Gupta D. (Ed.). Diffusion Processes in Advanced Technological Materials

Подождите немного. Документ загружается.

52 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS

estimated from Eq. (70) and its small temperature coefficient are in

agreement with the estimates that McLean

[85]

arrived at on the basis of

grain boundary melting.

1.5.4 Grain Boundary Solute Segregation Effects

Equations (67) and (68) are also applicable to the interface diffusion

in alloys. G. B. Gibbs pointed out

[86]

that in diffusion measurements in

alloys in the Harrison’s regime B, a triple product sdD

is measured

where s(T) is the solute segregation factor. The segregation factor has its

own Arrhenius temperature dependence. The alloy addition results in

reduction of the interface energy denoted by g

and increase in the free

energy for diffusion, as shown in Fig. 1.20. This is the well-known

Gibbs absorption effect.

[87]

To extract grain boundary segregation param-

eters, self-diffusion measurements of the solvent species are required in

both the lattice and the interface. From the difference of the interface

energies between the pure host and the alloy, (g

 g

), it has been

possible to compute the enthalpy and entropy for the solute segrega-

tion using Gibb’s adsorption isotherm

[87]

combined with the McLean

statistics.

[85]

The free energy for solute grain boundary segregation, ∆G,

is given by:

 g

)  ∆g

 RT ln [1  C

exp(∆GRT)]. (71)

From the interfacial free-energy decrement, ∆G, and the solute

concentration,C

, the segregation coefficient in the interface, s  C

C

can be readily obtained as a function of temperature according to McLean’s

statistics:

 C

exp(∆GRT)[1  C

exp(∆GRT)], (72)

where C

is solute concentration in the interface. Furthermore, the

Arrhenius parameters for solute segregation can also be written as:

∆G(∆HT∆S). (73)

Note that a monolayer coverage (C

) is implicit in Eq. (72), beyond

which it is not valid. The limiting value of s depends on the input of C

. It

also determines the magnitude of the entropy DS in Eq. (73), but the

enthalpy DH remains unaffected.

It is also possible to obtain s in Eq. (72) directly by measuring self-

diffusion in the Harrison’s B and C regimes separately in which sdD

and

Ch_01.qxd 11/30/04 8:36 AM Page 52

DIFFUSION IN BULK SOLIDS AND THIN FILMS, GUPTA 53

are measured. Comparing the two quantities and assuming a value for

d leads to the values of s as a function of temperature. Such an approach

has been used by Surholt et al.

[50]

for Ag and Au segregation in Cu grain

boundaries.

We have discussed the interrelationships among diffusion, solute seg-

regations, and interfacial energies in a number of diverse materials such

as dilute alloys, concentrated alloys, a lamellar eutectic system, and inter-

metallic compounds in a recent article.

[88]

In Sec. 1.4.3, the effect of Ta

solute addition in Au on the lattice and grain boundary diffusion were dis-

cussed qualitatively.

[89]

The data shown are examined further according to

Eq. (71) (see Fig. 1.22). The decrements of the grain boundary energies,

∆g

, from their values in pure Au upon addition of 1.2 at.% Ta, were first

Figure 1.22 Temperature dependence of grain boundary energies in pure Au bulk

and thin films and Au-1.2 at.% Ta alloy determined from the self-diffusion (

195

Au)

measurements. Note the signs of the slopes.

Ch_01.qxd 11/30/04 8:36 AM Page 53

computed according to Eq. (67). McLean’s statistical analysis, namely,

Eqs. (71) and (72), was then applied to the decrements to obtain s as a

function of temperature. Finally, the segregation parameters were com-

puted according to Eq. (73). The resulting parameters are listed in Table 1.5,

where data from a large number of alloys studied in a similar way are also

listed.

The solute segregation parameters in grain boundaries depend on

the elastic strain energy associated with the size difference between the

solute and the host atoms and on the electrochemical difference between

the two. The former has been examined by McLean

[85]

and by Wynblatt

and Ku.

[90]

Accordingly, the solute elastic strain contribution, ∆H

, is

given by:

∆H

 [24p k Gr

 r

)

(4G

 3kr

)], (74)

where k is the bulk modulus of the solute, G is the shear modulus of the

solvent, and r

and r

are the effective radii of the solvent and the solute

atoms, respectively. These values were obtained from Smithells and

Brandes.

[91]

The elastic strain enthalpies ∆H

for various alloys are listed

in Table 1.5 and displayed by a solid curve in Fig. 1.23 as a function of

the relative atomic size difference. The data points are for the ∆H values

obtained from diffusion data.

