grain boundary and triple junction migration

Materials Science and Engineering A302 (2001) 141–150

Grain boundary and triple junction migration

L.S. Shvindlerman a,b,*, G. Gottstein b

a Institute of Solid State Physics, Russian Academy of Sciences, Chernogolo6ka, Moscow District 142432, Russiab Institut fur Metallkunde und Metallphysik, RWTH Aachen, Kopernikusstr. 14, D-52074 Aachen, Germany

Abstract

The current status and latest achievements of grain boundary (GB) and triple junction (TJ) migration in metals are reviewed.The migration of 90° �112� planar tilt symmetrical and asymmetrical GB in specially grown Bi-bicrystals driven by magnetic forceand the dependence of GB mobility on temperature, driving force and the direction of motion are addressed. The motion of low-and high angle planar tilt �112� and �111� GBs moved by shear stresses and the peculiarities of such a motion are considered.In particular, the sharp transition from low- to high-angle boundaries was observed. In practice the motion of a straight GB isthe exception rather than the rule. The shape of a moving GB is a source of new and useful findings concerning GB migration.The experimentally derived shape of a GB in Al-bicrystals was compared with theoretical calculations in the Lucke–Detertapproximation. The experimental and theoretical results of a motion of grain boundary systems with triple junctions arepresented. Their impact on the kinetics of microstructure evolution and, in particular, on Von Neumann–Mullins relation isoutlined. © 2001 Elsevier Science B.V. All rights reserved.

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1. Magnetically driven grain boundary motion

A grain boundary (GB) migration occurs when theboundary displacement leads to the reduction of a totalenergy of the system. There are two ways by which thismay be accomplished. The first uses free energy of theGB itself, the other utilizes a free energy difference ofthe adjacent grains. The most frequently used method isthe displacement of a curved GB [1–4]. However, theobtained mobilities can, therefore, not be related to aspecific GB structure, while by using the second type ofdriving force a plane boundary can be forced to move.A bicrystal with grains that have some orientationdependent property like elastic constants or magneticsusceptibility can be utilized in this case. This drivingforce does not depend on boundary properties. Suchconditions, in particular, can be obtained by the actionof a magnetic field on a bicrystal of a material withanisotropic magnetic susceptibility [3,4].

The origin of the driving force for grain boundarymigration in a magnetically anisotropic material wasconsidered by Mullins [5]. The expression for the driv-ing force, as applied to bismuth, reads

P=m0

Dx

2H2(cos2 U1−cos2 U2) (1)

where H is the magnetic field strength, Dx is the differ-ence of the susceptibilities parallel and perpendicular tothe trigonal axis, U1 and U2 are the angles between themagnetic field and the trigonal axes in both grains ofthe Bi-bicrystal.

Three efforts were made in the past to utilize amagnetic field for the study of GB kinetics in Bi byGoetz, Mullins and Fraser [6,7], however, no specificboundary motion was investigated.

Our experiments were carried out on bicrystals ofhigh purity (99.999%) bismuth [8,9]. Symmetrical andasymmetrical (c=45°) pure tilt boundaries with 90°�112� misorientation were examined (Fig. 1, the devia-tion of asymmetrical GB from symmetrical positionequals c).

The experiments were carried out using the highmagnetic field facilities of the National High MagneticField Laboratory in Tallahassee, FL, USA. A resistive,steady-state 20-T bitter magnet with 50-mm bore di-ameter was used, and a field strength between 0.80×107 and 1.59×107 A m−1 was applied. The magneticfield was imposed on the samples at different tempera-tures ranging from 210 to 260°C.

* Corresponding author. Tel.: +7-95-9132324; fax: +7-64-412654.E-mail address: [email protected] (L.S. Shvindlerman).

0921-5093/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S0921 -5093 (00 )01366 -6

L.S. Sh6indlerman, G. Gottstein / Materials Science and Engineering A302 (2001) 141–150142

Fig. 1. Geometry of investigated bicrystals with 90° �112� tilt, (a) symmetrical; and (b) asymmetrical boundaries.

