Effect of oxygen on diffusion of manganese in α-titanium

~l-6~~,88 $3.04 + 0.00 CopyrIght C 1988 Pergamon Press plc

km merati. Vol. 36, No. 10, pp. 2787-2795. 1988 Printed in Great Britain. All rights reserved

EFFECT OF OXYGEN ON DIFFUSION MANGANESE TN a-TITANIUM Y. NAKAMURA,‘*t

H. NAKAJIMA,’

OF

S. ISHIOKA’ and M. KOIWA’

‘Institute for Materials Research, Tohoku University, Sendai 980. Japan and *Department of Metal Science and Technology, Kyoto University, Kyoto 606. Japan (Received 16 December 1987) Abstract-Diffusion coefficients of “Mn in single crystal Ix-Ti have been measured over the temperature range from 878 to 1135 K. It has been confirmed that manganese exhibits fast diffusion, which is about three orders of magnitudefaster than self-diffusion in a-Ti. Temperature dependence of “Mn diffusivities parallel and perpendicular to the c axis is, respectively, expressed as D,, = (4.9 4 1.9) x IO-&exp(- 160.5 + 2.7 kJmol_‘IRT) mrs-‘, D, = (6.0 It 1.3) x 10e5 exp( - 189.2 & 1.6 kJmol”-‘/RT) m’s_‘.

Effect of oxygen on Mn diffusion in polycrystal a-Ti has also been investigated. The addition of suppresses the Mn diffusion, which is interpreted as blocking effect of oxygen.

oxygen

Rbum&Les coeEicients de diffusion de MnY dans le titane K monocristallin ont ete mesures entre 878 et 1135 K. La diffusion du manganese est rapide (elle est sup&ieure d’environ trois ordres de grandeur a l’autodiffusion dans T&a). Les diffusivitts de Mn”, parallelement et perpendiculairement a l’axe, c, ont les expressions suivantes en fonction de la temperature D,, = (4,9 + 1,9) x IO-” exp( - 160,s + 2,7 kJmol_i/W) D,=(6,0+

1,3) x 10-5exp(-189,2+

m2 s-i,

1,6kJmol-1/RT)mZs-‘.

Nous avons egalement itudie l’effet de l’oxygene sur la diffusion de Mn dans Ti-a polycristallin. L’addition d’oxygine supprime la diffusion de Mn, ce que I’on interprete comme un effet de blocage du $ l’oxyg&e. Zusammenfassung--Die Diffusionskoeffizienten von “Mn in Einkristallen aus a-Ti wurden im Temperaturbereich zwischen 878 und 1135 K gemessen. Es bestatigte sich, dag Mangan ranch diffundiert, etwa drei GriiBenordnungen rascher als die Selbstdiffusion in a-Ti betriigt. Die Temperaturebhangigkeit parallel und senkrecht zur c-Achse 1lBt sich schreiben D,, = (4.9 + 1.9) x 10W6exp(- 160,5+ 2.7 kJMol-’ RT)m’s-i, D, = (60 + 1.3) x 10e5 exp( - 189,2 + 1.6 kJMol_‘RT) m’s_‘. Augerdem wurde der Enflug des Sauerstoffs auf die Mangandiffusion in a-Ti-Polykristallen untersucht. Zugabe von Sauerstoff unterdrijckt die Mangandiffusion. welches als Blockierungseffekt durch den Sauerstoff interpretiert wird.

series of experiments, we report the diffusivity manganese in a-Ti single crystals; a previous

I. INTRODUCTION

Determination of diffusion coefficients of various elements is important for the understanding of various processes occurring in materials. The present authors have made a series of measurements of diffusion coefficients of several elements such as Co [ 11, Fe [2], Ni f3] and P [4] in single crystal a-titanium. All these elements exhibit very fast diffusion in comparison with elf-di~usion. As an extension of this

tPresent address: Isohara Plant, Nippon Mining Co., Ltd. Kitaibaraki 319-15, Japan

of

inTi indicates that Mn

vestigation [S] on polycrystalline is a fast diffuser. Titanium is a reactive metal; commercially available titanium contains impurity oxygen of the order of a few thousand ppm. Therefore, it is important to know how oxygen affects the diffusivity. Previous results by the present authors [4] indicate that phosphorus diffusion is si~ificantly affected by oxygen, whereas nickel diffusion is not much affected [3]. it is interesting to investigate the effect of oxygen on the manganese diffusion in a-Ti in comparison with the previous

2787

data.

