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Image forces on edge dislocations: a revisit of the fundamental conceptwith special regard to nanocrystalsPrasenjit Khanikara; Arun Kumara; Anandh Subramaniama

a Department of Materials Science and Engineering, Indian Institute of Technology Kanpur, Kanpur208016, India

First published on: 04 December 2010

To cite this Article Khanikar, Prasenjit , Kumar, Arun and Subramaniam, Anandh(2011) 'Image forces on edgedislocations: a revisit of the fundamental concept with special regard to nanocrystals', Philosophical Magazine, 91: 5, 730— 750, First published on: 04 December 2010 (iFirst)To link to this Article: DOI: 10.1080/14786435.2010.529089URL: http://dx.doi.org/10.1080/14786435.2010.529089

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Philosophical MagazineVol. 91, No. 5, 11 February 2011, 730–750

Image forces on edge dislocations: a revisit of the fundamental

concept with special regard to nanocrystals

Prasenjit Khanikar, Arun Kumar and Anandh Subramaniam*

Department of Materials Science and Engineering, Indian Institute ofTechnology Kanpur, Kanpur 208016, India

(Received 3 April 2010; final version received 30 September 2010)

Two conditions under which image forces become significant are when adislocation is close to a surface (or interface) or when the dislocation is ina nanocrystal. This investigation pertains to the calculation of image forcesunder these circumstances. A simple edge dislocation is simulated usingfinite element method (FEM) by feeding-in the appropriate stress-freestrains in idealised domains, corresponding to the introduction of an extrahalf-plane of atoms. Following basic validation of the new model, theenergy of the system as a function of the position of the simulateddislocation is plotted and the gradient of the curve gives the image force.The reduction in energy of the system arises from two aspects: firstly, dueto the position of the dislocation in the domain and, secondly, due todeformations to the domain (/surfaces). The second aspect becomesimportant when the dislocation is positioned near a free-surface or innanocrystals and can be calculated using the current methodology withoutconstructing fictitious images. It is to be noted that domain deformationshave been ignored in the standard theories for the calculation of imageforces and, hence, they give erroneous results (magnitude and/or direction)whenever image forces play an important role. An important point to benoted is that, under certain circumstances, where domain deformationsoccur in the presence of an edge dislocation, the ‘image’ can be negative(attractive), zero or even positive (repulsive). The current model is extendedto calculate image forces based on the usual concept of an ‘imagedislocation’.

Keywords: dislocation stress fields; image forces; nanocrystals; finiteelement method

1. Introduction

A dislocation near a free surface feels a force towards the surface, which is called theimage force [1]. The term image force is used in literature because a hypotheticalnegative dislocation is assumed to exist on the other side of the free surface for thecalculation of the force. If the material with a dislocation is bonded with anothermaterial of a lower elastic modulus, the dislocation would still feel an attractiveforce towards the interface, which would be lower than that for a free surface.

*Corresponding author. Email: [email protected]

ISSN 1478–6435 print/ISSN 1478–6443 online

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If the material across the interface is elastically harder, then the force experiencedby the dislocation could become repulsive in nature (depending on the exactconfiguration). This could lead to an equilibrium position of a dislocation within thecrystal (instead of the free surface) [2]. Additionally, if the free surface is replaced bya surface with constraints, then the magnitude and nature of the force experiencedwould be altered. The term ’image force’ can be extended to these cases as well,keeping in mind the fact that the construction of an image dislocation may or maynot be possible, as in the case of a free surface-bounded crystal (wherein, the vacuumpart is replaced with the same material having a negative dislocation).

These cases, with dislocations near an interface, can be analysed as dislocationsin semi-infinite domains [2]. Head [3] has analysed the force experienced by adislocation near a surface film by using an infinite set of images. When thedislocation is near an interface, the interface deformations cannot be ignored andshould be taken into account for the calculation of the energy of the system [4,5]. Infree-standing nanocrystals, more than one surface will be at comparable distancesfrom the dislocation line, and net force experienced by the dislocation would then bea superposition of all the image forces. Additionally, with decreasing size of thecrystallite, the deformations to the domain shape become increasingly important.This implies that for free-standing nanocrystals, the standard theoretical formula-tions used in the analysis of semi-infinite domains do not yield correct results.

The image force experienced by a dislocation can be resolved into a glidecomponent parallel to the slip plane and a climb component perpendicular to the slipplane. When the glide component of the image force exceeds the Peierls force [6], itcan lead to depletion of dislocations from regions near the surface of large crystals[1,7] and, in the case of nanocrystals, these forces can lead to a completelydislocation-free crystal [8]. The climb component of the image force is expected toplay a role at high temperatures when dislocation climb becomes feasible.