The possibility to change the magnitude of the driv-ing force for boundary migration by exposing the sam-ples to magnetic fields of different strength yields theunique opportunity to change the driving force on aspecific grain boundary and thus, to obtain the drivingforce dependence of grain boundary velocity (Fig. 2).

The experiments unambiguously confirmed that grainboundaries in Bi-bicrystals actually move under theaction of a magnetic driving force. The observed lineardependence of boundary displacement on annealingtime proves the free character of its motion.

To prove that boundary motion was caused exclu-sively by the magnetic driving force, the experiment wascarried out in two different ways. First, a specimen wasmounted in a holder such that the c-axis (�111�) ofcrystal 1 was directed parallel to the field (Fig. 3a). The�111� axis in crystal 2 in this case was perpendicular tothe field, and the grain boundary moved in the direc-tion of the latter crystal due to its higher magnetic freeenergy. Second, a specimen was mounted in a positionwhere the axis �111� in crystal 2 was close to the fielddirection, and the corresponding axis in crystal 1 wasperpendicular to the field. The direction of boundarymotion in this case was opposite, from crystal 2 towardcrystal 1 (Fig. 3b).

This result provides unambiguous evidence that thegrain boundaries in the bicrystals were forced to moveby the magnetic driving force only. In addition, somebicrystals were annealed in a magnetic field in bothpositions, and boundary motion in opposite directionwas observed in the same specimen dependent on itsposition with regard to the magnetic field.

In the current experiments, we investigated the mi-gration of two differently inclined 90° �112� tilt grainboundaries, namely a symmetrical and an asymmetricalboundary (Fig. 1).

In contrast to the symmetric tilt boundary, for theasymmetric tilt boundary the measured boundary mo-bilities were found to be distinctly different for motionin opposite directions (Fig. 4). There are several poten-tial reasons for this anisotropy. First, there is an essen-tial difference in the distance between thecrystallographic planes on each side of the boundary.An estimation shows that this factor may change thevelocity of grain boundary motion; however, this factoris unlikely to change the velocity of grain boundary

motion by more than 20%, which is distinctly less thanthe observed effect. Second, because grain boundarymotion in Bi-bicrystals may be influenced by impuritydrag, the difference in the diffusivity of impurities intwo opposite directions in the anisotropic structure ofBi should be taken into consideration. In this respect itis interesting that the symmetric tilt boundary exhibiteda much higher mobility than the asymmetric tiltboundary and did not show a dependence of boundarymobility on the sense of motion (Fig. 4).

In any event, if this asymmetry of grain boundarymobility holds also for other metals, it would have aserious impact on our understanding of grain boundary

Fig. 2. Dependence of the velocity of a 90° �112� symmetrical tiltboundary on the magnetic driving force.

Fig. 3. Geometry of investigated bicrystals and sense of driving forceP with regard to direction of the magnetic field H.

L.S. Sh6indlerman, G. Gottstein / Materials Science and Engineering A302 (2001) 141–150 143

Fig. 4. (a) Normalized displacement of a grain boundary vs. annealing time for the same grain boundary moving in opposite directions; (b)temperature dependence of mobility of 90° �112� symmetrical () and asymmetrical (�, ) boundaries in Bi-bicrysytals: �–trigonal axis in thegrowing grain is parallel to the growth direction; –trigonal axis in the growing grain is perpendicular to the growth direction.

motion, since the mobility of a grain boundary iscommonly conceived as not dependent on its directionof motion.

2. Influence of external shear stresses on grainboundary migration

A method to activate and investigate the migration ofplanar, symmetrical tilt boundaries in aluminum bicrys-tals under the influence of an external shear stress wasintroduced. It was shown that low- as well as high-an-gle boundaries could be moved by this shear stress.From the activation parameters for grain boundarymigration, the transition from low- to high-angleboundaries can be determined. The migration kineticswere compared with results on curved boundaries, andit was shown that the kinetics of stress induced motionwere different from the migration kinetics of curvaturedriven boundaries.

In 1952, Washburn, Parker et al. [10,11] investigatedplanar low-angle boundaries in Zn under the influenceof an external shear stress and observed the motionwith polarized light in an optical microscope. Thecurrent study was aimed at probing the effect of amechanical, shear stress field on planar low- and high-angle boundaries [12].