2788

NAKAMURA Table Single

ef al.:

BLOCKING

1. Chemical

analysis

Ti

Ti (wt PPm)

Al

70 -

ND

V

-

alloys

Ti-1.52

at.%

(wt

PPm)

0

Ti-2.35

al.%

(wt

ppm)

0

ND

Ti

-

that

Cr

90

-

Mn

20

ND

Fe

40

Ni

40

4 10 -

:

Ti-0


<30

Co

and

Polycrystal

(wt ppm)

Cl

of Ti

crystal

Impurity

Si

EFFECT IN DIFFUSION

20

-

to.5


42

N

5

0

180

raw for

material

is of the same

stock

as

the polycrystals

30
2. EXPERIMENTAL

Single crystal rods about 100mm in length and 11 mm in diameter were grown from high-purity titanium by an electron beam floating zone melting technique. The single crystals were oriented with an accuracy better than 2” either parallel or perpendicular to the c axis by back-reflection Laue diffraction, and were cut into 3 mm thick discs using a multiwire saw. Titanium-oxygen alloys were prepared by arcmelting of appropriate mixtures of titanium metal and titanium oxide (TiO,, 99.999% pure). The resulting alloy ingots were remelted several times to insure homogeneity. The ingots were forged and swaged to rods 11 mm in diameter, which were then cut into discs of 3 mm thick. Chemical analysis of the single crystals and the Ti raw materials for the alloys is shown in Table 1 together with oxygen analysis of the alloys. The diffusion measurements were carried out by a standard sequential-sectioning technique using the radioisotope “Mn. The radioisotope was purchased from New England Nuclear Corporation in the form of MnCl, in a 0.5 N HCl solution. The solution containing “Mn was dropped onto the polished surface of the disc specimens whose lateral side was covered by Melcoat to prevent contamination, and then dried by an infrared lamp. A tantalum plate was attached to the deposited surface of the specimen. The specimen was first wrapped with titanium foils and then with zirconium foils, and was sealed in a quartz tube with high-purity argon of about 3 x lo4 Pa together with a. piece of manganese, which was used for suppression of the evaporation of the radioactive manganese from the specimen surface. The diffusion anneals were carried out at given temperatures controlled within f0.2 K. After the diffusion anneals the lateral surface of specimens was removed by a lathe in order to eliminate a possible contribution of surface diffusion or evaporation. The specimens were sectioned by the lathe and the chips were weighed on a precision torsion balance with an accuracy of kO.01 mg. The mass loss in the sectioning process was usually less than 1%. The chips were dissolved in 1 ml of the solution of 15% hydrofluoric

8000

5100

acid, 60% nitric acid and 25% distilled water. The “Mn activity was measured by a well-type Packard Auto Gamma Scintillation Spectrometer. At least IO4 counts were collected for each sample. 3. RESULTS

The diffusion condition is equivalent to an infinitely thin source diffusing into a semi-infinite cylinder. The diffusion-penetration profiles conform well to the thin film solution of the diffusion equation C(x, t) = M (nDr)-‘I’ exp( -x’/4Dt)

(1)

where C(x, t) is the tracer concentration at a depth x after a diffusion interval t, D is the tracer diffusivity, and M is the initial amount of a tracer at the surface. Figure 1 shows the diffusion-penetration profiles in the single crystals. All diffusion profiles were

0

5

1 0

1

2 xa

3

4

(lO-‘mP)

Fig. 1. Diffusion-penetration profiles of “Mn in single crystal a-Ti.

