- Email: [email protected]
Oxygen self-diffusion in non-stoichiometric rutile TiO2−x at high temperature
Solid State lonics 28-30 (1988) 1344-1348 North-Holland, Amsterdam
OXYGEN SELF-DIFFUSION
IN NON-STOICHIOMETRIC
RUTILE rio2_x
AT H I G H T E M P E R A T U R E F. M I L L O T a n d C. P I C A R D l.S. Ma., CNS, 8~timent 415, UniversitOParis-Sud, 91405 Orsay Cedex, France Received 2 July 1987; in revised ve~io~ 2 November 1987
Oxygen self-diffusion coefficients, D*, were determined in TiO:_x at 1373 K (8.4X 10-3>~x>~ 5X 10-4). D~ increases with increasing non-stoichiometry. It is concluded that oxygen vacancies contribute 20% to the non-stoichiometry at x = 8.4× 10-3 and it is suggested that both singly and doubly ionized vacancies are oresent in the approximate ratio of 1-2. The interpretation of the results also ir,dicate that the complete description of defect structure includes interstitial titanium ions with three positive effective charges as major defects for the more reduced oxide whereas interstitial titanium ions with four effective charges may become important in almost stoichiometric TiO,,.
1. Introduction
Rutile exhibits non-stoichiometry that increases with temperature. For instance, at 1 100 ° C, the homogeneity domain approaches TiO~ 99. The nature of defects accommodating non-stoichiometry has been the subject of numerous studies in the past with different techniques. Nevertheless, a recent critical review [ 1 ] on this oxide indicates that the detailed defect structure is still a matter of discussion. In this context, we have decided to determine oxygen selfdiffusion coefficients in reduced rutile at high temperatures, because in spite of a number of similar studies on stoichiometric TiP2 [2-7], doped TiP2 [6] and Anderson's phases Ti,O2,_ ~, which border TIP2_, [ 5 ], no data have been reported for TiO2_x. We have previously published a paper on the experimemal technique [8]. This is briefly summarizcd in the following.
gle crystals (2.2 X0.9 ×0.1 cm 3) oriented along the crystallographic "c" direction o f TIP2. We t h e n have approximately 35 cm 2 surface of which 96% is perp e n d i c u l a r to "c" a n d 4% parallel to "c". All crystals were obtained from a single boule by Verneuil's method and obtained from Djevahirdjian S.A. (Monthey/Switzerland). They were optically polished with 2 gm diamond paste and washed with distilled water, alcohol and ether in an ultrasonic recipient. Finally, they were placed in a recrystallized alumina crucible in such a way that contact surfaces between TiP2 and A1203 was minimized and each TiP2 plate was separated by 1 mm from neighbom ing plates. The crucible was suspended in a silica tube by a silica wire to the balance and heated by an Adamel furnace. During the initial heating at 1373 K, the sample was equi!ibra~ed in oxygen gas and thus essentially steichiemet.r~¢ ~ wa~ then rednced with a k n o ~ H , H2~60 mixture diluted in nitrogen and delivered by 7VU~IYtk.:I ! ~LIlII~3
2. E×perimentaJ The experimental method involves a gas/solid isotope exchange between '60 and ~sO. This is measured as the weight variation of the sample with a microbalance. The sample is made of eight parallelepipedic sin-
~I.llK.l VV~ILt~;I KJVO.~.~UIO.LUI. ~ a ' K . ~ a i l a
~o,~ ~ , ~ a
mixtures flowing ~.he apparatus and specific i~robterns of weight measure m~.~x" een be found i:: ou:: previous paper [ 8]. Once sample weight had been observed to be ,:on:;rant for an extended period of time, N2-H2-H2t60 mixture was replaced by an isocomposition N2-H2H.~sO mixture and we started recording the weight
0 167-2738/88/$ 03.50 © Elsevier Sciepce Publishers B.V. (North-Hoiiand Physics Pu~,~i~hing Divi~ivii)
F. Millot, C Picard/Oxyge;~selfld(~,~ion in non-stoichiometric 730:_,
every 5 min for approximately 3 h. Weight variations were then analyzed to calculate the self-diffusion coefficient of oxygen, DS, and the velocity constant, a, for the surface exchange between 'sO and ~sO. This procedure led in practice to a tao penetration distance of a few microns beneath the surface after 3h. If after this, one replaces the 'SO-containing mixture by an equivalent t~O mixture, t sO will leave the sample and, after some time, it will be possible to perform a new tso experiment and deduce D* and a having almost the same value than in the first experiment (detailed calculations are presented in section 3). The samples were regenerated by flowing N 2 H 2 - H 2 ~ 6 0 mixtures during 45 h between two 3 h ~sO runs, successive ~sO runs being done on more reduced sample.
