Gupta D. (Ed.). Diffusion Processes in Advanced Technological Materials
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The larger finite values of b imply better resolution between the GB
and lattice diffusion processes. Physically, it ensures the type-B kinetic
regime, as shown in Fig. 1.9.
LeClaire
[55]
and Cannon and Stark
[56]
have pointed out the discrep-
ancies between the Fisher and Whipple-Suzuoka analyses. Even for
large b, the Fisher solution underestimates the GB diffusivity by a fac-
tor of 1.5 to 2.7. For b 30, the ratio D
b
(Fisher)/D
b
(true) depends on
the value of b, which results in erroneous measurement of the activa-
tion energy as well. Consequently, the Fisher solution has rarely been
used in recent years. The Fisher solution, however, has the advantage
that it involves linear power of the penetration distance; hence the ori-
gin of the penetration is not needed. It is therefore very useful in thin-
film couples where the interface may not be well defined. Furthermore,
we have never experienced a problem keeping b large in thin films
since enough diffusant or radioactive tracer is trapped in the GBs due
to the fine grain structure even at low temperatures. The penetration
distance in thin-film diffusion is also small (1 mm) and measurements
are conducted at low temperatures over a limited range. Under these
conditions, the quality of linear or 65 power fit to the experimental
data is usually comparable, and the difference between the use of Fisher
or Whipple-Suzuoka solutions is only about 10 to 50%, with higher val-
ues obtained from the Fisher solutions. For the GB diffusion measure-
ments in polycrystalline bulk specimens, however, a large discrepancy
is found between the use of the two solutions. Use of Whipple-Suzuoka
is recommended only when the penetration distance does not exceed a
few percent of the average grain diameter.
In diffusion work in thin films, it is not always possible to maintain a
semi-infinite specimen thickness condition compared to the diffusant pen-
etration distance. Gilmer and Farrell
[57, 58]
have shown that a correction
must be made to account for the finite thickness ( ) of the thin-film spec-
imen to obtain true diffusivities. Figure 1.11 shows ratios of the effective
D
b
eff
to D
b
for various values of bh
o
2
where h
o
D
l
t
. True value of
D
b
occurs when bh
o
2
≈ 0.1. Physically, this implies that very large values
of b result in flat diffusion profiles, so that diffusant has reached the thin-
film substrate interface and may even have been reflected. Consequently,
both b and h
o
need to be optimized so that the profiles have sufficiently
large slope in the type-B kinetic regime.
The Arrhenius dependence of the grain boundary diffusion is similar
to that in the lattice [see Eq. (27)] and may be written as:
dD
b
∼
1
4
a
2
Z f v
o
exp[(S
bm
S
bf
)k] exp[(H
bf
H
bm
)kT]. (65)
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DIFFUSION IN BULK SOLIDS AND THIN FILMS, GUPTA 33
The entropy and enthalpy terms in grain boundaries are much smaller
than in the lattice so that the grain boundary diffusion coefficient is many
orders of magnitude larger. Section 1.4 discusses this difference.
1.4 Some Illustrative Experimental Data
This section uses some critical experimental results accumulated dur-
ing 50 years of research to illustrate the topics discussed in the preceding
sections. It is not possible to review this vast and complex field. The inten-
tion here is to provide insight into the diffusion processes of the engineering
materials. We will show examples of diffusion in single crystals, epitaxial
thin films, polycrystalline bulk and thin-film metals, and dilute alloys. We
will discuss the interplay of the various diffusion processes in specimens
with variable microstructures and the influence of impurity atoms.
Figure 1.11 Ratio of D
b
eff
to D
b
for various values of bh
o
2
.(After Gilmer and Farrell
[57,
58]
)
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1.4.1 Diffusion Profiles in Au Having Variable
Microstructure
1.4.1.1 Self-Diffusion Profiles in Au Single Crystals
We consider the case of self-diffusion in an Au single crystal meas-
ured by Makin et al. using an
198
Au radioactive tracer.
