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Numerical analysis of response time for resistive oxygen gas sensors
Sensors and Actuators B 87 (2002) 99–104
Numerical analysis of response time for resistive oxygen gas sensors Noriya Izu*, Woosuck Shin, Norimitsu Murayama Synergy Materials Research Center, National Institute of Advanced Industrial Science and Technology (AIST), 2266-98 Anagahora, Shimo-Shidami, Moriyama-ku, Nagoya 463-8560, Japan Received 14 March 2002; received in revised form 27 May 2002; accepted 29 May 2002
Abstract The response time (t90) for resistive-type oxygen gas sensors based on thick films formed with cerium oxide (CeO2d) powder can be calculated as a function of the diffusion coefficient (DV), surface reaction coefficient (kV) and particle size (R). In the case of large particle size the kinetics of the sensors were controlled by diffusion, while in the case of small ones the kinetics were controlled by surface reaction. For medium particle sizes, sensor kinetics were controlled both by diffusion and by surface reaction. It could be confirmed that the response time was directly proportional to R/kV and R2/DV when surface reaction and diffusion acted as rate-limiting step, respectively. The degree of surface reaction was the rate-limiting step for oxygen gas sensors, based on cerium oxide thick film with a particle size not exceeding 1 mm. It is necessary to increase kV and/or to decrease R in order to shorten the response time of such sensors using cerium oxide powder. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Oxygen gas sensor; Rate-limiting step; Diffusion; Surface reaction; Ceria
1. Introduction Oxygen gas sensors have come into wide use as exhaust gas sensors for gasoline-powered automobiles [1]. In addition, it is expected that they will be applied to motorcycles and diesel powered vehicles. Therefore, the market for oxygen gas sensors will continue to grow in future. Recently, resistive oxygen gas sensors are again attracting attention for such new applications because they are compact and have a simpler structure compared to conventional oxygen gas sensors using concentration cells comprising an oxygenion-conductor [2,3]. The disadvantage of the resistive sensors is the long response time to changes in oxygen concentration. In order to solve this problem, considerable research was carried out and various studies on the kinetics of resistive oxygen gas sensors have been reported [2,4–6]. Cerium oxide (CeO2d) is one material that can be used for fast-response sensors because of its high diffusion coefficient for oxygen. For example, Beie and Gno¨rich [2] reported that responsiveness of oxygen gas sensors made of cerium oxide improved when the particle size for the
* Corresponding author. Tel.: þ81-52-736-7108; fax: þ81-52-736-7244. E-mail address: [email protected] (N. Izu).
sensors was reduced from 7 to 2–3 mm. In their report, they concluded that sensor kinetics were controlled by surface reaction. They assumed that response time of cerium oxide sensors was approximately 10 ms when the surface reaction rate is higher the oxygen vacancy diffusion rate. Scho¨nauer [5] reported that when the particles were large particle the kinetics of densely sintered SrTiO3 ceramics (200–1000 mm thick) were controlled by diffusion and the response time was proportional to R2/DV, where R is the particle size and DV the diffusion coefficient for oxygen vacancy. He also reported that in cases where particle size was 1 mm or less the sensor kinetics were controlled by surface reaction and response time was proportional to R/kV, where kV is surface reaction coefficient for oxygen vacancy. As shown in the above description, the relationship between the particle size and the rate-limiting step of response has been clarified only qualitatively. However, the critical particle size where the rate-limiting step changes depends on the different diffusion coefficient and the surface reaction coefficient. Therefore, it is very important to understand how response time of resistive sensors and rate-limiting steps relate to diffusion coefficient, surface reaction coefficient and particle size quantitatively and systematically. In this paper, the response time for resistive oxygen sensors, based on cerium oxide (CeO2) is calculated as a
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function of the diffusion coefficient, surface reaction coefficient and particle size as well as critical particle size to clarify how the kinetics of sensors change with decreasing particle size. Then the calculation is compared with the experimental results. Further, how to shorten the response time of resistive oxygen sensors, based on cerium oxide powder, will also be discussed.
