The role of grain-boundary disorder in the generation and growth of antiphase domains during recrystallization of cold-rolled Cu3Au

Acta ma&~. Vol. 44, No. 9, pp. 3869-3880, 1996 Copyright 0 1996 Acta Metatlurgrca Inc.

1359-6454(95)00445-9

Published by Elsevier Science Ltd Printed

in Great

Britain. All rights reserved 1359-6454/96 $15.00 + 0.00

THE ROLE OF GRAIN-BOUNDARY DISORDER IN THE GENERATION AND GROWTH OF ANTIPHASE DOMAINS DURING RECRYSTALLIZATION OF COLD-ROLLED Cu3Au R. YANG?, Department

of Materials

G. A. BOTTON

and R. W. CAHNS

Science and Metallurgy, University of Cambridge, Cambridge CB2 342, England

Pembroke

Street,

(Received 5 August 1995) Abstract-An experimental study by transmission electron microscopy was made of the morphology of the antiphase domains formed when heavily rolled &Au is annealed at a temperature slightly below the critical temperature for ordering, T,. Domains are formed at the advancing grain boundary with extremely small size and they grow as recrystallization proceeds. From an early stage, domain walls show a preference for { 100) orientations. Diffraction experiments using a 1 nm probe on a scanning transmission electron microscope were conducted on a grain boundary 8.5” off the 23 coincident site lattice orientation. The results show that the superlattice reflection near to the boundary is markedly weaker than that away from it, suggesting the existence of an atomically disordered grain boundary zone l-2 nm thick. A theory was constructed for the genesis and growth of domains during recrystallization, taking into account the dragging pressure which newly formed domains exert upon a moving grain boundary, thereby diminishing the effective driving pressure for grain-boundary motion; a critical domain size is estimated which should completely inhibit grain-boundary motion. The intriguing fact that no domains at all are formed during the recrystallization of strongly ordered intermetalhcs such as N&Al is discussed and a reason is proposed. Copyright LO 1996 Acta Metallurgica Inc.

1. INTRODUCTION

It is firmly established order (LRO) inhibits subsequent annealed.

grain The

been

primary

growth

evidence

one of the authors has

that the presence when has with

a deformed

been

[l]. The most

performed

of long-range

recrystallization fully

detailed

CujAu,

and alloy

reviewed

is by

investigation which

has T,,

a

of 390°C. Hutchinson et al. [2] showed that for the heavily rolled alloys, the recrystallization rate just above T, is about 100 times faster than just below. A preliminary, unpublished investigation [3] showed that grains in deformed Cu,Au, recrystallized below 7’,, contain networks of antiphase domain boundaries (ABPs) on {loo}; some of the micrographs obtained were produced in Ref. [l]. It is also well established that strongly ordered phases such as N&Al (which has the same superlattice type as Cu,Au), when recrystallized, have grains without antiphase domain boundaries. The objective of the present research was to examine the mechanism of APB formation during recrystallization in Cu3Au below its T,, in the hope of getting some insight into the reasons for different behavior in different ordered phases. convenient

resent FP

critical

ordering

temperature,

address: Institute for Metals Research, Chinese Academy of Sciences, Shenyang, China. IAuthor to whom all correspondence should be addressed.

An associated, important question is whether the alloy has some degree of disorder at grain boundaries. When an (undeformed) alloy such as CU~AU is cooled from a high temperature, LRO and APBs form in the disordered phase as it cools through T,; Cahn et al. [4] have shown by experiments on off-stoichiometric Ni,Al of various compositions that APBs only form if T, is below freezing temperature, so that, during cooling, a disordered phase forms first and then orders. Extrapolating from this, it is a reasonable assumption that the antiphase domain array that is seen to form during recrystallization of deformed Cu,Au must originate in a disordered region, and the only place that this could be is at the boundary between an advancing new grain and the still-deformed structure. There has been a sharp flurry of research on the question of whether grain boundaries in (undeformed) boron-doped Ni,Al have a thin disordered zone, or phase: according to some investigators [5,6], for certain boundary types only, and compositions near stoichiometry in N&Al, such a disordered zone or “phase” exists in association with some nickel segregation stimulated by the presence of boron at the grain boundaries; others cannot find it (for example, Refs [7,8]). Some of the investigations have been reviewed by Baker [9], and the discrepancy in results has been discussed in terms of difference in experimental techniques and instrument sensitivity.

