Diffusion Solids Fundamentals Diffusion Controlled Solid State Episode 2 Part 10 pps

Large circles: I − ions; filled small circles: octahedral sites; filled squares: tetrahedral sites; filled triangles: trigonal sites.. Filled squares: tetrahedral sites; small filled circles

Fig 27.2 Crystal structure of α-AgI Large circles: I − ions; filled small circles: octahedral sites; filled squares: tetrahedral sites; filled triangles: trigonal sites Oc-

tahedral, tetrahedral, and trigonal sites can be used by Ag+ions

Fig 27.3 Probabiliy distribution of Ag in α-AgI at 300 ◦C according to Cava,Reidinger, and Wuensch[42]

tetrahedral sites The data also indicate that the Ag+ ions are preferentiallyfound in oblong ellipsoidal regions centered at the tetrahedral sites and ex-tending in the directions of the neighbouring octahedral sites This suggeststhat the motion of the Ag+ ions is not completely liquid-like and that thewords, diffusion of Ag ions occurs mainly by jumps between neighbouringtetrahedral sites

Besides α-AgI, the phases α-CuBr, α-Ag2S, α-Ag2Se, and α-Ag3SI havebcc anion structures The number of cations per bcc unit cell is two for

α-AgI and α-CuBr, three for α-Ag3SI, and four for α-Ag2S and α-Ag2Se In

27.1 Fast Silver-Ion Conductors 479

Fig 27.4 Cation pathway in an fcc anion sublattice according to Funke [19].

Filled squares: tetrahedral sites; small filled circles: octahedral sites

contradistinction to α-AgI and α-CuBr, which have bcc anion lattices, the anion lattice of α-CuI is fcc This is understandable from the ratios of the

cationic and anionic radii of these compounds, as the cation sites provided

by an fcc lattice are smaller in size when compared to those of a bcc lattice.The same systematic variation of the anion structure is observed in the case

of the Ag and Cu chalcogenides; while α-Ag2S and α-Ag2Se still exhibit the

bcc structure, α-Ag2Te, α-Cu2S and α-Cu2Se have fcc arrangements.Possible cation diffusion paths in fcc and hcp anion lattices have beendiscussed in [43] In an fcc unit cell, there are 8 tetrahedral and 4 octahedralinterstitial sites The cation diffusion paths consist of alternating octahedraland tetrahedral sites Cations jump from tetrahedron to octahedron to tetra-hedron etc An almost linear pathway is illustrated in Fig 27.4 Each aniontetrahedron shares four faces with four octahedra and each octahedron witheight tetrahedra This structure provides a large variety of pathways throughthe anion lattice

27.1.2 RbAg 4 I 5 and related Compounds

There have been several attempts to obtain better Ag ion conduction One

was to stabilise α-AgI at lower temperatures Another was to find new highly

conducting phases by substitution The most successful seems to be the

par-tial replacement of Ag by Rb in α-RbAg4I5 This material has still todayone of the highest ionic conductivities at room temperature (0.25 Scm−1) of

any known crystalline substance (see Fig 27.1) Its electronic conductivity isnegligibly small (about 10−9S cm−1) Some related compounds with similar

properties are MAg4I5, with M = K, Cs, and NH4

The crystal structure of α-RbAg4I5 and its isomorphs is different from

that of α-AgI and rather complex The arrangement of the 20 iodine ions

in the unit cell is similar to that of Mn atoms in the β-Mn structure and

provides 56 tetrahedral voids for the 16 Ag+ ions, while the 4 Rb+ ions areimmobilised at distorted octahedral environments of I−ions [29] Again there

are many more available sites than Ag+ions to fill them RbAg4I5undergoesphase transitions at 209 K and at 122 K The one at 209 K is a second orderphase transition with a discontinuity in the temperature derivative of the

conductivity, dσ/dT , while the conductivity is continuous The transition at

122 K is first order, which entails a sudden change in conductivity of severalorders of magnitude

A disordered α-AgI-type structure can also be stabilised at low

tempera-tures by a variety of cations, notably large alkalis, NH+4, and certain organiccations Some examples, all of which have room temperature conductivities