The utility finite element method (FEM) at the nanometre length-scale ishighlighted through the work of Benabbas et al. [9], Zhang and Bower [10],Rosenauer [11] and other researchers. FEM has also proved to be an important toolin understanding dislocations in materials and its interactions with other stress fields[12–15]. Belytschko and co-workers [16–19] have made significant contributions tothe understanding of dislocations; especially in the vicinity of interfaces (free surfacesand bimaterial interfaces). They have used techniques, such as enrichment of finiteelement space and J-integrals, to develop methodologies applicable to non-linear andanisotropic materials. Their contributions include calculating stress fields usingXFEM along with Peach–Koehler forces.

Schall et al. [20] have noted that, even on a scale of a few lattice vectors, thedislocation behaviour is well described by a continuum approach. In systems withcomplexity in: (i) distribution of dislocations or other internal stress fields, (ii)external loading and boundary conditions, (iii) geometry of the domain or (iv)material distribution, FEM becomes an indispensable tool. However, any modelbased on linear elasticity is not expected to be valid when the dislocation is very nearan interface and atomistic models taking in the core structure can yield good results[21,22]. Researchers have used different computational methods to calculate imagestress fields. Xin et al. [23] have used boundary element method for the investigationof dislocation–boundary interactions. Weinberger and Cai [24] have developed a

Philosophical Magazine 731


numerical method to compute image stress fields of defects in an elastic cylinder.Investigators studying dislocation dynamics (e.g. Van der Giessen and A. Needleman[25], Zaiser et al. [26], Weygand and coworkers [27]) have used alternatives to FEM[28] and also a hybrid of FEM and dislocation dynamics methodologies, to take intoaccount image forces [29, 30]. Yasin et al. [31] have considered size and boundaryeffects in their 3D discrete dislocation model to understand the deformation of singlecrystal metals. Zbib and de la Rubia [32] have developed a multiscale model ofplasticity wherein the effect of the interaction of interfaces with dislocations andpoint defects is taken into account. Verdier et al. [33] have reviewed the methods andtechniques used for the simulation of dislocation dynamics on a mesoscopic scale,including the effects of image forces. Moriarty et al. [34] have covered the broaderarea of atomistic simulations of dislocations and defects in their review. Due tolimitations of space, the review of literature presented here is but a sampling of thevast literature on the subject of dislocations. The reader can consult these referencesas a starting point to a detailed study.

The manuscript is intended to make the following fundamental contributions toour understanding of image forces, especially with reference to nanocrystals andtheir computation using FEM: (i) A dislocation near a free surface deforms it and,hence, alters the energy state of the system along with the image force experienced bythe dislocation. This effect is not taken into account in the standard formulationsand will be a topic of focus in this work. (ii) In nanocrystals, the surface and domaindeformations cannot be ignored, which will lead to an altered state of stress andenergy of the domain and, hence, the image forces (which could be drasticallyaltered). The current work considers these factors. (iii) In a fixed boundary scenario,it is easy to visualise the direction of the image force. In nanosized domains, whichcan bend or buckle, this cannot be taken for granted and the direction of the imageforce is a function of the position of the dislocation in the domain (i.e. it has to becalculated for a given position). The methodology developed is well suited to addressthis issue. (iv) In nanocrystals, due to proximity of more than one surface, the netimage force, to a first approximation, is a superposition of multiple images.The current work proposes a methodology wherein multiple images need not beconsidered (i.e. no fictitious images need to be constructed). (v) The work showsa systematic comparison between the concept of an image dislocation for thecalculation of image forces and the use of a single dislocation. In the context ofnanocrystals, this also brings out the limitations of the standard equation for thecalculation of the image force. (vi) This work brings out the concept of a positiveimage (via FEM), which brings about the convergence of the two and singledislocations models presented. (vii) The current methodology is easy to implementand extend to other configurations (e.g. in the presence of coherent or semi-coherentprecipitates in a free standing film). (viii) There are other minor contributions,including finding a bimaterial configuration, which can yield a nearly zero imageforce over most positions of the dislocation in the domain (implying regions ofstability even with low Peierls stress).

The current work, which uses methods different from the standard discretedislocation plasticity techniques, aims at the following tasks: (i) simulation of anedge dislocation in idealised domains using FEM, (ii) calculation of the glide andclimb components of the image force, using a single dislocation or by using two

732 P. Khanikar et al.


dislocations (one of which is the ‘image dislocation’) and compare the results of the

simulations with that of the standard equations, (iii) determination of the image

force experienced by an edge dislocation near a bi-material interface or near a

constrained boundary, (iv) utilise this methodology to analyse the case of

dislocations in nanocrystals, wherein domain deformations lead to a lowered net

energy of the system and hence affects the image force. Additionally, efforts will be

made to keep the procedures intuitive and simple; with suitable illustrations of the

models considered. Some of the configurations considered are similar to those

investigated by Sasaki et al. [13] and Belytschko and Gracie [18].

