GRAIN BOUNDARY SELF-DIFFUSION IN NICKEL: COMPARISON OF THEORETICAL AND EXPERIMENTAL DIFFUSIVITIES

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1 GRAIN BOUNDARY SELF-DIFFUSION IN NICKEL: COMPARISON OF THEORETICAL AND EXPERIMENTAL DIFFUSIVITIES Věra ROTHOVÁ a, Milan SVOBODA a, Jiří BURŠÍK a a IPM AS CR, v.v.i., Žižkova 22, Brno, Czech Republic rothova@ipm.cz, bursik@ipm.cz, svobm@ipm.cz Abstract An extremely wide spectrum of the experimental results of the grain boundary self-diffusivities in nickel was published in the literature. By using two approaches based on the empirical correlations and rules, the purpose of this study is to theoretically estimate the grain boundary (GB) self-diffusivities in nickel. The obtained dependences are compared with one another as well as with the literature experimental data. 1. INTRODUCTION Short circuit diffusion plays an important role in numerous processes involving mass transport such as Coble creep, sintering, diffusion induced grain boundary migration, recrystallization and grain growth. It controls also the evolution of microstructure during thermal treatment and has a significant effect on properties of engineering materials at elevated temperatures. Nickel and nickel-base alloys are vitally important to modern industry because of their ability to withstand a wide variety of severe operating conditions involving corrosive environments, high temperatures, high stresses, and combinations of these factors. Knowledge of basic diffusion data concerning nickel is essential to designing of new engineering alloys. Despite of large number of papers on GB self-diffusion in nickel published over the last half century, certain ambiguities concerning the large scatter in diffusion data still remain unresolved. Following the empirical rules and relationships, the aim of this paper is to estimate theoretically the GB self-diffusivities in nickel and to compare the result with the experimental data published in the literature. 2. ASSESSMENT OF SELF-DIFFUSIVITIES One of the methods of theoretical estimation of GB self-diffusion is based on the experimentally determined diffusion spectrum for fcc (face centered cubic) metals. We attempted to contextualize the GB self-diffusivities in nickel by assessing the remaining diffusion dependences. In this connection, the existing literature surveys were used to specify both the bulk and the surface diffusion. Regarding the self-diffusion in the liquid nickel, a complete summary of rare experimental data along with computed predictions is presented in this work. 2.1 Bulk diffusion The most precise measurements concerning bulk self-diffusion in metals have been carried out with single crystal specimens in a wide temperature range. Thus, selected from large number of results presented in the literature, the bulk diffusivities in nickel [1,2] listed in Tab. 1 were generally considered to be the most reliable, for complete overview see also [3].

2 Table 1. Arrhenius parameters D 0 and E for the effective bulk self-diffusion in nickel T [K] D 0 [m 2 s -2 ] E [kj mol -1 ] source [1] [2] Bulk self-diffusion in fcc metals occurs predominantly by a monovacancy mechanism, that is why the standard interpretation of tracer self-diffusion attributes the total (effective) diffusivity to monovacancy diffusion. Hovewer, at higher temperatures, a certain divacancy contribution may play an additional role. As a consequence of divacancy contribution, the Arrhenius diagram shows a slight upward curvature. Also in nickel in the temperature region K, the total bulk diffusivity D V of tracer atoms can be written as the sum of mono- and di-vacancy contributions D V 5 3 = D1 + D2 = exp( 278kJ / RT ) exp( 357kJ / RT ). (1) v v Equation (1) was proposed in [2] on the basis of treating both the high-temperature [1] and the lowtemperature data [2] together. 2.2 Diffusion in liquid state Compared to other mass transport properties, experimental data on diffusion in liquid nickel are scarce because of the technical difficulties of these measurements. By using classical techniques, diffusion experiments are hampered by convective flow and chemical reactions of the melt with the capillary especially at high temperatures. Nevertheless, the diffusion data obtained by neutron scattering experiments [4,5] are not affected by convection effects since dynamics are probed on significantly shorter time scales than the ones related to convective transport. Table 2. Diffusion coefficients D L in liquid nickel T [K] D L [m 2 s -2 ] note/description source neutron scattering Johnson77 [4] neutron scattering Chathoth04 [5] exp(-45kj/rt) (2) neutron scattering, levitated droplets T m = 1726 D L,melt = calculated from Eq. (2) Meyer08 [6] calculated, hard sphere model Protopapas [7] calculated, EAM average values from several methods Alemany98 [8]

