KINETICS OF DIFFUSION INDUCED GRAIN BOUNDARY

MIGRATION OF [100] TWIST BOUNDARIES IN THE Cu(Zn)

SYSTEM

Y. YAMAMOTO, M. MORIYAMA, M. KAJIHARA{ and T. MORI{Department of Materials Science and Engineering, Tokyo Institute of Technology, Midori-ku,

Yokohama 226-8502, Japan

(Received 29 June 1998; accepted 9 February 1999)

AbstractÐThe kinetics of di�usion induced grain boundary migration (DIGM) in the Cu(Zn) system wasexperimentally studied for [100] twist boundaries using Cu bicrystals annealed at 693 K for various timesbetween 5.4�104 and 1.73� 105 s. The experiment was carried out for bicrystal specimens with misorienta-tion angles of 15, 20, 23 (S13), 25, 28 (S17), 32, 37 (S5), 40, 44 (S29) and 458. During DIGM, the grainboundary migrates rather unidirectionally towards one of the crystal grains for most of the specimens andbecomes wavy with increasing annealing time. The migration distance at 5.4� 104 s is smaller for coinci-dence site lattice (CSL) boundaries with low energy than for random boundaries with high energy. The mi-gration rate of the moving boundary was observed to be almost constant independent of the annealingtime between 5.4� 104 and 1.73�105 s. The steady state migration rate is larger for low-energy CSLboundaries than for high-energy random boundaries. The experimental results were quantitatively analyzedusing the energy balance model proposed by Kajihara and Gust. In this analysis, the e�ective driving forcefor DIGM was calculated as a function of the migration rate. The calculation indicates that 75±93% ofthe chemical driving force is consumed by the volume di�usion of Zn in the untransformed Cu matrixahead of the moving boundary under the given experimental conditions. Considering the energy consump-tion, the mobility M of the moving boundary was evaluated for the steady state stage. The evaluationyields M � 6:6� 10ÿ17 m4=J s as the mobility of the random boundary for DIGM in the Cu(Zn) system at693 K. # 1999 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

Volume di�usion is considered to be frozen in butgrain boundary di�usion occurs su�ciently fast atintermediate temperatures in many polycrystalline

alloys. Under such conditions, di�usion inducedgrain boundary migration (DIGM) and/or di�usioninduced recrystallization (DIR) may take place.

DIR is the phenomenon that new grains with di�er-ent solute concentrations are produced behind mov-

ing grain boundaries due to recrystallizationcombined with di�usion of solute atoms along themoving grain boundaries. On the contrary, during

DIGM a region with di�erent composition is leftbehind a moving grain boundary owing to grainboundary migration combined with the di�usion of

solute atoms along the moving boundary. Many ex-periments on DIGM and DIR have been carriedout in binary systems, where Zn is the solute,

because it is rather easy to control the Zn activityusing a zinci®cation technique with an appropriate

Zn-source alloy.

The kinetics of DIGM in the Cu(Zn) system were

experimentally studied by Li and Hillert [1] using

polycrystalline Cu specimens zinci®ed at tempera-

tures between 523 and 873 K. Here, according to

convention, the notation A(B) means that a solute

B di�uses into a pure A metal or a binary A±B

alloy. In their experiment, DIGM was observed to

occur in specimens zinci®ed at 523±773 K. Their

observations indicate that the migration rate and

morphology of the grain boundary seem to vary

depending on the character of the boundary.

However, the relationship between the kinetics of

DIGM and the character of the grain boundary

was not determined in their experiment on polycrys-

talline material. Such a relationship will be obtained

by experiment utilizing appropriate bicrystals con-

taining a crystallographically well-de®ned grain

boundary.

King and co-workers [2±5] experimentally studied

DIGM in the Cu(Zn) system using Cu bicrystals

with [001] symmetric and asymmetric tilt bound-

aries. As to the symmetric boundaries, they exam-

ined the dependence of the kinetics of DIGM on

the misorientation angle at 653 and 673 K [2±4].

According to their results, the migration rate is

much smaller for the coincidence site lattice bound-

aries with low S values than for the nonspecial or

Acta mater. Vol. 47, No. 6, pp. 1757±1766, 1999# 1999 Acta Metallurgica Inc.

Published by Elsevier Science Ltd. All rights reservedPrinted in Great Britain

1359-6454/99 $20.00+0.00PII: S1359-6454(99)00053-1

{To whom all correspondence should be addressed.{Present address: Department of Materials Science and

Engineering, Northwestern University, Evanston, IL60208-3108, U.S.A.

1757

random boundaries. Recently, the relationshipbetween the boundary energy and the misorienta-

tion angle for the [001] symmetric tilt boundaries inCu was experimentally determined by Mori et al. [6]from the shape of boundary SiO2 particles. The

boundary energy is one of the most suitable par-ameters which represent the character of a grainboundary determined by the crystallographic

arrangement of atoms in the boundary. A higherdegree of disorder in the atomic arrangement alonga grain boundary leads to a larger boundary energy.