NAKAMURA Table 2.

et al.: Difiusivities

Diffusion direction

Temperature (K) 1134.5 1134.5 1074.0 1074.0 1029.2 1029.2 971.3 973.4 925.4 925.4 877.9 877.9

of %Mn

Annealing time(s) 8.629 8.629 3.239 3.239 1.027 1.027 5.869 1.272 3.866 3.866 4.202 4.202

1

I

BLOCKING

II

II 1 II 1

II

1

II

1 il

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

IO’ IO4 IO’ IO’ IO6 IO6 10’ 10’ IO’ I06 IO” IO’

2789

EFFECT IN DIFFUSION

measured

in single crystal

2-X

Diffusinty D(m’s-‘1 (I.171 *0.008)x (1.966~0.011) (4.104 i 0.084) (8.527 + 0.084) (1.443 &0.020) (3.081 f 0.046) (3.732 * 0.023) (I.151 +0.016)x (I.‘95 * 0.01 I) (4.;69 k 0.045) (3.454 + 0.0171 il.326 + O.Ol3j

DD x Y x x x x x x x x

lo-‘? IO-” 10 I4 IO I4 IO ‘4 IO ~I4 IO ‘( IO-” IO ‘! IO-” IO ‘6 IO-”

1.68 2.08 2.14 3.08” 3.68 3.84

“The value is nol for the same temperature

Gaussian without serious surface hold-up or noticeable non-Gaussian “tails”. The diffusivities obtained at each temperature are compiled in Table 2. The temperature dependence of the diffusivities parallel and perpendicular to the c-axis are shown in Fig. 2, together with the data for polycrystals. The diffusivities parallel to the c-axis are larger than those perpendicular to the axis by a factor of 1.7 to 3.8. The temperature dependence of the Mn diffusivities parallel and perpendicular to the c-axis can be expressed, respectively, as D,, = (4.9 f 1.9) x IO-6 x exp( - 160.5 f 2.7 kJmol-‘/RT)

m*s-’

D, = (6.0 + 1.3) x 1O-5 x exp( - 189.2 + 1.6 kJmol-‘/RT)

m’s_‘.

T (‘0 900 I’,

lo-‘2

800

r

700

I

Diffusivities determined for any polycrystalline specimens are expected to lie in between D, and DI. In fact, the two data points measured by Santos and Dyment [5] fall in between the two straight lines (D, and DL), as seen in Fig. 2. However, the present values determined for the polycrystalline specimens are very close to D,. By X-ray diffractometry, the specimens were found to have such a texture that the c-axes of most grains lie perpendicular to the direction of diffusion [6]. Oxygen-doped polycrystalline specimens have also similar texture. Figure 3 shows the diffusion profiles in the a-phase of a polycrystalline Ti-1.52at.O~ 0 alloy. The diffusivities measured in these polycrystalline specimens are listed in Table 3. The temperature dependence of the diffusivities are shown in Fig. 4. together with the data for oxygen diffusion in polycrystal cc-Ti [7]. The diffusivity of Mn decreases with increasing oxygen concentration. The temperature dependence of the diffusivities can be expressed as D(poly-Ti)

s

600

I

1

Parattet to c ws A Perpendicutar to c axis n .-TI polycrystal v Sontos and Dyment

l

= (1.7 * 1.1) x 10-S x exp( - 175.5 + 2.0 kJmol-‘/RT)

971.OK

1

._

6.0

9.0

11.0

10.0

10./T

12.0

(K-‘1

Fig. 2. Temperature dependence of diffusivities in a-Ti.

0

I

0.2

0.4

0.6

I

0.8

m’s_’

-

I 1.0

x2 (lO-sm’) Fig. 3. Diffusion-penetration profiles of “Mn in polycrystal Ti-1.52 at.% 0 alloy.

NAKAMURA et al.:

2790

BLOCKING EFFECT IN DIFFUSION

Table 3. Diffusivitics of %Mn measured in polycrystaflinc pl-Ti, Ti-1.52 at.% 0 and Ti-2.35 at.% 0 dlOYS

Temperature (W

Sample Polycrystal z-Ti

Ti-I.SZat.%

3.166 7.738 2.574 1.729 5.742 1.726 5.775 1.903 2.997 3.794 3.525 7.738 6.804 2.6% 4.934 4.202

1125.6 1071.6 1024. I 921.9 1123.6 1072.6 1024.3 971.0 922.2 872.0 1123.3 1071.6 1024.1 968.0 926.5 877.9

0

X-2.35 at.% 0

0(1.52at.%

Anneafing time(s)

0)

= (9.9 * 1.5) x 10-6 x exp( - 176.0 + 1.1 kJmol_*/RT) m2 s-’ D (2.35 at. % 0) = (8.3 f 0.7) x lO-6 x exp( - 175.5 f 0.7 kJmol-‘/RT)

mt s-l.

