High angle grain boundary diffusion of chromium in niobium bicrystals

PI1 S1359-6454(96)00016-X

Acra maw. Vol. 44, No. 9, pp. 3535-3541, 1996 Copweht _. - (0 1996 Acta Metallurgica Inc. Pulkhed by Elsevier Science Ltd Printed in Great Britain. All rights reserved

1359-6454/96 $15.00 + 0.00

HIGH ANGLE GRAIN BOUNDARY DIFFUSION CHROMIUM IN NIOBIUM BICRYSTALS

OF

X. M. LI and Y. T. CHOU Department

of Materials

Science and Engineering,

Lehigh

University,

Bethlehem,

PA 18015, U.S.A

(Received 9 August 1995; in revisedform 18 December 1995) Abstract-Grain boundary diffusion of Cr in [lOO]/[OOl]symmetrical Nb bicrystals with varying tilt angles, 0, was investigated in the temperature range of 958-l 197°C. For low angle boundaries, the grain boundary diffusion is dominated by the dislocation pipe mechanism, in which the activation energy Q’ is independent of 0 and the pre-exponential factor (sD’d)o varies linearly with sin(0/2). For high angle boundaries, the grain boundary diffusion coefficients, sD’6, fall into minima, whereas the activation energies Q’ attain

maxima at the CSL boundaries, 213 (22.6, 67.4’) Xl7 (28.1, 61.9’) and X5 (36.9, 53.1’). The pre-exponential factor (sD’6)0 also varies with 0, but the variation is random. The results of the study agree with those of the f.c.c. system, confirming that grain boundary diffusion is strongly boundary structure and associated energies. Cop_vright Q 1996 Acta Metallurgica Inc.

1. INTRODUCTION Grain boundaries play an important role in the processing and properties of polycrystalline materials, and a great number of studies have focused on understanding their structure. In f.c.c. metals, the grain boundary structure of bicrystals is fairly well understood from the consistency in experimental observations and theoretical predictions [ 11. However, in b.c.c. metals, ambiguity still exists [2,3]. Experimentally, an effective approach to exploring grain boundary structure is diffusion, although grain boundary diffusion itself is crucial to application of engineering materials; for example, the life prediction of a material part in service or the stability of a semiconductor device [4]. The measurement and analysis of grain boundary diffusion is well documented. Fisher first proposed his grain boundary diffusion model in 1951, assuming that the boundary was a high diffusivity thin slab embedded in a low diffusivity semi-infinite matrix [5]. However, his solution for concentration distribution was approximate, and, subsequently, Whipple [6] and Suzuoka [7] provided more exact solutions for constant and instantaneous source concentrations. These exact solutions are mathematically complex and not useful to experimentalists. A simplified expression was later derived by Le Claire [8], and has been successfully used for the experimental evaluation of grain boundary diffusion coefficients. The experimental verification of the slab grain boundary model started with the studies of selfdiffusion in Ag [9]. Subsequently self- and heterodiffusion using the well defined f.c.c. bicrystals

affected

by the

were conducted in Au, Al, Ni, Cu, Pb and Fe-Si systems [lo-141. More recently, the method of growing b.c.c. bicrystals has been developed [ 15,161, and facilitated studies of grain boundary diffusion in the b.c.c. system [17-191. Li and Chou [19] reported that the diffusion of Cr in Nb bicrystals with low angle symmetrical tilt boundaries (2-17”) was dominated by the dislocation-pipe mechanism [ 171. For high angle grain boundaries, Qian and Chou [ 181 measured the penetration depth of Cr in Nb bicrystals and found that the penetration depth increased with the tilt angle and with a minimum at the Cl3 boundary. In this paper, a detailed study of diffusion of Cr in Nb bicrystals with high-angle grain boundaries is presented in an attempt to provide experimental evidence for understanding the grain boundary structure and diffusion mechanism in b.c.c. metals.

3535

2. EXPERIMENTAL 2.1. Preparation

of Nb bicrystals

The Nb bicrystals were prepared by the floating zone melting method, in which each bicrystal was epitaxially grown from a bicrystal seed into locally molten Nb under a high vacuum of lo-” torr. Both Y-shaped seeds and matched-up seeds were used for this growth [ 15,161. The Y-shaped seed was made by splitting a single crystal rod along an assigned crystallographic plane into two arms of a Y shape. The matched-up seed was made by sectioning a single crystal rod along the required crystallographic plane into semi-cylindrical rods and then welding the

3536

LI and CHOU: Table Impurity !xrcentaee

Nitrogen 7LIlJ.m.

