Diffusion Solids Fundamentals Diffusion Controlled Solid State Episode 3 Part 5 pps

34.4 Diffusion in Nanocrystalline Metals 607 permanent interest, largely because material transport in nanostructured ma- terials belongs to the group of material properties differing most from their coarse-grained or single-crystalline counterparts. An overwiew of the early dif- fusion measurements on nanocrystalline metals and alloys was given in [40]. More recent developments can be found in a status report of W ¨ urschum and coworkers [28]. In the first diffusion studies in this field diffusivities in nanocrystalline Cu produced by inert-gas condensation and subsequent consolidation were found to be significantly faster than in grain boundaries of conventional polycrys- tals (see, e.g., [41, 42]). Soon after this initital era it was recognised that factors such as structural relaxation, grain growth, residual porosity, dif- ferent types of interfaces, and perhaps triple junctions must be taken into account to obtain an unambigous assessment of diffusion in nanocrystalline metals. More recent studies taking structural relaxation and grain growth into consideration came to the conclusion that diffusivities in relaxed interfaces of nanocrystalline metals are similar to or only slightly higher than grain- boundary diffusivities obtained from conventional bicrystals or polycrystals. Somewhat at variance with the finding that the grain boundary diffusivi- ties of nanocrystalline materials are siminlar to those obtained from conven- tional polycrystals are the observations of super-plasticity [43] and increased strength and ductility [44] of nanostructured materials processed by severe plastic deformation. These properties have been attributed to the formation of non-equilibrium grain-boundaries with enhanced diffusivity [45]. However, so far the existence of such grain-boundary structures has not been estab- lished by experiments. Most of the experimental techniques discussed in part II of this book have been applied to diffusion studies on nanocrystalline metals and alloys as well. These methods include radiotracer techniques, electron microprobe analysis, Auger electron and secondary ion mass spectrometry, Rutherford backscattering, and nuclear magnetic resonance. The nanostructured ma- terials studied were prepared by various synthesis routes discussed above including inert-gas condensation and consolidation, severe plastic deforma- tion, mechanical milling and compaction, and crystallisation of amorphous precursors. An overview of investigations available up to 2003 for metallic nanomaterials can be found in Table 1 of [28]. 34.4.2 Structural Relaxation and Grain Growth Since the conditions during the synthesis of nanocrystalline materials are far from thermodynamic equilibrium, the initial structure of grain-boundaries and interfaces of bulk samples may depend on their time-temperature history. For instance, for nanocrystalline metals prepared by inert-gas condensation and subsequent compaction or by severe plastic deformation structural relax- ation effects have been reported, which lead to a decrease of the self-diffusivity 608 34 Diffusion in Nanocrystalline Materials in the boundaries in nano-Fe [46] and nano-Ni [48]. In both cases the grain- boundary diffusion coefficients in the relaxed state are similar or only slightly higher than the values expected for conventional grain boundaries. The relaxed structure of nanocrystalline metals is prone to grain-boundary motion and grain growth. In this case the assessment of the diffusion be- haviour is affected by the concomitant grain-boundary migration. The oc- currence of grain growth during diffusion leads to a decrease of the interface fraction and, as a consequence of the growth-induced boundary migration to a slowing down of tracer diffusion, since tracer atoms are immobilised by incorporation in lattice sites of the crystallites. These complications may lead to deviations from diffusion profiles expected for type C kinetics. 