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GRAIN-BOUNDARY

There is no such experimental evidence relating to &Au, but a very recent computer simulation of grain boundaries for a C = 5 [OOl] twist boundary in this phase [lo] has predicted some composition change near the boundary and also a steep reduction of the degree of order over approximately 6 atomic layers at the grain boundary, in the temperature range T/T, = 0.880.9. In the present work, we have attempted to obtain some information on the state of atomic order in the vicinity of grain boundaries in Cu,Au, using a convergent beam electron diffraction (CBED) technique on a dedicated scanning transmission electron microscope (STEM).

DISORDER

allowing a sub-nanometer probe size. This instrument is equipped with a Gatan imaging filter (model UHV 678) allowing CBED patterns to be acquired on a charge-coupled device. Owing to limited tilting range of the specimen stage on the STEM, grain boundaries to be analysed were carefully chosen beforehand on the Philips CM30, to ensure that a favorable diffraction condition was obtainable and the boundaries could be oriented more or less edge-on when the specimen is transferred onto the STEM. Crystal orientations across the boundaries were determined from Kikuchi lines on pairs of CBED patterns, using an analysis similar to Ball’s [ 111. The grain-boundary plane normal was determined by orienting the boundary plane edge-on.

2. EXPERIMENTAL A button of exact composition CuiAu was made by repeated induction melting in a sealed silica crucible; the button was homogenised for 24h at 880°C followed by a quench. There was no appreciable weight loss during melting. Plates 2.2mm thick were sectioned and cold-rolled to 86% reduction, starting with the disordered state. Discs punched from the rolled sheet were annealed in sealed capsules at 380°C (about 10°C below T,) for up to 12 days. This annealing temperature was chosen because recrystallization at lower temperatures is impracticably slow. The Bragg order parameter just below T, is high, about 0.8. It has long been known that during electrolytic thinning of Cu,Au, irrespective of the choice of electrolyte, a layer of gold is redeposited on the thinned foil. Extensive experiments showed no way of avoiding this, but a short final period of ion-beam milling at glancing angle substantially removes the gold. The best quality was achieved with 8% perchloric acid in ethanol at 9OV and -5O”C, followed by ion-milling. Transmission electron microscopy (TEM) was performed on Philips EM400 and CM30 machines, at 100 and 300kV. STEM experiments were conducted on a VG HB501 microscope fitted with a field emission gun,

Fig. I. Antiphase

domain

morphology

3. OBSERVATIONS

3.1. Evolution time

of APD

morphology

with annealing

Specimens annealed for a long period often have stepped coherent (111) annealing twin boundaries [12]; detailed analysis showed that the domain boundaries are continuous across such an interface, in accordance with the findings of Tichelaar et al. [13]. This finding indicates that the domains were formed at an early stage of the recrystallization process, when the twins formed. Generally, domain boundries in the new grains were oriented approximately along { 100) planes. APDs form not only in the new grains but also in the (initially disordered) cold-worked grains. Figure 1 shows such APBs in a deformed grain after a 24 h anneal. The APBs are well defined and quite coarse but are not aligned along (100). Such APBs are often entangled with deformation dislocations in a complex manner, which probably has an effect on the rate of antiphase domain (APD) growth in the still-deformed grains. Figure 2 shows a key observation made during this research. In Fig. 2(a), relating to an anneal of 24 h

in the deformed

matrix

in a sample

annealed

for 24h at 380°C.

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DISORDER

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Fig. 2. (a) Recrystallized grains at an early stage of growth in a sample annealed for 24h at 380°C. (b) Dark-field micrograph showing APBs in the area framed in (a).

when recrystallization has just started, a number of small recrystallized grains are seen scattered in the deformed matrix. Most have annealing twins, which presumably formed at the nucleation stage: the key role of such twins in nucleating new grains in copper was established by Haasen and his coworkers [14]. Figure 2(b) shows, at higher magnification, the area delineated in Fig. 2(a). APD size is seen to increase continually from the grain boundary towards the grain interior. The antiphase domains near the grain boundary are very small, less than 5nm in diameter, whereas some of those in the interior of the new grain exceed 50 nm. A corresponding character can be seen in the new grains in Figs 3 and 4, in specimens given longer anneals (48 and 96h). These grains grew

respectively upwards and towards the right: the smallest antiphase domains are always behind the growing boundary. Many similar antiphase domain morphologies were seen in the new grains for various annealing times up to lOOh. The general conclusion is that in the early and intermediate stages of recrystallization, exceedingly small antiphase domains are nucleated at the growing boundary of the new grain, and these domains steadily coarsen as the growing boundary moves on. In samples annealed longer than lOOh, grainboundary motion slows down significantly. No tiny domains were observed at the boundary; antiphase domain size in the new grains becomes more uniform. After an anneal of 12 days, the samples have a