From a comparison between the results of DIGMand boundary energy for the [001] symmetric tiltboundaries, it is concluded that a high-energy grainboundary with a disordered atomic arrangement

migrates remarkably whereas a low-energy grainboundary with a less-disordered atomic arrange-ment moves slowly or even is immobile.

The kinetics of DIGM in the Cu(Zn) system wasalso experimentally studied by Gust and co-workers [7, 8] using Cu bicrystals with [011] sym-

metric and asymmetric tilt boundaries. The bicrys-tals were zinci®ed at temperatures between 623 and773 K and then DIGM was observed to occur. The

relationship between the migration rate and theboundary energy mentioned above holds good alsofor the [011] symmetric tilt boundaries.The orientation dependence of the migration

behavior of [100] twist boundaries due to DIGM inthe Cu(Zn) system was experimentally studied at693 K by the present authors in a previous paper [9].

The migration distance of the moving grain bound-ary was determined as a function of the misorienta-tion angle for an annealing time of 5.4�104 s. In

the case of Cu bicrystals with [001] and [011] sym-metric tilt boundaries, the Miller indices of bothside crystal grains along the boundary plane varydepending on the misorientation angle. This causes

a change not only in the atomic arrangement alongthe boundary but also in the elastic property ofeach grain along a plane parallel to the boundary.

The change in the elastic property may a�ect thedriving force for DIGM owing to the coherencystrain energy. On the other hand, in the case of the

Cu bicrystals with [100] twist boundaries, the elasticproperty is kept invariable independent of the mis-orientation angle. Therefore, twist boundaries are

suitable for examining only the e�ects of the atomicarrangement along the boundary on DIGM. In thepresent work, the kinetics of DIGM for the [100]twist boundaries in the Cu(Zn) system were exper-

imentally studied using Cu bicrystals zinci®ed at693 K for various times. The experimental resultshave been quantitatively analyzed using the energy

balance model proposed by Kajihara and Gust [10].

2. EXPERIMENTAL

Copper single crystals were prepared by theBridgman technique with a graphite crucible from

pure Cu with purity higher than 99.99%. Seed crys-tals with a size of 2� 2� 30 mm3 were carefully cut

from the single crystals by spark erosion. Copperbicrystals containing a grain boundary perpendicu-lar to the [100] twist axis with misorientation angles

of 15, 20, 23 (S13), 25, 28 (S17), 32, 37 (S5), 40, 44(S29) and 458 were prepared by the Bridgman tech-nique using the seed crystals. The misorientation is

de®ned as the angle y between the [001] directionsof the two single crystal grains on both sides of theboundary. After the preparation of the bicrystals,

the misorientation angles were checked by the Laueback re¯ection technique. The bicrystals with misor-ientation deviations greater than 28 were discarded.Rectangular specimens with a size of 3� 4� 6 mm3

were cut from the bicrystals by spark erosion. In allthe specimens, a grain boundary is located at thecenter of and perpendicular to the longitudinal

direction. A surface with an area of 3� 6 mm2 ofeach specimen was ®nished on 4000 emery paperand then chemically etched in an etchant consisting

of 20 vol.% of nitric acid and 80 vol.% of distilledwater. Each chemically etched specimen was separ-ately encapsulated in an evacuated silica tube

together with a Cu±35 wt% Zn alloy, a thin foil ofpure tantalum and a piece of graphite with highpurity, and then annealed at 693 K for varioustimes between 5.4� 104 and 1.73�105 s. The

amount of the Cu±35 wt% Zn alloy used as a Znsource was more than ten times greater than that ofthe specimen in each capsule. After the annealing,

the chemically etched surface of each specimen wasmechanically polished using diamond with a size of1 mm and then chemically etched in an etchant con-

sisting of 15 ml of distilled water, 4 ml of 25% aquaammonia, 0.5 ml of 3% hydrogen peroxide and 10drops of lactic acid. The chemical etching was car-ried out for 30±60 s using the freshly prepared etch-

ant. The microstructure was observed with anoptical microscope at various depths below the sur-face for each specimen. The depth was determined

using the Vickers hardness indentation method uti-lized by Gust and co-workers [7, 8]. According tothis method, the depth z below the surface is deter-

mined by

z � 0:143�d1 ÿ d2�; �1�where d1 and d2 are the diagonal lengths of a

Vickers hardness indentation on the specimen sur-face before and after polishing, respectively. Thechemical composition of the region alloyed with Zn

behind the moving grain boundary was determinedby electron probe microanalysis (EPMA).

3. RESULTS AND DISCUSSION

On the surfaces of the annealed specimens, ®negrains alloyed with Zn were observed to form dueto DIR. The thickness l of the ®ne-grain DIRregion increases with increasing annealing time t.