4. DISCUSSION

The present investigation has confirmed high diffusivity of Mn in a-Ti; the Mn diffusivity is about three orders of magnitude larger than the selfdiffusivity. In Fig. 5 is shown the diffusivities of various elements in a-phase measured in our laboratory, and those reported for p-phase. All the elements, Mn, Fe, Co, Ni and P exhibit fast diffusion. Although details of the mechanism of such fast diffusion are still controversial, some type of interstitial mechanism is believed to be operative on the basis of the considerations as given below. 4.1.1. Eflect of phase transformation. On transformation from the cr(HCP) to the @(BCC) phase, the diffusivity shows some change. The ratios, D,/D,, at the transformation temperature, 1155 K, by extrapolation are: Mn 1

Fe 0.4

Co 0.2

IO” 10’ IO6 10’ IO* IO’ Id 106 lo6 106 10’ 10J 10’ IO6 106 lo6

(9.241 (3.333 (1.398 (1.445 (7.097 (2.720 (1.132 (3.439 (1.084 (3.216 (5.813 (2.358 (9.023 (2.654

k 0.061) x + 0.022) x f 0.018) x +0.013)x * 0.074) x 2 0.017) x i: 0.018) x 2 0.067) x * 0.007) x it 0.073) x * 0.034) x * 0.032) x yO.194) x + 0.042) x (1.047& 0.004) x (3.088 t 0.4231 x

Ni 0.8

IO -I’ IO -‘I lo-” W’S lo- ” 10-‘+ IO-” IO-” 1o-‘5 IO-‘* 10-l’ IO-” IO-” lo-” lo-‘5 lo-l6

505 K. They regarded this as an indication of interstitial type diffusion of Au and Ag; if these impurities diffuse by the vacancy mechanism as in the case of self-diffusion, one should observe the increase in the diffusivity of the same order of magnitude as that for self-diffusion. The same reasoning also applies to the Mn diffusion in Ti. 4.1.2. Dzj&sion anisotropy. The diffusivity of Mn parallel to the c axis, L),, is larger than that perpendicular to the c axis, I), . The ratios, D!,/I), at 1073 K, are: Mn 2.0

4.1. Fast d&ion

Ti 310

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

Dihsivity L)(m’s-‘)

Fe 2.6

The trend is common

Co 3.6

Ni 1.7

P 0.66

for the transition

metal ele-

T (‘c)

l

=-Ti pdycrystot

a Ti-M2Ot.%O 8 Ti-2.35ot%O

P 1.2

where the values of Z&are evaluated as (ZD, + D, )/3. While the self-diffusion coefficient of Ti exhibits a large increase on transformation, the other elements show relatively small changes. Anthony et al. [8] have observed a similar trend for thallium with Au and Ag as fast diffusers; while the self-diffusivity increases by about 30 times on t~nsfo~ation, Au and Ag diffusivities are almost continuous or rather slightly decrease across the transformation temperature of

tO’/T

(K-‘1

Fig. 4. Temperature dependence of diffusivities of Mn and oxygen in a-Ti and oxygen-doped Ti.

NAKAMURA

1600 1400

1200

T (“Cl 1000

800

IO&/T

fK-‘)

Ed al.: BLOCKING

600

Fig. 5. Temperature dependence of self and impurity diffusion in Ti (see Ref. [I] and references therein).

ments. while it is opposite for P. Similar anisotropy, D, > D,. has been observed for noble metal elements in T1 and Cu in X-Zr. Hood and Schultz [9] favoured an interstitiat mechanism for Cu in a-Zr, and ascribed the above trend to the difference in the volume of saddle point configurations for interstitial jumps in the two direction, parallel and ~r~ndicular to the c axis. Since the c/a ratio of Ti. 1.586. is almost the same as that of Zr, 1.594, the interpretation also holds for titanium. The opposite trend observed for P may be related to the difference in the nature of bonds; semi-metallic phosphorus would interact with host titanium atoms in a different manner from that of metallic elements. 4.1.3. Atomic size qfert. Hood [IO. 111has demonstrated a correlation between atomic size and impurity diffusion in Pb and in r-Zr; smaller sized atoms diffuse faster. Similar correlation seems to hold as shown in Fig. 6, where the diffusivities of various impurities in z-Ti and r-Zr are plotted as a function of their atomic size. Here, we adopt, as representing atomic size, “metallic radii for the coordination number 12” after Teatum. Gschneider and Waber [ 121.Numerical values are given in tabulated form in the monograph by Pearson [13]. The radii were calculated by assuming a rigid sphere model from lattice spacings of the respective elements. For the case of non FCC crystals, the radius for the coordination number 12 was deduced by appropriate processing of original data. These radii are said to be the most suitable set to use for metallic alloys [13]. As seen in the figure, the trend that smaller atoms diffuse

