DISCLINATIONS AND ROTATIONAL DEFORMATION IN...

79Disclinations and rotational deformation in nanocrystalline materials

Corresponding author: M.Yu. Gutkin, e-mail: [email protected]

Rev.Adv.Mater.Sci. 4 (2003) 79-113

© 2003 Advanced Study Center Co. Ltd.

DISCLINATIONS AND ROTATIONAL DEFORMATION INNANOCRYSTALLINE MATERIALS

M.Yu. Gutkin and I.A. Ovid’ko

Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Bolshoj 61, Vas. Ostrov,St.Petersburg, Russia

Received: June 2, 2003

Abstract. Role of disclinations and rotational modes of plastic deformation in fine-grainedmaterials is discussed. First, we consider disclination models of generation and development ofmisorientation bands in severely deformed metals and alloys. The models predict the existenceof the critical external shear stress, above which nucleation of misorientation bands takes place.The further analysis demonstrates two main regimes of misorientation band development: stableand unstable propagation, and allows to find another critical stress that controls the transitionbetween these two regimes. We quote also some results of computer simulations of 2D dynamicsof dislocations in the stress field of a dipole of partial wedge disclinations to elucidate themicromechanisms of misorientation band propagation. Second, theoretical models of grainboundary disclination motion in fine-grained materials are considered. This motion leads tochanges in misorientations across the grain boundaries and may explain the rotation of graincrystalline lattice as a whole. It is demonstrated that motion of grain boundary disclinations mayoccur in fine-grained materials through emission of pairs of lattice dislocations into the adjacentgrains or through climb of grain boundary dislocations. We also consider a model of crossoverfrom grain boundary sliding to rotational deformation which is realized by the transformation of apile-up of gliding grain boundary dislocations stopped by a triple junction of grain boundaries,into two walls of climbing grain boundary dislocations (treated as the dipoles of partial wedgedisclinations). The conditions necessary for such a transformation are determined and discussed.

INTRODUCTIONAppearance in the early 1980s of the firstnanocrystalline materials (NCMs) [1, 2] has stimu-lated a great interest to disclinations as a powerfullmean to describe the structure and mechanicalbehavior of nano-objects. Geometric and elasticproperties of wedge disclinations were applied tomodel the pentagonal symmetry and strained statein nanoparticles [3-11], to explain the abnormal Hall-Petch relation [12-17] and various grain-boundaryphenomena [8, 9, 11, 13-27] in NCMs, to studypossible ways of misfit-strain accommodation inheterogeneous nano-layered structures [28-38], etc.

The aforementioned applications were based oncontinual description of disclinations in the frame-work of the classical theory of elasticity which al-lows to obtain the solutions of various (sometimesquite complicated) boundary-value problems for elas-

tic fields of disclinations localized in nano-volumes(e.g., see reviews [8, 39-41] and original papers [4,6, 42-49]). However, some components of thesefields are singular at the disclination lines, a factthat limits the applicability of the classical theory toconsider situations where it is important to knowthe strained state near disclination lines. This con-cerns, for example, disclination models for grainboundaries and their triple junctions in NCMs whereone deals with high-density ensembles ofdisclinations.

To avoid this problem, in recent years much ef-fort has been spent to describe both the wedge andtwist disclinations within non-classical theories suchas the nonlocal theory of elasticity [50, 51], gradi-ent theory of elasticity [52-55] and gauge theory ofelastoplasticity [56, 57], all of which allow to dis-pense with the classical singularities. By using the

80 M.Yu. Gutkin and I.A. Ovid’ko

gradient solutions, the short-range elastic interac-tions between disclinations in an infinite solid werestudied [52, 55], while the gauge solution gave non-singular elastic fields for a wedge disclination placedalong the axis of a thin cylinder [56]. It is worthnoting that the non-classical solutions obtainedwithin these quite different theories, totally coincide.Nevertheless, application of these new results tothe theory of mechanical behavior of NCMs is stillan open question.

Extensive investigations of structure, propertiesand fabrication methods of NCMs which have beencarried out during last two decades (e.g., see re-views [1, 2, 9, 16, 17, 58-75], monographs [27, 76-80] and collections of papers [81-93]), have shownthat NCMs qualitatively differ from conventional poly-crystals. First, this concerns the structure of grain(intercrystalline) boundaries whose thickness inNCMs may achieve 1-2 nm. It means that the grainboundaries (GBs) themselves are typical nano-ob-jects – the layers of material which often has theother atomic structure and sometimes is much moreporous than the material inside the grains (crystal-lites). Comparison of different experimental data fromMössbauer spectroscopy [94], positron lifetimespectroscopy [95, 96], X-ray diffraction [97], EXAFS[98], neutron diffraction [99] and HREM [100] al-lowed to conclude that a significant part of the GBsin NCMs have severely distorted near-boundary re-gions with smaller atomic density and higher levelof elastic strains [100]. For example, microstruc-ture studies [100] of nanocrystalline palladium withthe grain size of 4-9 nm, which was fabricated byhigh-pressure-compaction of nanocrystallites con-densed from the gas phase, demonstrated that itcontains ≈ 40 vol. % of undistorted crystalline ma-terial, ≈ 25% of stretched or amorphous-type GBlayers, ≈ 25% of highly strained material, and ≈ 10%of pores. Inside the grains, they observed lamellaeof twins, low-angle boundaries and dislocations lo-calized near the GBs. Under thermal annealing, thissystem was transformed into a conventional poly-crystalline structure with thin GBs and undistortedgrains.

Besides the GBs, their triple junctions play animportant role in the behavior of NCMs. Indeed, ifthe mean grain size is of some nanometers and theGB thickness equals 1-2 nm, then the volume frac-tion of the triple junction material is very high (up to50 % and even more) in such NCMs. In recent years,it has been definitely recognized that triple junc-tions of GBs have the structure and properties be-ing different from those of the GBs that they adjoin

[101]. From experimental data and theoretical mod-els [18-20, 101-108] it follows that the triple junc-tions act as enhanced diffusivity tubes, nuclei of thesecond phase segregation, strengthening elementsand sources of lattice dislocations during plasticdeformation, and drag centers of GB migration dur-ing re-crystallization processes. In particular, theoutstanding diffusional properties exhibited by NCMs[79, 109-112] are viewed to be related to the effectof the highly enhanced diffusion along triple junctiontubes [2].

These key features of NCMs (i.e., high densityof GBs and GB triple junctions with their genericdefects like GB dislocations and disclinations) whichmainly determine their mechanical properties, maystimulate, under special conditions, the generationand development of rotational mode of plastic flow.This mainly concerns the NCMs fabricated underhighly non-equilibrium conditions like ball milling andsevere plastic deformation. In the present review weconsider different models of rotational plastic defor-mation in nano- and polycrystalline materials. In thefirst part of the paper we discuss disclination mod-els for misorientation bands in severely deformedmetals and alloys. The models predict the exist-ence of the critical external shear stress, abovewhich nucleation of misorientation bands takesplace. The further analysis demonstrates two mainregimes of misorientation band development: stableand unstable propagation, and allows to find anothercritical stress that controls the transition betweenthese two regimes. We quote also some results ofcomputer simulations of 2D dynamics of disloca-tions in the stress field of a dipole of partial wedgedisclinations to elucidate the micromechanisms ofmisorientation band propagation. The second partof the paper is devoted to the theoretical models ofGB disclination motion which leads to changes inmisorientations across the GBs in NCMs and mayexplain the rotation of nanograin crystalline latticeas a whole. It is demonstrated that motion of GBdisclinations may occur in NCMs through emissionof pairs of lattice dislocations into the adjacent grainsor through climb of GB dislocations. We consider amodel of crossover from GB sliding to rotationaldeformation which is realized by the transformationof a pile-up of gliding GB dislocations stopped by atriple junction of GBs, into two walls of climbing GBdislocations (treated as the dipoles of partial wedgedisclinations). The conditions necessary for such atransformation are determined and discussed.


2. GENERATION ANDDEVELOPMENT OFMISORIENTATION BANDS

During the two last decades, the concept ofdisclinations has been broadly applied to treatmesoscopic substructures which are characteris-tic for metals and alloys under large deformation (e.g., see [8, 113-116] for a review). Misorientationbands (MBs) represent one of the typical elementsof such substructures. They are observed as longstraight strips of material having a crystallographicorientation different from that of neighbouring areasof the material [8, 113, 115-125]. The boundaries ofsuch MBs (i.e., the misorientation boundaries) areoften illustrated as low-angle dislocation tilt bound-aries, although they have a finite thickness of about0.1-0.5 µm depending on the deformation magni-tude [113], and consist in fact of high-density dislo-cation arrangements. Works [123, 124] give a re-view of experimental data on MBs and other differ-ent rotational structures in various high-strengthmaterials including submicrocrystalline metals andalloys. Based on these results as well as on recentdirect atomic-level observations of dipoles of partialdisclinations in mechanically milled, nanocrystallineiron [126], one can assume the possibility of MBgeneration in typical NCMs under large deforma-tion.

The appearance and development of MBs repre-sents one of the ways of rotational plastic flow inheavily deformed metals. MBs were found at thebeginning of the 1960s in TEM experiments [117,118] (see also [8, 114] for a review). In the 1970sand 1980s, they were extensively studied and di-rectly connected with the formation of specific dis-

ω

+ω

ω−V

2a

τ

Fig. 1. Dislocation-disclination model of a misorientation band propagation with the velocity

V under theaction of external shear stress τ. The front of the misorientation band is modelled as a two-axes dipole ofpartial wedge disclinations of strength ±ω.

location structures which may be described throughpartial disclinations (see [8, 113-116] and referencestherein). Indeed, the presence of two edges of themisorientation boundaries which border the MB area,allows to introduce a corresponding dipole of partialdisclinations whose strength is equal to themisorientation angle characterizing the MB [127].In Fig. 1, the edges of the tilt misorientation bound-aries are shown as a two-axes dipole of partial wedgedisclinations of strength ω. Such a disclination di-pole is geometrically related to the MB parametersthrough the equation [8], bρ =b/l=2tan(ω/2), for adislocation tilt boundary. Here b denotes the Burgersvector magnitude of the dislocations composing theboundary, ρ is their linear density along the bound-ary, l is the interdislocation spacing, and ω is theangle of misorientation across the boundary. In thecase of small misorientation angles, ω << 1, thisrelationship is transformed into ρ =1/l = ω/b. Also,the width of the MB is equal to the arm 2a of thedisclination dipole (Fig. 1).

In 1978, Vladimirov and Romanov [127] proposeda dislocation-disclination model to describe themechanism of MB propagation. The main idea ofthe model is that the elastic stresses created bythe disclination dipole (Fig. 1), divide a statisticallyarranged dislocation ensemble in front of the MBinto groups of “positive” and “negative” dislocations.The terms, “positive” or “negative”, are caught bythe positive or negative disclinations, respectively.Every event of capturing of a dislocation dipole bythe disclination dipole leads to an elementary act ofthe MB conservative motion. The mechanism [127]has been experimentally approved [8, 113] and usedin modeling the dislocation-disclination kinetics inmetals under large deformation [8, 114-116, 128-


131]. However, there is a number of questions whichare still open. In particular, details of dislocationcapture by disclination dipoles are still unknown.Moreover, computer simulations of elastic interac-tions between partial disclinations and edge dislo-cations [132] (see also Section 2.4 of the presentpaper) have shown that the simple scheme (Fig. 1)proposed in [127], can not provide complete descrip-tion of MB propagation and has to be elucidatedfurther.

Generally speaking, the partial disclinationswhich are used in describing MBs, are associatedby definition with terminated misorientation bound-aries in otherwise perfect crystals [8]. The nature ofsuch boundaries as well as their atomic structuremay be quite different. They can be low angle dislo-cation walls, high angle grain boundaries or twinboundaries. Anyway, the boundary edges may bedescribed (both geometrically and elastically) as thelines of partial disclinations of correspondingstrength. Therefore, the long-range elastic fields (farfrom the misorientation boundaries themselves) ofdisplacement, strain and stress of a terminatedmisorientation boundary may be calculated with thehelp of well-developmed mathematics of wedgedisclinations within the classical [8, 133] or non-classical [50-56] theories of elasticity.

It is well documented [8, 113-116, 119, 123, 124]that possible sites for MB generation in polycrys-talline metals are various faults (defects) of GBsincluding kinks, double and triple junctions of GBs.In nanocrystalline solids, such GB faults often con-tain disclinations, even in an initial as-sintered state[8, 9, 11-27, 126]. However, nowadays there is onlyone theoretical work [125] containing the modelsthat allows to describe the MB generation and pre-dicts appropriate critical conditions as well as re-gimes of MB propagation. The well-known disloca-tion-disclination model of MB propagation byVladimirov and Romanov [127] represents mainlygeometrical features and needs further development.

In paper [125] the models of initial disclinationconfigurations at GB kinks and junctions were pro-posed. It was shown these initial configurations mayserve as sources for MB generation when the ap-plied shear stress achieves a critical value. Furtherdevelopment of the generated MB may be stable orunstable, depending on the level of applied shearstress. If this level is lower than another critical value,the stable regime of MB propagation is realized, ifhigher – unstable. For the case of stable propaga-tion, an equilibrium length of the MB was introducedand studied. All of these results were obtained inthe framework of a quasiequilibrium thermodynamic

approach when only the necessary (not sufficient)conditions for MB generation and propagation wereanalyzed [125]. To obtain the sufficient conditions,one should investigate the dynamics of these pro-cesses which must be based on the dynamics ofcomplicated dislocation structures, first, at the placeof MB nucleation, and second, in front of the propa-gating MB. In a general three-dimensional (3D) casethis problem is very hard. However, in a 2D modelcase, when the lines of all dislocations and partialdisclinations are assumed to be straight and paral-lel to each other, it can be solved by means of com-puter simulation within coupled dislocation-disclination dynamics [125, 132]. It is worth notingthat computer simulation and modeling of discretedislocation ensembles represent nowadays one ofthe most popular topics in theoretical materials sci-ence. Since the late 1980s, 2D and 3D calculationsof the dynamics of interacting dislocations have in-tensively been developed (e.g., see reviews [134,135] and some recent papers on 2D [136, 137] and3D [138, 139] mesoscopic simulations). However,no other attempts but [125, 132] have been knownto the authors, which would be aimed at a correctsimulation of dislocation-disclination ensembles. Theformer computer models [114, 128-131] describingthe coupled evolutional kinetics of dislocations andpartial disclinations practically did not take into ac-count the elastic interactions between them. In con-trast, papers [125, 132] contain the first results of2D computer simulation of the coupled dynamics ofpartial disclnation dipoles and edge dislocationsaimed at studying peculiarities of elastic interac-tion between these defects. These results wereassumed to be used further for checking and refin-ing the existing theoretical (non-computer) modelsof MB development. In the following Sections wewill consider the models presented in [125, 132] inmore detail.

2.1. Initial disclination configurations at grain boundary junctions

Consider a simple scenario [125], shown schemati-cally in Fig. 2, which may be applied to initialdisclination configuration formation at a GB. Letsome gliding dislocations with Burgers vectors

b1

cross a flat GB (Fig. 2a) and move into theneighbouring grain where the gliding dislocationshave Burgers vectors

b2 (Fig.2b). As a result, a wall

of difference dislocations having Burgers vectorsδ

b =

b1-

b2 and interdislocation spacing l, appears

at the crossing site together with the GB kink. On amesoscale level, when the characteristic scale of


b1

b1

b2

ω−

δb

b2

GB δb

ω+

lb /δω =

l

(a) (b) (c)

Fig. 2. A simple model of the formation of an initial grain boundary disclination dipole with the strength ω.

consideration L is much larger than l, the geometryand resulting elastic fields for such a difference dis-location wall may be effectively described as thoseof a two-axes dipole of wedge partial disclinationswith strengths ±ω = ±δb/l [8, 133] (Fig.2c).

Such a simple scheme of GB disclination dipoleformation is possible to occur in NCMs with relativelylarge grains which are capable to deform by usualglide of lattice dislocations. This may concern, forexample, both the micro- and nanocrystalline met-als and alloys fabricated by intensive plastic defor-mation [23, 25, 27, 126, 140]. In a real conventional

Fig. 3. More realistic models of the formation ofinitial grain boundary quadrupole-like disclinationconfigurations.

polycrystalline material, the initial disclination con-figurations at GBs result from more complicatedprocesses which are typical for the stage whentranslational deformation modes are replaced byrotational ones [8, 113]. It is well established [113]that in the vicinity of GB kinks or junctions, the dis-location cells have smaller size than far from suchsites as is illustrated schematically in Fig.3a [125].This means that within these areas, the density ofGB difference dislocations must be much higher thanat straight or smooth segments of GBs. As a corol-lary, one can assume the formation of initial qua-drupole-like disclination configurations at GB kinksor junctions. In Fig. 3b, three possible quadrupole-like disclination configurations are shown where α,β, ω and θ denote the disclination strengths. Forsimplicity, to catch mostly qualitative features andmake rough estimates, the authors [125] consid-ered only wedge partial disclinations. It is worthnoting that the sum strength of any such disclinationconfiguration must be equal to zero or, in other words,the sum strength of positive disclinations must beequal to that of negative disclinations within a GBdisclination configuration (Fig.3b).