34.4.3 Nanomaterials with Bimodal Grain Structure In a number of nanocrystalline alloys, it has been possible to carry out diffu- sion measurements without complications caused by structural relaxation and grain growth. Despite the stable microstructure, the diffusion behaviour of nanocrystalline alloys may still be more complex than discussed in Sect. 34.3. One reason is the presence of several types of interfaces 1 . The existence of more than one type of boundaries may be a frequent feature of nanocrys- talline materials, particularly when bulk samples are prepared from powders consisting of agglomerates of nanograins. An interesting and well-studied example are nanocrystalline Fe-Ni al- loys produced during hydrogen reduction of ball-milled oxide powders (see Sect 34.2). After sintering the microstructure of these nanoalloys remains stable up to fairly high temperatures of about 1100 K. Their structure is bi- modal and consists of nanocrystalline grains of about 100 nm size clustered in agglomerates with an average size of 30 to 50 µm. In such a microstructure two types of interfaces exist: agglomerate boundaries and intra-agglomerate boundaries. Although this complexity was not included in the theoretical dis- cussion of Sect. 34.3, we illustrate the state-of-the-art below by radiotracer diffusion in Fe-Ni nanoalloys. The analysis of the diffusion experiments in nano-material with a hierarchical microstructure is a sophisticated task. For a detailed discussion of the diffusion kinetics, taking into account fluxes from the agglomerate to the intra-agglomerate boundaries, we refer to a paper of Divinski et al. [52]. Radiotracer experiments on nanocrystalline Fe-Ni alloys with bimodal microstructure are reported by Divinski et al. [50, 51]. The data cover a wide temperature range and encompass diffusion in type A, B, and C ki- netic regimes. Figure 34.5 shows examples of penetration profiles of 59 Fe 1 Further reasons for a more complex behaviour, not considered here, can be the presence of intergranular amorphous phases in materials obtained by crystallisa- tion of amorphous precursors and the occurrence of intergranular melting [28]. 34.4 Diffusion in Nanocrystalline Metals 609 self-diffusion in nanocrystalline Fe-40 % Ni. The profiles are plotted as func- tion of the penetration depth y either according to Gaussian penetration (y 2 axis, left part) or according to the Whipple-Suszuoka grain-boundary solution (y 6/5 axis, right part). The profile at the highest temperature corre- sponds to type A kinetics, the two profiles at lower temperatures reveal type B kinetics. For an unambiguous assessment of these profiles it is important to judge several parameters relevant for diffusion in polycrystals: using lattice diffusivities of conventional Fe-Ni alloys [49] (and s = 1 for self-diffusion) it can be shown that the parameter, α = sδ/(2 √ Dt), is always smaller or much smaller than unity [50]. This implies that considerable out-diffusion into the adjacent grains occurs for the profiles in Fig. 34.5 and excludes type C kinet- ics. Type A diffusion kinetics emerges when diffusion fringes from neighbour- ing boundaries overlap significantly, i.e. for d/ √ Dt < 1. Then, one expects diffusion profiles which are linear in a plot of logarithm of specific activity versus y 2 . This is indeed the case for the 1013 K profile of Fig. 34.5. From such profiles an effective diffusivity can be deduced. On the other hand, if the grain-boundary fringes do not overlap, i.e. for d/ √ Dt  1, type B kinetics is expected. Values between 40 and 80 are reported for the ratio between grain size and bulk diffusion length [50]. Under such conditions diffusion profiles should in general be composed of two parts (Chap. 32 and Sect. 34.3). The first part should correspond to direct in-diffusion from the surface. However, in the experiments shown in Fig. 34.5 the bulk penetration length is smaller Fig. 34.5. Penetration profiles of 59 Fe diffusion in Fe-40 % Ni nanoalloys represent- ing either type A or type B kinetics according to Divinski et al. [50]: Fe diffusion plotted as function of y 2 (left). Fe diffusion plotted as function of y 6/5 (right) 610 34 Diffusion in Nanocrystalline Materials than one µm. Since mechanical serial sectioning has been used, only the sec- ond part could be observed, which corresponds to boundary diffusion. The profiles at 852 and 751 K represent indeed Whipple-Suzuoka behaviour in the nanomaterial. The product δD gb can be deduced from such profiles and D gb is obtained if a value for δ ≈ 0.5 nm is assumed. The effect of the bimodal microstructure has been revealed in experiments under type C conditions for the same material [51]. Figure 34.6 shows exam- ples of penetration profiles of 59 Fe self-diffusion in a