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Transport of lanthanum ion and hole in LaCrO3 determined by electrical conductivity measurements
Solid State Ionics 164 (2003) 177 – 183 www.elsevier.com/locate/ssi
Transport of lanthanum ion and hole in LaCrO3 determined by electrical conductivity measurements Takaya Akashi a,*, Toshio Maruyama b, Takashi Goto a b
a Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan Department of Metallurgy and Ceramics Science, Graduate School of Science and Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
Received 14 November 2002; received in revised form 22 August 2003; accepted 24 August 2003
Abstract Effective diffusion coefficient in LaCrO3 was determined by transient electrical conductivity based on Fick’s second law. The diffusion coefficient of La3 + ion was calculated from the effective diffusion coefficient. The diffusion coefficient of La3 + ion and the activation energy was in agreement with the reported value. The diffusion coefficient of hole in LaCrO3 was also calculated from electrical conductivity and hole concentration. The hole concentration is estimated from the relation between the electrical conductivity and the amount of alkaline-earth in (La,A)CrO3 (A = Sr, Ca). The diffusion coefficient of hole in LaCrO3 was much higher than those of other components in LaCrO3. D 2003 Elsevier B.V. All rights reserved. Keywords: Lanthanum chromite; Diffusion coefficient; Lanthanum ion; Hole; Electrical conductivity; Fick’s second law; Nernst – Einstein’s equation
1. Introduction Alkaline-earth substituted LaCrO3 has high ptype conductivity [1– 5] and stability in wide range of oxygen pressures between 105 and 10 16 Pa at 1273 K [6]. Therefore, it has been used for interconnector in solid oxide fuel cells (SOFCs) [7– 10] and magneto-hydrodynamic (MHD) electrode [11]. Because these oxide materials are used under chemical potential gradient at elevated temperatures, constituents of the oxide are transported by the po-
* Corresponding author. Tel.: +81-22-215-2106; fax: +81-22215-2107. E-mail address: [email protected] (T. Akashi). 0167-2738/$ - see front matter D 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ssi.2003.08.050
tential gradient [9,10,12 –19]. These transport phenomena would be the problem for a long-term application. Various diffusion coefficients in alkaline-earth substituted lanthanum chromites have been widely investigated [9,10,12 – 17] to predict the problem of application. Diffusion coefficient of oxide ion in (La,Ca)CrO3 d was measured by 18O exchange and SIMS analysis [10,15], because the oxygen diffusion results in leak current in SOFC. Yasuda et al. [15] suggested the vacancy mechanism for lattice diffusion of oxide ion in polycrystalline (La,Ca)CrO3 d by a measurement of the oxygen pressure dependence of the lattice diffusion coefficient. Sakai et al. [10] reported that the oxygen vacancy diffusion coefficient was constant regardless of calcium content or oxygen-
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T. Akashi et al. / Solid State Ionics 164 (2003) 177–183
attain equilibrium. The specimen was identified as two phase of LaCrO3 and Cr2O3 by X-ray diffraction. The powder of LaCrO3 with 5 vol.% Cr2O3 addition was pressed into a tablet and sintered at 1710 K under 60 MPa for 1.2 ks. The relative density of the specimen was 96 %. The specimen was cut into 0.8 3.1 10.5 mm for electrical conductivity measurement. The electrical conductivity was measured isothermally from 1573 to 1673 K by D.C. four-probe technique. Oxygen partial pressure was controlled from 1.0 103 to 2.0 104 Pa in Ar – O2 gas. Fig. 1. Transient electrical conductivity of LaCrO3 with 5 vol.% Cr2O3 addition.
vacancy content for (La,Ca)CrO3 d. Cation diffusion coefficients in alkaline-earth substituted lanthanum chromites were also measured by SIMS [18,19]. Using these cation diffusion coefficients, the degradation rate, which would occur in the long-term operation of SOFC, can be evaluated [19]. In this paper, we measured the transient electrical conductivity to determine effective diffusion coefficient in LaCrO3. We also determined the diffusion coefficient of hole in LaCrO3 from the electrical conductivity. It is difficult to measure the electrical conductivity of undoped LaCrO3, because the chromium oxide evaporates at elevated temperatures. In this study, excess chromia was added to LaCrO3 to fix the activity of chromia at unity. This method enables one to measure the electrical conductivity with fixed activity.