3. E q u a t i o n s
3.1. D~ and a measurements
Solution of Fick's law corresponding to our diffusion problem can be found, for instance, in ref. [9]. We have generalized our former theoretical treatment [8] to include effecls of the ~60-~so surface exchange. These effects were observed in these new experiments because of increased precision of the weight measurements. (As the surface of the sample was three times larger than in ref. [8 ] and temperature 50 K higher, weight variations are significantly larger and consequently more precise.) Denoting Co the ~sO concentration in the sample before it comes in contact with ~80-containing gas mixture and C~ the concentration of ~80 at the surface of sample in the presence of ~80 mixture we have a f t e r t i m e : a w e i g h t v a r i a t i o n M'. per unit surface area: M, - Cl -h G~ [exp(h2D*t) erfc('~(D*¢)"2) - 1 + (2h/n ~/: ) ( D ' t ) ~/2] ,
where h = c d D ~ , which is the solution for semi-infin:re medium (~80 penetration is only a few microns whereas sample thickness is 1 mm).
1345
Table 1 Relative errors on weight variations during successive ~80 runs with 45 h ~60 regeneration.
1 2 3 4 10 100 1000
0 -6.80X -9.16× - 1.12X - 1.34× - 1.60X -- 1.68X
0 10 -4 10-" 10 -3 10 -3 10 -3 10 -~
- 3 . 9 1 x 10 -3 - 5.29X 10 -3 - 6 . 0 4 X 10 -3
-7.8 -9.43X
× 10 . 3 10 - 3
- 9 . 9 6 X 10 -3
From this, optimization of the experimental Mt (l) curve permits estimates of D* and a.
3.2. Regeneration o f sa;nple
The practical problem is to find how long one has to keep the sample in contact with ~60-containing gas mixture between two tso runs, to long, without introducing significant errors in successive determinations of DS. We will assume, for simplicity, that c~= + ~ , i.e. an infinitely fa~ ~60-t~O exchange at surface sample. Then, thcrc is no theoretical difficulty to calculate the concentration C(x,t) at any time g corresponding to various repetitive ~°O-~sO cycles. Denoting by :~ the time ~he sample is kept in contact with ~60-containing mixture, it can be demonstrated that, for the nth t80 run, the weight variation after time t (counted from the beginning of this last run) is n-- 1
M,=X t"2+
{-[e+it,+(: -l) to] i=!
+ (t+;t~ +ito) ~': +[;t~ + ( ; - ~ ) , o ] " - " - ~,l~ -f l~o)"- ;. )
and first run is M , = K t t':. Table 1 shows the errors in M, relative to the first run at t = 5 min and t = 3 h for various tsO runs (t0=3 b. ~ = 4 5 h). Error in Do* i~ twice this error. Given i" is alwa3s lower than 2°0, the procedure described :n section 2 is correct.
F. Millot, C. Picard/Oxygen selfdiffusion in non-stoichiometric TiO2_x
1346 Table 2 Experimenlai results. Iog,0Po,
x [ 11 ]
D~ (m2s -' )
o~ ( m s - ' )
!.32
1.45X 10-16 1.87X 10-16 2.70X 10-16
1.41X 10-9 8.8× 10 -m 7.1X 10-I°
1.60 1.76 1.91 2.05 2.11
2.8-3.1 × 10-'6 4.09 X 10-16
8 . 8 - 1 0 × 10-to 8.0X 10- io
5.01X 1 0 - " 6.85 X 10- t6 7.58X 10-16 9.20X 1 0 - "
9.5× 10-I° 7.8× 10- io 8.1X 10-I° 7.3× 10-I°
Sample weight variation aftcl 3 h (mg)
-9.48 -10.42 - 11.47
5 7
-
1.7X lO -~ 2.8X lO-3 4.1xlO -a 5.6X 10-3 7.0X 10 -3 8.4X 10 -3
12.42 13.47 14.27 14.87 15.38 15.77
× 1 0 -4 x l O -4 1.1X 10 -3
1.11 1.14
4. E x p e r i m e n t a l r e s u l t s
101
t
D~ a n d a were d e t e r m i n e d at 1373 K for nine different e q u i l i b r i u m p a r t i a l pressures o f o x y g e n which were calculated f r o m t h e relative p r e s s u r e s o f H2 a n d H 2 0 in the gas m i x t u r e [ 1 0 ] . These d a t a are r e p o r t e d in table 2 a n d fig. 1. Deviations f r o m s t o i c h i o m e t r y , x, at 1373 K, were est i m a t e d f r o m t h e work of M a r u c c o et al. [ 1 1 ] . Expected errors in x are 10 -4 a n d on D * a n d a o f the order o f 10% [ 8 ] . T h e e x p e r i m e n t at P o : = 10-t53~ a r m was d o n e twice, the p o i n t s o f the two e x p e r i m e n t s c o i n c i d i n g within the e x p e r i m e n t a l uncertainties. It is o b s e r v e d f r o m table 2 that, a p a r t for the first point, all values o f a are within ( 8 . 5 + 1 . 5 ) X 10 -~° m s -~, i n d i c a t i n g s o m e consistency b e t w e e n successive points (all d a t a were o b t a i n e d w i t h the s a m e PH:c~, viz. 4.58 m m H g ) .
