grain growth and collective migration of grain boundaries during plastic deformation of...

ISSN 1063-7834, Physics of the Solid State, 2008, Vol. 50, No. 7, pp. 1266–1279. © Pleiades Publishing, Ltd., 2008.Original Russian Text © M.Yu. Gutkin, K.N. Mikaelyan, I.A. Ovid’ko, 2008, published in Fizika Tverdogo Tela, 2008, Vol. 50, No. 7, pp. 1216–1229.

1266

1. INTRODUCTION

It is generally agreed that the unusually highmechanical properties of nanocrystalline materialshave been reliably established to be mainly due to thespecific deformation mechanisms closely related tograin boundaries (GBs) (see reviews and monographs[1–12]).

Among the GB-controlled mechanisms of plasticity,particular attention has been recently focused on thegrain growth during plastic deformation of ultrafine-grained [13–16] and nanocrystalline [13–27] metalsand alloys at room [13–27] and cryogenic [16–18] tem-peratures. Experimental studies on ultrafine-grainedpure Al [13–15] and Cu [16] and Al–Mg alloys [14] andon nanocrystalline pure Al [13–15, 22, 23], Cu [16–18],and Ni [19–21] and Co–P [24, 25] and Ni–Fe [25–27]alloys have shown that the grain growth is possible dur-ing nano- [13–15, 20] and microindentation [16–18] ofthin films and torsion of them under high pressure [19],during compression of micropillars [21] and macro-scopic samples [26], and, finally, during uniaxial ten-sion of thin films [22, 23] and bulk planar samples [24,25, 27].

For an understanding of the physical nature of thisphenomenon, the following experimental facts are ofimportance:

(i) At cryogenic temperatures, grains grow morerapidly than at room temperature [17].

(ii) The grain growth is the most intensive in theregions of a sample where the elastic stress and its gra-dient are the largest (e.g., under nanoindenter [13–15,20]) or in the immediate vicinity of a microindenter[16–18], near the tip of a slowly developing crack [22],and in the surface layer of a sample in the vicinity of aneck forming under tension [27].

(iii) The grain growth is completely suppressed [14]or reduced [24–27] in the presence of impurities.

(iv) During grain growth, not only the grain size butalso the character of the grain size distribution arechanged; the distribution is broadened and sometimesbecomes bimodal [17, 22, 25], with larger submicrongrains occupying up to 15% of the volume [17] and finenanograins surrounding them.

(v) The grain growth occurs at relatively lownanoindentation rates [20] and during microindentationin the creep mode [16–18], at relatively high compres-sion rates (~10

–3

–10

–1

s

–1

[21] and 10

–3

s

–1

[26]), and atwidely ranged tension rates (~10

–5

s

–1

[22, 23], 10

–3

s

–1

[23, 24], 10

–5

–10

–2

s

–1

[25], 10

–2

s

–1

[27]).(vi) The grain growth somewhat decreases the ulti-

mate strength, but it significantly increases the ultimatetensile strain (up to 25% in nanocrystalline pure Al [22]and up to 7.2% in the nanocrystalline Ni–Fe alloy [27]),which is accompanied by noticeable hardening and theformation of dislocation structures in coarse grains. Inmicropillars of pure nanocrystalline Ni under uniaxialcompression, ultrahigh plasticity (up to 200% of the

DEFECTS AND IMPURITY CENTERS, DISLOCATIONS,AND PHYSICS OF STRENGTH

Grain Growth and Collective Migration of Grain Boundaries during Plastic Deformation of Nanocrystalline Materials

M. Yu. Gutkin, K. N. Mikaelyan, and I. A. Ovid’ko

Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Vasil’evski

œ

Ostrov, Bol’sho

œ

pr. 61, St. Petersburg, 199178 Russiae-mail: [email protected], [email protected]

Received November 12, 2007

Abstract

—A theoretical model is proposed for the collective migration of two neighboring grain boundaries(GBs) in a nanocrystalline material under applied elastic stress. By analyzing the change in the energy of thesystem, it is shown that GBs can remain immobile or migrate toward each other depending on the values of theapplied shear stress and misorientation angles. The process of GB migration can proceed either in a stableregime, wherein the GBs occupy equilibrium positions corresponding to a minimum of the energy of the systemunder relatively small applied stress, or in an unstable regime, wherein the motion of GBs under relatively highstress is accompanied by a continuous decrease in the system energy and becomes uncontrollable. The stablemigration of GBs leads to a decrease of the grain bounded by them at the cost of growth of the neighbor grainsand can result in complete or partial annihilation of the GBs and the collapse of this grain. Unstable migrationleads either to annihilation of GBs or to passage of them through each other, which can be considered as thedisappearance of the grain and nucleation and growth of a new grain.

PACS numbers: 61.72.Bb, 61.72.Lk, 61.72.Mm, 62.20.Fe, 62.25.+g

DOI:

10.1134/S1063783408070135

PHYSICS OF THE SOLID STATE

Vol. 50

No. 7

2008

GRAIN GROWTH AND COLLECTIVE MIGRATION 1267

true strain) was observed at a flow stress of 2.0–2.4 GPa, which was accompanied by softening (due toequiaxial grain growth) followed by hardening causedby elongation of grown grains and accumulation of dis-locations and twins in them [21].

(vii) With increasing duration of holding of a sampleunder indenter, the low-angle GBs were observed toincrease in number, especially at cryogenic tempera-tures [18]. In submicron-sized grains grown duringsevere plastic deformation, subgrains were observed,which, in turn, were filled with dislocation cells [19].

The authors of [13–27] believe that the above resultsunambiguously indicate the athermic character of graingrowth, which occurs under elastic stresses forming atearlier stages of plastic flow. The process of graingrowth is inhomogeneous over a cross section of thesample; indeed, the grains that are located in places ofconcentrated stresses and, in addition, have a favorableorientation increase in size. When the growing grainsbecome a few hundred nanometers in size, the plasticitymechanisms typical of low-temperature deformation ofcoarse-grained metals begin to operate in them. Forexample, in copper, dislocation glide and the formationof dislocation pileups were observed at room tempera-ture and deformation twinning, at a cryogenic tempera-ture (77 K) [18]. Thus, the grain growth during plasticdeformation increases the plasticity of nanocrystallinemetals, while the flow stress remains high at low tem-peratures and relatively high loading rates, which isvery important for practical applications [15].

The grain growth at room temperature was alsoobserved earlier upon holding of nanocrystalline metalsamples for a few days after compaction [28–30] or dur-ing uniaxial pressing of a very pure nanocrystalline alu-minum powder [31] or uniaxial compression of ultrafine-grained copper samples [32] obtained by equal-channelangular pressing. The grain growth was considered athermally activated process favored by the specific fea-tures of the GBs in such materials [28–30, 32] or by theabsence of impurities capable of pinning GBs [31].

Recently, the significant grain growth has beenobserved during superplastic deformation of Si

3

N

4

nano-ceramics subjected to compression at 1500

°

C [33]. At alower strain rate (3

×

10

–5

s

–1

), the grain size distributionwas observed to become bimodal, with a considerablefraction of the initially equiaxial grains with a mean sizeof 68 nm growing to a mean size of 145 nm along a cer-tain direction. At a higher strain rate (10

–4

s

–1

), the graingrowth was less significant and the grain size distributionremained unimodal, with grains being elongated to amean size of 89 nm and becoming rod-shaped.