2.2. Generation of misorientation bands

2.2.1. Splitting of an initial GB disclination dipoleConsider the simplest initial GB disclination con-figuration that is a GB disclination dipole (Fig.4a)[125]. Let this dipole be under an external shearstress τ. It is also assumed that around the dipoleis typical dislocation-cell structure. Under the ac-tion of the internal shear stress (which is caused by


the GB disclination dipole) and external shear stressτ, the dislocations from the cell boundaries nearestto the disclinations have to glide to them thus form-ing two dislocation walls, i.e., a new disclinationconfiguration (Fig.4b) [125]. The latter may be con-sidered as produced by splitting of the initial GBdisclination dipole. This new split configuration ischaracterized by the split distance d, theinterdislocation spacing l, the dipole arm 2a andthe disclination strengths ±α and ±β satisfying therelationships β = b/l and ω = α + β, where b is themodule of Burgers vector for lattice dislocations andω is the initial strength of the dipole. In fact, the newsplit disclination configuration represents a modelfor a MB of finite length which consists of a newimmobile GB disclination dipole having the strengthα (α-dipole), a new mobile disclination dipole withthe strength β (β-dipole), and two misorientationboundaries of the length d.

To make possible the transition from the initialGB disclination dipole to the new split configura-tion, the total energy of the initial dipole must belarger than that of the new split configuration. Thus,to find critical conditions for such a transition, onehas to calculate and compare these energies.

The total energy of the initial GB disclination di-pole may be calculated as the work necessary togenerate this dipole in its proper elastic stress field[8]. As a result, the energy per unit length ofdisclinations reads

W D aR

a1

2 2 22

1= +

ω ln , (1)

where D=G/[2π(1-ν)], G is the shear modulus, ν isthe Poisson ratio, and R is a characteristic param-

d

2a α+

α−

β+

β− l

τ ω+

ω−

τ

W 1 W 2

βαω += , l = b / β

(a) (b) Fig. 4. Generation of a misorientation band by splitting of an initial grain boundary disclination dipole.

eter of screening of the disclination long-range elas-tic fields (e.g., the size of a sample).

The total energy of the new split configuration(per unit length of disclinations) was written in [125]as the following sum

W W W W d A2

2= + + + −−α β α β γ , (2)

where Wα and Wβ are the elastic energies of α- andβ-dipoles (Fig.4b), respectively, Wα-β is the energy oftheir interaction, γ the effective surface energy of thetwo new misorientation boundaries, and A the workby the external shear stress τ on the displacement dof the mobile β-dipole. The terms Wα and Wβare similarto that given by Eq. 1 with the replacement of ω by αand β, respectively. The energy of the dipole-dipoleinteraction Wα-β was calculated as the work duringthe generation of one dipole in the stress field of an-other dipole, thus resulting in [125]

W D a

R

a d

d

a

a d

d

α β αβ− =

+−

++

2

4 4

41

2

2

2 2

2

2

2 2

2ln ln . (3)

The work A was found in a similar way that gaveA=2τβad.

To estimate γ, the authors [125] used the well-known approximation for the energy of a dislocationcore [141] W

c ≈ Db2/2. Due to geometric reasons

[141], the linear density of “geometrically-necessary”dislocations [142] ρ

g within a misorientation bound-

ary is equal to β/b. However, it was necessary alsoto take into account the density of “statistically-stored” dislocations [142] which have different ori-entations of their Burgers vectors and do not create


any additional misorientation but give their incomeinto the effective surface energy of the boundary.Let the total dislocation density within themisorientation boundaries ρ

t=qρ

g , where q ≥ 1 is a

dimensionless parameter which accounts the pres-ence of “statistically-stored” dislocations. Hence,the number of dislocation cores, N, within amisorientation boundary is estimated as N = ρ

td =

qβd/b and the total core energy of a misorientationboundary is NW

c ≈ qDβbd/2. Thus, the approxima-

tion γ ≈ NWc/d ≈ qDβb/2 was found [125].

Let us consider now the energy difference [125]

∆W W W D a d qbD

d d d d

= − = − −

+ + −

2 1

2

2 2 2 2

4

2 1 1

βτ

α

~ ~

(~

) ln(~

)~

ln~

, (4)

where the dimensionless quantities ~d =d/2a and

~b =b/2a were introduced. The dependence of ∆Won the normalized displacement

~d is given in Fig. 5

for ~b =10-3, α = π/200, and different values of the

external shear stress τ and parameter q [125]. De-pending on τ and q, the character of the curves∆W(

~d ) changes drastically from monotonous de-

creasing (Fig. 5a) to non-monotonous one (Fig. 5b).In the first case, when q=1 while τ and d are smallenough (here 0 ≤ τ ≤ 0.007D and

~d <1), ∆W<0 for

~d >0. This means there is no energy barrier for thegeneration of a MB. In the second case, when q=10with the same values of τ and

~d , ∆W > 0 for

~d <

~d

c,

where ~d

c is determined by the equation

∆W(~d =

~d

c)=0, and ∆W<0 for

~d >

~d

c. This means

there is an energy barrier for the formation of a MBunder such a small τ. Following [125], let us intro-duce a characteristic critical value τ

g which is de-

termined by the equation ~d

c=~ , where ~

=l/2a isthe normalized spacing between the “geometrically-necessary” dislocations creating the tilt misorien-tation angle β. If τ <τ

g, the generation of a MB

nucleus (an initial GB disclination dipole plus onedislocation dipole joined to this dipole and localizedat the distance ~

from it) is energeticallyunfavourable; if τ > τ

g, it is favourable. For the situa-

tion illustrated in Fig.5b, ~ ≈ 0.1 was assumed andhence τ

g ≈ 0.005D for q=10.

Using Eq. 4, one can analytically estimate τg for

the case ~d <<1 as follows [125]

τα

α

ω

αgDb q

b≈ −

−−

−

~ ~

~ ln~

/~ ,

1

1

2 1 (5)

d~

(a)

)/( 2aDW β∆

d~

gτ

(b)

)/( 2aDW β∆

Dgτ

α~ (c) Fig. 5. Energy difference ∆W via the normalizedmisorientation band length

~d (a, b) and critical shear

stress τg/D via the normalized grain boundary

disclination dipole strength ~α (c) for the model of amisorientation band being generated at a grainboundary disclination dipole. The plots ∆W(

~d ) are

given for the external shear stress τ/D=0, 0.001,0.003, 0.005, and 0.007 (from top to bottom) for thetwo different values of the parameter q=1 (a) and 10(b). The plots τ

g(~α )/D are shown for the initial di-

pole strength ω =0.01, 0.02, 0.03, 0.04, and 0.05(from top to bottom).

(c)


where ~α =α/ω. The plots τg(~α ) are shown in Fig.5c

for q=10 and various values of the initial dipolestrength ω [125]. One can conclude that τ

g de-

creases when both ~α and ω increase that is in ac-cordance with physical intuition. When q=10, thenumerical estimate for τ

g gives the values of order

G/1000-G/400. The lower limit fits well with typicalexternal stresses at the end of Stage II of deforma-tion curves for BCC and FCC metals [113], whilethe upper limit corresponds to the level of deformingstress observed in NCMs [16, 17, 78].

2.2.2. Splitting of an initial quadrupole-likedisclination structure at a GB kinkConsider now a model which seems to be morerealistic than the previous one. It is a GB kink witha quadrupole-like disclination structure (Fig.6) [125].This configuration is characterized by the disclinationstrengths -α, +γ, -ω and +θ which satisfy the equa-tion: α + ω = γ + θ. As a result of the dislocationrearrangements which are suggested similar tothose considered above, the quadrupole-likedisclination structure issues a mobile two-axes di-pole of partial wedge disclinations having thestrengths ±β and moving under the action of theGB disclination stress field as well as an externalshear stress τ (Fig. 6) [125]. In a special simplecase, the new split configuration was suggested toconsist of three two-axes disclination dipoles, onemobile (β-dipole) and two immobile (α- and θ-dipole).Again, to find critical conditions for such a splittingtransformation, the total energy, W

1, of the initial

quadrupole-like disclination structure and that, W2,

of the new split configuration were compared.The energy difference ∆W=W

2-W

1 was calcu-

lated as shown above and resulted in [125]

y

α+

θ+ β−

β+

x x2

y2

y1

x 1

θ−

α− d

2a

τ

Fig. 6. Model of misorientation band generation bysplitting of an initial quadrupole-like disclination con-figuration at a grain boundary kink.

∆W D a d qbD

d d d d

x y d x y d

= − −

+ + + −

+ −

βτ

α θ

α θ

2

2 2 2 2

1 1 2 2

4

1 1

~ ~

( ) (~

) ln(~

)~

ln~

(~ , ~ ,~

) (~ , ~ ,~

) ,Ψ Ψ (6)

where

Ψ( , , ) ( ) ln( )

( ) ln ( )

( ) ln( ) ]

( ) ( ) ln ( ) ( ) ,

x y z x y x y

x y x y

x z y x z y

x z y x z y

= − + + +

+ − + − +

− + − + −

− + − − + −

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

1 1

1 1

(7)

~xi=x

i/2a and ~yi =y

i/2a, i=1,2.

The curves ∆W(~d ) are given in Fig.7 for

~b =10-3,

α = π/200, θ = π/300, ~x1= -~x

2 = ~y

2 = -1, ~y1=2 and

different values of τ and q [125]. Depending on τand q, the curves ∆W(

~d ) behave in quite different

manners. When q ≤ 5 (e.g., see Fig.7a for q=1),they are similar to those considered in the previoussection. When q ≥ 7 (e.g., see Fig. 7b for q=10),some of them increase monotonously. One canconclude that for the given values of the parameters,the critical external shear stress τ

g is higher than

for the case of GB disclination dipole splitting (Sec-tion 2.2.1).

Introducing (6) with (7) into the equation∆W(

~d =~

)=0, for ~d << 1 the authors [125] obtained

τ α θ

α θ

ω

α θ

α θω

αω

α θ

θω

α θ

g

Dbq

b b

bx y d

b

x y db

≈ −+

− −−

− −

+ − − =− −

−

=− −

~~

(~~

)

( ~ ~)

ln~

/~ ~

( ~ ~) ~

~ ~ , ~ ,~

~/~ ~

~ ~ , ~ ,~

~/~ ~ ,

4 11 2

1

14 1

1

2

1 1

2 2

Ψ

Ψ

(8)

where ~α =α/ω and ~θ =θ/ω. Three-dimensional plotsτ

g(~α , ~θ ) are shown in Fig. 7c for q=10, ~x

1= -~x

2 = ~y2

= -1, ~y1=2 and two values of the initial strength ω

=α +β + θ, ω = 0.01 and 0.05 [125]. It is seen that τg

decreases when ~α , ~θ and ω increase; that is againin accordance with intuition. When q=10, the nu-merical estimate for τ

g gave again the values of or-

der G/1000-G/400.


d~

(a)

)/( 2aDW β∆

d~

(b)

)/( 2aDW β∆

Dgτ

θ~

α~

(c)

Fig. 7. Energy difference ∆W via the normalizedmisorientation band length

~d (a, b) and critical shear

stress τg/D via the normalized grain boundary

disclination dipole strengths ~α and ~θ (c) for themodel of a misorientation band being issued by aquadrupole-like disclination configuration localizedat a grain boundary kink. The plots ∆W(

~d ) are given

for the external shear stress τ/D=0, 0.001, 0.003,0.005, and 0.007 (from top to bottom) for the twodifferent values of the parameter q=1 (a) and 10 (b).The plots τ

g( ~α ,~θ )/D are shown for the initial

disclination strength ω=0.01 (the upper surface) and0.05 (the lower surface).

2.2.3. Splitting of an initial quadrupole-likedisclination structure at a GB triple junctionAnother more realistic model is a GB triple junctionwith a quadrupole-like disclination structure of thegeometry shown in Fig.8 with d=0 [125]. Again thisdisclination configuration issues a mobile two-axesdipole of partial wedge disclinations having thestrengths ±β and moving under the action of the GBdisclination stress field as well as external shearstress τ. The new split configuration was suggestedto consist of four two-axes disclination dipoles, onemobile (β-dipole) and three immobile (α-, θ- and γ-dipole). These three immobile dipoles are formed bythe “central” negative triple-junction disclination hav-ing the strength -ω and three positive disclinationshaving the strengths +α, +θ and +γ and localized atthe joined GBs. The relation ω = α + θ + γ must bevalid.

The procedure of calculation of the difference inthe total energies of the initial quadrupole-likedisclination structure and new split configuration isabsolutely similar to that described in Section 2.2.1.The final result is [125]

∆W D a d qbD

d d d d

x y d x y d

= − −

+ + + + −

− −

βτ

α θ γ

θ γ

2

2 2 2 2

1 1 2 2

4

2 1 1

~ ~

( ) (~

) ln(~

)~

ln~

(~ , ~ ,~

) (~ , ~ ,~

) ,Ψ Ψ (9)

where the ΨΨΨΨΨ-function is given by (7). The depen-dences ∆W(

~d ) are plotten in Fig.9 for b=10-3, α = π/

300, θ = π/100, γ = π/200, ~x1= -~x

2 = ~y2 = ~y1=-1, and

different values of τ and q [125]. This set of param-eters gives the curves which are very similar to thoseconsidered in Section 2.2.1 (Fig.5) with the closevalues for the critical external stress τ

g when q ≥ 5.

In comparing these plots with those in Fig.7, onecan conclude that both ∆W(

~d ) and τ

g depend on

the concrete arrangement of the disclinations withinthe initial quadrupole-like disclination structure.

The substitution of (9) with (7) into the equation∆W(

~d =~

)=0 gives for ~d << 1 the following result

[125]

τ α θ γ

α θ γ α θ γ

α θ γ θα θ γ

γα θ γ

g

Dbq

b b

bx y d

b

x y db

≈ −+ +

− − −−

− − −−

− − − =− − −

+

=− − −

~~

( ~ ~ ~)

( ~ ~ ~)ln

~/

~ ~ ~

( ~ ~ ~) ~~ ~ , ~ ,

~~

/~ ~ ~

~ ~ , ~ ,~

~/

~ ~ ~ ,

2

4 11 2

1

14 1

1

2

1 1

2 2

Ω

ΩΨ

Ω

ΨΩ

(10)


where ~α =α/Ω, ~θ = θ/Ω, ~γ =γ/Ω, and Ω = α+β+θ+γis the strength of the initial quadrupole-likedisclination structure.

Using formula (10), the authors [125] investigatedthree-dimensional plots τ

g(~α ,~θ ) for different constant

values of ~γ , τg(~α , ~γ ) for those of ~θ , and τ

g(~θ , ~γ )

for those of ~α at the following parameter values: q=5,x

1 = -x

2 = y

1 = y

2= -1, and two values of the initial

disclination strength Ω. All the plots turned out to bevery similar to those shown in Fig.7c and hence wedo not represent them here. They demonstrated thatτ

g decreases when ~α , ~θ , ~γ and Ω increase [125].

When q=5 - 10, the numerical estimate for τg gave

again values of the order G/1000-G/400.

2.3. Regimes of misorientation band propagation

In the previous sections, we have considered threedifferent models of MB generation by using andanalysing energy expressions (4), (6) and (9) forthe limit of

~d << 1. To study further propagation of

MBs, the authors [125] used the same expressionsbut for

~d ≥ 1. Consider again the simplest model,

i.e., the splitting of an initial GB disclination dipole(see Section 2.2.1, Fig.4).

The characteristic example of graphical repre-sentation of Eq. 4 for the case

~d ≥ 1 is given in

Fig.10 for α = π/200, b = 10-3, q = 3, and differentvalues of the external shear stress τ [125]. Depend-ing on the value of τ, the curves ∆W(

~d ) behave in

different ways. When τ is smaller than some limit-ing quantity τ

p (e.g., τ

p ≈ 0.003D for q=3), the curves

∆W(~d ) are non-monotonous and achieve their

minima, which determine equilibrium values of theMB length ~d

eq. When τ >τ

p, the curves ∆W(

~d ) goes

with monotonous decrease and ~deq

is absent.To find an analytical estimate for τp, one can solve

the equation ∆W(τ)=0 for the limiting case ~d → ∞

that gives [125]

τp Dqb= ~

. (11)

Obviously this provides the linear relation betweenτ

p and q, where q characterizes the effective sur-

face energy of the misorientation boundaries.The equilibrium length ~d

eq is found from the stan-

dard equation ∂∆

∂

W

d~ =0 with

∂

∂

2

2

∆W

d~ >0 and

~d >1. The

final result reads [125]

~~

/,d

qb Deq

≈−

α

τ (12)

α+

ω−

1x

1y 2y

θ+ γ+

β+

β− 2x x

y

d

2a

τ

ωθγα =++

Fig. 8. Model of misorientation band generation bysplitting of an initial quadrupole-like disclination con-figuration at a triple junction of grain boundaries.