g
5. Discussion As m e n t i o n e d in section 1, d e v i a t i o n s from stoic h i o m e t r y at 1373 K do not exceed I 0 - ' in T i O 2 _ , . ., . .will .. P,u~ ..~L m c . ~ . . u. ,. c . . I I U I I I U.... V V I ! ~ : . . . . L IL I ~. . L. . . UA We assume in the e^,, .ygen mobility can be considered to be i n d e p e n d e n t of x. On this basis, we can look at the effective contributions o f o x y g e n vacancies to the defect structure of futile. The easiest w a y to do so consists o f c o m p a r i n g o u r D* values for s o m e value of x to t h o s e previously obtained with d o p e d crystals.
0
~
o
-
~
lO~x
Fig. I. ,Ox':gcn ,,,,elfdiffusion coefficicnl, D,*,. in TiO: , at 1373 K. =:~ ,,olving intentional d o p i n g have b e e n r e p o r t e d by A r i t a et al. [ 6 ] . T h e y o b t a i n e d D g (1373 K ) = 3 . 4 7 × 10- a6 m ~_s - t for a C r - d o p e d
F. Miliot, C Picard/Oxygen selfdiffusion in non-stoichiometric TiO,_,
TiO2 [ICcrystal where oxygen vacancy concentration was approximately 10 -3. Using the D~ anisotropy reported by these authors at this temperature D~.l.c/Fdllc= 1.5, one can deduce an approximate concentration of oxygen vacancies of 1.7 × 10- ~ for a deviation from the stoichiometry of 8.4 X 10-~. It can be deduced from this simple comparison that the description of the defect structure of TiO2_x implies introducing both oxygen and titanium defects in agreement with most previous works [ 1]. A second question to answer concerns the charge state of oxygen defects in TiO2_x or, in other words, the relative contributions of doubly ionized and singly ionized oxygen vacancies contributing to nonstoichiometry. To that end, reference to a point defect model is needed. This is generally done on considering that equilibrium of defects can be described with ideal mass action laws; for instance, we may write (KrSger's formalism [ 12 ] ):
1347
_Ilog D~ ( m2s-1 )
'A'
½0.~ *) + V~ + 2e' =O~ K(T) = [VG 1- t [e'] -z~-'/2*o., and deduce from some e!ectroneut_,'a!]ty relation be tween defects (for instance, [ e ' ] = 2 [ V S ] if V3 is the only defect) the variation of macroscopic properties with Po_,. Obviously, such a description is on!y approximate since a solution of electrically charged species requires activity coefficients of defects which may, in some cases, deviate strongly from 1. Nevertheless, the Deoye-Hiickel approximation [1] indicates rather high activity coefficients. Various authors [ 10,13-15 ] have described the variations of the deviation from stoichiometry and/ or the electronic conductivity with Po., with such models taking into account two or three of the four possible coexisting defects. Without commenting further on the physical foundations of such descriptions, we have tried to compare the previsions of two are Kofstad's model (k~vli-~-10-14, ~'Ti3'-'5.TX 10 -24, ~T,~-r . . . =.~,~, . . ~t.,'~-:7~ and Marucco's model (R'w~ = 6 . 4 X l0 -~s, KTi,~-=4.2X 10-z6). 200 ppm of extrinsic oxygen vacancies (from A1 and Fe impurities) in stoichiometric TiO2 were taken into account and mobility of oxygen vacancies was taken 5 X 10- ~3 m 2 s- ~ tool- ~ (corresponding to the
-1(
mog %~
Fig. 2. Comparison of the expcrimental variations of D~* with P~,: at 1373 K with various point defect models (see text).
one deduced from the work of A:i:a eta]. [~]). Results are shown in fig. 2 as curve 1 (Kofstad) and curve 2 (Marucco). It is observed that both models approximately predict correct values of DO*.However, neither is able to provide the correct shape of the curve In Do* versus In Po2, disagreement concerning mainly the most reduced TiO2_, oxides. Improvements of Marucco's and Kofstad's models imply the introduction of V8 in the modelisafion. Curve 3 of fig. 2 is an ameliorated Marucco's model with ~V,/. = 6.0 × I 0- ", mark that this last modelling respects the shape of In x versus ~n Po: o~ginally studied by Marucco el al. [11]. With this set of constants, we have [ V ~ ] - - - 1 . S X 1 0 -3, [ V o ] = S X l 0 -4a~ ¥ = 8 . 4 X i 0 -3.