2. Theoretical background

The �x component of the stress field of an edge dislocation in an infinite isotropic

medium [35] has been modified by Sasaki et al. [13] for a finite cylindrical domain of

radius r2:

�x ¼�Gb

2�ð1� �Þ

y

ðx2 þ y2Þ23 1�

r2

r22

� �x2 þ 1�

3r2

r22

� �y2

� �, ð1Þ

where G is the shear modulus, b is the modulus of the Burgers vector and � is the

Poisson’s ratio. When r2!1, the equation for the infinite domain case is obtained.

The energy of an edge dislocation per unit length (Edl) is given by [1]

Edl ¼Gb2

4�ð1� �Þ2þ ln

�

r0

� �� , ð2Þ

where � is the radius of a cylindrical crystal and r0 is the core radius (usually assumed

to be of the order of b [36]). The first term in the equation is due to the contribution

from the core of the dislocation, which is estimated to be a fraction of the total

energy [1]. The image force (Fimage) experienced by the edge dislocation at a distance

d from the surface of a semi-infinite domain is given by [1]

Fimage ¼�Gb2

4�ð1� �Þd: ð3Þ

For a finite domain of length L (Figure 1), the image force experienced by the

edge dislocation (towards the vertical surfaces) can be computed as a superimpo-

sition of two image forces (by assuming two image dislocations) as follows:

Fimage ¼�Gb2

4�ð1� �Þ

1

d�

1

L� d

� �¼�Gb2

�ð1� �Þ

2x

L2 � 4x2

� �, ð4Þ

where x is the position of the dislocation from the centre of the domain (Figure 1). It

is seen from Equation (4) that the dislocation does not experience any force at the

centre of the domain (x¼ 0), which is symmetrical position with respect to the two

surfaces.



3. Simulation methodology

An edge dislocation is simulated in aluminium (a0¼ 4.04 A; slip system: h110i{111},

b¼ffiffiffiffiffiffiffi2a0p

/2¼ 2.86 A, G¼ 26.18GPa, �¼ 0.34837) by imposing the stress-free strain

corresponding to the introduction of an extra half-plane of atoms. Stress-free strains

are also called eigen strains (or Eshelby strains) and the point to be noted is that on

the imposition of these strains, stresses would develop in the crystal (stress field of

the edge dislocation in the present case). The term stress-free strain implies that if the

body were unconstrained, then no stresses would develop in the body (e.g. thermal

expansion or phase transformation of an unconstrained crystal). Use of anisotropic

material properties is not expected to alter the methodologies developed in the

current work. However, this allows the use of simple theoretical equations for

comparison (Equations (1)–(4)). The value of G and � are calculated by Voigt

averaging single crystal data [35]. It is assumed that material properties of bulk

crystals are valid at the length-scales used in the simulations. The structure and the

energy of the core of the dislocation are ignored in the model. The energy of the core

is not expected to play an important role in the calculation of image forces, except

when the dislocation is at a distance of a few Burgers vectors from the an interface

(i.e. the core energy is not expected to be a strong function of the position of a

dislocation from the interface and, hence, would be nearly a constant additional term

to the total energy of the system). At this point, it should be noted that the current

model is amenable to modifications, to include a core model, as prescribed by Stigh

[12], by altering the shape of the region where strains are imposed to a wedge shape.

It is assumed that the dislocation has not split into partials. Isotropic, plane-strain

conditions are assumed and the stress-free (eigen) strain ("T) introduced as thermal

strains into a region (of width b) as shown in Figure 2 is

"T ¼a0 þ bð Þ � a0

a0¼

1

2, ð5Þ

where a0(¼2b) is the face diagonal of the cubic unit cell. The strain is imposed only in

a direction along the face diagonal of the cube (x-direction in Figure 2), as this

corresponds to the insertion of an extra half-plane of atoms (to simulate the stress

fields of an edge dislocation). This insertion implies that the additional half-plane is

Figure 1. Schematic illustration showing an edge dislocation in a finite domain experiencingtwo image forces. The ‘image dislocations’ are of opposite sign and lie on the extension of theslip plane.



added along the face diagonal making the total length (a0 þ b) and, hence, the strainis calculated to be ½ as in Equation (5). In the absence of symmetry, where a fulldomain will have to be considered (instead of the symmetric half), this region wherethermal strains are imposed to simulate an edge dislocation, will be of width 2b.Circular and rectangular domains were meshed with bilinear quadrilateral elementswith mesh size b� b (Figures 2 and 3). In the current simulations, L is taken to be500b. Two sets of boundary conditions imposed are also shown in Figure 2(the second set of boundary conditions is used to study the effect of deformations tothe domain shape on the energy of the system). This simple geometry has beenconsidered to make a comparison with the standard theoretical Equations (1) and (2)and to validate the FEM model of the edge dislocation. The numerical model wasimplemented using the Abaqus/standard (Version 6.7, 2007) FEM software. As thestandard equations for stress fields and energy are for circular domains (cylindricalin three dimensions), the circular domain is used for the comparison of the FEMcalculations with the theoretical equations [1,13] and to validate the finite elementmodel. The size of the circular domain is varied to get a plot of the energy of thedislocation as a function of the domain size. The rectangular domain is used forimage force calculations. For a comparison of the energy obtained from the finiteelement model, with the theoretical equation for energy (Equation (2)), the corecontribution term in the equation was ignored.