in the range of 0.02 Scm−1 to 0.2 Scm−1, are (NH4)Ag4I5, [(CH3)4]2Ag13I15

and PyAg5I6, where Py+ is the pyridinium ion (C5H5NH)+ A range of ions may partially substitute for iodine to form, e.g., Ag3SI, Ag7I4PO4 and

an-Ag6I4WO4

Fast fluor-ion conduction in PbF2, which has the fluorite structure type CaF2), was observed already by Michael Faraday Several halidesand oxides with the fluorite structure are very good anion conductors Otheralkaline earth fluorides, e.g., SrCl2, and β-PbF2 adopt this structure Theymay be classified as fast ion condcutors at high temperatures, where theyhave high halogen ion conductivity One of the best examples is PbF2 with

(proto-σ ≈5 Scm −1 at about 500◦C Above this temperature, the conductivity

in-creases slowly and there is little, if any, change in conductivity on melting at

822◦C.

The fluorite structure consists of simple cubes of anions, half of themoccupied by cations at the cube centers (Fig 27.5) The sites available forinterstitial F− ions are at the centers of the set of unoccupied cubes In

creating an interstitial F− ion, one corner F− ion must leave its corner site

and move into the body of the cube Defect complexes probably form, butthe details of the sites occupied are not fully known

At low to moderate temperatures, fluorite-structured halides are like mal ionic solids; they contain low concentrations of anion Frenkel pairs Onlythe anions are mobile Most fluorites and anti-fluorites exhibit a broad spe-

nor-cific heat anomaly which passes through a maximum temperature, Tc, a few

hundred degrees below the melting temperature In the same temperatureregime as the thermal anomaly, the ionic conductivity increases rapidly to

the extent that above Tc it reaches about 1 Scm−1 The high temperature

activation enthalpy is about 0.2 eV This behaviour is attributed to a tion, which involves disordering of the anion sublattice, a transition which is

transi-called the Faraday transition.

27.3 Stabilised Zirconia and related Oxide Ion Conductors 481

Fig 27.5 Fluorite structure (prototype CaF2): Filled circles represent anions and open circles cations Diamonds represent sites for anion interstitials

27.3 Stabilised Zirconia

and related Oxide Ion Conductors

The high-temperature cubic polymorph of zirconia (ZrO2) has the fluoritestructure as well At room temperature, pure ZrO2 is monoclinic However,the fluorite structure can be stabilised by additions of Y2O3 or CaO Suchstabilised zirconias (e.g., yttrium stabilised zirconia = YSZ) are good O2−

ion conductors at high temperatures This is because the formation of a solidsolution between ZrO2 and Y2O3 (or CaO) introduces vacant sites in theoxygen sublattice in order to preserve charge neutrality For example, lime-stabilised zirconia (CSZ) has the formula CaxZr(1−x)O(2−x) with 0.1 ≤ x ≤

0.2 One O2− ion vacancy is created for each Ca2+ ion that is introduced.Typical conductivities in stabilised zirconia (e.g., 85 mol % ZrO2, 15 %CaO) are about 5×10 −2S cm−1at 1000◦C with activation enthalpies around

1.3 eV At lower temperatures, stabilised zirconias have conductivities thatare many orders of magnitude smaller than those of good Ag+ and Na+

ion conductors The usefulness of zirconias stems from the fact that theyare refractory materials, which can be used to very high temperatures andhave good oxygen-ion conduction CeO2, HfO2, and ThO2may also be dopedheterovalently and are then good O2− ion conductors as well.

Increasing the point defect concentration increases the ionic conductivity

A compound in which this occurs naturally is bismuth oxide, Bi2O3 This

ma-terial has a solid-state phase transformation to a fluorite-structured δ-phase.

