5. Diffusion. Introduction. Models of diffusion. Fourier s law of heat flux. 5 Diffusion

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1 5. Dffuson Introducton Numerous chemcal reactons and mcrostructural changes n solds take place through sold state dffuson. In crystallne solds, the dffuson takes place because of the presence of defects; vacances and ntersttals. Dffuson also takes place along 1- and -dmensonal defects whch nclude dslocatons, gran boundares and surfaces. As dffuson along lnear, planar and surface defects s generally faster than n the lattce, they are also termed hgh dffusvty or easy dffuson paths. Another frequently used term s short-crcut dffuson. The relatve contrbutons of the dfferent types of dffuson n oxdes and other norganc compounds are functons of the temperature, partal pressures or actvtes of the consttuents of the compounds, the mcrostructure, gran sze, porosty etc. Gran boundary and dslocaton dffuson generally have smaller actvaton energes than lattce dffuson and as a result they become ncreasngly mportant the lower the temperature n solds wth a gven mcrostructure. In the lterature on dffuson and dffuson-controlled reactons or processes one encounters many dfferent terms that descrbe the dffusonal behavour under dfferent expermental condtons: tracer and self-dffuson of atoms and ons, dffuson of defects, chemcal dffuson, ambpolar dffuson, a.o. In the followng chapters these phenomena and terms wll be descrbed n more detal. Here we wll start out wth a few smple phenomenologcal descrptons that wll help us comprehend dffuson. Models of dffuson Fourer s law of heat flux It s a well known phenomenon that heat flows from hot to cold regons. Such a flow of heat n a one-dmensonal temperature gradent s descrbed by Fourer's law j q dt = κ (5.1) where j q s the heat flux densty,.e., the flow of heat per unt area of the plane dt through whch the heat traverses per second, s the temperature gradent, and 5.1

2 κ (kappa) s the thermal conductvty. It may be noted that the mnus sgn reflects that the heat flows from hgh to low temperatures (downhll). Fck's frst law The expresson for the flow of partcles from hgh to low concentratons s analogous to that for the flow of heat and s gven by Fck s frst law: dc j = D (5.) dc Here, j s the partcle flux densty, the concentraton gradent of the partcles, and D the dffuson coeffcent. As n the equaton for the heat flux, the mnus sgn reflects that the partcles flow from hgh to low concentraton of partcles (downhll). Ths relaton s named after A. Fck who frst formulated ths relaton. The partcle flux and gradent s llustrated schematcally n Fgure 5-1. Fgure 5-1. Schematc llustraton of Fck's frst law. The negatve of the partcle concentraton gradent s the "drvng force" of the dffuson. j represents the number (or moles) of partcles crossng a unt area (cm or m ) per unt tme (seconds). If the concentraton of partcles s expressed n number of partcles (or moles) per cm 3 and the dstance x n cm, the dffuson coeffcent D has the dmenson cm s -1. In SI unts the concentraton s expressed n number per m 3 and the dffuson coeffcent has the dmenson m s -1. dc In Fck's frst law n Eq. (5.) the negatve partcle gradent,, may be consdered to be an expresson of the "drvng force" for the partcle flux. The larger the concentraton gradent, the larger the partcle flux. When Fck s frst law s appled to uncharged (neutral) and ndependently dffusng partcles, t s vald n the sense that the coeffcent s a constant (ndependent of concentraton and gradent). Ths may apply e.g. to dlute solutons of neutral defects n solds. Such 5.

3 defects may be vacances, ntersttals, and mpurtes n metals. In onc solds they may comprse neutral ntersttal mpurtes, homovalent substtuents, or sotopc speces. The applcablty or napplcablty of Fck s frst law wll become clearer later n ths chapter and n the forthcomng chapters. Potental gradents as the drvng force More generally, the drvng force for the dffuson consttutes the chemcal potental gradent of the partcles that dffuse (provded that no other forces act on the partcles). Correspondngly, the drvng force for the transport of electrcal charges s the electrcal potental gradent. In the followng s gven a bref dervaton of Fck's frst law usng a potental gradent as the drvng force, n detal n the case of a chemcal potental gradent. Let us consder the transport of partcles of type "" across a plane under a drvng force F. The partcle flux densty through a plane s gven by the product of the volume concentraton c of the partcles at the plane and the average mgraton or drft velocty v of the partcles, e.g. j (number of partcles cm - s -1 )) = c (number of partcles cm -3 ). v (cm s -1 ) (5.3) For uncorrelated movements the drft velocty v of a partcle s proportonal to the drvng force F exerted on the partcle: v = B F (5.4) The proportonalty factor B s termed the moblty ("Beweglchket") of the partcles and s defned as the average drft velocty per unt drvng force. It s often referred to as mechancal moblty to dfferentate t from other types of moblty. It bascally says somethng about how easy t s to move the partcle. The drvng force s, n turn, gven by the negatve of the potental gradent normal to the cross-sectonal area n the plane F dp = (5.5) where P s a potental. The negatve sgn s, as above, due to the fact that the transport takes place from hgher to lower values of P, see Fgure

4 Fgure 5-. Schematc llustraton of flux through a plane n a potental gradent. When one combnes Eqs , the partcle flux densty becomes j dp = cv = c B (5.6) Chemcal potental gradent actng on neutral partcles If the partcle moves n a chemcal potental gradent, the potental P equals the chemcal potental of partcles of type : P = µ. Equaton 5.6 then takes the form j dµ = cv = c B (5.7) The chemcal potental µ s related to the chemcal actvty a of speces through µ = µ o + kt lna (5.8) If deal condtons can be assumed, the actvty can be expressed by the concentraton dvded by the concentraton n the reference (standard) state: c a = (5.9) c 0 and the chemcal potental gradent s then gven by 5.4

