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Disclinations in bulk nanostructured materials: their origin, relaxation and role in material

properties

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2013 Adv. Nat. Sci: Nanosci. Nanotechnol. 4 033002

(http://iopscience.iop.org/2043-6262/4/3/033002)

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IOP PUBLISHING ADVANCES IN NATURAL SCIENCES: NANOSCIENCE AND NANOTECHNOLOGY

Adv. Nat. Sci.: Nanosci. Nanotechnol. 4 (2013) 033002 (11pp) doi:10.1088/2043-6262/4/3/033002

REVIEW

Disclinations in bulk nanostructuredmaterials: their origin, relaxation and rolein material properties

Ayrat A Nazarov

Institute for Metals Superplasticity Problems, Russian Academy of Sciences, 39 Khalturin Street,Ufa, Russia

E-mail: [email protected]

Received 6 March 2013Accepted for publication 9 May 2013Published 4 June 2013Online at stacks.iop.org/ANSN/4/033002

AbstractThe role of disclinations in the processing, microstructure and properties of bulk nanostructured materials is reviewed. Models of grain subdivision during severe plasticdeformation (SPD) based on the disclination concept, a structural model of the bulk nanostructured materials processed by SPD are presented. The critical strength of triple junction disclinations is estimated. Kinetics of relaxation of triple junction disclinations andtheir role in the grain boundary diffusion are studied.

Keywords: disclination, nanostructured material, severe plastic deformation, grainsubdivision, triple junction, grain boundary diffusion

Classication number: 4.03

1. Introduction

Disclinations, along with dislocations, are well knownlinear defects of crystalline lattice [14]. The disclinationconcept is widely used in materials science to describethe structure and properties of grain boundaries, their triple junctions, the plastic deformation and fracture mechanisms,etc [38]. Disclinations play a particularly important rolein the structure and properties of bulk nanostructured (NS)materials processed by severe plastic deformation (SPD)methods [ 912]. The present paper is a review on the origin of disclinations in bulk NS materials, their role in the structuralmodeling of these materials and in the grain boundarydiffusion.

Invited talk at the 6th International Workshop on Advanced MaterialsScience and Nanotechnology IWAMSN2012, 30 October2 November, 2012,Ha Long, Vietnam.

C ontent from this work may be used under the terms of the Creative Commons Attribution 3.0 licence . Any further

distribution of this work must maintain attribution to the author(s) and thetitle of the work, journal citation and DOI.

2. Denitions and basic relationships

Disclinations, along with dislocations, were rst introduced inthe mechanics of deformed solids by Volterra as specic linearsources of internal stresses [ 13]. These defects are created

by making a cut in a cylindric body with a small inner holealong the axis and rotating the cut surfaces to a vector with respect to each other simultaneously lling the arisinggaps and eliminating excess material from overlaps. When thevector called Frank vector is parallel to the cylinder axis,the disclination is called a wedge one and if Frank vector isnormal to cylinder axis, it is called a twist one. The magnitudeof the Frank vector of a disclination, , is called the strengthof the disclination.

Wedge disclinations, which are most easily visualized,are obtained by inserting or removing a wedge of materialwith an angle ; respectively to these constructions, negativeand positive wedge disclinations are considered (gures 1(a)and (b)). Below, only wedge disclinations will be considered.

In crystalline lattice, perfect and partial disclinationsare distinguished; the Frank vectors of these disclinationsare, respectively, equivalent to a symmetry rotation and

2043-6262/13/033002+11$33.00 1 2013 Vietnam Academy of Science & Technology
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Adv. Nat. Sci.: Nanosci. Nanotechnol. 4 (2013) 033002 A A Nazarov

(a) (b)

Figure 1. Negative (a) and positive (b) wedge disclinations in acylindric body; the cylinders are innite along the direction normalto the paper plane. After inserting the wedge or closing the gap thematerial is allowed to relax.

(a) (b)

Figure 2. Equivalence between a disclination and terminated

dislocation wall (a) and disclination dipole and nite wall (b); whitetriangles designate negative wedge disclinations and the black onesthe positive disclinations.

not compatible with any symmetry operation. A partialdisclination necessarily bounds a grain boundary with amisorientation equal to the Frank vector of the disclination.