In recent computer simulation studies of the GBmigration [34–39] and grain growth [35–38] in nanoc-rystalline metals under stress at room [34–37] and zero[38, 39] temperatures, it has been shown that these pro-cesses are athermic in nature during plastic deforma-tion. The main mechanisms of grain growth have beenshown to be stress-stimulated migration of GBs and

their triple junctions, GB sliding, and rotation and coa-lescence of grains. In particular, all of these mecha-nisms were observed to operate simultaneously inmodel samples of nanocrystalline Ni [37] and Al [38]with a mean grain size of 5 and 7 nm, respectively,when nanograins rotated through GB sliding underuniaxial tension [37] or nanoindentation [38]. In thiscase, the misorientation of one of the boundaries of agrain decreases and the GB changes from the high-angle (with a misorientation angle

θ

≈

22

°

) to a low-angle GB (with

θ

≈

13

°

) and begins to migrate throughglide of its dislocations into the body of the grain [38].This process results in the absorption of these disloca-tions by the opposite GB and coalescence of the twograins into a single one. Analogous situations were alsoobserved in simulations of uniaxial tension of nanoc-rystalline aluminum with a mean grain size of 5 nm atroom temperature [40]. The rotation and coalescence ofnanograins were also studied in computer simulationsof deformation of nanocrystalline Pd with a mean grainsize of 15 nm at 1200 K [41].

Several continuum dislocation–disclination modelshave been used to theoretically study the GB migrationand grain growth in plastically deformed nanocrystal-line materials [42–45]. For example, the dynamics anddecay of a low-angle tilt boundary under an appliedshear stress

τ

were studied in [42, 43]. It was shownthat, as the stress

τ

increases, the tilt boundary is firstbent and then shifts to a new position corresponding tothe applied stress. At a certain critical stress propor-tional to the misorientation angle of the boundary, itbecomes unstable and glides irreversibly. The decay ofone such boundary significantly decreases the criticalstress for decay of neighboring low-angle boundaries.As a result, the chain decay of the neighboring bound-aries occurs and the grains separated by them coalesce.The static model of the escape of dislocations from aninfinite straight dislocation wall developed in [44] alsopermits one to estimate the critical stress for boundarydecay, which is proportional to the misorientation angleof the boundary. In spite of the obvious crudeness of themodel from [44], the stress at which an intrinsic dislo-cation of a low-angle boundary (with

θ

= 0.1(

≈

5.7

°

))breaks away from it (estimated from this model to beabout 2 GPa for Fe) is not far greater than that obtainedfrom dynamic calculations (1.53 GPa) [42, 43]. Themodel from [44] also gives lower values of the criticalstress for the decay of a boundary in the case where afew intrinsic or extrinsic dislocations break awaysimultaneously from it. However, the description of theboundary decay assuming that only single dislocationsmove while the positions of the other dislocationsremain unchanged seems incorrect.

In [45], a continuum disclination model was pro-posed for describing the migration of an arbitrary tiltboundary. A migrating boundary was approximated bya biaxial dipole of partial wedge disclinations capableof moving under an applied shear stress

τ

in the elasticfield of a similar disclination dipole of opposite sign

1268


Vol. 50

No. 7

2008

GUTKIN et

al.

that forms when the boundary breaks away from theneighboring GBs (i.e., at the moment when triple GBjunctions transform into double junctions). The mecha-nism of motion of a high-angle boundary was not con-sidered because it can be described only in terms ofatomic models [46–48]. It was shown that there are twomodes of GB migration. When the applied stressreaches the first critical value

τ

c

1

, the GB begins tomigrate in the stable mode in which its equilibriumposition is determined by a stress level

τ

≥

τ

c

1

. When thestress

τ

reaches the second critical value

τ

c

2

, the GBmigration becomes unstable; the equilibrium positionof the GB disappears, and the GB migration no longerdepends on

τ

. Analytical expressions and numericalestimates were obtained for the critical stresses

τ

c

1

and

τ

c

2

. For example, for pure nanocrystalline aluminumwith grain size

d

ranging from 30 to 100 nm, the stress

τ

c

1

ranges from 7.6 to 23.5 MPa for

θ

≈

5

°

and from46.5 to 144 MPa for

θ

≈

30

°

. The stress

τ

c

2

proves to befar greater, namely, 0.4 GPa for

θ

≈

5

°

and 2.5 GPa for

θ

≈

30

°

, irrespective of the grain size. We note that, inthin nanocrystalline aluminum films under tension,grains begin to grow intensively over the range of truetensile stresses from 130 MPa for

d

≈

90 nm to 190 MPafor

d

≈

40 nm [22, 23], which correspond to the maxi-mum values of

τ

from 65 to 95 MPa lying in the range

τ

c

1

<

τ

<

τ

c

2

for GBs with a misorientation angle

θ

≈

5

°

and in a part of this range for GBs with

θ

≈

30

°

. Com-puter simulations of nanoindentation of nanocrystallinealuminum films with a mean grain size of 7 nm showed[38] that the migration of a low-angle tilt boundary witha misorientation angle of 13.5

°

becomes unstable underlocal shear stresses coinciding with the estimated value0.7 GPa obtained from the formula for

τ

c

2

derived in [45].

The objective of this work is to generalize the discli-nation model from [45] to the case of collective migra-tion of the two opposite boundaries of a grain and todescribe (in terms of a continuum model) the graingrowth as a peculiar mechanism of plastic deformationin nanocrystalline materials.

2. MODEL

Let us consider a model of a nanocrystalline mate-rial subjected to a normal stress

σ

1

(Fig. 1a). The signof the stress is of no importance; however, for definite-ness, we consider the case of uniaxial tension. Out ofthe great number of nanograins, we separate four grainsdesignated by Roman numerals I–IV. These grains areassumed to be bounded by tilt boundaries with rela-tively small misorientation angles

ω

and

Ω

less thanthose of the other boundaries. For clarity, we assumethat these angles are less than 15

°

and, therefore, theboundaries can be considered to be walls of edge latticedislocations. In Fig. 1a, grains I and IV are bounded bydislocation walls with differing periods and, hence, dif-fering misorientation angles

ω

and

Ω

, and grains II andIII are bounded by dislocation walls with identical peri-

ods corresponding to the misorientation angle

Ω

. Weassume that all triple junctions of these walls with theadjacent GBs are compensated, i.e., they do not containjunction disclinations. In this work, we consider thecollective migration of GBs opposite in sign. In the caseof low-angle tilt boundaries, this means that the Burg-ers vectors of the dislocations in the opposite walls areoppositely directed. Moreover, we assume that the slipplanes of dislocations belonging to the opposite bound-aries of grains I and II coincide and that the slip planesof dislocations belonging to the opposite boundaries ofgrains III and IV do not coincide. The orientation ofdislocations in these boundaries is chosen so that, underthe applied tensile stress, the boundaries of one grainmigrate toward each other, which is favored by themutual attraction of the GBs of opposite sign.

If the applied stress

σ

1

is less than a certain criticalvalue, all GBs are in their initial positions (Fig. 1a).Under a stress

σ

2

exceeding this critical value, the GBsbegin to migrate under the action of the correspondingshear stresses that operate in the slip planes of the dis-locations belonging to these GBs. The GBs with thesmaller misorientation angle

ω

(

ω

boundaries) are thefirst to shift from their initial positions [42–45]. Then,as the stress

σ

2

increases further, the GBs with thelarger misorientation angle Ω (Ω boundaries) are dis-placed. Figure 1b schematically shows the situationwherein the migration of ω and Ω boundaries is stableand the equilibrium positions of these boundaries aredetermined by the resolved shear stress τ (which is thesame for all migrating boundaries in Fig. 1b due to thesymmetry of the figure). After the ω and Ω boundariesare displaced from their initial positions on triple junc-tions (which now became double), junction partialwedge disclinations are formed with strengths (Frankvector magnitudes) equal to ±ω and ±Ω , respectively[42, 43, 45]. A typical dipole–quadrupole structure ofjunction disclinations is formed in which each dipoleattracts “its boundary” back and repels the oppositeone. The balance between these attractive and repulsiveforces, the interactions between the migrating bound-aries, and the external stress determines the equilibriumposition of each migrating boundary.