These values of concentrations should be taken as indicative only since we could have choo~n to correct Kofstad's instead of Marucco's model with the
1348
F. Miilot, C. Picard/Oxygen selfdiffusion in non-stoichiometric TiO2_x
same credibility. The point is, in fact, that description of various regular curves (D*, tr, x versus Po: ) with a set ot adjustable parameters can be performed with various defect models (see, for instance, refs. [ 1~5,i 6]). in this context, it is a f interest te mention that Marucco's model, which has been reported to r e p r e s e n t correctly In x and In tr versus In Po, [ 11 ], In D~, versus In Po., [16] and with some modifications In D~ versus In Po., in this study, is obviously physically unreasonable since it does not take into account Ti~', which is known from very direct experimental evidence to be the fastest defect in TiO2_ x for x > 1.SX10 -3 [17,18]. This last remark stresses, in our opinion, the limits of the point defect model based on approximate ideal mass action laws, which does not permit to predict defect concentrations when various defects are contributing to non-stoichiometry. Defects occurrence has then to be deduced from experimental observations. In this context, TiO~_~ defect structure could be described with four defects. We have seen that V~ and V/~ contribute to the non-stoichiometry along with titanium defects. Electromigration and diffusion experiments by Ait-Younes et al. [ 17,19] permitted te deduce that Ti 3" was the fastest moving defect for x > 1.5 × 10 - 3 and that Ti,3" and/or Vo was the major defect for these concentrations in order to agree with the diffusion measurements of Akse and Whitehurst [20]. From V0 defect concentrations obtained in this study, it should be deduced that T 3" is the major defect for most reduced TiO2_x. Occurrence of Ti 4" is less obvious. However, recent data on self-diffusion of titanium in TiOE_x [ 16] show a change of anisotropy (D~-,llc/D*,±c) with Po.,, at least at 1273 and 1373 K, indicating a possible transition from a predominant Ti 4" defect ibr almost stoichiometric TiO2 to a predominant Ti,3 defect for the reduced oxides. This change in ien!c conduction is however difficult to correlate with defect concentrations because it is known, from heterodiffusion of metallic ions, M "+, in TiO: [21]. that D~, is a strongly decreasing function of the electrical charge of M.
6. Conclusian The experimental results for the self-diffusion of
oxygen in TiO:_x at 1373 K suggest the following conclusions: - Oxygen vacancies contribute significantly to the non-stoichiometry (about 20% at x = 8.4 × 10 - 3). - B o t h Vo and Vb, coexist ([V/:;]/[V~]--_2 at x = 8 . 4 X 10-3). - The results indicate that Ti~" is the major defect in most reduced TiO2_x, however, some data suggest the possible existence of Ti~" for almost stoichiometric TiO2.
Rei'erences
[ I ] F . Millot, M.G. Blanchin, R. Tetot, J.F. Marucco, B. Poumellec, C. Picard and B. Touzelin, Progr. Solid State Chem. 17 (1987) 263. [ 2 ] R. Haul and G. Dumbgen, Z. Elektrochem. 66 (1962) 636. [ 3 ] T.B. Gruenwald and G. Gordon, J. Inorg. Nucl. Chem. 33 (1971) ll51. [4] J.M. Calvert, D.J. Deny and D.G. Lees, J. Phys. D 7 (1974) 940. [ 5 ] A.N. Bagshaw and B.G. Hyde, J. Phys. Chem. Solids 37 (1976) 835. [ 6 ] M. Arita, M. Hosoya, M. Kobayashi and M. Someno, J. Am. Ceram. Soc. 62 ( 1979 ) 443. ~'~ V r~^~. t~ ~ y ~ [ 7, ,_,., . . . . 3, l~.',-,. ~ . . . ~. . d J.M. Calvert, J. Phys. Chem. Solids42 ~!981) 57. [ 8 ] F. Millot, C. Picard and J. Berthon, High Temp. High Press., 19 (1987) 337. [ 9 ] J. Crank, Mathematics of diffusion (Clarendon Press, Oxtbrd, 1956). [ 10 ] J.F. Elliott and M. Gleiser, Thermochemistry for steelmaking (Addison-Wesley, Reading, 1960). [ 11 ] J.F. Marucco, J. Gautron and P. Lemasson, J. Phys. Chem. Solids 42 (1981 ) 363. [ 12 ] F.A. Kr6ger, The chemistry of imperfect crystals (NorthHolland, Am~,.rdam, 1964). [ 13 ] R.N. Blumenthal, J. Coburn, J. Baukus and W.M. Hirthe, J. Phys. Chem. Solids 27 (1966) 643. [ 14] P. Kofstad, J. Less Common Metals 13 (1967) 635. [15] G.M. Raynaud and F. '~ " J. Phys. Chem. Solids 46 ,,~onn, ~IQR~ B~71 [ 16] K. Hoshino, N.L. Peterson and C.L. Wiley J. Phys. Chem. Solids 46 ( 198b ) 1397. [ 17] N. Ait-Younes, F. Millot and P. Gerdanian. Solid State lonics 12 (1984) 431. [ 18] F. Millot, J. Mater. Sci. Letters 4 (1985) 902. [ 19 ] N. Alt-Younes, F. Millot and P. Gerdanian, Solid State Ionics 12 (1984) 437. [20] J.R. Akse and H.B. Whitehurst, J. Phys. Chem. Solids 39 (1978) 457. [21] J. Sasaki. N.L. Pe~erson and K. Hoshino, J. Phys. Chem. Solids 46 (1985) 1267.