5 dµ = kt d ln c = kt c dc (5.10) When one substtutes the expresson for the chemcal potental gradent n Eq. 5.7, the partcle flux densty becomes j = c B dµ = B kt dc (5.11) B kt s termed the dffuson coeffcent D of partcles : D = B kt (5.1) By combnaton wth Eq. 5.1, Eq takes the form of Fck's frst law as gven n Eq. 5.: j dc = D (5.) It s mportant to realse that Fck s 1 st law, Eq. 5., s fully vald only for deal cases of dffuson of neutral partcles n chemcal potental gradents. If the partcles are charged, we need to take nto account both the electrcal potental gradent and requrements to the combnaton of fluxes to mantan a gven total current and electroneutralty. These cases wll be treated later on. In the followng, deal condtons wll be assumed and concentratons wll be used for actvtes of atoms, ons and varous types of defects. But t should be recalled that ths always represents an approxmaton. Smplfed model for one-dmensonal dffuson Followng Eq.5. the dffuson coeffcent represents the proportonalty constant between the partcle flux and the concentraton gradent. In order to descrbe the process of dffuson of partcles or atoms n solds we need to realse that the dffuson represents the sum of a large number of partcles or atoms that each make a large number of jumps. We wll therefore attempt to descrbe the dffuson coeffcent n terms of the number of jumps per unt tme (the jump frequency) and the dstance that each partcle or atom moves n each jump. For ths purpose let us consder a smplfed one-dmensonal model where partcles jump between parallel planes separated by a dstance s as llustrated n Fgure 5-3. The two neghbourng planes under consderaton are termed plane 1 5.5

6 and. The number of partcles per unt area n plane 1 and s termed n 1 and n, respectvely. Let us further assume that n 1 >n. Consder now that the partcles n plane 1 and may jump from one plane to a neghbourng plane at a jump frequency Γ (gamma). The partcles n plane 1 have an equal probablty of jumpng to plane and to the neghbourng plane n the opposte drecton. The total number of partcles jumpng out of a unt area of plane 1 per unt tme s equal to the product of number of partcles per unt area tmes the jump frequency: n 1 Γ. As the partcles may jump n opposte drectons, the number of partcles jumpng from plane 1 to s gven by ½ n 1 Γ. Correspondngly, the number of partcles jumpng from unt area of plane to plane 1 s gven by ½ n Γ. The dfference n the jump rates s equal to the net flux densty of partcles: j 1 1 = (n n )Γ (5.13) As all partcles have the same jump frequency, there s a net flow of partcles from plane 1 to because there are more atoms per unt area n plane 1 than n plane. Fgure 5-3. Schematc llustraton of smplfed model for one-dmensonal dffuson The number of partcles n 1 belongng to unt area of plane 1 s gven by the volume concentraton at plane 1, c 1, tmes the extenson n the x drecton,.e., the plane separaton s: n 1 = c 1 s and n = c s (5.14) By combnng Eqs.5.13 and 5.14 the partcle flux densty becomes j 1 1 = (c c )sγ (5.15) 5.6

7 dc The concentraton gradent normal to the planes s termed. The relaton between c 1 and c can then be expressed by seeng that a dfference s equal to the gradent tmes the length; dc c c 1 = s (5.16) By nserton of ths nto Eq we obtan dc j = 1 s Γ (5.17) Ths expresson apples to one-dmensonal dffuson of partcles, and the factor ½ reflects that only half of all jumps occur n the drecton we consder as the flux drecton. If the dffuson can take place n the three orthogonal drectons, only one thrd of the partcles jump n one drecton, and n the three-dmensonal case the net flux n one drecton s 1/3 of the flux of that when all atoms jump n one drecton only: dc j = 1 s Γ (5.18) 6 By comparng wth Fck's frst law, Eq. 5., t s seen that the dffuson coeffcent D n the three-dmensonal case s gven by D = s Γ (5.19) 1 6 If we consder a large number of jumps, n, whch occurs durng the tme t, then n Γ = (5.0) t and nsertng ths n Eq.5.19, one obtans ns = 6Dt (5.1) 5.7

8 The expresson for the dffuson coeffcent n Eqs and 5.1 provdes a qualtatve descrpton of how far dffuson proceeds. By way of example, the dffuson coeffcent for ntersttal dffuson of oxygen atoms n nobum metal at 800 C s approxmately D = cm s -1. The jump dstance can be assumed to be 1.65 Å ( cm), and then from Eq the jump frequency Γ s about s -1. Thus, each oxygen atom makes a tremendously large number of jumps per second. But t should then also be recalled that the atoms vbrate wth a (Debye) frequency of s -1, and thus only a small fracton - about 1 n 1000 or 10,000 - of the vbratons leads to a jump. We shall consder ths n more detal when the dfferent atomstc mechansms are treated below. Although the number of jumps s very large, the mean dsplacement of each atom s relatvely small most of the tme t moves back and forth. In the dffuson process t s not possble to observe the ndvdual jumps of the atoms, and t s necessary to fnd a relaton between the ndvdual atom jumps for large number of atoms and the dffuson phenomena whch may be observed on a macroscopc scale. The problem s to fnd how far a large number of atoms wll move from ther orgnal stes after havng made a large number of jumps. Such relatons may be derved statstcally by means of the so-called random walk approach. Random dffuson Let us consder that the jumps of the atoms are random,.e. that the jumps of the atoms are ndependent of all the prevous jumps and can occur n all drectons. In that case the dsplacement of a dffusng atom from the startng pont after n number of jumps, R n, s gven by the algebrac sum of the ndvdual jump vectors: R n n = s1 + s +... s = s (5.) n j= 1 j If the ndvdual jumps take place wth equal probablty n all drectons and the ndvdual jump dstances are equal, ths algebrac sum equals zero. Ths does not mean that the dffusng atom remans at ts startng pont after n jumps, but only that jumps n "postve " and "negatve" drectons are equally probable. In fact the total dsplacement may have any value between zero and ±ns. In order to obtan a value for the magntude (length) of the sum vector, one squares Eq. 5.: n j= 1 n 1 n R = R = s + s s (5.3) n n j j j= 1 k= j+ 1 k 5.8