HIGH

1. Tvuical Oxygen 130D.D.rn.

ANGLE

imouritv

2.2. Preparation

content

Hydrogen 50.0.m.

sectioned pieces together in a high vacuum to form a bicrystal seed of a preassigned angle. The bicrystal was epitaxially grown from the ends of the two arms of which the crystallographic orientations were preserved in the growth process. The details are given in Refs [15] and [16]. It was found that for bicrystals with a misorientation angle less than 15”, the Y-shaped seed growth method was very effective, while for high-angle boundary bicrystals, the matched-up method was more controllable. In the preparation of bicrystal seeds, the splitting or sectioning of a single crystal generates a pair of identical vectors spanning the tilt angle, which correspond to the two arms of the Y-shaped seed. This pair of identical vectors is denoted by the misorientation vector [Ml] [20,21]. For symmetrical bicrystals with varying tilt angles, 0, it is convenient to designate these crystals as 0[hkl]/[uvw] bicrystals, where vector [UUU’]is the rotation vector along the tilt axis. In the present investigation, the bicrystals have a [ 1001 misorientation vector and a [OOl] tilt ‘axis. Thus they are designated as the 0[100]/[001] bicrystals. A set of symmetrical @[lOO]/[OOl] bicrystals with 0 ranging from 2 to 83” were prepared for the study of grain boundary diffusion. Special attention was paid to the growth of CSL (coincident-site-lattice) bicrystals, i.e. the bicrystals with tilt angles of 22.6 (x13), 28.1 (x17), 36.9 (x5), 53.1 (x5), 61.9 (C17) and 67.4” (Cl 3). The tilt angle, misorientation vector and tilt axis of the bicrystals were accurately determined by the Laue back-reflection method on an X-ray unit and electron diffraction of thin foil specimens. The error of measurement in tilt angles is within f0.5”. The typical impurity content of the bicrystals is listed in Table 1. of difSusion couples

The diffusion couples were prepared by electroCr with a purity of plating a layer of -1OOpm 99.95% on a polished surface perpendicular to the [OOl] tilt axis of the Nb bicrystals. The electroplating was carried out at about 35°C in an agitated solution of CrO,, concentrated sulfuric acid (98%) and distilled water (2.50:3: 1000 by weight) at a voltage of 3V with a DC current density of 0.7A cmm2. After plating, the bicrystals were annealed in vacuum quartz capsules sealed at -5 x lo-’ torr. The annealing times and temperatures are listed in Table 2. After annealing, the diffusion couple was sectioned along the plane normal to the interface and the grain boundary. The sectioned surface was ground on 240-600 grid Carborundum papers and lapped with l-O.lpm AlzOi powders. The sample was sub-

GRAIN

BOUNDARY

in Nb bicrvstals Carbon 20D.D.rn.

Tantalum 4lOo.o.m.

Tungsten 650.0.m.

sequently subjected to a light chemical polishing with a solution of 70% HNO, and 30% HF to remove the strained layer. 2.3. Electron probe microanalysis The diffusion couples were quantitatively analyzed on the JEOL 733 superprobe equipped with spectrum (WDS) microwavelength-dispersive analyzers and the Tracer Northern X-ray-processing system. An incident electron beam of 15keV and - 30nA was used to generate characteristic X-rays of both Cr and Nb from the samples. LIF and PET crystals were selected to disperse the X-rays of Cr and Nb, respectively. The Cr K, and Nb L, lines were used for the quantitative analysis. The spatial resolution of the instrument is about l.Opm or better. The measurement was conducted along a line across the grain boundary with an interval of l.Oum between adjacent points. In the WDS microanalysis, the concentration of each element is obtained by comparing the characteristic X-ray intensity of the element in the specimen to that of its standard after the ZAF correction [22]. The Cr standard had a purity of 99.99% and the Nb standard was taken from the as-grown bicrystals of which the diffusion couples were made. Standard calibrations were carried out for each line measurement of IO-15 points.