coefficient for the oxygen vacancy (kV), particle size (R), and time (t). In this paper, the particle size (R) was defined as a particle radius. Supposing that equilibrium values of [VO ] for the particle are C0 and C1 for oxygen partial pressure of P0 and P1, respectively, the relationship between C0 (C1) and P0 (P1) can be expressed as follows: C0 ¼ K 0 P0 1=6
(7)
2. Calculation
C1 ¼ K 0 P1 1=6
(8)
First, the operation principle of resistive oxygen gas sensors, based on cerium oxide powder, is explained. In the Kro¨ ger–Vink notation [7], the defect reaction between cerium oxide and gaseous oxygen is
where K0 is constant. Assuming that oxygen partial pressure suddenly changes from P0 to P1 at t ¼ 0, where t is the time. The flow of oxygen vacancy is caused so that [VO ] in the particle corresponds to the equilibrium in oxygen vacancy C1. A basic differential equation is expressed as follows [8]: 2 @½VO @ ½VO 2 @½VO ¼ DV þ (9) @t @r 2 r @r
X 0 2CeX þ 12 O2 Ce þ OO ¼ 2CeCe þ VO
(1)
X where CeX Ce and OO represent the regular lattice ions while 0 CeCe and VO are the charged defects. Ce0Ce shows the electron trapped by CeX shows doubly ionized Ce and VO oxygen vacancy. Eq. (1) can be expressed as 2
K ¼ ½Ce0Ce ½VO P1=2
(2) [Ce0Ce ]
is where K is the equilibrium constant of the Eq. (1), the concentration of Ce0Ce , [VO ] is the concentration of VO and P is oxygen partial pressure. At this time, there is a state of electrical neutrality locally. ½Ce0Ce ¼ 2½VO
(3)
Therefore, the next relation is derived from Eqs. (2) and (3). ½Ce0Ce ¼ 2½VO / P1=6
(4)
where DV is diffusion coefficient for the oxygen vacancy. Particles are assumed to be spheres. The variable r is the position of the particle and the center of the particle is defined as r ¼ 0. When the surface reaction coefficient of oxygen vacancy on the particle surface is defined as kV, the following boundary condition is obtained: @½VO (10) kV ð½VO C1 Þr¼R ¼ DV @r r¼R Because the concentration of oxygen vacancy at the center of particle is limited, the following boundary condition is obtained: ð½VO Þr¼0 ¼ limited value
The total electrical conductivity s is the following relation:
An initial condition is written as follows:
s ¼ ½Ce0Ce eme þ 2½VO emo
ð½VO Þt¼0 ¼ C0
(5)
where me and mo are the mobility of electron and oxygen vacancy, respectively. When the variation of me and mo are negligible, the following equation is derived from Eqs. (4) and (5): s ¼ 2½VO eðme þ mo Þ / P1=6
(6)
Therefore, a value of P can be determined from the value of s. To calculate the response time, since a sample is porous and consists of particles connected to each other, only one particle need be focused on to calculate the response time. As seen from Eq. (6), the time to stabilize the output of sensor (s) after a sudden change in oxygen partial pressure is the same as the time to reach in the new equilibrium condition of the concentration of oxygen vacancy in cerium oxide particle. Therefore, the concentration of oxygen vacancy ([VO ]) was calculated as a function of diffusion coefficient for the oxygen vacancy (DV) surface reaction
(11)
(12)
The solution of differential Eq. (9) by using Eqs. (10)– (12) is " 1 X ½VO C1 fn2m þ ðhR 1Þ2 g1=2 ¼ 2hR C0 C1 n2m þ hRðhR 1Þ m¼1 # sin½nm ðr=RÞ 2 DV (13) exp nm 2 t nm ðr=RÞ R where h is kV/DV and vm is an eigenvalue with which tan n ¼ m=ð1 hRÞ is filled. As mentioned above, sensor output, which is s, is controlled by [VO ]. Therefore, response time of sensor (t90) was defined as the time when ð½VO C1 Þ=ðC0 C1 Þ is 0.1 at r ¼ 0. Since the numerical value for a term with the m value of 5 or more in the square brackets in Eq. (13) is negligibly small, we calculated the response time from a summation of terms with m value of 1–4 in Eq. (13).