YAMAMOTO et al.: KINETICS OF DIFFUSION1758

The average values of l were 15, 20 and 23 mm at

t � 5:4� 104, 1.08�105 and 1.73�105 s, respect-ively. Just below the DIR region, DIGM was found

to take place. Removing the DIR region from the

surface, the microstructure was observed with anoptical microscope. Typical cross sections parallel

to the surfaces with an area of 3� 6 mm2 are indi-cated on Fig. 1. Figures 1(a)±(c) show optical

micrographs of the specimens with a misorientationangle of y � 238 (S13) annealed at 693 K for

5.4�104, 1.08� 105 and 1.73�105 s, respectively, at

depths of z= 19±26 mm. In these ®gures, straighthorizontal lines correspond to the initial positions

of the moving grain boundaries, whereas wavycurves the ®nal positions of the grain boundaries.

The zone between the wavy curve and the horizon-

tal line is the region alloyed with Zn due to DIGM.

This region is hereafter called the DIGM region.

During DIGM, the grain boundary migrated rather

unidirectionally in most of the specimens within the

experimental annealing times. The concentration of

Zn in the DIGM region just below the DIR region

was observed to be around 1 wt% for most of the

specimens annealed at 693 K for 1.08�105 s. This

value is close to the Zn concentrations for the

DIGM regions in the Cu bicrystals with [011] sym-

metric tilt boundaries zinci®ed at 693 K for

1.74�106 s [7, 8], whereas smaller than those of 2±

8 wt% for the DIGM regions in the Cu bicrystals

with [001] symmetric tilt boundaries zinci®ed at

673 K for 1.73�105 s [3].

Fig. 1. Optical micrographs of cross sections parallel to the surfaces of the specimens with a misorienta-tion angle of y � 238 (S13) annealed at 693 K for (a) 5.4� 104, (b) 1.08� 105 and (c) 1.73� 105 s. The

depth z below the specimen surface is (a) 20, (b) 19 and (c) 26 mm.

YAMAMOTO et al.: KINETICS OF DIFFUSION 1759

According to the microstructures in Figs 1(a)±(c),

the migration distance of the moving grain bound-

ary increases with increasing annealing time. The

distances of the peak positions of the wavy curve

from the horizontal line were measured at more

than 20 positions for a determination of the average

value. The measurements were carried out at var-

ious depths below the surface for each specimen.

Typical results of the measurements are indicated as

open symbols in Fig. 2. In Figs 2(a) and (b), the

average migration distance w is plotted against the

depth z for the specimens with misorientation

angles of y � 28 (S17) and 448 (S29), respectively,annealed at 693 K for various times. According to

the results in these ®gures, the migration distance

monotonically decreases with increasing depth at

each annealing time whereas it increases with

increasing annealing time at a constant depth.

Furthermore, most of the experimental points exhi-

bit that the moving grain boundaries are slightly

curved with curvature convex towards the untrans-

formed matrix. Such curvature may be expressed

by [8]

w ��������������a� bzp

; �2�where a and b are ®tting parameters. Values of the

®tting parameters were obtained via a least-squaresmethod from the experimental points in Fig. 2. Using

these values of a and b, the migration distance w was

calculated as a function of the depth z. The resultsare shown as solid and dashed curves in Figs 2(a)

and (b). As can be seen, the experimental points at

each annealing time are well located on the corre-sponding solid curve within experimental uncer-

tainty. From the solid or dashed curve, the migration

distance was evaluated for various depths at eachannealing time. The evaluated values of w for the spe-

cimen in Fig. 2(a) are plotted against the annealing

time t as open symbols in Fig. 3. As shown in this®gure, the experimental points lie well on a set of par-

allel straight lines for each specimen. As the slope of

the line dw=dt is the migration rate v of the movinggrain boundary, the latter is constant regardless of

the chosen annealing times t and the depth z. From

the experimental points in Fig. 3, a value ofv � 1:28� 10ÿ1024:9� 10ÿ12 m=s was obtained for

the specimen with y � 288 (S17) by a least-squares

method. In a similar manner, values of v � 1:05�10ÿ10 27:5 � 10ÿ12; 6:02 � 10ÿ11 2 4:3 � 10ÿ13,2:98� 10ÿ1121:5� 10ÿ12 and 8:30� 10ÿ1123:5�10ÿ12 m=s were evaluated for y � 23 (S13), 37 (S5),40 and 448 (S29), respectively.At the beginning of the reaction, the grain

boundary is located at the initial position and thus

the migration distance w ought to be zero.However, if the straight lines are extrapolated to

t � 0 s in Fig. 3, we obtain positive values of w.

Consequently, it is concluded that the migrationrate should be higher at the early stages of the reac-

tion than at the steady state stage and will decrease

with increasing annealing time. After a certain

Fig. 2. The migration distance w vs the depth z below thespecimen surface for the [100] twist grain boundaries(GBs) with misorientation angles of (a) 28 (S17) and (b)448 (S29) due to DIGM during annealing at 693 K for

various times.