There is a close relationship between disclinations anddislocations. A terminated tilt boundary consisting of edgedislocations is modeled by a wedge disclination located atthe termination line (gure 2(a)). Correspondingly, a nitetilt boundary is modeled by a dipole of wedge disclinations

(gure 2(b)).The elastic energy of a wedge disclination located along

the axis of a cylinder with radius R per unit length is equalto [24]

E d =G 2 R2

16( 1 ), (1)

where G is shear modulus and the Poisson ratio.As one can see from this equation, the elastic energy

of disclinations diverges as the cylinder radius increases.Due to this, the existence of single disclinations is limitedto very small crystals (thin lms, nanoparticles, nanowires,

etc). For example, they are quite common in pentagonalnanoparticles [ 1416].

Screening of the stress elds greatly reduces the elasticenergy of disclinations. The energy of the disclination dipole

(a) (b)

Figure 3. Accumulation of dislocations in grain boundaries due toslip in grains: (a) the general case and (b) a particular case when onegrain is not deformed and one GB accumulates no dislocations.Dislocations forming the array modeled by junction disclination arerepresented in light gray color.

depicted in gure 2 is equal to [24]

E dd =G 2 L2

8( 1 )2 ln R

L+ 3 , (2)

that is, it diverges as the energy of a dislocation with theBurgers vector magnitude b = L .Similarly, the energy of a disclination quadrupoleconsisting of two dipoles of opposite signs having the length L and separated by the same distance L will be similar to theenergy of a dislocation dipole with the separation L :

E dq =G 2 L2

2( 1 )ln 2. (3)

Comparing equations ( 1) and (3) one can see that the energy of a disclination in a cylinder with radius R and in a quadrupolewith size L = R are approximately equal. As will be seenbelow, disclinations accumulated in the grain junctions of polycrystals during plastic deformation are coupled intoquadrupole congurations with the sizes approximately equalto the grain size. That is why the effects related to disclinationsin a polycrystal with grain size d and in a cylinder with thesame radius can be considered similar. This similarity arisingfrom the similarity of screening distances is important for thefurther analysis.

3. Origin of disclinations in deformed polycrystals

Accumulation of dislocations in grain boundaries (GBs) andthe formation of disclinations during plastic deformationof polycrystals have been explored by Rybin andco-authors [ 5,6,17,18]. Following their ideas, consider aGB that divides two grains s and s and at the two endsis bounded by triple junctions A and B (gure 3(a)). LetN12 be the unit vector of the GB plane normal. Under theapplied stress ms slip systems ( n p, b p) s ( p =1, 2, . . . , m s )are activated in the sth grain (s =1, 2) that leads to plasticstrain of the grain, which is considered to be uniform. Tomaintain the integrity of material during deformation, thetotal strains of crystallites, which consist of elastic andplastic parts, must be equal to the total strain of a sample.However, different orientations of slip systems (n p , b p) sin the grains with respect to the applied stress results in

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(a) (b)

Figure 4. A rectangular 2D probe grain surrounded by neighbor grains in a deforming polycrystal (a) and mesoscopic defects formed in theGBs and junctions of the probe grain (b).

a difference of the plastic strain tensors 1 and 2 and

plastic distortion tensors 1 and

2. Due to this difference,

deformation leads to an accumulation of jumps of the plasticstrain and distortion tensors across the GBs, [ ]12 and [ ]12 ,respectively. This incompatibility induces internal stresses,which, in combination with the applied stress, equalize thetotal strain rate in all grains and thus lead to an approximatefulllment of Taylors principle. The plastic incompatibilitiesaccumulated on GBs can be described in terms of dislocationswhose dimensionless surface density is given by the followingtensor:

B12 = N12 [ ]12 . (4)This tensor is decomposed into two parts. The former part

corresponds to the skew symmetric part of distortion jump, i.e.to the rotation tensor . The second corresponds to the straintensor that is equal to the symmetric part of the distortion jump:

B

12 = N12 [ ]12 , (5) B

12 = N12 [ ]12 . (6)The dislocations, which are described by the formerexpression, form compensated arrays causing an additionalmisorientation of adjoining grains; these arrays are stress-free.The dislocations described by the second equation form bothcompensated arrays of the rst type and non-compensated

dislocation arrays, which induce long-range internal stresses.The latter may be composed, for example, of dislocations witha Burgers vector tangential to the GB plane.