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3. Experimental Cerium oxide powder was prepared by mist pyrolysis of Ce aqueous solution 1.0 mol/dm3 at 973 K. The paste, made by mixing the cerium oxide powders with organic binder, was screen-printed on an Al2O3 substrate. The obtained thick films were calcined at 773 K for 5 h in air, and fired at 1573 K for 2 h in air. Finally, the platinum electrodes were attached to both ends of the thick film. The microstructure of thick films was characterized by X-ray diffraction (XRD) and scanning electron microscopy (SEM). The electrical conductivity of thick films as an output of sensor was measured in the range of 1073– 1273 K and in oxygen partial pressure from 1:0 103 to 1:0 105 Pa.
4. Results 4.1. Calculation Fig. 1 shows the calculation results for Eq. (13) in a case that the value of kV was fixed at 1:0 1010 m/s and the value of DV was a variable. Fig. 2 shows results in a case that the value of DV was fixed at 1:0 1012 m2/s and the value of kV was variable. The diagrams of t90 versus R are shown in Figs. 1(a) and 2(a), and the diagrams of n in t90 / Rn versus R are shown in Figs. 1(b) and 2(b). The value of n was about 1 in the smaller particle size range (region [I]), while the value was about 2 in the larger particle size range (region
Fig. 1. Calculation results in case that the value of kV was fixed at 1:0 1010 m/s and that of DV was varied: (a) t90; (b) n in t90 / Rn .
Fig. 2. Calculation results in case that the value of DV was fixed at 1:0 1012 m2/s and that of kV was varied: (a) t90; (b) n in t90 / Rn .
[III]). Region [II] was defined as representing a particle size range of 1 < n < 2. The particle size where n was 1.01, that is, the boundary of regions [I] and [II] was defined as Rk , and the particle size where n was 1.99, that is, the boundary of regions [II] and [III] was defined as RD . The value of t90 increased as R increased. In the region [I], t90 did not change when DV was varied, but changed when kV was varied. The value of t90 was directly proportional to R/kV in this region. In the region [III], t90 did not change when kV was varied, but changed when DV was varied and the value of t90 was directly proportional to R2/DV. Therefore, it was became apparent that sensor kinetics in region [I] and [III] were controlled by surface reaction and by diffusion, respectively. In the region [II], sensor kinetics were controlled both by surface reaction and diffusion, since the value of t90 changed when DV and/or kV were varied. The diagram of n in t90 / Rn versus R is important in order to define region [II] because this region was largely undefined in the diagram of t90 versus R. The region [II] shifted to larger particle size in the diagram with increasing DV and/or decreasing kV. In any kV and DV, the region [I], where sensor kinetics is controlled by surface reaction, was located at a smaller particle size compared to the region [III], where the kinetics are controlled by diffusion. That is to say, the ratelimiting step is diffusion in the case of large R, and it is surface reaction in the case of small R in any materials. Figs. 3(a) and (b) show Rk and RD as a function of kV and DV, respectively. These figures give the critical particle size for the boundary between diffusion-limiting step and surface-reaction-limiting step. For example, when kV and DV are 1010 and 1012 m2/s respectively, Rk is approximately
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controlled by surface reaction at the area where the line of t90 was vertical, while the kinetics were controlled by diffusion in the area where the line was horizontal. It was reported that the diffusion coefficient for oxygen in CeO2 was 2:6 1014 m2/s at 1081 K [9,10] or 6:6 109 m2/s at 1244 K [11]. Generally, the following relational expression is known [12]: DO CO ¼ DV CV
(14)
where DO is the diffusion coefficient of oxygen, CO is the oxygen concentration and CV is the oxygen vacancy concentration. The CV/CO ratio was 103 or less when oxygen partial pressure was more than 105 Pa at 1073 K [13,14]. From Eq. (14), the DV/DO ratio was more than 103. Therefore, the diffusion coefficient for oxygen vacancy in CeO2 seemed to be in the range of 1011 to 105 m2/s. It was reported that the surface reaction coefficient for oxygen in CeO2 was 5:5 1010 m/s at 1081 K [9,10]. We assumed for the moment that the surface reaction coefficient for oxygen was equal to that for oxygen vacancy. By using these values of the diffusion coefficient and the surface reaction coefficient for oxygen vacancy, it became clear that the calculated value of t90 was in the range of about 103 to 104 s from Fig. 4. When kV was 5 1010 m/s and DV was 2 1011 m2/s, it became clear that Rk was 103 m from Fig. 3(a). However, an actual kV may be 1000 times larger than assumed (please refer to Section 5). Therefore, it is more certain that the kinetics of sensor, with a particle size of 1 mm or less, are dependent on the degree of surface reaction. Fig. 3. Particle size where the rate-limiting step changed as a function of kV and DV. Particle size where the rate-limiting step changed from (a) surface reaction to surface reaction and diffusion (Rk ) and (b) diffusion to surface reaction and diffusion (RD ).