As the stress σ2 (and, accordingly, the stress τ)increases, the migrating boundaries approach eachother and finally meet at a certain stress σ3. As a result,the boundaries can annihilate partially (grain I) or com-pletely (grain II) if dislocations in them glide on thesame slip planes, or the boundaries can pass througheach other (grains III, IV) if the slip planes do not coin-cide (Fig. 1c). The dislocations that survived these pro-cesses are accumulated at the opposite boundaries ofenlarged grains I, III, and IV.

Further deformation of the sample stimulatessmoothing of the boundaries of grown grains I–IV (atthe cost of a decrease in the length of their commonboundary) and the corresponding change in the strengthof the remaining junction disclinations, which now

PHYSICS OF THE SOLID STATE Vol. 50 No. 7 2008


transformed into GB disclinations with strengths ±ω'and ±Ω' (Fig. 1d). The evolution of the defect structurecan include, first, the displacement of these disclina-tions toward each other through emission of perfect orpartial lattice dislocations into the adjacent grains [49]followed by complete or partial annihilation of thesedisclinations and, second, further growth and coales-cence of grains. For example, the dislocation wallsforced against the boundaries of grains III and IV cancause migration of these boundaries.

In this work, we consider in more detail the initialstages of the collective migration of two oppositeboundaries of one grain assuming that, in a first approx-imation, the processes occurring in this grain are inde-pendent of the processes in the adjacent growing grains.Figure 2 shows a simplified model of three neighboringrectangular grains G1–G3 separated by tilt boundariesG1/G2 and G2/G3. If the misorientation angles of theseboundaries are small and equal to Ω and ω, respectively(Fig. 2a), then these boundaries are finite periodic walls

Ω

Ω

Ω

Ω

Ω

Ω

σ1 σ2

σ2σ1

σ3

σ3 σ4

σ4

ω

ω

I

II

III

IV+Ω

+Ω

+Ω

+Ω

+Ω

+Ω

–Ω–Ω

–Ω

–Ω

–Ω–Ω

+ω

+ω–ω

–ω

+Ω

+Ω

+Ω

+Ω

+Ω

+Ω

+ω

–Ω

–Ω

–Ω

–Ω–Ω

–Ω–ω

–ω

+ω

III

IV

II

I

+Ω'

+Ω'

+Ω'+Ω'

+Ω'

+Ω'

–Ω'

–Ω'–Ω'

–Ω'

–Ω'

–Ω'

–ω'

–ω'

+ω'

+ω'

IV

III

II

I

(a) (b)

(d)(c)

Fig. 1. Grain growth through collective GB migration in a model nanocrystalline material under uniaxial tension: (a) the initial stateof tilt boundaries with misorientation angles ω and Ω at a low tensile stress σ1; (b) at a stress σ1 > σ2, the boundaries begin tomigrate into grains I–IV and partial wedge disclinations with strengths ±ω and ±Ω appear in the remaining double junctions andform dipole and quadrupole structures; (c) at a higher stress σ3 > σ2, some boundaries annihilate partially (grain I) or completely(grain II), while others pass through each other and stop only near the next boundaries (grains III, IV); (d) smoothing of the bound-aries of the enlarged grains I–IV at a stress σ4 ≥ σ3.

1270


GUTKIN et al.

of perfect lattice dislocations (Fig. 2b), which aredescribed in terms of the continuum disclination modelas biaxial dipoles of partial wedge disclinations whosestrengths ±Ω and ±ω are equal in magnitude to the mis-orientation angles of the dislocation walls (Fig. 2c). Ifthe misorientation angles are larger (Fig. 2d), theseboundaries can be considered as disclination dipoles ofgreater strength (Fig. 2c). Under an applied shear stressτ, boundaries G1/G2 and G2/G3 are displaced from ini-tial positions A and D to new positions B and C. This isaccompanied by plastic deformation associated withlattice dislocation glide (in the case of small values ofΩ and ω) or with motion of GB dislocations (for largeΩ and ω) [46–48]. Accordingly, the orientation of thecrystal lattice changes in regions AB and CD swept bythe migrating boundaries (Fig. 2). Thus, the stress-stim-ulated GB migration leads to rotational plastic defor-mation related to a rotation of the crystal lattice.

We assumed that, in the initial position, the triplejunctions of the G1/G2 and G2/G3 boundaries with theneighboring boundaries of grains G1–G3 are geometri-cally compensated; i.e., they do not contain junctiondisclinations. When boundaries G1/G2 and G2/G3 aredisplaced to new positions B and C, triple junctions Aand D become double and uncompensated, withuncompensated abrupt changes in the misorientationangles (and jumps in the GB dislocation density,

Fig. 2b) [42, 43, 45]. New triple junctions B and C arealso uncompensated. In terms of the theory of defects,such uncompensated tilt boundary junctions are partialwedge disclinations whose strengths are equal to themisorientation angle jumps (junction disclinations)[50, 51]. Therefore, in the continuum disclinationmodel, we have a system of four biaxial dipoles of par-tial wedge disclinations having strengths ±Ω and ±ω(Fig. 2c). Dipoles B and C are mobile, and dipoles Aand D are immobile. In order to analyze the possibleevolution of this system, let us calculate its energy andfind the dependence of this energy on the main param-eters of the model.

3. CHANGE IN THE TOTAL ENERGYOF THE SYSTEM

The change ∆W in the total energy of the system dueto the formation of the disclination structure shown inFig. 2c (per unit disclination length) can be written as

(1)

where and are the elastic self-energies of dis-clination dipoles A and C, which are equal to the elasticself-energies of dipoles B and D, respectively, a fact

taken into account by the factors 2 in Eq. (1); isthe elastic interaction energy between disclinationdipoles α and β (α, β = A, B, C, D); and A is the workdone by the applied shear stress τ in creating this discli-nation structure (i.e., the work done to produce the plas-tic deformation that accompanies this collective GBmigration). The terms on the right-hand side of Eq. (1)are given by [52]

(2)

(3)

(4)

(5)

∆W 2WsA

2WsC

W intAB

W intCD

W intAC

W intBD

+ + + + +=

+ W intBC

W intAD

A,–+

WsA

WsC

W intαβ

WsA

DΩ2a

22 R

2a------ln 1+⎝ ⎠

⎛ ⎞ ,=

WsC

Dω2a

22 R

2a------ln 1+⎝ ⎠

⎛ ⎞ ,=

W intAB

2DΩ2a

2 R2

4a2

p12

+--------------------ln

⎝⎜⎛

–=

–p1

2

4a2

--------4a

2p1

2+

p12

--------------------ln 1+⎠⎟⎞

,

W intCD

2Dω2a

2 R2

4a2

p22

+--------------------ln

⎝⎜⎛

–=

–p1

2

4a2

--------4a

2p2

2+

p22

--------------------ln 1+⎠⎟⎞

,

Fig. 2. Collective migration of (a, b) low-angle and (c, d)high-angle tilt boundaries separating grains G1–G3 underapplied shear stress τ: (a, d) geometrical models and (b, c)dislocation and disclination models, respectively.

