Discrete dislocation dynamics simulations to interpret plasticity...

International Journal of Plasticity 22 (2006) 2091–2117

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Discrete dislocation dynamics simulationsto interpret plasticity size and surface effects

in freestanding FCC thin films

H.D. Espinosa a,*, M. Panico a, S. Berbenni a,1, K.W. Schwarz b

a Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road,

Evanston, IL 60208-3111, USAb IBM Watson Research Center, Yorktown Heights, NY 10598, USA

Received 26 June 2005; received in final revised form 27 December 2005Available online 6 March 2006

Abstract

Strong size effects have been experimentally observed when microstructural features approach thegeometric dimensions of the sample. In this work experimental investigations and discrete disloca-tion analyses of plastic deformation in metallic thin films have been performed. Columnar grainsrepresentative of the film microstructure are here considered. Simulations are based on the assump-tions that sources are scarcely available in geometrically confined systems and nucleation sites aremainly located at grain boundaries. Especially, we investigated the influence on the mesoscopic con-stitutive response of the two characteristic length scales, i.e., film thickness and grain size. The sim-ulated plastic response qualitatively reproduces the experimentally observed size effects while themain deformation mechanisms appear to be in agreement with TEM analyses of tested samples.A new interpretation of size scale plasticity is here proposed based on the probability of activatinggrain boundary dislocation sources. Moreover, the key role of a parameter such as the grain aspectratio is highlighted. Finally, the unloading behavior has been investigated and a strong size depen-dent Bauschinger effect has been found. An interpretation of these phenomena is proposed based onthe analysis of the back stress distribution within the samples.� 2006 Elsevier Ltd. All rights reserved.

0749-6419/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijplas.2006.01.007

* Corresponding author. Tel.: +1 847 467 5989; fax: +1 847 491 3915.E-mail address: [email protected] (H.D. Espinosa).

1 Present address: Laboratory of Physics and Mechanics of Materials, CNRS, ENSAM, Technopole, 57078Metz Cedex, France.

mailto:[email protected]

2092 H.D. Espinosa et al. / International Journal of Plasticity 22 (2006) 2091–2117

Keywords: Size effects; Discrete dislocation dynamics; Thin films; Plasticity; Grain boundaries; Frank–Readsources; Bauschinger effect

1. Introduction

Thin film plasticity is currently an active field of research essentially for two reasons: (i)metallic thin films are widely used in electronic components; (ii) the plastic properties ofthin films differ significantly from the properties of the corresponding bulk materialsdue to the reduced length scales of the microstructure, which become comparable to thegeometric structural dimensions. Metallic thin films on substrates usually exhibit a flowstress which is an order of magnitude higher than the same material in bulk form and thisflow stress increases with the decrease of the film thickness (Arzt, 1998; Spaepen, 2000;Arzt et al., 2001).

So far, thin film plasticity has mostly dealt with films (essentially Al and Cu) deposited onSi substrates. Following the pioneering work of Nix (1989) for single crystalline films, sev-eral authors analyzed the response of polycrystalline thin films; they focused on the effect offilm thickness, grain size and orientation, on the stress/strain response and other importantfeatures like the level of thermal stresses as a function of the thickness and the effect of pas-sivation layers (Venkatraman and Bravman, 1992; Yu et al., 1997; Keller et al., 1998; Bakeret al., 2001; Hommel and Kraft, 2001). Recently, Baker et al. (2003) found an ‘‘anomalous’’large Bauschinger effect for thin Cu films on substrates when the films were exposed to air.

For polycrystalline free-standing thin films, the experimental data and setups are morelimited (Keller et al., 1996; Huang and Spaepen, 2000; Espinosa et al., 2003; Haque andSaif, 2004). Espinosa and co-workers (Espinosa and Prorok, 2001; Espinosa et al., 2003,2004), using the membrane deflection experiment (MDE), identified major size effects inthe mechanical properties of free-standing submicron FCC thin films (electron beam evap-orated Au, Cu, and Al). These films were polycrystalline in nature with thicknesses rang-ing from 200 nm to 1 lm and were tested by applying macroscopic pure homogeneous

uniaxial tension, i.e., in the absence of any macroscopic strain gradient. The average grainsize (about 200 nm) was independent of the film thickness. This feature is quite importantto eliminate the so-called Hall–Petch effect. The reader should note that for the case ofelectron-beam evaporated Cu, similar results concerning the deviation from linear elastic-ity and onset of plastic flow were obtained using a micro-tensile device with in situ TEMby Keller et al. (1996). These authors did observe dislocation nucleation and emissionfrom grain boundaries consistent with current understanding of plastic flow in metals.

Haque and Saif (2004) also explored size effects in FCC metal films such as sputteredpure Al and Au, by measuring the tensile stress–strain response of submicronic (downto a thickness of 0.1 lm) free-standing films subjected to loading and unloading. However,in their experiments, the grain size varied with film thickness, which makes the interpreta-tion more complex. The average grain size in the investigation was 100 nm or smaller.Upon performing in situ TEM observations, they did not notice dislocation activity foran average grain size lower than 50 nm, and they concluded that a grain boundary-basedmechanism was the dominant contribution to deformation. These observations are inagreement with the results obtained for ‘‘bulk’’ nanocrystalline materials, either throughin situ TEM, or by means of large scale molecular dynamics simulations, which are usuallyperformed as a ‘‘guide’’ to experiments (Kumar et al., 2003).

H.D. Espinosa et al. / International Journal of Plasticity 22 (2006) 2091–2117 2093

Balk et al. (2001) used in situ TEM to study the thermomechanical behavior of poly-crystalline Cu thin films with a grain size on the order of 200 nm. They observed in thecooling phase emission of dislocations from a grain boundary triple junction and subse-quent glide on a (111) plane parallel to the film surface.

These experimental results tend to prove the idea that dislocation sources are scarcelyavailable in geometrically confined systems. In this paper, we discuss new results fromboth the experimental and the modeling point of view. A new interpretation of size scaleplasticity is proposed based on the reduced number of available dislocation sources whensize is decreased. When the film thickness and particularly the number of grains throughthe thickness are reduced, the mechanical behavior of thin films becomes more and moreaffected by the probability of finding dislocation sources.

Efforts to understand and to model plasticity size effects have emerged very rapidly inthe last decade. Two different approaches have been chosen to interpret and capture sizeeffects.