9 If, as above, the ndvdual jump vectors are equal,.e., s 1 = s... s j = s, as, for nstance, n crystals wth cubc symmetry, and f they are random and uncorrelated, then the second term on the rght hand sde of the equalty sgn n Eq. 5.3 wll approach zero for large numbers of jumps, as on an average s j and s k have an equal chance of beng postve and negatve. The frst term on the rgh hand sde, on the other hand, s always non-zero and postve, and thus represents the average dsplacement length for a large number of jumps: R = n n s j = j= 1 ns (5.4) The mean dsplacement s gven by the square root of R n and s termed the root mean square dsplacement: R n = R = n s (5.5) n From ths t s seen that the mean dsplacement s proportonal to the square root of the number of jumps tmes the ndvdual jump dstance. Fgure 5-4 llustrates the relatonshp between ndvdual jumps and total dsplacement. Fgure 5-4. Random dffuson by n jumps each of dstance s gves a long travelled dstance s n, but a relatvely short dsplacement R n from the startng pont. By combnng Eqs. 5.1 and 5.4 we may express the random dffuson n terms of the dffuson coeffcent we dealt wth n dffuson down a concentraton gradent n the 3-dmensonal cubc case: R = ns = D t (5.6a) n 6 r 5.9

10 R = D t (5.6b) n 6 r where t s the tme durng whch the mean-square-dsplacement takes place. One may note that we have now started termng the dffuson coeffcent D r (r for random walk). R n s the radus of the sphere that a dffusng atom on average wll dstance tself by from the startng pont after tme t. Let us now consder the dsplacement n a sngle dmenson (e.g. the x- drecton) as a result of ths three-dmensonal dsplacement n the cubc case. From smple geometry we have R n = 3x where x s the mean square dsplacement n each orthogonal drecton. Thus, the mean dffuson length n one drecton n a three-dmensonal cubc crystal s: x = D t (5.7) r Ths length s thus shorter than the dsplacement radus, snce dsplacements n the y and z drectons are wasted for dsplacement n the x drecton. As descrbed above and usng oxygen dffuson n nobum as an example, the oxygen atoms on an average exchange postons approxmately tmes per second at 800 C. From the same consderatons one may also estmate that an oxygen atom has randomly covered total jump dstances of 5.4 cm and m after 1 second and 1 hour, respectvely. But what s the mean dsplacement? From Eq 5.7 one may estmate that the one-dmensonal root-mean-square dsplacement after 1 hour only amounts to 0.0 cm. Thus, the mean dsplacement s very small and on an average the oxygen atoms spend most of ther tme jumpng "back and forth". What we dd above was to consder random jumps, and then we related jumps, tmes and dstances to a term we recognsed from earler, namely the dffuson coeffcent, and we named t the random dffuson coeffcent D r. The dffuson coeffcent was n turn somethng we recognsed whle consderng dffuson down a one-dmensonal concentraton gradent from Fck s frst law. One should note, however, that random dffuson and the random dffuson coeffcent can be consdered and expressed and quantfed also n the absence of a concentraton gradent and also for charged partcles. As wll be descrbed below, dffuson s often measured by usng tracer atoms and one then obtans values of the dffuson coeffcent of the tracer atoms. Dependng on the dffuson mechansm the tracer dffuson s n most cases not completely random, but s to some extent correlated wth prevous jumps. Ths wll be further dscussed later on. 5.10

11 Fck's second law As descrbed above, Fck's 1 st law assumes a fxed concentraton gradent across the plane through whch the flux of partcles take place. But n numerous practcal cases the concentraton and concentraton gradent changes wth tme. Such cases are covered by Fck's nd law. Ths s shown schematcally n Fgure 5-5 whch llustrates the change n the concentraton gradent through the sold. As shown n the fgure let us consder a regon wthn a sold, enclosed between planes separated by the dstance. The net partcle flux densty from the regon of hgher concentraton nto s j 1 and the net partcle flux densty out of towards lower concentraton s j. When j 1 s greater than j the partcle concentraton n ncreases wth tme. Ths requres that the concentraton gradent n plane 1 s larger than n plane, snce Fck's 1 st law apples n both planes (and at any poston) at any tme. The process s llustrated n Fgure 5-5. Fgure 5-5. Schematc llustraton of Fck s nd law; the concentraton gradent changes wth tme. The change n concentraton per unt tme at any poston s proportonal to the gradent n flux at that poston c j = - (5.8) t x Although the concentraton and concentraton gradent change wth tme, Fck's 1 st law s vald at any one tme and poston and thus c j = - t x c = (D ) x x (5.9) If D can be consdered to be ndependent of concentraton then 5.11

12 c j c = - = D (5.30) t x x The ntuton of the qualtatve arguments as well as the double dervatve of the concentraton n Eq tell us that partcles wll flow from convex to concave regons n terms of dstrbuton of partcles. Equatons 5.9 and 5.30 are representatons of Fck's nd law. It may be solved explctly under certan boundary condtons that may be closely approxmated expermentally (Crank 1956). A couple of examples of ths are gven n the followng. Measurements of dffuson coeffcents by tracer technques The use of sotopes or tracers s a common means of studyng dffuson. Tracer methods permt measurements of self-dffuson, that s, the dffuson of the crystal components n a crystal. Furthermore, they allow measurements of dffuson n homogenous materals, that s, wthout mposng chemcal gradents (when one dsregards the dfference n atomc weght between atoms n the crystal and of the tracer). A common technque s to depost a very thn flm of radoactve sotopes on a plane surface of a sample, and, after subsequent dffuson anneal, determne the actvty of dffuson speces as a functon of dstance from the plane surface. If the thckness of the sample s very much larger than the penetraton depth of the tracers, the sold can be consdered sem-nfnte. Furthermore, f the dffuson s homogenous (e.g. takng place by lattce dffuson), the concentraton of the dffusng tracers normal to the plane s through soluton of Fck's second law wth approprate boundary condtons gven by c( x) co ( πd t) = 1/ t x exp( ) 4D t t (5.31) c s the actvty (or concentraton) of the tracer at a dstance x from the surface, c o s the actvty orgnally present on the surface, and t s tme of the dffuson anneal. D t s the tracer dffuson coeffcent. Followng Eq.5.31 t s determned by plottng lnc vs x 1, n whch case the resultant straght lne has the slope -. 4D t t Plots of c vs x and of lnc vs x accordng to Eq.5.31 are llustrated n Fgure 5-6a and b, respectvely. At the pont where the actvty s half of the actvty at the surface, then x =.77D t t. Ths dstance corresponds approxmately to the rootmean-square penetraton dstance (Eq. 5.7). 5.1