3. RESULTS

3.1. The concentration tration C

profile

and average concen-

A typical concentration profile of Cr parallel to the diffusion interface is shown in Fig. 1, where the maximum concentration is located at the grain boundary as the characteristic of grain boundary diffusion. The Cr concentrations are expressed in units of weight per unit volume and were converted from the WDS data using the equation given in Ref. [19]. All the concentration profiles were taken from a distance to the interface sufficiently large so that the bulk diffusion from the interface was negligible. The concentration profile started from the grain boundary Table

2. Annealing diffusion Temperature (“C) 958 997 1040 1078 1121 1161 1197

temperatures and times for grain of Cr in [lOO]/[OOl] Nb bicrystals Annealing time (10%) 8.8640 1.5919 1.1988 0.6162 0.2112 0.0600 0.0150

boundary

LI and CHOU:

HIGH

ANGLE

GRAIN

BOUNDARY

3531

shown in Fig. 2. The lattice diffusion coefficient, was calculated from Pelleg’s expression [24]

D = 3.0 x 10m5exp

:I2

-8

DISTANCE

Fig.

1. A typical

-4

0

4

8

FROM GRAIN BOUNDARY

concentration

profile

12 (pm)

of Cr in a 22.6”

[lOO]/[OOl] Nb bicrystal annealed at 1078°C for 0.6162 x IOhs. Data were measured on a line across the grain boundary (g.b.) at a distance of 7 pm from the interface.

and extended into the grain interior on both sides until the Cr was not detectable. In other words, all Cr content in the slab with the depth of 1.Opm and width of l.Opm was included in the line concentration profile, which was equivalent to sectioning a l.O,~m layer of diffusion couple (l.O,nm by width) in the tracer diffusion method. The integration of concentration profiles results in the total amount of Cr content in the slab, which is also equivalent to the average concentration, C, in the same volume of the sectioning method. 3.2. Grain boundary

d$fiision

coefJicients

sD’6

In the present study the electroplated Cr layer was about lOOpm, which was sufficient to be considered as a constant source of Cr, and Whipple’s analysis was then applied to data processing. According to Whipple [6] and Le Claire [S], the grain boundary diffusion coefficient sD’6 can be calculated from the equation

(ml/s).

D,

(4)

According to Le Claire [8], the term, cl(ln?)/ a constant of a(qfl- “’) 6’5,in equation (1) approaches 0.78 when /I is greater than 10. Because both the grain boundary segregation factor, s, and the grain boundary width, 6, are difficult to evaluate quantitatively, the data for grain boundary diffusion coefficient, D’, are presented in terms of SD’S and listed in Table 3. The corresponding /II for each sD’6 was calculated and is presented in Table 4 using equation (3). 3.3. Activation

energy

Q’ and pre-exponential

,factor

(sD’6)o

The activation energy, Q’ and pre-exponential factor, (sD’G),,, of grain boundary diffusion of Cr in Nb bicrystals were obtained by the best fit of ln(sD’6) against l/T, i.e. by the Arrhenius equation

sD’b = (sD’b),exp

- $T

(

‘1

The calculated Q’ and (SD’@, for various 0 are presented in Table 5. The error propagation method [25] was used in the data analysis, in which the errors of the tilt angle measurement, annealing temperature fluctuation, concentration measurements and curve fitting were all contained in the final results. The range of error was determined on the 90% confidence interval of the Student’s t-distribution.

4. DISCUSSION

where s is the segregation factor [23], 6 is the width of the grain boundary, C is the average concentration in the sectioning method, _r is the distance from the interface, D is the lattice diffusion coefficient, t is the diffusion annealing time, and q and /I’are dimensionless parameters given by

v=+

(2)

As shown in Figs 3 and 4, the activation energy Q’ and the pre-exponential factor (SD’& are strongly dependent on fI in the high angle range of 15-74” with appearance of maxima and minima. However, in the low angle range, 0 < 15” and 0 > 74’, Q’ is independent of 0 and (SD/G)” varies linearly with sin(B/2). Evidently, the diffusion process and

4“f--+-Y’”

(3) In equation (1) F at a distance, _v, from the interface in the sectioning method is equivalent to the integrated concentration profile at the same distance from the interface in the electron microprobe analysis method. Thus, the term, a(lnC)/8,r6’S, in equation (1) was determined by the slope of In? against yb5, as

51v

’

24 y“5

(pm6’5)

Fig. 2. Plot of In C versus yh’5 for a diffusion couple with a 22.6” tilt boundary annealed at 1078°C for 0.6162 x lobs.