103 m. In this case, the kinetics of sensors with a particle size of 103 m or less are controlled by surface reaction. The response time, t90, as a function of DV and kV for R ¼ 1 mm is shown in Fig. 4. The sensor kinetics were
Fig. 4. The response time t90 of resistive-type sensors as a function of DV and kV with R fixed at 1 mm.
4.2. Experiment results All of the XRD peaks were identified as CeO2. The SEM observation for the sensors obtained after firing at 1573 K showed there was neck growth and it revealed the threedimensional network structure of cerium oxide. An average particle size was 1 mm. Fig. 5 shows typical dynamic change
Fig. 5. Dynamic change of normalized output of sensors with an average particle size of 1 mm after a sudden change from 1:0 105 to 1:0 103 Pa at 1081 K. The variable s(t) is the electrical conductivity at the arbitrary selected time t, and s1 and s100 denotes the stable electrical conductivity in oxygen particle pressure of 1:0 103 and 1:0 105 Pa, respectively.
N. Izu et al. / Sensors and Actuators B 87 (2002) 99–104 Table 1 Response time of the sensors using cerium oxide powder having an average particle size of 1 mm Temperature (T, K)
Response time (t90, s)
1081 1130 1178 1226 1275
35 22 15 11 8
in normalized output of sensors after a sudden change of oxygen partial pressure from 1:0 105 to 1:0 103 Pa at 1081 K, where s(t) is the electrical conductivity at an arbitrary point in time t, and s1 and s100 are the stable electrical conductivity in oxygen partial pressure of 1:0 103 and 1:0 105 Pa, respectively. An output signal was very stable in state of the equilibrium state and the measurement results had reproducibility. The t90 value obtained from the experimental results was 35 s. The response times measured at various temperatures are summarized in Table 1.
5. Discussion We firstly discuss the difference between calculations and the experimental results of the response time. There are two possibilities as this reason. One possibility is that the assumption that the surface reaction coefficient of oxygen, kO, is equal to that of oxygen vacancy, kV, is false. Like the diffusion coefficient, a difference between the surface reaction coefficient of oxygen and that of oxygen vacancy may exist. If kV/kO ratio is equal to about 103 like DV/DO ratio, the t90 value of calculation is in the range of about 1–10 s and similar to the t90 value of the experiment. Another possibility is the difference in kV due to different sample shape. The sample in the present study was a porous thick film consisting of small particles, while a bulk sample was used in previous study [9,10]. The radius of curvature on the surface of cerium oxide particles used in this study was very small, while the radius of curvature on the surface of the bulk cerium oxide was very large [9,10]. The surface energy of small particles must be larger than that of bulk. The activated surface may increase the surface reaction coefficient. The t90 for the sensors with the particle radius of 100 and 1000 nm at 888 K were about 32 and 330 s [15]. As described in Section 4, the kinetics of sensor based on cerium oxide with the particles size of 1 mm or less are dependent on the degree of surface reaction and t90 is directly proportional to R. This calculated relation is consistent with the experiment result that reducing R by 1/10 reduced t90 by 1/10. In the future, it will be necessary to develop a processing method capable of making nano-particulate porous thick films in order to decrease R. The value of k will be increased
103
by the surface modification through a catalyst. In addition, it may be increased depending on the shape when R decreases.
