8, the temperature dependence of the tracer self-diffusion coefficient, D ∗, is often, but by no means always, de-scribed by an Arrhenius relation1 D ∗ = D0exp − ∆H kBT with a pre-expone
Trang 117 Self-diffusion in Metals
17.1 General Remarks
Self-diffusion is the most fundamental diffusion process in a solid This is themajor reason in addition to application-oriented motives why self-diffusionstudies have consumed energies of many researchers Self-diffusion in a metal-
lic element A is the diffusion of A atoms In practice, in most cases tagged atoms A ∗ – either radioactive or stable isotopes – are used as tracers (see
Chap 13), which are chemically identical to the atoms of the base metal
As already mentioned in Chap 8, the temperature dependence of the
tracer self-diffusion coefficient, D ∗, is often, but by no means always,
de-scribed by an Arrhenius relation1
D ∗ = D0exp
− ∆H kBT
with a pre-exponential factor D0 and an activation enthalpy ∆H The
pre-exponential factor can usually be written as
D0= gf ν0a2exp∆S
where ∆S is called the diffusion entropy, g is a geometrical factor of the order of unity (e.g., g = 1 for the vacancy mechanism in cubic metals),
f the tracer correlation factor, ν0 an attempt frequency of the order of
the Debye frequency, and a the lattice parameter For a diffusion process
with a temperature-independent activation enthalpy, the Arrhenius diagram
is a straight line with slope −∆H/kB From its intercept – for T −1=⇒ 0 – the pre-exponential factor D0 is obtained The physical meaning of the ac-tivation parameters of diffusion depends on the diffusion mechanism and onthe lattice geometry (see also Chap 8)
Self-diffusion in metals is mediated by vacancy-type defects [1–6] Strongevidence for this interpretation comes from the following observations:
1 The Kirkendall effect has shown that the diffusivities of different kinds ofatoms in a substitutional metallic alloy diffuse at different rates (see also
1 We use in this chapter again the upper index * to indicate tracer diffusivities.
Trang 2298 17 Self-diffusion in Metals
Chaps 1 and 10) Neither the direct exchange nor the ring mechanismcan explain this observation It became evident that vacancies are respon-sible for self-diffusion and diffusion of substitutional solutes in metals inpractically all cases
2 Vacant lattice sites are the dominating defect in metals at thermal librium Studies which permit the determination of vacancy propertieswere discussed in Chap 5 These studies are based mainly on differentialdilatometry, positron-annihilation spectroscopy, and quenching experi-ments
equi-3 Isotope-effect experiments of self-diffusion (see Chap 9) are in accordancewith correlation factors which are typical for vacancy-type mechanisms [5,6]
4 Values and signs of activation volumes of self-diffusion deduced fromhigh-pressure experiments (see Chap 8) are in favour of vacancy-typemechanisms [7]
5 Formation and migration enthalpies of vacancy-type defects add up to theactivation enthalpies observed for self-diffusion (see, e.g., [5, 6, 8–10]).Self-diffusion of many metallic elements has been studied over wide tempera-ture ranges by the techniques described in Chap 13 As an example, Fig 17.1displays the tracer diffusion coefficient of the radioisotope63Ni in Ni single-crystals A diffusivity range of about 9 orders of magnitude is covered by
Fig 17.1 Diffusion of63Ni in monocrystalline Ni T > 1200 K: data from grinder sectioning [11]; T < 1200 K: data from sputter sectioning [12]
Trang 317.2 Cubic Metals 299
the combination of mechanical sectioning [11] and sputter-sectioning
tech-niques [12] The investigated temperature interval ranges from about 0.47 Tm
to temperatures close the melting temperature Tm For some metals, data
have been deduced by additional techniques For example, nuclear magneticrelaxation proved to be very useful for aluminium and lithium, where nosuitable radioisotopes for diffusion studies are available A collection of self-diffusion data for pure metals and information about the method(s) employedcan be found in [8]
A convex curvature of the Arrhenius plot – i.e deviations from Eq (17.1) –may arise for several reasons such as contributions of more than one diffusionmechanism (e.g., mono- and divacancies), impurity effects, grain-boundary
or dislocation-pipe diffusion (see Chaps 31–33) Impurity effects on solventdiffusion are discussed in Chap 19 Grain-boundary influences are completelyavoided, if mono-crystalline samples are used Dislocation influences can beeliminated in careful experiments on well-annealed crystals
17.2 Cubic Metals