3 Some encouraging results have been published only recently [6] concerning determination of selfdiffusion coefficients by quasielastic neutron scattering measurements of levitated Ni droplets. The absence of a sample holder makes it not only possible to extend the accessible temperature range up to 2300 K, but also into the metastable regime of the undercooled liquid several hundreds of K below the equilibrium melting point (due to the avoidance of nucleation at crucible walls). A comparison of all experimental results known to date with those calculated [7,8] using embedded atom model (EAM) is also shown in Tab. 2 and Fig. 1. It stands to reason that diffusion coefficients are in very good agreement with each other. Thus, the value of diffusivity D L,melt in nickel at melting temperature T m (see Tab. 2) is widely reliable and can be used in assessment of the rest of diffusivities as a fixed point value. D L [m 2 s -1 ] Fig. 1. Arrhenius diagram of experimental (full symbols and line) and computed (open symbols) diffusivities in liquid nickel. For the literature sources of plotted data see Table 2. Fig. 2. Arrhenius diagram of mass transfer surface diffusivities in nickel in comparison with diffusivity in liquid nickel. For the literature sources of plotted data see Tables 2 and Surface diffusion Two basic types of surface diffusion exist: intrinsic, where the number of mobile particles remains constant with temperature, and mass transfer, where the surface releases mobile particles as the temperature increases [9]. The last one only is useful for our purpose because the mass transfer surface diffusion describes transport of material across large distances (generally of the order of several µm) whereas the intrinsic diffusion is defined by movement of particles over very short distances across a surface of uniform potential. Compilation of a large number of surface diffusion data was published earlier by Gjostein [9], Bonzel [10] and Seebauer [11]. Concerning the mass transfer surface diffusion, the two-regime behavior was observed frequently in a few specific self-diffusion systems such as Ni, W, Si, Ge and Al 2 O 3. Here, the high-temperature regime is characterized by an activation energy and a prefactor many times larger

4 than those in the low-temperature regime. In Table 3 and Fig. 2, experimental data on mass transfer surface self-diffusion are presented which include the results for both regimes. Table 3. Arrhenius parameters of the mass transfer surface diffusion D S = D 0 exp(-e/rt) on Ni surface T [K] D 0 [m 2 s -2 ] E [kj mol -1 ] note/description source random surface orientation Mills69 [12] random surface orientation Bonzel68,69 [13,14] Ni (110) surface Bonzel78 [15] Following the diffusion spectrum schemes published earlier (see the next paragraph) for fcc metals it is reasonable to expect that the low-temperature surface diffusivity extrapolated to the melting point approaches the diffusivity in the liquid state. Thus, in the present work, the data of Mills [12] was selected for construction of the nickel diffusion spectrum. 2.4 Diffusion spectrum In Fig. 3, the diffusion spectrum was compiled using bulk, liquid and surface Ni self-diffusivities assessed in the previous part of this paper. Fig. 3. Diffusion spectrum in Ni. Bulk diffusivity D V [2], diffusivity in liquid D L [8], surface diffusivity D S [12]. Considering empirical rules (see the text), experimental data of GB diffusivity D B [16] should be located in the hatched area defined by Eq. (2) Fig. 4. Arrhenius diagram of the experimental GB diffusivities (black lines, [16 19]) in comparison with those compiled in this work (hatched area and blue lines, [20, 21]).

5 In consequence, a plausible values of GB diffusivities were estimated on the basis of empirical trends observed for activation energies, pre-exponential factors, and T m : i) activation energy of GB diffusion can be calculated as 40 60% of the activation energy in bulk diffusion, ii) extrapolated GB diffusivities and these in liquid Ni intersect at T m. Thus, possible values of GB diffusivity are restricted by the two relations (see the hatched area in Fig. 3) exp( 168kJ / RT ) DB exp( 112kJ / RT ). (2) Also, for comparison, our experimental results of the triple product P = δ D B [16] are presented in Fig. 3 by using the commonly accepted value of GB width δ = 0.5 nm. It stands to reason that our presented data can be satisfactorily found in the bordered area of constructed GB diffusivities. 3. EMPIRICAL RELATIONS The second method of theoretical estimation of grain boundary self-diffusion in nickel is based on the well-known empirical correlations [20, 21] between grain-boundary self-diffusion for fcc metals and the melting temperature T m. Nevertheless, these correlations were confirmed experimentally for other crystal structures in a variety of metals. Table 4. Arrhenius parameters of the GB self-diffusion in Ni calculated from empirical correlations between GB self-diffusion for fcc metals and the melting temperature for T m = 1726 K. D 0 [m 2 s -2 ] E [kj mol -1 ] source Brown80 [20] Gust85 [21] Shown in the Table 4 and Fig. 4 are the GB self-diffusivities in nickel calculated by reverse substitution for T m = 1726 K in the empirical equations describing fcc metals [20, 21]. Note that pre-exponential factors are very close, whereas a substantial difference occurs in the values of activation energy. While the data according to Gust [21] are well consistent with Eq. 2 (see fig. 4), the data from Brown [20] occur quite outside of the hatched area. Moreover, in Fig. 4, a comparison is presented of GB self-diffusivities in Ni compiled in this work with the selected (typical) experimental data. It stands to reason that the experimental data published by Wazzan, Bokstein and Rothova are consistent with the compiled dependences, while the results of Neuhaus lie mostly outside the plausible location. 4. CONCLUSION Based on the empirical correlation and rules, the theoretical dependences of the GB self-diffusivity in Ni were introduced, assessed against one another and compared with the experimental data. It was found that i) the data according to Gust are well consistent with the diffusion spectrum assessed in this work in contrast to the data from Brown, ii) the experimental data published by Wazzan, Bokstein and Rothova are consistent with the compiled dependences, while the results of Neuhaus lie mostly outside the plausible location.