Ω ω

A D

G1 G2 G3

Ω ω

G3G2G1τ

A B C D

d

G1 G2 G3

A D

G1 G2 G3

A D

d

Ω ω

A D

G1 G2 G3

p1 p2

2aτ

A B C D

+ω–ω+Ω–Ω

A B C D

p1 p22a

Ω ω

A B C D

G1 G2 G3

(a)

(b)

(c)

(d)

τ

τ



(6)

(7)

(8)

(9)

(10)

where D = G/2π(1 – ν); G is the shear modulus; ν is thePoisson ratio; 2a is the arm of a disclination dipole (A,B, C, D); R is the screening parameter of the long-rangeelastic field of a dipole [50]; p1 and p2 are the distancestraveled by migrating boundaries G1/G2 and G2/G3,respectively; and d is the initial length of grain G2. Sub-stituting Eqs. (2)–(10) into Eq. (1), we find the changein the system energy ∆W to be

(11)

W intAC

2DωΩa2 R

2

4a2

d p2–( )2+

------------------------------------ln⎝⎜⎛

=

–d p2–( )2

4a2

---------------------4a

2d p2–( )2

+

d p2–( )2------------------------------------ln 1+

⎠⎟⎞

,

W intBD

2DωΩa2 R

2

4a2

d p1–( )2+

------------------------------------ln⎝⎜⎛

=

–d p1–( )2

4a2

---------------------4a

2d p1–( )2

+

d p1–( )2------------------------------------ln 1+

⎠⎟⎞

,

W intBC

–2DωΩa2 R

2

4a2

d p1– p2–( )2+

------------------------------------------------ln⎝⎜⎛

=

–d p1– p2–( )2

4a2

---------------------------------4a

2d p1– p2–( )2

+

d p1– p2–( )2------------------------------------------------ln 1+

⎠⎟⎞

,

W intAD

2DωΩa2 R

2

4a2

d2

+--------------------ln⎝

⎛–=

–d

2

4a2

-------- 4a2

d2

+

d2

--------------------ln 1+⎠⎟⎞

,

A 2aτ Ω p1 ω p2+( ),=

∆W 2Da2Ω2

x2

1+( ) x2

1+( )ln x2

x2 ---ln–

⎩⎨⎧

=

+ λ2y

21+( ) y

21+( )ln y

2y

2ln–[ ]

+ λ z2

1+( ) z2

1+( )ln z2

z2

ln–

+ z x– y–( )21+[ ] z x– y–( )2

1+[ ]ln

– z x– y–( )2z x– y–( )2

ln

– z x–( )21+[ ] z x–( )2

1+[ ]ln z x–( )2z x–( )2

ln+

– x y–( )21+[ ] z y–( )2

1+[ ]ln

+ z y–( )2z y–( )2 ln

2τDΩ--------- x λy+( )–

⎭⎬⎫

.

Here, x = p1/2a, y = p2/2a, z = d/2a, and λ = ω/Ω aredimensionless model parameters.

4. DESCRIPTION OF GRAIN GROWTH

Since the change in energy ∆W is a function of twovariables (x, y), it is convenient to study its behaviorusing isolines ∆W(x, y) = const. For definiteness, weconsider the case of initially uniaxial grain G2 (z = 1).

4.1. Collective Migration of Boundaries with Equal Misorientation Angles (λ = 1)

Figures 3 and 4 show isolines ∆W(x, y) = const con-structed for the applied stress τ varying from 0.3DΩ to2.5DΩ for the case of boundaries G1/G2 and G2/G3having equal misorientation angles (λ = 1). The valuesof ∆W are given in units of 2Da2Ω2. The insets toFigs. 3 and 4 show the equilibrium configurations ofthe system of three neighboring grains correspondingto the given values of the stress τ. The thick arrowsdrawn from the origin along the ∆W gradient indicatethe path of system evolution in the (x, y) plane. As canbe seen from Figs. 3 and 4, the function ∆W(x, y) andthe evolution of the system depend strongly on thevalue of the stress τ.

If the stress τ is very low (τ < 0.3DΩ), the function∆W(x, y) is positive and increases monotonically with xand y (the corresponding contour maps are not pre-sented here). Therefore, the formation of disclinationconfiguration ABCD is energetically unfavorable forthese values of τ. In other words, the stress τ is insuffi-cient for stimulating the migration of boundariesG1/G2 and G2/G3 and the growth of grains G1 and G3in the model under study.

At larger (but still relatively small) values of τ, from0.3DΩ to 1.3DΩ (Fig. 3), the function ∆W(x, y) is neg-ative at small values of x and y and is positive at largevalues of these variables. The minimum values of thisfunction lie on the line x = y and correspond to the equi-librium states of the system (these values are indicatedby the thick arrows in Fig. 3). Therefore, as the stress τincreases, boundaries G1/G2 and G2/G3 migrate grad-ually toward each other shifting from their initial posi-tions A and D to equilibrium positions B and C. In thecase of λ = 1 under discussion, boundaries G1/G2 andG2/G3 are identical and their displacements are equal.Therefore, the contour maps are symmetrical about thediagonal passing through points (0, 0) and (1, 1). Forexample, at τ/DΩ = 0.3, the equilibrium dimensionlessdisplacements of the boundaries are x = y ≈ 0.05(Fig. 3a). At larger stresses τ/DΩ = 0.5, 0.7, and 1.3, wehave x = y ≈ 0.13, 0.28, and 0.48, respectively (Figs. 3b,3c, 3d). It can be seen from the insets to Fig. 3 that theside grains grow toward each other through absorbingthe central grain G2 (bounded by lines B and C). Thecentral grain disappears when boundaries G1/G2 and

1272


GUTKIN et al.

G2/G3 meet (at x = y = 0.5), which occurs at τ = τm ≈1.6DΩ (not shown in Fig. 3).

Using Eq. (11), we can find the critical stress τc atwhich the migration of boundaries G1/G2 and G2/G3becomes possible and the characteristic stress τm atwhich these boundaries meet. In order to find τc, we setx = y, z = 1, λ = 1, and x 1 in Eq. (11). The result is

(12)

∆W x = y z = 1 λ = 1 x 1, , ,( )

≈ –8Da2Ω2

x x x xln τDΩ---------+ +⎝ ⎠

⎛ ⎞ .

An elementary GB displacement by one interatomicdistance b is possible if the change in energy is nega-tive, ∆W(x = b/2a) < 0. The critical stress τc correspondsto ∆W(x = b/2a) = 0 and can be found to be

(13)

If we drop the unity in parentheses in Eq. (13), weobtain the approximate expression derived in [45] forthe first critical stress τc1 at which a single boundarybegins to migrate. Thus, in a first approximation, the

τcDΩb

2a------------ 2a

b------ln 1–⎝ ⎠

⎛ ⎞ .≈

Fig. 3. Contour maps of the change in energy ∆W in the plane of the reduced displacements x = p1/2a and y = p2/2a of migratingboundaries with equal misorientation angles (Ω = ω) at various values of the applied stress τ/DΩ: (a) 0.3, (b) 0.5, (c) 0.7, and (d) 1.3.The values of ∆W are given in units of 2Da2Ω2. The insets schematically show the equilibrium configuration of the system of threeneighboring grains at the given stress τ. The thick arrows indicate an approximate path of the stable evolution of the system in the(x, y) plane.