The energy of the system (per unit depth) is calculated from the model and thenplotted as a function of the position of the dislocation. The direction of depth isparallel to the dislocation line, which is perpendicular to the xy-plane (Figure 3).Hence, the energy computed is the energy of the dislocation per unit length of the

Figure 2. Symmetrical half of a circular domain showing the regions where stress-free strainsare imposed to simulate an edge dislocation. Two sets of boundary conditions are used tostudy the effect of deformations to the domain shape on the energy of the system: (a) lessconstrained system, and (b) more constrained system to restrict parts of circumference fromdeforming. The strain imposed in this case and future cases for the simulation of the edgedislocation are according to Equation (5).



dislocation line. The image force (per unit depth) experienced by the dislocation isthe slope of the energy versus distance plot (negative slopes correspond to attractiveforces towards the boundary). By positioning the dislocation at various points alongthe x-direction, the glide component of the image force is computed.

To make a direct comparison with the theoretical notion of an image of adislocation, the configuration shown in Figure 4 is considered. It should be notedthat the theoretical image dislocation annuls surface stresses in the case of a screwdislocation, but does not annul the �xy (shear) stresses in the case of the edgedislocation [35]. Furthermore, the image force calculated using the image(edge) dislocation gives a correct result; but, the stresses have to be modified toannul the residual shear stresses. More on this aspect will be considered along withthe results based on this model. As shown in the figure, the domain is an extended

Figure 3. Rectangular domain showing regions where stress-free strains are imposed, alongwith the boundary conditions used, for the simulation of an edge dislocation at a distance xfrom the centre of the domain along the x-direction.

Figure 4. Domain, boundary conditions and the regions where stress-free strains are imposedto simulate a positive and a negative edge dislocation. This configuration is consistent with thetheoretical concept of an ‘image dislocation’ and is used for the calculation of glide componentof the image force.



one containing a simulated edge dislocation of the opposite sign. The energy of thesesystems as the dislocations are brought towards each other (symmetrically) is plottedas a function of their position and the slope of this curve gives an alternatecalculation of the image force; which is consistent with the concept of an imagedislocation.

The methodologies adopted for calculating the glide component of theimage force can be repeated to determine the climb component of the image force.This is done using the models shown in Figure 5. The distance y (distance from thecentre of the domain) is varied to position the dislocation at various points alongthe y-axis.

As mentioned in Section 1, the concept of image force experienced by adislocation can be applied to three distinct cases for the dislocation in the vicinity of:

(i) a free-surface, (ii) an interface with another material, and (iii) a constrainedsurface. In the case of the interface with a harder material and the case of aconstrained surface, the image force could be repulsive (this depends on the exactconfiguration). The case of a constrained surface is illustrated here with an example,wherein the nodes on the boundary are locked in the x-direction. In this model, theimage developed is a positive dislocation and, hence, the force is repulsive in nature.It is worthwhile to note that the concept of an ‘image dislocation’ can directly beextended to the case of crystal bound by constrained surface (of the kind described),by making the image a positive one. To investigate these additional cases, theconfigurations shown in Figure 6 are used. Without loss in generality, the hardermaterial can be made a softer one, which would lead to purely attractive force on theedge dislocation. Nickel is chosen as the harder material for the simulations(G¼ 94.66GPa, �¼ 0.2767). Aluminium will now play the role of the softer material.

Figure 5. FEM model used for the calculation of the climb component of the image force:(a) single-dislocation model, and (b) two-dislocation model (one of the dislocations being an‘image dislocation’). Symmetrical half of the domain used in the FEM analysis is shown inboth the figures.



Having used the aforementioned models to validate the applicability of the newmethodology; three further models (Figure 7) are considered, which illustrate thosecases wherein the analytical formulation clearly breaks down (with respect to thedirection and magnitude of the image force). These models will be used to highlight

Figure 7. FEM models which highlight the power of the new methodology developed in thecurrent investigation: (a) inclined slip plane of length ‘s’ in a rectangular domain of aluminiumwith length (L) to breadth/height (h) ratio of 1:3, (b) a thin plate of aluminium with ahorizontal slip plane (b¼ 2.86 A), and (c) an interfacial misfit edge dislocation in anheteroepitaxial system (Si/SiGe, b¼ 3.84 A).