OXYGEN SELF-DIFFUSION
IN NON-STOICHIOMETRIC
RUTILE rio2_x
AT H I G H T E M P E R A T U R E F. M I L L O T a n d C. P I C A R D l.S. Ma., CNS, 8~timent 415, UniversitOParis-Sud, 91405 Orsay Cedex, France Received 2 July 1987; in revised ve~io~ 2 November 1987
Oxygen self-diffusion coefficients, D*, were determined in TiO:_x at 1373 K (8.4X 10-3>~x>~ 5X 10-4). D~ increases with increasing non-stoichiometry. It is concluded that oxygen vacancies contribute 20% to the non-stoichiometry at x = 8.4× 10-3 and it is suggested that both singly and doubly ionized vacancies are oresent in the approximate ratio of 1-2. The interpretation of the results also ir,dicate that the complete description of defect structure includes interstitial titanium ions with three positive effective charges as major defects for the more reduced oxide whereas interstitial titanium ions with four effective charges may become important in almost stoichiometric TiO,,.
1. Introduction
Rutile exhibits non-stoichiometry that increases with temperature. For instance, at 1 100 ° C, the homogeneity domain approaches TiO~ 99. The nature of defects accommodating non-stoichiometry has been the subject of numerous studies in the past with different techniques. Nevertheless, a recent critical review [ 1 ] on this oxide indicates that the detailed defect structure is still a matter of discussion. In this context, we have decided to determine oxygen selfdiffusion coefficients in reduced rutile at high temperatures, because in spite of a number of similar studies on stoichiometric TiP2 [2-7], doped TiP2 [6] and Anderson's phases Ti,O2,_ ~, which border TIP2_, [ 5 ], no data have been reported for TiO2_x. We have previously published a paper on the experimemal technique [8]. This is briefly summarizcd in the following.
gle crystals (2.2 X0.9 ×0.1 cm 3) oriented along the crystallographic "c" direction o f TIP2. We t h e n have approximately 35 cm 2 surface of which 96% is perp e n d i c u l a r to "c" a n d 4% parallel to "c". All crystals were obtained from a single boule by Verneuil's method and obtained from Djevahirdjian S.A. (Monthey/Switzerland). They were optically polished with 2 gm diamond paste and washed with distilled water, alcohol and ether in an ultrasonic recipient. Finally, they were placed in a recrystallized alumina crucible in such a way that contact surfaces between TiP2 and A1203 was minimized and each TiP2 plate was separated by 1 mm from neighbom ing plates. The crucible was suspended in a silica tube by a silica wire to the balance and heated by an Adamel furnace. During the initial heating at 1373 K, the sample was equi!ibra~ed in oxygen gas and thus essentially steichiemet.r~¢ ~ wa~ then rednced with a k n o ~ H , H2~60 mixture diluted in nitrogen and delivered by 7VU~IYtk.:I ! ~LIlII~3
2. E×perimentaJ The experimental method involves a gas/solid isotope exchange between '60 and ~sO. This is measured as the weight variation of the sample with a microbalance. The sample is made of eight parallelepipedic sin-
~I.llK.l VV~ILt~;I KJVO.~.~UIO.LUI. ~ a ' K . ~ a i l a
~o,~ ~ , ~ a
mixtures flowing ~.he apparatus and specific i~robterns of weight measure m~.~x" een be found i:: ou:: previous paper [ 8]. Once sample weight had been observed to be ,:on:;rant for an extended period of time, N2-H2-H2t60 mixture was replaced by an isocomposition N2-H2H.~sO mixture and we started recording the weight
0 167-2738/88/$ 03.50 © Elsevier Sciepce Publishers B.V. (North-Hoiiand Physics Pu~,~i~hing Divi~ivii)
F. Millot, C Picard/Oxyge;~selfld(~,~ion in non-stoichiometric 730:_,
every 5 min for approximately 3 h. Weight variations were then analyzed to calculate the self-diffusion coefficient of oxygen, DS, and the velocity constant, a, for the surface exchange between 'sO and ~sO. This procedure led in practice to a tao penetration distance of a few microns beneath the surface after 3h. If after this, one replaces the 'SO-containing mixture by an equivalent t~O mixture, t sO will leave the sample and, after some time, it will be possible to perform a new tso experiment and deduce D* and a having almost the same value than in the first experiment (detailed calculations are presented in section 3). The samples were regenerated by flowing N 2 H 2 - H 2 ~ 6 0 mixtures during 45 h between two 3 h ~sO runs, successive ~sO runs being done on more reduced sample.