13 Fgure 5-6. Graphcal presentaton of relatonshp between actvty, c, and penetraton dstance, x, (Eq.5.31) for homogeneous dffuson of tracer ntally deposted as a thn flm on the surface the sold. a) c vs. x; b) lnc vs. x. An alternatve way of performng the experment s to have a constant surface composton of the dffusng speces. By solvng Fck's second law under proper boundary condtons the penetraton of the dffusng speces s then descrbed by the relatonshp c c c s 0 c 0 = 1 erf ( x D t t ) 1/ (5.3) where c s the concentraton of the dffusng speces at penetraton dstance x at tme t, c s s the constant surface concentraton, and c o s the orgnal concentraton n the sold. D s s the dffuson coeffcent of the dffusng speces. Fgure 5-7 shows a graphcal penetraton of the dffuson profle accordng to Eq.5.3 n a c c0 case where c 0 = 0. It may be noted that at the pont where = ½, that s, at cs c0 the pont where the concentraton of the dffusng speces s mdway between the surface composton and the orgnal composton, then x/ D t t = Thus at ths pont x ~ D t t. It may be noted that Eqs and 5.3 nvolve the dmensonless parameter x/( D t t ), and accordngly the penetraton and the amount of the dffusng speces dssolvng n the sold are proportonal to the square root of tme. 5.13

14 Fgure 5-7. Graphcal presentaton of dffuson profle when the surface concentraton of the dffusng speces remans constant wth tme. It s assumed that co=0 (Eq.5.3). The penetraton by the dffusng speces may be measured by means of the socalled sectonng method, that s cuttng, grndng or etchng off thn sectons of layers of the sample parallel to the plane surface and subsequently determnng the concentraton of the dffusng speces n each secton. By cuttng the specmen normal to or at an angle to the plane surface, the penetraton of radoactve tracers may also be measured by means of so-called autoradography. Electron mcroprobe analyss and secondary on mass spectroscopy (SIMS) combned wth sputterng technques also provde excellent tools for studyng penetraton and dffuson of foregn ons. As oxygen does not have a radoactve sotope sutable for tracer studes, SIMS s partcularly useful for studyng oxygen dffuson employng the stable 18 O sotope. In the sotope exchange method the vapour of the dffusng component surroundng the sample s enrched wth ether a radoactve or an nactve sotope, and the dffuson s measured by followng the exchange of the sotope wth the sample. The dffuson coeffcent may be evaluated f dffuson s the slower process and processes at the surface are rapd. Alternatvely, the sample tself may be sotopcally enrched, and the ncrease n concentraton of the sotope n the vapour phase may be measured. Dffuson rates may, n prncple, also be determned from any property or reacton whch depends on atomc moblty. By way of llustraton, onc conductvty of the anon s drectly proportonal to the anon dffuson coeffcent (see Ch. 6 Electrcal conductvty). From hgh temperature sold state reactons, snterng, oxdaton of metals etc. dffuson coeffcents may be evaluated provded the detaled mechansm of the processes are known. Examples of ths wll be gven n Ch. 7. Dffuson mechansms Lattce dffuson takes place through the movement of pont defects. The presence of dfferent types of defects gves rse to dfferent mechansms of 5.14

15 dffuson. These are llustrated schematcally for elemental solds n the followng descrptons. But they also apply to metal oxdes and other norganc compounds when the dffuson s consdered to take place n the sublattces of the catons or anons. Vacancy mechansm The dffuson s sad to take place by the vacancy mechansm f an atom on a normal ste jumps nto an adjacent unoccuped lattce ste (vacancy). Ths s llustrated schematcally n Fgure 5-8. It should be noted that the atoms move n the drecton opposte the vacances. Fgure 5-8. Schematc llustraton of vacancy dffuson n solds. Intersttal mechansm If an atom on an ntersttal ste moves to one of the neghbourng ntersttal stes, the dffuson occurs by an ntersttal mechansm. Ths s schematcally shown n Fgure 5-9. Such a movement or jump of the ntersttal atom nvolves a consderable dstorton of the lattce, and ths mechansm s probable when the ntersttal atom s smaller than the atoms on the normal lattce postons. Dffuson of ntersttally dssolved lght atoms, e.g. H, C, N, and O n metals provdes the best known examples of ths mechansm. Fgure 5-9. Schematc llustraton of ntersttal dffuson n solds. 5.15