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Fig. 3. APBs in a growing

et al.:

GRAIN-BOUNDARY

recrystallized

recrystallized fraction of 0.95. For these samples, measurements of domain size were made in 14 grains, using lineal analysis on dark-field micrographs, with test lines along (100) and taking into account the

Fig. 4. APBs in a growing

recrystallized

DISORDER

grain in a sample

annealed

for 48h at 380°C.

existence of invisible APBs. No domain overlap was assumed since the domain size is much larger than the foil thickness (< 100nm). The average value of the domain size was found to be 294nm.

grain in a sample

annealed

for 96h at 380°C

YANG et al.:

GRAIN-BOUNDARY

3.2. The state of atomic order at the grain boundary 3.2.1. Grain-boundary geometry. Ideally, STEM studies of a boundary between a recrystallized and an unrecrystallized grain would better serve the purpose of the present work, since they provide the most direct information of local state of order at the recrystallization front. However, the large curvature at these boundaries makes it impossible to orient them edge-on, and the presence of dense networks of APBs due to overlapping tiny ordered domains complicates the interpretation of results. It was therefore decided to examine a better-equilibrated boundary between two recrystallized grains. A boundary with suitable orientation was found in a sample annealed for 48 h during the preliminary TEM survey, and was subjected to a detailed study on the STEM. Figure 5 shows this boundary in [OOl] orientation of the lower grain (grain 1). This boundary actually consists of several slightly misoriented facets, which becomes evident if one views the micrograph at a glancing angle, and the leftmost facet is edge-on in Fig. 5. The foil normal is at 11” with respect to [OOl] of grain 1, and, since the boundary runs approximately perpendicular to the edge of the perforation, the foil thickness increases continually from left to right on the micrograph. The orientation matrix across the boundary was determined as

i which

- 0.749229 -0.219229 0.624976

corresponds

0.550271 -0.731195 0.403183 to a rotation

0.368590 0.649582 0.668467

i

of 154.96” around

DISORDER

3873

[-0.286777, -0.302825, -0.9088761. The closest coincident site lattice (CSL) orientation is the 146.44” on (311) variant of the C3: the rotation axis is only 0.9” away from [113], while the rotation angle exceeds 146.44” by 8.5”. Since the trace of the leftmost microfacet of the boundary seen in Fig. 5 is inclined at 36” to [loo] of grain 1, the boundary normal is estimated as [sin36”,-cos36”,0], which is 2.3” away from [230]. This is far away from low-energy boundary orientations of a C3 CSL, (111) or 1211). 3.2.2. CBED experiments on STEM. In order to estimate experimentally the probe size, we made use of a slightly-twisted (100) APB as a reference (Fig. 6) to estimate the width of the sharpest feature in the image (e.g. Ref. [15]). For the experimental conditions used to obtain the image (the same as the patterns), this method provides an upper estimate of the probe size due to the finite width of the boundary. The projected domain-wall thickness at the crossover point of the twisted APB [as arrowed in Fig. 6(a)] can be resolved to within lnm [Fig. 6(b)]. Our estimation includes the thickness-dependent diffraction. The 110/220 reflections of grain 1 were used to probe the ordered state in the vicinity of the boundary under a strong two-beam condition. The tilting away from exact [OOl] to achieve such a two-beam condition causes the boundary microfacet which was originally edge-on to be slightly inclined to the electron beam. CBED patterns were recorded when the electron probe was positioned immediately adjacent to the boundary on the side of grain 1 (this is referred to as position B), as well as away from the

Fig. 5. Bright-field micrograph showing a grain boundary analysed. The lower grain (grain 1) is oriented exactly along [OOl].