The additional misorientation of the grains 1 and 2 due tothe dislocation accumulation in their GB can then be writtenas follows:

N = N + N = [ ]12 N12 [ ]12N12 , (7)where [ ] is the vector of relative rotation of the crystallites.

If one now considers a triple junction, the misorientationmismatch, which is accumulated around this junction, iscalculated by summing up equation (7) written for each of the three GBs. The rst term gives nil sum, thus the total

misorientation mismatch is equal to

= 3

i=1N i [ ]i N i . (8)

Here the index i refers to ith GB. This equation shows

that the incompatibilities of the plastic strains of differentlyoriented grains lead to the accumulation of disclinations inthe junctions of grains.

The above analysis is clearly illustrated by a particularexample of the junction of two grains, when one grain is notdeformed ( 2 =0) and one GB is parallel to the slip planeof the rst grain and does not accumulate dislocations duringdeformation (gure 3(b)). In this case only one facet of the GBcontains an array of trapped dislocations that can be dividedinto the arrays of dislocations with tangential and normalBurgers vectors; the latter will be equivalent to a positivewedge disclination lying on the junction of the facets.

Rybin and co-workers [5,6,17,18] have shown that theabove described process is a quite common feature of largeplastic deformation and is an underlying process for thesubdivision of the grains into fragments. They have shownthat disclinations accumulated in junctions of grains induceinternal stresses that activate an accommodation slip nearthe junctions to generate low-angle dislocation boundariessubdividing the grains into fragments. This process isdescribed as the movement of disclinations across the grains.The process of fragmentation plays a key role in theprocessing of bulk NS materials by means of SPD methods.

4. Disclination modeling of grain subdivision duringplastic deformation

In [19,20] two-dimensional (2D) disclination based modelsfor the grain subdivision during SPD were proposed. Theauthors considered energetic criteria for the formation of newgrain nuclea due to the accumulation of junction disclinations.

Consider a two-dimensional rectangular probe grainsurrounded by other grains as shown in gure 4(a). Due todifferent orientations of the grain and its neighbor, arraysof dislocations are accumulated on the GBs. As shownabove in section 3, these arrays can be divided into theplanar mesodefects consisting of tangential dislocations and junction disclinations (gure 4(b)). Thedisclinations will havedifferent strengths denoted , q , g , and k , where 0 d 2 ln 2/ 2( 1 ) per one grain of the 2Dpolycrystal. The excess specic elastic energy of GBs relatedto disclinations then will be equal

ex = E 2d =

G 2 d ln 216( 1 ) (14)

Assuming 2 1/ 2 2, i.e. half of the critical strengthfor d =200nm calculated above ( c 4), equations (13)6

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(a) (b)

Figure 11. (a) A model of relaxation of a junction disclination dipole; the arrows indicate the direction of dislocation movement;(b) dependence of the relaxation rate of disclination dipole on the number of dislocations left; initial number of dislocations is 5 for curve 1,10 (curve 2), 20 (curve 3), and 50 (curve 4).

and (14) yield the following estimates: i 0.3% and ex 0.4 J m

2

. These estimates are comparable to thecharacteristics of NS metals processed by SPD [ 32,33].

7. Grain boundary diffusion controlled relaxation of triple junction disclinations

As extrinsic defects induced during plastic deformation,the triple junction disclinations relax and disappear at hightemperatures. A model for the relaxation of a junctiondisclination dipole was analyzed in [ 34] and that for therelaxation of a disclination quadrupole in [35]. The resultsof these analyses are similar, and below we will consider the