1.0

0.8

0.6

0.4

0.2

0

1.0

0.8

0.6

0.4

0.2

00 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0

p1/2ap1/2a

p 2/2

ap 2

/2a

AB CD CDAB

BCA DAB CD

4

3

2

1

0.50.3

0.1–0.02

–0.0260

τ = 0.3DΩ, d = 2a, Ω = ω τ = 0.5DΩ, d = 2a, Ω = ω

τ = 1.3DΩ, d = 2a, Ω = ωτ = 0.7DΩ, d = 2a, Ω = ω 2.5

2

1.5

1

0.5

0

–0.15–0.23

–0.256

3

2

1

0.4

0.10

–0.07

–0.097

0.2

0

–0.3

–0.6

–0.9

–1.1

–1.1–0.9

–0.6–0.3 –1.227

–1.21

–1.227

(a) (b)

(d)(c)

1.0

0.8

0.6

0.4

0.2

0

p 2/2

a1.0

0.8

0.6

0.4

0.2

0

p 2/2

a

0 0.2 0.4 0.6 0.8 1.0p1/2a

0 0.2 0.4 0.6 0.8 1.0p1/2a



critical stresses τc and τc1 coincide. A more accuratecalculation based on the formulas from [45] gives

(14)

The difference ∆τc = τc1 – τc ≈ 3DΩb/4a character-izes the effect of the elastic interaction between migrat-ing boundaries on the critical stress for the onset of GBmigration. Obviously, this effect becomes significantwhen the size of an initially uniaxial grain is small. Forexample, for pure nanocrystalline aluminum with elas-tic moduli G = 27 GPa and ν = 0.31, an interatomic dis-tance b ≈ 0.25 nm, and a grain size d = 2a = 10 nm, the

τc1DΩb

2a------------ 2a

b------ln 1

2---+⎝ ⎠

⎛ ⎞ .≈

difference between the critical stresses for the onset ofmigration of tilt boundaries with misorientation anglesΩ = 0.085 ≈ 5° and 0.52 ≈ 30° is approximately 19.9and 121.4 MPa, respectively, with the critical stress τc

being 35.6 and 217.7 MPa, respectively. Thus, if theelastic interaction between migrating boundaries isignored for such small grain sizes, the critical stress forthe onset of GB migration is overestimated by 56%. Forthe tenfold greater size of uniaxial grains and the samevalues of the other parameters, the excess is 2.0 and12.2 MPa, respectively, and τc is 6.60 and 40.4 MPa,respectively; i.e., the error decreases to about 30%.

Fig. 4. Contour maps of the change in energy ∆W in the plane of the reduced displacements x = p1/2a and y = p2/2a of migratingtilt boundaries with equal misorientation angles (Ω = ω) at various values of the applied stress τ/DΩ: (a) 1.7, (b) 2.2, and (c) 2.5.The values of ∆W are given in units of 2Da2Ω2. The insets schematically show the equilibrium configuration of the system of threeneighboring grains at the given stress τ. The thick arrows indicate an approximate path of the unstable evolution of the system witha bifurcation point in the (x, y) plane.

5

4

3

2

1

00 1 2 3 4 5

p1/2a

p 2/2

a–24

–21

–18

–15

–12

–10–8

–6–4

–2.2–1.3

–1.8

–2

–1.8–1

AB D

A CD

τ = 1.7DΩ, d = 2a, Ω = ω

0 1 2 3 4 5p1/2a

0 1 2 3 4 5p1/2a

5

4

3

2

1

0

p 2/2

a

5

4

3

2

1

0

p 2/2

a

τ = 2.2DΩ, d = 2a, Ω = ω

τ = 2.5DΩ, d = 2a, Ω = ω

–33

–30

–27

–24

–21

–18–15–12–9–6–4–3–2–1

–3.085–3.015

–3.15

A D

–39

–36

–33

–30

–27

–24

–21–18–15–12–9–6–3–1

–4–3.8

A D

(a) (b)

(c)

1274


GUTKIN et al.

The characteristic stress τm under which the equilib-rium position of the two grain boundaries is the pointx = y = 1/2 at which they meet can be found from thecondition that the function ∆W(x, y) have a minimum atx = y, z = 1, λ = 1, and x = 1/2

(15)

It follows from Eq. (15) that τm = DΩln5 ≈ 1.6DΩ. It isinteresting that this value is twofold greater than thesecond critical stress τc2 ≈ 0.8DΩ under which themigration of a single boundary becomes unstable in themodel developed in [45]. This result is due to the factthat, in the model from [45], at the same point x = 1/2,the mobile disclination dipole representing a migratingboundary overcomes the maximum attractive forceexerted on it by the immobile disclination dipole repre-senting the double GB junctions into which the triplejunctions transformed when the migrating boundarybroke away from them. In the model studied in thispaper, at z = 1 and λ = 1, each of the mobile dipolesovercomes a twofold greater maximum force at thispoint, which is the sum of the attractive force exerted onit by its parent immobile dipole (i.e., by the doublejunctions from which it broke away) and the repulsiveforce exerted by the opposite immobile dipole (oppo-site double junctions). Since these two forces are equalat x = 1/2 by symmetry (z = 1, λ = 1) and the attractiveforce between the mobile dipoles vanishes at this point,the critical stress is twofold greater in this case.

For the parameter values used above for pure nanoc-rystalline Al, we have τm ≈ 0.85 GPa if Ω = 0.085 ≈ 5°and τm ≈ 5.2 GPa if Ω = 0.52 ≈ 30°. The first estimateof the stress agrees well with the values of local (up to1.4 GPa) [38] and even averaged (≈0.8 GPa) [40] shearstresses in computer simulations of nanoindentationtests at T = 0 [38] and uniaxial tension tests at T = 300 K[40] for pure nanocrystalline aluminum with a meangrain size of 5–7 nm. As mentioned above, the authorsof [38, 40] observed the migration of low-angle tiltboundaries and growth of grains through absorption ofother grains under such stresses. The second estimategives of course an extremely high stress. Therefore, wecan conclude that high-angle tilt boundaries areunlikely to meet under usual deformation conditions,although this was observed in situ directly under ananoindenter tip in experiments on films of ultrafine-grained and nanocrystalline pure aluminum at roomtemperature [13–15]. We can assume that, in thoseexperiments, the contact shear stresses reached 4–5 GPa, which is of the order of the theoretical shearstrength (≈G/2π). The collective migration and mer-gence of low-angle boundaries, on the contrary, requirefar lower shear stresses and can be observed underusual deformation conditions.

∂∆W x z = 1 λ = 1, ,( )∂x

---------------------------------------------------x 1/2=

= 8Da2Ω2

5lnτm

DΩ---------–⎝ ⎠

⎛ ⎞ 0.=

Returning to our model, we note that we have so farconsidered the case where migrating boundariesapproach each other in the stable regime until they meetand where the position of the boundaries is determinedby the magnitude of the applied stress (Fig. 3). Now, letus consider the case of higher stresses (Fig. 4). Asalready discussed in Section 2, when migrating bound-aries with equal misorientation angles (λ = 1) meet,they either annihilate completely or simply passthrough each other as dislocation walls gliding towardeach other on differing slip planes. In the former case,grains G1 and G3 merge into one grain. The latter casecan be interpreted as the nucleation and growth of anew grain with the same orientation of the crystal lat-tice as in grain G2. In this case, in diagrams showingthe most probable paths of the evolution of the systemin the (x, y) plane, the point of mergence of the bound-aries is a bifurcation point at which the path splits intotwo equiprobable branches (Fig. 4) corresponding tothe following two possible situations:

(i) The G1/G2 boundary (line B in the left-hand insetto Fig. 4a) passes through the G2/G3 boundary (line Cin the right-hand inset to Fig. 4a) and “captures” it,forcing the boundary to follow it. The distance betweenthem remains finite; it is the thickness of a new grain.Thus, for some time, the new grain migrates as a whole(see the segment of the thick curve with an arrow cov-ering the range 0.5 < x < 1.0 in Fig. 4a). The drivingboundary B and the driven boundary C migrate togetheruntil boundary B reaches the initial position of bound-ary C at point D and moves further into grain G3 andboundary C stops at a certain distance from its initialposition at point D (Fig. 4a, the section of the thickcurve with an arrow covering the range 1 ≤ x ≤ 5). Thepoint at which boundary C stops is its equilibrium posi-tion; the attractive force exerted on it by its parent dou-ble junctions at point D and the repulsive force exertedby the opposite double junctions at point A are balancedby the applied stress τ = 1.7DΩ. This equilibrium is sta-ble. Indeed, in order to force boundary C to migrateback to point A, one should considerably increase thestress τ. This occurs at τ = 2.2DΩ (Fig. 4b); boundaryC reaches point A and then moves further into grain G1.As a result, grains G1 and G3 merge into one grain andthe orientation of the crystal lattice in them changes andbecomes similar to that in the central grain G2. At a stillhigher stress τ = 2.5DΩ (Fig. 4c), boundaries B and Cmigrate toward each other and, passing through eachother, continue to move (without changing direction)with the same final result. The boundaries do not uniteand do not migrate in one direction under such a highapplied stress.