6 ACKNOWLEDGEMENTS This work was supported by the Academy of Sciences of the Czech Republic under the Institute research plan No. AV0Z REFERENCES [1] BAKKER, H. A Curvature in ln D versus 1/T Plot for self-diffusion in nickel at temperatures from 980 to 1400 C. Physica Status Solidi, 1968, Vol. 28(2), pp [2] MAIER K. et al. Self-diffusion in nickel at low temperatures. Physica Status Solidi, 1976, Vol. 78 (2), pp [3] Diffusion in Solid Metals and Alloys, Vol. 26 of Landolt-Börnstein, New Series, Group III, edited by MEHRER, H. Springer-Verlag, Berlin 1990, pp [4] JOHNSON, M. W. et al. Neutron scattering and atomic dynamics in liquid nickel. Physics and Chemistry of Liquids, 1977, Vol. 6(4), pp [5] CHATHOTH, S. M. et al. Atomic diffusion in liquid Ni, NiP, PdNiP, and PdNiCuP alloys. Applied Physics Letters, 2004, Vol. 85 (21), pp [6] MEYER, A. et al. Determination of self-diffusion coefficients by quasielastic neutron scattering measurements of levitated Ni droplets. Physical Review B, 2008, Vol. 77(9), pp [7] PROTOPAPAS, P., ANDERSEN, H. C., PARLEE, N. A. D. Theory of transport in liquid metals.1. Calculation of Self- Diffusion Coefficients. Journal of Chemical Physics, 1973, Vol. 59(1), pp [8] ALEMANY, M.M.G., REY, C., GALLEGO, L.J. Computer simulation study of the dynamic properties of liquid Ni using the embedded-atom model. Physical Review B, 1998, Vol. 58(2), pp [9] GJOSTEIN N.A. Short circuit diffusion, pp in: Diffusion, American Society of Metals, Metals Park, Ohio [10] BONZEL, H.P. Surface diffusion on metals, pp , in: Diffusion in Solid Metals and Alloys, Vol. 26 of Landolt- Börnstein, New Series, Group III, edited by MEHRER, H. Springer-Verlag, Berlin 1990, 747 pages. [11] SEEBAUER, E.G., ALLEN, C.E. Estimating Surface Diffusion Coefficients. Progress in Surface Science, 1995, Vol. 49(3), pp [12] MILLS, B., DOUGLAS, P., LEAK, G.M. Transactions of the Metallurgical Society of AIME, 1969, Vol. 245(6), [13] BONZEL, H.P., GJOSTEIN, N.A. Diffraction theory of sinusoidal gratings and application to in situ surface self-diffusion measurements. Journal of Applied Physics, 1968, Vol. 39(7), p [14] BONZEL, H.P., GJOSTEIN, N.A. p. 533 in Molecular processes on solid surfaces, New York: McGraw-Hill, 1969, edited by Drauglir, E., Gretz, R.G., Jaffee, R.I, 651 pages. [15] BONZEL, H.P., LATTA, E.E. Surface self-diffusion on Ni(110): Temperature dependence and directional anisotropy. Surface Science, 1978, Vol. 76(2), pp [16] ROTHOVÁ, V., BURŠÍK, J., SVOBODA, M., ČERMÁK, J. Grain boundary self-diffusion in nickel 3. The effect of sample purity. Conference proceeding on CD ROM Metal Ostrava: Tanger, 2007, no. 81, pp [17] BOKSTEIN, S. Z. et al. Theory and experimental verification of the method for separate determination of the boundary diffusion coefficient and diffusion width of GBs. Doklady Akademii Nauk SSSR, 1985, Vol. 280(5), pp [18] WAZZAN, A. R. Lattice and grain boundary self-diffusion in Ni. Journal of Applied Physics, 1965, Vol. 36(11), p [19] NEUHAUS, P., HERZIG, C. The temperature dependence of grain boundary self-diffusion in nickel. Zeitschrift fur Metallkunde, 1988, Vol. 79(9), pp [20] BROWN, A.M., ASHBY, M.F. Correlations for diffusion constants. Acta Metallurgica, 1980, Vol. 28(8), [21] GUST, W. et al. Generalized representation of grain-boundary self-diffusion data. Journal de Physique, 1985, Vol. 46(NC4),