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

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¨urschumand coworkers

permanent interest, largely because material transport in nanostructured terials belongs to the group of material properties differing most from theircoarse-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¨urschumand coworkers[28].

ma-In the first diffusion studies in this field diffusivities in nanocrystalline Cuproduced by inert-gas condensation and subsequent consolidation were found

to be significantly faster than in grain boundaries of conventional tals (see, e.g., [41, 42]) Soon after this initital era it was recognised thatfactors such as structural relaxation, grain growth, residual porosity, dif-ferent types of interfaces, and perhaps triple junctions must be taken intoaccount to obtain an unambigous assessment of diffusion in nanocrystallinemetals More recent studies taking structural relaxation and grain growth intoconsideration came to the conclusion that diffusivities in relaxed interfaces

polycrys-of nanocrystalline metals are similar to or only slightly higher than 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 increasedstrength and ductility [44] of nanostructured materials processed by severeplastic deformation These properties have been attributed to the formation

grain-of non-equilibrium grain-boundaries with enhanced diffusivity [45] However,

so far the existence of such grain-boundary structures has not been lished by experiments

estab-Most of the experimental techniques discussed in part II of this bookhave been applied to diffusion studies on nanocrystalline metals and alloys

as well These methods include radiotracer techniques, electron microprobeanalysis, Auger electron and secondary ion mass spectrometry, Rutherfordbackscattering, and nuclear magnetic resonance The nanostructured ma-terials studied were prepared by various synthesis routes discussed aboveincluding inert-gas condensation and consolidation, severe plastic deforma-tion, mechanical milling and compaction, and crystallisation of amorphousprecursors An overview of investigations available up to 2003 for metallicnanomaterials 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 farfrom thermodynamic equilibrium, the initial structure of grain-boundariesand interfaces of bulk samples may depend on their time-temperature history.For instance, for nanocrystalline metals prepared by inert-gas condensationand subsequent compaction or by severe plastic deformation structural relax-ation effects have been reported, which lead to a decrease of the self-diffusivity

in the boundaries in nano-Fe [46] and nano-Ni [48] In both cases the boundary diffusion coefficients in the relaxed state are similar or only slightlyhigher than the values expected for conventional grain boundaries.

grain-The relaxed structure of nanocrystalline metals is prone to grain-boundarymotion 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 interfacefraction and, as a consequence of the growth-induced boundary migration

to a slowing down of tracer diffusion, since tracer atoms are immobilised byincorporation 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 sion measurements without complications caused by structural relaxation andgrain growth Despite the stable microstructure, the diffusion behaviour ofnanocrystalline alloys may still be more complex than discussed in Sect 34.3.One reason is the presence of several types of interfaces1 The existence ofmore than one type of boundaries may be a frequent feature of nanocrys-talline materials, particularly when bulk samples are prepared from powdersconsisting of agglomerates of nanograins

diffu-An interesting and well-studied example are nanocrystalline Fe-Ni loys produced during hydrogen reduction of ball-milled oxide powders (seeSect 34.2) After sintering the microstructure of these nanoalloys remainsstable 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

al-in agglomerates with an average size of 30 to 50µm In such a microstructuretwo types of interfaces exist: agglomerate boundaries and intra-agglomerateboundaries Although this complexity was not included in the theoretical dis-cussion of Sect 34.3, we illustrate the state-of-the-art below by radiotracerdiffusion in Fe-Ni nanoalloys The analysis of the diffusion experiments innano-material with a hierarchical microstructure is a sophisticated task For

a detailed discussion of the diffusion kinetics, taking into account fluxes fromthe agglomerate to the intra-agglomerate boundaries, we refer to a paper ofDivinski et al.[52]

Radiotracer experiments on nanocrystalline Fe-Ni alloys with bimodalmicrostructure are reported by Divinski et al [50, 51] The data cover

a wide temperature range and encompass diffusion in type A, B, and C netic regimes Figure 34.5 shows examples of penetration profiles of 59Fe

ki-1 Further reasons for a more complex behaviour, not considered here, can be thepresence of intergranular amorphous phases in materials obtained by crystallisa-tion of amorphous precursors and the occurrence of intergranular melting [28]

self-diffusion in nanocrystalline Fe-40 % Ni The profiles are plotted as

func-tion of the penetrafunc-tion depth y either according to Gaussian penetrafunc-tion (y2 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 tojudge 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 theadjacent 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 y2 This is indeed the case for the 1013 K profile of Fig 34.5 Fromsuch 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 grainsize and bulk diffusion length [50] Under such conditions diffusion profilesshould in general be composed of two parts (Chap 32 and Sect 34.3) Thefirst 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 of59Fe diffusion in Fe-40 % Ni nanoalloys ing either type A or type B kinetics according to Divinski et al [50]: Fe diffusion