In this structure, 25 % of the anion sites are vacant It is hardly surprising thatdue to the structural vacancies this compound has a very high O2− conduc-

tivity [44] The highest oxygen-ion conductivities are found in Bi2O3-basedmaterials However, most of these are readily susceptible to reduction, thusbecoming mixed electron-ion conductors Therefore, they cannot be used assolid electrolytes in reducing atmospheres or at low oxygen partial pressure

Fig 27.6 Perovskite structure

There have been attempts to stabilise the high-temperature phase at lowertemperatures by doping, e.g., with zirconia and vanadium oxide

27.4 Perovskite Oxide Ion Conductors

Perovskites have the general formula ABO3 The perovskite structure is trated in Fig 27.6 The structure prototype is CaTiO3 and has a primitivecubic unit cell It contains one Ca2+ion per unit cell, e.g., at the cube edges,one Ti4+ ion in the cube center, and O2− ions at the face centers.

illus-Perovskite type oxides based on LaGaO3 are of considerable interest cause of their high oxygen-ion conductivity As for other materials, doping

be-is a convenient strategy to increase the ionic conductivity of perovskite-typeoxides Lanthanum gallates doped with Sr on La sites and with Mg on Gasites, La(1−x)SrxGa(1−y)MgyO[3−(x+y)/2] (LSGM), reach higher oxygen-ion

conductivities than YSZ [47] After optimising the single-phase composition

of LSGM an oxide-ion conductivity of 0.15 Scm−1 at 800◦C is stable over

time at any oxygen partial pressures between 10−23and 1 atm [48] This

con-ductivity is comparable to that of YSZ at 1000◦C Therefore, LSGM appears

to be a more promising electrolyte than YSZ for solid oxide fuel cells ating below 800◦C Cation diffusion in perovskites is known to be very slow.

oper-Nevertheless, one long term degradation effect may be due to a demixing ofthe electrolyte because of different cation diffusivities [49]

A family of phases with the general formula M2OnX2O3, where n is in the

range of 5 to 11, is denoted as β-alumina M is a monovalent cation (alkali+,

Cu+, Ag+, Ga+, In+, Tl+, NH+4, H2O+) and X is a trivalent cation (Al3+,

Ga3+, or Fe3+) The most important member of this family is sodium

β-alu-mina with M = Na+and X = Al3+, which has been long known as a

byprod-uct of the glass-making industry Interest in the β-aluminas began in the

27.5 Sodium β-Alumina and related Materials 483

Fig 27.7 Sites for Na+ions in the conduction plane of β-alumina m: mid-oxygen position, br: Beevers-Ross site, abr: anti-Beevers-Ross site Open circles: O 2− , grey circles: O 2−spacer ions

1960s with the pioneering work at the Ford Motor Company when Yao andKummer detected that the Na+ ions are very mobile at room temperatureand above [5]

The high conductivity of monovalent ions in β-alumina is a consequence

of its unusual crystal structure It is built of close-packed layers of oxygenions, stacked in three dimensions Every fifth layer has three-quarters of itsoxygens missing The Na+ions reside in these oxygen-deficient layers and areeasily mobile, because their radius is smaller than that of the O2− ions β- aluminas exist in two structural modifications, called β and β , which differ inthe stacking sequence of the layers The β form occurs with Na-rich crystalswhere n ≈ 5 − 7, whereas the β-form occurs for n ≈ 8 − 11 Both structures

are closely related to that of spinel (MgAl2O4) and may be regarded as beingbuilt of ‘spinel blocks’ The blocks are four oxide layers thick and their oxygenlayers are in cubic stacking sequence, separated by the oxygen-deficient layers

of the conduction planes

The atomic structure within the conduction plane has been the subject

of much crystallographic work The present understanding is as follows: sitesavailable for Na+ ions in the conduction plane of the β-modification are

shown in Fig 27.7 The conduction plane consists of close-packed layers of

O2− ions separated by pairs of O2− ions The ‘spacer’ O2− ions (grey) are

located in the conduction plane Only one quarter of the available O2− sites

in the conduction plane are occupied, i.e for every grey O2− ion there are

three empty sites Na+ions can occupy three different sites: the ‘mid-oxygen’

positions (m), the ‘Beevers-Ross’ sites1(br), and the ‘anti-Beevers-Ross’ sites (abr) It appears that Na+ ions spend most of their time in m and br sites,

1 These sites were favoured in the original structure determination of Beevers

Fig 27.8 Conductivities of some single crystal β-aluminas according to West [45]

but in order to undergo long-range migration they must pass through the abr sites, which are much smaller than the m and br sites.