(a)

d~

)/( 2aDW β∆

d~

(b)

)/( 2aDW β∆

Fig. 9. Energy difference ∆W via the normalizedlength

~d of a misorientation band being issued by a

quadrupole-like disclination configuration localizedat a triple junction of grain boundaries under theexternal shear τ/D=0, 0.001, 0.003, 0.005, and 0.007(from top to bottom) for the two different values ofthe parameter q=1 (a) and 10 (b).


d~

)0(~ =τeqd

pτ

)/( 2aDW β∆eqd

~

D/τFig. 10. Energy difference ∆W via the normalizedlength

~d of a misorientation band being generated

at a grain boundary disclination dipole and propa-gating far from it under the external shear stressτ/D=0, 0.001, 0.003, 0.005, and 0.007 (from top tobottom) for the parameter q=3.

Fig. 11. Normalized equilibrium length, ~d

eq, of a

misorientation band via the external shear stressτ/D for the different values of the parameter q=1, 3,5, 7, and 10 (from top to bottom).

where τ<τp. The dependence of ~

deq

on τ is illus-trated in Fig. 11 for α=π/200,

~b = 10-3, and different

q. One can see that ~deq

increases with increasing τand decreasing q.

It was thus shown in [125] that depending onthe external shear stress τ, two main regimes ofMB propagation are possible: stable and unstablepropagation. When τ<τ

p, the MB propagation is

stable and may be characterized by the equilibriumlength ~

deq

. When τ>τp, the MB propagation is un-

stable and there is no equilibrium length. In reality,this means that the MB will propagate until it meetsan obstacle like a GB or another MB (or some otherdefect configurations playing the role of obstacles)in which case the question of its further propagationmust be considered again.

2.4. Computer simulation of dislocation- disclination interactions

To check and refine the models of MB propagationthrough an ensemble of edge dislocations as wellas to calculate some important parameters of dis-location-disclination interactions (e.g., the effectivelength of dislocation capturing by a disclination di-pole; this length was treated as the distance from apartial disclination line at which the corresponding

edge dislocation must be stopped to provide theconservative motion “ahead” of the partialdisclination), the method of 2D dislocation-disclination dynamics was used in [132]. This ap-proach and some results were also quoted in [125].

The computer code objects were straight edgedislocations and straight wedge disclinations whichcould move within a two-dimensional rectangular boxof an infinite elastically isotropic medium (Fig.12).Periodic boundary conditions were realized. The boxsizes was chosen as 1x1 mm2. The defect lineswere normal to the box plane. The dislocations werecharacterized by their Burgers vectors b

x or b

y, co-

ordinates (x(i),y(i)) and velocities( , ( ) ( )x yi i ), where i=1… n and n is the number of dislocations. Alldisclinations were arranged in dipole configurationswhich were assumed to be immobile and consid-ered as sources of elastic fields. The dipoles werecharacterized by their strengths ω(j), by the sizeand orientation of their arms and by the coordinatesof the arm central points (X(j),Y(j)), where j = 1 … Nand N is the number of disclination dipoles.

In such a computer model [132], the disloca-tions can move by gliding or climbing under the ac-tion of the total force due to external loading, elas-tic fields of other defects and dynamic friction. Thedislocation dynamics is than ruled by Newton’s law


m x Fi i

x

i( ) ( ) ( ) ,= (13)

m y Fi i

y

i( ) ( ) ( ) ,= (14)

where m(i) is the effective mass of the i-th disloca-tion, ( )x i and ( )y i are the x- and y-components of itsacceleration, respectively. The p-component (p=x,y)of the resulting force on the i-th dislocation, F

p(i), is

assumed to be a superposition

F F F Fp

i

p

def i

p

fr i

p

ext i( ) ( ) ( ) ( )= + + (15)

with F e b sp

def i

mpl lk

i

k

i

m

i( ) ( ) ( ) ( )= σ being the elastic forcedue to all other defects, F v b

p

fr i

p

i( ) ( )( )= −τ the dy-namic friction force, and F b

p

ext i ext

p

i( ) ( )= τ the externaldriving force. Here e

mpl denotes the permutation sym-

bol, σk

i( ) is the resulting elastic stress due to allother defects that is measured at the point wherethe ith dislocation is located, b

k

i( ) is the k-compo-nent of its Burgers vector, and sm

i( ) is the m-compo-nent of the unit vector tangent to the dislocationline. All indexes p, m, l and k may denote x- or y-components. The shear stress τ(v) characterizesthe crystalline lattice friction and depends on thedislocation velocity v. The external stress τext is cre-ated by an external load applied to the solid.

Giving the initial coordinates and velocities ofevery defect, the system of motion equations (13)-(14) with (15) was solved numerically and the de-pendences of the coordinates x, y and the velocitycomponents v

x and v

y on the time t were found.

cl

)(ix

ω+

ω−

xb

yb

y

x

)(iy

)( jX

)( jY

Fig. 12. 2D box for computer simulation of dislocation-disclination ensemble. The parameter lc denotes the

length of dislocation capturing by the disclination dipole.

The above approach was used to consider theelastic interaction of a gliding edge dislocation witha two-axes wedge disclination dipole in pure cop-per [132]. The module of the Burgers vector wastaken as b

x = 0.256 nm, and hence the correspond-

ing dislocation mass (per unit length of the disloca-tion) follows as m = ρ b

x2/2 ≈ 1.4.10-9

kg.m-1 (see [143],

p.73). The position of a disclination dipole was fixedat the central point (X=500 µm, Y=500 µm) of thesimulation box. In obtaining the following results,the force F

p

ext ( was neglected to catch the main fea-tures of elastic dislocation-disclination interactionsas they are. The elastic force F

p

def (=σxy

bx was calcu-

lated with the disclination elastic stress field σxy

taken from [8]. The dynamic friction force Fp

fr ( wastaken as F

p

fr ((t)=-Bv(t), where B =1.7.10-5 Pa s thatis characteristic for pure copper [144] (see also[143], p.76).

Some typical situations were studied for differ-ent orientations of the dipole arm, initial positionsand velocities of the dislocation [125, 132]. It wasshown that the dislocation behavior may stronglyvary depending on the problem parameters. How-ever, it is ruled mainly by the elastic stress field ofthe disclination dipole.

For example, consider a disclination dipole hav-ing the strength ω =0.01 and the arm 2a = 100 nm.Let the dislocation move with the initial velocityv0=0.01 m s-1 quite far from the disclination dipole inthe manner shown in Fig. 13 [125, 132]. The corre-sponding plots for the dislocation coordinate x(t)(dashed line) and velocity v(t)= x (t) (solid line) are


given in Fig.13a. In Fig.13b, the distribution of thedipole shear stress σ

xy in units of Dω is shown. The

empty and black circles schematically denote inFig.13b the initial and final dislocation positions,respectively. In fact, the dislocation starts to glidebeing under the action of positive shear stress σ

xy of

the disclination dipole (the initial dislocation posi-tion is at the upper left corner in Fig.13b). As a re-sult, the dislocation glides with an acceleration andits velocity becomes very high when it passes overthe disclination dipole (Fig.13a), in the field of stron-gest disclination stresses (the upper central regionin Fig.13b). At the same time, the friction force alsoachieves it maximum value. Therefore, when thedislocation has passed the region of maximumstress values over the dipole, its velocity starts todecrease fast. As a result, the dislocation movesaway from the dipole with a negative accelerationuntil the point where the dipole stress turns to zero(the upper right region in Fig.13b). One can see there

is no effect of dislocation capturing by thedisclination dipole in this case.

The question arises: what are the problem con-ditions at which the dislocation capturing by thedisclination dipole would be possible for a givendefect configuration? The calculations [125, 132]demonstrated that the dislocation capturing is onlypossible when its initial position is just over the cen-ter of the disclination dipole, at a very small dis-tance of it, and the disclination strength is very small.Thus, let the dislocation begin its motion just nearthe disclination dipole having the ten times lowerstrength ω =0.001 and the same arm (Fig.14). Thedislocation is accelerated from v

0=0 within the re-

gion of the relatively higher positive shear stressesof the dipole (the initial dislocation position is at thecentral part of Fig.14b) and stopped after at the zero-value stress contour above the dipole, just near thepositive disclination (see Fig.14b). This means thatthe disclination dipole has captured the dislocation.

b

0

.03 .05 .07

-.03

-.05

x/2a

y/2a

.2

0 50 100 150 200 250 300 350

499,0

499,5

500,0

500,5

501,0

501,5

X (µm)

t (ns)

ω+ ω−

xb

0V

s

l

2a

a

0

50

100

150

200

250

300

350

V (m/s)

b

.8 .6 .4 .2 .1 0 -.1

-.2

y/2a

x/2a -.4

t (ns)

0,0 0,1 0,2 0,3 0,4 0,5 0,6

500,05

500,06

500,07

500,08

500,09

500,10

500,11

X (µm)

a

-100

0

100

200

300

400

V (m/s)

ω+ ω−

00 =V

l

Fig. 13. Accelerated glide of an edge dislocationalong the arm of a disclination dipole. The dashedand solid curves in plot (a) represent the dislocationposition x(t) and velocity v(t), respectively, when thedislocation glides in the field of the dipole positivelong-range shear stress σ

xy (b). The calculations have

been carried out for the following values of param-eters: ω =0.01, b

x=0.256 nm, 2a=100 nm, s=1000

nm, l =1100 nm, x0=499 µm, and v

0=0.01 m.s-1.

Fig. 14. Accelerated glide of an edge dislocationalong the arm of a disclination dipole and capturingof the dislocation by the dipole. The dashed andsolid curves in plot (a) represent the dislocationposition x(t) and velocity v(t), respectively, when thedislocation glides in the field of the dipole positiveshort-range shear stress σxy (b). The calculationshave been carried out for the following values of pa-rameters: ω=0.001, b

x=0.256 nm, 2a=100 nm, s=50

nm, l =1nm, x0=500.05 µm, and v

0=0.


However, the capturing occurs at very small dis-tances from the dipole only. This distance (here l =1nm) is much smaller than the spacing l

c ≈ b

x/ω =256

nm between dislocations in the low-angle tilt wallswhose edges are described by the disclination di-pole (see Fig.12). Therefore, such a small capturinglength can not provide the mechanism of conserva-tive motion of a disclination dipole in direction nor-mal to its arm by capturing or issuing edge disloca-tions.

The computer simulations [125, 132] show thatthe dynamics of the edge dislocation is totally ruledby the elastic field of the disclination dipole. Thedislocation is accelerated when it appears in theregion of increasing disclination stress, while fur-ther, when this field decreases, the dislocation ishampered by the force of dynamic friction and al-ways stopped at the line of zero-level disclinationshear stress. Thus, the dislocation behavior is de-termined by its initial position with respect to thedisclination dipole and does not depend, in fact, onits initial velocity (at least for those velocity valueswhich were used in simulations). The computermodel [125, 132] approved that the two-axes dipoleof wedge disclinations can move conservatively alongthe direction parallel to the dipole arm by means ofcapturing edge dislocations. However, the dipolemotion along the normal to its arm cannot be ex-plained correctly within the existing theoretical mod-els and needs further investigation.

Based on the results of the theoretical modelsfor MB generation and propagation considered inSection 2, one can make the following conclusions.· Disclination models of MB generation at GB

faults like kinks and GB junctions predict theexistence of a critical external shear stress τ

g

which is necessary for the MB generation eventstake place. The numerical estimate for the criti-cal stress gives values of the order G/1000-G/400; the lower limit is in a good accordance withtypical external stresses at the end of Stage II ofdeformation curves for conventional polycrystal-line BCC and FCC metals, while the upper limitcorresponds to the level of deforming stress ob-served in NCMs.

· The critical shear stress τg depends strongly onthe geometry and strengths of initial GBdisclination configurations, on the misorientationangle as well as on the effective surface energyof arising misorientation boundaries. It increaseswhen the initial disclination strength decreasesand the misorientation angle increases. The criti-cal stress varies in direct proportion to the effec-tive surface energy of misorientation boundaries.

· Disclination models of MB propagation predictthe existence of a limiting external shear stressτ

p, which separates two main regimes of MB

propagation: stable and unstable propagation.When the external stress is lower than the limit-ing stress, the MB propagation is stable and maybe characterized by the equilibrium MB lengthwhich increases when the external stress in-creases and the effective surface energy ofmisorientation boundaries decreases. If the ex-ternal stress is higher than the limiting stress,the MB propagation is unstable. The limitingexternal stress varies in direct proportion to theeffective surface energy of misorientation bound-aries.

· Computer simulations by means of a 2D dislo-cation-disclination dynamics code have shownthat the existing models of the disclination di-pole motion must be reconsidered with takinginto account the conclusion that the disclinationdipole cannot move concervatively along the nor-mal to its arm by capturing edge dislocations.

3. MOTION OF GRAIN BOUNDARYDISCLINATIONS

Transformations of GBs often strongly influence boththe structure and the properties of polycrystallineand nanocrystalline materials, e.g. [8, 20, 26, 74,75, 78, 107, 114, 145-173]. In particular, changes ofmisorientation parameters of GBs, that are capableof resulting in grain rotations, have been experimen-tally detected in polycrystalline and nanocrystallinematerials under (super)plastic deformation (see, e.g.[152-156]). Ke et al. [168, 169] observed in situ thatplastic deformation of nanocrystalline gold films withthe grain size d<25 nm occured through GB slidingand grain rotation near the tips of opening cracks.Noskova et al. [170-172] have also reported aboutin situ TEM observation of GB sliding and grain ro-tation in nanocrystalline pure metals (Cu, Ni and Tiwith grain size d=20-40 nm) and Fe

73.5Cu

1Nb

3Si

13.5B

9

alloy (with grain size d ≈ 10 nm) under active unidi-rectional tension with strain rate 10-5 s-1. It is impor-tant that rotation of grains does not always need toapply an external mechanical load. Sometimes it isenough to carry out a special thermal treatment.For example, grain rotations have been observedexperimentally in thin films of gold under thermaltreatment [173].

According to contemporary theoretical represen-tations of GBs, changes of their misorientation pa-rameters occur via motion of GB disclinations [8,26, 174]. Such disclinations are intensively gener-


ated in polycrystals and NCMs under highlynonequilibrium conditions of their fabrication. Theseconditions are typical for the technologies of inten-sive plastic deformation and ball milling that givesubmicro- and nanostructured materials withnonequilibrium GBs containing GB and triple junc-tion disclinations [23, 25, 27, 126, 140] (see alsoSection 2). It is quite natural that in such NCMs,like in conventional polycrystals under large defor-mation, the processes of rotational deformation areassumed to be realized by means of concervativemotion of disclination dipoles [125, 175-177].

Motion of GB disclinations in plastically deformedmaterials is commonly treated as that associatedwith absorption of lattice dislocations (that are gen-erated and move in grains under the action of me-chanical load) by GBs [8, 156]. This micromecha-nism, according to paper [156], is responsible forexperimentally observed grain rotations in fine-grained materials during (super)plastic deformation.However, the consequent motion of a GB disclinationalong a GB requires processes of the dislocationabsorption to be well ordered in space and time. Inparticular, lattice dislocations with certain Burgersvectors have to reach the GB in only vicinity of thedisclination that moves along the boundary due toacts of absorption of these dislocations. This is inan evident contradiction with the fact that sourcesof lattice dislocations in plastically deformed mate-rials are commonly distributed in a rather irregularway within a grain and, therefore, are not capable ofproviding the regular flow of dislocations to thedisclination moving along a GB. Moreover, in thesituation with grain rotations in thin films of goldunder thermal treatment [173], absorption of latticedislocations by GBs hardly plays an important rolein changes of GB misorientation parameters, be-cause the dislocation density in grain interiors istoo low to cause grain rotations.

Based on these conclusions, the alternative theo-retical models [176-179] have been proposed. Thesemodels have used the idea that GB disclinationscan move along GBs due to emission (in contrastto absorption [156]) of lattice dislocations from GBsinto adjacent grain interiors. It has also been as-sumed that the GB disclinations are partial, i.e. theymay be represented as ragged walls of edge dislo-cations [8, 133] (this assumption is always valid inpractice), and accordingly wedge in nature. In thiscase, a GB disclination can emit a pair of edge lat-tice dislocations with the Burgers vectors corre-sponding to the glide systems in the adjacent grains.The sum of these lattice Burgers vectors are sup-posed to be equal to the Burgers vector of a GB

dislocation from the ragged wall (strictly speaking,this condition is not necessary and has been takenfor the sake of simplicity; one could also includeinto consideration a difference GB dislocation whoseBurgers vector would compensate a possible dis-parity). The emission of dislocation pairs permitsthe GB disclination to move conservatively alongthe GB, and the events of dislocation emission aredetermined by the conditions at the GB disclination.We consider below the main results of the models[176-179] which describe the rotational mode of plas-tic deformation in fine-grained materials as that whichis realized through conservative motion of dipoles ofGB disclinations emitting pairs of lattice disloca-tions.