model for disclination dipole in more detail.Consider a dipole of junction disclinations which consistsof a nite wall of edge grain boundary dislocations (GBDs)on a probe GB (central vertical boundary in gure 11). Let theinitial strength of disclination dipole be 0 =b/ h 0, where bis the Burgers vector magnitude of GBDs and h 0 their initialspacing. The arm of the disclination dipole is equal to theGB length: L =2a . When the temperature is high enoughand diffusion is allowed, the GBDs, under their mutuallyrepulsive interaction forces, will climb to the junctions. Theycan leave theGB through the junction, if a dislocation splittingat the junction occurs. If the dihedral angle between theadjoining GBs is obtuse, that is the case in real polycrystalsdue to the equilibrium of interfacial tensions, energeticallyfavorable splitting of a GBD 1 will result in two dislocations2 and 3, which can glide in corresponding GBs. The lattercan easily quit the junction by glide and annihilate withopposite dislocations gliding from the opposite junctions. If the latter are absent, the dislocations can glide towards theopposite junctions and GBs (e.g., reaction 3 + 4 5). Inthis case a gliding GBD 5 will again be formed and quitthe corresponding junction without any diffusion. Therefore,dislocation 1 can be excluded from a consideration of theprocess as soon as it reaches the junction. The same processessymmetrically occur near the bottom junction of the probeGB. As a result of the disappearance of one GBD near eachof the two junctions the strength of the dipole diminishes toa value b/ a . Then, the next head dislocations approach the junction by climb and the process is repeated.

From the above consideration it is clear that the whole

process of relaxation is controlled by the climb of GBDs to the junctions. Since this occurs in a high-angle GB, the diffusioncoefcient of which, Db , is much higher than the latticediffusion coefcient; this process is completely controlled byGB diffusion.

The process can be analyzed in two approaches:the continuum approach (dislocations have innitesimalBurgers vectors) and the discrete dislocation approach (thedislocations have a nite Burgers vector). The results obtainedin the two approaches are similar [ 34], which is why thephysically most clear discrete dislocation model is consideredbelow.

Let the initial GBD wall contain N

=2n + 1 dislocations

with Burgers vector magnitude b with positions yi =0, h 0, 2h0, . . . , nh 0. For the climb rate of ithdislocation one can write [ 36] vi =( J l J u)V a/ bn , where J lis the vacancy ux from the lower, (i 1)st dislocation to thei th one and J u the ux from the i th dislocation to the upper,( i + 1)st one. According to Ficks rst law, the vacancy uxdensity is equal to j = Db / V akT . In the present case,the gradient of chemical potential for vacancies is relatedto the gradient of normal stresses by an equation = xx V a [36], and the vacancy ux will be equal to J = j = Db ( xx / x ) / kT , where is the GB diffusion width.Substituting this into the equation for dislocation velocity, oneobtains

vi = Db V abn kT

i +1 xx i x x yi+1 yi

i xx i1 xx yi yi1

(15)

for each i =1, 2, . . . , n 1, where i xx is a normal stressnear the core of the i th dislocation. For i =n equation (15)contains only a ux from the (n 1)st dislocation, that is thesecond term in the brackets. Due to symmetry, the centraldislocation ( i =0) is immobile. The stresses in equation (15)are calculated by simply summing the stresses of individualdislocations except for the self-stresses:

i xx = Gb n

2( 1 )n

j=i

1 yi y j

+ 1 yi

+n

j=1

1 yi + y j

.

(16)

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To solve equations (15) numerically, dimensionlesscoordinates and time have been introduced as follows:

yi = yi / h 0 and =[4 DbGV a /( 1 ) kT L3] t .When a dislocation approaches a junction, it is excludedfrom the system and n is decreased by 1. The relativerelaxation rate for the misorientation angle of the wall,

d / dt , is calculated from the time interval t in whichtwo neighbor dislocations successively approach a junction:d / dt =1/( n + 0.5) t .Figure 11(b) plots the dependence of the relativerelaxation rate d / dt on the number of dislocationsremaining in the wall for four initial values of this number2n0 + 1. One can see from the gure that for each n0there is a transition stage of relaxation of different durationfollowed by a stationary stage. For sufciently large n0 thisstage is characterized by a constant value of the relaxationrate, [( 1 ) kT L3/ 4 DbGV a](d / dt ) 8. Therefore,for the misorientation angle of a discrete GBD wall oneobtains the following equation:

ddt =

32 DbGV a( 1 ) kT L3 =

125 DbGV akT d 3

= t d .(17)

In the continuum approach a numeric coefcient 163 wasobtained instead of 125 [34]. Similar calculations for thedisclination quadrupole give the numeric coefcient 100 [ 35].Thus, very similar results are obtained in different approaches.For further estimates, the value 100 will be used.