3538

HIGH

LI and CHOU: Table 3. Values of sD’6 (d/s)

at various

ANGLE

GRAIN

temperatures

BOUNDARY

and tilt angles of [lOO]/[OOl] Nb bicrystals

TCCl

$0

2.0 8.0 15.0 20.0 22.6(X13) 26.0 28.1(X17) 31.0 36.9(25) 43.0 46.0 53.1(X5) 56.0 61.9(X:17) 67.4(2:13) 69.0 74.0 78.0 83.0

1040 3.92 5.64 6.40 8.73 7.58 7.47 1.49 6.66 3.07 1.57 6.75 3.94 5.66 4.61 4.69 4.50 1.93 1.27 1.45

x x x x x x x x x x x x x x x x x x x

IO-:’ IO-:’ lo-: lo-:’ IO-‘” IO-” IO-” IO-‘1 IO-?J IO-‘? IO I’ IO-‘” IO-? IO-?’ IO-?’ 10-Z lo-” IO z1 IO-”

6.08 8.82 9.31 1.42 1.33 1.13 2.47 1.22 5.13 2.67 1.35 7.91 7.67 7.86 7.87 8.88 4.32 1.85 2.28

x x x x x x x x x x x x x x x x x x x

IO 27 10~” 10-1’ 10-l? lo-” IO-” IO-?’ IO -?2 IO-?” IO-?? 10-Z? IO-“4 IO 21 IO-‘” 10-‘J IO-” IO-?’ IO-? 10mz’

I.01 1.56 1.58 2.22 1.70 1.84 4.55 1.54 9.33 3.91 1.66 I.18 1.09 1.67 1.36 1.58 4.76 2.53 3.28

x x x x x x x x x x x x x x x x x x x

1078

IO-‘? lo-‘? IO -X 10-X’ 10-2’ IO-?? 10-X’ lo-‘? IO-‘4 10-z: IO-?? IO-? IO-2 IO-? IO-” IO-*? IO-” IO-” IO-”

mechanisms for low angle and high angle boundaries are different and should be discussed separately.

1.58 2.07 2.43 3.01 2.45 2.50 7.03 2.56 1.77 5.70 2.76 1.59 2.21 I.58 2.11 1.92 4.47 5.01 5.04

Dzjiision

in low angle boundaries,

0 = 2-15”

In the range of (3 = 2-15, the activation energy is a constant of 162.3kJ/mol (Fig. 5) which is about 46% of the activation energy for the lattice diffusion, while the pre-exponential factor varies linearly with sin(B/2), as shown in Fig. 6. Such relationships indicate that the diffusion of Cr in [lOO]/[OOl] Nb bicrystals with low-angle boundaries is controlled by the dislocation-pipe mechanism [9]. This result agrees with the previous conclusion on Cr diffusion in low-angle [ilO]/[ 1121Nb bicrystals [19]. The difference in the activation energy values, 162.3 and 181.2kJ/mol, between the two systems may be explained by the difference in Burgers vectors of the boundary dislocations. In [lOO]/[OOl] bicrystals, b = a[0101 which is greater in magnitude than that in [ilO]/[l12] bicrystals (b = a/2[111]). The increase of Table 4. Values of fi at various

IO-?’ IO-?? lo-?? IO-‘? IO-” IO-?? IO-2 IO-j? IO-2 IO-?? IO ?? IO-” lo-” IO-” IO-” IO-” 10-z IO-” IO-”

II21 2.21 3.06 4.00 5.56 8.71 4.43 1.08 4.40 3.37 I.10 3.64 3.32 3.01 3.49 3.90 3.93 9.78 7.02 8.36

x x x x x x x x x x x x x x x x x x x

IO-‘2 IO-” lo-:? IO-‘? lo-” IO IZ IO-” IO 22 IO-” IO-?’ lo-?? lo-” 10~21 IO I1 IO-” IO 22 IO-2 IO-:’ IO-:’

II61 3.57 5.28 5.78 9.63 1.07 7.39 1.75 7.82 6.29 1.49 8.77 7.36 4.78 6.39 7.32 5.37 2.26 1.25 I.58

x x x x x x x x x x x x x x x x x x x

1197

IO-2 IO-?” IO-?” ION?: lo-= IO-‘? IO 12 IO-” IO-” IO 2’ IO-” 10mzl IO-‘? IO-” IO-” lo-‘? 10 Iz IO x IO x