It is clear that most binary species, such as Pb-Sn, Pb-In, Cu-Ag,

Cu-Au, Cu-Bi, and Cu-Sb, follow the computed elastic strain energy

curve shown by the solid curve in Fig. 1.23. In the Cu-Bi and Cu-Sb

alloys, although the atomic size difference is large, their electronic char-

acteristics are similar. Consequently, their binding energies deviate little

from the computed elastic strain energies where the actual atomic size dif-

ference is implicit. Briefly, the factors that promote solid solution in an

alloy also prevent grain boundary segregation such as the close proximity

of the solute and solvent species in the Periodic Table, similarity of the

atomic size, the electronic structure in various atomic shells, and the tran-

sition state of the elements and the electro-negativity. On the other hand,

the binary species with large valence differences show large deviations

from the elastic strain energy curve. Hondros et al.

[92, 93]

have provided a

relatively complete treatment of this difference; its discussion is beyond

the scope of this chapter. Contributions arising from variable electronic

configurations can be clearly seen in the cases of the Au-Ta and Cu-Cr

alloys. In both alloys, the difference in atomic size is negligible but the

binding energy is extremely large, owing to the addition of the Ta and Cr

transition metals, which are also almost totally immiscible in Au and Cu

solvents, respectively.

54 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS

Ch_01.qxd 11/30/04 8:36 AM Page 54

Table 1.5. Arrhenius Parameters for Diffusion and Solute Segregation in Some Alloys

(10

4

/sec) sdD

(10

16

/sec)

S HH

Alloy Q

(kJ/mol) D

(kJ/mol) sdD

(mJ/m

) (R) (kJ/mol) (kJ/mol) Ref.

Pb-5In 85.8 0.15 31.8 1.7 135  0.094T 0.9 3.5 1.0 a

Pb-5Sn 80.9 0.0027 41.4 51 130  0.125T 3.4 16 2.3 a, b

Pb-5In-1Au 84.8 0.0071 43.4 23 125  0.106 0.7 9.0 1.0–2.3 a

Pb-2.7In-3.7Sn 69.4 0.00015 31.8 1.6 144  0.06T 0.7 9.0 1.0–2.3 a

Pb-62Sn 100 0.16 80 7  10

150 ... ... ... c

(Lamellae) (T  400K)

Pb-62Sn 100 0.16 ~40 ~50 233 ... ... ... c

(equiaxed) (T  400K)

Au-1.2Ta 142.5 0.0013 121.4 5000 106  0.39T 1.3 32 15.4 d

Cu-1Cr 160 0.021 115.4 9500 314  0.364T 4 83.6 0.3 e

Cu-0.13Ti 195 0.63 121.2 13000 353  0.315T 1.3 62.7 22.3 e

Cu-0.08Zr 213.2 3.9 121.2 13000 353  0.315T 1.6 65.3 55 e

Cu-Sb (AES) ... ... ... ... ... 1–3 66 55 f

Cu-Bi (AES) ... ... ... ... ... ~1 107 88 f

Cu-Sn 199.4 0.16 133.9 180 418  0.043T 1.6 42.3 58 g

Cu-Au 191 0.08 81.24 21.1 ... 0.9 9.7 22.3 h

Cu-Ag 194.4 0.61 69.1 14 ... 4.1 39.5 15.4 i

Al 303 3.59 168 3270 888  0.028T ... ... ... j

Al(B) 303 3.59 187 12400 754  0.107T ... ... ... j

a. D. Gupta and J. Oberschmidt, in Diffusion in Solids: Recent Developments (M. A. Dayananda and G. E. Murch, eds.), Metall. Soc AIME, Warrendale, PA (1985), p. 121

b. K. K. Kim, D. Gupta, and P. S. Ho, J. Appl. Phys., 53:3620 (1982)

c. D. Gupta, K. Vieregge, and W. Gust, Acta Mater., 47:5 (1999)

d. D. Gupta, Philos. Mag., 33:189 (1976)

e. G. Barreau, G. Brunel, G. Cizeron, and P. Lacombe, Mem. Sci. Rev. Metall., 68:357 (1971)

f. E. D. Hondros, in Proc. Interfaces Conf., Australian Inst. Metals, Butterworths, London (1969), p. 77

g. D. Gupta, C.-K. Hu, and K. L. Lee, Defect Diffusion Forum, 143–147:1397 (1997)

h. T. Surholt, Y. M. Mishin, and C. Herzig, Phys. Rev., B50:3577 (1994)

i. S. Divinski, M. Lohmann, and C. Herzig, Acta Mater., 49:249 (2001)

j. S. Frank, J. Rusing, and C. Herzig, Intermetall, 4:601 (1996)

Ch_01.qxd 11/30/04 8:36 AM Page 55

1.6 Diffusion in Quasicrystalline and

Amorphous Alloys

This section discusses diffusion in some noncrystalline solids to show

that long-range diffusion is possible and much of the formalism discussed

in the context of crystalline solids is applicable to these meta-stable sys-

tems as well. Discussion about diffusion in these alloys is an introduction

to the subject rather than a comprehensive review. Quasicrystalline and

amorphous phases are relatively new materials discovered in the middle

to late twentieth century by Schechtman et al.