3. E q u a t i o n s
3.1. D~ and a measurements
Solution of Fick's law corresponding to our diffusion problem can be found, for instance, in ref. [9]. We have generalized our former theoretical treatment [8] to include effecls of the ~60-~so surface exchange. These effects were observed in these new experiments because of increased precision of the weight measurements. (As the surface of the sample was three times larger than in ref. [8 ] and temperature 50 K higher, weight variations are significantly larger and consequently more precise.) Denoting Co the ~sO concentration in the sample before it comes in contact with ~80-containing gas mixture and C~ the concentration of ~80 at the surface of sample in the presence of ~80 mixture we have a f t e r t i m e : a w e i g h t v a r i a t i o n M'. per unit surface area: M, - Cl -h G~ [exp(h2D*t) erfc('~(D*¢)"2) - 1 + (2h/n ~/: ) ( D ' t ) ~/2] ,
where h = c d D ~ , which is the solution for semi-infin:re medium (~80 penetration is only a few microns whereas sample thickness is 1 mm).
1345
Table 1 Relative errors on weight variations during successive ~80 runs with 45 h ~60 regeneration.
1 2 3 4 10 100 1000
0 -6.80X -9.16× - 1.12X - 1.34× - 1.60X -- 1.68X
0 10 -4 10-" 10 -3 10 -3 10 -3 10 -~
- 3 . 9 1 x 10 -3 - 5.29X 10 -3 - 6 . 0 4 X 10 -3
-7.8 -9.43X
× 10 . 3 10 - 3
- 9 . 9 6 X 10 -3
From this, optimization of the experimental Mt (l) curve permits estimates of D* and a.
3.2. Regeneration o f sa;nple
The practical problem is to find how long one has to keep the sample in contact with ~60-containing gas mixture between two tso runs, to long, without introducing significant errors in successive determinations of DS. We will assume, for simplicity, that c~= + ~ , i.e. an infinitely fa~ ~60-t~O exchange at surface sample. Then, thcrc is no theoretical difficulty to calculate the concentration C(x,t) at any time g corresponding to various repetitive ~°O-~sO cycles. Denoting by :~ the time ~he sample is kept in contact with ~60-containing mixture, it can be demonstrated that, for the nth t80 run, the weight variation after time t (counted from the beginning of this last run) is n-- 1
M,=X t"2+
{-[e+it,+(: -l) to] i=!
+ (t+;t~ +ito) ~': +[;t~ + ( ; - ~ ) , o ] " - " - ~,l~ -f l~o)"- ;. )
and first run is M , = K t t':. Table 1 shows the errors in M, relative to the first run at t = 5 min and t = 3 h for various tsO runs (t0=3 b. ~ = 4 5 h). Error in Do* i~ twice this error. Given i" is alwa3s lower than 2°0, the procedure described :n section 2 is correct.
F. Millot, C. Picard/Oxygen selfdiffusion in non-stoichiometric TiO2_x
1346 Table 2 Experimenlai results. Iog,0Po,
x [ 11 ]
D~ (m2s -' )
o~ ( m s - ' )
!.32
1.45X 10-16 1.87X 10-16 2.70X 10-16
1.41X 10-9 8.8× 10 -m 7.1X 10-I°
1.60 1.76 1.91 2.05 2.11
2.8-3.1 × 10-'6 4.09 X 10-16
8 . 8 - 1 0 × 10-to 8.0X 10- io
5.01X 1 0 - " 6.85 X 10- t6 7.58X 10-16 9.20X 1 0 - "
9.5× 10-I° 7.8× 10- io 8.1X 10-I° 7.3× 10-I°
Sample weight variation aftcl 3 h (mg)
-9.48 -10.42 - 11.47
5 7
-
1.7X lO -~ 2.8X lO-3 4.1xlO -a 5.6X 10-3 7.0X 10 -3 8.4X 10 -3
12.42 13.47 14.27 14.87 15.38 15.77
× 1 0 -4 x l O -4 1.1X 10 -3
1.11 1.14
4. E x p e r i m e n t a l r e s u l t s
101
t
D~ a n d a were d e t e r m i n e d at 1373 K for nine different e q u i l i b r i u m p a r t i a l pressures o f o x y g e n which were calculated f r o m t h e relative p r e s s u r e s o f H2 a n d H 2 0 in the gas m i x t u r e [ 1 0 ] . These d a t a are r e p o r t e d in table 2 a n d fig. 1. Deviations f r o m s t o i c h i o m e t r y , x, at 1373 K, were est i m a t e d f r o m t h e work of M a r u c c o et al. [ 1 1 ] . Expected errors in x are 10 -4 a n d on D * a n d a o f the order o f 10% [ 8 ] . T h e e x p e r i m e n t at P o : = 10-t53~ a r m was d o n e twice, the p o i n t s o f the two e x p e r i m e n t s c o i n c i d i n g within the e x p e r i m e n t a l uncertainties. It is o b s e r v e d f r o m table 2 that, a p a r t for the first point, all values o f a are within ( 8 . 5 + 1 . 5 ) X 10 -~° m s -~, i n d i c a t i n g s o m e consistency b e t w e e n successive points (all d a t a were o b t a i n e d w i t h the s a m e PH:c~, viz. 4.58 m m H g ) .
g
5. Discussion As m e n t i o n e d in section 1, d e v i a t i o n s from stoic h i o m e t r y at 1373 K do not exceed I 0 - ' in T i O 2 _ , . ., . .will .. P,u~ ..~L m c . ~ . . u. ,. c . . I I U I I I U.... V V I ! ~ : . . . . L IL I ~. . L. . . UA We assume in the e^,, .ygen mobility can be considered to be i n d e p e n d e n t of x. On this basis, we can look at the effective contributions o f o x y g e n vacancies to the defect structure of futile. The easiest w a y to do so consists o f c o m p a r i n g o u r D* values for s o m e value of x to t h o s e previously obtained with d o p e d crystals.