The β-aluminas are two-dimensional conductors Alkali ions can move

easily within the conduction planes but cannot penetrate the dense spinel

blocks Most other monovalent ions also prefer the br and m sites in

β-alumina, with the exception of Ag+and Tl+which prefer the abr sites This

is understandable, because Ag+ and Tl+ prefer covalent binding and sites

of low oxygen coordination The conductivities of various β-alumina single

crystals (Fig 27.8) parallel to the conduction plane fit Arrhenius equationsover wide ranges of temperature The conductivity is highest and the activa-tion enthalpy lowest for Na+and Ag+β-alumina With increasing cation size

(K+, Tl+) the conductivity becomes lower, since the larger cations cannotmove as easily in the conduction planes

There are other layered materials in which the conductivity is dimensional On the whole they have not been as thoroughly studied as the

two-β-aluminas An example of a three dimensional conductor is the Na+ductor Na3Zr2PSi2O12 [46], which is now referred to as NASICON (Na su-

-con-perionic conductor) Like β-alumina it is a ceramic material, but at 300 ◦Cits conductivity is higher than that of β-alumina.

27.6 Lithium Ion Conductors

Materials that have high Li+-ion conductivity are used as electrolytes inlithium batteries The enormous, world-wide interest in such devices arises

27.7 Polymer Electrolytes 485

because cells containing Li anodes generally have a higher emf than

corre-sponding cells containing, e.g., Na anodes Thus, commercial lithium ies currently have 4 V single cells with an anode containing Li metal and anintercalation cathode based on LiCoO2 or on the spinel LiCoMnO4 Solid-state lithium batteries have important applications in a variety of consumerand medical products The batteries consist of cathodes that are crystalline

batter-or nanocrystalline oxide-based lithium intercalation compounds At present,most Li cells still work with liquid, non-aqueous electrolytes such as LiPF6disolved in an organic solvent (see, however, Sect 27.7) Sometimes the elec-trolyte is a glassy lithium phosphorous oxynitride (‘Lipon’) [50]

Conductivity data of some solid Li+ion conductors are shown in Fig 27.1

Li2SO4 undergoes a phase transition at 572◦C and has a high conductivity

around 1 Scm−1 in its high-temperature phase Above that temperature,

many substituted sulphates have been studied in attempts to reduce thetemperature of the phase transition and thus preserve the fast-conducting

α-polymorph even at lower temperatures It seems that the α-polymorph

cannot be stabilised at room temperature

An example of a binary compound that exhibits two-dimensional ionicconductivity is lithium nitride (Li3N) Its anisotropy is the result of the crys-tal structure [52] It has a layered structure with sheets of ‘Li2N’ alternatingwith layers of Li Conductivity appears to occur primarily in the ‘Li2N’ sheets

by a Li vacancy mechanism The conductivity of impure, H-containing Li3N

is higher than that of pure Li3N Hydrogen is tightly bound to N, forming

NH units and leaving Li sites vacant in Li3−xNHx.

A family of Li-containing perovskites has high Li+ ion conductivitiesaround 10−3Scm−1 at room temperature These perovskites are based on

Li0.5La0.5TiO3, which does not exist in stoichiometric form but only as deficient compound It is formed by substitution of La3+ for 3Li+ to form

Li-Li0.5 −3xLa0.5+xTiO3

27.7 Polymer Electrolytes

Since their discovery in 1973 by Wright and coworkers [51], polymerelectrolytes have attracted much attention because of their promising ap-plications as ion-conducting materials Polymer electrolytes are mixtures ofpolymers and salts, which are ionic conductors at moderate temperatures.The technological interest in polymer electrolytes stems from the work ofArmand and coworkers, who studied polyethylene oxide (PEO) andpolypropylene oxide (PPO) salt complexes and highlighted the potential ofthese materials for battery applications [53] The electrolyte is the heart ofany battery It must allow the passage of the ions, while blocking electronconduction between the active components of the battery Indeed, Li-ion bat-teries, nowadays commonly used in laptop computers and in cellular phones,are based on polymer electrolytes containing a suitable Li salt [54]