(ii) The other situation is radically different. Undera relatively low stress τ, boundary C captures boundaryB and they migrate together to point A (Fig. 4a, theupper branch of the thick curve with an arrow). Then,the system evolves in the same manner as in situation(i) with substitutions B C, A D, and G1 G3. Naturally, the final result is the same; namely,


Vol. 50

No. 7

2008


grains

G

1 and

G

3 coalesce and the orientation of thecrystal lattice in them changes and becomes similar tothe initial lattice orientation in grain

G

2.

Thus, in the case of the initially equiaxial grain

G

2(

z

= 1) with identical side boundaries (

λ

= 1), one mightexpect the following evolution of the system under theapplied shear stress

τ

. At relatively low stresses exceed-ing a certain critical value (τ > τc), boundaries B and Cmigrate synchronously toward each other and the thick-ness of grain G2 decreases with increasing τ. The equi-librium thickness of grain G2 is a single-valued func-tion of τ. At a certain, relatively high stress τ = τm ≈1.6DΩ , boundaries B and C meet and can annihilate; asa result, grain G2 disappears and the neighboring grainsmerge. Estimates show that, in the case of low-angleboundaries (dislocation walls) B and C, this can occurunder usual deformation conditions and, in the case ofhigh-angle boundaries, only under ultrahigh shearstresses of the order of the theoretical shear strength. Ifthe boundaries do not annihilate, then, at still higherstresses (τ > τm), they pass through each other. If thestress τ remains close to τm, then one boundary capturesthe other and they migrate together in one direction(i.e., the central grain migrates as a whole) in an unsta-ble mode until the leading boundary reaches the initialboundary line of grain G2. Passing this line, the leadingboundary breaks away from the driven boundary andcontinues to migrate in the unstable mode, while thedriven boundary remains in the equilibrium positionnear the initial boundary line of grain G2. At still highervalues of τ, the driven boundary becomes unstable andmigrates in the opposite direction. If the stress τ is farhigher than τm, then the boundaries pass through eachother and migrate in opposite directions in the unstablemode. In the last two cases, the result is the mergenceof grains G1 and G3 into one grain with lattice orienta-tion identical to the initial lattice orientation of grainG2. This result can also be considered as an unstablegrowth of grain G2 at the expense of grains G1 and G3.

4.2. Collective Migration of Boundaries Differing in Misorientation Angle (λ = 2)

Now, we consider the case where the side bound-aries G1/G2 (B) and G2/G3 (C) of the initially equiaxialcentral grain (z = 1) have differing misorientationangles, Ω = ω/2 or λ = 2. So far, we have not indicatedwhether the tilt boundaries are symmetric or nonsym-metric. Now, however, we note that at least one of thetwo boundaries should be nonsymmetric. For example,in the case of a low-angle tilt boundary, this means thatthe periodic wall of edge dislocations should be accom-panied by a periodic row of edge dislocations withBurgers vectors lying in the plane of the boundary. Dueto the symmetry of the chosen model about the horizon-tal axis (Fig. 2), the addition of this row of dislocationsto one of the vertical dislocation walls (Fig. 2b) or toone of the mobile disclination dipoles (Fig. 2c) does not

influence Eq. (11) for the change in the total energy ofthe system, ∆W. Therefore, in our model based on ananalysis of the behavior of the function ∆W(x, y), theasymmetry of one of the migrating boundaries does notinfluence the results obtained. If both boundaries arenonsymmetric, the results may change. However, ana-lyzing this situation is beyond the scope of this study.

Figure 5 shows isolines ∆W(x, y) = const con-structed for relatively low values of the applied stress τ.It can be seen that the isolines are no longer symmetricas in the case of λ = 1 (Figs. 3, 4). Boundaries B and Cmigrate in the stable mode and occupy the equilibriumpositions corresponding to a given value of τ (thickcurves with arrows in Fig. 5). Boundary B has a smallermisorientation angle and its mobility is significantlyhigher. As the stress τ increases, this boundary travels alarger fraction of the distance d between the initial posi-tions A and D of the boundaries, while boundary C isdisplaced only a small distance. For example, atτ/DΩ = 0.3, 0.7, and 1.0, the reduced displacements ofthe boundaries are x ≈ 0.05 and y ≈ 0.02 (Fig. 5a), x ≈0.28 and y ≈ 0.07 (Fig. 5b), and x ≈ 0.75 and y ≈ 0.10(Fig. 5c), respectively. As a result, the growth of grainG1 is dominant and grain G3 grows only slightly at thecost of a contraction of grain G2 with an increase in τ.

The critical stress τc for the onset of GB migrationin this case can be estimated from Eq. (14). Indeed, atsmall values of τ, boundary C is displaced only slightlyand, hence, a disclination quadrupole does not form.Therefore, the problem reduces to determining the crit-ical stress τc1 for the onset of migration of a singleboundary [45]. For example, in pure nanocrystallinealuminum with elastic moduli G = 27 GPa and ν = 0.31,an interatomic distance b ≈ 0.25 nm, and a grain sized = 2a = 10 nm, the critical stress for migration of tiltboundaries with misorientation angles Ω = 0.085 ≈ 5°and 0.52 ≈ 30° is approximately 55.5 and 339.1 MPa,respectively.

As the stress τ increases further, the GB migrationbecomes unstable. For example, at τ = 1.3DΩ , bound-ary B reaches boundary C and either annihilates (thiscase is not discussed here) or passes through it and cap-tures it. Then, the boundaries migrate together to theinitial position D of boundary C, where boundary Bbreaks away from boundary C and migrates in an unsta-ble mode deep into grain G3 (Fig. 6a), while boundaryC remains in its initial position D. As a result, grain G1occupies the entire initial area of grain G2 and grain G2grows at the expense of grain G3. At a still higher stress(τ = 2DΩ in Fig. 6b), the evolution of the system is thesame, but boundaries B and C meet and boundary Bcaptures boundary C near the center of the initial grainG2 and boundary C first reaches line D and thenmigrates back toward line A. At the next value τ =2.4DΩ (Fig. 6c), the point at which boundaries B andC meet becomes a bifurcation point; the boundarieseither migrate together to the equilibrium position (x ≈0.57, y ≈ 0.43) corresponding to a local minimum

1276


GUTKIN et al.

∆W/2Da2Ω2 ≈ –3.803 (shown in the inset to Fig. 6c onan enlarged scale) or boundary C captures boundary Band they migrate together toward line A. Near this line,boundary C breaks away from boundary B and migratesdeep into grain G1, while boundary B remains veryclose to line A. Thus, in the former case, grain G2 dis-appears and boundaries B and C annihilate partly, withthe formation of a stable (difference) boundary BCbetween enlarged grains G1 and G3. In the latter case,grain G3 grows and occupies almost the entire areabetween lines A and D initially occupied by grain G2,while grain G2 increases without limit and occupies the

area of grain G1. Finally, at a still higher stress τ =2.7DΩ (Fig. 6d), the point at which boundaries B and Cmeet is no longer a bifurcation point. Boundary B isimmediately captured by boundary C, and they migratetogether to line A as in the previous case and with thesame result.