plot of the logarithm of the specific activity versus penetration distance squared. The existence of two types of interfaces – agglomerate and intra-agglomerate boundaries – mani- fests itself in two-stage diffusion profiles. Diffusivities in the grain-boundaries inside the agglomerates and diffusivities in the boundaries between the ag- glomerates have been deduced therefrom. Figure 34.7 summarises grain-boundary diffusivities of Fe-Ni nanoalloys under type A and B [50], and type C kinetics conditions [51]. The results cover a relatively wide temperature interval. Data obtained in different dif- Fig. 34.6. Penetration profiles of 59 Fe diffusion in Fe-40 % Ni nanoalloys as function of y 2 representing type C kinetics according to Divinski et al. [51]. Two types of grain boundaries contribute to the diffusion profiles 34.4 Diffusion in Nanocrystalline Metals 611 Fig. 34.7. Arrhenius diagram of Fe grain-boundary diffusion in Fe-40 % Ni nanoal- loys according to Divinski et al. [51]. Open circles and solid line: D gb for agglom- erate boundaries. Filled circles and solid line: D gb for intra-agglomerate boundaries. For comparison, grain-boundary diffusion in conventional polycrystals is shown as dashed lines:NiinFe-Ni[53];Feinγ-iron [54] fusion regimes are consistent, when a value of δ ≈ 1nm is assumed for the grain-boundary width. The grain-boundary diffusivity along well-relaxed intra-agglomerate boundaries has an activation enthalpy of about 190 kJ/mol and the diffusivities in the boundaries between the agglomerates is faster by about two orders of magnitude than that in the boundaries between the nanograins. Grain-boundary diffusion of Ni has been measured in coarse-grained poly- crystals of Fe-Ni alloys [53] and is also shown in Fig. 34.7. For Fe diffusion no data for grain-boundary diffusion in conventional polycrystals of Fe-Ni alloys are available. Therefore, the results on Fe-Ni nanoalloys are also compared with grain-boundary diffusion in coarse-grained γ-Fe [54]. This comparison seems to be justified, since bulk diffusion in γ-Fe and in conventional γ-Fe-Ni alloys are not much different [55]. The comparison indicates that the atomic 612 34 Diffusion in Nanocrystalline Materials mobilities in the intra-agglomerate boundaries of Fe-Ni nanoalloys are similar to those in large-angle boundaries of conventional polycrystals. This coinci- dence indicates that the grain-boundaries between the nanocrystallites had sufficient time to relax into a quasi-equilibrium state during the production process. 34.4.4 Grain Boundary Triple Junctions In nanocrystalline materials a further aspect is the presence of many triple junctions. A triple junction is a linear defect that is formed when three grain boundaries join (Fig. 34.1). With decreasing grain size of nanocrystalline materials both the fractions of atoms located in grain boundaries as well as those located in triple junctions increase. It is well recognised that grain boundaries act as rapid diffusion paths in metals and can dominate mass transport at lower temperatures. The rˆole of diffusion along triple junctions is not yet completely settled. It is, however, not unlikely that they can make an appreciable contribution to mass transport due to their more open structure compared to grain boundaries. A mathematical model of triple junction diffusion analogous to the Fisher model for grain boundaries is available in the literature [56, 57]. Unfortu- nately, the rˆole of triple junctions so far has been almost overlooked in the experimental diffusion literature. This is perhaps connected with the diffi- culty of separating the contribution of triple junction diffusion from the total diffusion flux. To the author’s knowledge, only very few systematic studies are available. An example is diffusion of Zn in triple junctions of aluminium studied by Peteline et al. [58]. The authors conclude that diffusivity along triple junctions at 280 ◦ C is about three orders of magnitude faster than in grain boundaries. An enhanced mobility at triple junctions might also be important for mechanisms of plastic deformation of nanostructured materials that are based on grain-boundary sliding [59]. Such mechanims, especially a rigid body ro- tation of nanograins under the application of an external shear stress, have been observed in molecular dynamics simulations [60]. For steric reasons, nanograin rotations need to involve considerable atomic transport, especially near triple junctions. 