3. Results Fig. 1 shows the transient electrical conductivity of LaCrO3 with 5 vol.% Cr2O3 addition during the experiment. After changes of oxygen pressure or temperature, the electrical conductivity approached to the new equilibrium value. The electrical conductivity increased after the oxygen partial pressure or the temperature increased. The electrical conductivity decreased after the oxygen partial pressure or the temperature decreased. More than 4 months were required to measure the electrical conductivity at
2. Experimental procedure Powders of La2O3 (Rare Metallic 99.9% purity) and Cr2O3 (Shin-Etsu Chemical 99.9% purity) were mixed in the composition to form stoichiometric LaCrO3 with excess 5 vol.% Cr2O3 to fix the activity of Cr2O3 at unity. The amount of Cr2O3 was determined as minimum amount detectable for X-ray diffraction. The mixed powder was pressed at 100 MPa into a disk-shaped tablet and reacted at 1673 K at 259 ks. After the reaction, the specimen was pulverized into powder. The powder was pressed, reacted and pulverized in the same procedure to
Fig. 2. Oxygen pressure dependence of the electrical conductivity of LaCrO3 with 5 vol.% Cr2O3 addition.
T. Akashi et al. / Solid State Ionics 164 (2003) 177–183
equilibrium state, involving a interruption by meltdown of the SiC furnace. Fig. 2 shows the oxygen pressure dependence of the electrical conductivity for LaCrO3 with 5 vol.% Cr2O3 addition at 1673 K, which was obtained from the equilibrium state in Fig. 1. The electrical conductivity was proportional to PO3/16 . This oxygen pressure 2 dependence was in good agreement with the oxygen pressure dependence of the diffusion coefficient of La3 + ion in LaCrO3 [20]. This indicates that the electrical neutrality is maintained by the hole and lanthanum vacancy and the carrier is the hole in LaCrO3. The electrical conductivity of LaCrO3 is plotted as a function of temperature in Fig. 3. This figure also shows the reported electrical conductivities of LaCrO3s [1 –3]. The present electrical conductivity was in agreement with the value by Webb et al. [2] and that by Karim and Aldred [3], but higher than that by Meadowcroft [1]. The apparent activation energy of 58 kJ mol 1 was higher than the reported values.
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4. Discussion 4.1. Effective diffusion coefficient in LaCrO3 The specimen of dimensions a1 < x1 < a1, a2 < x2 < a2, a3 < x3 < a3 is in equilibrium. The initial electrical conductivity r0 is proportional to the initial uniform concentration of vacancies c0. At time t = 0, the new condition is established and the surface concentration is changed to cf. As t approaches infinite, the electrical conductivity r approaches a new equilibrium conductivity rf, which is proportional to cf. For these boundary condition, the following solution of Fick’s second law was derived by Newman [21 – 23]. For the average concentration of vacancies cave(t) in the sample at a given time, the fractional saturation is given by cave ðtÞ cf ¼ s1 ðtÞ s2 ðtÞ s3 ðtÞ c0 cf
ð1Þ
where
( ) 2 l ˜ n þ 12 p2 Dt 2 X 1 sj ðtÞ ¼ 2 : exp p n¼0 n þ 1 2 a2j 2
ð2Þ ˜ is the effective diffusion coefficient and t is Here, D the time. In this experiment, a1 is much smaller than a2 and a3. Therefore, Eq. (1) can be arranged as rðtÞ rf ¼ s1 ðtÞ: r 0 rf
ð3Þ
Here, r(t) is the electrical conductivity at time t. ˜ t/a12 vs. t plots for the data in Fig. 4 shows the D Fig. 1. The linear relationship in Fig. 4 demonstrates that the electrical conductivity changed based on Eq. (3). The effective diffusion coefficient was calculated from the slope in Fig. 4. Fig. 5 shows the temperature dependence of the effective diffusion coefficient. The apparent activation energy of the diffusion coefficient was 387 kJ mol 1. A mechanism of the change in the electrical conductivity in Fig. 1 is proposed in Fig. 6. As shown in Fig. 6(a), after oxygen pressure or temperature increases, chromia and oxygen generate lanthanum vacancy and hole at the surface of LaCrO3 by Fig. 3. Temperature dependence of the electrical conductivity of LaCrO3 with 5 vol.% Cr2O3 addition.