Thus, the fundamental distinction between the caseof boundaries with different misorientation angles andthe case of boundaries with identical misorientationangles is that, in the former case, the symmetry of thesystem is destroyed in the course of the evolution of thesystem. When the misorientation angles are different

Fig. 5. Contour maps of the change in energy ∆W in the plane of the reduced displacements x = p1/2a and y = p2/2a of migratingboundaries with differing misorientation angles (Ω = ω/2) at various values of the applied stress τ/DΩ: (a) 0.3, (b) 0.7, and (c) 1.0.The values of ∆W are given in units of 2Da2Ω2. The insets schematically show the equilibrium configuration of the system of threeneighboring grains at the given stress τ. The thick arrows indicate an approximate path of the stable evolution of the system in the(x, y) plane.

τ = 0.3DΩ, d = 2a, Ω = ω/21.0

0.8

0.6

0.4

0.2

00 0.2 0.4 0.6 0.8 1.0

p 2/2

a

CD

1.0

0.8

0.6

0.4

0.2

00 0.2 0.4 0.6 0.8 1.0

0 0.2 0.4 0.6 0.8 1.0

1.0

0.8

0.6

0.4

0.2

0

τ = 0.7DΩ, d = 2a, Ω = ω/2

τ = DΩ, d = 2a, Ω = ω/2

AB A B CD

A BCD

10

8

6

4

2

1

0.50.2

0 –0.023

7

5

3

1.5

0.5

0

–0.205 –0.18 –0.1

6

5

4

3

2

1

0.4

0

–0.2

–0.4

–0.5

–0.6

–0.631

p1/2a

p 2/2

a

p 2/2

a

p1/2a p1/2a

(a) (b)

(c)



and the applied stress is not high, the boundary with asmaller misorientation angle (“weak” boundary) is farmore mobile than that with a larger misorientationangle (“strong” boundary). Indeed, the former bound-ary requires a lower external stress τ to overcome theattractive force exerted on it by its parent double junc-tions because they have a lower strength. On the otherhand, this stress τ is insufficient for noticeably displac-ing the strong boundary from its initial position. Thesmall displacement of this boundary causes strongscreening of the elastic interaction between the weakboundary and the strong quadrupole disclination con-

figuration (strong boundary plus strong double junc-tions). Therefore, when the stress τ is not high, themigration of the weak boundary is mainly controlled byτ. At somewhat higher values of τ, we have a seeminglyparadoxical situation wherein the weak boundary cap-tures the strong boundary. This apparent paradox isexplained by the fact that the applied stress τ is suffi-cient for the strong boundary to be displaced a signifi-cant distance from its initial position but is insufficientfor overcoming the progressively increasing forces ofattraction by the strong double junctions and repulsionby the weak double junctions. At the same time, the

Fig. 6. Contour maps of the change in energy ∆W in the plane of the reduced displacements x = p1/2a and y = p2/2a of migratingboundaries with differing misorientation angles (Ω = ω/2) at various values of the applied stress τ/DΩ: (a) 1.3, (b) 2.0, (c) 2.4, and(d) 2.7. The values of ∆W are given in units of 2Da2Ω2. The insets schematically show the equilibrium configuration of the systemof three neighboring grains at the given stress τ. The thick arrows indicate an approximate path of the unstable evolution of the sys-tem in the (x, y) plane. The additional inset to panel (c) shows a fragment of the contour map of ∆W plotted on an enlarged scalenear the bifurcation point. In this fragment, there is a local minimum ∆W/2Da2Ω2 = –3.803 at the point with x ≈ 0.57 and y ≈ 0.43.

5

4

3

2

1

0

5

4

3

2

1

0

5

4

3

2

1

0

5

4

3

2

1

0

p 2/2

ap 2

/2a

p 2/2

ap 2

/2a

0 1 2 3 4 5 0 1 2 3 4 5

0 1 2 3 4 50 1 2 3 4 5p1/2a p1/2a

p1/2ap1/2a

τ = 1.3DΩ, d = 2a, Ω = ω/2 τ = DΩ, d = 2a, Ω = ω/2

τ = 2.7DΩ, d = 2a, Ω = ω/2τ = 2.4DΩ, d = 2a, Ω = ω/2

A CD CDA

AB DAB D

A BC D

–14

–12

–10

–8

–8

–10

–6–2

2

4

4.9

0

–0.7–1.15

–30

–26

–22

–18

–14–10–6

–3

–2–1

0

0.47

–2.5–2

–1

–2.7

–46

–42

–38

–34

–30

–26

–22

–18–14–10–6–2–3

–3–3.8

–2–1 –4

0.6

0.4

0.4 0.6

–3.3 –3.7 –3.8

–3.7

–3.3–3.803

–54

–50

–46

–42

–38

–34

–30

–26

–22–18–14–10–4

–6

–5

–4.8

–4.6–3

–1

(‡) (b)

(c) (d)

1278


GUTKIN et al.

weak boundary is still controlled by the applied stress τthat is sufficient for the weak boundary to pass throughthe middle of the initial grain G2, which is the region ofthe maximum attraction by the parent weak doublejunctions and the maximum repulsion by the oppositestrong double junctions. Under these conditions, theelastic attraction between the strong and weak bound-aries is sufficient to overcome the stress τ acting on thestrong boundary. As a result, the strong boundary iscaptured by the weak one and they migrate together. Ata sufficiently high stress τ, the situation is reversed; thestrong boundary passes through the middle of the initialgrain G2 and captures the weak boundary and theyeither occupy the equilibrium position near the centerof the initial grain G2 or migrate together to the weakdouble junctions. In the latter case, the strong boundarypasses by these junctions and migrates further, whilethe weak boundary stops in front of them and occupiesa position close to the initial one.

5. CONCLUSIONS

The proposed continuum disclination modeldescribes the grain growth stimulated by the appliedshear stress as a result of the collective migration ofneighboring grain boundaries (GBs). This model makesit possible to estimate the critical stresses τc and τm atwhich the boundaries begin to migrate and meet,respectively. It has been shown that these criticalstresses are determined by the elastic properties of thematerial and are proportional to the misorientationangles of the boundaries. The stress τc varies approxi-mately in inverse proportion to the grain size, while thestress τm is independent of the grain size. Quantitativeestimates have shown that the stress τc can be easilyattained under usual deformation conditions for nanoc-rystalline materials, while the stress τm is attainableonly for low-angle tilt boundaries. The high-angleboundaries can meet only under ultrahigh stresses ofthe order of the theoretical shear strength. Using theproposed model, we have described the characteristicmodes of GB migration, in which GBs migrate towardeach other (at moderate stresses τ > τc), one boundarycaptures the other, and they migrate together in anunstable mode in one direction (at relatively highstresses τ > τm) or one boundary breaks away from theother and then they migrate in opposite directions (atstill higher stresses). In any mode of GB migration,grains grow at the expense of other grains. The jointmigration of two boundaries in one direction can beinterpreted as the migration of the grain as a whole. Thecollision and partial annihilation of migrating bound-aries should result in the accumulation of reaction prod-ucts in growing grains, namely, low-angle boundariesand single dislocations, which, in turn, can cause plas-ticization and hardening of the material by hamperingexcessive localization of plastic deformation and subse-quent fracture of the sample, as was observed in exper-

iments [18, 19, 21, 22, 27]. On the whole, the GBmigration and the corresponding grain growth are anefficient mechanism of rotational plastic deformationof nanocrystalline materials.

ACKNOWLEDGMENTS

This work was supported by the Federal Agency forScience and Innovation (contract no. 02.513.11.3190,program “Industry of Nanosystems and Materials”),the Foundation for Support of Leading ScientificSchools (project no. NSh-4518.2006.1), the RussianScience Support Foundation, the program “StructuralMechanics of Materials and Members of Construc-tions” of the Russian Academy of Sciences, CRDF(grant no. RUE2-2684-ST-05), and the St. PetersburgScientific Center of the Russian Academy of Sciences.