16 Oxdes wth close-packed oxygen lattces and only partally flled tetrahedral and octahedral stes may also facltate dffuson of metal ons n the unoccuped, ntersttal postons. Fnally, even large anons may dffuse ntersttally f the anon sublattce contans structurally empty stes n lnes or planes whch may serve as pathways for ntersttal defects. Examples are rare earth sesquoxdes (e.g. YO3) and pyrochlore-type oxdes (e.g. LaZrO7) wth fluorte-derved structures and brownmllerte-type oxdes (e.g. CaFeO5) wth perovskte-derved structure. Intersttalcy mechansm If the dstorton becomes too large to make the ntersttal mechansm probable, ntersttal atoms may move by another type of mechansm. In the ntersttalcy mechansm an ntersttal atom pushes one of ts nearest neghbours on a normal lattce ste nto another ntersttal poston and tself occupes the lattce ste of the dsplaced atom. Ths mechansm s llustrated schematcally n Fgure In the ntersttalcy mechansm one may dstngush between two types of movements. If the atom on the normal lattce ste s pushed n the same drecton as that of the ntersttal atom, the jump s termed collnear (Fgure 5-10). If the atom s pushed to one of the other neghbourng stes so that the jump drecton s dfferent from that of the ntersttal atom, the jump s termed non-collnear. Fgure Schematc llustraton of ntersttalcy dffuson n solds. Collnear jump. Other mechansms. In elemental solds also other mechansms have been proposed. The crowdon s a varant of the ntersttalcy mechansm. In ths case t s assumed that an extra atom s crowded nto a lne of atoms, and that t thereby dsplaces several atoms along the lne from ther equlbrum postons. The energy to move such a defect may be small, but t can only move along the lne or along equvalent drectons. For metals t has also been proposed that dffuson may take place through a so-called rng mechansm, but ths mechansm s less probable n most oxdes or other norganc compounds. A partcular feature of ths mechansm s that t can 5.16

17 nvolve exchange of stes and thus would contrbute to tracer dffuson, but that t would not lead to net transport of atoms n any drecton. If the atoms were charged (ons) ths would then not gve rse to net charge transport. Dffuson of protons n metal oxdes. Protons that dssolve n metal oxdes assocate wth the oxde ons and form hydroxde ons. As the proton has no electron shell, t nteracts strongly wth the electron cloud of the oxde on, and n ts equlbrum poston n the hydroxde on t s embedded n the valence electron cloud. The O-H bond that s formed has a bond length less than 100 pm; ths may be compared wth the onc radus of 140 pm of the oxde on. In prncple the protons may move by two dfferent mechansms: ) the free transport mechansm, whch s also alternatvely termed the Grotthuss mechansm or ) the vehcle mechansm. The free transport s the prncpal mode of transport of protons n oxdes, and n ths mechansm protons jump from one oxde on to a neghbourng one. After each jump the proton n the hydroxde rotates such that the proton reorents n the electron cloud and becomes algned for the next jump. Ths s llustrated schematcally n Fgure The rotaton and reorentaton s beleved to nvolve a small actvaton energy and the jump tself s consdered to be the ratedetermnng step. Fgure Schematc llustraton of free transport of protons n metal oxdes (Grotthuss mechansm). In the vehcle mechansm the proton s transported as a passenger on an oxde on. Thus ths mechansm may be consdered to consttute transport of hydroxde ons. The hydroxde on may n prncple move by an oxygen vacancy mechansm or as an ntersttal hydroxde on. It may be noted that the hydroxde on has a smaller radus and charge than the oxde on and may as such be expected to have a smaller actvaton energy for dffuson than the oxde on. Also other speces such as water molecules and hydronum ons, H 3 O +, may serve as vehcles for protonc dffuson, notably n relatvely open structures. 5.17

18 Factors that affect the dffuson coeffcent n crystallne solds In Eq we saw that the random dffuson coeffcent can be expressed n terms of the jump dstance and the number of jumps per unt tme: 1 1 n = 6 s Γ = 6 s (5.19) t D r We wll now further characterse D r n dfferent crystal structures (lattces or sub-lattces) and subsequently derve expressons for the temperature and oxygen pressure dependence of dffuson n metal oxdes. Vacancy dffuson Let us consder vacancy dffuson n an elemental sold or n a caton or an anon sub-lattce. The jump frequency (number of jumps per unt tme) Γ depends on several factors. Frst, t depends on the frequency of suffcently energetc jump attempts ω towards an adjacent ste. Furthermore, t s also proportonal to the number of adjacent stes Z to whch the atom may jump,.e., the number of nearest neghbour postons of the atom. Fnally, the atom may only jump f a vacancy s located on an adjacent ste, and ths probablty s gven by the fracton (concentraton) of vacances n the crystal, N d. Thus, Γ s n ths case gven by Γ = ωzn d (5.33) In crystallne solds the jump dstance s a functon of the crystal structure and may be expressed as a functon of the lattce parameter. In a body centered cubc (bcc) crystal of an elemental sold, for nstance, each atom has 8 nearest-neghbour postons or atoms, and thus n ths case Z = 8. From smple geometrcal consderatons of the crystal structure t may further be 3 shown that the jump dstance s gven by s = a 0, where a o s the lattce parameter. When nsertng these values of Z and s n Eqs and 5.19 one obtans D = a ω (5.34) r 0 N d In general, D r for a cubc structure s wrtten 5.18

19 D = a (5.35) r α 0ωN d where α s a geometrcal factor nvolvng the factor 1/6 (from Eq. 5.1), the factor Z from Eq and the relaton between the jump dstance and the lattce parameter. For vacancy dffuson n a bcc lattce α s thus equal to unty. From the same consderatons t may also be shown that α = 1 for vacancy dffuson n fcc lattces. Intersttalcy dffuson Consder an atom on a normal lattce ste of a caton or anon sub-lattce. If ths atom s to move by the ntersttalcy mechansm, an atom on a nearest neghbour ntersttal ste has to push the atom on the normal ste to a neghbourng ntersttal ste. Thus for ths dffuson mechansm an atom may only dffuse when t has an ntersttal atom on a neghbourng ste, and as for vacancy dffuson the dffuson coeffcent of the atoms s proportonal to the fracton (concentraton) of defects, n ths case ntersttal atoms or ons n the sub-lattce. Intersttal dffuson When one consders ntersttal dffuson of an ntersttally dssolved foregn speces n dlute sold soluton, essentally all the nearest neghbour ntersttal stes of the same type are unoccuped and avalable for occupancy by the dffusng ntersttal atoms. Thus the ntersttal atom may jump to any of the nearest neghbour ntersttal stes and n ths case N d s equal to unty. The dffuson coeffcent of the ntersttally dssolved foregn speces s then gven by D r 6 1 = s Zω (5.36) As mentoned above the best known examples of ths mechansm s dffuson of O, N, C, and H atoms ntersttally dssolved n metals. By way of example oxygen and ntrogen atoms n bcc metals, e.g. n the group 5 metals V, Nb, and Ta, occupy octahedral stes, and n ths case each ntersttal atom has 4 nearest neghbour octahedral stes to whch they may jump, thus Z= 4. Furthermore, the jump dstance s s equal to a o. Insertng these values n Eq the dffuson coeffcent for ntersttal dffuson between octahedral stes n a bcc lattce becomes D 1 r = a ω (5.37) 6 0 Thus n ths case α =1/