Since, according to equations ( 1)(3), the energy of disclinations and its multipoles is proportional to the square of

the strength of disclinations, E el

2, one has d E el / E el dt =2d / dt , and the characteristic time for the relaxation

of elastic energy will be half of that for the strength of defects. Therefore, we will have the following equation forthe relaxation of elastic energy of a junction disclinationquadrupole in deformed polycrystals:

E el = E 0el exp t

t enq (18)

with the relaxation time

t enq

kT d 3

200 Db GV a . (19)

This estimate will be used in the next section to analyze therelationship between the relaxation of junction disclinationsand the measured GB diffusion coefcient in nanostructuredmaterials.

8. Triple junction disclinations and grain boundarydiffusion in nanostructured materials

8.1. Enhancement of grain boundary diffusion due to junction disclinations

Since triple junction disclinations induce high internalstresses, via these stresses they can have a signicant effecton the grain boundary diffusion coefcient in nanostructuredmaterials. This effect was studied in [ 37].

Consider a single semi-innite grain boundary whoseplane is normal to the surface of a polycrystal. Let the y-axisof the coordinate system lie along a normal to the surfacewith the positive direction inside material. On the surface,there is a layer of radio-tracer atoms. The GB is subjectto non-uniform internal stresses whose derivatives along the

y-axis are assumed to be constant. In the presence of a gradientof the hydrostatic stress, / y =0 , where =( x x + yy + zz)/ 3, the ow of tracer atoms will be given by a relation j = Db[c/ y + (V a/ kT ) c (/ y)], where Db is the GBdiffusion coefcient, c the concentration of tracer atoms andV a the atomic volume. The diffusion process will then bedescribed by an equation

c( y, t ) y = Db

2c( y, t ) y2

DbV akT

y

c( y, t ) y

. (20)

For initial conditions c( y, 0) =0 ( y > 0) and c( y, 0) =c1( y < 0) the solution to equation ( 20) is [38]

c( y, t ) =c(0, 0) erfc y v t

Lef , (21)

where c(0, 0) =c1 / 2, Lef =2 Dbt and v =( DbV a/ kT )(/ y) is the average drift speed of atomsin the constant stress gradient.

As follows from equation (21), a positive gradient of thestress shifts the concentration front to the right to a value of vt thus leading to an increase of the penetration depth. Oppositestress gradients shift this front to the left only for positivevalues of y , since the stresses act only in the material and thelayer of diffusate is considered to be stress free. That is, the

effect is asymmetric with respect to the sign of / y.Consider now a model polycrystal in which the GBsare considered to form chains approximately normal to thesurface. The stress gradient in each chain is assumed to beconstant, but there is a random distribution of the stresses inthe ensemble of chains such that the parameter v can assumeany value between maximum opposite and maximum positiveones:

DbV a

kT y max = vm < v < v m =

DbV akT

y max

.

(22)Assuming an even distribution of the drift speeds in this

interval, one can calculate the average concentration of traceratoms in any section of the polycrystal parallel to the surface:

c( y, t )c(0, t ) =

vm t

vm t erfc

x u Lef

d u vm t

vm t er f c

u Lef

d u .

(23)

The graph of this function for vm t =10 Lef is presented ingure 12(a) (curve 1). For a comparison, the concentrationproles for a non-stressed grain boundary (curve 2) and for theGB chain experiencing the maximum positive stress gradient(v t

=+10 Lef ) (curve 3) are given too.

The distribution of stress gradients results in a muchmore at concentration prole as compared to the sharpproles characteristic for the diffusion along GBs subject to aconstant or zero stress gradient. Furthermore, the stresses can

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(a) (b)

Figure 12. (a) Average concentration of tracer atoms according to equation ( 23) at vm t =10 Lef (curve 1), to compare with theconcentration proles along non-stressed grain boundaries (curve 2) and the fastest grain boundary chain (curve 3); (b) average penetrationproles plotted in terms of the logarithm of concentration versus the square of distance; vm t =0 (curve 1), vm t =4 Lef (2), and vm t =10 Lef (3). The dashed curves correspond to a stress free GB with an effective diffusion coefcient equal to 10 D

b (4) and 100 D

b (5).

signicantly increase the penetration depth. This increase isdetermined by the value of the ratio vm t / Lef . The higher thisratio is, the more is the effect of stresses on the penetrationdepth.

