hsiung& Tomas Diaz de la Rubia (2001) Dislocation multiplication in bcc molybdenum: Adislocation dynamics simulation, Philosophical Magazine Letters, 81:9, 595-605,DOI: 10.1080/09500830110069594

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PHILOSOPHICAL MAGAZINE LETTERS, 2001, VOL. 81, NO. 9, 595±605

Dislocation multiplication in bcc molybdenum: adislocation dynamics simulation

Moono Rheey, David H. Lassila, Vasily V. Bulatov, Luke Hsiungand Tomas Diaz de la Rubia

Lawrence Livermore National Laboratory, Livermore, California 94550, USA

[Received 30 April 2001 and accepted 29 May 2001]

AbstractPlastic deformation of Mo single crystals is examined by direct simulation of

dislocation dynamics under stress. Initial dislocation populations are made tomimic real dislocation microstructures observed in transmission electronmicroscopy cross-sections of pure annealed Mo single crystals. No a priorisources for dislocation multiplication are introduced, and yet multiplicationtakes place through a sequence involving aggregation of grown-in superjogs,bowing of screw dislocation segments and fast lateral motion of edge segments,producing a large number of elongated loops and a characteristic cross-gridpattern of screw dislocations.

} 1. BackgroundSince the initial discovery of dislocations over 65 years ago (Orowan 1934,

Polanyi 1934, Taylor 1934), it has been realized that crystal plasticity propertiesare derivable, in principle, from the aggregate behaviour of large collections ofcrystal dislocations. However, a quantitative connection between dislocation theoryand the phenomenology of crystal plasticity is yet to be established, owing to theimmense computational complexity of having to deal with large groups of disloca-tions and the multitude of their physical behaviours. Starting in the early 1990s, therealization that larger faster computers could be used for crystal plasticity simula-tions has led to the development of dislocation dynamics (DD), a powerful compu-tational methodology (Kubin et al. 1992). The predictive power of various DDimplementations available today is based on their ability to incorporate, in a com-putationally tractable manner, multiple and complex mechanisms of dislocationbehaviour.

On one hand, the current trend in DD simulations is to let dislocations behavemost naturally (Schwarz and TersoŒ1996, Rhee et al. 1998, Zbib et al. 1998). On theother hand, computational limitations often dictate that simpli®ed rules for dis-location unit processes be adopted (Kubin et al. 1992). One of the key uncertaintiesin DD simulations is how dislocations multiply. Up to now, several ingeniousmechanisms for dislocation multiplication have been suggested and, in some cases,experimentally veri®ed. Typically, multiplication is thought to take place when adislocation attains a certain special con®guration. In particular, the famous

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DOI: 10.1080 /0950083011006959 4

y Author for correspondence. E-mail: Rhee1@llnl.gov

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Frank±Read (FR) mechanism requires the ends of an otherwise mobile dislocationto be pinned (constrained) to the lattice, leaving the nature of such pinning pointsunspeci®ed. In view of such uncertainties, dislocation sources are usually introducedin the DD simulations in an ad hoc manner, by creating appropriate constraineddislocation con®gurations by hand. Such simpli®cations, while often unavoidable,hide yet another important issue, namely how dislocation sources themselves formduring deformation. Stated alternatively, one may ask what features in the original(undeformed) dislocation microstructure and which dislocation mechanisms con-spire to create the necessary multiplication sources.

} 2. IntroductionIn this letter, we report the ®rst results of direct DD simulations of crystal

plasticity in bcc Mo, in which dislocations are initially arranged to mimic salientfeatures of the real grown-in dislocation microstructures as revealed by transmissionelectron microscopy (TEM) analysis. In our simulations, no ad hoc sources areintroduced; instead, the dislocations are allowed to evolve naturally under uniaxialcompressive stress. Progressive straining of the simulated specimen is then observedto produce multiplication sources through a sequence involving the motion, coales-cence and mutual pinning of superjogs. Formed as a result of dislocation multi-plication and glide, the resulting dense network of dislocations features a `cross-grid’ pattern characteristic of experimental transmission electron micrographs ofplastically deformed single-crystal Mo. At the same time, the predicted net plasticityrate is similar to that observed experimentally.