6. Conclusion The response time of resistive-type oxygen sensors based on cerium oxide can be calculated as a function of diffusion coefficient, surface reaction coefficient and particle size. The kinetics of resistive-type oxygen gas sensors can be controlled by diffusion in the case of large particle size and by surface reaction in the case of small size. The rate-limiting step was surface reaction for oxygen gas sensors based on cerium oxide thick film with a particle size of 1 mm or less. Therefore, in order to shorten the response time of these sensors it is necessary to increase k and/or decrease R.
References [1] M. Ogita, K. Higo, Y. Nakanishi, Y. Hatanaka, Ga2O3 thin film for oxygen sensor at high temperature, Appl. Surf. Sci. 175/176 (2001) 721–725. [2] H.-J. Beie, A. Gno¨ rich, Oxygen gas sensors based on CeO2 thick and thin films, Sens. Actuators B 4 (1991) 393–399. [3] J. Sheng, N. Yoshida, J. Karasawa, T. Fukami, Platinum doped titania film oxygen sensor integrated with temperature compensating thermistor, Sens. Actuators B 41 (1997) 131–136. [4] M. Fleischer, H. Meixner, Fast gas sensors based on metal oxides which are stable at high temperatures, Sens. Actuators B 43 (1997) 1–10. [5] U. Scho¨ nauer, Response times of resistive thick-film oxygen sensors, Sens. Actuators B 4 (1991) 431–436. [6] C.H. Tragut, K. Ha¨ rdtl, Kinetic behavior of resistive oxygen sensors, Sens. Actuators B 4 (1991) 425–429. [7] F.A. Kro¨ ger, H.J. Vink, Relations between the concentration of imperfections in crystalline solids, in: F. Seitz, D. Turnbull, Solid State Physics, Vol. 3, Academic Press, New York, 1956, p. 307. [8] J. Crank, The Mathematics of Diffusion, Oxford University Press, Oxford, London, 1956. [9] M. Kamiya, E. Shimada, Y. Ikuma, M. Komatsu, H. Haneda, S. Sameshima, Y. Hirata, Oxygen self-diffusion in cerium oxide doped with Nd, J. Mater. Res. 16 (1) (2001) 179–184. [10] M. Kamiya, E. Shimada, Y. Ikuma, M. Komatsu, H. Hanada, Intrinsic and extrinsic oxygen diffusion and surface exchange reaction in cerium oxide, J. Electrochem. Soc. 147 (3) (2000) 1222–1227. [11] F. Millot, P. De Mierry, A new method for the study of chemical diffusion in oxides with application to cerium oxide CeO2x, J. Phys. Chem. Solids 46 (7) (1985) 797–801. [12] Y.-M. Chiang, D. Birnie III, W. D. Kingery, Physical Ceramics, Wiley, New York, 1997, p. 106. [13] H.L. Tuller, A.S. Nowick, Defect structure and electrical properties of non-stoichiometric CeO2 single crystals, J. Electrochem. Soc. 126 (2) (1979) 209–217. [14] R.J. Panlener, R.N. Blumenthal, J.E. Garnier, A thermodynamic study of non-stoichiometric cerium dioxide, J. Phys. Chem. Solids 36 (1975) 1213–1222. [15] N. Izu, W. Shin, N. Murayama, Resistive oxygen gas sensors based on fine powders of cerium oxide prepared by mist pyrolysis, in: Proceeding of Fifth East Asian Conference on Chemical Sensors, Huis Ten Bosch, Sasebo-shi, Nagasaki, Japan, 4–7 December 2001, p. 151.