Self-diffusion in metallic elements is perhaps the best studied area of state diffusion Some useful empirical correlations between diffusion and bulkproperties for various classes of materials are already discussed in Chap 8.Here we consider self-diffusivities of cubic metals and their activation param-eters in greater detail
solid-17.2.1 FCC Metals – Empirical Facts
Self-diffusion coefficients of some fcc metals are shown in Fig 17.2 as rhenius lines in a plot which is normalised to the respective melting temper-atures (homologous temperature scale) The activation parameters listed inTable 17.1 were obtained from a fit of Eq (17.1) to experimental data Thefollowing empirical correlations are evident:
Ar-– Diffusivities near the melting temperature are similar for most fcc metalsand lie between about 10−12 m2s−1 and 10−13 m2s−1 An exception is
self-diffusion in the group-IV metal lead, where the diffusivity is aboutone order of magnitude lower and the activation enthalpy higher2.– The diffusivities of most fcc metals, when plotted in a homologous tem-perature scale, lie within a relatively narrow band (again Pb provides anexception) This implies that the Arrhenius lines in the normalised plothave approximately the same slope Since this slope equals−∆H/(kB T m)
2 The group-IV semiconductors Si and Ge have even lower diffusivities than Pb atthe melting point (see Chap 23)
Trang 4300 17 Self-diffusion in Metals
Fig 17.2 Self-diffusion of fcc metals: noble metals Cu, Ag, Au; nickel group
metals Ni, Pd, Pt; group IV metal Pb The temperature scale is normalised to the
respective melting temperature T m
Table 17.1 Activation parameters D0and ∆H for self-diffusion of some fcc metals
– The pre-exponential factors lie within the following interval:
several 10−6m2
s−1 < D0
< several 10 −4m2
s−1 . (17.4)The factor gf ν0a2in Eq (17.2) is typically about 10−6m2s−1 Hence therange of D0 values corresponds to diffusion entropies ∆S between about
1 kB and 5 kB
– Within one column of the periodic table, the diffusivity in homologoustemperature scale is lowest for the lightest element and highest for theheaviest element For example, in the group of noble metals Au self-
Trang 517.2 Cubic Metals 301
diffusion is fastest and Cu self-diffusion is slowest In the Ni group, Ptself-diffusionm is fastest and Ni self-diffusion is slowest
17.2.2 BCC Metals – Empirical Facts
Self-diffusion of bcc metals is shown in Fig 17.3 on a homologous temperaturescale A comparison between fcc and bcc metals (Figs 17.2 and 17.3) revealsthe following features:
– Diffusivities for bcc metals near the melting temperature lie betweenabout 10−11 m2s−1 and 10−12 m2s−1 Diffusivities of fcc metals near
their melting temperatures are about one order of magnitude lower.– The ‘spectrum’ of self-diffusivities as a function of temperature is much
wider for bcc than for fcc metals For example, at 0.5 Tm the differencebetween the self-diffusion of Na and of Cr is about 6 orders of magnitude,whereas the difference between self-diffusion of Au and of Ni is only about
Fig 17.3 Self-diffusion of bcc metals: alkali metals Li, Na, K (solid lines); group-V
metals V, Nb, Ta (dashed lines); group-VI metals Cr, Mo, W (solid lines) The temperature scale is normalised to the respective melting temperature T
Trang 6302 17 Self-diffusion in Metals
– A common feature of fcc and bcc metals is that within one group ofthe periodic table self-diffusion at homologous temperatures is usuallyslowest for the lightest and fastest for the heaviest element of the group.Potassium appears to be an exception
Group-IV transition metals (discussed below) are not shown in Fig 17.3, cause they undergo a structural phase transition from a hcp low-temperature
be-to a bcc high-temperature phase The self-diffusivities in the bcc phases β-Ti, β-Zr and β-Hf are on a homologous scale even higher than those of the alkali metals On a homologous scale self-diffusion of β-Ti – the lightest group-IV
transition element – is slowest; self-diffusion of the heaviest group-IV
transi-tion element β-Hf is fastest In additransi-tion, β-Ti and β-Zr show upward ture in the Arrhenius diagram β-Hf exists in a narrow temperature interval,
curva-which is too small to detect curvature
Using Eqs (4.29) and (6.2) the diffusion coefficient of tracer atoms due
to monovacancies in cubic metals can be written as
D ∗ 1V = g 1V f1V a2C 1V eq ω1V (17.5)
g 1V is a geometric factor (g 1V = 1 for cubic Bravais lattices), a the lattice
pa-rameter, and f 1V the tracer correlation factor for monovacancies The atomic
fraction of vacant lattice sites at thermal equilibrium C 1V eq (see Chap 5) isgiven by