3 Cr2 O3 þ O2 ! 2VLa j þ 2Crcrx þ 6Oox þ 6h : 2
S
ð4Þ
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˜ t/a21 and time for curves in Fig. 1. Fig. 4. Relation between D
The lanthanum vacancy and hole diffuse to the inside of LaCrO3. On the other hand, after oxygen pressure or temperature decreases opposite transport and reaction occur as shown in Fig. 6(b). Because the diffusion of hole is much higher than that of lantha-
Fig. 6. Mechanism of the electrical conductivity change for (a) oxidation and (b) reduction.
num vacancy in LaCrO3 and these concentrations are fix by 3[VLa UV]=[h.], the flux of hole also much higher than that of lanthanum vacancy. Therefore, the diffusion of lanthanum vacancy determined the change of the electrical conductivity in Fig. 1. 4.2. Diffusion of hole in LaCrO3 Fig. 5. Temperature dependence of the effective diffusion ˜ ) LaCrO3 with 5 vol.% Cr2O3 addition. coefficient in (D
The diffusion coefficient of hole Dh is calculated from the electrical conductivity of hole rh and hole
T. Akashi et al. / Solid State Ionics 164 (2003) 177–183
181
concentration [h .] using the Nernst – Einstein’s equation: kT Dh ¼ rh : ð5Þ ½h e2
S
The electrical conductivity can be expressed as
S
rh ¼ elh ½h
ð6Þ
where e is the elementary electric charge and lh is the mobility of hole in LaCrO3. The mobility of the hole in LaCrO3 can be estimated from the electrical conductivity of La1 xAxCrO3 (A = Sr, Ca) as a function of the composition, x. The concentration of hole in La1 xAxCrO3 is fixed by [h.]x=[ALa V ]x = xNA/Vm(x), because the electrical conductivity was measured at higher temperatures. The electrical conductivity of La1 xSrxCrO3 becomes rh ðxÞ ¼ elh ðxÞ
xNA Vm ðxÞ
Hall
Hall
ð7Þ
Here, NA is the Avogadro’s number, lh(x) is the mobility of hole and Vm(x) is the volume of one mole of La1 xAxCrO3. The mobility of hole in LaCrO3 is determined by the slope of the plot of x and rh(x) at x = 0. Fig. 7 shows the reported electrical conductivity of La1 xAxCrO3 (A = Sr, Ca) as a
Fig. 8. Mobility of hole in LaCrO3 calculated from the electrical conductivity.
function of the composition x at 1273 K. The mobility of hole was calculated from the slope at x = 0 and plotted in Fig. 8. This figure also shows the Hall mobility for LaCrO3 and that for La0.95Sr0.05CrO3 determined by Bansel et al. [24] The extrapolated values of the mobility was in agreement with the Hall mobility for LaCrO3 and smaller than the Hall mobility for La0.95Sr0.05CrO3. Substituting r in Fig. 3 and lh in Fig. 8 into Eq. (6), one can obtain the hole concentration. Fig. 9 shows the hole concentration as a function of temperature. This figure also shows the concentration of lanthanum vacancy based on electroneutrality 3[VLaUV]=[h.]. Because the concentration of hole were expressed as DHð4Þ j 1=8 3=16 ½h ¼ AaCr2 O3 PO2 exp ð8Þ 8RT
S
Fig. 7. Compositional dependence of the electrical conductivity of La1 xAxCrO3 at 1273 K.
the standard enthalpy change of the defect reaction (4), DH(4)j was determined to be 360 k mol 1 from the slope in Fig. 9. Here, A is a constant independent on temperature, including the standard entropy change of the defect reaction (4), the activity coefficient of hole and the activity coefficient of lanthanum vacancy. As shown in Fig. 3, the apparent activation energy for the electrical conductivity of LaCrO3 in this study
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efficient. Yasuda and Hishinuma [25] suggested the following relation of chemical diffusion coefficient ˜ ) and diffusion coefficient of oxygen vacancy (DV ) (D o for transport of the oxygen and hole transport. SS ˜ ¼ DVo 1 þ 4½Vo
D ð9Þ ½h
S
For the transport of lanthanum vacancy and hole in this experiment, Eq. (9) can be arranged as follows. 9½VLa j
˜ D ¼ DVLa 1 þ ð10Þ ½h
S
The diffusion coefficient of lanthanum vacancy was obtained from Eq. (10) and the electroneutrality of j ]=[h.]. 3[VLa ˜ DVLa ¼ D=4 ð11Þ
Fig. 9. Concentration of lanthanum vacancy and hole in LaCrO3 as a function of temperature.