2791

EFFECT IN DIFFUSION

faster is evident. In particular, Co, Fe and Ni are the smallest atoms and the fastest diffusers in a-Ti. In discussion of fast metallic impurity diffusion in metals, Bakker [14] pointed out that the atomic size estimated from the molar volume of pure metal is not sufficient description. He applied the MiedemaNiessen model fl5] of the volume effect on alloying to the interpretation of fast diffusion in Pb and several other metals. By adopting the same model, we have calculated the molar volume of various elements when alloyed in n-Ti or a-Zr, as shown in Fig. 7. Unlike the case of fast diffusion in Pb, the two figures, Figs 6 and 7, exhibit qualitatively similar trends. However, the correlation in the latter is approximately linear, indicating some intrinsic relationship. 4.1.4. Solubilityt‘s diffusicity.It is known that fast diffusers have generally very low solubility. In Fig. 8 is plotted the solubility of various elements in X-Ti and a-Zr at 1100 K versus the diffusivity at 1100 K, indicating some correlation between the two quantities. It is interesting to recall the 15% empirical rule for the formation of a solid solution proposed by Hume-Rothery et al. 1161;an extended solid solution can only be formed between elements if the difference in atomic sizes is less than about 15%. In terms of molar volume, the critical value is 38.6%; solute atoms with a relative volume less than 61.4% of the solvent would not form an extensive solid solution. In a-Ti, the volumes of elements Co, Ni and Fe are smaller than the critical value, as seen in Fig. 7. It is tempting to speculate that these elements are too small to occupy the stable substitutional positions. and some fraction would occupy interstitial sites.

10-s

. , -

I

*

1

I

.

1

*co 10-a

Nt *

P l

in a-Ti

0 in a-i!r

1W” 4 Ni$ 1E ‘”

1U” i

k MnO

h

,048 -

0.11 0~2 0.13 0.14 0.15 Od6 0.17 Metallic Radius (nm)

Diffusivity vs atomic radius at 1100K in a-Ti and a-Zr. The single crystal data are indicated by the bars; the upper limit value is for diffusion parallel to the c axis and the lower Limit value for that ~~ndicular to the c axis. Fig. 6.

2792

NAKAMURA et 01.: BLOCKING EFFECT IN DIFFUSION

lOa

.

4 co rNi

10’

in a-Ti(l156K)

f FQ

A in o-fr(ll35K)

E Fe

t

I

CU

crI

ICO

Mn A

fNi

IFe fMn

i

* Nb inokTi 0 in&r l

lOOf *

’ 0.4

’

’

0.5

*

’

0.6

*

f

0.7

’

10

Fig. 7.

Diffusivity vs molar volume in a-Ti and a-Zr. &,,,,

and D,t are, respectively, impurity and self-diffusivity. I’=,, and V,, are, respectively, partial molar volume of impurity and molar volume of pure metal. The singi: crystal data are indicated by the bars; the upper limit value is for diffusion parailei to the c axis and the lower limit value for that perpendicular to the c axis.

10’ 103 10‘ 10s IO6 Soiubility (at.ppmJ

Fig. 8. Diffusivity vs solubility of various elements in a-E and a-Zr (7’ = 1100 K). The single crystal data are indicated by the bars; the upper limit value is for diffusion parallel to the c axis and the lower limit value for that ~r~ndicular to the c axis, except for P. For the latter, D, 1 I),.

has by far the largest value, being consistent with the observation of the strong trapping effect in diffusion.