3.1. Changes in grain boundary misorientation

Following the theory of GBs, two fragments of a GBthat are characterized by different values of tiltmisorientation parameter are divided by the line of aGB disclination [8, 26, 114, 174]. More precisely,the line that separates the two boundary fragmentswith tilt misorientation parameters θ

1 and θ

2, respec-

tively, is described as the line of a GB wedgedisclination with strength ω=θ1-θ2 (Fig.15). In theframework of the discussed representations, evolu-tion (in time) of tilt misorientation along GBs istreated as that related to the motion of GBdisclinations.

In the framework of the model [178, 179], an el-ementary act of transfer (by distance l) of a GBdisclination with strength ω is accompanied byemission of two lattice dislocations with Burgersvectors

b1 and

b2 from the GB into the adjacent

grains I and II, respectively (Fig.16). The disclination

Fig. 15. Grain boundary disclination (black triangle)separates boundary fragments characterized by dif-ferent values, θ

1 and θ

2, of tilt misorientation.


with strength ω can be treated as that terminating aragged wall of periodically spaced GB dislocationswith identical Burgers vectors

b and spacing (pe-riod) l. This dislocation representation is relevant toboth small- and large-angle GBs with GB disloca-tions having a “large” crystal lattice Burgers vectorin the case of small-angle GBs and a “small” DSC-

σ

1τ

2φ 1φ

2τ

b2

ω−

ω+

p

y1

y

x1

y2

x

p

L

l

x2

L1

b1

σ

Grain II Grain I

Fig. 16. Displacement of the wedge disclination(black triangle) with the strength ω from its initialposition (dashed triangle) by the distance l is ac-companied by the emission of two lattice dislo-cations with Burgers vectors

b1 and

b2. The ω-

disclination moves along the grain boundary plane(which is perpendicular to the figure plane and inter-sects it along the y-axis) towards another disclination(white triangle) with the strength -ω. The x-axis isnormal to the grain boundary plane. The x

1y

1 and

x2y

2 coordinate systems are associated with the

gliding planes of emitted dislocations. φ1 and φ2 arethe angles between the normal to the grain bound-ary plane and the gliding planes of respectively thefirst and the second dislocations. τ1 and τ2 are theshear stresses acting along the gliding planes ofthe first and second dislocation, respectively. L andL1 are the distances between the disclinations be-fore and after displacement of the ω-disclination.

lattice Burgers vector in the case of large-angle GBs[146, 180]. With the spacing l between such dislo-cations assumed to be the distance of an elemen-tary transfer of the disclination, Burgers vectors ofthe GB dislocations and strength ω of the GBdisclination obey the equations:

b1 +

b2 =

b and|

b |=b ≈ lω. From the former equation, one finds thatthe Burgers vector magnitude b is in the followingrelationship with the magnitudes, b

1 and b

2, of

Burgers vectors of the emitted dislocations:

b bb

b

b

= + =

+

1 1

2

1

2

1

1 2

2

cos cos

sin( )

sin,

φ φ

φ φ

φ

(16)

where φ1 and φ

2 are the angles between the normal

to the GB plane and the gliding planes of the re-spectively first and second emitted lattice disloca-tions (Fig.16). Notice that b < b

1, b

2 in the case of

large-angle GBs (containing dislocations with DSC-lattice Burgers vectors).

The consequent emission of lattice dislocationpairs (Fig.16) causes change of GB misorientationalong large fragments of the GB plane. This pro-cess is capable of giving rise to the rotation of agrain as a whole.

The above model is approximate. First, the au-thors [178, 179] have restricted their considerationto the z-independent situation with a disclinationdipole at a tilt boundary characterized by two mac-roscopic geometric parameters, the angles φ

1 and

φ2 (Fig.16). However, tilt boundaries which are ef-

fectively described in terms of disclinations [8], rep-resent the most widespread type of GBs in real ma-terials, in which case this model covers most realGBs. Second, the choice of the disclination dipoleas a subject of the theoretical analysis (addressedthe GB disclination motion) in this model has beenrelated to the two following aspects: (i) the long-range stress field of the moving disclination shouldbe screened, and (ii) there is an experimental evi-dence (see reviews [8, 114] and references therein)that disclinations form dipole, quadrupole and mul-tipole configurations in real materials under largedeformation. In principle, one could consider an in-dividual GB disclination whose elastic fields wouldbe screened by outer boundaries of the solid aswas the case in works [20, 181-183]. Such a con-sideration is sometimes reasonable but always sig-nificantly more complicated. That is why the au-thors [178, 179] have chosen the simplest way to


screen the long-range stress field of the moving (first)disclination by using the simplest self-screeneddisclination configuration (disclination dipole) amongothers which are observed in real materials. Third, ithas been supposed that the disclination dipole con-sists of mobile (first) and immobile (second)disclinations (Fig.16). As with the first disclination,the second disclination may also represent the dis-continuity of misorientation across the GB, or maybe a triple junction disclination, etc. It may also beas mobile as the first disclination is. This would notchange results of the model, because the authors[178, 179] have analysed only the energetic possi-bility for an elemental displacement of the firstdisclination. It would hardly be expected that boththe disclinations must make such elemental“jumps” simultaneously. Therefore, one can treat oneof them as “mobile” while another one as “immo-bile”.

Following papers [178, 179], let us consider en-ergetic characteristics of the GB disclination mo-tion under discussion (Fig.16). The dipole of the GBdisclinations can be treated as the defects termi-nating the GB dislocation wall of finite extent (Fig.17),in which case an elementary transfer of the movingdisclination occurs via the splitting of one GB dislo-

Fig. 17. Disclinations composing a dipole terminatea ragged wall of periodically (with period l) spacedgrain boundary dislocations with either a crystal-lattice or DSC-lattice Burgers vector in respectivelysmall- and large-angle boundaries. An elementarytransfer of a moving disclination, shown in Fig. 16,is accompanied by the splitting of one of the grainboundary dislocations into the two lattice disloca-tions.

cation belonging to the wall into two lattice disloca-tions. The motion of the GB disclination occurs un-der the action of external mechanical load which, inthe framework of the model, causes uniaxial stressparallel with the GB plane and promotes the motionof the emitted dislocations (Fig.16). The elemen-tary transfer of a GB disclination by distance l, ac-companied by emission of two lattice dislocations,is energetically favourable, if the energy (per unitdisclination length) W

2 of the defect configuration

resulted from the transfer is lower than the energyW

1 of the pre-existent configuration (before the trans-

fer): ∆W = W2-W

1 <0.

The pre-existent configuration represents a di-pole of GB disclinations with the distance L (thedipole arm) between them. The energy W

1 of this

system is given by

W ED L R

LNE

d b

c

1

2 2

2

1

2= = + +

ωln , (17)

where the first term is the strain energy of thedisclination dipole (see formula (1) in Section 2.2.1)and the second term is the sum energy of cores ofthe N GB dislocations which compose the raggedwall (Fig.17). The core energy is approximated as[141] E

b

c ≈ Db2/2.The energy density W

2 of the dipole configura-

tion resulted from the elementary transfer (Fig.16)can be written as follows [178, 179]:

W E E E E E

E E E

d b b d

b

d

b

b

b b b

2 1 2

1 2

2

1 1 2

= + + + + +

+ +

'

.σ σ

(18)

Here Ed

' denotes the strain energy of the resultantdisclination dipole characterized by the dipole armL

1=L-l, Eb1

(Eb2) the self energy of the first (second)

emitted dislocation, Ed

b1 (Ed

b2 ) the energy that char-acterizes elastic interaction between the first (sec-ond) emitted dislocation and the disclination dipole,E

b

b

2

1 the energy that characterizes elastic interac-tion between the emitted dislocations, and E b

σ1 (E b

σ2 )

the work of the external stress σ spent to transferthe first (second) dislocation to its position shownin Fig.16.

The strain energy Ed

' of the disclination dipolewith the arm L

1 is given by formula (17) with the

substitution of L for L1 and N for (N-1).The self energies of the dislocations read [141]

EDb R

rb

i

c

i

= +

2

21ln , (19)


where i=1,2, and rc is the dislocation core radius

(which is assumed to be the same for both the dis-locations under consideration).

Calculation of the other terms of the sum (18) isgiven in paper [179]. The final expressions for theenergies E

d

bi , Eb

b

2

1 , and E biσ are as follows:

ED

b LR L RL

p L pL

R R

p p

d

b

i i

i

i

i

i

i =+ +

+ +−

+ +

+ +

2

2

2

2

2

2 2

2 2

2 2

2 2

ω φφ

φ

φ

φ

cos lnsin

sin

lnsin

sin,

(20)

ED

b b

p R pR

ep

pR

p R pR

b

b

2

1

21

2

2 1

2

2

1 2

1 2

2 2

1 2

2

1 2

2

1 2

2 2

1 2

= +

++ + +

+ +

−+

+ + +

cos lncos

cos

sin

cos,

φ φφ φ

φ φ

φ φ

φ φ

(21)

E b pb

i ii

σ

σφ= −

22sin , (22)

where e is the base of the natural logarithm and σ isthe external normal stress (Fig.16). With formulae(17)-(22), the authors [178, 179] have found the dif-ference ∆W:

∆WD

b bR

r

bD

p b b

b bp R pR

ep

pR

p R pR

LR

L

L

L

R

L

b LR

i i

i

= + + −

− + +

+ ++ + +

+ +

−+

+ + +

+ + − + +

+

=

∑

21

2 2

12

2 1

2

2

1

2

1

2

1

2

2

2

2

1 1 2 2

1 2 1 2

2 2

1 2

2

1 2

2

1 2

2 2

1 2

2

1

2

1

2

1

2

1

2 2

ln

sin sin

cos lncos

cos

sin

cos

ln ln

cos ln

σφ φ

φ φφ φ

φ φ

φ φ

φ φ

ω

ω φL RL

p L pL

R R

p p

i

i

i

i

2

2 2

2 2

2 2

2

2

2

2

+

+ +−

+ +

+ +

sin

sin

lnsin

sin.

φ

φ

φ

φ

Formula (23) has allowed to numerically investi-gate the dependences of ∆W on parameters, p, φ

1,

(23)

φ2, σ, R, and L, of the system under consideration.

Thus, the dependences ∆W(p) are shown in Fig.18for various values of characteristic angles φ

1 and φ

2

related to the crystallography of the adjacent grains[179]. These dependences indicate that angles φ

1

and φ2 crucially influence elementary transfer of aGB disclination. In fact, the disclination transfer isenergetically facilitated at low angles and hamperedwith rising φ1 and φ2. At large values of φ1 and φ2

(tentatively >50 °) there exists an energetic barrierfor motion of the emitted lattice dislocations, andthe disclination transfer is energetically unfavourable.In the range of φ

1 and φ

2 from 0° to tentatively 20 °,

the dislocation motion has a barrier-less characterwith ∆W (<0) decreasing with rising the dislocationpath p (Fig.18). In the range of φ

1 and φ

2 from tenta-

tively 20° to 50°, the disclination transfer is eitherfavourable or unfavourable, depending on other pa-rameters (L, ω, σ) of the system. In this situation,the characteristic energy difference ∆W(p=b) at thestarting point of the dislocation motion is highlysensitive to both the distance L between thedisclinations and their strength magnitude ω (seeFigs.19 and 20) [178, 179].

At the same time, ∆W(p=b) weakly depends onthe external stress σ (see Fig.21) [179]. The influ-ence of the stress σ on the characteristic energydifference ∆W(p) increases with rising the disloca-tion path p at some intermediate values (close to45°) of angles φ1 and φ2. The energy difference∆W(p) at low values of angles φ

1 and φ

2 is weakly

20 40 60 80 100 120 140

0.5

0.4

0.3

0.2

0.1

0.1

0.2

∆W / Gb2

p/b

1 3 4 2

Fig. 18. Dependence of ∆W on the distance p movedby each of the emitted lattice dislocations, forL=30b, σ =10-3G, R=105b, ω=0.1, and the followingvalues of the characteristic angles: φ

1=φ

2=45° (curve

1), φ1=30° and φ

2=40° (curve 2), φ

1=20° and φ

2=30°

(curve 3), and φ1=φ

2=2° (curve 4).


sensitive to σ, as is shown in Fig. 22 [179]. Thisnaturally follows from the geometry of the system(Fig.16).

Thus, probability of splitting of a GB dislocationinto a pair of lattice dislocations, their emission andcorresponding motion of the GB disclination in-creases with decreasing values of angles φ

1 and φ

2,

and with growth of the dipole arm L and disclinationstrength ω. The GB disclination motion has beenproved to be an energetically favourable process inrather wide ranges of parameters that characterizethe defect configuration under consideration. In con-trast to the previously considered situation withdisclination motion associated with (ordered inspace and time) absorption of dislocations by GBs[8, 156], the model of disclination motion associ-ated with emission of dislocation pairs [178, 179]does not require any correlated flux of dislocationsfrom grain interiors to GBs. The suggestedmicromechanism for the disclination motion (Fig.16)can be responsible for experimentally observed ro-tations of grains in fine-grained materials under(super)plastic deformation and thermal treatment.

3.2. Motion of dipole of grain boundary disclinations

The model of motion of a GB disclination [178, 179](discussed in the previous Section) is easily to ex-tend to the case of conservative motion of a dipoleof such disclinations under the action of external

∆W(p=b) / Gb2

L/b

40 60 80 100

0.2

0.15

0.1

0.05

0.05

0.1

0.15

0.2

1

2

3

4

Fig. 19. Dependence of ∆W on the distance L be-tween grain boundary disclinations at the initial stageof the dislocation emission (at p=b), for σ = 10-3G,R=105b, ω=0.1, and the following values of the char-acteristic angles: φ

1=φ

2=45° (curve 1), φ

1=30° and

φ2=40° (curve 2), φ

1=20° and φ

2=30° (curve 3), and

φ1=φ

2=2° (curve 4).

0.12 0.14 0.16 0.18 0.2

0.2

0.15

0.1

0.05

0.05

0.1

∆W(p=b) / Gb2

ω

1

2

3 4

Fig. 20. Dependence of ∆W on the disclinationstrength ω at the initial stage of the dislocation emis-sion (at p=b), for σ=10-3G, R=105b, L=30b, and thefollowing values of the characteristic angles:φ

1=φ

2=45° (curve 1), φ

1=30° and φ

2=40° (curve 2),

φ1=20° and φ

2=30° (curve 3), and φ

1=φ

2=2° (curve 4).

20 40 60 80 100

0.5

0.4

0.3

0.2

0.1

0.1

∆W / Gb2

p/b

1 2 3

Fig. 21. Dependence of ∆W on the distance p movedby the emitted dislocations, for φ

1=30° and φ

2=40°,

R=105b, L=30b, ω=0.1, and the following values ofthe applied stress σ/G=10-4, 10-3 and 10-2 (curves 1,2 and 3, respectively).

20 40 60 80 100

0.5

0.4

0.3

0.2

0.1

0.1

∆W / Gb2

p/b

1 3

2Fig. 22. Dependence of ∆W on the distance p movedby the emitted dislocations, for φ

1=φ

2=2°, R=105b,

L=30b, ω=0.1, and the following values of the ap-plied stress σ/G=10-4, 10-3 and 10-2 (curves 1, 2 and3, respectively).


loading [176, 177]. Motion of dipoles of the GBdisclinations may be considered as similar to themotion of (super)dislocations and essentially con-tributes to high-strain deformation (Fig.23) [8]. Re-member that experimental evidence of the existenceof disclination dipoles in typical NCMs has recentlybeen demonstrated in direct atomic-level HREMobservations by Murayama et al. [126].

Following the model [176, 177], one can assumethat the rotational deformation in a fine-grained ma-terial occurs via the stress-induced motion of a di-pole of GB disclinations. In the framework of themodel, the dipole consists of two GB disclinationswith the strengths +ω and -ω, respectively, and hasthe arm L. The disclination of the strength +ω (+ω-disclination) transfers by the distance l due to theemission of two lattice dislocations with the Burgersvectors

b and

b1 into the adjacent grains (Fig.24)

[176, 177]. The dislocation with the Burgers vector

b moves along its gliding plane towards thedisclination of the strength -ω (-ω-disclination). Thisdisclination moves by the distance l in the samedirection as the +ω-disclination due to the emis-sion of two lattice dislocations with the Burgers vec-tors -

b and

b2 into the abutting grains. The disloca-

tions with the Burgers vector

b and -

b emitted bythe +ω- and -ω-disclinations, respectively, meet eachother and annihilate.