[59]
This has been the
basis of comparison for a variety of subsequent studies of the kinetics of
point defects, notably, quenched-in resistance,
[10, 15]
positron annihilation,
[22]
and nuclear magnetic resonance.
[9]
In Fig. 1.12, a tracer distribution curve
in an Au single crystal annealed at 772°C for 1.033 10
6
sec is shown
yielding a diffusion coefficient of 1.80 10
14
m
2
/sec using the Gaussian
solution [Eq. (10)]. The serial sectioning was carried out on a lathe, which
required long annealing times and limited the lowest temperature of the
measurements to 600°C. The parameters for self-diffusion are listed in
Table 1.1.
1.4.1.2 Self-Diffusion Profiles in Au Polycrystalline
Bulk Specimens
A sputtering technique was introduced to measure small diffusion
coefficients in bulk as well as thin-film specimens.
[60, 61]
In Fig. 1.13(a)
and (b), typical
195
Au tracer profiles are shown in polycrystalline bulk
Au specimens. Figure 1.13(a) shows a
195
Au tracer profile in the first
3 mm depth in an Au specimen annealed at 500°C for 4 hours, yielding
a diffusion coefficient of 2.61 10
17
m
2
/sec. This distribution is within
the grains and free from contributions of grain boundaries and even dis-
locations because of the small diffusion length in the lattice. It will be
seen in Sec. 1.4.2 that the diffusion coefficients extracted from such
tracer profiles agree well with the data obtained in a single Au crystal.
Note that diffusion could be measured at very low temperatures (250°C)
down to 10
22
m
2
/sec, which was considered to be impossible before the
introduction of the sputtering technique. Because bulk Au specimens in
general contain large-angle grain boundaries and subgrains within the
grains, in addition to the lattice defects, the penetration profiles typified
by Fig. 1.13(b) show three segments relating to diffusion along these
paths. As the profile is traversed from right to left, the diffusivities along
these paths are progressively smaller because of the larger activation
energies involved. In thin films, however, one or more diffusion paths
may be absent, depending on their microstructure. The diffusion coeffi-
cients extracted from the three regions are discussed in Sec. 1.4.2.
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DIFFUSION IN BULK SOLIDS AND THIN FILMS, GUPTA 35
Figure 1.12
198
Au tracer profile in an Au single crystal at 772°C annealed for
48 hours over a depth of 400 mm. (After Makin et al.
[59]
)
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36 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS
Figure 1.13(a)
195
Au tracer diffusion profile in a polycrystalline Au specimen
annealed at 500°C for 4 hours; obtained by sputtering.The total profile is obtained
over a depth of 3 mm. (Gupta and Tsui
[60]
)
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DIFFUSION IN BULK SOLIDS AND THIN FILMS, GUPTA 37
1.4.1.3 Self-Diffusion in Au Epitaxial Films
Figure 1.14 shows
195
Au tracer penetration profiles for epitaxial Au
films grown on (001) MgO.
[62]
Despite very long annealing times, the pro-
files maintain steep slopes because the diffusion process is very slow.
Furthermore, the epitaxial films display two regions of diffusion: At small
penetration distance, the diffusion is related to the lattice diffusion; at
deeper depths, to dissociated dislocations.
1.4.1.4 Self-Diffusion in Polycrystalline Au Films
As shown in Fig. 1.15, the
195
Au profiles in polycrystalline Au films
grown on fused quartz substrates
[63]
exhibit extremely fast diffusion at
much lower temperatures and shorter annealing times compared to the
single-crystalline bulk or epitaxial films of Au.
The penetration profiles in epitaxial and polycrystalline Au films were
analyzed according to the Whipple-Suzuoka asymptotic solution [Eq. (64)]
to extract the combined diffusivity dD
b
for dissociated dislocations and
Figure 1.13(b)
195
Au tracer profile in a polycrystalline Au specimen showing three
regions of diffusion in the lattice, sub-boundaries (dissociated dislocations), and
grain boundaries. (Gupta
[61]
)
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high-angle GBs, respectively. The activation energies for diffusion along
these two paths were determined to be 111 kJ/mol (1.16 eV) and 85 kJ/mol
(0.88 eV), respectively. The former is noticeably larger, which has rami-
fications on the core structure of dissociated dislocations and grain
boundaries.