The specimens for mechanical testing and TEM analysis were cut from singlecrystals of Mo grown along the [011] direction using the ¯oat-zone method. Prior totesting, the specimens were annealed in ultrahigh vacuum for 1 hour at 15008C. Thetotal impurity content was determined to be 200 wtppm, with approximately50 wtppm in interstitials (O, C, N, and H). To test the net plastic response, a com-pressive load along the [118] axis was applied. The tests were performed at roomtemperature, at a constant strain rate of 0.001 s¡1. Thin foils for TEM analysis werecut from the puri®ed and annealed crystals, before and after deformation. A frag-ment of a typical transmission electron micrograph obtained for a [0·111Š-sliced foil ofan as-annealed specimen is shown in ®gure 1 (a). As is typical for Mo and otherrefractory metals, the observed microstructure is dominated by long segments ofscrew character, although dislocation tangles were also observed on occasion.Somewhat less common is that the screw dislocations contain a signi®cant numberof edge steps, superjogs and superkinks formed during growth. Based on the TEMobservations the total dislocation density was estimated at 107 cm¡2.

The ultimate test of the predictive capabilities of DD methodology is a simula-tion of the deformation response of real dislocation microstructures under stress.Ideally, one would want to compare directly the evolution of the same microstruc-ture in a DD simulation and under TEM. However, this is not an option, as it isnearly impossible to index and position accurately all the dislocations using in-situTEM, even in the relatively small volumes amenable to computational modelling.The next best thing that TEM analysis can do is to provide statistical parameters ofdislocation microstructures, which can then be used to set up DD simulations. Basedon examination of a large number of transmission electron micrographs obtained forvarious cross-sections of the annealed specimens, we accumulated and analysed

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Dislocation multiplication in bcc Mo 597

(b)

Figure 1. (a) TEM dislocation microstructure in the annealed Mo. (b) A sectional view of thesimulated microstructure.

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several statistical characteristics of the dislocation shapes in the initial dislocationmicrostructures.

As a ®rst step towards direct simulation of realistic dislocation microstructures inMo, we used these parameters to generate initial dislocation con®gurations for DDsimulations. The initial con®gurations were produced using a biased random walkprocedure made to conform to the TEM analysis data as follows:

(i) The total dislocation density was set at 107 cm¡2 and was distributed equallyover eight groups of dislocations with 1

2h111i Burgers vectors.

(ii) Each dislocation was represented by a random-walk trajectory consisting ofalternating screw steps (along the h111i direction) and edge steps (along sixh112i directions).

(iii) The lengths of the steps were assigned randomly from an exponentialdistribution with the average length of 550 nm for the screw steps, ten timeslonger than that for the edge steps, at 55 nm.

(iv) Directions of adjacent edge steps were correlated. The probability ofstepping in the same h112i direction was made 12 times higher than theprobability of stepping in a diŒerent h112i direction.

This procedure leaves aside the mechanisms by which these features are formedon the initial grown-in and annealed dislocations. It is only designed to reproducethese features, regardless of their origin.

To ensure computational e� ciency of the subsequent DD simulations we trun-cated both the screw and the edge exponential distributions, eliminating all thesegments shorter than 100b, or 27 nm. A thin slice through one of the so-generateddislocation microstructures is shown in ®gure 1 (b) and should be compared with thetransmission electron micrograph in ®gure 1 (a). Prior to straining, the simulateddislocations were allowed to move in response to the interaction forces until theseforces were relaxed to an acceptable level below the lattice frictional stress. Such`pre-annealed’ simulated specimens displayed noticeable but minor changes in theirstatistical parameters. The resulting relaxed con®gurations were then regarded asstatistically equivalent to annealed TEM microstructures.

To simulate plastic response in the bulk of single-crystal Mo we used themicro3D code, an e� cient oŒ-lattice implementation of the DD approach (fordetails, see Rhee et al. (1998) and Zbib et al. (1998)). An initial pre-annealed dis-location microstructure was generated in a 10 mm £ 10 mm £ 10 mm box using thetwo-stage procedure described above. Then, uniaxial compression along the [118]axis was applied at a constant strain rate of 1 s¡1, at 300 K. The total Peach±Koehlerforce acting on a given dislocation segment includes two major contributions: theline tension force which opposes any increase in the dislocation line length, and theforce exerted on the segment by the local elastic stress. The latter is a combination ofthe external (uniform) stress and the internal (non-uniform) stress due to elasticinteraction with all the other dislocation segments in the simulation volume.Evaluation of these long-range elastic interaction terms presents the bottleneck ofDD simulations. To accelerate stress updates, we used a fast multipole method (Zbibet al. 1998).

} 3. GlideDislocation glide is the most common manifestation of plastic deformation.

There are a number of constitutive relationships that describe glide velocity vg as

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a function of stress (Johnson and Gilman 1959, Kocks et al. 1975, Hirth and Lothe1992). In a number of cases of pure phonon±electron damping control, or of glideover the Peierls barrier, a linear form for the glide velocity vg predicts the results verywell:

vg ˆ Mb…½ ¡ ½f†; …1†

where M is the temperature-dependen t dislocation glide mobility and climb mobilityrespectively, ½ is the eŒective shear stress, ½f is the frictional stress and b is themagnitude of the Burgers vector. We adopt this simple form for initial calculations,although an extension to a power-law form can be readily implemented (Prekel andConrad 1968, Leiko et al. 1975). To account for the anisotropy of dislocation motionin bcc Mo, we used two diŒerent sets of mobility parameters for screw and non-screwdislocations. Speci®cally, the frictional stress ½f for screw dislocations was set at26 MPa, 20 times higher than the frictional stress for the non-screw dislocations.Likewise, the mobility coe� cient M for the screw dislocations was set at 1 Pa¡1 s¡1,100 times lower than that for non-screw dislocations (Urabe and Weertman 1975).To handle the events of dislocation exit from and re-entry to the simulation box,re¯ection boundary conditions were applied (Zbib et al. 1998). Connection betweenthe stress and the strain is through the rate of plastic strain increase produced bydislocation unit processes. The observed stress±strain curve is, then, a result of thedynamic equilibrium between the external straining conditions and the internal rateof dislocation multiplication and motion. In the present simulations, we decidedagainst introducing any pre-existing sources, hoping to examine what kind of plasticresponse the initial dislocation microstructure itself can produce.

The dislocation arrangement attained during the DD simulation is shown in®gure 2 (a), and the overall stress±strain response in ®gure 2 (b). The latter showswell-resolved plastic yield behaviour, with the yield stress of 130 MPa, in reasonableagreement with experimental observations (Lassila et al. 1999). The characteristiccross-grid pattern is also clearly observed and should be compared with the trans-mission electron micrograph of the deformed specimen given in ®gure 2 (c). Thecross-grid consists of two sets of intersecting screw dislocations with 1

2‰111Š and

12‰11·11Š Burgers vectors respectively, viewed plane on along the ‰·1101Š direction. In

the course of the deformation, the net dislocation density increased fourfold. What isremarkable is that no dislocation sources were introduced in the initial microstruc-ture and yet multiplication has clearly taken place.

To examine the multiplication mechanisms in more detail, we performed a seriesof simpler simulations in which all but one dislocation, randomly selected from thesame initial microstructure, were removed from the simulation box. The remainingdislocation was then subjected to a constant uniaxial stress applied along the [001]axis. The result shown in ®gure 3 shows that, under a stress of 200 MPa, multi-plication sources develop even on a single dislocation. The sequence begins withthe motion of edge superjogs along the dislocation line. The superjogs moving inthe same glide plane and in the same direction merge, occasionally forming largersuperjogs. At the same time, superjogs gliding on two diŒerent planes can move inopposite directions, collide and form topologically constrained con®gurations con-sisting of two superjogs unable to glide past each other. These pinning points chopdislocations up into smaller segments and facilitate further superjog pile-up, coales-cence and extension. The pinning points can move along the line in response to theunbalanced force exerted by the pile-ups pushing against each other. Figure 3 shows

Dislocation multiplication in bcc Mo 599

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that these pinning points persist over the entire simulation, which suggests that thelooping process around the sources is continuous. The superjog clusters formed bycollision and coalescence are in a quasi-equilibrium state under constant loadingconditions, where dislocations lower internal energy by interaction and motion.

At a lower stress (50 MPa), superjog motion and pile-up proceed unabated butmultiplication does not occur (®gure 4). The reason for this is that, at such a lowstress, the force acting on the screw segments is below the threshold of frictionalstress. On the other hand, superjogs can still move along the direction of screwdislocation. This is due to the low frictional stress for edge character superjogs,which continue to move until they are pinned by interaction with other superjogs

600 M. Rhee et al.

Figure 2. (a) A snapshot of the simulated microstructure. (b) The simulated stress±strainresponse in the inset. (c) TEM micrograph showing characteristic cross-grids of screwdislocations.

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moving from the opposite direction. By themselves, accumulation and coalescence ofsuperjogs in the pile-ups appear to be insu� cient to reach critical conditions for thesource initiation.

That screw motion is important for dislocation multiplication is con®rmed by theresults obtained in a high stress simulation as shown in ®gure 5. Here, dislocationsbegin to bow almost immediately after the stress is applied and continue to bow andwrap around the superjogs. Consequently, a large number of small dislocation loopsare produced by several multiplication sources activated almost simultaneously,

Dislocation multiplication in bcc Mo 601

Figure 3. Snapshots of dislocation multiplication sources at 200 MPa along [001].

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602 M. Rhee et al.

Figure 4. Behaviour of a single dislocation at 50 MPa.

Figure 5. Snapshots of behaviour of a single dislocation under stress at 1 GPa.

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resulting in a higher rate of multiplication. The rate at which superjogs move andcollide at higher stress is much less than the case at lower stress (200 MPa). Ingeneral, at all stress levels, multiplication begins if and when superjog pile-up andcoalescence and screw dislocations glide combine to produce superjogs long enoughfor the applied stress to overcome the frictional stress, line tension and mutualattraction of dislocation segments, as shown schematically in ®gure 6.

} 4. Mechanistic understandingMechanistic understanding of dislocation multiplication involves two distinctly

diŒerent issues. On the one hand, to begin producing fresh dislocations, it is neces-sary to examine critical con®gurations for dislocation sources. On the other hand,understanding the scenarios in which such critical con®gurations can form is just as

Dislocation multiplication in bcc Mo 603

Figure 6. Formation of a pinning point for multiplication source by superjogs under stress.

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important. Although the multiplication behaviour observed here may be viewed asbeing similar to the `wandering’-source mechanism considered earlier by Louchetand Kubin (1975) and Louchet et al. (1979), the pathways leading to source forma-tion are very diŒerent. In the wandering source mechanism, the superjog of criticalheight required for the source to operate was assumed to be formed by cross-slip.However, our simulations suggest a diŒerent scenario in which stress-inducedmigration and coalescence of like-sign superjogs result in gradual build-up of largersuperjogs, similar to the mechanism of elementary jog aggregation considered byLouat and Johnson (1962) . In both cases, the con®guration from which the sourcesbegin to operate consists of two dislocation arms bowing under stress, separated by asuperjog whose length exceeds the critical value dc.

The analysis given by Louchet and Kubin (1975) and Lochet et al. (1979) showedthat the major source in Mo at low temperatures is due to cross-slip at large strains,omitting the detailed information about the initial microstructures. The dynamicsource generated by the motion of clusters of superjogs mentioned in this letter isunique because of the ability of superjogs to coalesce by moving along the dislocationline. This mechanism is distinctly diŒerent from the FR or other `®xed’ sources fordislocation multiplication. It is possible that, if the occasional dense tangles of disloca-tions were accounted for in our simulations, multiplication sources of a diŒerent naturewould have appeared, including the ®xed sources. On the other hand, the featuresresponsible for the occurrence of the wandering sources are realistic for the as-annealed specimens of Mo. This, combined with the fact that the simulated yieldstress is close to the experimental value, is signi®cant in indicating that the dynamicmultiplication sources formed by the clusters of superjogs may alone be su� cient fordislocation multiplication in the deformation conditions explored here.