5.28 7.82 8.02 1.25 1.16 I.17 2.96 9.89 9.22 2.30 1.02 1.04 8.69 1.05 I.01 8.47 4.15 2.19 2.34

x x x x x x x x x x x x x x x x x x x

IO-‘? IO-‘? IO-?’ lo-?’ IO 12 10-I’ IO-?? IO-” 10-l’ 10-Z’ lo-?’ IO 22 IO-” IO-” IO 22 IO-:’ IO-‘? IO-:? IO-”

(SD’@” with 0 is caused by the increase in dislocation density at the grain boundary. 4.2.

4.1.

x x x x x x x x x x x x x x x x x x x

Dijiision

in low angle boundaries,

0 = 74-83”

In the range 8 = 74-W (7716” from 90’) the activation energy Q’, ranging from 174.6 to 177.8kJ/mol, can also be considered to be independent of 8, as shown in Fig. 4. The pre-exponential to sin[(90 - @)/ factor (SD’S), is linearly proportional 21 as shown in Fig. 7. It is obvious that the diffusion in this angle range follows the dislocation-pipe diffusion mechanism. The[lOO]/[OOl] bicrystal at 90” has a (100) twin plane in which all the atoms are bonded and the grain boundary is equivalent to that found in 0” [liO]/[OOl] bicrystal, i.e. a single crystal. Therefore, the 74, 78 and 83” [lOO]/[OOl] bicrystals are equivalent to the 16, 12 and 7’ [liO]/[OOl] bicrystals, respectively. The grain boundaries of these bicrystals are composed of dislocations with a Burgers vector b = a[llO], which is $ times greater than that for 2215” [lOO]/[OOl] bicrystals (b = a[OlO]). However,

annealing

temperatures

and tilt angles

T (‘C) 0 (“I

958

997

1040

1078

II21

1161

II97

2.0 8.0 15.0 20.0 22.6(X13) 26.0 28.1(X17) 31.0 36.9(X5) 43.0 46.0 53.1(Z) 56.0 61.9(X17) 67.4(X 13) 69.0 74.0 78.0 83.0

714 1028 II66 1591 138 1361 272 1214 56 2861 1230 72 1013 84 85 820 352 231 264

542 786 830 1266 I19 1008 220 1088 46 2381 1204 71 684 70 70 792 385 165 203

204 315 319 449 34 372 92 311 I9 790 335 24 220 34 27 319 96 51 66

115 I51 177 220 18 I82 51 187 I3 416 201 I2 I61 I2 I5 140 33 37 37

65 90 II8 164 26 131 32 130 IO 325 107 IO 89 10 12 116 29 21 25

56 83 91 I51 I7 116 27 123 IO 234 I38 I2 75 IO II 84 35 20 25

56 84 86 134 12 I25 32 106 IO 246 109 II 93 II II 90 44 23 25

LI and CHOU:

HIGH

ANGLE

GRAIN

BOUNDARY

3539

Table 5. Values of the activation energy, Q’, and pre-exponential factors, (SD’S), for grain boundary diffusion of Cr in [lOO]/[OOl] symmetrical Nb bicrystals in the temperature range of 958-l 197°C (sD’d)o

(m’ls) 2.0 8.0 IS.0 20.0 22.6(X13) 26.0 28.1(217) ’ 31.0‘ 36.9@5) 43.0 46.0 53.1(X5) 56.0 61.9(X17) 67.4(X13) 69.0 74.0 78.0 83.0

162.0 162.1 162.8 170.1 187.4 171.7 183.7 171.7 220.6 167.4 169.0 204.0 171.8 190.8 196.7 178.5 174.6 177.8 175.0

i 79.5 x 65.4 k 81.0 + 88.6 f 90.8 f 67.3 f 82.9 -i_ 75.2 k 98.7 * 56.8 f 94.3 + 88.6 k 76.5 f 86.1 * 90.7 k 72.1 k 68.0 f 83.2 f 68.7