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Biographies Noriya Izu received his BS, MS and Dr. Eng. degrees in materials science and processing from Osaka University in 1995, 1997, and 2001, respectively. He is currently a research scientist in the environmental sensors team at Synergy Materials Research Center, National Institute of Advanced Industrial Science and Technology (AIST) in Nagoya, Japan. His research interests include environmental sensors and defect chemistry. Woosuck Shin studied material science and engineering and finished his master’s course in 1994 at KAIST in Korea. After receiving doctorate 1998 in applied chemistry from the Nagoya University in Japan, he has been
employed at AIST, in Nagoya, Japan. His research activity includes optical properties of nano-structured materials, electronic structures of wide-bandgap materials, and oxide thermoelectrics. His scientific interests are particularly in the gas sensor technology with microsystems. Norimitsu Murayama is a leader in the environmental sensors team at Synergy Materials Research Center, AIST. He received his BS, and MS degrees in electronics from Kyoto University in 1982, and 1984, respectively, and his Dr. Eng. degree in materials science and engineering from Tokyo Institute of Technology in 1992. His research interests focus on novel processing for conductive ceramics including oxide superconductors, oxide thermoelectrics, and gas sensors.
Numerical analysis of response time for resistive oxygen gas sensors Noriya Izu*, Woosuck Shin, Norimitsu Murayama Synergy Materials Research Center, National Institute of Advanced Industrial Science and Technology (AIST), 2266-98 Anagahora, Shimo-Shidami, Moriyama-ku, Nagoya 463-8560, Japan Received 14 March 2002; received in revised form 27 May 2002; accepted 29 May 2002
Abstract The response time (t90) for resistive-type oxygen gas sensors based on thick films formed with cerium oxide (CeO2d) powder can be calculated as a function of the diffusion coefficient (DV), surface reaction coefficient (kV) and particle size (R). In the case of large particle size the kinetics of the sensors were controlled by diffusion, while in the case of small ones the kinetics were controlled by surface reaction. For medium particle sizes, sensor kinetics were controlled both by diffusion and by surface reaction. It could be confirmed that the response time was directly proportional to R/kV and R2/DV when surface reaction and diffusion acted as rate-limiting step, respectively. The degree of surface reaction was the rate-limiting step for oxygen gas sensors, based on cerium oxide thick film with a particle size not exceeding 1 mm. It is necessary to increase kV and/or to decrease R in order to shorten the response time of such sensors using cerium oxide powder. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Oxygen gas sensor; Rate-limiting step; Diffusion; Surface reaction; Ceria
1. Introduction Oxygen gas sensors have come into wide use as exhaust gas sensors for gasoline-powered automobiles [1]. In addition, it is expected that they will be applied to motorcycles and diesel powered vehicles. Therefore, the market for oxygen gas sensors will continue to grow in future. Recently, resistive oxygen gas sensors are again attracting attention for such new applications because they are compact and have a simpler structure compared to conventional oxygen gas sensors using concentration cells comprising an oxygenion-conductor [2,3]. The disadvantage of the resistive sensors is the long response time to changes in oxygen concentration. In order to solve this problem, considerable research was carried out and various studies on the kinetics of resistive oxygen gas sensors have been reported [2,4–6]. Cerium oxide (CeO2d) is one material that can be used for fast-response sensors because of its high diffusion coefficient for oxygen. For example, Beie and Gno¨rich [2] reported that responsiveness of oxygen gas sensors made of cerium oxide improved when the particle size for the
* Corresponding author. Tel.: þ81-52-736-7108; fax: þ81-52-736-7244. E-mail address: [email protected] (N. Izu).
sensors was reduced from 7 to 2–3 mm. In their report, they concluded that sensor kinetics were controlled by surface reaction. They assumed that response time of cerium oxide sensors was approximately 10 ms when the surface reaction rate is higher the oxygen vacancy diffusion rate. Scho¨nauer [5] reported that when the particles were large particle the kinetics of densely sintered SrTiO3 ceramics (200–1000 mm thick) were controlled by diffusion and the response time was proportional to R2/DV, where R is the particle size and DV the diffusion coefficient for oxygen vacancy. He also reported that in cases where particle size was 1 mm or less the sensor kinetics were controlled by surface reaction and response time was proportional to R/kV, where kV is surface reaction coefficient for oxygen vacancy. As shown in the above description, the relationship between the particle size and the rate-limiting step of response has been clarified only qualitatively. However, the critical particle size where the rate-limiting step changes depends on the different diffusion coefficient and the surface reaction coefficient. Therefore, it is very important to understand how response time of resistive sensors and rate-limiting steps relate to diffusion coefficient, surface reaction coefficient and particle size quantitatively and systematically. In this paper, the response time for resistive oxygen sensors, based on cerium oxide (CeO2) is calculated as a
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function of the diffusion coefficient, surface reaction coefficient and particle size as well as critical particle size to clarify how the kinetics of sensors change with decreasing particle size. Then the calculation is compared with the experimental results. Further, how to shorten the response time of resistive oxygen sensors, based on cerium oxide powder, will also be discussed.