kB
exp
kB
exp
1V denote the Gibbs free energy, the enthalpy and
the entropy of vacancy migration, respectively ν0
1V is the attempt frequency
of the vacancy jump
Trang 7
. (17.10)
Then, according to Eq (17.9) the activation enthalpy ∆H 1V equals the sum
of formation and migration enthalpies of the vacancy The diffusion entropy
∆S ≈ ∆S1V = S F
1V + S M
equals the sum of formation and migration entropies of the vacancy Typical
values for ∆S are of the order of a few kB As discussed in Chap 7, the
correlation factor f 1V accounts for the fact that for a vacancy mechanism thetracer atom experiences some ‘backward correlation’, whereas the vacancyperforms a random walk The tracer correlation factors are temperature-
independent quantities (fcc: f 1V = 0.781; bcc: f 1V = 0.727; see Table 7.2)
17.2.4 Mono- and Divacancy Interpretation
In thermal equilibrium, the concentration of divacancies increases morerapidly than that of monovacancies (see Fig 5.2 in Chap 5) Even moreimportant, individual divacancies, once formed, will avoid dissociating andthereby exhibit extended lifetimes in the crystal In addition, divacancies
in fcc metals are more effective diffusion vehicles than monovacancies sincetheir mobility is considerably higher than that of monovacancies [2, 9] Attemperatures above about 2/3 of the melting temperature, a contribution
of divacancies to self-diffusion can no longer be neglected (see, e.g., the view by Seeger and Mehrer [2] and the textbooks of Philibert [3] andHeumann[4]) The total diffusivity of tracer atoms then is the sum of mono-and divacancy contributions
+ D02V exp
− ∆H 2V kBT
D ∗ 2V
The activation enthalpy of the divacancy contribution can be written as
∆H 2V = 2H F − H B + H M (17.13)
Trang 8bution Near 2/3 Tmthe divacancy contribution is not more than 10 % of the
total diffusivity and below about 0.5 Tmit is negligible
As a consequence, the Arrhenius diagram shows a slight upward curvatureand a well-defined single value of the activation enthalpy no longer exists
Fig 17.4 Self-diffusion in single-crystals of Ag: squares [15], circles [16],
trian-gles [17] Mono- and divacancy contributions to the total diffusivity are shown as dotted and dashed lines with the following Arrhenius parameters: D01V = 0.046 ×
10−4m2 −1 , ∆H 1V = 1.76 eV and D02V = 2.24 × 10 −4m2 −1 , ∆H
2V = 2.24 eV
according to an analysis of Backus et al [17]
Trang 9D ∗ 1V + D ∗ 2V
Equation (17.15) is a weighted average of the individual activation enthalpies
of mono- and divacancies
Additional support for the monovacancy-divacancy interpretation comesfrom measurements of the pressure dependence of self-diffusion (see Chap 8),
from which an effective activation volume, ∆V ef f, is obtained For the taneous contribution of the two mechanisms we have
simul-∆V ef f = ∆V 1V D
∗ 1V
D ∗ 1V + D ∗ 2V + ∆V 2V D
∗ 2V
D ∗ 1V + D ∗ 2V
which is a weighted average of the activation volumes of the individual
activation volumes of monovacancies, ∆V 1V , and divacancies, ∆V 2V Since
∆V 1V < ∆V 2V and since the divacancy contribution increases with perature, the effective activation volume increases with temperature as well.Figure 17.5 displays effective activation volumes for Ag self-diffusion An in-crease from about 0.67 Ω at 600 K to 0.88 Ω (Ω = atomic volume) near themelting temperature has been observed by Beyeler and Adda [21] andRein and Mehrer [22]
tem-Fig 17.5 Effective activation volumes, ∆V ef f , of Ag self-diffusion versus ature in units of the atomic volume Ω of Ag: triangle, square [21], circles [22]
Trang 10temper-306 17 Self-diffusion in Metals
Fig 17.6 Experimental isotope-effect parameters of Ag self-diffusion: full
cir-cles [15], triangles [25], full square [26], triangles on top [27], open circir-cles [28]
Isotope-effect measurements can throw also light on the diffusion anism, because the isotope-effect parameter is closely related to the corre-sponding tracer correlation factor (see Chap 9) If mono- and divacanciesoperate simultaneously, measurements of the isotope-effect yield an effectiveisotope-effect parameter:
mech-E ef f = E 1V D ∗
1V
D ∗ 1V + D ∗ 2V + E 2V D ∗
2V
D ∗ 1V + D ∗ 2V
E ef f is a weighted average of the isotope-effect parameters for
monovacan-cies, E 1V , and divacancies, E 2V The individual isotope effect parameter are
related via E 1V = f 1V ∆K 1V and E 2V = f 2V ∆K 2V to the tracer tion factors and kinetic energy factors of mono- and divacancy diffusion (seeChap 9) Fig 17.6 shows measurements of the isotope-effect parameter for Ag
correla-self-diffusion According to Table 7.2, we have f 1V = 0.781 and f 2V = 0.458.