Figure 6. Domains used for the simulation of image forces experienced by an edge dislocationin the vicinity of: (a) an interface with a material of higher elastic modulus, and (b) a surfacelocked in x-direction (constrained surface).



the power of the current method. The objectives behind the choice of these cases areoutlined next. In the first model (Figure 7a), the slip plane is inclined, which impliesproximity of the free surface. However, the domain is longer than this length,implying that the self energy of the dislocation will be high. Additionally, thisconfiguration implies only a small bending of the domain. In the second model(Figure 7b), the bending is larger, leading to considerable deviation in energy(and stress plots), compared to a constrained domain. When the models shown inFigures 3 and 7a,b are put together, a broad range of domain deformations arecovered. These represent cases wherein Equation (4) becomes progressivelyinapplicable. The third model (Figure 7c) corresponds to an interfacial misfit edgedislocation in a heteroepitaxial system (Si/Ge0.5Si0.5). As discussed later, the misfitedge dislocation represents a considerable deviation from the conventional edgedislocation and, hence, there are no available analytical expressions to compute theimage force experienced by such a dislocation. The details of the simulation of theepitaxial system (model Figure 7c) along with the material properties are givenelsewhere [14].

4. Results and discussion

Figure 8 shows the plot of �x stress contours in a circular domain of 125b radius,determined from the current FEM simulations, along with those calculated using thetheoretical Equation (1). A comparison of the two sets of contours in Figure 8 showsa close match. Unlike the case of the infinite domain, wherein the compressive andtensile regions are separated by a single horizontal zero stress line, the finite circulardomain shows a stress reversal. A comparison of the variation in the energy of thedislocation (per unit length) in a circular domain, as a function of the domainsize, determined from the FEM calculations, with that from the theoreticalEquation (2) is shown in Figure 9 (these simulations correspond to the

Figure 8. Plot of the FEM-simulated �x stress contours and its comparison with thetheoretical equation [13] in a symmetrical half of a circular domain: a close match is seenbetween the two values.



configurations shown in Figures 2a and b). Similar trends, along with a good matchare observed between the theoretical results and the FEM computed values. It is alsoseen that the energy of the system increases with additional constraints to thedeformation of the domain, with the energy of the more constrained domain beingcloser to the theoretical values. The FEM determined values are expected to be lowerthan the theoretical values, as relaxations are allowed in the FEM model (i.e. thesurface is allowed to deform, except at points on which it has been constrained(Figure 2)). The shape of the deformed and the undeformed domains have beencompared in Figure 10. It should be noted that slight geometrical relaxations couldlead to large energetic relaxations.

Figure 11 shows the plot of the FEM calculated energy values of the system as thesimulated edge dislocation is positioned at various points along the x-axes. Twoconfigurations of the system with edge dislocation(s) have been used to determine theenergy as shown in Figures 3 and 4. These two configurations represent domains inwhich one domain is double the size of the other and a comparison in the energyvalues is made possible by taking half the energy determined from the double-sizeddomains. The image forces (glide component) calculated from these curves are alsoshown in Figure 11, along with the plot of the theoretical equation (Equations (3)and (4)). Positive image force values are indicative of an attraction towards thesurface. Figure 12 shows the deformations to the free-surface, along with the �xstress contours, when the edge dislocation is positioned at a distance of 20b from it.It is important to note that small geometric deformations can lead to considerablerelaxation in energy and, hence, can have a profound effect on the image forceexperienced by the dislocation. It is to be noted that, in the model with two-dislocations (Figure 4), these deformations would be restricted, compared to the

Figure 9. Energy of the edge dislocation (per unit length of the dislocation line) as a functionof the circular domain size: FEM-calculated values and its comparison with the theoreticalvalues [1].



model with a single dislocation (Figure 3). This implies the energy of the system with

a single dislocation would be lower, which is seen in Figure 11. It is seen from the

plots in Figure 11 that there is a reasonable match between the theoretical plot of

image force (Equation (4)) and the values calculated from the FEM simulation.

However, it is seen that the values obtained from the simulation are higher than that

from theory for positions of the dislocation away from the centre. This is as expected

and is due to a steeper fall in the energy of the system as the dislocation moves away

from the centre of the domain, which is consistent with the relaxations allowed by a

free-standing crystal.

Figure 11. Energy of an edge dislocation (per unit length of the dislocation line) and the imageforce (glide component) experienced by it as a function of its position along the x-axis from thecentre of the rectangular domain.

Figure 10. Deformation to the shape of a circular domain (symmetrical half shown) ofdiameter 100b, due to the dislocation stress fields (plot of �x contours shown on the deformeddomain). The deformed shape has been shown exaggerated by a scale factor of six.



An important point worthy of reiteration here is the fact that theoreticalcalculation of stress fields using an image edge dislocation need corrections [35];unlike the stress fields computed using an image screw dislocation. However, theimage force calculated from the image dislocation model turns out to be right [35].This aspect further highlights the importance of single-dislocation model developedin the current work; wherein the construction of an image dislocation need not beconsidered and stress fields arise naturally from the model (hence, is an additionalmotivation for the current work). In light of these discussions, it becomes clear thatthe FEM model with an image dislocation has been considered for making aninternal comparison and also to emphasise the importance of the single dislocationmodel (main theme of the current work).