Fig. 3. The migration distance w vs the annealing time tfor the [100] twist grain boundary (GB) with a misorienta-tion angle of 288 (S17) at various depths below the speci-

men surface due to DIGM at 693 K.

YAMAMOTO et al.: KINETICS OF DIFFUSION1760

annealing time shorter than 5.4� 104 s, the mi-gration rate reaches a constant value. Such anneal-

ing time dependence of the migration rate due toDIGM was also reported by Gust and co-workers [7, 8] for the [011] symmetric tilt boundaries

in the Cu(Zn) system. On the other hand, accordingto the observations by King and co-workers [2±4],the migration rate monotonically decreases with

increasing annealing time during DIGM in the caseof the [001] symmetric tilt boundaries.The misorientation angle dependence of the mi-

gration distance w of DIGM for the [100] twistboundaries reported in a previous paper [9] is rep-resented as open circles in Fig. 4. The results in this®gure were obtained at a depth of z � 20 mm for

the specimens annealed at 693 K for 5.4� 104 s. Inthe case of the boundary with y � 458, the mi-gration distance was negligibly small even at the

longest annealing time of 1.73�105 s and thuscould not be determined with su�cient accuracywith an optical microscope. Hence, the experimental

point of this boundary is omitted in Fig. 4. In this®gure, the relationship between the boundaryenergy gB and the misorientation angle y deter-

mined by Mori et al. [11] for the [100] twist bound-aries in Cu from the shape of boundary SiO2

particles is also shown as open squares. The ordi-nate on the right-hand side of Fig. 4 indicates the

ratio of the boundary energy gB of Cu to the inter-face energy gI between Cu and SiO2. Since there isno anisotropy and thus no misorientation angle

dependence in the interface energy gI, the gB/gI±ydiagram exhibits the same type of misorientationangle dependence as the gB±y diagram. Therefore,

the gB/gI±y diagram is hereafter merely referred toas the gB±y diagram. According to the gB±y dia-gram in Fig. 4, clear cusps exist at y � 23 (S13), 28

(S17), 37 (S5) and 448 (S29). This means that thesecoincidence site lattice (CSL) boundaries are low-

energy boundaries with less disordered atomicarrangements whereas the boundaries with y � 15,20, 25, 32 and 408 are high-energy boundaries with

disordered atomic arrangements. Such a high-energy disordered boundary is hereafter called arandom boundary. As can be seen in Fig. 4, clear

cusps appear at y � 23 (S13), 28 (S17), 37 (S5) and448 (S29) also in the w±y diagram. Hence, it is con-cluded that a random boundary with high energy

migrates fast whereas a less-disordered CSL bound-ary with low energy moves sluggishly. However,there is an exception that the boundary energy isvery large at y � 168 whereas the migration distance

is very small at y � 158 according to the results inFig. 4. In order to ®nd the reason why the 158boundary cannot move fast though this boundary is

considered to be a random boundary, more exper-imental information is needed.As already shown in Fig. 3, the migration rate v

is kept constant independent of the annealing time tand the depth z within the experimental annealingtimes for most of the specimens. The values of v at

this steady state stage are plotted against the misor-ientation angle y in Fig. 5. In this ®gure, the resultsfor the specimens with y � 23 (S13), 28 (S17), 37(S5), 40 and 448 (S29) are shown as open circles.

At early stages of the reaction, the less-disorderedCSL boundaries migrate slowly whereas the randomboundaries move fast as mentioned above. Such a

relationship may be expected to hold good also atthe steady state stage. However, the steady-state mi-gration rate is smaller for the random boundary

with y � 408 than for the CSL boundaries with y �23 (S13), 28 (S17), 37 (S5) and 448 (S29) as can beseen in Fig. 5. At very early stages of the reaction,the moving boundary is considered to be almost ¯at

and thus its character may be maintained invariableduring DIGM. Therefore, at the early stages therandom boundaries migrate fast whereas the CSL

boundaries move slowly. As the annealing timeincreases, the moving boundary becomes wavy asshown in Fig. 1. The curvature of the boundary

causes a pulling-back force due to the boundaryenergy and thus decelerates the migration. The fas-ter the boundary migrates, the earlier the boundary

becomes wavy. Therefore, the deceleration occursmore remarkably for faster moving boundaries.This may be the reason why the random boundariesmigrate faster at the early stages whereas at the

steady state stage they move slower than the lessdisordered CSL boundaries.As mentioned earlier, the boundary tends to

migrate unidirectionally during DIGM in most ofthe specimens. Such unidirectional boundary mi-gration was also observed by King and co-

workers [2±4] for DIGM in the Cu(Zn) systemusing the Cu bicrystals with [001] symmetric tiltboundaries prepared by the Bridgman technique.