20 Temperature and oxygen pressure dependence of dffuson n metal oxdes We wll now address how dffuson coeffcents vary wth temperature and actvty of the components of the compound, manly the oxygen partal pressure over oxdes. In order to evaluate these dependences one must consder the temperature and actvty dependences of N d (e.g. cf. Eq.5.35) and the temperature dependence of ω. We start by analysng Nd for some cases. Dependences related to the concentraton of defects Vacances n an elemental sold For the sake of smplcty let us frst consder the case of dffuson by a vacancy mechansm n a pure elemental sold. The dffusng atoms may only make a jump when a neghbourng ste s vacant. Thus the jump frequency n/t s proportonal to the vacancy concentraton and as descrbed n Chapter 3 the fracton of vacances may be wrtten N d Gd S d H d = exp( ) = exp( )exp( ) (5.38) RT R RT where G d, S d, and H d denote the Gbbs energy, entropy, and enthalpy of formaton of the defects here vacances. In elemental solds Hd s postve and the vacancy concentraton ncreases wth ncreasng temperature. Vacances n an oxygen defcent oxde. In a nonstochometrc oxde the concentraton of the predomnatng pont defects wll be a functon of temperature but also of the oxygen pressure. By way of llustraton, let us consder an oxygen defcent oxde, M a O b-δ, n whch doubly charged oxygen vacances are the predomnatng pont defects. If ntrnsc onsaton and effects of mpurtes can be neglected, the concentraton of the oxygen vacances s as descrbed n prevous chapters (Eq. 3.57) gven by 0 1 / 1/ 3 1/ 6 1 / 1/ 3 H vo 1/ 6 Nd = [vo ] = ( K 4 vo ) po = ( K 4 0,vO ) exp( )po (5.39) 3RT / where K vo represents the equlbrum constant for the formaton of doubly 0 charged oxygen vacances, and H vo the enthalpy of formaton. 5.0

21 In such an oxde the oxygen vacancy concentraton may also be determned by the presence of dopants or a suffcently large level of mpurtes wth negatve effectve charge. If the dopant s doubly negatvely charged (as, for nstance, Ca +, an acceptor dopant, n ZrO ), then [ v ] = [A O // M ] (5.40) In ths case [ v O ] wll be ndependent of oxygen pressure and most often temperature (see Chapter 4). In a smlar manner one may obtan the temperature and oxygen pressure dependences of the concentraton of any defect (majorty or mnorty) when the defect structure s known. Of course the concentratons of acceptors or donors wll enter when they domnate the defect stuaton and the water vapour partal pressure may enter when protons are domnatng defects. Temperature dependence of the frequency of suffcently energetc jump attempts ω When atoms jump or move between defnte stes n the crystal, they have to surmount energy barrers. A large part of ths energy barrer nvolves the stran energy requred to dsplace neghbourng atoms to create a suffcently large openng between the atoms to permt the atom jump. The potental energy of the atom dffusng from one ste to another may be qualtatvely llustrated as shown n Fgure 5-1. The potental barrer heght s H m and represents the actvaton energy whch the atom has to surmount durng the jump. Each atom vbrates n ts poston, and only durng a fracton of tme gven by the Boltzmann dstrbuton factor exp(- H m /RT), t possesses suffcent energy to overcome the energy barrer. The frequency of suffcently energetc jump attempts s thus proportonal to exp(- H m /RT). Fgure 5-1. Potental energy of atom dffusng n a sold. Hm s the actvaton enthalpy. A more complete analyss based on the theory of actvated complexes and on statstcal mechancs has been gven by Zener (1951,195). He consdered the 5.1

22 system or an atom n ts ntal equlbrum condton and n the actvated state at the top of the potental barrer whch separates the ntal poston from ts neghbourng equlbrum poston. The rate of transton from one equlbrum ste to another s gven by Gm Sm H m ω = ν exp = ν exp exp (5.41) RT R RT where G m, S m, and H m represent the Gbbs energy, entropy and enthalpy change, respectvely, connected wth the movement of the atom from the equlbrum poston to the top of the potental barrer and ν ( nu ) represents the vbraton frequency. ν s often assumed to equal the Debye frequency,.e. about Hz, as an order of magntude approxmaton. S m s often assumed to be a small, postve term (below, say, 10 J/molK). As a more detaled analyss of ν, Zener suggested that ν may be approxmated as ν = α/a Hm / M, where α s a structure- and mechansmdependent factor, a s a lattce parameter, and M s the reduced mass of the oscllator. Intutvely, a partcle vbratng n an energy valley wll vbrate faster (υ ncreases) when the walls become steeper ( Hm ncreases). One may also vew the partcle as vbratng on a sprng; the frequency becomes hgher when the sprng s shorter (α/a decreases), when the sprng s stffer ( H m ncreases) or the partcle becomes lghter (M decreases). Thus, the temperature-ndependent (preexponental) term n the dffuson coeffcent ncreases when Hm ncreases, so that the two tend to counteract each other. Expermental observatons of ths s sometmes referred to as the Meyer-Neldel effect (Meyer and Neldel (1937)). Resultng analyss of the dffuson coeffcent D r Vacancy dffuson n an elemental sold. From Eq t s seen that the temperature dependence of D r for vacancy dffuson n an elemental sold s determned by that of N d and ω. For an elemental sold wth cubc structure, D r s thus obtaned by combnng Eqs. 5.35, 5.38 and 5.41: D r Sd + Sm ( H d + H m ) = αa0ν exp exp (5.4) R RT Expermentally determned values of dffuson coeffcents are usually obtaned as Q D = D0 exp (5.43) RT 5.