[94]

and Duwez et al.,

[95]

respectively. Discovery of alloys with five-fold crystallographic symme-

try was very tantalizing to the physics and material community because it

contradicted the classical laws of crystallography. The quasicrystals are

actually quite stable and display many interesting properties. In 1960,

Duwez and co-workers successfully made the first metallic glass consist-

ing of metal and metalloid atomic species, notably the Pd

glass, by

melt-spinning accompanied by rapid quenching. Metallic glasses have

x-ray diffraction patterns similar to their silicate counterparts, showing

only two halos of short-range order and absence of peaks for long-range

56 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS

Figure 1.23 Grain boundary binding energy parameters from diffusion measure-

ments and elastic strain energies computed from Eq. (74) in some alloys. (See

also Table 1.5.)

Ch_01.qxd 11/30/04 8:36 AM Page 56

DIFFUSION IN BULK SOLIDS AND THIN FILMS, GUPTA 57

order. They show high tensile strengths exceeding those of crystalline

materials, and they are magnetically soft. They can be produced continu-

ously in ribbons and tapes so that many and varied applications are possi-

ble. Another innovation of great importance was the production of bulk

metallic glasses

[96]

so that the alloy can be cast in any shape and size. The

state of defects and diffusion in these meta-stable phases has been intriguing

because of their aperiodic atomic arrangement or absence of translational

symmetry. Obviously, their recrystallization behavior, phase separation,

and ultimate stability would be diffusion controlled. This section dis-

cusses diffusion processes in these materials and compares them with the

self-diffusion processes in crystalline solids, particularly the metals.

1.6.1 Diffusion in Quasicrystalline Alloys

Quasicrystalline alloys display five-fold symmetry, contrary to the

accepted laws of classical crystallography, where it is forbidden.

Quasicrystals are classified according to the dimensionality of their qua-

siperiodic order: three-dimensional, icosahedral; two-dimensional,

decagonal; and one-dimensional, packing. The AlPdMn and AlNiCo ter-

nary alloys are two common examples of the icosahedral and decagonal

quasicrystalline systems. The two systems are denoted by the prefixes I

and D, respectively. Another classification of the quasicrystal is based on

the constituting elements: the Al-transition-metal group and the Frank-

Kaspar group.

[97]

The quasicrystalline phase in the former is represented

by a group of Zn-Mg-RE (HoY) alloys. Diffusion of several radioactive

tracers (such as

Ni,

Co,

Mn,

103

Pd, and

Zn) has been studied in the

I-AlPdMn, I-ZnMgHo, and D-AlNiCo quasicrystals by investigators in

Germany and Japan. We will discuss their important results and conclu-

sions for diffusion processes operative in these novel phases.

Nakajima and Zumkley

[98]

have studied diffusion of

Ge,

Ni,

Co,

Cr, and

Mn in I-Al

single quasicrystals. Diffusion of

Zn,

114m

In,

Fe, and

Mn in I-Al

, D-Al

72.6

10.5

16.9

, and

I-ZnMgHo quasicrystals has been reviewed recently.

[98100]

For our dis-

cussion, the data for

Zn,

Co, and

Ni tracers, representing fast and

slow diffusion of the majority and minority atomic species in I and D

types of quasicrystals, have been selected. These data are shown in Fig. 1.24,

and their diffusion parameters are listed in Table 1.6. It is seen that diffu-

sion coefficients of all diffusing tracers follow linear Arrhenius relation-

ships over wide temperature ranges. The diffusion parameters are within

the permissible limits typical for metals and alloys; that is, the activation

energy Q ∼ 142T

(kJ/mol) ± 20% and the pre-exponentials factors

∼10

4

/sec within a factor of 100 (see Table 1.1). The signs and the

Ch_01.qxd 11/30/04 8:36 AM Page 57

58 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS

Figure 1.24 Diffusion of radioactive tracers in several icosahedral (I) and decago-

nal (D) quasicrystals. (See Table 1.6 for compositions and diffusion parameters.)