Figure 13 shows the plot of the climb component of the image force, along withtheoretical plots of Equations (3) and (4), using the models shown in Figure 5. As inthe plots in Figure 11, it is seen that the model with a single dislocation yields a lowerenergy for the system, compared to a model with two dislocations (one of the edgedislocations being an image dislocation as before). One difference to be notedbetween Figures 11 and 13 is the residual climb component of the image force presentfor the position of the dislocation at the centre of domain, using a single-dislocationmodel (in Figure 13). This is due to the asymmetry in the boundary conditions used(Figure 5a). This was necessitated by the requirement to leave a free-surface, towardswhich the dislocation can feel the image force. The lower surface could also havebeen left free (with the numerical requirement of locking just the central point on thisline along the y-direction); which would lead to further relaxation of the system, bybuckling of the domain maintaining the x¼ 0 line mirror symmetry.

In Figures 11 and 13, the theoretical plot of the image force obtainedfrom Equation (3) has been included; which is valid only for semi-infinite domains.

Figure 12. Deformations to a free surface due to the presence of a FEM-simulated edgedislocation at a distance of 20b from the surface. Plot of �x contours shown on the deformeddomain (the deformed shape has been shown exaggerated by a scale factor of six).



The inclusion of this plot is for the estimation of the distance of the dislocation fromthe surface, at which the domain seems like an infinite one, with regard to the imageforces experienced. The rule of thumb, which can be evolved by analysing the plots,is that, when d/L5 0.1, the finite domain behaves like an semi-infinite one withinan error of 1%.

At this point, it should be noted that the FEM simulations are expected to bevalid for dislocation positions which are large compared to the core dimension (i.e.approximately, when d4 5b). In the examples considered, the method has been usedto calculate the image force along the x- and y-directions and can readily be extendedto force calculations along any arbitrary direction.

Figure 14 shows FEM calculations of image force (glide component) in thepresence of an elastically harder material (according to the model in Figure 6a) ofthree thicknesses (t¼ 5b, 50b and 100b). For t¼ 5b, the edge dislocation feels anattractive force towards the interface in the validity regime of the simulations.For t¼ 50b, there is an equilibrium position for the dislocation at a distance of about20b from the interface. If the thickness of the harder material is made larger, thedislocation feels only a repulsive force from the interface. This is seen for the case oft¼ 100b; wherein, it is additionally observed (Figure 14) that the repulsive forceexperienced by the dislocation is very small and nearly a constant, till the dislocationis at a distance of 50b from the interface. This implies that choosing a harder materialof appropriate modulus and thickness, one can ‘taylor’ the image force to be nearlyzero for most part of the domain. Head [2,37] had analysed the case of the imageforce experienced by a screw dislocation near a bi-material interface using an infiniteset of images. It was pointed out in Head’s work that the equilibrium position ofa screw dislocation in the presence of a harder material would be of the order of thethickness of the harder material, for the case of a semi-infinite domain. Additionally,

Figure 13. Energy of an edge dislocation (per unit length of the dislocation line) and the imageforce (climb component) experienced by it as a function of its position along the y-axis fromthe centre of the rectangular domain.



Head [2,38] had pointed out the difficulty in analysing the case of an edge dislocationnear a bi-material interface. As seen from the example considered, the currentmethodology can avoid the use of images and can be used for finite domains, whereinthe image forces can be considerably altered with respect to an infinite domain.

A comparison of a single-dislocation model (model shown in Figure 6b) with atwo-dislocations model, for the case of a dislocation approaching a constrainedsurface, is shown in Figure 15. The surface has been constrained in the x-direction.The model with two dislocations consists of two positive edge dislocations, unlike thecase of a dislocation near a free-surface, wherein the image dislocation was a negativeone. It is seen from the figure that a very good match is obtained between the models.This is to be expected, as the additional boundary condition used, acts like a ‘mirror’(reflecting the domain on the left) and there are no free-surface deformations todifferentiate the two models. It is interesting to note that the concept of an ‘imagedislocation’ can be extended to the case of a constrained boundary, with a change inthe sign of the image dislocation and with a reversal in the direction of the forceexperienced.