For the investigations bicrystals of different purities(1 and 7.7 ppm impurity content) with �112� and�111� tilt grain boundaries with misorientation anglesin a range from 4 to 32° were grown. The grainboundary motion was measured in situ with an X-rayinterface continuous tracking device (XICTD).

Symmetrical low angle tilt boundaries consist of peri-odic arrangements of a single sets of edge dislocations.An external shear stress perpendicular to the boundaryplane will cause a force on each dislocation and insummary a driving force on the boundary. The sampleswere exposed to a shear stress ranging from 10−1 to

10−3 MPa. In aluminum (purity 99.999%) the yieldstress is 15–20 MPa, hence the applied shear stress isdefinitely in the elastic range.

High angle symmetrical tilt boundaries also can beformally described as an arrangement of a single set ofedge dislocations except that the dislocation cores over-lap and the identity of the dislocations gets lost in therelaxed boundary structure.

First of all we want to show that irrespective of themagnitude of the angle of rotation, grain boundariescan be moved under the action of the applied shearstress. Fig. 1 shows the dependence of the grainboundary velocity on the applied mechanical shearstress for two different �112�-tilt boundaries. FromFig. 5 we can see that obviously, both the low- and thehigh-angle grain boundary move under the influence ofthe shear stress and the grain boundary velocitychanges in proportion to the stress in both cases.

Fig. 6 shows the dependence of the activation en-thalpy on misorientation angle for different tilt axesand impurity content. For low angle grain boundarieswe find a constant activation enthalpy of DH=1.28 eVand for high angle grain boundaries DH=0.85 eV. Thetransition from low- to high-angle grain boundaries is

Fig. 5. Dependency of the grain boundary velocity on the externalshear stress for two symmetrical �112� tilt boundaries.


Fig. 6. Activation enthalpy vs. misorientation angle for �112� and�111� tilt boundaries with different impurity content.

see a strong dependency of the activation enthalpy onthe misorientation angle, i.e. on the grain boundarystructure. There is also a clear difference between theactivation enthalpies for the stress induced motion ofthe planar high angle grain boundaries and the curva-ture driven migration of the curved high angle grainboundaries.

Obviously, a dislocation in a high angle grainboundary does not relax completely its strain field andcorrespondingly, a biased elastic energy density inducedby an applied shear stress will induce a force on alldislocations that comprise the grain boundary.

The results prove that grain boundaries can be drivenby an applied shear stress irrespective whether low- orhigh-angle boundaries. Obviously, the motion of thegrain boundary is caused by the movement of thedislocations, which compose the grain boundary. Theactivation enthalpy for the low angle grain boundariesamounts to DH=1.28 eV and is comparable with theactivation enthalpy of bulk self-diffusion in aluminum.For the high angle grain boundaries we found anactivation enthalpy of DH=0.85 eV, which is compara-ble to the activation enthalpy for grain boundary diffu-sion in aluminum.

The motion of an edge dislocation in a fcc crystal inreaction to an applied shear stress ought to be purelymechanical and not thermally activated. Obviously, theobserved grain boundary motion is a thermally acti-vated process controlled by diffusion. To understandthis, one has to recognize first that grain boundarymotion is a drift motion since it experiences a drivingforce that is smaller compared with thermal energy.Moreover, real boundaries are never perfect symmetri-cal tilt boundaries but always contain structural dislo-cations of other Burgers vectors. These dislocationshave to be displaced by nonconservative motion tomake the entire boundary migrate. The climb processrequires diffusion, which can only be volume diffusionfor low angle grain boundaries but grain boundarydiffusion for high angle grain boundaries according tothe observed activation enthalpies.

The different behavior of curvature driven grainboundaries is not due to the curvature of theboundaries rather than due to a different effect of therespective driving force. While an applied shear stresscouples with the dislocation content of the boundary ina curved grain boundary each individual atom experi-ences a drift pressure to move in order to reducecurvature.