The first approach based on continuum mechanics has been initiated by phenomeno-logical strain gradient plasticity (Aifantis, 1992; Fleck and Hutchinson, 1993; Fleck andHutchinson, 1997) and later on by ‘‘mechanism-based’’ strain gradient plasticity (Nixand Gao, 1998; Gao et al., 1999; Huang et al., 2000). Both theories have been appliedsuccessfully to non-uniform macroscopic straining experiments (micro-indentation, tor-sion or bending) for systems with characteristic dimensions from 0.1 to 100 lm. How-ever, further insight is still needed to use strain gradient plasticity for describing thesize dependent mechanical behavior of thin films in the submicronic range in the absenceof macroscopic strain gradient (Hutchinson, 2000). Recently, Bazant et al. (2004) havealso proposed a continuum-based model which suggests the existence of a boundary layerof constant thickness located at the film surfaces and originated by the deposition processon the substrate. The ‘‘boundary layer model’’ is a phenomenological model based on anon uniform initial yield stress distribution in the film due to a gradient of dislocationdensity in the boundary layer. It has been shown that this analytical model based on sim-ple assumptions captures the experimental results of Espinosa et al. (2004) in pure ten-sion reasonably well. However, the experimental identification of such boundary layerremains elusive.

The second approach to plasticity size effects is to perform large scale simulationsbased on molecular dynamics or on dislocation dynamics. This is the method adoptedin this paper for identifying the relevant mechanisms through which the microstructureand the film characteristic dimensions lead to the observed size effects in pure tension.Following the conclusions of Espinosa et al. (2004), the misfit dislocation strengtheningmechanism (Nix, 1989) is not present in polycrystalline free-standing unpassivated FCCthin films. This proves the existence of another mechanism, as previously observed byVenkatraman and Bravman (1992). Emission and motion of dislocations are key fea-tures to explain the observed experimental results as discussed in Espinosa et al.(2005). Thus, from the modeling point of view, there is a need to account for the spa-tio-temporal occurrence and evolution of discrete events such as the ones given by dis-locations in confined samples. The self-organization of dislocations within a smallstructure is well captured using discrete dislocation dynamics (DDD) (e.g. Liu et al.,2000; Madec et al., 2003). Likewise, the influence of image stresses on this self-organi-zation was studied by Khraishi and Zbib (2002a) and Ohashi (2004). ExperimentalTEM observations can be conducted to identify the mechanisms and validate the


DDD calculations. The numerical technique is computationally intensive, but we believeit may allow determining the relevant mechanisms responsible for size and grain bound-ary effects. Among the different codes recently developed (Kubin et al., 1992; Van derGiessen and Needleman, 1995; Verdier et al., 1998; Zbib et al., 1998; Schwarz, 1999;Ghoniem et al., 2000; Weygand et al., 2002; Bulatov et al., 2004; Deshpande et al.,2005; Nicola et al., 2003), the PARANOID code (Schwarz, 1999) is used in this paper.The code has been modified to account for the presence of grain boundaries, to computethe plastic strain, and to define different initial dislocation source configurations in orderto interpret grain boundary and size effects. Note that boundary effects in DDD imple-mentations were also explored by Fivel et al. (1996), Fivel and Canova (1998), Ghoniemand Sun (1999), Khraishi et al. (2001), Yasin et al. (2001), Khraishi and Zbib (2002b),Yan et al. (2004), and Nicola et al. (2005).

This paper is based on the grain boundary source model first proposed by Espinosaet al. (2005). Here we provide additional computational details and examine the modelin the context of grain size and Bauschinger effects. The paper is organized as follows: Sec-tion 2 discusses the new original experimental investigations performed to characterize themechanical behavior and plasticity size effects observed in free-standing thin films. In Sec-tion 3 the DDD method is presented in the framework of the fully parallelized codePARANOID. The numerical procedure, the problem definition, and the assumptions usedare presented and discussed. Section 4 concerns numerical results obtained by DDD. Inparticular, we examine the effect on the material mesoscopic response of the system char-acteristic dimensions, i.e. film thickness, h, and grain size, d. This section also illustratesthe presence of size scale effects during unloading. Finally, conclusions of the present studyare drawn in Section 5.

2. Experimental results and TEM observations

2.1. MDE experiments and TEM/SEM observations

As mentioned before, the MDE experiment is a recently developed technique able tocharacterize the mechanical behavior of free-standing thin films. Using this experimentalprocedure Espinosa et al. (2003, 2004) showed that for electron beam evaporated Au theonset of plasticity is a strong function of film thickness, with thinner films exhibiting amuch higher yield stress. Furthermore, a major transition in failure mode was observedwhen the thickness decreased from 1.0 to 0.5 lm, which implies different strengtheningmechanisms. For electron beam evaporated Cu and Al membranes, engineering stress/strain curves obtained for specimen thicknesses of 0.2 and 1.0 lm also showed strongthickness dependence, like for Au. In all these experiments, the dimensional constraintappears clearly to be the thickness because the grain size is kept approximately the samefor all the investigated film thicknesses. As previously stated, Keller et al. (1996) obtainedsimilar results regarding the flow stress and the rate of hardening for Cu specimens withthicknesses of 0.2 and 1.7 lm.

Postmortem transmission electron microscopy (TEM) and scanning electron micros-copy (SEM) observations on MDE tested films were performed to gain insight into the ori-gin of the size scale experimental observations. Samples for plan view TEM observationswere prepared by placing of the membranes on TEM grids followed by ion milling of thefilm from both top and bottom sides. Cross-sectional TEM samples were prepared by


focused ion beam (FIB) processing. For this purpose, the membrane specimen was firstglued to an approximately 50-lm-wide Si block to provide a handling base, and thenplaced into a dual SEM/FIB chamber. Platinum (Pt) was deposited on top of the film sur-face to protect it from excessive FIB etching. The area to be observed was thinned withFIB on both sides to an electron-transparent thickness following the method of Ishidaand Sato (2003). Both top and cross-sectional samples were observed in a JOEL2010TEM at room temperature.

SEM observation of the 0.3-lm-thick Au films revealed no major plastic activity as evi-dent from the smoothness of the surface of the specimen near the fracture plane (Fig. 1A).

Fig. 1. Study of the plastic activity of 0.3-lm-thick Au films (A) (SEM picture showing the fracture surface) and1-lm-thick Au films, (B) (SEM picture showing large deformation slip bands close to the fracture edge), (C)Cross-sectional TEM view of fracture region, (D) high aspect ratio grain exhibiting dislocation network.


The TEM cross-sectional view displays some residual dislocation patterns within a few iso-lated grains, with a decrease in dislocation activity as we depart from the fracture surface.TEM images before testing show the initial grain structure of the film, with grains contain-ing annealing twins and basically free of dislocations. For 1-lm-thick Au films, a signifi-cant plastic activity is evident in the SEM images of the specimen surface (Fig. 1B). Severaldeformation bands are also noted. This is consistent with the observed softening in thestress–strain behavior Espinosa et al. (2004). The TEM cross-sectional views of the samespecimen (Figs. 1C and D) show residual dislocation patterns only within grains with highaspect ratio h/d. This suggests that dislocations have moved mainly to the free surfaces orgrain boundaries and that they have formed networks only when the probability of form-ing dislocation junctions was increased. We will come back to this feature later in the dis-cussion of the DDD simulation results. Again, our TEM observations are in agreementwith the ones reported by Keller et al. (1996). In their in situ TEM studies, the films exhib-ited limited plastic activity with dislocation nucleation at grain boundaries. Dislocationswere found to sweep the grains with very short glide distances, which did not allow signif-icant dislocation interaction.

2.2. Self-similar experiments

To further investigate the plasticity size scale effect, e.g., in Cu films, we tested self-

similar specimens until failure in tension occurred. The experiments were conducted usinga miniature tensile stage (Fullam, Inc., Latham, NY) equipped with a 2000 lb load cell.The specimens were machined out of a copper foil and processed to have all the charac-teristic dimensions (thickness, grain size) 1000 times larger than the specimens used inthe MDE. As a result, the number of grains through the thickness was about the sameas in the thin films but the area of grain boundaries was increased 1 million times! Thesetup was placed under an optical microscope and pictures were taken throughout thestretching process to keep track of deformations at the grain level. The specimen surfaceroughness was used to determine local strains by means of digital image correlation. Fig. 2

0

200

400

600

800

1000

0 0.1 0.2 0.3 0.4 0.5

True Strain

Tre

u S

tres

s (M

Pa)

Coarse-Grained Cu(200µm)

Thin film Cu (200nm)

1 mm

0.3 mm

1 μm

0.3 μm

Fig. 2. Self-similar experiments in pure tension for Cu films: comparison between the mechanical response ofsubmicron thin films (MDE) and millimeter thick coarse-grained films (Fullam tensile stage).


compares the stress–strain responses for coarse-grained films with thicknesses of 0.3 and1 mm (using the Fullam tensile stage) with the ones for thin films with thicknesses of 0.3and 1 lm (using the MDE). No size effect is observed in tension for the coarse-grained

films, with an average grain size of 200 lm, whereas a dramatic size effect is observed inthe submicronic thin films as previously stated. In Fig. 3 some snapshots of the deforma-tion process for the coarse-grained films are displayed. Fig. 3A shows the initial grainstructure, while Fig. 3B is taken at a deformation of approximately 10%. The featuresare similar to plastic deformation within bulk polycrystalline metals where dislocationsnucleate and propagate at relatively low levels of deviatoric stress, forming multiple sliplines across the grains. Figs. 3C and D show the measured local strain field, by meansof digital image correlation, at an applied external strain of 10% and at the end of the test,respectively. One may notice the large values of local strain and some degree of grain rota-tion, which are typical for bulk polycrystalline plasticity. In the case of millimeter-thickfilms, due to the increased grain boundary area and the enlarged probability to find intra-granular dislocation sources, defect nucleation does not control the plastic activity anymore. Furthermore, dislocation multiplication leading to deformation bands within thegrains is clearly observed. For this reason the macroscopic bulk response is achievedand no significant statistical variations in the sample behavior occur.

Fig. 3. Deformation of coarse-grained films: (A) initial grain structure, (B) grain configuration for an externalapplied strain of approximately 10%, (C) local strain field, exx, corresponding to an external applied strain ofapproximately 10%, (D) local strain field, exx, at failure.


3. Discrete dislocation dynamics

3.1. General description of the code

In this work we have employed the PARANOID code to perform discrete dislocationdynamics (DDD) simulations. PARANOID is a 3D code that follows the motion of dis-location lines modeled as smooth curves passing through a sequence of tracking points(Schwarz, 1999). There are two main limitations in our calculation of dislocation motion:(i) we consider only the glide of perfect dislocations in a linear, isotropic medium; (ii) theproblems of dislocation self-interaction and interaction with free surfaces are treated in anapproximate way which introduces errors on the order of 10–20% (Schwarz, 1999). To thislimit of accuracy, the code has previously provided quantitatively accurate predictions ofdislocation configurations in small epitaxial islands (Liu et al., 2000), and of the residualmisfit dislocation structure in plastically relaxed single-crystal layers (Schwarz et al., 2004).The polycrystal line films considered have not yet been as cleanly characterized in terms ofa well-defined computational model, and our present aim is the more qualitative one ofunderstanding the main deformation mechanisms rather than achieving an accurate com-parison with experimental results.

Like the other mainstream DDD codes mentioned earlier, PARANOID is able to cap-ture 3D effects like line tension effects, and to include the full 3D interactions between dis-location lines. Moreover, the code is able to run in a highly resolved manner, in which thepoint spacing can be reduced down to atomic dimensions. Recently, Schwarz (2003) intro-duced approximation rules for strong interactions such as at junctions, line–line crossingregions, and dipoles. The results obtained through these approximations have been foundin good agreement with fully resolved calculations.

PARANOID is a parallel, load-balanced program that runs on SP-type or Linux-clus-ter-type distributed memory machines. It is written in Fortran-77 and uses MPI calls forcommunication between processors.

3.2. Dislocation dynamics

The evolution of the dislocation configuration is obtained by tracking the motion of thediscrete meshing nodes. An explicit numerical scheme is used. The local stress tensor isgiven by the stress contribution due to the applied strain, stresses arising from the dislo-cation themselves, and corrections due to boundary effects. Once the stress tensor has beencomputed at each node, the Peach–Koehler equation is used to determine the force perunit length acting locally on the dislocation at that position. The Peach–Koehler force,fg, to move a dislocation element dl in the glide pane is given by:

fg ¼ ðbirijnjÞn� dl; ð1Þwhere b is the Burgers vector of the dislocation, n is the unit normal to the glide planeand r is the local stress tensor. The singularity in the self-interaction stress is regularizedby means of the Brown splitting procedure (Brown, 1964), which has been modified inorder to avoid the unstable behavior pointed out by Schwarz (1999). Each node is thenmoved in the direction perpendicular to the local line tangent following the simplest re-sponse model, i.e., with a velocity proportional to the local force component given by(1):


v ¼ fg

dl B. ð2Þ

B is a phonon viscous drag coefficient and a commonly used value for copper at room tem-perature B = 1.5 · 10�5 Pa s is here employed following Verdier et al. (1998). Only a vis-cous drag type equation of motion is considered through the constant B. The inertialforces, sound velocities and internal forces arising from interactions of dislocations withobstacles like the Peierls barriers are neglected. A complete thermodynamic descriptioncan be found in Hirth et al. (1998) and Zbib et al. (2002).

3.3. Problem definition and boundary conditions

In contrast with plastic deformation in bulk materials, where interactions betweenmany (forest) dislocations may lead to self-organization such as cell structures, the presentsimulations involve a much smaller number of dislocations.