1.4.2 Self-Diffusion Data in the Au Lattice
In Fig. 1.16, self-diffusion coefficients in the Au lattice measured by
a variety of techniques
[59, 61, 62, 64–66]
are shown over 10 orders of magni-
tude. Within the experimental errors, all the data agree with the mono-
vacancy model analyzed by Seeger and Mehrer.
[67]
However, the data in
38 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS
Figure 1.14
195
Au tracer profile in epitaxial Au films held on MgO substrates over
a depth of 0.7 mm. Note the two regions of diffusion in the lattice and dissociated
dislocations. (Gupta
[62]
)
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DIFFUSION IN BULK SOLIDS AND THIN FILMS, GUPTA 39
Figure 1.15
195
Au tracer profiles in polycrystalline Au films held on fused SiO
2
substrates over a depth of 1 mm. Note short periods of annealing and diffusion
in grain boundaries only. The blank run on the extreme right shows a shallow
and steep profile to which no diffusion process can be assigned. (Gupta and
Asai
[63]
)
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40 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS
Figure 1.16 Self-diffusion in Au from various investigations. The monovacancy
analysis of Seeger and Mehrer
[67]
is also shown.
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DIFFUSION IN BULK SOLIDS AND THIN FILMS, GUPTA 41
the Au epitaxial films
[62]
are consistently higher because of the high den-
sity of dissociated dislocations, ∼10
14
lines/m
2
, so that diffusion of the
lattice atoms also gets accelerated, in accordance with Eq. (60). Similarly,
diffusion in the lattice of the polycrystalline bulk specimens remains
unperturbed because of the large grain size (∼100 µm) and spacing of
dissociated dislocations in relation to the diffusion length 2D
l
t
of
∼100 nm. (Bulk metals commonly have a dislocation density on the
order of 10
10
/m
2
at the most.)
1.4.3 Self-Diffusion in the Au and Au-1.2 at.% Ta
Alloy Grain Boundaries
In Fig. 1.17(a) and (b), self-diffusion in polycrystalline bulk Au and
thin films is shown in the lattice and grain boundaries. The effect of 1.2
at.% Ta addition on the two self-diffusion processes is also shown.
[68]
Ta enhances solvent lattice diffusion according to the expectation of the
five-frequency model discussed in Sec. 1.2.2 [Eq. (32)]. The change in
the activation energy by about 27 kJ/mol (0.28 eV) may be attributed to
the sum of vacancy-solute binding (E
b
VS
) and the vacancy motion energy
difference (B). The former may be ∼10 kJ/mol.
The effect of adding Ta to Au grain boundary diffusion is even more
complex because it shows enhancement at high temperature and sup-
presses diffusion at lower temperatures. This is due to the solute segrega-
tion effect common to most polycrystalline metallic alloys and results in
lowering of the grain boundary energy. The details of this process are
discussed in Sec. 1.5.
In Fig. 1.17(b), self-diffusion along dissociated dislocations, the sub-
grain boundaries, is also displayed on a comparative basis in pure Au and
the Au-1.2 at.% Ta alloy. The two data are similar, within experimental
errors, with respect to the activation energies. This is because dissociated
dislocations constituting the sub-boundaries are in the lowest energy
configuration with a shorter Burger’s vector of (a/6)[21
1
], and any further
relaxation on addition of solute is unlikely. Furthermore, as seen in
Fig. 1.17(b), the pre-exponential factor for diffusion in the sub-boundaries
of the Au-1.2 at.% Ta alloy is about 8 times larger than that in the epitaxial
Au films. The difference may be attributed to Ta segregation at the sub-
boundaries, similar to the high-angle grain boundaries discussed in Sec. 1.5.4.
Indeed, in grain boundary diffusion measurements, a triple-product sdD
b
instead of dD
b
in Eq. (64) is measured in alloys, where s is the grain
boundary segregation factor.
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