4.2. Oxygen effect on dz#iion The Mn diffusion coefficient is decreased by the addition of oxygen. In the previous investigations, the present authors have reported the effect of oxygen addition on the diffusivities of Ni [3] and P [4]; the trend for the Ni diffusivity is more or less similar to that for the Mn diffusivity. We shall discuss the results of the three experiments below. The P diffusivity is significantly reduced by the oxygen addition particularly at lower temperatures; the Arrhenius plot of the diffusion coefficient exhibits a marked curvature. Such a trend is usually explained in terms of the trapping theory. In a simple model, the apparent diffusion coefficient L$ can be expressed as R = D/II -I; +J; exp(BlkV1

7

? =” At *

f

0.8

Vinp/Vmel

0Ti 0Al

(2)

where D is the normal diffusion coefficient in the absence of trapping sites, f; is the fraction of the number of trapping sites, and B the binding energy [t7]. The experimental data of the diffusion coefficients of P were analyzed by using equation (2); the binding energy is estimated to be about 230 kJmol-‘. For Mn and Ni, the Asrhenius plot of the diffusivity is virtually linear, indicating that the binding energy of these elements to oxygen is smaller than that of P to oxygen. It seems interesting to compare such a trend with the Gibbs energy of formation of respective oxides, per metal atom, as listed in Table 4. Apart from the matrix atom, Ti, phosphorus

4.3. The blocking effect in dz~~~Q~ As mentioned above, the addition of oxygen only slightly reduces the diffusivity of Mn and Ni. The diffusion coefficient in the Arrhenius plot seems to be shifted downwards with increasing oxygen concentration. In other words, the activation energy remains unchanged, but the value of D, decreases. Such a trend may be understood in terms of the blocking effect. The idea is as follows: Oxygen and diffusers (Mn or Ni) share a common lattice formed by the interstices of the hexagonal close-packed lattice of titanium. Oxygen exerts no chemical attraction for diffusers, and yet reduces the diffusivity by its own presence; the sites occupied by oxygen are no longer available for diffusers. The blocking ef%ct in diffusion has been discussed by Koiwa and Ishioka [19] and previous works are

Table 4. The Gibbs energies of formation of metal oxides at 873 K. Data an taken from Kubaschewsk and Alcock [ISI

Reaction Ni+ 1/20,=NiO l/2 p,+5/4 4=1/2 P*O, Mn + l/2 Oz = MnO Ti + 0, = TiO,

-165 - 543 - 322 - 74s

NAKAMURA

ef al.:

BLOCKING

EFFECT

2193

IN DIFFUSION

1.0'

0.2

0.3

0.6 Critical p%ability.

&j

-

Fig. 10. Blocking factor vs critical probability percolation.

D(C)/D(O) as a function of C for the square (SQ) lattice, the simple cubic (SC) lattice, the body centered cubic (BCC) lattice, and the face centered cubic (FCC) lattice. The estimated values of b are listed in Table 5. Alternatively the value of b can be calculated by applying the method employed in a spin-wave theory of ferromagnetic crystals containing impurities developed by Izyumov [22]. The values thus calculated (Method II, Ishioka and Koiwa [23]) are expressible in terms of the lattice Green functions; the result is listed in Table 5 in comparison to those by the primitive method I. The values of critical probability of site percolation, taken from Shante and Kirkpatrick [24], are also shown in the table. The salient features to be observed from the table are: (I) The blocking factors calculated by the two different methods are in good agreement. (2) The blocking factor is larger for a lower dimensional lattice; incidentally for the one dimensional lattice b = co, because one blocked site is sufficient to suppress the long range diffusion. This trend is understandable, because in a higher dimensional lattice a random walker can find a larger number of paths to by-pass a blocked site. (3) The blocking effect is smaller for lattices with a larger coordination number; the same reasoning as in item (2) holds. (4) The blocking factors and the critical percolation probabilities show an excellent correlation (see Fig. 10). In addition,

.961

I

0

.Ol

I

I

c

i

.03

.02

L

Fig. 9. Ratio of diffusion coefficients of defective and perfect lattices, D(C)/D(O), as a function of the concentration, C, of blocked sites.

reviewed by Haus and Kehr in a recent paper [20]. For consistency’s sake, it seems appropriate to give here a brief description of the theory of the blocking effect. Consider the diffusion of a random walker in a lattice with inaccessible sites or blocked sites. When the concentration of the blocked sites, C, is higher than a certain critical value, long range diffusion becomes no longer possible. This is a well known problem of site percolation. For smaller concentrations of blocked sites, the diffusion coefficient, D(C), may be written as D(C) = D(0) (1 - bC +.