For each simulation, a single FCC crystal representing a grain within a polycrystallinefilm of copper is considered. In the simulations the following material parameters for cop-per have been employed: l = 42.3 GPa (shear modulus), m = 0.3 (Poisson ratio),a = 0.36148 nm (lattice parameter), b = 0.2556 nm (magnitude of Burgers vector). Sincethe films tested by MDE have a pronounced Æ111æ fiber texture, the orientation of the sin-gle crystal is chosen so that the crystallographic h�1�11i-direction is parallel to the global Z-axis (axis normal to the film plane) as described in Fig. 4B. The two other crystallographicaxes respectively parallel to the global X- and Y-axes are chosen arbitrarily. The grain iseither represented by a cubic box with edges equal to the average grain size d = 0.2 lm orby a rectangular box (corresponding to a columnar grain, i.e., only one grain through thethickness) with a square base (0.2 lm · 0.2 lm) and a varying height (i.e., the thickness ofthe film h) in the h�1�11i-direction (Fig. 4A).

In our simulations grain boundaries are modeled as impenetrable surfaces, which is afirst order approximation since dislocation transmission from one grain to another may

Fig. 4. Problem definition for DDD calculations. (A) Columnar grains simulated by rectangular boxes withvarious heights. (B) Present study: single crystal (grain) considered as a rectangular box (thickness: h, grain size: d)with a crystallographic direction h�1�11i perpendicular to the film plane (consistent with the strong Æ111æ texture ofthe film) and with an applied tensile loading parallel to the Æ112æ direction (arbitrary).


occur (De Koning et al., 2002). Top and bottom faces of the box are attractive in order torepresent the effect of free surfaces and image stresses on dislocations.

The box is subjected to a pure tensile loading in the Y-direction at a prescribed compu-tational strain rate _e. In order to limit the strain rate effects inherent to the dislocationdynamics scheme and to run calculations with a reasonable CPU time, we chose a strainrate _e ¼ 6:66� 105 s�1. The effect of the strain rate on the material response has been para-metrically investigated as we will report later.

The computation of the local stress tensor is subjected to corrections when free surfacesand other interfaces are present. For sufficiently thin free standing films, two kind of cor-rections must normally be accounted for. The first one concerns the interactions of dislo-cations with the free surfaces and the second takes into account the interactions betweenthe two free surfaces (top and bottom). This problem has been presented and solved byHartmaier et al. (1999) using the so-called Boussinesq operator and the principle of super-position for boundary conditions (Van der Giessen and Needleman, 1995). Differentapproaches to the more general problem of finding image corrections have been discussedby several authors (Fivel et al., 1996; Fivel and Canova, 1998; Ghoniem and Sun, 1999;Liu et al., 2000; Khraishi et al., 2001; Yasin et al., 2001; Khraishi and Zbib, 2002a,b;Yan et al., 2004; Liu and Schwarz, 2005). In all cases, an accurate accounting for imageeffects, in particularly when dislocations can terminate on some of the surfaces, requiresa massive increase in the computational load. However, a recent detailed study by Liuand Schwarz (2005) has demonstrated that for systems like that considered here, in whichthe dislocations are curved on the scale of the geometry, it is possible to obtain resultsaccurate to the 10–20% level using the following simple approximations: (a) surfaces thatrepel dislocations are treated as impenetrable, while dislocations are allowed to reconnectto surfaces that attract dislocations, such as a free surface; (b) wherever a dislocation inter-sects the free surface, it locally feels the force induced by a free surface on a straight dis-location line (Lothe et al., 1982). The latter assures that the dislocation enters the freesurface at the correct angle. DDD simulations in general approximate the detailed struc-ture of the dislocation core in terms of some simplified continuum model, and are thusthemselves accurate only to the 10–20% level. There is little point, therefore, in goingbeyond these simple approximations, unless one is treating nearly 2D systems where thedislocations have significant curvatures, systems which develop buildups of large-scale cor-related dislocation arrays leading to large mean fields, like the misfit array in a single-crystal strained layer. A good illustration of how well this seemingly oversimplifiedapproach can reproduce observed dislocation behavior in a small structure similar to agrain is given in Liu et al. (2000).

3.4. Plastic strain tracking and dislocation densities

The formula to compute the plastic strain tensor due to a dislocation at any point x isgiven by Mura (1987, p. 46):

epijðxÞ ¼ �

1

2ðbinj þ bjniÞdðS� xÞ; ð3Þ

where S is the dislocation slip plane and d(S � x) is the one-dimensional Dirac delta func-tion in the normal direction of S, which is unbounded when x is on S and zero elsewhere.The sign ‘‘�’’ is due to the use of the right-handed screw convention for the normal direction


of S. Using (3) and the same sign convention, the total macroscopic plastic strain increment

denoted by depij yields:

depij ¼ �

1

2V

XN

k¼1

ðbðkÞi dSðkÞj þ bðkÞj dSðkÞi Þ; ð4Þ

where V is the volume of the crystal, and dS(k) is the algebraic increment of area swept outby the segment of dislocation between mesh points k and k + 1. Initially and at each step,the total dislocation density is also computed. Its expression is:

�qtot ¼P

I lðIÞ

V; ð5Þ

where l(I) is the length of the dislocation segment (I). To identify which are the most activeslip systems for the different initial configurations, the dislocation density per slip system isalso computed for the 12 primary slip systems of the FCC structure for which the Schmidand Boas notation is used in the following.

4. Numerical results and discussion

4.1. Activation stress of a Frank–Read source

As a preliminary analysis we examine the stress needed to activate a Frank–Readsource within a grain (volume source). Such activation stress is a function of two internallength scale parameters, defined in Fig. 5A, i.e., s and l, the Frank–Read source size and itsdistance from the grain boundary towards which it is propagating, respectively. For thispurpose a cubic grain with d = h = 0.2 lm is considered and both parameters, l and s,are normalized to the film thickness. We define the activation stress as the stress needed

0 0.01 0.02 0.03 0.04 0.05 0.060

20

400

600

800

1000

1200

1400

Plastic Strain (%)

Str

ess

(Mp

a)

5*104 s-1

6.66*105 s-1

5*106 s-1

5*107 s-1s h

l

A B

0

Fig. 5. (A) Definition of two material length scale parameters: source size s and source distance to the oppositeboundary l. (B) Strain rate effect on the stress/strain curve for the case of the activation of a Frank–Read sourcein the presence of impenetrable boundaries.