34.5 Diffusion and Ionic Conduction in Nanocrystalline Ceramics Diffusion and ionic conduction in nanocrystalline ceramics has been reviewed by Heitjans and Indris [6] and by Chadwick [7]. In this section, we focus on some selected diffusion and conductivity measurements in nanocrystalline ceramics. These examples comprise the classical oxygen ion conductor ZrO 2 , the anion conductor CaF 2 , and some composite materials. 34.5 Diffusion and Ionic Conduction in Nanocrystalline Ceramics 613 Ionic Conduction: The interest in nanocrystalline ion-conducting materi- als dates back to an observation of Liang [61]. This author discovered for the composite LiF:Al 2 O 3 that, when the insulator Al 2 O 3 is added to the ion conductor LiF, the conductivity of the material increases by more than one order of magnitude (Fig. 34.8). In such systems, denoted as dispersed ionic cinductors (DIC), the enhanced conductivity has been attributed to conduction along interfacial regions between the ion-conducting grains and the grains of the insulator. Conventional DIC’s are composites of microcrys- talline materials, partially with sub-micrometer grains of the insulator. In principle, the conductivity enhancement may have different origins, such as the formation of space charge layers, an enhanced dislocation density, or the formation of new phases (see [6] for references). Similar results were reported for the composite CuBr:TiO 2 by Knauth and associates [62–64]. These studies also showed that the conductivity enhancement is larger for 3 µm CuBr grains than for 5 µmgrains. An attractive explanation for a high conductivity along the interfaces has been suggested by Maier in terms of the formation of a space-charge layer [65]. As discussed in Chap. 26, in ionic crystals the concentration of defects, e.g., cation and anion vacancies in the case of Schottky disorder, is equal in the bulk due to the constraint of charge neutrality even though the formation enthalpies of the defects are different. Near the grain-boundary or near an interface, this constraint is relaxed due to grain-boundary or interface charges and the concentrations of cation and anion vacancies can be different. Fig. 34.8. Conductivity of LiI:Al 2 O 3 composites according to Liang [61] 614 34 Diffusion in Nanocrystalline Materials This leads to the formation of a space charge layer. The unbalanced defect concentrations decay away in moving from the interface to the interior of the solid. The space charge layer can be treated by the classical Debye-H¨uckel theory [65]. This leads to a Debye screening length, L D ,givenby L D =   0  r k B q 2 C b T , (34.11) where  0 and  r are the permittivities of vacuum and sample, respectively. C b is the concentration of the majority carrier in the bulk and q its charge. For an ionic solid with  r = 10 and a bulk carrier concentrationm of 10 22 m −3 the Debye length is about 50 nm at 600 K. Thus, the effective space charge region is many times larger than the width of the boundary core, which for a grain boundary is typically 0.5 nm (see Chap. 32). The effect on the carrier concentration as the grain size decreases is illustrated in Fig. 34.9. The enhanced carrier concentration in material with grain sizes comparable or smaller than the Debye length translates into enhanced diffusivity and conductivity. There are a number of investigations on dispersed ion conductors. We refrain from discussing all of them, since the results are far from beeing conclusive. Instead let us in the rest of this section focus just on the effect of particle size on diffusion and conduction. A clearcut result has been reported by Heitjans and associates [66, 67] for conductivity studies in nanocrystalline CaF 2 , which is a model substance for anionic conductors. The nanocrystalline material was prepared by inert- gas condensation with a particle size of 9 nm. As seen in Fig. 34.10, the overall conductivity in the nanocrystalline material was found to be four Fig. 34.9. Defect concentration profiles in nanostructures of ionic materials with dimension d. L D is the Debye screening length 34.5 Diffusion and Ionic Conduction in Nanocrystalline Ceramics 615 orders of magnitude higher than in polycrystals. As indicated by the solid line, the conductivity in the nanocrystalline material fits well to a space charge enhancement