Fig. 4. The migration distance w vs the misorientationangle y for the [100] twist grain boundaries (GBs) at adepth of z � 20 mm due to DIGM during annealing at693 K for 5.4� 104 s reported in a previous paper [9]. Themisorientation angle dependence of the boundary energyfor Cu determined by Mori et al. [11] is also shown as

open squares.

YAMAMOTO et al.: KINETICS OF DIFFUSION 1761

On the other hand, Gust and co-workers [7, 8] uti-

lized a di�usion bonding technique to prepare theCu bicrystals with [011] symmetric tilt boundariesand observed bi-directional boundary migrationowing to DIGM in the Cu(Zn) system. If the

bicrystal is completely symmetric, the boundaryshould move towards both side crystal grains withequal probability. From this point of view, the uni-

directional boundary migration may be due to asmall degree of asymmetry of the bicrystal aspointed out by Gust and co-workers. In order to

draw any conclusions, more detailed experimentalinformation is needed.

4. ANALYSIS

4.1. Energy balance model

Many investigators assume that the compo-sitional discontinuity between the regions aheadand behind the moving boundary, i.e. the chemical

driving force, provides the most important drivingforce for DIGM [12±16]. On the basis of this idea,Hillert and Purdy proposed a chemical driving

force (CDF) model [13]. However, there are severalpossibilities to dissipate the chemical drivingforce [17±19]. For instance, if the migration rate ofthe moving boundary is su�ciently small, solute

atoms will penetrate into the untransformed matrixahead of the moving boundary. Such a process maycompletely consume the driving force [17]. This can

be avoided by coherency stresses building up in thepenetration zone due to the compositional depen-dence of the lattice parameter for the matrix [18].

In order to obtain the driving force under suchconditions, a coherency strain (CS) model wasproposed by Hillert [12]. The CS model is valid

at small migration rates, whereas the CDF modelbecomes applicable at large migration rates.However, there is no quantitative criterion thatdetermines the lower limit of the migration rate for

the CDF model and the upper limit of that for the

CS model. Recently, an energy balance (EB) model

was proposed by Kajihara and Gust [10] to evaluate

the driving force for DIGM during alloying in

binary systems as a function of the migration rate.

This model is applicable to a rather broad range of

migration rates and provides the upper and lower

limits of the migration rate for the CS and

CDF models, respectively. The EB model will be

explained brie¯y below.

In the EB model, the following assumptions have

been adopted: (a) a grain boundary is a thermo-

dynamic phase with a thickness d and its own molar

Gibbs energy; (b) local equilibrium is established

at the interface between the grain boundary and

the matrix at each moment during the boundary

migration; (c) the di�usion coe�cients of the solute

and solvent atoms in the boundary are almost the

same. Let us consider DIGM in the A(B) system,

where elements A and B indicate the solvent and

solute, respectively. If one mole of the DIGM region

with composition x cB is produced from �x c

B ÿ x 0B�=

�1ÿ x 0B� moles of the solute atoms and �1ÿ x c

B�=�1ÿ x 0

B� moles of the untransformed matrix with

composition x 0B, the molar Gibbs energy change

DcGm due to this reaction can be calculated as

DcGm�x cB� �

1

1ÿ x 0B

f�1ÿ x cB�G a

m�x 0B�

� �x cB ÿ x 0

B�G aB�x c

B�g ÿ G am�x c

B� �3�according to the EB model. Here, the composition xBshows the mole fraction of the solute. G a

m�x 0B� and

G am�x c

B� are the molar Gibbs energies of the untrans-

formed matrix and the DIGM region, respectively,

and G aB�x c

B� is the chemical potential of the solute in

the DIGM region. The molar Gibbs energy change

DcGm is considered to be the molar chemical driving

energy for this reaction. Equation (3) indicates that

DcGm takes a positive value when the reaction occurs

spontaneously. In the case of DIGM, there are four

kinds of possibilities to dissipate the molar chemical

driving energy. The ®rst one is the consumption of

the energy DvdGm due to volume di�usion of the

solute in the penetration zone ahead of the moving

boundary. The second and third ones are the intrinsic

and solute drag energies, DidGm and DsdGm, respect-

ively. The last one is the boundary energy contri-

bution DcvGm due to the curvature of the moving

boundary. If the boundary migrates with a constant

steady state rate, the following relationship holds

among these types of energies:

DcGm ÿ �DvdGm � DidGm � DsdGm � DcvGm� � 0: �4�All the terms in parentheses in equation (4) are

drag energies and take positive values like DcGm.

The EB model gives the following mathematical

expression of DvdGm for a constant migration rate

of the boundary

Fig. 5. The migration rate v at the steady state stage vsthe misorientation angle y for the [100] twist grain bound-

aries (GBs) due to DIGM at 693 K.