To extract the diffusion coefcient from experimentalconcentration proles, the latter are usually plotted in termsof logarithm of concentration versus the square of the distancefrom the surface. For several values of the ratio vm t / Lef suchgraphs are presented in gure 12(b). If one does not takeinto account the internal stresses, the diffusion coefcient willbe determined by tting equation ( 21) with vt

=0 to the

concentration proles. This results in an effective diffusioncoefcient D ef b that is different from the intrinsic GB diffusioncoefcient Db. The latter should be determined by tting theconcentration proles to equation ( 23). Figure 12(b) illustratesthe relation between the two coefcients. The concentrationprole for a stress free GB with D ef b =10 Db is similar to thatfor the stressed polycrystal in which vm t / Lef =4. If the ratiovmt / Lef is equal to 10, the corresponding effective diffusioncoefcient amounts to 100 times the GB diffusion coefcient.Thus, orders of magnitude in excess of the GB diffusioncoefcient as compared to that in equilibrium boundaries canbe detected when processing the experimental data by the

solution of an ordinary diffusion equation.For the strength of junction disclinations =3 attemperatures about T =400K the parameter vm t / Lef canhave a value up to 6 [37]. Therefore, at least one order of

magnitude increase of the apparent GB diffusion coefcientcan be caused by the stress gradients due to junctiondisclinations. As has been suggested in [ 37], a randomdistribution of the strength of disclinations can result in aneven larger enhancement of the GB diffusion coefcient.

8.2. Recovery of the grain boundary diffusion coefcient

In order to measure the GB diffusion coefcientexperimentally, diffusion annealing is carried out ata sufciently high temperature that would provide anappreciable penetration of tracer atoms into the material understudy. Under these conditions, the junction disclinations relax

as discussed in section 7. This relaxation is accompanied by agradual recovery of the grain boundary diffusion coefcient.Thus, an actual diffusion experiment with nanostructuredmaterials is necessarily accomplished with a time-dependentdiffusion coefcient. This can signicantly affect theexperimentally measured diffusion coefcient. The relationbetween the recovery of nonequilibrium GB structure causedby disclinations and GB diffusion coefcient was analyzedin [39].

Phenomenologically, the relationship between the GBdiffusion coefcient and excess energy of GBs was given byBorisovs equation [40]

Dneb = Deqb exp

kT

, (24)

where Dneb and Deqb are, respectively, diffusion coefcients

along nonequilibrium and equilibrium GBs, and is theexcess GB energy per atom.

Considering the excess energy of GBs related todisclination quadrupoles, equations (18) and (19) can bere-written for the recovery of

= 0 exp t

t 0 , t 0 d 3kT

200 Deqb GV a . (25)

If the gradient of diffusing matter has the direction along the x -axis lying on the GB plane, the GB diffusion equation thencan be re-written as follows:

c t = D

eqb exp

E 0kT

e t t 0

2c x 2

. (26)

This equation can be solved by the following substitution:

= t

0exp

E 0kT

e t t 0 dt =t 0

t / t 0

0exp

E 0kT

e z d z .(27)

As a result, we obtain the standard diffusion equation

c = D

eqb

2c x 2

. (28)

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(a) (b)

Figure 13. (a) Time dependence of the ratio / t , which indicates the ratio of effective and intrinsic diffusion coefcients; (b) inversetemperature dependence of diffusion coefcient: (1) equilibrium GBs (high temperatures); (2) nonequilibrium GBs (low temperatures);(3) relaxing GBs (intermediate temperatures).

For the diffusion regime C mostly used during themeasurements of diffusion coefcient in nanostructuredmaterials [ 41], the solution to equation ( 28) is

c( x , ) =c0 erfc x

4 Deqb . (29)

Taking account of equation (27), equation ( 29) can betransformed into a function of the coordinate x and time t .