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DISORDER

Fig. 6. (a) STEM image of a slightly-twisted (100) APB in grain 1 which was used to estimate image resolution on the STEM. The width of the thinnest part of its projection, as indicated by the arrow, can be clearly resolved down to lnm (b).

bounc iary in both grain 1 (position A) and grain 2 (posit ion C). A series of 30 recordings were made along the boundary. In thicker parts of the specimen, there was no visible difference between diffraction

patterns recorded at positions B and A, but the intensity ratio of 110/220 in general decreases with decrease in foil thickness. Figure 7 shows a group of diffraction patt .erns

YANG

et al.:

GRAIN-BOUNDARY

Figures

recorded at the thinnest part of the foil. It can be seen that the 110 superlattice reflection has a fairly strong intensity at position A [Fig. 7(a)], but is barely visible at position B [Fig. 7(b)] although the main reflection 220 is still fairly strong. Figure 7(c) shows a corresponding diffraction pattern in grain 2 across the boundary. The two weak discs (which can be discerned above and to the right of the 000 disc) seen in Fig. 7(c) are 201 and lil, their low intensity being due to the fact that grain 2 is at no particular

DISORDER

7(a and b).

diffracting orientation: its beam normal was close to [122] when grain 1 was at exact [OOl]; the beam normal is now near to [132] in grain 2, after tilting around 1 lo/220 axis of grain 1 to achieve a two-beam diffraction condition. These two reflections of grain 2 produced some diffuse intensity on the diffraction patterns from grain 1 recorded at positions A and B, as can be seen from Figs 7(a) and (b), respectively, suggesting that the path of the electron probe at these positions includes part of grain 2 (because the

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Figure

DISORDER

7(c).

Fig. 7. A group of CBED patterns recorded across the grain boundary. (a) About 2nm from the boundary in grain 1; (b) immediately adjacent to the boundary in grain 1; (c) about 5nm away from the boundary in grain 2.

boundary plane is not exactly edge-on). This accounted for the slight reduction of the 220 intensity seen in Fig. 7(b) relative to that seen in Fig. 7(a), because a larger volume of grain 2 contributes to diffraction when the electron beam is at position B than when it is at position A.

disordered. The observation of tiny ordered domains continually being generated from the grain boundary area suggests that the layer interface (with the ordered grain interior), in these samples at least, is not sharply defined. (The morphology of this interface can be seen below, in the schematic drawing of Fig. 8). A working thickness can be estimated by

4. INTERPRETATION

4.1. The disordered zone near the grain boundary The CBED experiments on STEM presented above clearly show a difference in the state of atomic order between regions near to and away from a grain boundary in CU~AU below its T,, in the form of an evidently weakened superlattice reflection near to the boundary. However, this superlattice reflection never completely disappears. The questions to be answered are: (1) is the zone adjacent to the boundary disordered or nearly disordered (i.e. ordered, but with a low value of LRO parameter), (2) is the interface between this zone and the grain interior sharp or difiuse, and (3) how thick is this zone, or layer? The deterioration of spatial resolution caused by electron-beam broadening and the slight inclination of the boundary plane with respect to the beam direction make it difficult to answer these questions in a definitive manner. Nevertheless, considering that the faint 110 reflection in Fig. 7(b) might be due to a neighbouring ordered region, we tentatively conclude that a layer adjacent to the boundary is

0

D

0

5nm

I

I

Fig. 8. Schematic diagram of grain boundary migration and of antiphase domain formation in a disordered grainboundary zone. The left-hand grain is recrystallized, the right-hand grain is the deformed matrix.

YANG et al.:

GRAIN-BOUNDARY

comparing diffraction patterns recorded at different positions. Because the diffraction pattern shown in Fig. 7(a), obtained at a position about 2nm from the boundary, is similar to those obtained far away from the boundary, we conclude that the disordered zone is no more than 2nm wide. Since the orientation relation between the two grains is far from any CSL orientations, the existence of a disordered zone seems to be in accordance with the general observation that a disordered layer is only seen at some high-angle boundaries [9]. 4.2. The formation