An important and still unresolved issue in the dislocation theory of crystalplasticity is how much the stress±strain behaviour depends on the nature of initialdislocation sources. In a typical DD simulation, sources are introduced in the formof dislocation segments pinned at the ends. The origin of such pinning points is leftunspeci®ed, although presumably they can be associated with the nodes of disloca-tion junctions or impurity precipitates. Sometimes, to avoid the ambiguity associatedwith unphysical termination of the segments inside the crystal, prismatic loops areintroduced (Ghoniem 2000). In both cases, the sources are introduced ad hoc withthe purpose of generating large number of dislocations during the early stages ofplastic deformation. The argument in support of such an arbitrary construct is thateventually some other natural multiplication mechanisms should take over in thelater stages and the simulated system would have no memory of the initial arti®cialsources.

Ironically, the maximum plastic strain a DD simulation can produce has beenlimited to fractions of 1% and, at these low strain values, the overall behaviour isstill sensitive to the nature of the initial dislocation sources. This was exactly why inthe present simulations we decided to avoid any ad hoc sources but attempted toreproduce faithfully the initial grown-in dislocation microstructure. Still, the dislo-cations found a way to multiply, producing overall initial yield behaviour similar tothe experiment. This observation does not preclude other types of dislocation sourcefrom becoming active in appropriate conditions, such as FR sources pinned atprecipitate particles. What our results do show is that, when properly accountedfor, the initial dislocation microstructure contains the features (superjogs) fromwhich dislocation sources can develop.

604 M. Rhee et al.

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} 5. ConclusionIn summary, we have examined the deformation behaviour of bcc metals at room

temperature by direct DD simulations on the 10 mm scale. The initial dislocationarrangements used in our simulations closely match the statistics of grown-in dis-location microstructures in pure annealed Mo single crystals. Contrary to commonDD simulation practice, we did not introduce any ad hoc multiplication sources, andyet the dislocation density increased fourfold during straining. Careful analysisrevealed that sources form in the initial stages of deformation from the grown-insuperjogs and can move, coalesce and pin each other under stress. Source operationbegins once a critical con®guration for line extension is reached, depending on theapplied stress. Screw dislocations produced by such sources form a cross-grid micro-structure characteristic of bcc transition metals deformed at low temperatures.Likewise, the overall yield behaviour agrees closely with experimental data.Dislocation con®gurations become more isotropic as the temperature increases.The behaviour of the dynamic source at high temperature is not studied in thispaper and remains to be investigated. However, one must introduce other importantingredients which aŒect the dislocation motion at high temperatures, that is non-conservative motion caused by diŒusion process.

ACKNOWLEDGEMENTS

One of the authors (M.R.) would like to thank H. M. Zbib and J. P. Hirth forhelpful discussions. This work was performed under the auspices of the USDepartment of Energy by the Lawrence Livermore National Laboratory undercontract W-7405-ENG-48.

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solid St., 17, 1814.Louchet, F., and Kubin, L. P., 1975, Phys. Stat. sol. (a), 56, 169.Louchet, F., Kubin, L. P., and Vesely, D., 1979, Phil. Mag. A, 39, 433.Louat, N., and Johnson, C. A., 1962, Phil. Mag., 7, 2051.Orowan, E., 1934, Z. Phys., 89, 604.Polanyi, M., 1934, Z. Phys., 89, 660.Prekel, H. L., and Conrad, H., 1968, Dislocation Dynamics, edited by A. R. Rosen®eld, G.

T. Hahn, A. L. Bement, Jr, and R. I. JaŒe (New York: McGraw-Hill), pp. 431±451.Rhee, M., Zbib, H. M., Hirth, J. P., Huang, H., and Diaz de la Rubia, T., 1998, Modelling

Simulation Mater. Sci. Engng, 6, 467.Schwarz, K. W., and Tersoff, T., 1996, Appl. Phys. Lett., 69, 1220.Taylor, G. I., 1934, Proc. R. Soc. A, 14, 362.Urabe, N., and Weertman, J., 1975, Mater. Sci. Engng, 18, 41.Zbib, H. M., Rhee, M., and Hirth, J. P., 1998, Int. J. Mech. Sci., 40, 113.

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