Experimentally, the diffusion coefcient is normallydetermined from constructed concentration proles, i.e. fromthe dependences of the tracer concentration on the diffusiondepth at the instant of time corresponding to the time of diffusion annealing. As follows from equation ( 29), uponchanging over from the variable to the variable t , thecharacter of dependence of the concentration on the depth x remains unchanged. Consequently, the time dependence of the diffusion coefcient does not affect the functional form of the concentration proles. However, from the analysis of theexperimental proles constructed in the ct coordinates, onewill determine the effective (apparent) diffusion coefcient,which is related to the intrinsic diffusion coefcient by theequation D eqb = Def t , that is

Def / Deq

b = ( t )/ t . (30)The time dependence of the ratio ( t )/ t for the case,

when the initial enhancement of the diffusion coefcientis equal to Dneb / D

eqb =102 , is presented in gure 13(a).From this gure one can see that for annealing times much

exceeding the relaxation time t 0 , t and Def Deqb , i.e.

diffusion experiments yield the intrinsic diffusion coefcient.However, if t t 0 and exp( 0/ kT ) t t , Def D

eqb ,

i.e. for too short diffusion times one will measure aneffective diffusion coefcient much exceeding the intrinsicone. This will happen for too short a time of annealingat low temperatures. In the intermediate cases the diffusiontime is of the order of relaxation time, and with increasingtemperature and/or annealing time the ratio Dneb / D

eqb will

decrease. As one can see from gure 13(a), for Dneb / Deqb =100 even after annealing in time interval t 3t 0 the effective

diffusion coefcient is still one order of magnitudehigher thanthe true coefcient.The considered behavior of GB diffusion coefcient is

conrmed by the experimental observations. For example,in GBs of nanocrystalline iron the measured self-diffusioncoefcient decreases one order of magnitude when theannealing time at the same temperature 473 K is increasedfrom 1.5h to 3 days [42].

Recovery of the GB diffusion coefcient duringannealing can result in a strong decrease of the apparentactivation energy for GB diffusion. If the activation energyfor diffusion along equilibrium GBs is equal to E b, fornonequilibrium GBs it will be E b

. Figure 13(b)

illustrates schematically the dependences lg Db 1/ T withthe activation energies E b (line 1) and E b (line 2).When increasing the temperature of diffusion experiments,the dependence of apparent diffusion coefcient on inversetemperature will rst follow line 2, then, after full relaxationof GBs, line 1. At intermediate temperatures the relaxationdecreases the GB diffusion coefcient as compared to line2, so the latter will follow line 3. The slope of this lineis less than that of line 2, so the experimentally measuredactivation energy is even less than E b . This explainsthe experimentally established fact that in NS materials theactivation energy of GB diffusion is often twice less than in

ordinary polycrystals [ 43]. For some specic combinations of the relaxation and diffusion times and temperature, even theobservation of negative activation energy may be possible.

9. Conclusions

The present short review shows that disclinations arenecessary elements of the structure in NS materials processedby SPD methods. They can accumulate at triple junctionsof grains and play a key role in the subdivision of grainsduring plastic deformation. After SPD, they are inheritedby the as-processed material and play important role in itsstructure and properties. The disclination approach allows oneto calculate the internal elastic strains, excess GB energyin nanomaterials and also to explain the enhancement of GB diffusion in these materials. At elevated temperaturesdisclinations relax causing the recovery of GB energy and

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diffusion coefcient. In order to measure the true GB diffusioncoefcient, the diffusion annealing time must be much longerthan the relaxation time for junction disclinations.

Acknowledgment

The author is grateful to the Russian Foundation for BasicResearch for support under grant no. 120893001-Viet a.

References

[1] De Wit R 1977 Continual Theory of Disclinations (Moscow:Mir) in Russian

[2] Romanov A E and Vladimirov V I 1992 Dislocations in Solidsvol 9 ed F R N Nabarro (Amsterdam: North-Holland)pp 191402

[3] Romanov A E and Kolesnikova A L 2003 Eur. J. Mech. A22 727

[4] Romanov A E and Kolesnikova A L 2009 Prog. Mater. Sci.54 740

[5] Rybin V V and Zhukovskii I M 1978 Sov. Phys. Solid State20 1056

[6] Rybin V V, Zisman A A and Zolotarevskii N Yu 1985 Sov.Phys. Solid State 27 105

[7] Mullner P 2002 Solid State Phenom. 87 227[8] King A H 1993 Mater. Sci. Forum 126128 221[9] Nazarov A A, Romanov A E and Valiev R Z 1993 Acta Metall.

Mater. 41 1033[10] Nazarov A A, Romanov A E and Valiev R Z 1995 Nanostruct.

Mater. 6 775[11] Nazarov A A, Romanov A E and Valiev R Z 1996 Scr. Mater.

34 729[12] Nazarov A A and Mulyukov R R 2002 Nanoscience,

Engineering and Technology Handbook ed S Lyshevski,

D Brenner, J Iafrate and W Goddard (Boca Raton, FL: CRCPress) pp 22-1[13] Volterra V 1907 Ann. Ecole Normale Sup Paris 24 401[14] De Wit R 1973 J. Res. Natl Bur. Stand. A77 607[15] Galligan J M 1972 Scr. Met. 6 161[16] Gryaznov V G, Heydenreich J, Kaprelov A M, Nepijko S A,

Romanov A E and Urban J 1999 Cryst. Res. Technol.34 1091

[17] Rybin V V 1986 Large Plastic Deformations and Fracture of Metals (Moscow: Metallurgia) in Russian

[18] Rybin V V, Zisman A A and Zolotarevsky N Yu 1993 Acta Metall. Mater. 41 2211

[19] Orlova T S, Nazarov A A, Enikeev N A, Alexandrov I V,Valiev R Z and Romanov A E 2005 Phys. Solid State47 845

[20] Orlova T S, Romanov A E, Nazarov A A, Enikeev N A,Alexandrov I V and Valiev R Z 2005 Tech. Phys. Lett.31 1015

[21] Nazarov A A, Enikeev N A, Romanov A E, Orlova T S,Alexandrov I V, Beyerlein I J and Valiev R Z 2006 Acta Mater. 54 985

[22] Molinari A, Canova G R and Ahzi S 1987 Acta Metall.35 2983

[23] Hansen N 1990 Mater. Sci. Technol. 6 1039[24] Hughes D A and Hansen N 2000 Acta Mater. 48 2985[25] Xue Q, Beyerlein I J, Alexander D J and Gray G T III 2007

Acta Mater. 55 655[26] Gutkin M Yu and Ovidko I A 1994 Phil. Mag. A 70 561[27] Zhou K, Wu M S and Nazarov A A 2006 Phys. Rev. B

73 045410[28] Foiles S M, Baskes M I and Daw M S 1986 Phys. Rev. B

33 7983[29] Wu M S, Zhou K and Nazarov A A 2007 Phys. Rev. B

76 134105[30] Iskendarov A M and Nazarov A A 2008 Deformation Fracture

Mater. 7 2 (in Russian)[31] Zhou K, Nazarov A A and Wu M S 2007 Phys. Rev. Lett.

98 035501[32] Valiev R Z, Islamgaliev R K and Alexandrov I V 2000 Prog.

Mater. Sci. 45 103[33] Valiev R Z and Alexandrov I V 2007 Bulk Nanostructured

Metallic Materials: Production, Structure and Properties(Moscow: Akademkniga)

[34] Nazarov A A 2000 Interface Sci. 8 315[35] Bachurin D V and Nazarov A A 2003 Phil. Mag. 83 2653[36] Arzt E, Ashby M F and Verrall R A 1983 Acta Metall. 31 1977[37] Nazarov A A 2000 Phil. Mag. Lett. 80 221[38] Manning J R 1968 Diffusion Kinetics for Atoms in Crystals

(Toronto: Van Nostrand)

[39] Nazarov A A 2003 Phys. Solid State 45 1166[40] Borisov V T, Golikov V T and Shcherbedinsky G V 1964Phys. Metals Metall. 17 881

[41] Wurschum R, Brossmann U and Schaefer H-E 2001 Nanostructured Materials Processing, Properties and Potential Applications ed C C Koch (New York: WilliamAndrew) 267

[42] Herth S, Michel T, Tanimoto H, Egersmann M, Dittmar R,Schaefer H-E, Frank W and W urschum R 2001 Def. Diff.Forum 194199 1199

[43] Kolobov Yu R, Grabovetskaya G P, Ivanov M B, Zhilyaev A Pand Valiev R Z 2001 Scri. Mater. 44 873

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