qf antiphase domains

Regarding the origin of the antiphase domains, one possible interpretation can be eliminated at once. In principle, if atoms move across a boundary in a coordinated manner, the domains present in a deformed grain (as in Fig. 1) could “imprint” themselves on a new grain nucleated within the deformed grain, creating domain walls continuous across a grain boundary. This possibility is excluded by the fact that the new domains at the growing boundary of a recrystallizing grain are much smaller than those already present in the deformed grain. The only alternative model is that antiphase domains are nucleated in the disordered zone which travels along with the advancing boundary between a new, recrystallized grain and the deformed matrix. Two questions then arise: (1) why do such boundaries move much more sluggishly (by a factor of about 100 times) just below T, than just above it, if the boundary zone itself is disordered both below and determines whether above T,? (2) What consideration domains form, or not, when one compares chemically different ordered phases? Both these questions can be answered by a calculation of energy balance involving domain boundaries. Figure 8 shows the geometrical features: a disordered zone travels along with an advancing grain boundary. On the deformed side, APBs are spaced at about 50nm (they are drawn much smaller than this); immediately behind the advancing boundary, where full order is being re-established, the APBs are spaced at less than 1Onm (say, 2nm). These tiny domains require an energy input because of the finite specific energy of the domain boundaries; the APB energy per unit volume behind the advancing boundary can easily be shown to be inversely proportional to the domain size and is thus more than 10 times that ahead of the boundary. This means that the pull forward on the boundary due to the domains destroyed in the deformed structure as it is consumed is small compared with the drag backwards from the fine domain structure immediately behind the boundary. (The fact that these fine domains subsequently coarsen does not affect the argument.) We will neglect the small forward pull. The backwards drag arising from the APB energy in newly formed fine domains can partially account for the observed sluggish motion of grain boundaries

DISORDER

3877

regions are below T,, even though these boundary disordered. We start from the familiar expression for grain-boundary velocity, v = Mp, where M is the mobility of the grain boundary block, or “atmosphere”, involving a disordered zone on either side of the boundary, and p is the driving “force”. p is the difference between a forward (true driving) pressure and a backwards, or drag, pressure. pfOrW= pGb’, where p is the dislocation density, G is the relevant shear modulus and b, the Burgers vector of the dislocations. The forward pull from the coarse APBs in the deformed structure is, as stated, ignored. The drag pressure pdrag= 3y/D, where ‘/ is the specific energy of the (100) domain boundaries and D is the mean domain diameter. When the two pressures are equal, the boundary is unable to advance; this happened for a critical domain size, D,. Allowing for the fact that the order parameter just below T, is only about 0.8, using a theoretical expression for y following the arguments outlined in Ref. [4] (the value of y for Cu,Au is calculated by this method to be about 28mJ/m’) and taking 4 x lO“MN/m* for G (from an old paper by Kbster [16]) and 10’hm-’ for p, one obtains a critical domain size, D,, of roughly 0.771.0nm. (The greatest uncertainty is in the value of p.) If domains larger than D, can be formed within the disordered grain boundary zone, then the boundary is free to migrate, with the production of a domain population. On the basis of our STEM experiments and Polatoglou’s simulation [lo], we can reasonably assume that the thickness of the disordered boundary layer is between 1 and 2nm. Since D, is smaller than this, domains can form and grow above this size still within the mostly disordered layer, boundary motion is possible and a domain population can form in the new grain, as observed. With recrystallization progressing, the forward driving force is expected to reduce gradually as the deformed microstructure recovers, leading to increasingly larger values of the critical domain size D,. When D, finally exceeds the thickness of the disordered zone, antiphase domains can no longer form within this zone. This happens at a late stage of recrystallization. Domains nucleated earlier would grow at the slowly moving boundary, leading to a much more uniform APD size distribution than during the early stage of recrystallization. This is consistent with our experimental observations. 4.3. Comparison

with NiiAl

In N&Al, the APB energy on { lOO} is much higher than for CuXAu; according to Liu [17], measured values are in the range 9&170mJ/m2; a mean value of 140mJ/m2 is about 5 times larger than the CmAu value. The drag force is therefore much larger than in Cu,Au. Given the same lattice type as CmAu, deformed NijA1 is expected to have a similar dislocation density, and D, works out at 335nm. This