REFERENCES

1. A. K. Mukherjee, Mater. Sci. Eng., A 322, 1 (2002).2. K. S. Kumar, H. van Swygenhoven, and S. Suresh, Acta

Mater. 51, 5743 (2003).3. M. Yu. Gutkin and I. A. Ovid’ko, Physical Mechanics of

Deformed Nanostructures, Vol. 1: NanocrystallineMaterials (Yanus, St. Petersburg, 2003) [in Russian].

4. S. C. Tjong and H. Chen, Mater. Sci. Eng., R 45, 1(2004).

5. V. A. Pozdnyakov and A. M. Glezer, Fiz. Tverd. Tela(St. Petersburg) 47 (5), 793 (2005) [Phys. Solid State 47(5), 817 (2005)].

6. D. Wolf, V. Yamakov, S. R. Phillpot, A. K. Mukherjee,and H. Gleiter, Acta Mater. 53, 1 (2005).

7. M. A. Meyers, A. Mishra, and D. J. Benson, Prog. Mater.Sci. 51, 427 (2006).

8. M. Dao, L. Lu, R. J. Asaro, J. T. M. de Hosson, andE. Ma, Acta Mater. 55, 4041 (2007).

9. C. C. Koch, I. A. Ovid’ko, S. Seal, and S. Veprek, Struc-tural Nanocrystalline Materials: Fundamentals andApplications (Cambridge University Press, Cambridge,2007).

10. G. A. Malygin, Fiz. Tverd. Tela (St. Petersburg) 49 (6),961 (2007) [Phys. Solid State 49 (6), 1013 (2007)].

11. F. A. Mohamed, Metall. Mater. Trans. A 38, 340 (2007).12. A. V. Sergeeva, N. A. Mara, and A. K. Mukherjee,

J. Mater. Sci. 42, 1433 (2007).13. M. Jin, A. M. Minor, E. A. Stach, and J. W. Morris, Jr.,

Acta Mater. 52, 5381 (2004).14. W. A. Soer, J. Th. M. de Hosson, A. M. Minor, J. W. Mor-

ris, Jr., and E. A. Stach, Acta Mater. 52, 5783 (2004).15. M. Jin, A. M. Minor, and J. W. Morris, Jr., Thin Solid

Films 515, 3202 (2007).16. K. Zhang, J. R. Weertman, and J. A. Eastman, Appl.

Phys. Lett. 85, 5197 (2004).17. K. Zhang, J. R. Weertman, and J. A. Eastman, Appl.

Phys. Lett. 87, 061 921 (2005).18. P. L. Gai, K. Zhang, and J. Weertman, Scr. Mater. 56, 25

(2007).



19. X. Z. Liao, A. R. Kilmametov, R. Z. Valiev, H. Gao,X. Li, A. K. Mukherjee, J. F. Bingert, and Y. T. Zhu,Appl. Phys. Lett. 88, 021 909 (2006).

20. D. Pan, T. G. Nieh, and M. W. Chen, Appl. Phys. Lett.88, 161 922 (2006).

21. D. Pan, S. Kuwano, T. Fujita, and M. W. Chen, NanoLett. 7, 2108 (2007).

22. D. S. Gianola, S. van Petegem, M. Legros, S. Brandstet-ter, H. van Swygenhoven, and K. J. Hemker. Acta Mater.54, 2253 (2006).

23. D. S. Gianola, D. H. Warner, J. F. Molinari, andK. J. Hemker, Scr. Mater. 55, 649 (2006).

24. G. J. Fan, L. F. Fu, D. C. Qiao, H. Choo, P. K. Liaw, andN. D. Browning, Scr. Mater. 54, 2137 (2006).

25. G. J. Fan, L. F. Fu, H. Choo, P. K. Liaw, and N. D. Brow-ning, Acta Mater. 54, 4781 (2006).

26. G. J. Fan, Y. D. Wang, L. F. Fu, H. Choo, P. K. Liaw,Y. Ren, and N. D. Browning, Appl. Phys. Lett. 88,171914 (2006).

27. G. J. Fan, L. F. Fu, Y. D. Wang, Y. Ren, H. Choo, P. K. Liaw,G. Y. Wang, and N. D. Browning, Appl. Phys. Lett. 89,101 918 (2006).

28. B. Günther, A. Kumpmann, and H.-D. Kunze, Scr. Met-all. Mater. 27, 833 (1992).

29. A. Kumpmann, B. Günther, and H.-D. Kunze, Mater.Sci. Eng., A 168, 165 (1993).

30. V. Y. Gertsman and R. Birringer, Scr. Metall. Mater. 30,577 (1994).

31. J. A. Haber and W. E. Buhro, J. Am. Chem. Soc. 120,10847 (1998).

32. R. Z. Valiev, E. V. Kozlov, Yu. F. Ivanov, J. Lian,A. A. Nazarov, and B. Baudelet, Acta Metall. Mater. 42,2467 (1994).

33. X. Xu, T. Nishimura, N. Hirosaki, R.-J. Xie, Y. Yama-moto, and H. Tanaka, Acta Mater. 54, 255 (2006).

34. A. Hasnaoui, H. van Swygenhoven, and P. M. Derlet,Acta Mater. 50, 3927 (2002).

35. J. Schiotz, Mater. Sci. Eng., A 375–377, 975 (2004).

36. D. Farkas, A. Frøseth, and H. van Swygenhoven, Scr.Mater. 55, 695 (2006).

37. J. Monk and D. Farkas, Phys. Rev. B: Condens. Matter75, 045 414 (2007).

38. F. Sansoz and V. Dupont, Appl. Phys. Lett. 89, 111901(2006).

39. F. Sansoz and J. F. Molinari, Thin Solid Films 515, 3158(2007).

40. T. Shimokawa, A. Nakatani, and H. Kitagawa, Phys.Rev. B: Condens. Matter 71, 224 110 (2005).

41. A. J. Haslam, D. Moldovan, V. Yamakov, D. Wolf,S. R. Phillpot, and H. Gleiter, Acta Mater. 51, 2097(2003).

42. S. V. Bobylev, M. Yu. Gutkin, and I. A. Ovid’ko, ActaMater. 52, 3793 (2004).

43. S. V. Bobylev, M. Yu. Gutkin, and I. A. Ovid’ko, Fiz.Tverd. Tela (St. Petersburg) 46 (11), 1986 (2004) [Phys.Solid State 46 (11), 2053 (2004)].

44. J. C. M. Li, Phys. Rev. Lett. 96, 215 506 (2006).45. M. Yu. Gutkin and I. A. Ovid’ko, Appl. Phys. Lett. 87,

251 916 (2005).46. J. W. Cahn, Y. Mishin, and A. Suzuki, Acta Mater. 54,

4953 (2006).47. L. Zhou, N. Zhou, and G. Song, Philos. Mag. 86, 5885

(2006).48. H. Zhang, D. J. Srolovitz, J. F. Douglas, and J. A. War-

ren, Acta Mater. 53, 4527 (2007).49. M. Yu. Gutkin, I. A. Ovid’ko, and N. V. Skiba, Fiz.

Tverd. Tela (St. Petersburg) 46 (11), 1975 (2004) [Phys.Solid State 46 (11), 2042 (2004)].

50. V. V. Rybin, Severe Plastic Deformations and Fractureof Metals (Metallurgiya, Moscow, 1986) [in Russian].

51. V. I. Vladimirov and A. E. Romanov, Disclinations inCrystals (Nauka, Leningrad, 1986) [in Russian].

52. M. Yu. Gutkin, K. N. Mikaelyan, A. E. Romanov, andP. Klimanek, Phys. Status Solidi A 193, 35 (2002).

Translated by Yu. Epifanov

grain growth and collective migration of grain boundaries during plastic deformation of...

Documents