Of course, this model is very idealistic, even morethan that considered in Section 3.1. In a real NCMfabricated by severe deformation method, the GB

y3

φ

φ2

φ1

y y1

x1

x

x3

y2

x2

l

d

p2 p1

− p − L+p

−b

b b2

b1

τ2

τ

τ

τ

σ

σ

−ω

+ω

B

−B

− L

Fig. 23. Motion of a disclination dipole may be con-sidered as similar to the motion of a (super)dislo-cation.

Fig. 24. Motion of a dipole of grain boundarydisclinations is accompanied by emission of latticedislocations from the grain boundaries into the ad-jacent grains. The dislocation slip planes are inclinedat the angles φ, φ

1 and φ

2 to the x-axis.

disclinations with various values of strength are dis-tributed in a rather disordered manner along non-equilibrium GBs and the emitted lattice dislocationsdo not necessary annihilate. In addition, grain inte-riors of a fine-grained material contain many latticedislocations generated during severe plastic defor-mation. These dislocations are capable of interact-ing and annihilating with the dislocations emittedby GB disclinations. Thus, real processes occur-ring during motion of GB disclinations can be verycomplicated and different from those described bythe model [176, 177]. However, this simplified modelis convenient for a strict mathematical analysis andeffective for understanding the key peculiarities ofthe GB disclination motion accompanied by emis-sion of lattice dislocations. Also, it can serve as abasis for further detailed consideration of evolutionof GB defects in plastically deformed NCMs.

The cooperative motion of the GB disclinationscomposing the dipole causes the plastic deforma-tion of a NCM. The dipole motion is energeticallyfavourable, if the difference between the systemenergies (per unit disclination length) after (W

2) and

before (W1) the dipole elementary transfer is nega-

tive: ∆W=W2-W

1<0. The corresponding calculation


of ∆W is very similar to that described in Section3.1, so we give below only several intermediate for-mulae and the final result. The full calculation pro-cedure is represented in paper [176].

The energy of the system in its initial state (be-fore the dipole transfer) W

1 is the strain energy of

the disclination dipole which is given by the firstterm in formula (17).

The disclination dipole motion is accompaniedby the emission of the lattice dislocations, in whichcase the energy density W

2 of the defect system

consists of the five following terms [176, 177]:

W E E E E Ed b b bb2

1 2

= + + + +int . (24)

Here Ed=W

1 is the strain energy of the disclination

dipole which does not change during its correlatedmotion, Eb1

and Eb2 are the self energies of the emit-

ted dislocations with the Burgers vectors

b1 and

b2,

respectively; Ebb

is the self energy of the disloca-tion dipole consisting of the emitted dislocationswith Burgers vectors

b and -

b , and Eint

is the sumenergy that characterizes all the interactions be-tween the defects composing the system (Fig.24)(except between both the disclinations whose in-teraction energy is included in E

d, and between the

b - and

b -dislocations whose interaction energy isincluded in E

bb), and between the defects and the

applied stress σ. The last component can be decom-posed into nine terms as follows [176]:

E E E E E

E E E E E

d

b

d

b

d

bb

b

b

bb

b

bb

b b b bb

int

.

= + + + +

+ + + +

1 2

2

1

1 2 1 2

σ σ σ

(25)

Here Ed

b1 , E

d

b2 and E

d

bb denote the energies thatcharacterize the interaction of the disclination di-pole with respectively the

b1- and

b2-dislocations,

and the dipole of ±

b -dislocations; Eb

b

2

1 is the energyof the interaction between the

b1- and

b2-disloca-

tions; Ebb

b1 and Ebb

b2 denote the energies that charac-terize the interaction of the dislocation dipole withthese dislocations; E b

σ1

, Eb

σ2 and E

bb

σ are the ener-gies that characterize the interaction of the appliedstress σ with respectively the emitted

b1- and

b2-

dislocations, and the dislocation dipole. Considerthe energies figuring on the r.h.s. of formula (24)with account for (25).

The self energies of the

b1- and

b2-dislocations

are given by formula (19).The self energy of the dipole of ±

b -dislocationsis calculated as the work spent to generate thesedislocations in their sum elastic field that finally gives[176]

E DbL p r

rbb

c

c

=− −

+

2 21ln , (26)

where the last term “1” is added to take into ac-count the contributions of the dislocation cores.

The energies of interaction between thedisclination dipole and the lattice dislocations arecalculated in a similar way, as the work spent togenerate these dislocations in the stress field ofthe disclination dipole. In doing so, one has to usethe different coordinate systems shown in Fig.24.These quite cumbersome calculations result in [176]

ED b

pd

b1 1

1 12

=ω

φ φΨ , , , (27)

ED b

pd

b2 2

2 22

= − −ω

φ φΨ , , , (28)

ED b

L p L p

p p

d

bb = − ×

− + + −

+ +

ω

φφ

φ

2

2 2

2 2

4 4 2 2

4 4 2 2cos ln

cos

cos,

(29)

where the following function

Ψ p L

L R LR L R

L p Lp L x

R R

p p

1 1 1 1

2 2 2

1 1

2

1

2 2

1 1 1

1

2 2

1

1

2 2

1 1

2 2 2

2 2 2

2

2

, , cos sin

lncos sin sin

cos sin sin

cos lnsin

sin

φ φ φ φ φ

φ φ φ φ

φ φ φ φ

φφ

φ

= + + ×

+ + + + + −

+ + + + + −

−+ −

+ −

is introduced. Here l denotes the distance betweenthe GB dislocations which compose the ragged dis-location walls with the disclinations at the edges(Fig.24). This distance may be written through themagnitude of the Burgers vector

B of the GB dislo-cations and the disclination strength ω as

≈ B ω . (31)

In its turn, the Burgers vector

B is connected withother geometric parameters of this defect configu-ration by the relations: B =b

1cosφ

1+bcosφ and b

1/

b =sinφ/sinφ1. Their substitution into equation (31)

gives

≈+b sin

sin.

φ φ

ω φ1

1

(32)

(30)


The energy of interaction Eb

b

2

1 between the

b1- and

b2-dislocations is also calculated as the work spent

to generate one dislocation in the stress field of theother. The result is [176]

E Db bS S

P

R

Pb

b

2

1

1 2

1 2

2 1 2= − + −

cos ln ,φ φ (33)

where the following denotations

(34)

are used.

The energy of interaction Ebb

bi of the

bi-disloca-

tion with the dipole of ±

b -dislocations is determinedin a similar way that results in [176]

EDbb

pbb

b i

i ii =

2Φ , , ,φ φ (35)

where i=1, 2 and

Φ p

L p p L p p

p p pp

p L p p p Lp

L p p L p p p p pp

i i i

i i i

i i i

i i i

i i i i i i

, , cos

lncos

cos

sin

cos cos

φ φ φ φ

φ φ

φ φ

φ φ

φ φ φ φ

= +

×− + + − +

+ + +−

− + − +

− + + − + + + +

2 2

2 2

2 2 2

2 2 2 2

2

2

2 2

2 2

P L p p p p

Lp Lp

2 2

1

2

2

2

1 2 1 2

1 1 2 2

2

2 2

= + + + − +

+ + +

cos

cos cos

φ φ

φ φ φ φ

S p L1 1 1 2 2

= − − +sin sin ,φ φ φ φ S p L

2 2 1 2 1= − + +sin sin ,φ φ φ φ

(36)

φ (degrees)

5 15 25 35 45 55

1

11

3

5

7

9

φ1 = φ2 (degrees) ∆Wmin /Gb2

L/b 1000 200 400 600 800

- 0.4

- 0.3

- 0.2

- 0.1

0

0.1

Lc/b

2

1

3

Fig. 25. φ-dependence of the angles φ1 and φ

2(=φ

1)

corresponding to the minimum of the energy differ-ence ∆W, for the following parameters of the model:ω=0.1, p

1=p

2=p=b, σ=5.10-3G, R=107b and

L=1000b.

Fig. 26. Dependence of the energy difference ∆Won the disclination dipole arm L, for the followingparameters of the model: ω=0.1, p

1=p

2=p=b,

σ=5.10-3G, R=107b and φ=30° (curve 1), 40° (2) and50° (3).

The energies of interaction of the lattice disloca-tions with the external stress σ are given by simpleformulae

E b pb

i i ii

σ

σφ= −

22sin , (37)

E bpb

σ

σφ= −

22sin . (38)

Thus, all terms of sums (24) and (25) are ob-tained. The energy difference ∆W=W

2-W

1 may then

be written in the following final form [176]:

∆

Φ Φ

Ψ Ψ

WD

b bR

r

bL p r

r

b bS S

P

R

P

Dbp b p b p

b b p b p

b p b p

blL p

c

c

c

= + + +

− −+

− + − −

+ + +

+ +

+ − − −

− +

21

22

1

2

2 2 2 2

1

2

2

2

2

1 2

1 2

2 1 2

1 1 1 2 2 2

1 1 1 2 2 2

1 1 1 2 2 2

4 4

ln

ln

cos ln

sin sin sin

, , , ,

, , , ,

cos ln( )

φ φ

σφ φ φ

φ φ φ φ

ω φ φ φ φ

ω φ + −

+ +

2 2

2 2

2 2

4 4 2 2

( ) cos

cos.

L p

p p

φ

φ

(39)


The results of numerical calculations carried outby using formula (39) are presented in Figs. 25 and26 [176, 177]. Thus, the dependences of the char-acteristic angles φ

1 and φ

2 on the angle φ are shown

in Fig.25, in the case with φ1=φ

2. These depen-

dences, for the given values of φ, indicate the anglesφ

1 and φ

2 at which the disclination dipole motion

(Fig.24) is characterized by the lowest value ∆Wmin

(where ∆W<0) of the energy difference (39) that cor-responds to the start of the disclination dipole mo-tion. (In other words, ∆W

min is the largest energy

gain due to the start of motion of emitted latticedislocations, associated with the process in ques-tion). As it follows from Fig.25, these angles φ

1 and

φ2(=φ

1) increase with rising φ from 0° to ≈15°, have

the constant value of ≈11° in the range of φ from≈15° to ≈30°, and decrease from ≈11° to ≈4° withrising φ from ≈30° to ≈55°.

The angles φ1 and φ

2(=φ

1), which correspond to

the minimum of the energy difference ∆W, dependweakly on other parameters L, σ, and ω of the sys-tem. At the same time, the minimum value ∆W

min

of this characteristic energy difference is sensitiveto the parameters discussed. In Fig. 26, the depen-dences of ∆W

min on the disclination dipole arm L

are presented for various values of the angle φ [177].It follows from Fig. 26 that the disclination dipolemotion is energetically favourable (∆Wmin<0) if thedipole arm L exceeds a critical value L

c, which is φ

dependent and close to about 150b ≈ 50 nm. In therange of φ from 1° to tentatively 30°, the depen-dences ∆W

min(L) coincide; they are presented as

curve 1 in Fig. 26. The dependences ∆Wmin

(L) atφ=40° and 50° (see curves 2 and 3 respectively inFig.26) are different from those at φ≤30°. They indi-cate that ∆W

min(L) slightly decreases with rising φ

from 30° to 50°. To summarize, ∆Wmin<0, that isthe disclination dipole motion (Fig.24) is an ener-getically favourable process within rather wideranges of parameters that characterize the defectconfiguration under consideration.

Thus, in papers [176, 177], it has been theoreti-cally revealed that the rotational mode of plastic flowin the fine-grained materials produced by severe plas-tic deformation can effectively occur via the GBdisclination dipole motion associated with the emis-sion of lattice dislocation pairs into the adjacentgrains (Fig.24). In contrast to the previously consid-ered [8, 156] situation with the disclination motionassociated with the absorption (ordered in spaceand time) of lattice dislocations by GBs, thedisclination dipole motion associated with the emis-sion of lattice dislocation pairs does not require anycorrelated flux of dislocations from grain interiors to

GBs. This micromechanism for the disclinationmotion can be responsible for the rotations of grainsexperimentally observed [152-156, 168-173] in thefine-grained materials under (super)plastic deforma-tion and thermal treatment.

3.3. Crossover from grain boundary sliding to rotational deformation

In the previous Sections we have considered themotion of partial disclinations which is realized byabsorption (Section 2) or emission (Sections 3.1and 3.2) of lattice dislocations. Thus, cooperativeaction of the translational (dislocation gliding) androtational (disclination motion) modes of plasticdeformation in NCMs has been assumed. However,as we have already mentioned above, with grain re-finement below a critical grain size d* the gliding oflattice dislocations stops to play any significant rolein NCMs, and new micromechanisms of plasticityas GB sliding and diffusion mass transfer are acti-vated. The question arises how the rotational modeof plasticity can develop in these conditions. In thepresent Section we discuss the results of recenttheoretical investigation of relation between themicromechanisms of GB sliding and diffusion, androtational deformation which occurs through gen-eration and motion of partial GB disclinations bymeans of the climb of GB dislocations along GBsin NCMs [184]. It is demonstrated that under spe-cial conditions of plastic deformation, the crossoverfrom the translational micromechanism of plasticity(GB sliding) to the rotational one (grain rotation) ispossible.

3.3.1. Splitting of gliding grain boundary dislocations at triple junction into climbing dislocations (small-scale view)GB sliding which is treated to be the dominant modeof superplasticity in nano- and microcrystallinematerials occurs via motion of gliding GB disloca-tions. They have Burgers vectors that are parallelwith corresponding GB planes along which thesedislocations glide. Triple junctions of GBs, whereGB planes change their orientations, serve as ob-stacles for the GB dislocation motion. In these cir-cumstances, splitting of gliding GB dislocations canoccur at triple junctions, resulting in the formationof sessile dislocations and gliding dislocations pro-viding the further GB sliding along the adjacent GBs[185]. However, in general, GB dislocations stoppedat a triple junction are also capable of being splitinto climbing GB dislocations (Fig.27) [184]. Whenthis process repeatedly occurs at a triple junction,it results in the formation of two walls of GB dislo-


cations climbing along the GBs adjacent to the triplejunction. The climbing GB dislocation walls causethe rotational deformation, in which case the repeat-edly occurring splitting of gliding GB dislocations atthe triple junction provides the crossover from theGB sliding to the rotational deformation mode. Thisprocess can be spread over the grain which has torotate on an angle as a whole. Thus, the stoppedGB sliding can stimulate plastic rotation of theneighbour grain. Obviously this mechanism may onlybe effective under the condition of intensive GB dif-fusion of vacancies which must be capable to pro-vide the necessary velocity of GB dislocation climb.

The reality of the model [184] may be estimatedfrom the viewpoint of thermodynamics (by analysingthe energetic favoritism of this process and obtain-ing only the necessary conditions of its realization)or kinetics (which allows to find its sufficient condi-tions). In the framework of the first approach, whichhas already been demonstrated many times throughthis review, the problem is solved relatively simplyand gives new results (see below). The second ap-proach means the necessity to state and solve avery complicated problem of GB diffusion for vacan-cies in a system with many sources and sinks, withaccount for highly inhomogeneous field of elasticstresses of the dislocation-disclination structure thatevolves in space and time. Such a problem haveneither been stated nor solved yet. Using the firstapproach, the authors [184] have considered thevery beginning of the transition from the GB slidingto the formation of final disclination structure, whenthe “head” GB dislocation in a pile-up splits into a

2φ

j n c i

y

x''

O

b

τ

xj − xi

x

y'' p

p

b2

b1

b O''

x'

y'

j 1 n c i

y

x O

a

τ

xj − xi b

2φ

Fig. 27. Combined action of grain boundary slidingand rotational deformation mode. (a) Nanocrystallinespecimen in a non-deformed state. (b) Grain bound-ary sliding occurs via motion of gliding grain boundarydislocations under shear stress action. (c) Gliding dis-locations split at triple junction O of grain boundariesinto climbing dislocations. (d) The splitting of glidinggrain boundary dislocations repeatedly occurs caus-ing the formation of walls of grain boundary disloca-tions whose climb is accompanied by crystal latticerotation in a grain. (e) Climbing dislocations reach triplejunction O’ where they converge into gliding disloca-tions causing further grain boundary sliding.

Fig. 28. Splitting of the (a) head dislocation of a grain boundary dislocation pile-up at a triple junction into (b)two dislocations that climb along the adjacent grain boundaries. Coordinate systems Oxy, Ox’y’ and O’’x’’y’’are shown which have been used in calculations.

xi-xj

xi-x

j


pair of new GB dislocations (small-scale view; seethe current Section 3.3.1), and the very end of thistransition, when two pile-ups of GB dislocations atthe opposite sides of a grain are totally transformedinto a system of GB disclination dipoles (large-scaleview; see Section 3.3.2).