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is so large that no domain of this size can form in the narrow disordered zone (about 2nm according to Ref. [6]) even at the beginning of recrystallization when D, is the smallest; the consequence would be that no stable array of domains can form at all and the boundary advances without domain formation. The above argument can explain why Ni,Al forms no antiphase domains during recrystallization, but there is still a problem concerning the recrystallization kinetics. This is another aspect of the behavior of Ni3Al, and also of another Liz compound, (Co,Fe),V which was investigated by Cahn et al. [18]. The form of (Co,Fe),V used in that study has a critical temperature of 910°C and is thus considerably more strongly ordered than Cu,Au. The factor of disparity between the recrystallization kinetics just above and just below T, is here as high as 300 times. In the case of Ni,Al, it is not possible to compare recrystallization kinetics above and below T,,because T,almost coincides with the melting temperature, but a critical comparison between the recrystallization kinetics of rolled NixAl and of rolled pure nickel [19] has indicated quite clearly that the recrystallization of N&Al is also substantially retarded by the presence of LRO. The problem is then that neither N&Al nor indeed (Co,Fe),V contains any antiphase domains after recrystallization, and so the retarding mechanism for grain-boundary migration based on antiphase domain drag cannot in fact apply. If all the grain boundaries in these alloys have a disordered zone, on the face of it they should migrate as freely as boundaries in the fully disordered alloy. The conclusion is that in these strongly ordered alloys and in contradiction to the weakly ordered CuiAu, either no boundaries at all, or else only a minority of the boundaries, have disordered zones, and so most boundaries are subject to the necessary slowing-down associated with the presence of LRO right up to the boundary plane and the consequent enhanced difficulty the two atomic species experience in migrating to their correct sites in the new lattice as a moving boundary passes by. If only some high-angle boundaries are disordered, as seems to be the case for Ni?Al [6,9], and the rest are not, then the majority of ordered boundaries should set the kinetics for the entire specimen.

DISORDER

present work, and to compare the computed results with experimental measurements. We make three simplifying assumptions: (1) the nucleation of new grains is site-saturated, i.e. all nucleation events take place at the beginning of recrystallization; (2) growth of grains is isotropic; and (3) the critical APD size D, as defined in Section 4.2, does not vary with time. Limitations of these assumptions will be discussed later. The theoretical grains are therefore spherical in shape, and they have to intersect each other when the recrystallization factor f approaches unity, in order that the total volume of new grains equals the specimen volume. The growth of these grains obeys experimentally measured kinetics. The number of grains is determined by the mean grain size atf= 1. Antiphase domain growth inside a new grain is illustrated in Fig. 9: we take the grain nucleus as containing one domain with size Da, and allow successive shells of domains with critical size D, to nucleate inside a disordered layer (not drawn in Fig. 9). Growth of these domains obeys the familiar parabolic law, with an experimentally measured time constant c, (here the subscript n stands for “new grain”). Consider that the ith shell of domains start to form at time r, and take time t, to complete. In accordance with the parabolic growth law, the domain size of different shells at time (r + t!) can be expressed as ,-I

Do = [(Dz + C,(T + t,)]“‘,

wherer

= 1 t,;

,=i

1:2

D, =

D,'+ c, i \

D,

=

k=,+l

tk

, j=ltoi-1;

1

(1)

/

D,.

5. ANTIPHASE DOMAIN GROWTH Having obtained a coherent picture of how antiphase domains were generated at advancing grain boundaries in Cu,Au, we now present a model of antiphase domain growth. Antiphase domain size in Cu,Au at various annealing times in the early stage of recrystallization was measured by Ward and Mikkola [20], using X-ray diffraction techniques. Our aim is to predict the average APD size at different stages of annealing, using a model based on the mechanism of APD formation established in the

Fig. 9. Diagram illustrating a model of antiphase domain growth.

YANG et al.:

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The time constant c, in the above equations has been experimentally determined as 339nm2/h at 375°C [21]. In the spherical grain model, recrystallization fraction f equals (r,/R0)3 at time (7 + t,), where Y, is the grain radius at time (7 + t,) and R. is the grain radius at f = 1. R, = 1500nm at 380°C [2]. Expressing f by the Avrami equation, one obtains Y,= R,( 1 - exp( - A(7 + I,)~o)),

CD,=r,. ,=I

(3)

Substituting equation (1) and (2) into equation (3), one can solve equation (3) for successive t, using a numerical procedure. The obtained t, can then be used to compute the domain size and the number of domains in each shell, and an average domain size in a new grain, D,. As described in Section 3.1, at the late stage of recrystallization, grain boundary motion becomes so slow that no domains can be generated from the disordered layer at the boundary; the only event in the new grain is APD growth. This fact is reflected in the model when equation (3) no longer has a solution for t,. From this point onwards, we allow domains of the inner shells in Fig. 9 to grow at the expense of the outermost shell. We can assume that equation (3) no longer has a solution for t, after the nth iteration, when thejth shell has a domain size D,,. As the mth shell of domains disappears (m is counted inwards from the grain boundary), equation (1) can be rewritten as I2

D, =

D;,, + c, i k=n+l

>

th

,

j=Oton--,andm=l and equation

ton

(4)

(3) now becomes

Do/Z+

c D,=r,,

m=

1 ton.