by the source to overcome a critical configuration from which dislocation multiplicationbecomes possible. In order to compute the activation stresses, we have applied a strain rateto the sample. The activation stress corresponds to a plateau in the stress vs. plastic strainresponse of the material. Since this stress value may be affected by the applied strain rate,we investigated the effect of different strain rates on the computed activation stress. In thisstudy a Frank–Read source of size 0.16 lm has been positioned in the center of the grainwith all impenetrable surfaces. The source is located on the (111) plane with a Burgersvector in the h0�11i direction. For this initial configuration the stress vs. plastic strain issimulated for several values of the applied strain rate (Fig. 5B). This curve is characterizedby three different parts which correspond to different phases in the evolution of the source.In the first stage of the simulation, the source is increasing the total swept area withincreasing applied stress; this corresponds to the first part of the curve in which significantplastic strain is produced. Then, the dislocation encounters the impenetrable surfaces andthe stress is rising without any significant increase in the plastic strain (second part of thestress–plastic strain curve). This status is preserved until the so called activation stress isreached and allows the dislocation to overcome a critical configuration beyond which dis-location multiplication becomes possible. This discrete event corresponds to a plateau inthe stress–plastic strain curve, since a sudden large increase in the plastic strain occurs.Fig. 5B shows that the plastic behavior of the crystal is converging for decreasing strainrates. This feature was also identified by Rhee et al. (1998). Therefore, in the calculationsreported in the following, a strain rate of 6.66 · 105 s�1 has been applied in order to reducethe CPU time and at the same time minimize the effect arising from viscosity.

The activation stress has been parametrically investigated by considering three values ofthe ratio s/h (0.1, 0.3 and 0.8) and several values of the distance l. In Fig. 6A, we plot across-section of the grain in which the left and right surfaces are modeled as impenetrable(grain boundaries) and the top and bottom surfaces are modeled as attractive (free sur-faces). In Fig. 6B the activation stress to reach the final configuration displayed inFig. 6A (where only half loop is generated) is plotted. Fig. 6B indicates that for largel/h, the stress needed to activate the dislocation is source-size dependent and, as in VonBlanckenhagen et al. (2001a,b), the smaller the source, the higher the activation stress.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x 10-5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1 x 10-5

0 0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500

l/h

s/h=0.1

s/h=0.3

s/h=0.8

A B

Fig. 6. Plot of the activation stress needed to reach the final configuration in (A) as a function of s/h and l/h (B).


Fig. 6A shows that forming dislocation arms, close to free surfaces, are attracted by thesurfaces further assisting their glide; this was also shown by Khraishi et al. (2001). Onthe contrary, when the source is very close to an impenetrable grain boundary, the latteracts as a strong obstacle and a high stress, independent of s/h, is needed to activate thesource. This result shows that in geometrically confined systems, the yield stress and thehardening rate will be affected by the statistical distribution of sources in terms of bothlength scale parameters l/h and s/h. When the microstructural characteristic dimensions,e.g. source size, are comparable to the geometrical features, such as film thickness andgrain size, confinement exerts a strong constraint and only favorably located sourcesmay be activated. Therefore, the scarce availability of dislocation sources and the reducedprobability of nucleating defects are likely to dominate the plastic process. This may alsoexplain the limited ductility exhibited by these systems in the aforementioned experiments.

While the picture emerging from the analysis of volume sources is somewhat consistentwith the experimentally observed plasticity size effects, there is no TEM evidence of vol-ume sources being active in ultrafine-grained FCC materials, Cheng et al. (2003). It isworth mentioning that in the context of 2D discrete dislocation calculations, Biner andMorris (2003) considered grain boundary sources as the only sources for nucleation ofthe dislocations. In the following we use a statistical model based solely on grain boundarysources as proposed by Espinosa et al. (2005). The reader should not confuse grain bound-ary sources with Frank–Read sources in the sense that the latter are typically volumesources capable of developing a whole glide loop around them and consequently multiplyat relative constant stress. This feature is not present in grain boundary sources as high-lighted by molecular dynamic simulations.

4.2. Effect of film thickness

In this section the focus is to examine the mesoscale constitutive response of a singlecrystal with multiple interacting grain boundary sources. Specifically, we analyze the effectof a change in the film thickness at constant grain size. As previously mentioned, the filmstested experimentally exhibited a few grains through the thickness, as illustrated inFig. 4A. However, in order to overcome the current limitations of DDD, only columnargrains are here considered. Therefore, the computational domain is made of a rectangularbox with a square base of area d2 (with d being the grain size: 0.2 lm in the simulations ofthis section) and a varying height h, representing the thickness of the polycrystalline film(Fig. 4B). This columnar geometry is actually exhibited by some grains of the tested films(Fig. 1C), which will be referred to as ‘‘high aspect ratio grains’’. Samples with four thick-nesses have been considered: 0.2, 0.4, 0.6 and 1 lm, while the grain size was constant andequal to 0.2 lm. Thus, the four samples had aspect ratios, respectively, equal to 1, 2, 3and 5.

In the present DDD simulations of size scale plasticity in thin films, dislocation sourceshave been randomly distributed only at grain boundaries. Therefore, the sources are not ofthe Frank–Read type in the classical sense, because they are not able to generate circularor elliptical pile ups. They are rather characterized by a one-time activation process lead-ing to the formation of one dislocation line at the opposite grain boundary. The simula-tions clearly show that when grain boundary dislocation sources are activated, they shearthe slip plane and are either absorbed by free surfaces or stored at grain boundaries. Thisappears to be consistent with a number of TEM observations and atomistic simulations


reported in the literature. For a grain size of 200 nm, like the one here considered, grainboundary dislocation sources are expected to control the plastic process. As a matter offact, Cheng et al. (2003) proposed the existence of four ranges of grain size in FCC metalsfor which distinct plastic deformation mechanisms exist. In the so-called ‘‘ultrafine’’regime, which is the case of the tested freestanding thin films, grain boundary dislocationsources appear to be predominant. They also suggested that the upper limit of this mate-rial behavior may be up to 1 lm. Only for larger average grain sizes are intragranular dis-location sources (volume sources) expected to contribute significantly to the plasticactivity.

A constant initial dislocation density 2.4 · 104 cm�1 has been used to characterize thegrain boundary, the concept of a constant density of boundary sources being consistentwith atomistic simulations of nanostructured metals (Farkas and Curtin, 2004). We wantto highlight that for a change in the film thickness, a constant area density on the grainboundary also implies a constant volume density, which allows us to compare the averageplastic strain over the volume. For each sample three different runs with different initialconfigurations of defects have been performed. The stress–strain curves for the three cal-culations have been averaged in order to reduce the scatter arising from the statistical nat-ure of the sample. The physical meaning of this averaging procedure consist of taking intoaccount three different grains within the polycrystalline film. The source lengths have beengenerated by means of a normal distribution with a mean value of 0.08 lm and a standarddeviation of 0.02 lm. The parameters of this distribution have been kept constant for allthe thicknesses.