23 where Q s termed the actvaton energy and D o the pre-exponental factor. By comparng Eqs. 5.4 and 5.43 t s seen that the actvaton energy, Q, n ths case comprses Q = H d + H m (5.44) Correspondngly, D o s gven by Sd + Sm D 0 = αa0ν exp (5.45) R If experments are carred out under such condtons that the concentraton of defects, N d, s constant and ndependent of temperature, e.g. at suffcently low temperatures that the defect concentraton s frozen n, then D r s gven by D r Sm H m H m = αa0ν N exp d,frozen exp = D exp (5.46) R RT 0 RT and correspondngly the actvaton energy s under these condtons gven smply by Q = H m (5.47) Oxygen vacancy dffuson n oxygen-defcent oxdes. In an oxygen-defcent oxde n whch oxygen vacances predomnate and for whch effects of mpurtes can be neglected, the oxygen vacancy concentraton s gven by Eq Correspondngly, the oxygen dffuson coeffcent for random oxygen vacancy dffuson n the oxde n equlbrum wth the ambent oxygen gas at a partal pressure becomes D r 0 H vo ( + H S m ) 1 / 1/ 3 1 / 6 m = αa ( K vo ) p 3 0ν 4 0, O exp exp (5.48) R RT D r thus ncreases wth decreasng oxygen pressure. The actvaton energy for the dffuson s n ths case gven by 5.3

24 0 HvO Q = + H m (5.49) 3 If the concentraton of oxygen vacances s determned by lower valent mpurtes or dopants, e.g. when Eq.5.40 apples, then D r s gven by D r // Sm H m = αa0ν [AM ] exp exp (5.50) R RT Thus n ths case the actvaton energy s equal to that of the moblty of the oxygen vacances: Q = H m. A smlar stuaton would arse f the vacancy concentraton was frozen n rather than determned by acceptor dopng. Such a transton from ntrnsc to extrnsc dffuson may take place when the temperature s lowered from hgh temperatures, where the natve pont defects predomnate, to low temperatures, where the pont defect concentratons are determned by the mpurty concentraton or, n other cases, frozen n. The temperature dependences and the correspondng change n actvaton energy of the random dffuson n such a case are llustrated n Fgure The above stuatons represent deal cases. As the temperature s decreased defect nteractons may become ncreasngly mportant. Ths may be treated as formaton of assocated defects. In the non-stochometrc (ntrnsc) case one may for nstance have assocaton between charged vacances and electrons formng sngly charged or neutral vacances. Ths wll change the oxygen pressure dependency of Nd and probably the temperature dependences of Nd and ω, but probably not dramatcally. Of larger effect, and more frequently observed, are the assocatons between the moble vacances and the relatvely statonary acceptors n the extrnsc regme: The assocated vacances can be regarded as mmoblsed, and the concentraton of moble vacances (Nd) starts to decrease wth decreasng temperature. Ths s seen as an ncreasng actvaton energy of dffuson wth decreasng temperature n many heavly doped oxdes. One may note that nstead of expressng the effect as a changng concentraton of free vacances one may express t as a changng moblty n that the actvaton energy for dffuson s ncreased by the trappng energy exerted by the acceptor. However, the smple model of free and statonary (assocated) pont defects appears capable of explanng most behavours farly well. 5.4

25 Fgure The dffuson coeffcent for oxygen dffuson by the vacancy mechansm n an oxygen defcent oxde n whch oxygen vacances are the predomnant natve pont defects. At hgh temperatures the oxde exhbts ntrnsc behavour and at reduced temperatures extrnsc behavour (.e. the oxygen vacancy concentraton s determned by the concentraton of lower valent catons). Intersttal dffuson of solute. From Eq.5.36 t s seen that N d does not enter nto the expresson for the dffuson coeffcent for ntersttal dffuson n dlute solutons, thus n ths case the actvaton energy, Q, represents that of the moblty of the dffusng ntersttal atoms: H m = Q. For ntersttal dffuson between octahedral stes n bcc metals D o s by combnaton of Eqs and 5.41 gven by D 1 Sm = a0 exp (5.51) 6 R 0 ν Assumng that ν ~ s -1 and a o = cm, and as t s probable that S m > 0, one may estmate a lower lmt of D o of 1 4 s 1 a 6 0 = 4 10 cm D o > ν (5.5) The dffuson of protons by the free transport mechansm s another case of ntersttal dffuson of a solute. 5.5

26 Intersttal dffuson of a consttuent The dffusvty of a consttuent such as the host metal or oxde ons by an ntersttal mechansm s not only proportonal to the probablty that the ntersttal defect jumps, but also to the probablty that a consttuent on s ntersttal,.e., the fractonal concentraton of ntersttals. Thus the dffuson coeffcent of the consttuent contans the temperature and oxygen pressure dependences of the concentraton of ntersttals n addton to the temperature dependency of the dffusvty of the defects. As n the case of vacancy dffuson, the fxaton of the defect concentraton by dopng or freezng as well as assocaton and trappng of defects apply also to ntersttal dffuson. Dffuson coeffcents of pont defects In the above treatment of vacancy dffuson, only the dffuson coeffcents of the atoms have been consdered. For many purposes t may be convenent to consder the dffuson coeffcents of the vacances themselves. When an atom dffuses by the vacancy mechansm, t can only jump f a vacancy s located on an adjacent ste, and the number of jumps per unt tme s thus proportonal to N d (Eq. 5.33). However, the vacancy tself can jump to any one of the occuped nearest neghbour postons, provded t s occuped by an atom. Accordngly the vacancy dffuson coeffcent D v for a cubc system s gven by (cf. Eq.5.35) D v = α a o ω Ν (5.53) where N s the fracton of occuped atom postons. In dlute solutons of vacances, N~1, and the dffuson coeffcent of the vacances s then not dependent on N d. From Eqs and 5.53 D v for vacances s related to D r for the atoms through the relaton D r N = D v N d (5.54) where, as stated above, N denotes the fracton of stes occuped by atoms, often approxmated as ~1. Ths relaton can be generalsed to be a very mportant and useful approxmaton for any pont defect: Dr N = Dd Nd (5.55) 5.6

27 where Dd s the defect dffuson coeffcent. It proves to be a good approxmaton for component dffuson by the ntersttalcy mechansm. It s also a useful approxmaton for component dffuson by ntersttal dffuson n the case of small defect concentratons (N~1) but as the concentraton of defects ncreases the term N must reflect the number of unoccuped ntersttal stes. Eq stll holds for ntersttal dffuson of a dlute soluton of an ntersttally dssolved solute, such as lght elements (H, C etc. n metals or protons n oxdes), but n ths case the number of solute atoms or ons and the number of defects s of course the same, so that n these cases the dffuson coeffcent for the solute and for the (ntersttal) defects s the same. Dffuson of protons n oxdes; sotope effects Transport of protons n an oxde (cf. Fg. 5.11) may be consdered accordng to the prncples above. For dlute solutons essentally all nearest neghbour oxde ons are avalable, and thus n ths case N d s unty. However, the specfcatons of Z, s and ω are not straghtforward n ths case. The dynamcs of free proton dffuson n oxdes are complcated by 1) the multstep process (jump+rotaton), ) the dependency on the dynamcs of the oxde on sublattce, and 3) the quantum mechancal behavour of a lght partcle such as the proton. The unquely large ratos between the masses of the sotopes of hydrogen gve rse to a number of strong sotope effects n the case of dffuson of protons. (These are also n prncple operatve for dffuson of hydrogen atoms or hydrde ons, but they would be essentally neglgble for dffuson of protons on a heaver vehcle, such as n OH-.) The sotope effects for proton dffuson can be classfed as follows: The attempt frequency ν (n the pre-exponental of ω) s gven as the O-H stretchng frequency, and t s gven by the nverse of the square root of the reduced mass of the harmonc oscllator. The reduced mass equals (m O +m H )/m O m H and s roughly nversely proportonal to the mass of the hydrogen speces. Therefore, the ratos of the pre-exponentals of the dffuson coeffcents of protons, deuterons, and trtons are approxmately related by D 0H : D 0D : D 0T = 1 : 1/ : 1/ 3. Ths s called the classcal effect. Furthermore, the oscllators have dfferent ground-state or zero-pont energes, such that dffuson of lghter sotopes may be expected to have a slghtly smaller actvaton energy of jumpng. Accordngly, proton dffuson typcally has ev lower actvaton energy than deuteron dffuson. Ths s called the non-classcal effect. However, there are more factors nvolved, connected to the fact that the lght proton/deuteron/trton must be treated quantum mechancally and to ther dynamcs n a dynamc lattce of much heaver oxde ons. For nstance, the lghter sotope has a lower stckng probablty after an otherwse successful jump. In effect, ths reduces the effectve dffusvty of all hydrogen sotopes and t can to a varyng degree counteract or even seemngly nverse the classcal effect. The neglectance of the latter have n general made many nvestgators ascrbe hgher dffusvtes for protons compared 5.7

28 to deuterons to the classcal effect, whle n realty the non-classcal zero-pont energy dfference appears to be the man contrbutor to the observed effect. The possblty of tunnellng as a major component of dffuson s not expected to apply to protons except at very low temperatures, and the orders-ofmagntude sotope effects that would be expected for proton vs deuteron or trton dffuson have not been reported for oxdc materals at elevated temperatures. Lterature Crank, J. (1956) The mathematcs of dffuson, The Clarendon Press, Oxford. Meyer, W., Neldel, H. (1937), Z. Tech. Phys., 1, 588. Zener (1951,195) Problems 1. Random (self) dffuson a) The self-dffuson coeffcent of a metal wth cubc structure can be expressed as 1 n D = s 6 t where n/t represents the jump frequency (.e. number of jumps n over a tme t). Close to the meltng pont most fcc and bcc metals have D 10-8 cm /s. ) If the jump dstance s 3 Å, what s the jump frequency near the meltng pont? ) What s the relaton between ths frequency and the vbratonal frequency? ) How far has one atom traveled after 1 hour? v) What s the root mean square dsplacement after one hour? v) What s the root mean square dsplacement n one dmenson after one hour? b) For a metal wth cubc structure the dffuson coeffcent can also be expressed as D = a ω α 0 N d where α s a geometrc factor, a 0 the lattce constant, ω s the jump frequency, and N d s the defect concentraton. Derve the value for α for vacancy dffuson n a metal wth fcc structure. 5.8

29 . Defect and self dffuson TO s doped wth 1 mol% Al 3+ acceptors substtutng the T 4+ ons. Ths s compensated by oxygen vacances. The dffusvty of oxygen vacances s at a gven temperature equal to cm /s. TO. a) Fnd the concentraton (rato) of oxygen vacances n the doped TO. b) Fnd the self dffuson coeffcent of oxygen (oxde ons) n the doped 5.9

30 Answers and hnts to selected Problems, Ch a) ) s -1 ) to be compared to ca s -1 ) 75.6 m v) cm v) cm b) α = 1. a) (ste fracton) b) cm /s 5.30