represent-plotted as function of y2 (left) Fe diffusion plotted as function of y 6/5 (right)

than oneµm Since mechanical serial sectioning has been used, only the ond part could be observed, which corresponds to boundary diffusion Theprofiles at 852 and 751 K represent indeed Whipple-Suzuoka behaviour in the

sec-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 experimentsunder type C conditions for the same material [51] Figure 34.6 shows exam-ples of penetration profiles of59Fe 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 – fests itself in two-stage diffusion profiles Diffusivities in the grain-boundariesinside the agglomerates and diffusivities in the boundaries between the ag-glomerates have been deduced therefrom

mani-Figure 34.7 summarises grain-boundary diffusivities of Fe-Ni nanoalloysunder type A and B [50], and type C kinetics conditions [51] The resultscover a relatively wide temperature interval Data obtained in different dif-

Fig 34.6 Penetration profiles of59Fe diffusion in Fe-40 % Ni nanoalloys as function

of y2 representing type C kinetics according to Divinski et al [51] Two types ofgrain boundaries contribute to the diffusion profiles

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 gbfor

agglom-erate boundaries Filled circles and solid line: D gbfor intra-agglomerate boundaries.For comparison, grain-boundary diffusion in conventional polycrystals is shown as

dashed lines: Ni in Fe-Ni [53]; Fe in γ-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-relaxedintra-agglomerate boundaries has an activation enthalpy of about 190 kJ/moland the diffusivities in the boundaries between the agglomerates is faster byabout two orders of magnitude than that in the boundaries between thenanograins

Grain-boundary diffusion of Ni has been measured in coarse-grained crystals of Fe-Ni alloys [53] and is also shown in Fig 34.7 For Fe diffusion nodata for grain-boundary diffusion in conventional polycrystals of Fe-Ni alloysare available Therefore, the results on Fe-Ni nanoalloys are also compared

poly-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

mobilities in the intra-agglomerate boundaries of Fe-Ni nanoalloys are similar

to those in large-angle boundaries of conventional polycrystals This dence indicates that the grain-boundaries between the nanocrystallites hadsufficient time to relax into a quasi-equilibrium state during the productionprocess

coinci-34.4.4 Grain Boundary Triple Junctions

In nanocrystalline materials a further aspect is the presence of many triplejunctions A triple junction is a linear defect that is formed when three grainboundaries join (Fig 34.1) With decreasing grain size of nanocrystallinematerials both the fractions of atoms located in grain boundaries as well

as those located in triple junctions increase It is well recognised that grainboundaries act as rapid diffusion paths in metals and can dominate masstransport at lower temperatures The rˆole of diffusion along triple junctions isnot yet completely settled It is, however, not unlikely that they can make anappreciable contribution to mass transport due to their more open structurecompared to grain boundaries

A mathematical model of triple junction diffusion analogous to the Fishermodel for grain boundaries is available in the literature [56, 57] Unfortu-nately, the rˆole of triple junctions so far has been almost overlooked in theexperimental diffusion literature This is perhaps connected with the diffi-culty of separating the contribution of triple junction diffusion from the totaldiffusion flux To the author’s knowledge, only very few systematic studiesare available An example is diffusion of Zn in triple junctions of aluminiumstudied by Peteline et al [58] The authors conclude that diffusivity alongtriple junctions at 280◦C is about three orders of magnitude faster than in

ro-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 nanocrystallineceramics These examples comprise the classical oxygen ion conductor ZrO2,the anion conductor CaF , and some composite materials

Ionic Conduction: The interest in nanocrystalline ion-conducting

materi-als dates back to an observation of Liang [61] This author discovered forthe composite LiF:Al2O3 that, when the insulator Al2O3 is added to theion 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 andthe grains of the insulator Conventional DIC’s are composites of microcrys-talline materials, partially with sub-micrometer grains of the insulator Inprinciple, the conductivity enhancement may have different origins, such asthe formation of space charge layers, an enhanced dislocation density, or theformation of new phases (see [6] for references) Similar results were reportedfor the composite CuBr:TiO2 by Knauth and associates [62–64] Thesestudies also showed that the conductivity enhancement is larger for 3µmCuBr grains than for 5µm grains

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, isequal in the bulk due to the constraint of charge neutrality even though theformation enthalpies of the defects are different Near the grain-boundary ornear an interface, this constraint is relaxed due to grain-boundary or interfacecharges and the concentrations of cation and anion vacancies can be different

Fig 34.8 Conductivity of LiI:Al2O3 composites according to Liang [61]