Polymer electrolytes contrast sharply with the fast ion conducting als based on ceramics, glasses, or inorganic crystals discussed above Polymerelectrolytes transport charge well only above their glass transition temper-ature The conductivity of polymer electrolytes is of the order of 10−4 to

materi-10−3S cm−1 and thus two to three orders of magnitude lower than the best

fast ion conductors (see Fig 27.1) This disadvantage is countered by theirease of processing as very thin films of only a few microns thickness Inaddition, they have the advantage of being flexible The flexible nature ofthese materials allows a space-efficient battery design of variable dimensions.The polymer electrolyte flexibility has the important advantage that volumechanges in the cell can be accommodated during cycling without degradation

of the interfacial contacts, which is often observed for crystalline or vitreoussolid electrolytes [55]

Polymer electrolytes may be categorised into several classes according

to electrolyte composition and morphology [56] In what follows, we focus onPEO–salt systems, which belong to the most thoroughly investigated polymerelectrolytes [55, 57, 58] The state-of-the-art knowledge is restricted to a fewestablished features [58]:

1 High ionic conductivity is observed in the amorphous phase of the mer electrolyte This relates to the fact that pure PEO (partially) crys-tallises at temperatures below about 65◦C Similar crystallisation prop-

poly-erties are also found in PEO–salt systems with not too high salt trations (≈ one salt molecule per 30 O-atoms).

concen-2 Long-range ionic motion is coupled to local motions of the polymer chainsegments This coupling is most prominent for the cations since these ionsare usually coordinated by four to five ether oxygens In fact, the cation-oxygen interaction is responsible for the main enthalpy contribution tothe solvation of the salt in the polymer matrix The cation translationalmotion is illustrated in Fig 27.9 This schematic conveys the notion thatcation motion proceeds through the ‘making and breaking of bonds’ be-tween the cation and oxygen atoms of one or two locally mobile polymerchains

3 Anions move faster than cations The higher mobility of anions can beunderstood from their higher degree of freedom: they are not directlybound to the polymer chains (Fig 27.9)

Despite numerous studies related to ionic conductivity, the understanding ofthe diffusion mechanisms in these electrolytes is still unsatisfactory A majorreason for this unsatisfactory situation is that conductivity measurementsonly yield the net effect of all mobile species Only few publications in thisfield report the use of ion-specific techniques, by which the diffusion prop-erties of cations and anions can be determined individually One such tech-nique is the pulsed-field nuclear magnetic resonance (see, e.g., [59]) Anotherpowerful ion-specific method is radiotracer diffusion, which has been em-ployed only on few polymer-salt systems [60–63] Both techniques have pro-

27.7 Polymer Electrolytes 487

Fig 27.9 Schematic illustration of ion solvation and migration in amorphous

polymer electrolytes according to [62]

Fig 27.10 Tracer diffusion coefficients of22Na and125I in an amorphous PEO–NaI

polymer electrolyte compared to the charge diffusivity, D σ, according to Stolwijkand Obeidi [62, 63] The dashed line is shown for comparison: it represents the

sum D(22Na) + D(125I)

vided unambiguous evidence that the anion is moving at least as fast as thecation

As a typical example, we present results of Stolwijk and Obeidi on

a polymer-salt system consisting of PEO and NaI [62, 63] These authorsperformed measurements of 22Na and 125I tracer diffusion and of the over-

all ionic conductivity They also deduced the charge diffusivity, Dσ, from

the dc conductivity via the Nernst-Einstein relation Figure 27.10 compares

the tracer diffusivities of both ions, D(22Na) and D(125I), with the chargediffusivity The latter exhibits a downward curvature, characteristic of Vogel-Fulcher-Tammann behaviour frequently observed in the (supercooled) liquidstate The charge diffusivity falls below the sum of the tracer diffusivities To

explain this discrepancy the authors propose a model, which considers butions from isolated cations and anions and neutral cation-anion pairs Thelatter contribute to tracer transport of both ions but not to charge trans-port The authors conclude that diffusivities increases in the order cation,anion, and ion pair This sequence reflects the decreasing degree of coupling

contri-to the polymer matrix This model reveals some analogy contri-to the diffusion viacation and anion vacancies and neutral vacancy pairs in alkali halide crystalsmentioned in Chap 26

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