YAMAMOTO et al.: KINETICS OF DIFFUSION1762

DvdGm�x cB� �

�1ÿ x cB�

�1ÿ xnfB ��1ÿ x 0

B�f�1ÿ xnf

B �G am�x 0

B�

� �xnfB ÿ x 0

B�sG aB�xnf

B �

ÿ �1ÿ x 0B�sG a

m�xnfB �g: �5�

Here sG am�xnf

B � and sG aB�xnf

B � are the molar Gibbs

energy and the chemical potential of the solute,respectively, in the thin ®lm with coherency stressformed in the penetration zone, and xnf

B is the

composition of this thin ®lm. For a steady stateboundary migration with a migration rate of v,the composition xnf

B is calculated as [17]

xnfB � x 0

B � �xpfB ÿ x 0

B�exp�ÿvl=D�; �6�if the volume di�usion coe�cient D of the solutein the penetration zone is constant independent of

the composition at the reaction temperature.Here, xpf

B is the composition in the penetrationzone at the interface between the boundary and

the untransformed matrix, and l is the intera-tomic distance. When the boundary migration iscontrolled by boundary di�usion along theboundary over long distances and the di�usion

coe�cients along and across the boundary arealmost the same, the composition xpf

B is deter-mined from the parallel-tangent construction

method [17]:

sG aB�xpf

B � ÿ G aB�x c

B� �sG aA�xpf

B � ÿ G aA�x c

B�: �7�

4.2. E�ective driving force

According to the kinetic analysis for DIGM in

binary systems using the EB model [10], the e�ec-tive molar driving energy DefGm is de®ned as

DefGm � DcGm ÿ DvdGm: �8�From equations (4) and (8), the following relation-

ship is obtained:

DefGm � DidGm � DsdGm � DcvGm: �9�equation (9) indicates that the e�ective molar driv-ing energy is ®nally consumed to keep the steady

state migration of the boundary against the intrinsicand solute drag-forces and the pulling-back forcedue to the boundary curvature.

The migration rate v of a moving boundary isusually related to the driving force DG by

v �MDG �M

�DGm

Vm

�; �10�

where M is the mobility of the moving boundaryand DG has the dimension of force per unit area orenergy per unit volume. DGm and Vm are the molar

driving energy and the molar volume, respectively.The mobility M may vary depending on the mi-gration rate v or the driving force DG, and thusequation (10) will result in a rather complicated

mathematical expression. Supposing that the mi-gration rate v is correlated with the e�ective molar

driving energy DefGm by the same formula asequation (10), we obtain the following relationship:

v �MDefGm

Vm

!�MDefG: �11�

The term DefG de®ned as DefG � DefGm=Vm is here-after called the e�ective driving force.

4.3. Thermodynamic functions

For a substitutional solution a phase in thebinary A±B system, the molar Gibbs energy G a

m is

usually expressed as

G am � xA

oG aA � xB

oG aB � RT�xA ln xA

� xB ln xB� �E G am: �12�

Here oG ai (i � A,B) is the molar Gibbs energy of

each pure element, and EG am is the excess molar

Gibbs energy. According to the subregular solution

model by Hillert and WaldenstroÈ m [20], EG am is

expressed as

EG am � xAxBf0La

A,B �1LaA,B�xA ÿ xB�

�2LaA,B�xA ÿ xB�2g; �13�

where the parameter nLaA,B (n � 0,1,2) represents

a thermodynamic interaction between elementsA and B.The molar Gibbs energy sG a

m of the penetrationzone is described by

sG am � G a

m �el G am; �14�

where elG am is the coherency strain energy. If there

is anisotropy in the elasticity of the a phase, thecoherency strain energy elG a

m is obtained by the fol-

lowing equation [21, 22]:

elG am � YVmZ2�xB ÿ x 0

B�2: �15�Here Y is the biaxial elastic modulus of the a phasealong a plane parallel to the grain boundary, and Zis the mis®t parameter. For cubic crystals like face-centered cubic (f.c.c.) Cu, Y is expressed as [22]

Y � c11 � 2c122

��3ÿ c11 � 2c12

c12 � c44

��

3�Bÿ 1�2c� 2o �Bÿ 1�f� o 2

�Bÿ 1�2�B� 2�c� o �B2 ÿ 1�f� o 2�B� o �

���16�

in which

c � l2m2n2

YAMAMOTO et al.: KINETICS OF DIFFUSION 1763

f � l2m2 �m2n2 � n2l2

B � �c11 ÿ c44�=�c12 � c44�and

o � c44=�c12 � c44�:Here c11, c12 and c44 are the elastic constants, and l,m and n are the direction cosines of the unit vector

normal to the grain boundary. For the [100] twistboundaries, l � 1 andm � n � 0, and thus c � f � 0independent of the misorientation angle y. Con-

sequently, equation (16) reduces to

Y � �c11 � 2c12��1ÿ c12

c11

�: �17�

equation (17) indicates that no misorientation angle

dependence of the elastic property exists in bothside crystal grains along the planes parallel to theboundary in the case of the f.c.c.-Cu bicrystals with

the [100] twist boundaries.