0
~
o
-
~
lO~x
Fig. I. ,Ox':gcn ,,,,elfdiffusion coefficicnl, D,*,. in TiO: , at 1373 K. =:~ ,,olving intentional d o p i n g have b e e n r e p o r t e d by A r i t a et al. [ 6 ] . T h e y o b t a i n e d D g (1373 K ) = 3 . 4 7 × 10- a6 m ~_s - t for a C r - d o p e d
F. Miliot, C Picard/Oxygen selfdiffusion in non-stoichiometric TiO,_,
TiO2 [ICcrystal where oxygen vacancy concentration was approximately 10 -3. Using the D~ anisotropy reported by these authors at this temperature D~.l.c/Fdllc= 1.5, one can deduce an approximate concentration of oxygen vacancies of 1.7 × 10- ~ for a deviation from the stoichiometry of 8.4 X 10-~. It can be deduced from this simple comparison that the description of the defect structure of TiO2_x implies introducing both oxygen and titanium defects in agreement with most previous works [ 1]. A second question to answer concerns the charge state of oxygen defects in TiO2_x or, in other words, the relative contributions of doubly ionized and singly ionized oxygen vacancies contributing to nonstoichiometry. To that end, reference to a point defect model is needed. This is generally done on considering that equilibrium of defects can be described with ideal mass action laws; for instance, we may write (KrSger's formalism [ 12 ] ):
1347
_Ilog D~ ( m2s-1 )
'A'
½0.~ *) + V~ + 2e' =O~ K(T) = [VG 1- t [e'] -z~-'/2*o., and deduce from some e!ectroneut_,'a!]ty relation be tween defects (for instance, [ e ' ] = 2 [ V S ] if V3 is the only defect) the variation of macroscopic properties with Po_,. Obviously, such a description is on!y approximate since a solution of electrically charged species requires activity coefficients of defects which may, in some cases, deviate strongly from 1. Nevertheless, the Deoye-Hiickel approximation [1] indicates rather high activity coefficients. Various authors [ 10,13-15 ] have described the variations of the deviation from stoichiometry and/ or the electronic conductivity with Po., with such models taking into account two or three of the four possible coexisting defects. Without commenting further on the physical foundations of such descriptions, we have tried to compare the previsions of two are Kofstad's model (k~vli-~-10-14, ~'Ti3'-'5.TX 10 -24, ~T,~-r . . . =.~,~, . . ~t.,'~-:7~ and Marucco's model (R'w~ = 6 . 4 X l0 -~s, KTi,~-=4.2X 10-z6). 200 ppm of extrinsic oxygen vacancies (from A1 and Fe impurities) in stoichiometric TiO2 were taken into account and mobility of oxygen vacancies was taken 5 X 10- ~3 m 2 s- ~ tool- ~ (corresponding to the
-1(
mog %~
Fig. 2. Comparison of the expcrimental variations of D~* with P~,: at 1373 K with various point defect models (see text).
one deduced from the work of A:i:a eta]. [~]). Results are shown in fig. 2 as curve 1 (Kofstad) and curve 2 (Marucco). It is observed that both models approximately predict correct values of DO*.However, neither is able to provide the correct shape of the curve In Do* versus In Po2, disagreement concerning mainly the most reduced TiO2_, oxides. Improvements of Marucco's and Kofstad's models imply the introduction of V8 in the modelisafion. Curve 3 of fig. 2 is an ameliorated Marucco's model with ~V,/. = 6.0 × I 0- ", mark that this last modelling respects the shape of In x versus ~n Po: o~ginally studied by Marucco el al. [11]. With this set of constants, we have [ V ~ ] - - - 1 . S X 1 0 -3, [ V o ] = S X l 0 -4a~ ¥ = 8 . 4 X i 0 -3.