The decrease of the effective isotope-effect parameter with increasing perature has been attributed to the simultaneous contribution of mono- anddivacancies in accordance with Fig 17.4
tem-17.3 Hexagonal Close-Packed and Tetragonal Metals
Several metallic elements such as Zn, Cd, Mg, and Be crystallise in the onal close-packed structure A few others such as In and Sn are tetragonal
Trang 11hexag-17.3 Hexagonal Close-Packed and Tetragonal Metals 307
Fig 17.7 Self-diffusion in single crystals of Zn, In, and Sn parallel and
perpen-dicular to their unique axes
According to Chap 2, the diffusion coefficient in a hexagonal or tetragonalsingle-crystal has two principal components:
D ∗
⊥ : tracer diffusivity perpendicular to the axis ,
D ∗
: tracer diffusivity parallel to the axis
Figure 17.7 shows self-diffusion in single-crystals of Zn, In, and Sn for both
principal directions In hexagonal Zn we have D ∗
> D ⊥ ∗; i.e diffusion parallel
to the hexagonal axis is slightly faster For the tetragonal materials In and Sn
D ∗
< D ∗ ⊥ holds true For all of these materials the anisotropy ratio D ⊥ ∗ /D ∗
is small; it lies in the interval between about 1/2 and 2 in the temperatureranges investigated Diffusivity values in hcp Zn, Cd, and Mg reach about
10−12m2s−1 near the melting temperature Such values are similar to those
of fcc metals This is not very surprising, since both lattices are close-packedstructures
Let us recall the atomistic expressions for self-diffusion in hcp metals.The hcp unit cell is shown in Fig 17.8 Vacancy-mediated diffusion can be
expressed in terms of two vacancy-atom exchange rates The rate ωaaccounts
for jumps within the basal plane and ωbfor jumps oblique to the basal plane.The two principal diffusion coefficients can be written as
3
4c2
C V eq ω b f b . (17.18)
Trang 12308 17 Self-diffusion in Metals
Fig 17.8 Hexagonal close-packed unit cell with lattice paranmeters a and c
Indi-cated are the vacancy jump rates: ω a is within the basal plane and ω boblique to it
Here a denotes the lattice parameter within the basal plane and c the lattice parameter in the hexagonal direction fa ⊥ , f b ⊥ and f b are correlation factors.The anisotropy ratio is then:
A ≡ D ∗ ⊥
D ∗
= 23
For the ideal ratio c/a =
8/3 and ω a = ω b , one finds A = 1; this remains correct if correlation is included The correlation factors and A vary with the ratio ω a /ω b For details the reader is referred to a paper by Mullen [29]
17.4 Metals with Phase Transitions
Many metallic elements undergo allotropic transformations and reveal ferent crystalline structures in different temperature ranges Such changesare found in about twenty metallic elements Allotropic transitions are first-order phase transitions, which are accompanied by abrupt changes in physicalproperties including the diffusivity Some metals undergo second-order phasetransitions, which are accompanied by continuous changes in physical prop-erties A well known example is the magnetic transition from the ferromag-netic to paramagnetic state of iron In intermetallic compounds (considered
dif-in Chap 20) also order-disorder transitions occur, which can be second der In what follows we consider two examples, which illustrate the effects ofphase transitions on self-diffusion: