Diffusion Solids Fundamentals Diffusion Controlled Solid State Episode 2 Part 3 pdf

17 Self-diffusion in Metals 17.1 General Remarks Self-diffusion is the most fundamental diffusion process in a solid. This is the major reason in addition to application-oriented motives why self-diffusion studies 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 relation 1 D ∗ = D 0 exp  − ∆H k B T  , (17.1) with a pre-exponential factor D 0 and an activation enthalpy ∆H. The pre- exponential factor can usually be written as D 0 = gfν 0 a 2 exp ∆S k B , (17.2) 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, ν 0 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/k B . From its intercept – for T −1 =⇒ 0– the pre-exponential factor D 0 is obtained. The physical meaning of the ac- tivation parameters of diffusion depends on the diffusion mechanism and on the lattice geometry (see also Chap. 8) . Self-diffusion in metals is mediated by vacancy-type defects [1–6]. Strong evidence for this interpretation comes from the following observations: 1. The Kirkendall effect has shown that the diffusivities of different kinds of atoms 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. 298 17 Self-diffusion in Metals Chaps. 1 and 10). Neither the direct exchange nor the ring mechanism can explain this observation. It became evident that vacancies are respon- sible for self-diffusion and diffusion of substitutional solutes in metals in practically all cases. 2. Vacant lattice sites are the dominating defect in metals at thermal equi- librium. Studies which permit the determination of vacancy properties were discussed in Chap. 5. These studies are based mainly on differential dilatometry, positron-annihilation spectroscopy, and quenching experi- ments. 3. Isotope-effect experiments of self-diffusion (see Chap. 9) are in accordance with correlation factors which are typical for vacancy-type mechanisms [5, 6]. 4. Values and signs of activation volumes of self-diffusion deduced from high-pressure experiments (see Chap. 8) are in favour of vacancy-type mechanisms [7]. 5. Formation and migration enthalpies of vacancy-type defects add up to the activation 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. 17.1 displays the tracer diffusion coefficient of the radioisotope 63 Ni in Ni single- crystals. A diffusivity range of about 9 orders of magnitude is covered by Fig. 17.1. Diffusion of 63 Ni in monocrystalline Ni. T>1200 K: data from grinder sectioning [11]; T<1200 K: data from sputter sectioning [12] 17.2 Cubic Metals 299 the combination of mechanical sectioning [11] and sputter-sectioning tech- niques [12]. The investigated temperature interval ranges from about 0.47 T m to temperatures close the melting temperature T m . For some metals, data have been deduced by additional techniques. For example, nuclear magnetic relaxation proved to be very useful for aluminium and lithium, where no suitable radioisotopes for diffusion studies are available. A collection of self- diffusion data for pure metals and information about the method(s) employed can 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 diffusion mechanism (e.g., mono- and divacancies), impurity effects, grain-boundary or dislocation-pipe diffusion (see Chaps. 31–33). Impurity effects on solvent diffusion are discussed in Chap. 19. Grain-boundary influences are completely avoided, if mono-crystalline samples are used. Dislocation influences can be eliminated in careful experiments on well-annealed crystals. 17.2 Cubic Metals Self-diffusion in metallic elements is perhaps the best studied area of solid- state diffusion. Some useful empirical correlations between diffusion and bulk properties 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. 17.2.1 FCC Metals – Empirical Facts Self-diffusion coefficients of some fcc metals are shown in Fig. 17.2 as Ar- rhenius lines in a plot which is normalised to the respective melting temper- atures (homologous temperature scale). The activation parameters listed in Table 17.1 were obtained from a fit of Eq. (17.1) to experimental data. The following empirical correlations are evident: – Diffusivities near the melting temperature are similar for most fcc metals and lie between about 10 −12 m 2 s −1 and 10 −13 m 2 s −1 . An exception is self-diffusion in the group-IV metal lead, where the diffusivity is about one order of magnitude lower and the activation enthalpy higher 2 . – The diffusivities of most fcc metals, when plotted in a homologous tem- perature scale, lie within a relatively narrow band (again Pb provides an exception). This implies that the Arrhenius lines in the normalised plot have approximately the same slope. Since this slope equals −∆H/(k B T m ) 2 The group-IV semiconductors Si and Ge have even lower diffusivities than Pb at the melting point (see Chap. 23). 300 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 D 0 and ∆H for self-diffusion of some fcc metals Cu Ag Au Ni Pd Pt Pb [14] [16] [18] [12] [19] [20] [23] ∆H [kJ mol −1 ] 211 170 165 281 266 257 107 D 0 [10 −4 m 2 s −1 ] 0.78 0.041 0.027 1.33 0.205 0.05 0.887 ∆H/k B T m 18.7 16.5 14.8 19.5 17.5 15.2 21.4 a correlation between the activation enthalpy ∆H and the melting tem- perature T m exists (see also Table 17.1). This correlation can be stated as follows: ∆H ≈ (15 to 19) k B T m (T m in K) . (17.3) Relations like Eq. (17.3) are sometimes referred to as the rule of van Liempt [13] (see also Chap. 8). – The pre-exponential factors lie within the following interval: several 10 −6 m 2 s −1 <D 0 < several 10 −4 m 2 s −1 . (17.4) The factor gfν 0 a 2 in Eq. (17.2) is typically about 10 −6 m 2 s −1 . Hence the range of D 0 values corresponds to diffusion entropies ∆S between about 1 k B and 5 k B . – Within one column of the periodic table, the diffusivity in homologous temperature scale is lowest for the lightest element and highest for the heaviest element. For example, in the group of noble metals Au self- 17.2 Cubic Metals 301 diffusion is fastest and Cu self-diffusion is slowest. In the Ni group, Pt self-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 temperature scale. A comparison between fcc and bcc metals (Figs. 17.2 and 17.3) reveals the following features: – Diffusivities for bcc metals near the melting temperature lie between about 10 −11 m 2 s −1 and 10 −12 m 2 s −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 T m the difference between 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 3 orders of magnitude. – Self-diffusion is slowest for group-VI transition metals and fastest for alkali metals. – The Arrhenius diagram of some bcc metals shows clear convex (upward) curvature (see, e.g., Na in Fig. 17.3). 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 m 302 17 Self-diffusion in Metals – A common feature of fcc and bcc metals is that within one group of the periodic table self-diffusion at homologous temperatures is usually slowest 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, be- cause they undergo a structural phase transition from a hcp low-temperature 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 addition, β-Ti and β-Zr show upward curva- ture in the Arrhenius diagram. β-Hf exists in a narrow temperature interval, which is too small to detect curvature. 