This leads to the formation of a space charge layer The unbalanced defectconcentrations decay away in moving from the interface to the interior of thesolid The space charge layer can be treated by the classical Debye-H¨uckel

theory [65] This leads to a Debye screening length, L D, given by

L D=

&

0 r kB

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 1022

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 onthe 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 andconductivity

There are a number of investigations on dispersed ion conductors Werefrain from discussing all of them, since the results are far from beeingconclusive Instead let us in the rest of this section focus just on the effect ofparticle size on diffusion and conduction

A clearcut result has been reported by Heitjans and associates [66, 67]for conductivity studies in nanocrystalline CaF2, which is a model substancefor anionic conductors The nanocrystalline material was prepared by inert-gas condensation with a particle size of 9 nm As seen in Fig 34.10, theoverall 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 is the Debye screening length

orders of magnitude higher than in polycrystals As indicated by the solidline, the conductivity in the nanocrystalline material fits well to a spacecharge enhancement model [65, 67] The enhanced conductivity is caused bythe high number of grain boundaries Analogous results have been obtained

on nanocrystalline BaF2prepared 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 CaF2 and BaF2performed by Maier and coworkers [69] The CaF2–BaF2 heterostruc-tures were produced by molecular beam epitaxy, with layers in the nanometerregime In agreement with the space charge model, the conductivity increases

as the layer thickness decreases as shown in Fig 34.11 For distances largerthan 50 nm the conductivity is proportional to the number of interfaces Whenthe distance becomes smaller than the Debye screening length in the system(50 nm), the space charge layers of neighbouring interfaces overlap, whichleads to an even stronger increase of the conductivity At this point sin-gle interfaces loose their individual character and a nanoionic material withanomalous transport properties is generated

Diffusion and Ionic Conduction in ZrO2 and Related Materials:

A number of oxides shows fast oxygen ion conduction Such materials haveapplications 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 CaF2(circles) and of microcrystalline material (diamonds) according to Heitjans and associates [66, 67] The solid

has been calculated from the space charge layer model

Fig 34.11 Conductivity of CaF2-BaF2 layered heterostructures parallel to the

layers of thickness L according to Maier and coworkers [69]

a number of studies of nanocrystalline zirconia Common SOFC membranesusually consist of cubic stabilised ZrO2 Pure ZrO2 is monoclinic at normaltemperatures 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 cubicphase, the dopants are compensated by oxygen vacancies, which increase theconductivity

Diffusion of oxygen in nanocrystalline monoclinic ZrO2has 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 Thediffusion of 18O has been investigated by SIMS profiling The profiles could

be attributed to three contributions: (i) diffusion in the grains, (ii) diffusionalong the grain-boundaries, and (iii) diffusion due to residual pores in the

sample The grain-boundary diffusivity, D gb, is reported to be 3 to 4 orders

of magnitude larger than the diffusivity inside the grains, D A comparison

of the 18O diffusion in the lattice and grain-boundary diffusivities in ZrO2with that of other oxide ceramics is shown in Fig 34.12

The available data for ZrO2 are, however, not clearcut Firstly, tivity studies of bulk ZrO2showed that the grain-boundary diffusivity is less

conduc-than the bulk diffusivity (see, e.g., [71, 72]) This has been attributed to thesegregation of impurities into the grain boundaries forming blocking phases.However, blocking has also been proposed due to oxygen vacancy depletion

Fig 34.12 Oxygen diffusion in ZrO2 and YSZ n-ZrO2: nanocrystalline zirconia

(squares: bulk diffusion, diamonds: interface diffusion); m-ZrO2: microcrystalline

zirconia; YSZ: yttrium stabilised zirconia (dashed-dotted lines) After Schaefer

and associates[70]

in the grain-boundary space charge layers [73] Nanocrystalline YSZ with 30

to 50 nm grain size has been prepared by inert-gas condensation and the bulkand grain-boundary conductivities turned out to be similar to those of nor-mal ceramics [74] Similar results have been reported for nanocrystalline YSZwith a grain size of 90 nm [75]

Conclusion: Diffusion and ionic conduction in nanocrystalline ceramics is

far from being well understood This is mainly due to a lack of knowledgeabout the detailed microstructure, which is less well known than for nanocrys-talline metals The rˆole of sample preparation has not been resolved for manysystems The complexity of these systems is determined, for example, by thenumber of phases involved, the deviation from purely cationic or anionic con-duction, the average grain size and the width of the grain size distribution.More work is needed to avoid some of the complications found in early pa-pers Results of diffusion and conductivity are partly incompatible For thecase of ionic materials there appears to be a need for more studies of diffusionrather than conductivity measurements

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