Following the model [184], consider a pile-up ofn

c GB dislocations having the Burgers vectors

bnear a triple junction of GBs under the action ofexternal shear stress τ (Fig.28a). Assume that thehead GB dislocation splits into the two new GB dis-locations with Burgers vectors

b1 and

b2, which can

climb along the adjacent GBs (Fig.28b). The split-ting process may be characterized by the difference∆W=W

2-W

1 between the energies of the final (W

2,

Fig.28b) and initial (W1, Fig.28a) states of the de-

fect configuration. The splitting is energetically fa-vorable (unfavorable), if ∆W<0 (∆W>0, respectively).The equation ∆W=0 gives a set of critical values ofparameters for the defect configuration, at which thesplitting becomes energetically favorable.

The total energy W1 of the initial defect configu-

ration (per unit dislocation length) consists of twoterms [184]

W E Epu b b

1 1 1= + −

Σ , (40)

where E pu

1 is the sum of the self energies of n

c GB

dislocations composing the pile-up, and E b b

1

−

Σ , is theenergy that characterizes pair interactions betweenall these dislocations.

The first term is written at once as [141]

E nDb R

b

pu

c1

2

21= +

ln . (41)

In order to find the second term, let us first cal-culate the energy E

ijb-b that characterizes elastic

interaction between the ith and jth GB dislocationsof the pile-up, with the distance (xi-xj ) between them(Fig.28a). The positions x

i and x

j of these disloca-

tions may be calculated [186] with the help of theLaguerre polynomials Ln(x) which are determined assolutions of the differential equation

xd y

dxx

dy

dxny2

2

21 0+ − + = . (42)

The first derivative of the Laguerre polynomial L’n(x)

is a solution of the equation

xd y

dxx

dy

dxn y

2

22 1 0+ − + − = (43)

and is given by the following formula

L xn x

k k n kn

k

k

n' !

! ! !.

= −−

+ − −=

−

∑1 10

1

(44)

Eshelby et al. [186] showed that the roots of thefirst derivative of the Laguerre polynomial L’

n(x) de-

termine the equilibrium positions of dislocations ina discrete dislocation pile-up. Thus, by calculatingnumerically the positions of the GB dislocations inthe pile-up for the given values of n

c and τ (Fig.28a),

one can also find the energy of elastic interactionbetween two GB dislocations as the work spent totransfer one dislocation from a free surface of a solidto its current position in the stress field created bythe other dislocation. A similar approach has beenused for analysing dislocation interactions within aGB dislocation pile-up in work [185]. In doing so,the energy E

ijb-b of interaction between the i-th and

j-th GB dislocations has been found as [184]

E DbR

x xij

b b

i j

− =−

2 ln . (45)

The energy E b b

1

−

Σ , that characterizes pair interac-tions between all GB dislocations belonging to thepile-up (Fig.28a) is the sum of energies E

ijb-b over

the GB dislocation indices i and j (i ≠ j) [184]:

E E DbR

x x

b b

ij

b b

j i

n

i

n

i jj i

n

i

ncc cc

Σ1

11

12

11

1− −

= +=

−

= +=

−

= =−

∑∑ ∑∑ ln . (46)

The energy of the defect configuration (Fig.28b)resulted from the splitting of the head dislocationbelonging to the GB dislocation pile-up consists ofnine terms [184]:

W E E E E

E E E E E

pu b b

s

b

s

b

b b b b b b b b

2 2 2

2 2

1 2

1 2 1 2 1 2

= + + + +

+ + + +

−

− − − − −

Σ

Σ Σ

τ τ, (47)

where E pu

2 is the sum of all self energies of the GBdislocations belonging to the pile-up after the split-ting, E b b

2

−

Σ the energy of pair interactions between all

GB dislocations composing the pile-up, Es

b1 and

Es

b2 are the self energies of the two GB disloca-

tions resulted from the splitting of the head disloca-tion, E b b

21−

Σ and E b b

22−

Σ are the energies that charac-

terize interaction between GB dislocations of thepile-up and the GB dislocations resulted from thesplitting and characterized by Burgers vectors

b1

and

b2, respectively; E b b

1 2− is the energy of interac-

tion between these latter (climbing) dislocations; andE b

1−τ and E b2−τ, are the effective works of the shear


stress τ, spent to transfer these two GB disloca-tions along the GBs adjacent to the triple junction.

The pile-up after the splitting of its head disloca-tion (Fig.28b) contains (n

c-1) GB dislocations. To

simplify the calculations of the energy characteris-tics of these dislocations, the authors [184] haveassumed that the dislocation positions remain un-changed during the splitting of the head disloca-tion. This assumption is reasonable in at least thesituation where the GB dislocations resulted fromthe splitting are close to the triple junction (Fig.28b).Then the first term of the sum (47), E pu

2 , is directlyobtained from formula (41) through replacement ofn

c by (n

c-1). The second term, E b b

21−

Σ, yields from for-

mula (46) with substitution of i=1 for i=2 in the lowerlimit of the first sum.

The self energies of the new climbing GB dislo-cations, Es

b1 and Es

b2 , are well known [141]. They

are given by the r.h.s. of equation (19) with replace-ment of the cut-off radius r by b

1 and b

2, respec-

tively.Calculation of the energies of interaction between

the GB dislocations of the pile-up and the climbingGB dislocations has been carried out in the follow-ing way [184]. First, the energy of interaction be-tween the climbing dislocation and an arbitrary dis-location of the pile-up has been found. This proce-dure is quite cumbersome because it includes thedetermination of the shear stress created by theclimbing dislocation, which acts in the plane of thepile-up, with further calculation of the work spent togenerate a dislocation from the pile-up in this shearstress field. For example, the energy of interactionof the b1-dislocation with the i-th dislocation fromthe pile-up has been obtained as [184]

E Dbbp Rp

R Rp p

p x p

x x p p

R Rp p

x x p p

i

b b

i

i i

i i

− =+

+ +−

+

+ ++

+ +

+ +

1

1

2

2 2

2

2 2

2 2

2 2

2

2

1

2

2

2

sincos

cos

cos

cos

lncos

cos,

φφ

φ

φ

φ

φ

φ

(48)

where 2φ is the angle between the GBs along whichthe

b1- and

b2-dislocations climb, p the path of the

b1-dislocation, and x

i the coordinate of the i-th dis-

location in the pile-up (Fig.28b). In a similar way,the energy E

ib-b2 of interaction of the

b2-dislocation

with the i-th dislocation in the pile-up is calculated.Therefore, the total energy that charactrizes inter-

action of both the

b1- and

b2-dislocations with GB

dislocations of the pile-up reads [184]

E E E Eb b b b

i

b b

i

b b

i

nc

Σ Σ2 2

1

1 2 1 2− − − −

=

+ = +∑ . (49)

The energy E b b1 2− of interaction between the climb-

ing dislocations is also calculated as the work ongeneration of one dislocation in the stress field ofthe other. Omitting intermediate results, we give thefinal formula [184]:

EDb b R

R Rp p

R Rp p

p

b b1 2 1 2

2

2 2 2

2 2 2

2 2

2

2

4 4

22 2 4

4

− = −− +

+

− +

cosec ctg

cos lnsin sin

sin.

φ φ

φφ φ

φ

The work E bk−τ spent to transfer the

bk-dislocation

by distance p under the shear stress τ is equal to

E b pb

kkτ τ φ− = − sin .2 (51)

Thus, we have all terms figuring in the energydifference ∆W=W

2-W

1 that characterizes the split-

ting (Fig.28) considering as an elementary act ofthe crossover from the GB sliding to the rotationaldeformation mode in NCMs. The numerical resultsfor the dependences ∆W(p), which have been ob-tained for b

1=b

2=b/(2sinφ), n

c=5, R=107b, τ=G/200

and different values of the angle 2φ, are presentedin Fig.29 [184]. These curves show that the split-ting - an elementary act of the crossover from theGB sliding to the rotational mode – is energeticallyfavorable at large values (80°-120°) of the angle 2φ.

(50)

20 40 60 80 100 120 140

− 2.5

− 2

− 1.5

− 1

− 0.5

0.5

p/b

∆ W/Gb 2

Fig. 29. Dependences of ∆W (in units of Gb2) ondistance p (in units of b) moved by climbing

b1- and

b2-dislocations, for 2φ =40, 60, 80 and 120° (from

top to bottom).


In paper [185] it has been demonstrated that thesplitting of the head GB dislocation of a pile-upstopped at a triple junction into two, gliding andsessile, GB dislocations is energetically favorableat low values of the angle 2φ. That version of thesplitting serves as an elementary act of the GB slid-ing at triple junctions.

Thus, the crossover from the GB sliding to therotational deformation (Figs.27 and 28) occurs ef-fectively at triple junctions with large values of theangle 2φ, while the GB sliding itself occurs effec-tively at triple junctions with comparatively low val-ues of the angle 2φ.

3.3.2. Cooperative action of grain boundarysliding and rotational deformation modein nanocrystalline materials (large-scaleview)

Consider, how the system of defects described inthe previous Section can develop in time under anexternal shear stress [184]. After the head GB dis-location has split into the

b1- and

b2-dislocations

(Fig.28b), their stress fields prevent movement ofthe second GB dislocation of the pile-up towardsthe triple junction where the splitting has occurred.When the climbing

b1- and

b2-dislocations move far

from the triple junction, their effect on the secondGB dislocation of the pile-up becomes weak. In thiscase the second dislocation of the pile-up moves tothe triple junction where it splits into two GB dislo-cations climbing along the adjacent GBs. Such asplitting process repeatedly occurs transforming GBdislocations of the pile-up into the climbing GB dis-locations (Fig.27). These climbing dislocations formdislocation walls of finite extent at the two GBs ad-jacent to the triple junction. The geometric and elasticproperties of these two ragged dislocation walls aresimilar to those of the corresponding two-axes di-poles of partial wedge disclinations [8, 133] (seealso Sections 2 and 3.1).

Now assume that similar splitting processesoccur at two opposite triple junctions of GBs sur-rounding a nano-sized grain. The GB dislocation pile-ups located near the opposite triple junctions con-sist of dislocations with the Burgers vectors of op-posite signs, which tend to move under the shearstress action towards each other and are stoppedby the triple junctions. Such a pile-up consisting ofn

c dislocations in its initial state may be represented

effectively as a superdislocation with the Burgersvector ±

B=±nc

b (Fig.30a). Following the model[184], these two superdislocations split and aretransformed into four GB dislocation walls of finiteextent, which are effectively represented as four di-poles of disclinations (Fig.30b).

2φ

S y

P O

τ

L

x

ncb − ncb 2φ

a

2φ

τ

2φ

b

−ω

+ω

−ω

−ω

−ω

+ω

+ω

+ω

− p

y x

O

−L 1

2 4

3

x'

y'

O' p

L

Fig. 30. Large-scale model of crossover from grainboundary sliding to rotational deformation mode. (a)Two superdislocations (models of pile-ups of grainboundary dislocations) are stopped at the oppositetriple junctions. (b) Four disclination dipoles (mod-els of climbing dislocation walls) are located at thegrain boundaries adjacent to the triple junctions.

In calculating the energy characteristics of evo-lution of the GB dislocation ensemble, causing grainrotation, for definiteness and simplicity, the authors[184] have made the following model assumptions:(i) The grain is a hexagon with the angles 2φ char-acterizing the splitting processes at the oppositetriple junctions being the same. (ii) Magnitudes ofthe Burgers vectors of all GB dislocations belong-ing to the dislocation pile-ups are the same. (iii)The plane of the shear stress action is parallel withplanes of GBs where the GB dislocation pile-upsare located. (iv) All the disclinations modeling thewalls of the climbing GB dislocations have the samestrength magnitude ω. Also the total number 2nc ofGB dislocations participating in the transformationshave believed to be constant. It is worth noting thatall these assumptions are not necessary and maybe omitted in a more general model.

Following the authors [184], consider the condi-tions which are necessary for realization of the totaltransformation of two initial pile-ups of GB disloca-tions (Fig.30a) into four ragged walls of climbing GBdislocations modeled by four dipoles of GBdisclinations (Fig.30b). The transformation of the GBdislocation ensemble may be characterized by the


difference ∆ ~W =

~W2

-~

W1 between the energies of the

final (~

W2) and initial (

~W1

) states of the ensemble.The transformation is energetically favorable (unfa-vorable), if ∆ ~

W <0 (∆ ~W >0, respectively).

In the framework of the model discussed, theenergy of the dislocation ensemble in its initial state(Fig.30a) consists of the three terms [184]:

~,intW E E Epu b b B B

1 1 12 2= + +− −

Σ (52)

where the two first terms are already known — theydescribe the sum of self energies of the GB disloca-tions in the pile-ups (2Epu

12 ) and the sum of all ener-

gies of pair interactions between the GB disloca-tions within their pile-ups ( E b b

12 −

Σ). They are deter-

mined by formulae (41) and (46), respectively. Thethird term, ,

intEB B− , is the energy that characterizes

pair interactions between GB dislocations belong-ing to the different pile-ups. In the first approxima-tion, this energy may be estimated as the energy ofelastic interaction of two superdislocations with theBurgers vectors ±

B [184]:

E DBR

P

B B

int ln ,− = − 2 (53)

where P is the distance between the opposite triplejunctions.

Thus, the energy (52) of the system in its initialstate is approximated by the sum of three terms.The two first terms strictly take into account thefine structure of the dislocation pile-ups (i.e., thespecific distributions of dislocations within the pile-ups), while the third one is approximated by equa-tion (53), which does not account for this fine struc-ture. However, this approximation (when a pile-up istreated as a superdislocation) gives rather good re-sults if the distance P (Fig.30a) is much larger thanthe pile-up length [141]. The latter must be smallerthan S/2, and S is assumed to be always smaller(two or more times, see below) than P in the model[184]. Therefore, the pile-up length must be four ormore times smaller then P, and the approximationof superdislocations in calculating the term (53) isthus proved.

The energy of the GB dislocation ensemble inits final state (Fig.30b) consists of the six terms[184]:

~

,

int

int int

W E E E E

E E

s

pu

2

1 2

1 3 1 4

4 4 2 2

2 2

= + + + +

+

−

− −

∆ ∆

τ τ

(54)

where Es

4 ∆ denotes the self-energy of a disclinationdipole, E4 ∆

τ the work spent to transfer the GB dislo-

cations resulted from the splitting under the shear

stress τ, E pu2 τ the work spent to transfer all GB dis-

locations belonging to the pile-ups from their initialpositions up to the corresponding triple junctionsunder the shear stress τ, and

intE i j2 − the energy of

interaction between the i-th and j-th disclination di-poles (i ≠ j; i=1,2,3; j=2,3,4).

The total self energy Es

4 ∆ of the disclination di-pole with the arm (L-p) and strength ω is given byformula (17) with replacement of L by (L-p), N by n

c,

and b by b1.

The shear stress τ acts on the climbing GB dis-locations which compose the disclination dipoles.The work E4 ∆

τ spent to climb of n

c dislocations within

one disclination dipole is [184]

E b pn n b

c c

τ τ φω

∆ = − +−

1

121

2sin .

(55)

This work is the same for all the disclination dipolesunder consideration (Fig.30b).

The GB dislocations of each pile-up conse-quently move under the shear stress τ action fromtheir initial positions x=x

i, to the new positions at

the triple junctions (x=0). The work E pu2 τ spent to the

transfer of all the GB dislocations of a pile-up is[184]

E b xpu

i

i

nc

τ τ= −=

∑1

. (56)

The energy int

E i j2 − of elastic interaction betweenthe i-th and j-th disclination dipoles is calculated asthe work spent to generate one dipole in the stressfield of the other. Omitting cumbersome intermedi-ate calculations, we give the final results [184]:

ED

x L y

x p yx

x R

y p

y L

int

'

'

'

'

', , , , ',

', , , , ', ,

1 2

2

0

40 0

0 0

−

−

− =

=

=

=

= −ω

φ

φ

Ψ

Ψ

(57)

ED

x L T T y

x p T T yx

x R

y p

y L

int

'

'

'

'

' , , , ', ',

', , , ', ', ,

1 3

2

0

4

−

+

+ =

=

=

=

= −ω

φ

φ

Ψ

Ψ

(58)

ED

x T T L y

x T T p yx

x R

y p

y L

int

'

'

'

'

', ', , , ',

', ', , , ', ,

1 4

2

0

40

0

−

+

+ =

=

=

=

= −ω

Ψ

Ψ

(59)

where,


Ψ± =

+ + + − ± −

+

× + + + − ± −

+

x t t y

x t y t y t

x t

x t y t y t

x t

' , , , ', ',

' ' ' ' ' cos

' sin

ln ' ' ' ' cos

' sin

φ

φ

φ

φ

φ

2 2 2

2 2 2

2 2

2 2

2 2

2 2

(60)

and T=Psinφ, T ’=Pcos2φ.Thus, all terms figuring in formulae (52) for the

energy ~

W1 and (54) for the energy

~W2

, which com-pose the characteristic difference ∆ ~

W =~

W2-

~W1

, havebeen found. If ∆ ~

W is negative (positive), the trans-formation of the defect configuration is energeticallyfavorable (unfavorable, respectively). The authors[184] have analysed numerically the dependence of∆ ~

W on the disclination dipole strength ω (Fig.31).The calculations have been carried out for b

1=b

2=b/

(2sinφ), nc=5, R=107b, τ =G/200, L=200b, P=400b

and different values of the angle 2φ that character-izes geometry of the triple junction. The plots∆ ~

W (ω) show that the transformation is most favor-able in the range of 2φ from 100° to 160°. Also, themagnitude of ∆ ~

W (that is, the energy gain due tothe transformation when ∆ ~

W <0) decreases with anincrease of the disclination strength ω (ranging from0.05 to 0.1).