4.2.1. Strain rate analysis

As already mentioned, the computational samples are loaded in pure tensile mode inthe Y-direction by prescribing a strain rate _e. Therefore, before analyzing and compar-ing the results for the samples with different thicknesses, the effect of the adopted com-putational strain rate is investigated. In this analysis the 0.2-lm-thick sample has beenconsidered and tested for several values of the applied strain rate. Fig. 7A shows thestress–strain curves for these computational tests; a high strain rate produces an over-estimation of the activation stresses at which boundary sources are activated (discreteevents which correspond to the sudden drops in the stress–strain behavior). It maybe noticed that the two results at _e ¼ 6:66� 105 s�1 and 5 · 104 s�1 are not very differentfrom both a qualitative and quantitative point of view. This suggests that the materialresponse is converging for diminishing values of the strain rate as discussed by Rheeet al. (1998). In order to validate this hypothesis, another simulation was performedin a ‘‘quasi-static’’ fashion, which was obtained by applying a stepped strain to the sam-ple (step strain of amplitude 0.1) and relaxing it at constant applied strain for suffi-ciently long time intervals. In this fashion, the quasi-static stress corresponding to thevalue of the applied strain was determined. This result is also reported in Fig. 7A inthe form of dots. These dots are in full agreement with the continuous stress–straincurve obtained for _e ¼ 5� 104 s�1. However, simulations performed at this strain raterequire very large computational time while the objective of the present study is to iden-tify the main deformation mechanisms underlying size scale plasticity in small volumes.Therefore, in the following simulations a strain rate of 6.66 · 105 s�1 has been adoptedin order to compromise between reduction of strain rate artifacts and reasonable com-putational time.

0 0.2 0.4 0.6 0.8 10

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Strain (%)

5*106 s-1

6.66*105 s-1

5*104 s-1

Quasi static

0 0.2 0.4 0.6 0.8 10

100

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300

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500

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Strain (%)

h=0.2 μm (h/d=1)h=0.4 μm (h/d=2)h=0.6 μm (h/d=3)h=1 μm (h/d=5)

A

B

Str

ess

(Mp

a)S

tres

s (M

pa)

Fig. 7. (A) Stress–strain response computed for several values of the applied strain rate. The dots represent therelaxation stresses corresponding to the applied strain values. (B) Stress–strain curves for samples with differentthicknesses and aspect ratios. Initial grain boundary dislocation density: 2.4 · 104 per cm. The error barsrepresent the statistical scatter of different runs at the end of the simulation for each sample.


4.2.2. Stress–strain responseFig. 7B displays the averaged stress–strain curves for the samples with varying thickness

and aspect ratio. The error bars represent the statistical scatter of the three runs at the endof the simulation for each sample. The stress drops in the stress–strain curves correspondto dislocation emissions from boundary sources. The results in Fig. 7B, which wereobtained for an initial source density of 2.4 · 104 cm�1, reproduce the experimentally-observed size effects for the thicknesses h = 0.2 lm, h=0.4 lm and h = 0.6 lm. For thesethicknesses ‘‘thinner is stronger’’ and the simulations clearly predict differences in the yield


stress and hardening rate. This feature is not manifested for the 1-lm-thick sample, which,in the early stages of the simulation, presents a response very similar to the 0.6-lm-thicksample and exhibits a larger hardening rate after a deformation of approximately 0.6%.

The following interpretation of the aforementioned results is proposed. Thinner filmspresent less grain boundary area and, consequently, a lower probability to activate bound-ary sources at a certain stress level. This is consistent with our self-similar experiments,which do not exhibit any size effect as a consequence of the large grain boundary areaavailable for defect nucleation. Moreover, for smaller thicknesses, there is a higher prob-ability for dislocations to encounter free surfaces and to be partially adsorbed; thus, theavailable free path for dislocations is reduced by the presence of free surfaces and onlya limited amount of plastic strain may be produced. However, the effect of these two fea-tures is diminished when the film thickness and the grain aspect ratio are increasing. Infact, for larger aspect ratios (h = 1 lm, h/d = 5), the probability for dislocations to pileup at grain boundaries and intersect each other by forming networks and junctions ishigher. We refer to this characteristic behavior as the ‘‘aspect ratio effect’’, and it reflectsan increase of the internal stress (back stress) during the deformation process, whichcauses an obstacle to further dislocation nucleation and leads to an increase in hardeningrate. Fig. 8 shows snapshots of the dislocation structure for an applied strain of 0.8% inthree different samples: 0.2, 0.6, and 1 lm. Only active dislocation sources are plotted.These snapshots show the formation of junctions in the 0.6-lm-thick and the 1-lm-thicksamples. This finding is consistent with our TEM observations (Fig. 1D), which presentevidence of networks formation only in the grains with highest aspect ratio.

In order to confirm our interpretation, similar simulations were performed for a smallerinitial dislocation density of about 0.8 · 104 cm�1. The results are illustrated in Fig. 9. Asexpected, a smaller initial density decreases the possibility of junctions and network forma-tion; therefore, in this case, the saturation of the size effect does not occur. It is evidentfrom this analysis that the initial source density is the main parameter governing the crit-ical aspect ratio for which a reversal in the hardening rate is manifested.

In the following two sections a more detailed analysis of the thickness effect for the casewith density 2.4 · 104 cm�1 will be performed. To achieve this we will consider the evolu-tion of the dislocation density on the 12 slip systems of the FCC structure and the historyof the internal stresses during the straining process. In this regard we have selected for eachsample the most representative run, i.e., the simulation that was closest to the averageresponse of the sample.

4.2.3. Analysis of dislocation density evolution

The dislocation density may be computed at any time during the simulation by theexpression in (5). The same definition may be applied to the 12 slip systems by defininglocal densities which evolve independently in the simulation. In order to identify the 12 slipsystems we have adopted the Schmid and Boas notation which is recalled in Table 1. In thesame table the Schmid factors have also been computed with regard to the loading direc-tion prescribed in our calculations. In Fig. 10 the dislocation densities on each slip systemare plotted as functions of the applied total strain, for varying thickness. In all four cases,the most active slip systems are the A3 and D1, as expected, having the highest Schmidfactor. Two main features may be noticed: (i) thicker samples exhibit a higher degree ofactivation of the primary slip systems (A3,D1); (ii) in thicker samples, secondary slip sys-tems (A6,B4,D6), with Schmid factor 2/3, participate earlier in the plastic process. This is

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A

Fig. 8. Final dislocation configurations for an applied strain of 0.8% (left image: top view, right image: 3D view).(A) 0.2 lm sample; (B) 0.6 lm sample; (C) 1 lm sample.


0 0.2 0.4 0.6 0.8 10

100

200

300

400

500

600

700

800

Strain (%)

Str

ess

(MP

a)

h=0.2 μm (h/d=1)h=0.4 μm (h/d=2)h=0.6 μm (h/d=3)h=1 μm (h/d=5)

Fig. 9. Stress–strain curves for samples with different thicknesses and aspect ratios. Initial grain boundarydislocation density: 0.8 · 104 per cm. The error bars represent the statistical scatter of different runs at the end ofthe simulation for each sample.

Table 1Schmid and Boas notations to identify the 12 slip systems of the FCC structure

Schmid and Boas notation Slip plane: (hk l) Slip direction: Æuvwæ Schmid factor

A2 ð�111Þ h0�11i 1/3A3 ð�111Þ Æ101æ 1A6 ð�111Þ Æ110æ 2/3B2 (111) h0�11i 2/3B4 (111) h�101i 2/3B5 (111) h�110i 0C1 ð�1�11Þ Æ011æ 0C3 ð�1�11Þ Æ101æ 0C5 ð�1�11Þ h�110i 0D1 ð1�11Þ Æ011æ 1D4 ð1�11Þ h�101i 1/3D6 ð1�11Þ Æ110æ 2/3


caused by the higher probability of nucleating defects in larger samples. However, asalready noticed, this effect is saturated for the 1-lm-thick sample. This sample exhibits lessplastic activity than the 0.6-lm-thick one, due to the significant raise in internal stressesmanifested in grains with high aspect ratios. This feature is analyzed quantitatively inthe next section.

4.2.4. Back stress analysisThe objective in this section is to analyze the stress field within our samples and espe-

cially to differentiate the contribution coming from the applied external strain and the dis-location themselves (stress that we refer to as ‘‘back stress’’). For this purpose, we computethe local stress field at any fixed point inside the computational box. In particular, we ana-lyze the contribution of the back stress to the Peach–Kohler force acting per unit length ona dislocation on the A3 slip system (Schmid factor 1) in proximity of one of the grain

0 0.2 0.4 0.6 0.8 10

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3

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6

Strain (%)

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rmal

ized

Dis

loca

tio

n D

ensi

ty A2A3A6B2B4B5C1C3C5D1D4D6

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Strain (%)

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loca

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ensi


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4

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6

Strain (%)

No

rmal

ized

Dis

loca

tio

n D

ensi


A B

C D

Fig. 10. Dislocation density evolution on each slip system: (A) 0.2 lm sample; (B) 0.4 lm sample; (C) 0.6 lmsample; (D) 1 lm sample.


boundaries. The evolution during straining of the average profile of this force along thethickness of the film for the four samples is reported in Fig. 11. As expected, the force aris-ing from dislocations is negative, in the sense that it opposes the external stress field.Moreover, the average absolute value of this back stress force is growing with the appliedstrain as a consequence of the increased dislocation density. This result is also summarizedin Fig. 12 where the average absolute value of the force profile is plotted as a function ofthe applied total strain for the four samples. This plot represents the average effect of theback stress on the grain boundary sources for different thicknesses at different appliedstrains.

The average magnitude of the back force per unit length is very similar for all thethicknesses in the early stages of the simulation. However, increasing the deformationproduces an obvious rise in back stress which is more marked in the case of thick sam-ples. As mentioned before, this is a consequence of the high aspect ratio of thicker sam-ples, which results in a higher probability for dislocations to encounter grain boundariesrather than free surfaces, and also to intersect each other and form junctions. Theseevents produce the observed significant increase of the back stress, which is an obstacleto subsequent dislocation nucleation. This explains the fact that the 1-lm-thick sample

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A B

C D

Fig. 11. Evolution of the back stress force profile along the thickness during straining: (A) 0.2 lm sample;(B) 0.4 lm sample; (C) 0.6 lm sample; (D) 1 lm sample.


possesses a larger hardening rate than the 0.6-lm-thick one at the assumed dislocationdensity of 2.4 · 104 cm�1 (Fig. 7B).

4.3. Effect of grain size

In this section, we investigate the influence of the grain size d on the mesoscale con-stitutive response of the columnar grain. For this purpose, a grain with h = 0.2 lm andd = 0.4 lm is simulated and the computed stress–strain response is compared with theone of the cubic sample, i.e., h = d = 0.2 lm. In these calculations, it is not possibleto keep constant both the initial density of sources per unit of grain boundary area,and the initial volume density of sources. Since plastic strain is an average volume quan-tity, we have chosen to preserve the initial volume source density of the samples (4 · 1014

per m2) which is the only way to compare the plastic activity in the two tests. Again,three simulations with different initial configurations have been performed. The averagestress–strain results are compared in Fig. 13, which also shows the scatter of the resultsfor an applied strain of 0.8%. The response of the crystal with larger grain size is signif-icantly softer, and, particularly, the yield stress and hardening rate are much smaller

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.94

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Strain (%)

Fo

rce

per

un

it le

ng

th (

GN

/cm

)

h= 0.2 μm (h/d=1)h= 0.4 μm (h/d=2)h= 0.6 μm (h/d=3)h= 1 μm (h/d=5)

Fig. 12. Average magnitude of the back force per unit length as a function of the applied strain for the foursamples.

0 0.2 0.4 0.6 0.8 10

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800

Strain (%)

Str

ess

(MP

a)

d= 0.2 μmd= 0.4 μm

Fig. 13. Stress–strain curves for samples with different grain sizes. Initial volume dislocation density: 4 · 1014 perm2. The error bars represent the statistical scatter of different runs at the end of the simulation for each sample.


than in the sample with d = 0.2 lm. The significant increase in plastic activity may havetwo explanations: (i) as in the case of a change in the thickness a larger grain boundaryarea produces a higher probability of dislocation nucleation; (ii) the aforementionedeffect is more marked than in the case of an equivalent change in the thickness, becausewhen the grain size is increased, opposite grain boundaries are moved farther apart. Thisimplies a less significant internal stress generated during straining, because the disloca-tion walls are more distant from each other. Moreover, for larger grain sizes there isa bigger surface area available for dislocation adsorption; therefore, dislocations are


more likely to encounter free surfaces. This causes a release of the internal stresseswithin the grain.

4.4. Bauschinger effect

As we mentioned before, there is experimental evidence in the literature of a strong‘‘anomalous’’ Bauschinger effect for thin Cu films on substrates (Baker et al., 2003; Xiangand Vlassak, 2005). Recently, Xiang and Vlassak (2005) proposed an interpretation forthis material behavior. The authors suggested that in the loading phase of the tensile test,a large number of dislocation pile-ups are formed at the interface between the film and thesubstrate. This produces a strong back stress in the film, which, upon unloading, leads dis-locations to glide in the opposite direction. The early sweeping of dislocations in the oppo-site direction may cause an unloading nonlinear behavior.

In order to investigate this phenomenon in free-standing thin films, we have tested theunloading behavior of the samples with h = 0.2 lm and h = 0.4 lm. We have chosen toperform the unloading tests quasi-statically, i.e., by applying a stepped strain and relaxingthe sample to reach a stable configuration. This allows computing the equilibrium relax-ation stress corresponding to the applied strain. The results are illustrated in Fig. 14 inthe form of points connected by straight segments. Two main features may be noticed.First, a strong Bauschinger effect is indeed manifested in the simulations. Reverse plasticstrain begins very early, when the overall stress in the sample is still in tension. Second, thiseffect appears to be size-dependent, since, for the sample with h = 0.4 lm, the deviationfrom linearity starts much earlier in the unloading phase. Again, an explanation may beadvanced based on back stress arguments as shown by Leger et al. (2004). As we under-lined in previous sections, even in free-standing films, a strong back stress at grain bound-aries arises from piling up of dislocations and formation of junctions within the sample.This is more marked in the case of high aspect ratio grains, see Section 4.2.4, which causes

0 0.2 0.4 0.6 0.8 10

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800

Strain (%)

Str

ess

(MP

a)

h= 0.2 μm

h= 0.4 μm

Fig. 14. Unloading behavior of the samples with different thicknesses. Significant size dependent Bauschingereffect is manifested.

1 x 1

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Fig. 15. Reverse gliding of dislocations upon unloading in the case of the sample with h = 0.4 lm. The twosnapshots show the dislocation configuration during unloading at an applied strain of (A) 0.7% and (B) 0.6%.


a significant nonlinearity during unloading (Fig. 14). Simulations clearly show that reversegliding of dislocations starts earlier in the case of the grain with h = 0.4 lm, i.e., withhigher aspect ratio. In order to illustrate this process, Fig. 15 compares the two dislocationconfigurations for the thicker sample at an applied strain of 0.7% and 0.6% upon unload-ing. The arrows show the reverse motion of some of the dislocations segments. It is alsoobserved the motion of junctions along sessile segments. All these features provide an indi-cation of the mechanism for reversed plasticity during unloading.

5. Concluding remarks and future challenges

In this work experimental investigations and DDD simulations have been performed inorder to study thin film plasticity and the relevant deformation mechanisms. An interpre-tation of size scale plasticity has been proposed based on the idea of a limited initial num-ber of defects with most probable sites for dislocation nucleation at grain boundaries. Thisassumption is well supported by a large number of in situ TEM observations as well as ourpostmortem TEM observations, which show that the grains are basically defect-free in theinitial state and with few residual dislocation networks in isolated grains after straining.Especially, residual networks were observed in the coarsest grains and in the ones withhighest aspect ratio.

In order to validate the idea of a dislocation-source-dependent plasticity, self-similarexperiments were performed by comparing the results for submicron Cu thin films andcoarse-grained millimeter-thick films. As expected, coarse-grained films did not exhibitany size effect and their deformation behavior appeared to be very similar to the one ofbulk materials.

This statistical concept was used to implement a computational model in DDD in orderto simulate the activation and the evolution of grain boundary dislocation sources in acolumnar grain of a polycrystalline thin film. By changing the thickness of the columnar


grain and, hence, the available surface for dislocation nucleation, we were able to capturequalitatively the same trend than the experimentally observed size effects. However, thesefeatures gradually disappeared for a critical value of the aspect ratio of the columnargrain, due to the increase in internal stresses generated by junction formation and pileup at the impenetrable boundaries. This critical value of the aspect ratio has been foundto be strongly dependent on the initial density of sources.

An even stronger size effect has been obtained through DDD simulations when thegrain size of the crystal was varied. This has been attributed to the larger available areaof free surfaces, which attract dislocations partially adsorbing them and releasing theinternal stresses. In this situation, a strong increase in the number of activated grainboundary sources was observed.

Finally, the statistical grain boundary source model captured a strong size-dependentBauschinger effect, which can also be interpreted based on the significant back stress gen-erated during the loading stage.

We have shown that the approach adopted here is able to capture the essential featuresof the discrete mechanisms involved. However, strong assumptions were required in orderto simplify the computational problem. For instance, we have assumed a density ofsources distributed at grain boundaries with sizes randomly generated by means of a nor-mal distribution. This starting configuration can only qualitatively represent the real nat-ure and complexity of a grain boundary. This structure, at least for low angle grainboundaries, can be seen as arrays of dislocations, which under an appropriate stress con-dition may be pulled out and propagate into the interior of the grain. On the other hand,grain boundaries have also manifested, to a certain extent, the ability to adsorb disloca-tions approaching the boundary by modifying its structure. This emission-adsorption pro-cess requires a deeper modeling effort where atomistic simulations are likely to improveour knowledge of modifications in the grain boundary structure and to formulate an ener-getic activation criterion to be incorporated into our DDD model.

In this contribution, local discrete phenomena have been described efficiently in aqualitative way by an adequate numerical approach, underscoring the important rolesof (i) the characteristic internal length parameters which are grain size and film thicknessand (ii) the boundary conditions prescribed (here free surfaces, impenetrable boundariesrepresenting grain boundaries). However, the description does not allow us to directlysimulate the tensile response of the polycrystalline thin films, since the sample usedfor DDD simulations is not representative enough of the peculiarities of the microstruc-ture. These are: a given grain size distribution (here a bimodal distribution) with a givencrystallographic orientation distribution function of the grains which can significantlymodify internal stresses (i.e. the back stress profile) within the grains. Moreover, grainshape and the presence of triple junctions may also modify the spatial dislocation sourcedistribution so that the probability density function of source size is different from agrain to another when considering a polycrystalline sample. We have already foundthrough this study that the aspect ratio of the grain is important. Lastly, mechanismslike intergranular dislocation slip transmission (see e.g. De Koning et al., 2002) insteadof pile-up may also affect the profile of back stresses and the propagation of mobile dis-locations at the local scale. While this complexity can be incorporated in future DDDmodels, we expect the statistical grain boundary source and other features of the modeladvanced in this work will be the dominant reasons for the experimentally observed sizescale effects.


Acknowledgements

The support of the NSF under grants number CMS-00304472, CMS-0120866 andDMR-0315561 is gratefully acknowledged. We express our appreciation to Bei Peng,Yong Zhu and Francois Barthelat for obtaining the experimental results reported in thiswork. The authors are also grateful to the Supercomputer Centers at University of Illinoisat Urbana Champaign (NCSA resources) and at University of California at San Diego(NPACI resources) for providing allocations to run the calculations. The SEM andTEM work was carried out in the Center for Microanalysis of Materials, University of Illi-nois, which is partially supported by the U.S. Department of Energy under GrantDEFG02-96-ER45439. We are particularly thankful to James Mabon, Michael Marshaland Ivan Petrov for their input and assistance in the SEM and TEM work.

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