(2.86 (4.15 (4.91 (1.35 (6.17 (1.29 (9.02 (1.23 (6.28 (1.92 (1.00 (1.66 (9.18 (5.43 (9.62 (1.79 (4.55 (3.74 (3.40

k & + * f f k + + f f f + f k * i & *

1.56) 1.89) 2.03) 1.21) 2.84) 0.88) 3.26) 0.65) 1.12) 0.96) 0.45) 0.89) 4.02) 2.11) 4.56) 1.02) 2.48) 1.90) 2.01)

x x x x x x x x x x x x x x x x x x x

lo-l6 IO-lb IO-‘” IO-15 10-‘L 1O-‘1 IO-‘” 1O-‘5 lO-‘5 IO-” 10-15 IO-” 10-‘L IO-10 IO-‘” lo-” lo-l6 IO-16 lo-‘6

,501 0

I 30

I 45

I 75

I 60

I 90

B (deg)

Fig. 4. Orientation dependence of the activation energy of grain boundary diffusion on the tilt angle 0 of [lOO]/[OOl] Nb bicrystals.

the average Q’ (1758kJ/mol) is greater than that in the angle range of 2-15” (q’ = 162.3kJ/mol). The reason for the increase in the activation energy in 74-83” [lOO]/[OOl] bicrystals is not clear. 4.3. DifSusion in high angle boundaries,

I I5

Q = 15-74”

In the angle range of 15-74”, neither the grain sD’6, nor the actiboundary diffusion parameter, vation energy, Q’, nor the pre-exponential factor, (SD’& follow any defined relationships with 8. The values of sD’6 fall into minima, whereas Q’ reaches maxima at special tilt angles of 22.6, 28.1, 36.9, 53.1, 61.9 and 67.4, as shown in Figs 3 and 4. These special angles correspond to CSL boundaries of Cl 3, Cl7 and C5. 4.3.1. Diffusion in random high angle boundaries. Excluding the CSL boundaries, the grain boundaries with f3 ranging from 15 to 74” are random high angle boundaries. The associated activation energies Q’ appear to be constant (170kJ/mol) (Fig. 4), whereas sD’6 varies with 0 in the same way as the diffusion penetration depth measurement in Ni bicrystals [26] and as the grain boundary diffusion measurements in Au/Ag [27,28], as shown in Fig. 3. The low and

constant activation energies of these random boundaries indicate that they are less densely packed with similar structures and the diffusion processes are, therefore, controlled by the same mechanism. 4.3.2. Pre-exponential factor, (sD’~)~. The variation of pre-exponential factor, (sD’G)~, with 0 is within an order of magnitude, ranging from lo-l5 to 10-‘6. This indicates that the change in grain boundary entropy is limited. Because of the lack of knowledge about the segregation parameter, s, and the grain boundary width S, it would be difficult to describe quantitatively the effect of grain boundary entropy on 0;. 4.3.3. The activation energy of grain boundary difSusion, Q’. The variation of Q’ with 0 follows the same trend as the grain boundary diffusion of Zn in Al bicrystals [29-311. The present data also indicate a linear relationship between Q’ and Cm’:‘, as previously observed in the Zn/Al system by Aleshin et al. [30]. On the other hand, the peaks of the measured activation energy, Q’, are consistent with the results of grain boundary corrosion in [ 1OO]/[OO l] Nb bicrystals [32]. For C = 13 and Cl7 boundaries, the high Q’ values correspond to the shallow penetration depths of corrosion along these CSL

-23-

-241 0

I

I

15

30

I

I

45

60

I

75

I 90

8 (deg)

-51.61 0.67

Fig.

3.

parameter,

The dependence of grain boundary diffusion sD’6, on tilt angle, 0, for [lOO]/[OOl] Nb bicrystals.

I 071

1

1000/T

Fig. 5. Arrhenius

I

0.75

0.79

I 0.83

(I/K)

plot of 2, 8 and 15” diffusion

couples.

LI and CHOU:

3540

HIGH ANGLE GRAIN BOUNDARY Table 6. Comparison between the activation energy, Q’, of grain boundary diffusion in [lOO]/[OOl] symmetrical Nb bicrystals, the calculated grain boundary surface energy, ‘/. and the free volume, VF, for b.c.c. metals[34]. VF IS in units of lattice parameter, a x 5 13

21 0

I 0.04

I 0.08

I 0.12

1 0.16

17

0 (deg) 53.1 36.9 22.6 61.4 28. I 61.9

G.B. plane

Q’ (kJ/mol)

(120) (130) (150) (230) (140) (350)