3.3.3. Critical stress of crossover from grainboundary sliding to rotational deformation

The equation ∆ ~W =0 allows one to calculate the

critical shear stress τc at which the crossover from

Fig. 31. Dependences of ∆ ~W (in units of Gb2) on

disclination strength ω in the range of values of thetriple junction angle 2φ =60, 80, 100, 120 and 160°(from top to bottom).

GB sliding to rotational deformation (Fig.30) occurs.This equation may be written as follows [184]:

τ

τ

τ φω

c

c

i c

i

n

c c

W W

b x b pn n bc

=

−

+ +−

=

∑

~ ~

sin

,'

2 1

1

1

12 4 21

2

(61)

where W~ '

2 is given by formula (54) for

~W2

with terms

4Eτ

∆ and 2E pu

τ removed.

Let us consider how τc changes depending on

the values of the angle 2φ and grain size P. Theauthors [184] have used the following characteristicvalues of parameters of a NCM and defect configu-ration under consideration. The Poisson ratio ν hasbeen equal to 0.3. Moduli of the GB dislocationBurgers vectors and disclination strength have beentaken as follows: b=0.1 nm and ω=0.1 (≈5.7°). Thenumber n

c has been varied from 3 to 20 together

with the distance L to keep the disclination strengthω constant. The distance p has also been assumedto be constant and equal to l. These assumptionsare considered reasonable based on the informa-tion available. As we will see later, actual form ofthe curves is not too sensitive on the values.

Now express the distance L through S and P,whose ratio q=S/P<1. Then we have [184]

Lq

P=−1

2cos.

φ (62)

With formulas under discussion, for the abovecharacteristic values of parameters, the depen-dences of the critical shear stress τ

c on the grain

size (diameter) P have been calculated in [184].These dependences are shown for q=0.5 (Fig.32a)and q=0.1 (Fig.32b), for different values of the triplejunction angle 2φ. The curves τc (P) lie, in general,in different ranges of P because the disclinationstrength ω is taken constant here. Therefore, in-crease in P leads to increase in both L and nc, andthis relation depends on the angle 2φ. For the sakeof convenience, the points corresponding to n

c = 3

(triangles), 5 (pentagons) and 15 (circles) are at-tached to every curve. The upper ends of the curvescorrespond to n

c=20. As a result, one can addition-

ally trace how τc depends on the angle 2φ whenboth the L and n

c keep constant.

From Fig.32 it follows that τc decreases with the

decrease of P (grain refinement) [184]. This is themain result of this section. It demonstrates that thesmaller grains are rotated easier (they are rotated

0.06 0.07 0.08 0.09 0.1

10

10

20

30

∆ W/Gb 2

ω

~


under the action of smaller critical stress τc) than

the larger ones. Moreover, τc strongly depends on

the grain shape (i.e., on q and 2φ). Increase in qleads to decrease in τ

c because larger q (for the

fixed values of P and 2φ) needs smaller number nc

of GB dislocations to spread along the GB (or smallerdipole arm L-p). The dependence of τ

c on the angle

2φ is more complicated. For the fixed L and nc, in

the range of relatively small angles, 2φ < 100°, thecritical stress τ

c decreases with increasing 2φ and

achieves its minimum at 2φ ≈ 100° . In the range ofrelatively large angles, 2φ

> 100°, the critical stress

τc increases with rising 2φ.

Thus, one can conclude that the smaller grainsthat are characterized by the larger q and 2φ

≈100°,

need the smaller critical stress τc to rotate [184].

The theoretical models for motion of GBdisclinations considered in Section 3 allow us toconclude the following:

Fig. 32. Dependence of the critical shear stress τc

on the grain size P, (a) for q=0.5 and (b) for q=0.1,for different values of the triple junction angle 2φgiven by figures at the curves. The numbers of grainboundary dislocations n

c are shown by the triangles

(nc=3), pentagons (n

c=5) and circles (n

c=15).

· Motion of GB disclinations is an effective mecha-nism of rotational deformation in NCMs.

· Motion of GB disclinations may be realizedthrough the micromechanisms of emission oflattice dislocations or climb of GB dislocations.Realization of these micromechanisms are pos-sible in wide ranges of values of the main geom-etry and force parameters of the system.

· Motion of GB disclinations by means of latticedislocation emission is effective until the usualdislocation plasticity is possible in nanograins.When the glide of lattice dislocations is replacedby the GB sliding, appearance of GB disloca-tion pile-ups at the GB triple junctions and fur-ther transformation of these pile-ups into thewalls of climbing GB dislocations (the dipoles ofGB disclinations) serve as a possible way toactivate the rotational mode of plastic deforma-tion in NCMs.

· The transition from the GB sliding to the rota-tional deformation becomes energetically favor-able when the external shear stress achievesits critical value which depends on the elasticproperties of the NCM, structure of its GBs, grainsize and shape. The smaller grains need thesmaller critical stress to rotate.

4. GENERAL CONCLUSIONS

The rotational mode of plastic deformation in NCMsoccurs through nucleation and development of spe-cific (rotational) defect structures which consist oflattice and/or GB dislocations and may be describedeffectively by means of partial disclinations. Rota-tional structures are generated in NCMs at variousimperfections of GBs (kinks, double and triple junc-tions, etc.) where the misorientation angle sharplychanges. Development of rotational plasticity de-pends on the mechanisms of translational plastic-ity which dominate in the given NCM. Domination oflattice gliding may result in appearance ofmisorientation bands (or other disclination structureswhich are typical for conventional metals and al-loys, but have not been considered here) inside thegrains or motion of GB disclinations through emis-sion of lattice dislocations. Domination of GB slid-ing may lead to formation and motion of GBdisclinations through climb of GB dislocations. Inboth the latter cases, the motion of GB disclinationsalong their GBs changes the misorientation of crys-talline lattices across the GBs and may serve as amechanism of grain rotation in the process of exter-nal mechanical loading.

b


ACKNOWLEDGEMENTS

This work was supported, in part, by the Office ofUS Naval Research (grant N00014-01-1-1020), theRussian Fund of Basic Research (grant 01-02-16853), Russian State Research Program on Solid-State Nanostructures, RAS Program “StructuralMechanics of Materials and Constructions”,St.Petersburg Scientific Center, and “Integration”Program (grant B0026).

REFERENCES

[1]. R. Birringer and H. Gleiter, In: Advances inMaterials Science, Encyclopedia of MaterialsScience and Engineering, v. 1, ed. byR.W.Cahn (Pergamon Press, Oxford, 1988), p.339.

[2]. H. Gleiter // Progr. Mater. Sci. 33 (1989) 223.[3]. V.G. Gryaznov, A.M. Kaprelov and A.E.

Romanov, In: Disclinations and RotationalDeformation of Solids, ed. by V.I.Vladimirov(Ioffe Physico-Technical Institute, Leningrad,1988), p. 47, in Russian.

[4]. V.G. Gryaznov, A.M. Kaprelov, I.A. Polonskiiand A.E. Romanov // Phys. Stat. Sol. (b) 167(1991) 29.

[5]. V.G. Gryaznov, A.M. Kaprelov, A.E. Romanovand I.A. Polonskii // Phys. Stat. Sol. (b) 167(1991) 441.

[6]. I.A. Polonsky, A.E. Romanov, V.G. Gryaznovand A.M. Kaprelov // Phil. Mag. A 64 (1991)281.

[7]. L.I. Trusov, M.Yu. Tanakov, V.G. Gryaznov,A.M. Kaprelov and A.E. Romanov // J. Cryst.Growth 114 (1991) 133.

[8]. A.E. Romanov and V.I. Vladimirov, In: Disloca-tions in Solids, v. 9, ed. by F.R.N.Nabarro(North-Holland, Amsterdam, 1992), p. 191.

[9] V.G. Gryaznov and L.I. Trusov // Progr. Mater.Sci. 37 (1993) 290.

[10]. A.E. Romanov, I.A. Polonsky, V.G.Gryaznov, S.A. Nepijko, T. Junghaus, N.I.Vitrykhovski // J. Cryst. Growth 129 (1993)691.

[11]. A.E. Romanov, In: Nanostructured Materials:Science and Technology, ed. by G.-M.Chowand N.I.Noskova (Kluwer Academic Publish-ers, Dordrecht/Boston/London, 1998), p.207.

[12]. M.Yu. Gutkin and I.A. Ovid’ko // Nanostruct.Maters. 2 (1993) 631.

[13]. M.Yu. Gutkin and I.A. Ovid’ko, In: Strengthof Materials, ed. by H.Oikawa, K.Maruyama,

S.Takeuchi and M.Yamaguchi (The JapanInstitute of Metals, Sendai, 1994), p. 227.

[14]. D.A. Konstantinidis and E.C. Aifantis //Nanostruct. Maters. 10 (1998) 1111.

[15]. S.G. Zaichenko and A.M. Glezer // Phys.Solid State 39 (1997) 1810.

[16]. M.Yu. Gutkin, I.A. Ovid’ko and C.S. Pande //Rev. Adv. Mater. Sci. 2 (2001) 80.

[17]. M.Yu. Gutkin, I.A. Ovid’ko and C.S. Pande,In: Nanoclusters and Nanocrystals, Ch. 8,ed. by H.S.Nalwa (American ScientificPublishers, 2003).

[18]. G. Palumbo, U. Erb and K.T. Aust // Scr.Metall. Mater. 24 (1990) 2347.

[19]. A.V. Osipov and I.A. Ovid’ko // Appl. Phys. A54 (1992) 517.

[20] M.Yu. Gutkin and I.A. Ovid’ko // Phil. Mag. A70 (1994) 561.

[21] M.Yu. Gutkin, K.N. Mikaelyan and I.A.Ovid’ko // Phys. Solid State 37 (1995) 300.

[22] M.Yu. Gutkin, I.A. Ovid’ko and K.N.Mikaelyan // Nanostruct. Maters. 6 (1995)779.

[23] A.A. Nazarov, A.E. Romanov and R.Z. Valiev// Nanostruct. Maters. 6 (1995) 775.

[24] M.Yu. Gutkin, K.N. Mikaelyan and I.A.Ovid’ko // Phys. Stat. Sol. (a) 153 (1996)337.

[25] A.A. Nazarov, A.E. Romanov and R.Z. Valiev// Scr. Mater. 34 (1996) 729.

[26] P. Müllner and W.-M. Kuschke // Scr. Mater.36 (1997) 1451.

[27] R.Z. Valiev and I.V. Alexandrov,Nanostructured Materials Obtained byIntensive Plastic Deformation (Logos, Mos-cow, 2000), in Russian.

[28] F.K. LeGoues, M. Copel and R. Tromp //Phys. Rev. Lett. 63 (1989) 1826.

[29] F.K. LeGoues, M. Copel and R. Tromp //Phys. Rev. B 42 (1990) 11690.

[30] C.S. Ozkan, W.D. Nix and H. Gao // Mater.Res. Soc. Proc. 399 (1996) 407.

[31] C.S. Ozkan, W.D. Nix and H. Gao // Appl.Phys. Lett. 70 (1997) 2247.

[32] P. Müllner, H. Gao and C.S. Ozkan // Phil.Mag. A 75 (1997) 925.

[33] H. Gao, C.S. Ozkan, W.D. Nix, J.A.Zimmerman and L.B. Freund // Phil. Mag. A79 (1999) 349.

[34] I.A. Ovid’ko // J. Phys.: Condens. Matter 11(1999) 6521.

[35] I.A. Ovid’ko // Phys. Solid State 41 (1999)1500.


[36] A.G. Sheinerman and M.Yu. Gutkin // Phys.Stat. Sol. (a) 184 (2001) 485.

[37] A.L. Kolesnikova, A.E. Romanov and I.A.Ovid’ko // Solid State Phenomena 87 (2002)265.

[38] I.A. Ovid’ko, A.G. Sheinerman and N.V.Skiba // J. Phys.: Condens. Matter 15 (2003)1173.

[39] A.E. Romanov, In: Experimental Investigationand Theoretical Description of Disclinations,ed. by V.I.Vladimirov (Ioffe Physico-TechnicalInstitute, Leningrad, 1984), p. 110, in Rus-sian.

[40] A.E. Romanov // Mater. Sci. Eng. A 164(1993) 58.

[41] M.Yu. Gutkin, A.L. Kolesnikova and A.E.Romanov // Mater. Sci. Eng. A 164 (1993)433.

[42] A.L. Kolesnikova, In: Experimental Investiga-tion and Theoretical Description ofDisclinations, ed. by V.I.Vladimirov (IoffePhysico-Technical Institute, Leningrad,1984), p. 194, in Russian.

[43] A.L. Kolesnikova, N.D. Priemskii and A.E.Romanov, Preprint No. 869, (Ioffe Physico-Technical Institute, Leningrad, 1984), inRussian.

[44] V.I. Vladimirov, A.L. Kolesnikova and A.E.Romanov // Phys. Metals Metallogr. 60 (1985)58.

[45] A.L. Kolesnikova and A.E. Romanov, PreprintNo. 1019 (Ioffe Physico-Technical Institute,Leningrad, 1986), in Russian.

[46] A.G. Zembilgotov and N.A. Pertsev, In:Disclinations and Rotational Deformation ofSolids, ed. by V.I.Vladimirov (Ioffe Physico-Technical Institute, Leningrad, 1988), p. 158,in Russian.

[47] A.Yu. Belov // Phil. Mag. Letters 64 (1991)207.

[48] A.Yu. Belov // Phil. Mag. A 65 (1992) 1429.[49] A.L. Kolesnikova and A.E. Romanov // Phys.

Solid State 45 (2003) 1626.[50] Y.Z. Povstenko // Int. J. Engng. Sci. 33

(1995) 575.[51] Y.Z. Povstenko // J. Physique IV 8 (1998)

309.[52] M.Yu. Gutkin and E.C. Aifantis // Phys. Stat.

Sol. (b) 214 (1999) 245.[53] M.Yu. Gutkin and E.C. Aifantis // Phys. Solid

State 41 (1999) 1980.[54] M.Yu. Gutkin and E.C. Aifantis, In:

Nanostructured Films and Coatings, NATO

ARW Series, High Technology, v. 78, ed. byG.-M.Chow, I.A.Ovid’ko and Y.Tsakalakos(Kluwer, Dordrecht, 2000), p. 247.

[55] M.Yu. Gutkin // Rev. Adv. Mater. Sci. 1(2000) 27.

[56] M. Lazar // Phys. Lett. A 311 (2003) 416.[57] M. Lazar // J. Phys.: Condens. Matter (2003),

in press.[58] H. Gleiter // Phys. 47 (1991) 753.[59] H. Gleiter // Nanostruct. Mater. 1 (1992) 1.[60] U. Erb, A.M. El-Sherik, G. Palumbo and K.T.

Aust // Nanostruct. Mater. 2 (1993) 383.[61] R.W. Siegel // Mater. Sci. Eng. A 168 (1993)

189.[62] R.W. Siegel // Nanostruct. Mater. 4 (1994)

121.[63] N.I. Noskova, In: Contemporary Problems of

Physics and Mechanics of Materials (Saint-Petersburg State University, Saint-Peters-burg, 1997), p. 333, in Russian.

[64] A.I. Gusev // Physics-Uspekhi 41 (1998) 49.[65] R.A. Andrievskii and A.M. Glezer // Phys.

Metals Metallogr. 88 (1999) 45.[66] R.A. Andrievskii and A.M. Glezer // Phys.

Metals Metallogr. 89 (2000) 83.[67] H. Gleiter // Acta Mater. 48 (2000) 1.[68] I.A. Ovid’ko // Rev. Adv. Mater. Sci. 1 (2000)

61.[69] R.A. Andrievskii // Perspektivnye Materialy

No. 6 (2001) 5, in Russian.[70] R.A. Andrievski // Mater. Trans. 42 (2001)

801.[71] R.A. Andrievski and G.V. Kalinnikov // Surf.

Coat. Technology 142-144 (2001) 573.[72] J. Eckert, A. Reger-Leonhard, B. Weiss,

M. Heilmaier and L. Schultz // Adv. Eng.Mater. 3 (2001) 41.

[73] D.V. Shtanskii and E.A. Levashov //Izvestiya VUZov. Tsvetnaya Metallurgiya No.3 (2001) 52, in Russian.