Ch_01.qxd 11/30/04 8:36 AM Page 58

magnitude of the activation volumes of Zn (0.74Ω at 776K) and Mn

(0.74Ω at 1023K) in quasicrystals

[99100]

are similar to those observed in

metals (see Chapter 2, Table 2.4). These observations suggest diffusion by

a vacancy mechanism in quasicrystals as well. Because quasicrystals are

essentially intermetallic compounds, the diffusion mechanism may have

similar characteristics in view of maintaining the long-range degree of

DIFFUSION IN BULK SOLIDS AND THIN FILMS, GUPTA 59

Table 1.6. Diffusion in Some Noncrystalline Alloys

Alloy Diffusant D

(10

4

/s) Q (kJ/mol) References/Remarks

Quasicr

ystals

(a) I-AlPdMn

Zn 0.27 121.3 a; fast diffuser

(b) I-ZnMgHo

Zn 2.86 150.0 b; fast diffuser

Ni 53.0 209.0 c; slow diffuser

(d) D- AlNiCo

Ni 2.2 266.5 d; slow diffuser

(e) I-AlPdMn

Co 3900.0 266.5 e; slow diffuser

(f) D-AlNiCo

Co 28.0 251.7 f; slow diffuser

Amorphous

(a) Pd

110m

Ag 2  10

6

125.3 f

(b) Co

76.7

14.3

Co,

Co 1.0  10

3

221.6 g; ∆VV

∼ 0, ∆E ∼ 0

Co,

Co 1.1  10

3

159.0 h; ∆E ∼ 0

(d) Co

Zr ... ... i; ∆VV

∼ 0.9

(e) Ni

Hf (SIMS) 7.4  10

–13

79.6 j; ∆VV

∼ 0.5

(f) Ni

Cu (SIMS) 3.3  10

3

149.3 k; ∆VV

∼ 0.2

(g) ZrTiCuNiBe B, Fe . . . . . . l; curved plot, ∆E ∼ 0

a. H. Mehrer, T. Zumkley, M. Eggersmann, R. Galler, and M. Salamon, Mater. Res. Soc. Proc.,

527:3 (1998)

b. R. Galler, R. Sterzel, Wolf Assmus, and H. Mehrer, Defect Diffusion Forum, 194–199:867

(2001)

c. T. Zumkley, H. Nakajima, and T. A. Lograsso, Philos. Mag., A80:1065 (2000)

d. C. Khoukaz, R. Galler, M. Feurerbacher, and H. Mehrer, Defect Diffusion Forum,

194–199:873 (2001)

e. W. Sprengel, H. Nakajima, and T. A. Lograss, Proc. 6th Int. Conf. Quasicrystals ICQ6

(S. Takeuchi and T. Fujiwara, eds.), World Scientific, Tokyo, Japan (1998), p. 429

f. D. Gupta, K. N. Tu, and K. W. Asai, Phys. Rev. Lett., 35:796 (1975)

g. F. Faupel, P. W. Hüppe, and K. Rätzke, Phys. Rev. Lett., 65:1219 (1990)

h. A. Heesmann, K. Rätzke, F. Faupel, J. Hoffmann, and K. Heinmann, Europhys. Lett., 29:221

(1995)

i. P. Klukgist, K Rätzke, and F. Faupel, Phys. Rev. Lett., 81:614 (1998)

j. A. Grandjean and Y. Limoge, Defect Diffusion Forum, 143–147:711 (1997)

k. Y. Loirat, Y. Limoge, and J. L. Bocquet, Defect Diffusion Forum, 194–199:827 (2001)

l. T. Zumkley, V. Naundorf, M.-P. Macht, and G. Frohberg, Defect Diffusion Forum,

194–199:801 (2001)

Ch_01.qxd 11/30/04 8:36 AM Page 59

order. Divinski and Larikov

[101]

have reviewed the mechanisms of diffu-

sion in quasicrystals at length. They have favored a diffusion mechanism

of collective cyclic movement mediated by a vacancy similar to the six-

jump cycle in B2 intermetallic compounds (see Chapter 4), with the dif-

ference that a sequence of 10 or 6 jumps may be involved in the A

B type

of atomic packing in quasicrystals. While cyclical atomic diffusion is self-

correcting to maintain order, it is an inefficient process of mass transport.

Consequently, diffusion coefficients are likely to be 2 to 4 orders of mag-

nitude smaller by virtue of a smaller pre-exponential factor rather than a

larger activation energy. The experimental data support such a hypothesis.

Kalugin and Katz

[102]

have suggested the alternate mechanism of diffusion

by phason flips of the atomic packing in quasicrystals. Such a mechanism

has not been ruled out for noble metal diffusion at low temperatures.