.)

(3)

where b is referred to as the blocking factor. A straightforward way of estimating the b value is to calculate directly D(C) for a defective lattice with a regular array of the blocked sites with the method described by Koiwa and Ishioka [19] and in Ishioka and Koiwa [21]. By performing calculations for various concentrations one can evaluate the value of b by extrapolation (Method I). Figure 9 shows

Table 5. Blocking factors and critical probabilities for site percolation

Blocking factor Coordination

h

Lattice

Dimension

number Z

Method I

Method II

Square

2 3 3 3 3

4 4 6 8 12

2.14

2.1416 2(exact) I.531 I I.3751 I .2796

Diamond SC BCC FCC

for site

I .54 I .37 I .28

‘Taken from the review by Shante and Kirkpatrick [24].

Critical probability’ for site ocrcolation 0.581 0.436 0.325 0.243 0.199

0.569 0.425 0.307 0.195

0.590

2794

NAKAMURA

et al.:

BLOCKING

Mn: b = 12-16

“i

Ni: b = S-10.

1.0

0.6. 0

EFFECT IN DIFFUSION

..‘.,..‘.,..’ 1.0

‘\

1124K

2.0

3.0

Oxyge;oCpm;entration

Fig. 11. D(C)/D(O) of Mn and Ni diffusion in cx-Ti vs oxygen concentration.

These values are considerably larger than those listed in Table 5; for three dimensional lattices. b = 1.28-2.0. Although the theoretical value of b for interstitial diffusion in the HCP lattice is not known, it is considered to be not much different from those for other lattices. The diffusivity of oxygen is about two-orders of magnitude smaller than Mn or Ni. so that we.have regarded the sites occupied by oxygen as immobile blocking sites. One might argue that the mobility of blocked sites, i.e. oxygen, would increase the blocking effect. However, according to the analysis of TahirKheli (261, the mobility of blocked sites rather decreases the blocking factor b, and b -P 1, irrespective of the lattice, in the limit of extremely fast movement of the blocked sites. Thus, in order to explain the experimental values, we have to assume that some neighbouring sites surrounding the oxygen site are not available for a diffuser. 5. SUMMARY AND CONCLUSIONS

(5) The blocking factor b is the reciprocal of the correlation factor,f, for self-diffusion via the vacancy mechanism in the respective lattice, b = l/f. This relation is exact for the lattices tabulated in Table 5: the correlation factor can also be expressed in terms of the lattice Green function (see, e.g. Koiwa and Ishioka [25]). However, it is not clear at present whether the relation holds for other complex nonBravais lattices. The most advanced theoretical description of random walk of a particle on a lattice with blocked sites has been developed by Tahir-Kheli [26]. His theory describes diffusion of tagged atoms in a multicomponent alloy consisting of several species of atoms with different transition rates, and vacancies. The case of one immobile species with concentration C, one mobile tagged particle, and the rest vacancies, which is exactly the problem we are concerned with here, is obtained as a special case in his theory. The diffusion coefficient D(C) can be expressed as (see Haus and Kehr [20]) D(C)=D(O)(l =D(O)[l

-,,[I +fz]’ -f-‘C-t...].

(4)

where f is the correlation factor defined previously, which is consistent with the result by the present authors. 4.4. Experimental values of the blocking factor

By adopting the blocking model, the value of b is estimated from the experimental values of D as a function of oxygen concentration, for Mn and Ni, as shown in Fig. 11. Although the scatter is large, the values are:

The salient points of the present investigation of Mn diffusion are summarized as follows. (I) Manganese exhibits fast diffusion, which suggests some type of interstitial mechanism. (2) The diffusivity of manganese is smaller than that of Fe, Co and Ni by a factor of about 100. This difference is consistent with the correlations of the diffusivity with the atomic size and the molar volume. (3) The addition of oxygen suppresses the manganese diffusion. This is interpreted as the blocking effect of oxygen. Acknow,ledg~~e~fs-The authors wish to express their appreciation to Messrs S. Ono, T. Yamada and K. Sasaki for the help in specimen preparation. We also wish to thank the staff of the Cyclotron Subcenter for the use of the facility.

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