The plot of state of stress (�x) for the models considered in Figure 7, are shown inFigure 16. Figures 7a and b also show the deformed shape of the domains in thepresence of the dislocation. Table 1 gives a comparison of the magnitude anddirection of the image force, between the analytical results (Equation (4)) and thecurrent work. It is seen that when large domain deformations occur in the presenceof the dislocation (leading to considerable relaxation of energy of the system), amajor alteration to the image force takes place (both in magnitude and in direction).The important point to be noted is that ‘image’ can be negative (attractive but withaltered direction and magnitude), zero (no image construction possible) or evenpositive (repulsive). From Figures 16a and b, it is seen that the compression–tension

Figure 14. FEM calculations of image force (glide component) in the presence of anelastically harder material (according to the model in Figure 6a) of three thicknesses (t¼ 5b,50b and 100b).



inversion symmetry present in the stress plot of a usual edge dislocation is lost.This is an additional point to be noted and will occur when there is buckling(additionally, geometrical configuration of the system can lead to this asymmetry,as in Figure 16a).

Special attention should be focussed on the case of the misfit edge dislocation in aheteroepitaxial film. An interfacial edge dislocation represents a very different case ascompared to a normal edge dislocation, in the following aspects: (i) the compression–tension inversion symmetry present in the stress plot of an usual edge dislocation isbroken (firstly due to the asymmetry in the geometry), (ii) the material propertiesabove (film) and below (substrate) the interface are different, (iii) the interaction ofcoherency stresses of the system with the dislocation stress fields significantlymodifies the latter. Hence, there are no analytical expressions for comparison withthe results of the current work (Table 1). The interfacial misfit dislocation feels aforce towards the centre of the domain from the position indicated in Figure 7c(99.5b), as the energy of the system can be lowered by its position at the centre of thedomain. As for the case considered in Figure 7b (climb force), this aspect is reflectedby the negative sign for the image force in Table 1. This in not an ‘image-force’ in theusual sense of the phrase and arises primarily due to interaction of coherency stresseswith the dislocation stress fields. Nonetheless, the dislocation feels a force which canbe calculated from the current method and does not lead to the erroneous conclusionthat the dislocation is attracted towards the free surface. When the dislocation is‘sufficiently close’ the surface the image force will again be positive (attractivetowards the surface).

Even in the case of the thin plate with horizontal slip plane (Figure 16b), theapplicability of the analytical formula (Table 1) is questionable, as in the analytical

Figure 15. Comparison of single- and two-dislocation finite element models for theimage force experienced by a dislocation in the vicinity of a constrained surface. In themodel with two dislocations, two positive edge dislocations are used for the determinationof the image force.



model the domain is assumed to be infinite in the y-direction. A thin plate deviatesconsiderably from this assumption. The error in direction (angles) calculated inTable 1 are with respect to the domain where distortions are not considered; wherein,the direction of the force would be along the x-axis (horizontal). Additionally, theangles calculated in Table 1 are approximate as the deformations are non-uniform.

The cases illustrated in Figure 7 are just a few of the cases where the currentmodel can be used. Further configurations include image forces experienced bydislocation loops (normal and misfit), dislocations in the presence of otherdislocations, dislocations in epitaxial superlattices, etc.

Although, on the one hand, the FEM models considered suffer from thefollowing assumptions: (i) material properties of bulk crystals have been used for thenano-sized crystals, (ii) Voigt averaged material properties have been used tosimplify the comparisons between theory and simulations, (iii) structure and energyof the core of the edge dislocation have been ignored in the simulation, (iv)dissociation of the dislocation into partials has not been considered, (v) certainidealised domain shapes have been used instead of real shapes of crystals, and (vi)calculations are for 300 K (room temperature). On the other hand, it offers thefollowing advantages: (i) can be used in cases standard equations of stress fields andimage forces are not valid (e.g. small domain sizes of any shape), (ii) the currentmodel can account for surface deformations as a result of the internal stress fields,

Figure 16. �x plot for the models considered in Figures 7a–c, respectively. x¼ 20b (y¼ 10b) inpart (b) of the figure. The shape of the deformed domains in the presence of edge dislocationare to be noted (a) and (b). Dimensions correspond to the undistorted domains.



Table

1.

Comparisonbetweenanalytical(Equation(4))andresultsfrom

thecurrentwork

forthemodelsshownin

Figure

7.‘Errorin

direction’

refers

totheerrorin

theanalyticalmodelwithrespectto

theFEM

calculations(see

Figure

16).Thenegativesignsin

column3are

tobenoted,which

implies

aforceawayfrom

thesurface

(detailsare

explained

inthetext).

Model

Imageforce(A

nalytical)

(N/m

)(%

Error)

Imageforce(FEM)(N

/m)

Errorin

direction

Inclined

slip

planein

aplate

(Figure

7a)

2.95�10�3(Errorof�85%

)4.4�10�4

�4�

Plate

withhorizontalslip

planealongthe

length

oftheplate

(Figure

7b)

(forx¼40.5b,y¼10b)

Fx¼8.61�10�2(G

lide)

(Errorof�89.5%

)

9.0�10�3

�3.5�

(forx¼20.5b,y¼10b)

Fx¼1.8�10�2(G

lide)

(Errorof�100%

)

0�4�

(forx¼0,y¼14.5b)Fy¼1.03�10�1

(Climb)(Errorof�346%

)�2.53�10�1(towardsthecentre)

180�

Epitaxialfilm

withmisfitedge

dislocation(Figure

7c)

Noneavailable

forcomparison

�5.28�10�2(towardsthecentre)

(180�)



which leads to a lowered state of energy and the deformed shape can be computed(this aspect is not considered in the standard theoretical formulations), (iii) it is verysimple to implement and extend to three-dimensions and to complicated configu-rations (e.g. presence of multiple dislocations, splitting of the dislocation intopartials, presence of coherent precipitates etc.) and material properties(e.g. multiphase materials, anisotropy etc.). Additionally, when domain deforma-tions are small, an approximate estimate of the image force experienced by a purescrew dislocation can be determined by considering a multiplication factor of (1� �)(i.e. ð1� �ÞF edge

image ¼ F screwimage [1]). The biggest advantage perhaps, is the simplicity of the

model and its ease of implementation using any standard FEM software.A comparison of the current work with the state of the art techniques in

dislocation dynamics is perhaps due at this stage; even though the two methodsemploy entirely different approaches and have different goals. As stated before,studies using discrete dislocation dynamics techniques have provided considerableinsight into the mechanical behaviour of materials starting with dislocations andother defects at the microscale. The evolution of the system in the presence ofinternal and external stresses (and constraints) at multiple length scales has beensuccessfully implemented using an elasto-viscoplastic continuum framework [32].In these techniques, the well known solutions of the dislocation fields are imposed asin an infinite domain and the fields are corrected for (surface tractions [28]) thepresence of surfaces (and interfaces) using the superposition principle [31].Computational efficiency (using FEM) is obtained by replacing the defectedinhomogeneous volume element with a homogenous body with eigenstress [32].The aim of the current work is to simulate dislocation stress fields using eigen strainsand further calculate image forces using a FEM model. In this model, a prioriknowledge of dislocation stress fields either in a finite or infinite body is not required.Hence, image forces naturally arise out of the model and not as a correction to theinfinite body solution. Though the model can handle complex material andgeometric, configurations (including the presence of other defects in the material);it lacks any dynamics capability and is meant for static calculations.

5. Summary and conclusions

An edge dislocation is simulated using stress-free strains corresponding to theinsertion of an extra half-plane of atoms. The stress state and energy of the system iscalculated using the finite element model and are validated by comparison with thestandard theoretical equations (for cases where these expressions are valid). Byvarying the position of the simulated dislocation, the variation of energy of thesystem is calculated. The glide and climb components of the image force, experiencedby the edge dislocation are determined from these plots (as a slope of the curves). Itis seen that the image force experienced by the edge dislocation is a naturalconsequence of its asymmetric position in the domain and can be calculated using theFEM model without externally applying any forces or invoking the concept offictitious images. Surface/domain-shape relaxations are also taken into account inthe model, which are ignored in the standard theoretical formulations. An addedbenefit of the method is its computational simplicity and efficiency.



An alternate methodology for the determination of the image force is alsodeveloped based on the concept of an ‘image dislocation’; which is in accord with thetheoretical notion of an ‘image force’. Good match is seen for the image forcedetermined by both the methodologies with the standard theoretical formulations.However, it is seen that the model with a single dislocation, which takes into accountthe surface/domain deformations (leading to some relaxation in energy), is a betternumerical model for dislocations in nanocrystals or for dislocations near a free-surface. These are the two cases where image forces become important, but thetheoretical formulation breaks down (gives erroneous results). Further, an importantpoint seen is that under certain circumstances where domain deformations occur the’image’ can be negative (attractive but with altered magnitude and direction), zero(no image construction is possible) or even positive (repulsive). Hence, this aspectleads to a radical deviation from the conventional concept of an ‘image’.

The numerical model for the calculation of image forces is extended for the casesof the edge dislocation near a bimaterial interface and near a constrained surface.It is to be noted that for these cases the traditional concept of an ‘image dislocation’is no longer valid. For a dislocation near an interface with a harder material theequilibrium position of the dislocation is determined using the model (which is notthe surface as in the case of a dislocation near a free-surface or near an interfacebordering a softer material). The case of a dislocation near the type of constrainedsurface considered can be analysed using a ‘positive image dislocation’ rather thanthe usual ‘negative image dislocation’. The simplicity, versatility and power of thesimulation methodologies are brought forth via the examples considered. Themethodology developed in the current work can be extended to various complicatedcases, like the presence of multiple edge dislocations, anisotropic material properties,systems with multiple materials, presence of coherent precipitates etc., withoutrecourse to additional concepts.

Acknowledgements

The authors would like to acknowledge the financial assistance from the Department ofScience and Technology, Government of India (SR/S3/ME/048/2006). The authors also thankProfessor Sumit Basu for his invaluable discussions.

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