3. Shape of the moving grain boundaries

The principal parameter which controls the motionof a grain boundary is the grain boundary mobility. Inpractically all relevant cases the motion of a straight

Fig. 7. Dependency of the activation enthalpy on misorientationangle for curvature driven and planar �111� grain boundaries, andplanar �112� grain boundaries (open symbols [14]; filled [12]).

revealed by a conspicuous step of the activation en-thalpy at a misorientation angle of 13.6°.

There exists no evidence, that the deviations from theactivation enthalpy levels for low- and high-angleboundaries show a dependence of the activation en-thalpy on grain boundary structure. From Fig. 6 weconclude that �112�- and �111�-tilt boundaries movewith the same activation enthalpies when exposed to amechanical stress. This holds for low angle as well asfor high angle symmetrical tilt boundaries.

Previous experiments on curvature driven �111� tiltboundaries in aluminum bicrystals showed a strongmisorientation dependence of the activation enthalpies[13]. For comparison, we conducted curvature drivenboundary migration experiments on �111� tiltboundaries. The driving force was a constant capillaryforce, p=s/a where a is the width of the shrinkinggrain (quarter-loop technique). The typical values of adriving force in both types of our experiments arenearly the same, p�103 J m−3.

In Fig. 7 the dependence of the activation enthalpyon the misorientation angle for the curved and theplanar grain boundaries is shown. For the curvaturedriven grain boundaries our results are in good agree-ment with previous experimental data [14] and one can


grain boundary is the exception rather than the rule.That is why the shape of a moving grain boundary is ofinterest and it will be shown that the grain boundaryshape is a source of new interesting and useful findingsof grain boundary motion, for the interaction of amoving grain boundary with mobile particles, in partic-ular. The experimentally derived shape of grainboundary ‘quarter-loop’ [15] in Al-bicrystals of differ-ent purity was compared with theoretical calculationsin the Lucke–Detert approximation.

The shape of a moving GB quarter-loop was deter-mined analytically under the assumption of uniformGB properties and quasi-two-dimensionality [15]:

y(x)

=

ÁÃÍÃÄ

− (bF−bL)arc cos�sin U

ex*/bF

�+

a2

−bL

p

2+

bF arc cos(e bF ln(sin U)−x/bF)

05x5x*a2−bL

p

2

+bL arc cos(ebL ln(sin U)−x*((bL/bF)−1)−x/bL)

x]x* (2)

where bL is:

bL=bF(arc cos(sin U/ex*/bF+U− (p/2)−a/2))

arc cos(sin U/ex*/bF)− (p/2)(3)

The parameters in Eq. (2) are the width of theshrinking grain a/2, the angle U of the grain boundarywith the free surfaces in the triple junction, the criticalpoint x*, and bF and bL: bL=mLs/V ; bF=mFs/V,where mL and mF are the GB mobilities for ‘loaded’and ‘free’ GB, respectively, s is GB surface tension, Vis a velocity of a quarter-loop. The first two parameterscan be measured directly in the experiment. The lattertwo have to be chosen in an approximate way to fit theexperimentally derived grain boundary shape. Thepoint x* is the point of intersection ‘free’ and ‘loaded’segments of the GB.

The value mL/mF is a measure for how drastic thechange between the ‘free’ and the ‘loaded’ part in thepoint of intersection will be. Corresponding experi-ments were carried out on aluminum bicrystals with a40.5°�111� tilt boundary. The samples differed by theamount of dissolved impurities. Bicrystals with a totalimpurity content of 0.4, 1.0, 3.6, 4.9 and 7.7 ppm werestudied. The investigation proves that the influence ofthe impurity atoms on grain boundary properties andbehavior is rather strong even in very pure materials.The experimentally measured shape of a moving grainboundary (Al with 1.0 ppm impurities) and the shapecalculated according to equation which does not takeinto consideration impurity drag, are compared in Fig.8. The large discrepancy is obvious and apparently dueto the neglect of boundary–impurity interactions. How-ever, with mL and mF determined as explained abovethe measured boundary shape can be successfully fittedby Eq. (2)) using only x* as a free fit parameter (Fig.9).

The same holds for the reversed-capillary technique.Any attempt to fit the shape by assuming a freelymoving boundary fails, but good agreement betweenexperiment and theory can be observed when impuritydrag and thus, different mobilities are taken into ac-count (Fig. 10) [16].

As mentioned above, the shape of a moving grainboundary is a new source of information on grainboundary migration. One example is given in Fig. 11,where the value of the critical distance x*, normalizedby the driving force (in terms of the quarter-loop widtha) is plotted versus the impurity content. In accordancewith the Lucke–Detert theory the critical velocity 6*(and rigidly bound to it the position of the critical point

Fig. 8. Comparison between experimental data (solid line) and calcu-lated shape (dotted line) disregarding segregation.

Fig. 9. Experimentally observed (solid line) and calculated (Eq. (3),dotted line) grain boundary shape for Al with an impurity content of(a) 1.0; (b) 3.6 ppm (quarter-loop technique).


Fig. 10. Experimentally observed and calculated shape accounting for drag effect, (shaded area); neglecting the drag effect (dashed line) (Fe–3%Si, reversed-capillary technique).

Fig. 11. Dependence of critical point x*/a on (a) impurity content; (b) reciprocal impurity content.

x* on the quarter-loop) is determined by the balancebetween the maximum force of interaction of the impu-rity atoms with the boundary and the force, which isimposed by the energy dissipation caused by boundarymotion across the matrix. The difference of the impu-rity drag for grain boundaries in samples with differentamount of impurities is caused by the adsorption ofimpurities at the grain boundary. According to theory,the velocity should decrease proportionally to the in-verse of the concentration of adsorbed atoms. There-fore x* should increase with decreasing impuritycontent, as observed qualitatively (Fig. 11) [15]. How-ever, a linear relation between the inverse of the impu-rity concentration and 6*, i.e. x*, is not observed overthe whole concentration range, which indicates a morecomplicated interaction of adsorbed atoms with thegrain boundary. In such a case, x*/a should increasemore strongly with decreasing impurity content than

it does linearly. This tendency is indeed observed(Fig. 11).

4. Dragging effect of triple junction on grain boundarymotion

In spite of the fact that a line (or column) of intersec-tion of three boundaries constitutes a system with spe-cific thermodynamic properties was realized more than100 years ago (by Gibbs), the kinetic properties of thissubject, in particular the mobility of triple junctions,and their influence on grain growth and relevant pro-cesses were ignored up to now. Although the number oftriple junctions in polycrystals is comparable in magni-tude with the number of boundaries, all peculiarities inthe behavior of polycrystals during grain growth weresolely attributed to the motion of grain boundaries sofar. It was tacitly assumed in theoretical approaches,


computer simulations and interpretation of experimen-tal results that triple junctions do not disturb grainboundary motion and that their role in grain growth isreduced to preserve the thermodynamically prescribedequilibrium angles at the lines (or the points for 2-Dsystems) where boundaries meet. The most prominentexample of how this assumption determines the funda-mental concepts of grain structure evolution gives theVon Neumann–Mullins relation [17,18].

No doubt this relation forms the basis for practicallyall theoretical and experimental investigations as well ascomputer simulations of microstructure evolution in2-D polycrystals in the course of grain growth. Thisrelation is based on three essential assumptions,namely, (i) all grain boundaries possess equal mobilitiesand surface tensions, irrespective of their misorientationand crystallographic orientation of the boundaries; (ii)the mobility of a grain boundary is independent of itsvelocity; (iii) the third assumption relates directly to thetriple junctions, namely, they do not affect grainboundary motion; therefore, the contact angles at triplejunctions are in equilibrium and, due to the first as-sumption, are equal to 120°.

As it was shown in [17,18], for 2-D grain, the rate ofchange of the grain area S can be expressed by

dSdt

= −Ab7

d8 (4)

where Ab=mbs ; mb being the grain boundary mobility,s is the grain boundary surface tension.

If the grain were bordered by a smooth line, theintegral in Eq. (4) would equal 2p. However, owing tothe discontinuous angular change at every triple junc-tion, the angular interval D8=p/3 is subtracted fromthe total value of 2p for each triple junction.Consequently,

dSdt

= −Ab�

2p−np

3�

=Abp

3(n−6) (5)

where n is the number of triple junctions for eachrespective grain, i.e. the topological class of the grain.Thus, the rate of area change is independent of theshape of the boundaries and determined by the topo-

logical class n only. Grains with n\6 will grow andthose with nB6 will disappear [18].

The existence of triple junctions drastically affects thekinetics of grain growth. To discuss this problem quan-titatively the mobility of a triple junction should bemeasured. However, the steady-state motion of a grainboundary system with a triple junction is only possiblein a very narrow class of geometrical configurations.Two of these special boundary systems were investi-gated in [19–21] under three main assumptions. Two ofthem comply with the assumptions (1) and (2) of theVon Neumann–Mullins consideration, while the thirdone is determined by Eq. (2): the normal GB displace-ment rate 6 is proportional to the GB curvature K.

As shown in [19], the model grain boundary system(Fig. 12) can move steadily, and the analysis of itsmotion permits us to understand the influence of thefinite mobility of a triple junction on the migration ofGBs.

The considered grain boundary system (Fig. 12) con-sists of three grain boundaries, two of them are curvedwith a common triple junction. During steady-statemotion of the system the velocity V parallel to thex-axis (Fig. 12) is related to the rate of normal displace-ment 6 :

6=V cos 8=Vy %

[1+ (y %)2]1/2 (6)

where y(x) is the shape of the positive part (upper partin Fig. 12) of the curved boundary. Due to the mirrorsymmetry of the problem relative to the x-axis, theshape of the lower part boundary is the negativeequivalent.

Then the equation for the steady-state shape of themoving grain boundary

y¦= −V

mbsy %(1+ (y %)2) (7)

Eq. (7), restricted by three boundary conditions, per-mits us to find the desired shape y(x) and the velocityV of the moving grain boundary (Fig. 12)

y(0)=0

y(�)=a2

y %(0)= tan U (8)

The meaning of the length a and the angle U is clearfrom Fig. 12. A driving force s(2 cos U−1) acts on thetriple junction from the curved boundaries. Introducingthe mobility of the triple junction mTj, its velocity reads

VTj=mTjs(2 cos U−1) (9)

Due to the fact that the driving force acting on thegrain boundary is a pressure and the driving force on

Fig. 12. Configuration of grain boundaries at a triple junction duringsteady-state motion for nB6.


Fig. 13. Angle U as a function of L, (a) for nB6 (Eq. (12)); (b) forn\6 (Eq. (18)).

j=a

2U

c1= ln(sin U)

c2= −�p

2−U

�(10)

The velocity V of steady-state motion of the system is

V=2Umbs

a(11)

The steady-state value for the angle U can be foundfrom the equation.

2U

2cos U−1=

mTjamb

=L (12)

If a triple junction is mobile and does not drag grainboundary motion, the criterion L�� and U�p/3,i.e. the equilibrium angular value at a triple junction inthe uniform grain boundary model. In contrast, how-ever, when the mobility of the triple junction is rela-tively low (strictly speaking, when mTja�mb) thenU�0 (Fig. 13). It should be stressed that the angle U

is strictly defined by the dimensionless criterion L,which, in turn, is a function of not only the ratio oftriple junction and grain boundary mobility, but of thegrain size as well.

Experimental investigations [20] were based on theconsidered grain boundary system (Fig. 12). It wasshown that triple junctions do possess a finite mobility.It was found that the vertex angle U at the triplejunction could deviate distinctly from the equilibriumvalue, when a low mobility of the triple junction hin-ders the motion of the grain boundaries. In fact, atransition from triple junction kinetics to grainboundary kinetics was observed (Figs. 14 and 15).

For grains with topological class greater than six letus consider the steady-state motion of a grain boundarysystem shown in Fig. 16 with the same set of assump-tions applied to the previous boundary system, namely,uniform grain boundary properties and quasi-two-di-mensionality [21,22].

The steady-state motion of this system is determinedby the system of Eqs. (7) and (8) only with differentboundary and initial conditions

y %(0)=�

y %(x0)= tan U

y(0)=0 (13)

The velocity of the triple junction motion can beexpressed as (Fig. 3)

VTj=mTjs(1−2 cos U) (14)

Like in the previous case, Eqs. (5) and (13) define theconsidered problem completely.

Fig. 14. The angles in the tip of the tricrystal half-loop at differenttemperatures. (Zn tricrystal, misorientation angles of the tilt grainboundaries are 46° �1011�, 43° �1011� and 3°).

Fig. 15. The temperature dependence of the criterion L of theinvestigated Zn tricrystal.

the triple junction is a force, the dimensions of grainboundary and triple junction mobility are different, sothat their ratio mb/mTj has the dimension of a length.

For the configuration in Fig. 12, Eqs. (7) and (8)define the problem completely. The solution can beexpressed as [19]:

y(x)=j arc cos(e−x/j+c1)+c2


Fig. 16. Configuration of grain boundaries at triple junctions during steady-state motion for n\6 [20].

y(x)= −x0

ln sin Uarc cos(ex/x0 ln sin U) (15)

The velocity of steady-state motion of the system is

V= −mbs

x0

ln sin U (16)

The length x0 replaces the role of the grain size a inthe previous case (Fig. 16) or

y0=y(x0)= −x0

ln sin Uarc cos(eln sin U)

= −x0

ln sin U

�p

2−U

�(17)

From Eqs. (14) and (16) we obtain L, which de-scribes the influence of the triple junction mobility ongrain boundary migration

−ln sin U

1−2 cos U=

mTjx0

mb

=L (18)

Obviously, for L�1, when the boundary mobilitydetermines the kinetics of the system, the angle U tendsto its equilibrium value (p/3).

Again, the angle U changes when a low mobility ofthe triple junction starts to drag the motion of theboundary system. However, as can be seen from Eq.(18) and Fig. 16, in this case the steady state value ofthe angle U increases as compared with the equilibriumstate (otherwise the triple junction would move in thenegative direction of the x-axis, increasing the freeenergy of the system).

For L�1 the angle U (Eq. (18)) tends to approachp/2. The dependency U=U(L) for both nB6 andn\6 are shown in Fig. 13.

The rate of area change for a grain with nB6 can beexpressed as

dSdt

= −mbs7

d8= −Ab(2p−n(p−2U))

=Ab(n−2U)�

n−2p

p−2U

�(19)

Since the limited mobility of the triple junction re-duces the steady-state value of the angle U as comparedwith the equilibrium angle, the shrinking rate of grainswith nB6 decreases, as obvious for the case when the

mobility of the triple junction becomes very low. Inother words, for grains with nB6, the influence of thetriple junction mobility slows down the process of grainstructure evolution, decreasing the vanishing rate ofgrains with small topological class (nB6).

For grains with topological class greater than six letus refer to the considered steady-state motion of a grainboundary system with a large number of triple junc-tions (Fig. 13) [19,20]. The dimensionless parameter L,which describes the influence of the triple junctionmobility on grain boundary migration for such a sys-tem is given by Eq. (18). When a low mobility of thetriple junction starts to drag the motion of theboundary system, the angle U changes. However, inthis case, the steady-state value of the angle U increasesas compared with the equilibrium state. Such an in-crease of the angle U also decreases the magnitude of(p−2U) in 31, in other words, it decreases the ‘effec-tive’ magnitude of the topological class of the growinggrain with n\6. Consequently, microstructural evolu-tion will slow down due to triple junction drag for anyn-sided grain.

The only exception holds for n=6, since a hexagonalgrain structure becomes unstable when the contact an-gle 2U"2p/3. Since the actual magnitude of U isdetermined by the triple junction and grain boundarymobility as well as the grain size and is independent ofthe number of sides of a grain, there is no uniquedividing line between vanishing and growing grainswith respect to their topological class anymore, liken=6 in the Von Neumann–Mullins approach.

Acknowledgements

The authors express their gratitude to the DeutscheForschungsgemeinschaft (DFG Grant 436 RUS113/539/0) and to the Russian Foundation for FundamentalResearch under contract RFFI-DFG 99 02 for financialsupport for their collaboration.

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grain boundary and triple junction migration

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