There does not appear to be a universal diffusion mechanism in qua-

sicrystal. The mechanism can change from case to case depending on the

type of bonding, composition, temperature, pressure, and so forth.

1.6.2 Diffusion in Amorphous Alloys: Metallic Glasses

This discussion of diffusion and related kinetic is confined to amor-

phous metallic alloys. The other groups of amorphous solids – the amor-

phous semiconductors and silicate glasses – will not be covered. The

amorphous metallic alloy group itself has become very large: It now com-

prises metal-metalloid glasses, metal-metal glasses, the recently discov-

ered bulk metallic glasses, and super-cooled liquids. We provide here a

few examples of diffusion measurements and their implications in terms

of the diffusion mechanisms that best explain the results.

Long-range diffusion in meta-stable glasses was considered unlikely

in the early years following their discovery because it would initiate crys-

tallization in these meta-stable alloys. Only short-range diffusion was

considered possible. However, in 1975, the existence of long-range diffu-

sion of

110m

Ag radioactive tracer in the Pd

metallic glass produced by

the Duwez group was demonstrated.

[103]

The x-ray diffraction patterns

taken before and after diffusion showed that the metallic glass remained

amorphous except for a little phase separation at high temperatures

approaching the glass transition temperature (T

). The diffusion coeffi-

cients, while small, could be extracted from the broadening of the

110m

profiles at the respective temperatures and yielded the activation energy

and the pre-exponential factor with reasonable precision. In addition to

the thermal broadening, the

110m

Ag profiles were also found to be dis-

placed, indicating structural relaxation of the host material. Based on the

observation of an unusually small pre-exponential factor, 2  10

10

/sec,

60 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS

Ch_01.qxd 11/30/04 8:36 AM Page 60

DIFFUSION IN BULK SOLIDS AND THIN FILMS, GUPTA 61

compared to the normal value of 10

5

/sec in close packed metals (see

Table 1.1), a cooperative atomic diffusion mechanism involving a group

of atoms was proposed without the benefit of any point defect. It is simi-

lar to the relaxion diffusion mechanism shown in Fig. 1.2(e). The pro-

posed mechanism was consistent with the dense random packing of the

metal atoms analogous to the hard-sphere packing model of Bernal, with

metalloid species filling the larger voids. It was envisioned that an atomic

volume, approximately the size of a vacancy, would be distributed among

a group or cluster of an unspecified number of atoms. It was hypothesized

that thermal vibrations below the glass transition temperature would result

in continual redistribution of the free volume, thereby permitting atomic

diffusion. The small probability of such events taking place was reflected

in a small pre-exponential factor. At the same time, diffusive jumps would

be easier because of the looser atomic packing and lower saddle-point

energy, which would result in larger diffusion coefficients than what could

be construed from a similar crystalline alloy or the viscosity data.

Diffusion data in some metallic glass systems, consisting of the metal-

metalloid, the metal-metal, and the bulk metallic glasses, are shown in

Fig. 1.25 and listed in Table 1.6. In view of very small diffusion length

(∼100 nm), ingenious experimental techniques were used. These included

radioactive tracer profiling by ion-sputtering, use of stable isotopes and

detection by secondary ion mass spectroscopy (SIMS), Rutherford back

scattering (RBS), and monitoring of the x-ray satellite in a composition-

ally modulated amorphous film package similar to that described in

Sec. 1.2.5. Rothman has described these experimental techniques.

[30]

Diffusion coefficients and the attendant parameters (Q and D

) have been

determined with good precision in most cases, and Arrhenius behavior is

observed except for the bulk metallic glass and undercooled liquids. In

addition to the determination of the diffusion parameters, activation vol-

umes (∆VV

) and the isotope effects (∆E) have been measured in some

systems to enable full characterization of the diffusion mechanism in the

respective host amorphous alloy. These parameters are listed in Table 1.6.

Frank,

[104]

Mehrer and Hummel,

[105]

and Faupel et al.

[106]

have criti-

cally reviewed diffusion in a variety of amorphous metallic alloys. There

is general agreement that (1) diffusion coefficients follow Arrhenius

behavior in the all-metallic glasses with the exception of the bulk metal-

lic glasses, and (2) there is a change from a vacancy diffusion mechanism

in the as-produced metallic glasses, by techniques such as splat-quenching,

melt-spinning or thin-film deposition, to a cooperative atomic process

mentioned above in the relaxed state.

[103]

Diffusion coefficients in the unre-

laxed state are about one order of magnitude larger compared to the

relaxed state, which is obtained within a few hours, depending on the tem-

perature of annealing. Contribution of a higher diffusion coefficient in

Ch_01.qxd 11/30/04 8:36 AM Page 61