These values of concentrations should be taken as indicative only since we could have choo~n to correct Kofstad's instead of Marucco's model with the
1348
F. Miilot, C. Picard/Oxygen selfdiffusion in non-stoichiometric TiO2_x
same credibility. The point is, in fact, that description of various regular curves (D*, tr, x versus Po: ) with a set ot adjustable parameters can be performed with various defect models (see, for instance, refs. [ 1~5,i 6]). in this context, it is a f interest te mention that Marucco's model, which has been reported to r e p r e s e n t correctly In x and In tr versus In Po, [ 11 ], In D~, versus In Po., [16] and with some modifications In D~ versus In Po., in this study, is obviously physically unreasonable since it does not take into account Ti~', which is known from very direct experimental evidence to be the fastest defect in TiO2_ x for x > 1.SX10 -3 [17,18]. This last remark stresses, in our opinion, the limits of the point defect model based on approximate ideal mass action laws, which does not permit to predict defect concentrations when various defects are contributing to non-stoichiometry. Defects occurrence has then to be deduced from experimental observations. In this context, TiO~_~ defect structure could be described with four defects. We have seen that V~ and V/~ contribute to the non-stoichiometry along with titanium defects. Electromigration and diffusion experiments by Ait-Younes et al. [ 17,19] permitted te deduce that Ti 3" was the fastest moving defect for x > 1.5 × 10 - 3 and that Ti,3" and/or Vo was the major defect for these concentrations in order to agree with the diffusion measurements of Akse and Whitehurst [20]. From V0 defect concentrations obtained in this study, it should be deduced that T 3" is the major defect for most reduced TiO2_x. Occurrence of Ti 4" is less obvious. However, recent data on self-diffusion of titanium in TiOE_x [ 16] show a change of anisotropy (D~-,llc/D*,±c) with Po.,, at least at 1273 and 1373 K, indicating a possible transition from a predominant Ti 4" defect ibr almost stoichiometric TiO2 to a predominant Ti,3 defect for the reduced oxides. This change in ien!c conduction is however difficult to correlate with defect concentrations because it is known, from heterodiffusion of metallic ions, M "+, in TiO: [21]. that D~, is a strongly decreasing function of the electrical charge of M.
6. Conclusian The experimental results for the self-diffusion of
oxygen in TiO:_x at 1373 K suggest the following conclusions: - Oxygen vacancies contribute significantly to the non-stoichiometry (about 20% at x = 8.4 × 10 - 3). - B o t h Vo and Vb, coexist ([V/:;]/[V~]--_2 at x = 8 . 4 X 10-3). - The results indicate that Ti~" is the major defect in most reduced TiO2_x, however, some data suggest the possible existence of Ti~" for almost stoichiometric TiO2.
Rei'erences
[ I ] F . Millot, M.G. Blanchin, R. Tetot, J.F. Marucco, B. Poumellec, C. Picard and B. Touzelin, Progr. Solid State Chem. 17 (1987) 263. [ 2 ] R. Haul and G. Dumbgen, Z. Elektrochem. 66 (1962) 636. [ 3 ] T.B. Gruenwald and G. Gordon, J. Inorg. Nucl. Chem. 33 (1971) ll51. [4] J.M. Calvert, D.J. Deny and D.G. Lees, J. Phys. D 7 (1974) 940. [ 5 ] A.N. Bagshaw and B.G. Hyde, J. Phys. Chem. Solids 37 (1976) 835. [ 6 ] M. Arita, M. Hosoya, M. Kobayashi and M. Someno, J. Am. Ceram. Soc. 62 ( 1979 ) 443. ~'~ V r~^~. t~ ~ y ~ [ 7, ,_,., . . . . 3, l~.',-,. ~ . . . ~. . d J.M. Calvert, J. Phys. Chem. Solids42 ~!981) 57. [ 8 ] F. Millot, C. Picard and J. Berthon, High Temp. High Press., 19 (1987) 337. [ 9 ] J. Crank, Mathematics of diffusion (Clarendon Press, Oxtbrd, 1956). [ 10 ] J.F. Elliott and M. Gleiser, Thermochemistry for steelmaking (Addison-Wesley, Reading, 1960). [ 11 ] J.F. Marucco, J. Gautron and P. Lemasson, J. Phys. Chem. Solids 42 (1981 ) 363. [ 12 ] F.A. Kr6ger, The chemistry of imperfect crystals (NorthHolland, Am~,.rdam, 1964). [ 13 ] R.N. Blumenthal, J. Coburn, J. Baukus and W.M. Hirthe, J. Phys. Chem. Solids 27 (1966) 643. [ 14] P. Kofstad, J. Less Common Metals 13 (1967) 635. [15] G.M. Raynaud and F. '~ " J. Phys. Chem. Solids 46 ,,~onn, ~IQR~ B~71 [ 16] K. Hoshino, N.L. Peterson and C.L. Wiley J. Phys. Chem. Solids 46 ( 198b ) 1397. [ 17] N. Ait-Younes, F. Millot and P. Gerdanian. Solid State lonics 12 (1984) 431. [ 18] F. Millot, J. Mater. Sci. Letters 4 (1985) 902. [ 19 ] N. Alt-Younes, F. Millot and P. Gerdanian, Solid State Ionics 12 (1984) 437. [20] J.R. Akse and H.B. Whitehurst, J. Phys. Chem. Solids 39 (1978) 457. [21] J. Sasaki. N.L. Pe~erson and K. Hoshino, J. Phys. Chem. Solids 46 (1985) 1267.