204.0 220.6 187.4 196.7 183.7 190.8

($I’) I.135 I.068 I.025 1.162 1.142 1.142

G 0.19 0.16 0.19 0.17 0.29 0.17

sin (e/2)

Fig. 6. The dependence of the pre-exponential factor of grain boundary diffusion on the tilt angle Bin [lOO]/[OOl] Nb bicrystals with 0 ranging from 2 to 15”.

boundaries. For random boundaries, Q’ values are low in agreement with the large corrosion depths. mechanisms in CSL boundaries. 4.3.4. Dijiision The variation of activation energy with tilt angle is related to the atomic structure of the grain boundaries. It is known that the grain boundary structure is characterized by the intergranular energy, y. Theoretically, the higher the activation energy, the lower the value of y [33]. Using the central force potential, Vitek et al. [34] calculated the integranular energies for the CSL boundaries in b.c.c. metals, as given in Table 6. By comparison, it is found that the measured activation energies of grain boundary diffusion are inconsistent with the calculated values of y. For example, Q’ for the 22.6” Cl3 boundary is smaller than that for 67.4” El3 boundary, which indicates that y for the former should be larger than that for the latter; however, the calculated y values show the opposite trend. The inconsistency may be caused by the special electron structure of Nb atoms. Because of the partially filled 4d electron band, the central force potential may not be sufficient to describe the interaction of Nb atoms. Other interatomic potentials were also used for the calculation of intergranular energies of pure metals 135-401, and, unfortunately, Nb was not included. Grain boundaries may also be described in terms of the grain boundary free volume, which is the expansion in volume caused by the creation of a grain boundary in a perfect crystal [41]. Such an expansion

provides excess opening for grain boundary diffusion. Thus the greater the VF, the smaller the Q’. By comparison, it is found that the calculated VF [34] is consistent with the measured Q’ (Table 6). This indicates that the grain boundary free volume may play a dominant role in grain boundary diffusion in high angle bicrystals. It should be noted that the conclusions drawn above are merely indicative of a trend because both comparisons are based on different temperature ranges (958-l 197°C for Q’ and - 273°C for y and VF). It is anticipated that a more conclusive discussion will be available with detailed data of y or VF. In particular Borisov’s formula [33] for D’/D and y could be verified.

5. CONCLUSIONS Grain boundary diffusion of Cr in [lOO]/[OOl] Nb bicrystals was investigated in the temperature range of 95%1197°C by electron microprobe analysis. It was found that the grain boundary diffusion coefficients vary with the tilt angle, 8. In the low angle range of 2-l 5” and 74-83”, the diffusion is dominated by the dislocation-pipe mechanism and the activation energy is independent of 8, while the pre-exponential factor is linearly proportional to sin(8/2). In the high angle range of 15-74”, the activation energies are maxima at X13, X17 and X5 CSL boundaries. The activation energies for the random high angle boundaries are nearly constant at about 170kJ/mol. Acknowledgement-The work was supported in part through the New Century Professorship Fund of Lehigh

University.

REFERENCES

2.8 002

I 0.06

I 0.10

I 0.14

I 0.18

sin [(90’-8)/2]

Fig. 7. The dependence of the pre-exponential factor of grain boundary diffusion on tilt angle, 8, in [lOO]/[OOl]Nb bicrystals. 0 ranges from 74 to 83”.

M. S. Daw and M. I. Baskes, Phys. Rev. Lett. 50, 1985 (1983); P&s. Rev. B 33, 7983 (1986); 39, 7441 (1989). G. H. Campbell, W. E. King, S. M. Foiles, P. Gumbsch and M. Riihle, in Structure and Properties ofInterface in Materials (edited by W. A. T. Clark, U. Dahmen and C. L. Briant), pp. 163-169. Materials Research Society, Pittsburgh, PA (1992). G. H. Campbell, P. Gumbsch, W. E. King and M. Riihle, Z. Metallkd. 83, 472 (1992). T. Kwok and P. S. Ho, in D$kion Phenomena in Thin Films and Microelectronic Materials (edited by D. Gupta and P. S. Ho), p. 369. Noyes Publications, New Jersey (1988).