(5)

,=I

Equation solve for grain, D,,, D., of the

(4) can be substituted

into equation (5) to domain size in a new can be computed. The average domain size whole sample can be obtained from

tkr and the average

D, = jDn + (I -f)D,

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ICC0 0

(2)

where A and k. are constants in the Avrami equation, and can be obtained by fitting available experimental data [2]. It is clear from Fig. 9 that the sum of thicknesses of different domain shells equals the grain radius

0,/2+

DISORDER

(6)

where D,, is the domain size in the deformed matrix and obeys the parabolic growth law: its time constant c is unknown and is the only adjustable parameter in this treatment.

Annealing

time,

h

Fig. 10. Logarithmic plots of computed average domain size against annealing time, assuming a different time constant c (in nm’/h) for the deformed grains. X-ray results by Ward and Mikkola [20] and TEM measurements made in the present work are included for comparison.

Figure 10 shows the computed average domain D, plotted against annealing time, assuming different values for the time constant c of the deformed matrix. The effect of a domain size distribution in new grains is to slow down the increase of D, (because of the generation of tiny domains), causing the logarithmic plots to deviate from a straight line. This drop in D, recovers later on when domain nucleation is prohibited at grain boundaries. In Fig. 10, X-ray measurements by Ward and Mikkola [20] are also plotted along with a TEM result of the present work. Agreement between experimental data and theoretical value of D, is achieved by assuming a smaller time constant c in the deformed grains than in recrystallized grains (we use the time constant c,, measured at 375”C, 339nm’/h [21], for our 380°C case). This suggests that cold deformation hinders antiphase domain growth in CU~AU. Experimentally, this is an unresolved issue: while Ward and Mikkola [20] suggested that cold working accelerates APD growth, earlier measurements by Roessler et al. [22] concluded that the reverse is true. While the assumption of site-saturated nucleation prevents this treatment from obtaining highly polished results, we have shown that a straightforward model based on such an assumption seems to account for the main features of the experimental findings. The assumption of isotropic grain growth does not describe an individual grain correctly, since boundary motion during recrystallization of CulAu is highly anisotropic. When averaged over the whole sample, however, the effect of such an anisotropy on the average domain size is cancelled out. The use of a fixed critical domain size D, has caused a larger and sharper dip on the logarithmic plots in Fig. 10 (at around 100h). 6. CONCLUSIONS

A TEM study of antiphase in recrystallized CU~AU just

domain morphology below its r, reveals

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GRAIN-BOUNDARY

that domains are formed in the early stage of recrystallization, at the advancing grain boundary with extremely small size and grows as recrystallization proceeds. The observations indirectly suggest the existence of an atomically disordered grain boundary zone. Direct diffraction experiments on a STEM using a 1nm probe show that the superlattice reflection is markedly weaker in the vicinity of a well-characterized high-angle grain boundary than away from it. The results of the present work indicate that a disordered zone of about l-2nm thick exists at grain boundaries in Cu3Au below its T,. The drag due to the newly generated APDs are shown to be partially responsible for the remarkably slower recrystallization rate of CuzAu just below its T, than just above. A critical domain size, which should completely inhibit boundary motion, was estimated based on arguments of energy balance at advancing grain boundaries. The introduction of this concept explains why no tiny domains are nucleated at the late stage of recrystallization in Cu3Au, and why antiphase domains are not formed at all during recrystallization of Ni,Al. A model of antiphase domain growth, based on the mechanism of APD formation established in the present work, is presented. Comparison of the computed results with available experimental data suggests that plastic deformation retards domain growth in CuiAu.

Acknowledgements-We are grateful to the Royal Society for a grant and EPSRC for a ROPA award, to Professor C. J. Humphreys for the provision of laboratory facilities, and to Dr J. A. Leake for helpful discussions. We thank Dr P. Schumacher for providing assistance in melting the materials used in this work. R.Y. and G.A.B. are indebted to St. John’s and Darwin College, respectively, for the award of Research Fellowships,

DISORDER REFERENCES

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