17.2.3 Monovacancy Interpretation Self-diffusion in most metallic elements is dominated by the monovacancy mechanism (see Fig. 6.5) at least at temperatures below 2 3 T m .Athigher temperatures, a certain divacancy contribution, which varies from metal to metal, may play an additional rˆole (see below). Let us first consider the monovacancy contribution: 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 f 1V a 2 C eq 1V ω 1V . (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 eq 1V (see Chap. 5) is given by C eq 1V =exp  − G F 1V k B T  =exp  S F 1V k B  exp  − H F 1V k B T  , (17.6) where the Gibbs free energy of vacancy formation is related via G F 1V = H F 1V − TS F 1V to the pertinent formation enthalpy and entropy. The exchange rate between vacancy and tracer atom is ω 1V = ν 0 1V exp  − G M 1V k B T  = ν 0 1V exp  S M 1V k B  exp  − H M 1V k B T  , (17.7) where G M 1V , H M 1V ,andS M 1V denote the Gibbs free energy, the enthalpy and the entropy of vacancy migration, respectively. ν 0 1V is the attempt frequency of the vacancy jump. 17.2 Cubic Metals 303 The standard interpretation of tracer self-diffusion attributes the total diffusivity to monovacancies: D ∗ ≈ D ∗ 1V = D 0 1V exp  − H F 1V + H M 1V k B T  . (17.8) In the standard interpretation, the Arrhenius parameters of Eq. (17.1) have the following meaning: ∆H ≈ ∆H 1V = H F 1V + H M 1V (17.9) and D 0 ≈ D 0 1V = f 1V g 1V a 2 ν 0 1V exp  S F 1V + S M 1V k B  . (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 ≈ ∆S 1V = S F 1V + S M 1V (17.11) equals the sum of formation and migration entropies of the vacancy. Typical values for ∆S are of the order of a few k B . As discussed in Chap. 7, the correlation factor f 1V accounts for the fact that for a vacancy mechanism the tracer atom experiences some ‘backward correlation’, whereas the vacancy performs 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 more rapidly than that of monovacancies (see Fig. 5.2 in Chap. 5). Even more important, individual divacancies, once formed, will avoid dissociating and thereby exhibit extended lifetimes in the crystal. In addition, divacancies in fcc metals are more effective diffusion vehicles than monovacancies since their mobility is considerably higher than that of monovacancies [2, 9]. At temperatures above about 2/3 of the melting temperature, a contribution of divacancies to self-diffusion can no longer be neglected (see, e.g., the re- view by Seeger and Mehrer [2] and the textbooks of Philibert [3] and Heumann [4]). The total diffusivity of tracer atoms then is the sum of mono- and divacancy contributions D ∗ = D 0 1V exp  − ∆H 1V k B T     D ∗ 1V + D 0 2V exp  − ∆H 2V k B T     D ∗ 2V . (17.12) The activation enthalpy of the divacancy contribution can be written as ∆H 2V =2H F 1V − H B 2V + H M 2V . (17.13) 304 17 Self-diffusion in Metals Here H M 2V and H B 2V denote the migration and binding enthalpies of the diva- cancy. For fcc metals the pre-exponential factor of divacancy diffusion is D 0 2V = g 2V f 2V a 2 ν 0 2V exp 2S F 1V − S B 2V + S M 2V k B . (17.14) f 2V is the divacancy tracer correlation factor, g 2V a geometry factor, ν 0 2V the attempt frequency, S M 2V and S B 2V denote migration and binding entropies of the divacancy. Measurements of the temperature, mass, and pressure dependence of the tracer self-diffusion coefficient have proved to be useful to elucidate mono- and divacancy contributions in a quantitative manner. As an example of divacancy-assisted diffusion in an fcc metal, Fig. 17.4 shows the Arrhenius daigram of silver self-diffusion according to [15–17]. A fit of Eq. (17.12) to the data, performed by Backus et al. [17], resulted in the mono- and di- vacancy contributions displayed in Fig. 17.4. Near the melting temperature both contributions in Ag are about equal. With decreasing temperature the divacancy contribution decreases more rapidly than the monovacancy contri- bution. Near 2/3 T m the divacancy contribution is not more than 10 % of the total diffusivity and below about 0.5 T m it is negligible. As a consequence, the Arrhenius diagram shows a slight upward curvature and 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: D 0 1V =0.046 × 10 −4 m 2 s −1 ,∆H 1V =1.76 eV and D 0 2V =2.24 × 10 −4 m 2 s −1 ,∆H 2V =2.24 eV according to an analysis of Backus et al. [17] 17.2 Cubic Metals 305 Instead an effective activation enthalpy, ∆H eff , can be defined (see Chap. 8), which reads for the simultaneous action of mono- and divacancies ∆H eff =∆H 1V D ∗ 1V D ∗ 1V + D ∗ 2V +∆H 2V D ∗ 2V D ∗ 1V + D ∗ 2V . (17.15) Equation (17.15) is a weighted average of the individual activation enthalpies of mono- and divacancies. Additional support for the monovacancy-divacancy interpretation comes from measurements of the pressure dependence of self-diffusion (see Chap. 8), from which an effective activation volume, ∆V eff , is obtained. For the simul- taneous contribution of the two mechanisms we have ∆V eff =∆V 1V D ∗ 1V D ∗ 1V + D ∗ 2V +∆V 2V D ∗ 2V D ∗ 1V + D ∗ 2V , (17.16) 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 tem- 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 the melting temperature has been observed by Beyeler and Adda [21] and Rein and Mehrer [22]. Fig. 17.5. Effective activation volumes, ∆V ef f ,ofAgself-diffusionversus temper- ature in units of the atomic volume Ω of Ag: triangle, square [21], circles [22] 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 circles [28] Isotope-effect measurements can throw also light on the diffusion mech- anism, because the isotope-effect parameter is closely related to the corre- sponding tracer correlation factor (see Chap. 9). If mono- and divacancies operate simultaneously, measurements of the isotope-effect yield an effective isotope-effect parameter: E eff = E 1V D ∗ 1V D ∗ 1V + D ∗ 2V + E 2V D ∗ 2V D ∗ 1V + D ∗ 2V . (17.17) E eff 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 correla- tion factors and kinetic energy factors of mono- and divacancy diffusion (see Chap. 9). Fig 17.6 shows measurements of the isotope-effect parameter for Ag 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 tem- perature has been attributed to the simultaneous contribution of mono- and divacancies in accordance with Fig. 17.4. 17.3 Hexagonal Close-Packed and Tetragonal Metals Several metallic elements such as Zn, Cd, Mg, and Be crystallise in the hexag- onal close-packed structure. A few others such as In and Sn are tetragonal. [...]... 57, 22 5 (19 73) 17 J.G.E.M Backus, H Bakker, H Mehrer, Phys Stat Sol (b) 64, 151 (1974) 18 M Werner, H Mehrer, in: DIMETA 82 – Diffusion in Metals and Alloys, F.J Kedves, D.L Beke (Eds.), Trans Tech Publications, Switzerland, 19 83, p 39 3 19 N.L Peterson, Phys Rev A 136 , 568 (1964) 20 G Rein, H Mehrer, K Maier, Phys Stat Sol (a) 45, 25 3 (1978) 3 12 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 17... (Eds.), University of Tokyo Press, 19 82, p 558 J Mullen, Phys Rev 124 (1961) 1 7 23 M L¨bbehusen, H Mehrer, Acta Metall Mater 38 , 28 3 (1990) u Y Iijima, K Kimura, K Hirano, Acta Metall 36 , 28 11 (1988) F.S Buffington, K.I Hirano, M Cohen, Acta Metall 9, 434 (1961) C.M Walter, N.L Peterson, Phys Rev 178, 922 (1969) J.F Murdock, T.S Lundy, E.E Stansbury, Acta Metall 12 1 033 (1964) F Dyment, Proc 4th Int Conf... D⊥ 2 a2 (3 a fa⊥ + ωb fb⊥ ) ∗ = 3 c2 D ωb fb (17.19) If correlation effects are negelected, i.e for fa⊥ = fb⊥ = fb = 1, we get from Eq (17.19) 2 a2 ωa A ≈ +1 (17 .20 ) 3 2 3c ωb For the ideal ratio c/a = 8 /3 and ωa = ωb , one finds A = 1; this remains correct if correlation is included The correlation factors and A vary with the ratio ωa /ωb For details the reader is referred to a paper by Mullen [29 ]... Metals M Beyeler, Y Adda, J Phys (Paris) 29 , 34 5 (1968) G Rein, H Mehrer, Philos Mag A 45,767 (19 82) J.W Miller, Phys Rev 181, 10905 (1969) R.H Dickerson, R.C Lowell, C.T Tomizuka, Phys Rev A 137 , 6 13 (1965) P Reimers, D Bartdorff, Phys Stat Sol (b9 50, 30 5 (19 72) N.L Peterson, L.W Barr, A.D Le Claire, J Phys, C6, 20 20 (19 73) Chr Herzig, D Wolter, Z Metallkd 65, 27 3 (1974) H Mehrer, F Hutter, in: Point... influence of Fig 17.9 Self-diffusion in the α-, γ- and δ-phases of Fe: full circles [30 ]; open circles [31 ]; triagles [ 32 ] ; squares [33 ] 31 0 17 Self-diffusion in Metals ferromagnetic order on diffusion The simplest one correlates the variation of the activation enthalpy according to 0 D∗ = Dp exp − ∆Hp (1 + αS 2 ) kB T (17 .21 ) with some ferromagnetic order parameter S The long-range order parameter is connected... 69/70, 3 (1978) 6 H Mehrer, J Nucl Mat 69/70, 38 (1978) 7 H Mehrer, Defect and Diffusion Forum 129 – 130 , 57 (1996) 8 H Mehrer, N Stolica, N.A Stolwijk, Self-diffusion in Solid Metallic Elements, Chap 2 in: Diffusion in Solid Metals and Alloys, H Mehrer (Vol.-Ed.), Landolt-B¨rnstein, Numerical Data and Functional Relationships in Science o and Technology, New Series, Group III: Crystal and Solid State Physics,... 11 H Bakker, Phys Stat Sol 28 , 569 (1968) 12 K Maier, H Mehrer, E Lessmann, W Sch¨ le, Phys Stat Sol (b) 78, 689 u (1976) 13 B.S Bokstein, S.Z Bokstein, A.A Zhukhovitskii, Thermodynamics and Kinetics of Diffusion in Solids, Oxonian Press, New Dehli, 1985 14 S.J Rothman, N.L Peterson, Phys Stat Sol 35 , 30 5 (1969) 15 S.J Rothman, N.L Peterson, J.T Robinson, Phys Stat Sol 39 , 635 (1970) 16 N.Q Lam, S.J... and D0 ≡ ga2 ν 0 exp (18 .2) SM kB , (18 .3) where H M and S M denote the migration enthalpy and entropy of the intertitial With reasonable values for the migration entropy (between zero and several kB ) and the attempt frequency ν 0 (between 10 12 and 10 13 s−1 ), Eq (18 .3) yields the following limits for the pre-exponential factors of interstitial solutes: (18.4) 10−7 m2 s−1 ≤ D0 ≤ 10−4 m2 s−1 The experimental... soluTable 18 .2 Activation parameters of H diffusion in Pd, Ni, and Fe according to ¨ Alefeld and Volkl [ 13] Pd 0 2 −1 Ni (T < TC ) Ni (T > TC ) Fe (above 30 0 K) −7 D /m s 2. 9 × 10 ∆H/eV (or kJ mol−1 ) 0 . 23 (22 .1) 4.76 × 10−7 0.41 (39 .4) 6.87 × 10−7 0. 42 (40.4) 4.0 × 10−4 0.047 (4.5) ... details to Part II of this book: – – – The radiotracer method can be applied using the isotope tritium (3 H) Its half-life is t1 /2 = 12. 3 years and it decays by emitting weak β-radiation Nuclear reaction analysis (NRA) can be used to measure concentration profiles (see Chap 15) For example, the 15 N resonant nuclear reaction H(15 N, αγ) 12 C has been used to determine H concentration profiles In steady-state . Pb [14] [16] [18] [ 12] [19] [20 ] [ 23 ] ∆H [kJ mol −1 ] 21 1 170 165 28 1 26 6 25 7 107 D 0 [10 −4 m 2 s −1 ] 0.78 0.041 0. 027 1 .33 0 .20 5 0.05 0.887 ∆H/k B T m 18.7 16.5 14.8 19.5 17.5 15 .2 21.4 a correlation. 19 82, p. 558 29 . J. Mullen, Phys. Rev. 124 (1961) 1 7 23 30 . M. L¨ubbehusen, H. Mehrer, Acta Metall. Mater. 38 , 28 3 (1990) 31 . Y. Iijima, K. Kimura, K. Hirano, Acta Metall. 36 , 28 11 (1988) 32 . . is D 0 2V = g 2V f 2V a 2 ν 0 2V exp 2S F 1V − S B 2V + S M 2V k B . (17.14) f 2V is the divacancy tracer correlation factor, g 2V a geometry factor, ν 0 2V the attempt frequency, S M 2V and S B 2V denote