4.4. Migration rate dependence of e�ective drivingforce

In the experiment of the present work, the mi-gration rate v was found to be constant independentof the annealing time t and the depth z within the

experimental annealing times. At this steady statestage, equation (4) holds good and thus the EBmodel becomes valid. Hence, the experimental

results were quantitatively analyzed using the EBmodel. The analysis gives the relationship betweenthe e�ective driving force DefG and the migration

rate v at the steady state stage.According to the observations, the mole fraction

x cZn of Zn in the DIGM region just below the DIR

region is around 0.01 for most of the specimens

annealed for 1.08�105 s. The mole fraction x cZn

will decrease with increasing depth z and maychange with annealing time t. However, such

changes in x cZn could not be determined with su�-

cient accuracy by EPMA due to the small values ofx cZn. In the present analysis, attention was focused

on the relationship between DefG and v for a givenvalue of x c

Zn. Therefore, the e�ective driving forceDefG was calculated as a function of the migrationrate v using a typical experimental value of x c

Zn by

equations (3), (5)±(8), (12)±(15) and (17). The calcu-lation was carried out choosing a value ofx cZn � 0:0089, which was obtained for the specimen

with y � 408 annealed at 693 K for 1.08�105 s. Thefollowing parameters were adopted for the calcu-lation: c11 � 168:4 and c12 � 121:4 MPa [23];

Vm � 7:2� 10ÿ6 m3=mol [24]; Z � 0:05809 [25];0LCu,Zn � ÿ41661:1� 12:83485T, 1LCu,Zn � ÿ6160:7ÿ2:9274T and 2LCu,Zn � 14034:4ÿ 7:30663T J=mol

[26]; D0 � 3:4� 10ÿ5 m2=s and Q � 191 kJ=mol forD � D0 exp�ÿQ=RT � [27]; l � 0:3 nm; and x 0

Zn � 0.The result is shown as a solid curve in Fig. 6. As canbe seen in this ®gure, the e�ective driving force DefG

monotonically increases with increasing migration

rate v and then reaches a maximum value of

DefG max � 7:24 MPa at v > 4� 10ÿ9 m=s. The maxi-

mum value DefG max is obtained by inserting

DvdGm � 0 into equation (8) and thus coincides with

the chemical driving force DcGm=Vm. This means

that the whole chemical driving force is e�ectively

utilized for DIGM at migration rates higher than

4�10ÿ9 m/s. Consequently, the value v �4� 10ÿ9 m=s is the lower limit of the migration rate

for the CDF model under the present conditions. On

the other hand, DefG takes the minimum value of

DefG min � 30:2 kPa at v<10ÿ14 m=s. At such small

migration rates, most of the chemical driving force is

consumed by the volume di�usion of the solute in

the penetration zone ahead of the moving boundary.

Merely 0.4% of the chemical driving force remains as

the e�ective driving force at v<10ÿ14 m=s. The value

v � 10ÿ14 m=s is the upper limit of the migration rate

for the CS model.

In the present analysis, the biaxial elastic mod-

ulus Y was evaluated from equation (17). This

implies that the moving boundary is assumed to be

¯at and parallel to the initial boundary plane.

However, the moving boundary is actually wavy

during DIGM at the steady state stage as shown in

Fig. 1. The migration rate v was obtained from the

average migration distance w for the peak positions

of the wavy moving boundary. At the peak pos-

itions, the moving boundary is considered to be

locally ¯at and almost parallel to the initial bound-

ary plane. Therefore, the evaluation of the elastic

modulus Y by equation (17) may be approximately

valid.

The experimental results in Fig. 5 indicate that

the migration rate takes values of v � 1:05� 10ÿ10,1.28�10ÿ10, 6.02� 10ÿ11, 2.98�10ÿ11 and

8.30�10ÿ11 m/s for the specimens with y � 23

(S13), 28 (S17), 37 (S5), 40 and 448 (S29), respect-

Fig. 6. The e�ective driving force DefG as a function ofthe migration rate v for the [100] twist grain boundaries(GBs) calculated for DIGM at 693 K. The results corre-sponding to the experimental migration rates are shown as

open symbols.

YAMAMOTO et al.: KINETICS OF DIFFUSION1764

ively, at the steady state stage. These values of v aremore than one order of magnitude smaller than the

lower limit for the CDF model, 4�10ÿ9 m/s.However, they are more than three orders of mag-nitude higher than the upper limit for the CS

model, 10ÿ14 m/s. At such intermediate migrationrates, the e�ective driving force changes remarkablydepending on the migration rate. From the solid

curve in Fig. 6, values of DefG � 1:53, 1.83, 0.94,0.50 and 1.25 MPa are obtained as the e�ectivedriving force at the experimental migration rates for

the specimens with y � 23 (S13), 28 (S17), 37 (S5),40 and 448 (S29), respectively. These values are alsoshown as open symbols in Fig. 6. According to theresults in this ®gure, DefG becomes 7±25% of

DefG max at the experimental migration rates. Thus,it is concluded that 75±93% of the chemical drivingforce is consumed by volume di�usion in the pen-

etration zone ahead of the moving boundary underthe experimental conditions.

4.5. Mobility of moving grain boundary

As mentioned earlier, the migration rate v maybe correlated with the e�ective driving force DefGby equation (11). Using this equation, the mobility

M was calculated from the values of v and DefGshown as the open symbols in Figs 5 and 6, respect-ively. The results are plotted against the misorienta-

tion angle y as open circles in Fig. 7. As can beseen in this ®gure, the mobility is almost constantindependent of the misorientation angle. The

calculation gives the average value of M �6:58� 10ÿ17 m4=J s. At very early stages of thereaction, the moving boundary is considered to bealmost ¯at as mentioned above. For such a ¯at

boundary, the atomic arrangement along theboundary and thus the character of the boundary isdetermined by the misorientation angle. However, if

a CSL boundary is curved, the atomic arrangementalong the boundary will become disordered. At the

steady state stage, the moving boundary becomeswavy and thus is considered to be a highly disor-dered random boundary even for a CSL boundary

with low energy at the beginning. Therefore, theaverage value M � 6:58� 10ÿ17 m4=J s indicates themobility of the random boundary for DIGM in the

Cu(Zn) system at 693 K.If DefG max and DefG min are used instead of DefG

for the calculation of the mobility M by equation

(11), we obtain the results shown as open squaresand triangles, respectively, in Fig. 7. Unlike thecalculation with DefG, misorientation angle depen-dencies appear in both the M±y diagrams obtained

from DefG max and DefG min as can be seen in this®gure. Since DefG max and DefG min are constant, themisorientation angle dependence of the v±ydiagram in Fig. 5 is re¯ected in these M±y diag-rams. According to the results in Fig. 7, thecalculation with DefG max gives M � 4:1�10ÿ18±1:8� 10ÿ17 m4=J s, whereas that withDefG min M � 9:9� 10ÿ16±4:3� 10ÿ15 m4=J s. In anevaluation of the driving force for DIGM, only the

chemical driving force or the coherency strainenergy may sometimes be considered. However, it isvery important for the evaluation to take the mi-gration rate dependence of the energy consumption

into consideration. Otherwise, we may obtain anoverestimated or an underestimated result.

5. CONCLUSIONS

The migration behavior of [100] twist boundariesdue to DIGM in the Cu(Zn) system was experimen-tally studied using Cu bicrystals with misorientationangles of 15, 20, 23 (S13), 25, 28 (S17), 32, 37 (S5),40, 44 (S29) and 458. The bicrystal specimens wereannealed at 693 K for various times between5.4�104 and 1.73� 105 s. On the surfaces of the

annealed specimens, the Zn-rich ®ne-grain regionwith thicknesses of 19±26 mm were found to formdue to DIR. Just below the ®ne-grain DIR region,

DIGM was clearly observed to occur except for the458 boundary. The grain boundary migrates ratherunidirectionally during DIGM for most of the spe-cimens. The migration distance at 5.4�104 s is

smaller for the 23 (S13), 28 (S17), 37 (S5) and 448(S29) CSL boundaries with low energy than for the20, 25, 32 and 408 random boundaries with high

energy. However, in the case of the 158 boundary,the migration distance is very small at this anneal-ing time, though the boundary energy is very large.

Within the experimentally covered annealing times,the migration rate at each misorientation angle isabout constant independent of the annealing time.

At this steady state stage, the migration rate ishigher for the low-energy CSL boundaries than forthe high-energy random boundaries. Consideringthe migration rate dependence of the e�ective driv-

Fig. 7. The mobility M vs the misorientation angle y forthe [100] twist grain boundaries (GBs) due to DIGM at693 K. The results calculated using DefG, DefG max andDefG min are shown as open circles, squares and triangles,

respectively.

YAMAMOTO et al.: KINETICS OF DIFFUSION 1765

ing force, the experimental results were quantitat-ively analyzed using the energy balance model pro-

posed by Kajihara and Gust [10]. The analysisindicates that 75±93% of the chemical driving forceis consumed by the volume di�usion of Zn in the

untransformed Cu matrix ahead of the movingboundary. The analysis also gives the mobility ofthe random boundary for DIGM under the present

experimental conditions.

AcknowledgementsÐThe present work was partially sup-ported by a Grant-in-Aid for Scienti®c Research from theMinistry of Education, Science, Sports and Culture ofJapan.

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YAMAMOTO et al.: KINETICS OF DIFFUSION1766