was higher than the reported values. The higher activation energy can be explained by the enthalpy change of the defect reaction (4). Because the activity of Cr2O3 was fixed in this study, the concentration of hole increased with increasing temperature by the defect reaction (4). On the other hand, the hole concentrations in the reported LaCrO3s were probably fixed by impurities. If the temperature dependence of the hole concentrations in reported LaCrO3s are small enough, the difference of the apparent activation energy is corresponding to DH(4)j/8. As shown in Fig. 3, the value of DH(4)j/8, 45 k mol 1, was in good agreement with the difference between the apparent activation energy in this study and the reported values, 40 –46 kJ mol 1. Substituting r in Fig. 3 and [h.] in Fig. 9 in Eq. (5), one can determine the diffusion coefficient of hole. The diffusion coefficients in LaCrO3 are summarized in Fig. 10. The diffusion of holes is the fastest, diffusion of oxide ions is slower, and cation diffusion is the slowest in LaCrO3.
The diffusion coefficient of La3 + ion is calculated from that of lanthanum vacancy by DLa3þ ¼ ðcVLa =cLa3þ ÞDVLa :
ð12Þ
Fig. 10 shows the diffusion coefficient of lanthanum vacancy and that of La3 + ion calculate from Eqs.
4.3. Diffusion of lanthanum ion in LaCrO3 The diffusion coefficient obtained by the transient conductivity measurement are chemical diffusion co-
Fig. 10. Diffusion coefficients of components in LaCrO3 as a function of temperature. LCC: (La,Ca)CrO3, LSC: (La, Sr)CrO3.
T. Akashi et al. / Solid State Ionics 164 (2003) 177–183
(11) and (12). The calculated La3 + diffusion coefficient and activation energy were in agreement with the La3 + diffusion coefficient determined by solid state reaction [20].
5. Conclusion Effective diffusion coefficient in LaCrO3 was determined by an transient electrical conductivity measurement. The diffusion of lanthanum vacancy determined the change of the electrical conductivity. The calculated diffusion coefficient of La3 + ion and the activation energy was in agreement with the reported value. The diffusion coefficient of hole in LaCrO3 was also determined by electrical conductivity measurement. The diffusion coefficients of hole is much higher than those of other components in LaCrO3. These diffusion coefficients are useful in understanding the transport behavior in LaCrO3 at elevated temperatures.
References [1] D.B. Meadowcroft, Br. J. Appl. Phys., J. Phys. D 2 – 2 (1969) 1225. [2] J.B. Webb, M. Sayer, A. Mansingh, Can. J. Phys. 55 (1977) 1725. [3] D.P. Karim, A.T. Aldred, Phys. Rev. B 20 (1979) 2255. [4] R. Koc, H.U. Anderson, J. Mater. Sci. 27 (1992) 5477. [5] I. Yasuda, T. Hikita, J. Electrochem. Soc. 140 (1993) 1699. [6] T. Nakamura, G. Petzow, L.J. Gauckler, Mater. Res. Bull. 14 (1979) 649.
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[7] V.E.J. van Dieten, J.P. Dekker, J. Schoonman, Solid State Ionics 53 – 56 (1992) 611. [8] I. Yasuda, M. Hishinuma, J. Electrochem. Soc. 143 (1996) 1583. [9] M. Suzuki, H. Sakaki, A. Kajimura, Solid State Ionics 96 (1997) 83. [10] N. Sakai, K. Yamaji, T. Horita, H. Yokokawa, T. Kawada, M. Dokiya, J. Electrochem. Soc. 147 (2000) 3178. [11] N. Ram Mohan, K. Thiagarajan, V. Sivan, Ceram. Int. 20 (1994) 143. [12] B.A. van Hassel, T. Kawada, N. Sakai, H. Yokokawa, M. Dokiya, Solid State Ionics 66 (1993) 41. [13] T. Kawada, T. Horita, N. Sakai, H. Yokokawa, M. Dokiya, Solid State Ionics 79 (1995) 201. [14] N. Sakai, T. Horita, H. Yokokawa, M. Dokiya, T. Kawada, Solid State Ionics 86 – 88 (1996) 1273. [15] I. Yasuda, K. Ogasawara, M. Hishinuma, J. Am. Ceram. Soc. 80 (1997) 3009. [16] N. Sakai, K. Yamaji, T. Horita, H. Yokokawa, T. Kawada, M. Dokiya, K. Hiwatashi, A. Ueno, M. Aizawa, J. Eletrochem. Soc. 146 (1999) 1341. [17] D.-K. Lee, H.-I. Yoo, J. Electrochem. Soc. 147 (2000) 2835. [18] T. Horita, M. Ishikawa, K. Yamaji, N. Sakai, H. Yokokawa, M. Dokiya, Solid State Ionics 108 (1998) 383. [19] N. Sakai, K. Yamaji, T. Horita, H. Negishi, H. Yokokawa, Solid State Ionics 135 (2000) 469 – 474. [20] T. Akashi, M. Nanko, T. Maruyama, Y. Shiraishi, J. Tanabe, J. Electrochem. Soc. 145 (1998) 2090. [21] A.B. Newman, Trans. Am. Inst. Chem. Eng. 27 (1931) 310. [22] L.C. Walters, R.E. Grace, J. Phys. Chem. Solids 28 (1967) 245. [23] W.L. George, R.E. Grace, J. Phys. Chem. Solids 30 (1969) 889. [24] K.P. Bansal, S. Kumari, B.K. Das, G.C. Jain, J. Mater. Sci. 18 (1983) 2095. [25] I. Yasuda, M. Hishinuma, Solid State Ionics 80 (1995) 141.
Transport of lanthanum ion and hole in LaCrO3 determined by electrical conductivity measurements Takaya Akashi a,*, Toshio Maruyama b, Takashi Goto a b
a Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan Department of Metallurgy and Ceramics Science, Graduate School of Science and Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
Received 14 November 2002; received in revised form 22 August 2003; accepted 24 August 2003
Abstract Effective diffusion coefficient in LaCrO3 was determined by transient electrical conductivity based on Fick’s second law. The diffusion coefficient of La3 + ion was calculated from the effective diffusion coefficient. The diffusion coefficient of La3 + ion and the activation energy was in agreement with the reported value. The diffusion coefficient of hole in LaCrO3 was also calculated from electrical conductivity and hole concentration. The hole concentration is estimated from the relation between the electrical conductivity and the amount of alkaline-earth in (La,A)CrO3 (A = Sr, Ca). The diffusion coefficient of hole in LaCrO3 was much higher than those of other components in LaCrO3. D 2003 Elsevier B.V. All rights reserved. Keywords: Lanthanum chromite; Diffusion coefficient; Lanthanum ion; Hole; Electrical conductivity; Fick’s second law; Nernst – Einstein’s equation
1. Introduction Alkaline-earth substituted LaCrO3 has high ptype conductivity [1– 5] and stability in wide range of oxygen pressures between 105 and 10 16 Pa at 1273 K [6]. Therefore, it has been used for interconnector in solid oxide fuel cells (SOFCs) [7– 10] and magneto-hydrodynamic (MHD) electrode [11]. Because these oxide materials are used under chemical potential gradient at elevated temperatures, constituents of the oxide are transported by the po-
* Corresponding author. Tel.: +81-22-215-2106; fax: +81-22215-2107. E-mail address: [email protected] (T. Akashi). 0167-2738/$ - see front matter D 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ssi.2003.08.050
tential gradient [9,10,12 –19]. These transport phenomena would be the problem for a long-term application. Various diffusion coefficients in alkaline-earth substituted lanthanum chromites have been widely investigated [9,10,12 – 17] to predict the problem of application. Diffusion coefficient of oxide ion in (La,Ca)CrO3 d was measured by 18O exchange and SIMS analysis [10,15], because the oxygen diffusion results in leak current in SOFC. Yasuda et al. [15] suggested the vacancy mechanism for lattice diffusion of oxide ion in polycrystalline (La,Ca)CrO3 d by a measurement of the oxygen pressure dependence of the lattice diffusion coefficient. Sakai et al. [10] reported that the oxygen vacancy diffusion coefficient was constant regardless of calcium content or oxygen-
178
T. Akashi et al. / Solid State Ionics 164 (2003) 177–183
attain equilibrium. The specimen was identified as two phase of LaCrO3 and Cr2O3 by X-ray diffraction. The powder of LaCrO3 with 5 vol.% Cr2O3 addition was pressed into a tablet and sintered at 1710 K under 60 MPa for 1.2 ks. The relative density of the specimen was 96 %. The specimen was cut into 0.8 3.1 10.5 mm for electrical conductivity measurement. The electrical conductivity was measured isothermally from 1573 to 1673 K by D.C. four-probe technique. Oxygen partial pressure was controlled from 1.0 103 to 2.0 104 Pa in Ar – O2 gas. Fig. 1. Transient electrical conductivity of LaCrO3 with 5 vol.% Cr2O3 addition.
vacancy content for (La,Ca)CrO3 d. Cation diffusion coefficients in alkaline-earth substituted lanthanum chromites were also measured by SIMS [18,19]. Using these cation diffusion coefficients, the degradation rate, which would occur in the long-term operation of SOFC, can be evaluated [19]. In this paper, we measured the transient electrical conductivity to determine effective diffusion coefficient in LaCrO3. We also determined the diffusion coefficient of hole in LaCrO3 from the electrical conductivity. It is difficult to measure the electrical conductivity of undoped LaCrO3, because the chromium oxide evaporates at elevated temperatures. In this study, excess chromia was added to LaCrO3 to fix the activity of chromia at unity. This method enables one to measure the electrical conductivity with fixed activity.
3. Results Fig. 1 shows the transient electrical conductivity of LaCrO3 with 5 vol.% Cr2O3 addition during the experiment. After changes of oxygen pressure or temperature, the electrical conductivity approached to the new equilibrium value. The electrical conductivity increased after the oxygen partial pressure or the temperature increased. The electrical conductivity decreased after the oxygen partial pressure or the temperature decreased. More than 4 months were required to measure the electrical conductivity at
2. Experimental procedure Powders of La2O3 (Rare Metallic 99.9% purity) and Cr2O3 (Shin-Etsu Chemical 99.9% purity) were mixed in the composition to form stoichiometric LaCrO3 with excess 5 vol.% Cr2O3 to fix the activity of Cr2O3 at unity. The amount of Cr2O3 was determined as minimum amount detectable for X-ray diffraction. The mixed powder was pressed at 100 MPa into a disk-shaped tablet and reacted at 1673 K at 259 ks. After the reaction, the specimen was pulverized into powder. The powder was pressed, reacted and pulverized in the same procedure to
Fig. 2. Oxygen pressure dependence of the electrical conductivity of LaCrO3 with 5 vol.% Cr2O3 addition.
T. Akashi et al. / Solid State Ionics 164 (2003) 177–183
equilibrium state, involving a interruption by meltdown of the SiC furnace. Fig. 2 shows the oxygen pressure dependence of the electrical conductivity for LaCrO3 with 5 vol.% Cr2O3 addition at 1673 K, which was obtained from the equilibrium state in Fig. 1. The electrical conductivity was proportional to PO3/16 . This oxygen pressure 2 dependence was in good agreement with the oxygen pressure dependence of the diffusion coefficient of La3 + ion in LaCrO3 [20]. This indicates that the electrical neutrality is maintained by the hole and lanthanum vacancy and the carrier is the hole in LaCrO3. The electrical conductivity of LaCrO3 is plotted as a function of temperature in Fig. 3. This figure also shows the reported electrical conductivities of LaCrO3s [1 –3]. The present electrical conductivity was in agreement with the value by Webb et al. [2] and that by Karim and Aldred [3], but higher than that by Meadowcroft [1]. The apparent activation energy of 58 kJ mol 1 was higher than the reported values.
179
4. Discussion 4.1. Effective diffusion coefficient in LaCrO3 The specimen of dimensions a1 < x1 < a1, a2 < x2 < a2, a3 < x3 < a3 is in equilibrium. The initial electrical conductivity r0 is proportional to the initial uniform concentration of vacancies c0. At time t = 0, the new condition is established and the surface concentration is changed to cf. As t approaches infinite, the electrical conductivity r approaches a new equilibrium conductivity rf, which is proportional to cf. For these boundary condition, the following solution of Fick’s second law was derived by Newman [21 – 23]. For the average concentration of vacancies cave(t) in the sample at a given time, the fractional saturation is given by cave ðtÞ cf ¼ s1 ðtÞ s2 ðtÞ s3 ðtÞ c0 cf
ð1Þ
where
( ) 2 l ˜ n þ 12 p2 Dt 2 X 1 sj ðtÞ ¼ 2 : exp p n¼0 n þ 1 2 a2j 2
ð2Þ ˜ is the effective diffusion coefficient and t is Here, D the time. In this experiment, a1 is much smaller than a2 and a3. Therefore, Eq. (1) can be arranged as rðtÞ rf ¼ s1 ðtÞ: r 0 rf
ð3Þ
Here, r(t) is the electrical conductivity at time t. ˜ t/a12 vs. t plots for the data in Fig. 4 shows the D Fig. 1. The linear relationship in Fig. 4 demonstrates that the electrical conductivity changed based on Eq. (3). The effective diffusion coefficient was calculated from the slope in Fig. 4. Fig. 5 shows the temperature dependence of the effective diffusion coefficient. The apparent activation energy of the diffusion coefficient was 387 kJ mol 1. A mechanism of the change in the electrical conductivity in Fig. 1 is proposed in Fig. 6. As shown in Fig. 6(a), after oxygen pressure or temperature increases, chromia and oxygen generate lanthanum vacancy and hole at the surface of LaCrO3 by Fig. 3. Temperature dependence of the electrical conductivity of LaCrO3 with 5 vol.% Cr2O3 addition.
3 Cr2 O3 þ O2 ! 2VLa j þ 2Crcrx þ 6Oox þ 6h : 2
S
ð4Þ
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T. Akashi et al. / Solid State Ionics 164 (2003) 177–183
˜ t/a21 and time for curves in Fig. 1. Fig. 4. Relation between D
The lanthanum vacancy and hole diffuse to the inside of LaCrO3. On the other hand, after oxygen pressure or temperature decreases opposite transport and reaction occur as shown in Fig. 6(b). Because the diffusion of hole is much higher than that of lantha-
Fig. 6. Mechanism of the electrical conductivity change for (a) oxidation and (b) reduction.
num vacancy in LaCrO3 and these concentrations are fix by 3[VLa UV]=[h.], the flux of hole also much higher than that of lanthanum vacancy. Therefore, the diffusion of lanthanum vacancy determined the change of the electrical conductivity in Fig. 1. 4.2. Diffusion of hole in LaCrO3 Fig. 5. Temperature dependence of the effective diffusion ˜ ) LaCrO3 with 5 vol.% Cr2O3 addition. coefficient in (D
The diffusion coefficient of hole Dh is calculated from the electrical conductivity of hole rh and hole
T. Akashi et al. / Solid State Ionics 164 (2003) 177–183
181
concentration [h .] using the Nernst – Einstein’s equation: kT Dh ¼ rh : ð5Þ ½h e2
S
The electrical conductivity can be expressed as
S
rh ¼ elh ½h
ð6Þ
where e is the elementary electric charge and lh is the mobility of hole in LaCrO3. The mobility of the hole in LaCrO3 can be estimated from the electrical conductivity of La1 xAxCrO3 (A = Sr, Ca) as a function of the composition, x. The concentration of hole in La1 xAxCrO3 is fixed by [h.]x=[ALa V ]x = xNA/Vm(x), because the electrical conductivity was measured at higher temperatures. The electrical conductivity of La1 xSrxCrO3 becomes rh ðxÞ ¼ elh ðxÞ
xNA Vm ðxÞ
Hall
Hall
ð7Þ
Here, NA is the Avogadro’s number, lh(x) is the mobility of hole and Vm(x) is the volume of one mole of La1 xAxCrO3. The mobility of hole in LaCrO3 is determined by the slope of the plot of x and rh(x) at x = 0. Fig. 7 shows the reported electrical conductivity of La1 xAxCrO3 (A = Sr, Ca) as a
Fig. 8. Mobility of hole in LaCrO3 calculated from the electrical conductivity.
function of the composition x at 1273 K. The mobility of hole was calculated from the slope at x = 0 and plotted in Fig. 8. This figure also shows the Hall mobility for LaCrO3 and that for La0.95Sr0.05CrO3 determined by Bansel et al. [24] The extrapolated values of the mobility was in agreement with the Hall mobility for LaCrO3 and smaller than the Hall mobility for La0.95Sr0.05CrO3. Substituting r in Fig. 3 and lh in Fig. 8 into Eq. (6), one can obtain the hole concentration. Fig. 9 shows the hole concentration as a function of temperature. This figure also shows the concentration of lanthanum vacancy based on electroneutrality 3[VLaUV]=[h.]. Because the concentration of hole were expressed as DHð4Þ j 1=8 3=16 ½h ¼ AaCr2 O3 PO2 exp ð8Þ 8RT
S
Fig. 7. Compositional dependence of the electrical conductivity of La1 xAxCrO3 at 1273 K.
the standard enthalpy change of the defect reaction (4), DH(4)j was determined to be 360 k mol 1 from the slope in Fig. 9. Here, A is a constant independent on temperature, including the standard entropy change of the defect reaction (4), the activity coefficient of hole and the activity coefficient of lanthanum vacancy. As shown in Fig. 3, the apparent activation energy for the electrical conductivity of LaCrO3 in this study
182
T. Akashi et al. / Solid State Ionics 164 (2003) 177–183
efficient. Yasuda and Hishinuma [25] suggested the following relation of chemical diffusion coefficient ˜ ) and diffusion coefficient of oxygen vacancy (DV ) (D o for transport of the oxygen and hole transport. SS ˜ ¼ DVo 1 þ 4½Vo
D ð9Þ ½h
S
For the transport of lanthanum vacancy and hole in this experiment, Eq. (9) can be arranged as follows. 9½VLa j
˜ D ¼ DVLa 1 þ ð10Þ ½h
S
The diffusion coefficient of lanthanum vacancy was obtained from Eq. (10) and the electroneutrality of j ]=[h.]. 3[VLa ˜ DVLa ¼ D=4 ð11Þ
Fig. 9. Concentration of lanthanum vacancy and hole in LaCrO3 as a function of temperature.
was higher than the reported values. The higher activation energy can be explained by the enthalpy change of the defect reaction (4). Because the activity of Cr2O3 was fixed in this study, the concentration of hole increased with increasing temperature by the defect reaction (4). On the other hand, the hole concentrations in the reported LaCrO3s were probably fixed by impurities. If the temperature dependence of the hole concentrations in reported LaCrO3s are small enough, the difference of the apparent activation energy is corresponding to DH(4)j/8. As shown in Fig. 3, the value of DH(4)j/8, 45 k mol 1, was in good agreement with the difference between the apparent activation energy in this study and the reported values, 40 –46 kJ mol 1. Substituting r in Fig. 3 and [h.] in Fig. 9 in Eq. (5), one can determine the diffusion coefficient of hole. The diffusion coefficients in LaCrO3 are summarized in Fig. 10. The diffusion of holes is the fastest, diffusion of oxide ions is slower, and cation diffusion is the slowest in LaCrO3.
The diffusion coefficient of La3 + ion is calculated from that of lanthanum vacancy by DLa3þ ¼ ðcVLa =cLa3þ ÞDVLa :
ð12Þ
Fig. 10 shows the diffusion coefficient of lanthanum vacancy and that of La3 + ion calculate from Eqs.
4.3. Diffusion of lanthanum ion in LaCrO3 The diffusion coefficient obtained by the transient conductivity measurement are chemical diffusion co-
Fig. 10. Diffusion coefficients of components in LaCrO3 as a function of temperature. LCC: (La,Ca)CrO3, LSC: (La, Sr)CrO3.
T. Akashi et al. / Solid State Ionics 164 (2003) 177–183
(11) and (12). The calculated La3 + diffusion coefficient and activation energy were in agreement with the La3 + diffusion coefficient determined by solid state reaction [20].
5. Conclusion Effective diffusion coefficient in LaCrO3 was determined by an transient electrical conductivity measurement. The diffusion of lanthanum vacancy determined the change of the electrical conductivity. The calculated diffusion coefficient of La3 + ion and the activation energy was in agreement with the reported value. The diffusion coefficient of hole in LaCrO3 was also determined by electrical conductivity measurement. The diffusion coefficients of hole is much higher than those of other components in LaCrO3. These diffusion coefficients are useful in understanding the transport behavior in LaCrO3 at elevated temperatures.
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