[74] I.A. Ovid’ko, In: Science and Technology ofInterfaces, International Symposium in Honorof Dr. Bhakta Rath, ed. by S. Ankem, C.S.Pande, I. Ovid’ko and S. Ranganathan (TMS,Warrendale, 2002), p. 245.

[75] I.A. Ovid’ko, In: Encyclopedia of Nanoscienceand Nanotechnology, Chapter 4, ed. byH.S.Nalwa (American Sci. Publ., California,2003).

[76] V.M. Fedosyuk, Multilayered MagneticStructures (Belorussian State University,Minsk, 2000), in Russian.


[77] A.I. Gusev and A.A. Rempel, NanocrystallineMaterials (Nauka, Moscow, 2000), in Rus-sian.

[78] M.Yu. Gutkin and I.A. Ovid’ko, Defects andMechanisms of Plasticity in Nanostructuredand Non-crystalline Materials (Yanus, Saint-Petersburg, 2001), in Russian.

[79] Yu.R. Kolobov, R.Z. Valiev, G.P.Grabovetskaya, A.P. Zhilyaev, E.F. Dudarev,K.V. Ivanov, M.B. Ivanov, O.A. Kashin andE.B. Naidenkin, Grain-Boundary Diffusionand Properties of Nanostructured Materials(Nauka, Novosibirsk, 2001), in Russian.

[80] V.M. Fedosyuk and T.A. Tochitskii,Electrolitically deposited nanostructures(Belorussian State University, Minsk, 2002),in Russian.

[81] Proceedings of the Second InternationalConference on Nanostructured Materials(Stuttgart, Germany, October 3-7, 1994), ed.by H.-E.Schaefer, R.Würschum, H.Gleiterand T.Tsakalakos // Nanostruct. Mater. 6(1995) 533.

[82] Nanomaterials: Synthesis, Properties andApplications, ed. by A.S.Edelstain andR.C.Cammarata (Institute of Physics Publ.,Bristol and Philadelphia, 1996).

[83] Structure, Phase Transformations andProperties of Nanocrystalline Alloys (UralDepartment of Russian Academy of Sci-ences, Ekaterinburg, 1997), in Russian.

[84] R&D Status and Trends in Nanoparticles,Nanostructured Materials, and Nanodevicesin the United States, ed. by R.W.Siegel,E.Hwu and M.C.Roco (International Technol-ogy Research Institute, Baltimore, 1997).

[85] Nanostructured Materials: Science andTechnology, ed. by G.-M.Chow andN.I.Noskova (Kluwer Academic Publishers,Dordrecht/Boston/London, 1998).

[86] Handbook of Nanostructured Materials andNanotechnology, Vols. 1-5, ed. by H.S.Nalwa(Academic Press, San Diego, 1999).

[87] Material Science of Carbides, Nitrides andBorides, ed. by Y.G.Gogotsi andR.A.Andrievski (Kluwer Academic Publish-ers, Dordrecht/Boston/London, 1999).

[88] Mechanical Behavior of NanostructuredMaterials // MRS Bulletin 24 (1999) 14.

[89] Nanostructured Films and Coatings, NATOARW Series, High Technology, v. 78, ed. byG.-M.Chow, I.A.Ovid’ko and T.Tsakalakos(Kluwer, Dordrecht, 2000).

[90] Structure and Mechanical Properties ofNanophase Materials - Theory and ComputerSimulations vs. Experiment, ed by D.Farkas,H.Kung, M.Mayo, J.Van Swygenhoven andJ.Weertman (MRS, Warrendale, 2001).

[91] Problems of Nanocrystalline Materials, ed. byV.V.Ustinov and N.I.Noskova (Ural Depart-ment of Russian Academy of Sciences,Ekaterinburg, 2002), in Russian.

[92] Nanophase and Nanocomposite Materials IV,MRS Symp. Proc., v. 703, ed. byS.Komarneni, R.A.Vaia, G.Q.Lu,J.- I.Matsushita and J.C.Parker (MRS,Warrendale, 2003).

[93] Nanomaterials for Structural Applications,MRS Symp. Proc., v. 740, ed. by C.C.Berndt,T.Fischer, I.A.Ovid’ko, G.Skandan andT.Tsakalakos (MRS, Warrendale, 2003).

[94] U. Herr, J. Jing, R. Birringer, U. Conser andH. Gleiter // Appl. Phys. Lett. 50 (1987) 472.

[95] H.-E. Schaefer and R. Würshum // Phys.Lett. A 119 (1987) 370.

[96] R. Würshum, M. Scheytt and H.-E. Schaefer// Phys. Stat. Sol. (a) 102 (1987) 119.

[97] X. Zhu, R. Birringer, U. Herr and H. Gleiter //Phys. Rev. B 35 (1987) 9085.

[98] T. Haubold, R. Birringer, B. Lengeler andH. Gleiter // Phys. Lett. A 135 (1989) 461.

[99] E. Jorra, H. Franz, J. Peisl, G. Wallner,W. Petry, T. Haubold, R. Birringer andH. Gleiter // Phil. Mag. B 60 (1989) 159.

[100] W. Wunderlich, Y. Ishida and R. Maurer //Scr. Metall. Mater. 24 (1990) 403.

[101] A.H. King // Interf. Sci. 7 (1999) 251.[102] V.B. Rabukhin // Poverkhnost’. Fizika,

Khimiya, Mekhanika No. 7 (1986) 126, inRussian.

[103] G. Palumbo and K.T. Aust // Mater. Sci.Eng. A 113 (1989) 139.

[104] K.M. Yin, A.H. King, T.E. Hsieh, F.R. Chen,J.J. Kai and L. Chang // Microscopy andMicroanalysis 3 (1997) 417.

[105] U. Czubayko, V.G. Sursaeva, G. Gottsteinand L.S. Shvindlerman, In: Grain Growth inPolycrystalline Materials III, ed. byH.Weiland, B.L.Adams and A.D.Rollett(TMS, Pittsburg, 1998), p. 423.

[106] G. Gottstein, A.H. King and L.S.Shvindlerman // Acta Mater. 48 (2000) 397.

[107] K. Owusu-Boahen and A.H. King // ActaMater. 49 (2001) 237.

[108] A.A. Fedorov, M.Yu. Gutkin and I.A. Ovid’ko// Scr. Mater. 47 (2002) 51.


[109] J. Horvath, R. Birringer and H. Gleiter //Solid State Comm. 62 (1987) 319.

[110] H.-E. Schaefer, R. Würschum,T. Gessmann, G. Stöckl, P. Scharwaechter,W. Frank, R.Z. Valiev, H.-J. Fecht andC. Moelle // Nanostruct. Mater. 6 (1995) 869.

[111] Yu.R. Kolobov, G.P. Grabovetskaya, I.V.Ratochka and K.V. Ivanov // Nanostruct.Mater. 12 (1999) 1127.

[112] Yu.R. Kolobov, G.P. Grabovetskaya, K.V.Ivanov, R.Z. Valiev and T.C. Lowe, In:Investigations and Applications of SeverePlastic Deformation, NATO Science Ser.,ed. by T.C.Lowe and R.Z.Valiev (Kluwer,Dordrecht, 2000), p. 261.

[113] V.V. Rybin, Large Plastic Deformations andFracture of Metals (Metallurgia, Moscow,1986), in Russian.

[114] M. Seefeldt // Rev. Adv. Mater. Sci. 2 (2001)44.

[115] P. Klimanek, V. Klemm, A.E. Romanov andM. Seefeldt // Adv. Eng. Maters. 3 (2001)877.

[116] Local Lattice Rotations and Disclinations inMicrostructures of Distorted CrystallineMaterials, ed. by P. Klimanek, A.E.Romanov, M. Seefeldt // Solid State Phe-nomena 87 (2002).

[117] U. Essmann // Phys. stat. sol. 3 (1963) 932.[118] I.W. Steeds // Proc. Roy. Soc. A 292 (1966)

343.[119] A.N. Vergazov, V.A. Likhachev and V.V.

Rybin // Fiz. Met. Metalloved. 42 (1976)146, in Russian.

[120] G.V. Berezhkova and P.P. Perstnev // Sov.Phys.-Doklady 24 (1979) 799.

[121] G.V. Berezhkova, P.P. Perstnev, A.E.Romanov and V.I. Vladimirov // Cryst. Res.Techn. 18 (1983) 139.

[122] E.V. Nesterova and V.V. Rybin // Fiz. Met.Metalloved. 59 (1985) 395, in Russian.

[123] A.D. Korotaev, A.N. Tyumentsev and V.F.Sukhovarov, Dispersion Hardening ofRefractory Metals (Nauka, Novosibirsk,1989), in Russian.

[124] A.D. Korotaev, A.N. Tyumentsev and Yu.P.Pinjin // Phys. Mesomechanics 1 (1998) 23.

[125] M.Yu. Gutkin, K.N. Mikaelyan, A.E.Romanov and P. Klimanek // Phys. Stat.Sol. (a) 193 (2002) 35.

[126] M. Murayama, J.M. Howe, H. Hidaka and S.Takaki // Science 295 (2002) 2433.

[127] V.I. Vladimirov and A.E. Romanov // Sov.Phys. - Solid State 20 (1978) 1795.

[128] B.K. Barakhtin, S.A. Ivanov, I.A. Ovid’ko,A.E. Romanov and V.I. Vladimirov // J.Phys. D 22 (1989) 519.

[129] A.E. Romanov and E.C. Aifantis // Scr.Metall. Mater. 29 (1993) 707.

[130] M. Seefeldt and P. Klimanek // Mater. Sci.Eng. A 234-236 (1997) 758.

[131] M. Seefeldt and P. Klimanek // Model.Simul. Mater. Sci. Eng. 6 (1998) 349.

[132] K.N. Mikaelyan, M. Seefeldt, M.Yu. Gutkin,P. Klimanek and A.E. Romanov // Phys.Solid State 45 (2003) 000.

[133] R. de Wit // J. Res. Nat. Bur. Stand. A 77(1973) 607.

[134] A. Needleman // Acta mater. 48 (2000) 105.[135] B. Devincre, L.P. Kubin, C. Lemarchand and

R. Madec // Mater. Sci. Eng. A 309-310(2001) 211.

[136] L. Nicola, E. Van der Giessen and A.Needleman // Mater. Sci. Eng. A 309-310(2001) 274.

[137] N. Argaman, O. Levy and G. Makov //Mater. Sci. Eng. A 309-310 (2001) 386.

[138] O. Politano and J.M. Salazar // Mater. Sci.Eng. A 309-310 (2001) 261.

[139] H. Yasin, H.M. Zbib and M.A. Khaleel //Mater. Sci. Eng. A 309-310 (2001) 294.

[140] P. Rodriguez, D. Sundararaman, R. Divakarand V.S. Raghunathan // Chem. Sustain-able Development 8 (2000) 69.

[141] J.P. Hirth and J. Lothe, Theory of Disloca-tions (John Wiley, New York, 1982).

[142] M.F. Ashby // Phil. Mag. 21 (1970) 399.[143] U.F. Kocks, A.S. Argon and M.F. Ashby //

Progr. Maters. Sci. 19 (1975) 1.[144] K.M. Jassby and T. Vreeland, Jr. // Phil.

Mag. 21 (1970) 1147.[145] H. Gleiter // Mater. Sci. Eng. A 52 (1982)

91.[146] A.P. Sutton and R.W. Balluffi, Interfaces in

Crystalline Materials (Clarendon Press,Oxford, 1995).

[147] J.P. Hirth // Acta. Mater. 48 (2000) 93.[148] L.G. Kornelyuk, A.Yu. Lozovoi and I.M.

Razumovskii // Philos. Mag. A 77 (1998)465.

[149] M.Yu. Gutkin and I.A. Ovid’ko // Phys. Rev.B 63 (2001) 064515.

[150] I.A. Ovid’ko, In: Nanostructured Films andCoatings, NATO ARW Series, High Tech-nology, v. 78, ed. by G.-M.Chow, I.A.Ovid’ko


and T.Tsakalakos (Kluwer, Dordrecht, 2000),p. 231.

[151] S.V. Bobylev, I.A. Ovid’ko and A.G.Sheinerman // Phys. Rev. B 64 (2001)224507.

[152] K.A. Padmanabhan and G.J. Davies,Superplasticity (Springer, Berlin, 1980).

[153] J. Pilling and N. Ridle, Superplasticity inCrystalline Solids (The Institute of Metals,London, 1989).

[154] O.A. Kaibyshev and F.Z. Utyashev, Super-plasticity, Structure Refinement and Treat-ment of Hard-Deforming Alloys (Nauka,Moscow, 2002), in Russian.

[155] M.G. Zelin and A.K. Mukherjee // Mater.Sci. Eng. A 208 (1996) 210.

[156] R.Z. Valiev and T.G. Langdon // Acta Metall.41 (1993) 949.

[157] M. Mayo, D.C. Hague and D.-J. Chen //Mater. Sci. Eng. A 166 (1993) 145.

[158] M. Jain and T. Christman // Acta Metall.Mater. 42 (1994) 1901.

[159] R.S. Mishra, R.Z. Valiev and A.K.Mukherjee // Nanostruct. Maters. 9 (1997)473.

[160] V.V. Astanin, O.A. Kaibyshev and S.N.Faizova // Scr. Metall. Mater. 25 (1991)2663.

[161] H.S. Yang and M.G. Zelin // Scr. Metall.Mater. 26 (1992) 1707.

[162] M. Seefeldt and P. Van Houtte // Mater.Phys. Mech. 1 (2000) 133.

[163] M.J. Mayo // Nanostruct. Mater. 9 (1997)717.

[164] B. Baudelet // Scr. Metall. Mater. 27 (1992)745.

[165] M.G. Zelin, R.Z. Valiev, M.V. Grabski, J.W.Wyrzykowski, H.S. Yang and A.K.Mukherjee // Mater. Sci. Eng. A 160 (1993)215.

[166] M.G. Zelin and A.K. Mukherjee // ActaMetall. Mater. 43 (1995) 2359.

[167] M.G. Zelin and A.K. Mukherjee // J. Mater.Res. 10 (1995) 864.

[168] M. Ke, W.W. Milligan, S.A. Hackney, J.E.Carsley and E.C. Aifantis // Mat. Res. Soc.Symp. Proc. 308 (1993) 565.

[169] M. Ke, S.A. Hackney, W.W. Milligan andE.C. Aifantis // Nanostruct. Maters. 5 (1995)689.

[170] N.I. Noskova and E.G. Volkova // FizikaMetallov i Metallovedenie 91 (2001) 100, inRussian.

[171] N.I. Noskova and E.G. Volkova // FizikaMetallov i Metallovedenie 92 (2001) 107, inRussian.

[172] N.I. Noskova, In: Problems ofNanocrystalline Materials, ed. byV.V.Ustinov and N.I.Noskova (Ural Depart-ment of Russian Academy of Sciences,Ekaterinburg, 2002), p. 159, in Russian.

[173] K.E. Harris, V.V. Singh and A.H. King //Acta Mater. 46 (1998) 2623.

[174] K.N. Mikaelyan, I.A. Ovid’ko, and A.E.Romanov // Mater. Sci. Eng. A 288 (2000)61.

[175] I.A. Ovid’ko // Science 295 (2002) 2386.[176] M.Yu. Gutkin, A.L. Kolesnikova, I.A. Ovid’ko

and N.V. Skiba // J. Metastable &Nanostruct.Maters 12 (2002) 47.

[177] M.Yu. Gutkin, A.L. Kolesnikova, I.A. Ovid’koand N.V. Skiba // Phil. Mag. Letters 82(2002) 651.

[178] M.Yu. Gutkin, I.A. Ovid’ko and N.V. Skiba //Techn. Phys. Lett. 28 (2002) 437.

[179] M.Yu. Gutkin, I.A. Ovid’ko and N.V. Skiba //Mater. Sci. Eng. A 339 (2003) 73.

[180] H. Grimmer, W. Bollmann and D.H.Warrington // Acta Crystall. A 30 (1974)197.

[181] M.Yu. Gutkin, K.N. Mikaelyan and I.A.Ovid’ko // Phys. Solid State 37 (1995) 300.

[182] M.Yu. Gutkin, I.A. Ovid’ko and K.N.Mikaelyan // Nanostruct. Maters. 6 (1995)779.

[183] M.Yu. Gutkin, K.N. Mikaelyan and I.A.Ovid’ko // Phys. Stat. Sol. (a) 153 (1996)337.

[184] M.Yu. Gutkin, I.A. Ovid’ko and N.V. Skiba //Acta Mater. 51 (2003) 4059

[185] A.A. Fedorov, M.Yu. Gutkin and I.A. Ovid’ko// Acta Mater. 51 (2003) 887.

[186] J.D. Eshelby, F.C. Frank, and F.R.N.Nabarro // Phil. Mag. 42 (1951) 351.

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