model [65, 67]. The enhanced conductivity is caused by the high number of grain boundaries. Analogous results have been obtained on nanocrystalline BaF 2 prepared by ball milling [68]. A fine example for the validity of the space charge model is provided by conductivity measurements on alternating thin films of CaF 2 and BaF 2 performed by Maier and coworkers [69]. The CaF 2 –BaF 2 heterostruc- tures were produced by molecular beam epitaxy, with layers in the nanometer regime. In agreement with the space charge model, the conductivity increases as the layer thickness decreases as shown in Fig. 34.11. For distances larger than 50 nm the conductivity is proportional to the number of interfaces. When the distance becomes smaller than the Debye screening length in the system (50 nm), the space charge layers of neighbouring interfaces overlap, which leads to an even stronger increase of the conductivity. At this point sin- gle interfaces loose their individual character and a nanoionic material with anomalous transport properties is generated. Diffusion and Ionic Conduction in ZrO 2 and Related Materials: A number of oxides shows fast oxygen ion conduction. Such materials have applications in solid electrolyte membranes in solid oxide fuel cells (SOFC) and as oxygen permeation membranes (see Chap. 27). Thus, there have been Fig. 34.10. Conductivity of nanocrystalline CaF 2 (circles) and of microcrystalline material (diamonds) according to Heitjans and associates [66, 67]. The solid has been calculated from the space charge layer model 616 34 Diffusion in Nanocrystalline Materials Fig. 34.11. Conductivity of CaF 2 -BaF 2 layered heterostructures parallel to the layers of thickness L according to Maier and coworkers [69] a number of studies of nanocrystalline zirconia. Common SOFC membranes usually consist of cubic stabilised ZrO 2 .PureZrO 2 is monoclinic at normal temperatures and transforms at high temperatures to a tetragonal and then to a cubic structure. Addition of aliovalent dopants, such as yttrium (YSZ) and calcium (CSZ), stabilise at low concentrations the tetragonal phase and at higher concentrations the cubic phase. In addition to stabilise the cubic phase, the dopants are compensated by oxygen vacancies, which increase the conductivity. Diffusion of oxygen in nanocrystalline monoclinic ZrO 2 has been studied by Schaefer and associates [70]. Nanocrystalline powders were prepared by inert-gas condensation and in situ consolidation at ambient temperature and pressures of 1.8 GPa and subsequent pressureless sintering. Samples with a mass density of 97 % and an average grain size of 80 nm were obtained. The diffusion of 18 O has been investigated by SIMS profiling. The profiles could be attributed to three contributions: (i) diffusion in the grains, (ii) diffusion along the grain-boundaries, and (iii) diffusion due to residual pores in the sample. The grain-boundary diffusivity, D gb ,isreportedtobe3to4orders of magnitude larger than the diffusivity inside the grains, D. A comparison of the 18 O diffusion in the lattice and grain-boundary diffusivities in ZrO 2 with that of other oxide ceramics is shown in Fig. 34.12. The available data for ZrO 2 are, however, not clearcut. Firstly, conduc- tivity studies of bulk ZrO 2 showed that the grain-boundary diffusivity is less than the bulk diffusivity (see, e.g., [71, 72]). This has been attributed to the segregation of impurities into the grain boundaries forming blocking phases. However, blocking has also been proposed due to oxygen vacancy depletion [...]... interdiffusion in Fe2 Al 20.12 Solute diffusion of Zn, In, Ni, Co, Mn, and Cr in Fe3 Al according to [ 23, 51 ] Fe self-diffusion in Fe3 Al is also shown for comparison 32 8 33 3 33 4 33 5 33 8 20.1 34 2 34 2 34 5 34 6 34 8 34 9 34 9 35 0 35 1 35 2 35 3 35 4 List of Figures 20. 13 Schematic illustration of the sublattice vacancy mechanism in the majority sublattice of... 629 35 5 35 6 35 7 35 8 35 9 36 1 36 2 36 3 36 4 36 5 36 6 37 2 37 3 37 5 37 7 630 List of Figures 21 .5 Self-diffusion of 65 Zn in icosahedral Zn64.2 Mg26.4 Ho9.4 and Zn60.7 Mg30.6 Y8.7 quasicrystals and in a related hexagonal phase (h-ZnMgY) according to Galler et al [34 ] Dashed lines: self-diffusion in Zn parallel and perpendicular to its hexagonal axis; dotted line: Zn diffusion in icosahedral Al-Pd-Mn 37 8... 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