LI and CHOU:

HIGH

ANGLE

5. J. C. Fisher, J. uppl. Phys. 22, 74 (1951). 6. R. T. P. Whipple, Phil. Mug. 45, 1225 (1954). 7. T. Suzuoka, Trans. Jpn. Inst. Met. 2,25 (1961); J. Phys. Sot. Jpn 19, 839 (1964). 8. A. D. Le Claire, Br. J. appl. Phys. 14, 351 (1963). 9. D. Turnbull and R. E. Hoffmann, Acta metall. 2, 419 (1954). 10. I. Kaur, W. Gust and L. Kozma, in Handbook of Grain and Intecface Boundary D$iusion Data. Ziegler Press, Stuttgart, (1989). 11. N. L. Peterson, Int. Met. Rea. 28, 65 (1983). 12. D. Gupta, D. R. Campbell and P. S. Ho, in Thin Film Interd@sion and Reactions (edited by J. M. Poate et al.), p. 161. Wiley, New York (1978). 13. R. W. Balluffi, Metall. Trans. A 13, 2069 (1982). 14. G. R. Purdy, in Diffusion in Solids: Unsolved Problems (edited by G. E. Murch), p. 131. Trans Tech Publications, Segantinistr. 216, CH-8049 Zurich, Switzerland (1992). 15. C. S. Pande, S. L. Lin, S. R. Butler and Y. T. Chou, J. Cryst. Growth 19, 209 (1973). 16. B. C. Cai, A. DasGupta and Y. T. Chou, J. less-common Metals 86, 145 (1982). 17. X. M. Li and Y. T. Chou, Phil. Mug. Lett. 66, 281 (1992). 18. X. R. Qian and Y. T. Chou, Phil. Mug. A 52, L13 (1985). 19. X. M. Li and Y. T. Chou, Phil. Mug. A. in press, (1996). 20. Y. T. Chou, B. C. Cai, A. D. Romig and L. S. Lin, Phil. Mug. A 47, 363 (1983). 21. B. C. Cai, A. DasGupta and Y. T. Chou, Phil. Mug. B 55, 67 (1987). 22. J. 1. Goldstein, D. E. Newbury, P. Echlin, D. C. Joy, C. Fiori and E. Lifshin, in Scanning Electron Microscopy and X-Ray Microanalysis, p. 305. Plenum Publishing Corp., NY (1981).

GRAIN

BOUNDARY

3541

23. G. B. Gibbs, Physica status solidi 16, K27 (1966). 24. J. Pelleg, Phil. Mug. 33, 165 (1976). 25. J. Mandel, The Statistical Analysis of E.xperimental Data, p. 13 1. Dover Publications, Mineola, NY (1984). 26. W. R. Upthegrove and M. J. Sinnott, Trans. ASM 50, 1031 (1958). 27. Q. Ma and R. W. Balluffi, Acta metall. mater. 41, 133 (1993). 28. Q. Ma, R. W. Balluffi, C. L. Liu and J. B. Adam, Acta metall. mater. 41, 143 (1993). 29. I. Herbeuval, M. Biscondi and C. Goux, Mem. Sci. Rev. Metall. 60, 39 (1973). 30. A. N. Aleshin, B. S. Bokshtein and L. S. Shvindlerman, Sov. Phys. Sol. State 19, 2051 (1977). 31. M. Biscondi, in Physical Chemistry qf the Solid States: Applications to Metals and Their Compounds (edited by P. Lacombe), p. 225. Elsevier Science Publishers, Amsterdam (1984). 32. X. R. Qian and Y. T. Chou, J. less-common Metals 134, 179 (1987). 33. V. T. Borisov, V. M. Golikov and G. V. Shcherbedinskii, in D@ision Processes in Metals, p. 122. English translation, Israel Program for Scientific Translations, Jerusalem (1970). 34. V. Vitek, D. A. Smith and R. C. Pond, Phil. Mug. A 41, 649 (1980). 35. R. A. Johnson, Phys. Rev. A 134, 1329 (1964). 36. D. Wolf, Acfa metall. mater. 38, 781 (1990). 37. M. W. Finnis and J. E. Sinclair, Phil. Mug. A 50, 45 (1984); 53, 161 (1986). 38. R. A. Johnson and D. J. Oh, J. Mater. Rev. 4, 1195 (1989). 39. J. A. Moriarty, Phys. Rev. B 38, 3199 (1988); 42, 1609 (1990). 40. D. Wolf, Phil. Mug. A 62, 447 (1990). 41. H. B. Aaron and G. F. Bolling, Surf. Sci. 31, 27 (1972).