coefficient for the oxygen vacancy (kV), particle size (R), and time (t). In this paper, the particle size (R) was defined as a particle radius. Supposing that equilibrium values of [VO ] for the particle are C0 and C1 for oxygen partial pressure of P0 and P1, respectively, the relationship between C0 (C1) and P0 (P1) can be expressed as follows: C0 ¼ K 0 P0 1=6
(7)
2. Calculation
C1 ¼ K 0 P1 1=6
(8)
First, the operation principle of resistive oxygen gas sensors, based on cerium oxide powder, is explained. In the Kro¨ ger–Vink notation [7], the defect reaction between cerium oxide and gaseous oxygen is
where K0 is constant. Assuming that oxygen partial pressure suddenly changes from P0 to P1 at t ¼ 0, where t is the time. The flow of oxygen vacancy is caused so that [VO ] in the particle corresponds to the equilibrium in oxygen vacancy C1. A basic differential equation is expressed as follows [8]: 2 @½VO @ ½VO 2 @½VO ¼ DV þ (9) @t @r 2 r @r
X 0 2CeX þ 12 O2 Ce þ OO ¼ 2CeCe þ VO
(1)
X where CeX Ce and OO represent the regular lattice ions while 0 CeCe and VO are the charged defects. Ce0Ce shows the electron trapped by CeX shows doubly ionized Ce and VO oxygen vacancy. Eq. (1) can be expressed as 2
K ¼ ½Ce0Ce ½VO P1=2
(2) [Ce0Ce ]
is where K is the equilibrium constant of the Eq. (1), the concentration of Ce0Ce , [VO ] is the concentration of VO and P is oxygen partial pressure. At this time, there is a state of electrical neutrality locally. ½Ce0Ce ¼ 2½VO
(3)
Therefore, the next relation is derived from Eqs. (2) and (3). ½Ce0Ce ¼ 2½VO / P1=6
(4)
where DV is diffusion coefficient for the oxygen vacancy. Particles are assumed to be spheres. The variable r is the position of the particle and the center of the particle is defined as r ¼ 0. When the surface reaction coefficient of oxygen vacancy on the particle surface is defined as kV, the following boundary condition is obtained: @½VO (10) kV ð½VO C1 Þr¼R ¼ DV @r r¼R Because the concentration of oxygen vacancy at the center of particle is limited, the following boundary condition is obtained: ð½VO Þr¼0 ¼ limited value
The total electrical conductivity s is the following relation:
An initial condition is written as follows:
s ¼ ½Ce0Ce eme þ 2½VO emo
ð½VO Þt¼0 ¼ C0
(5)
where me and mo are the mobility of electron and oxygen vacancy, respectively. When the variation of me and mo are negligible, the following equation is derived from Eqs. (4) and (5): s ¼ 2½VO eðme þ mo Þ / P1=6
(6)
Therefore, a value of P can be determined from the value of s. To calculate the response time, since a sample is porous and consists of particles connected to each other, only one particle need be focused on to calculate the response time. As seen from Eq. (6), the time to stabilize the output of sensor (s) after a sudden change in oxygen partial pressure is the same as the time to reach in the new equilibrium condition of the concentration of oxygen vacancy in cerium oxide particle. Therefore, the concentration of oxygen vacancy ([VO ]) was calculated as a function of diffusion coefficient for the oxygen vacancy (DV) surface reaction
(11)
(12)
The solution of differential Eq. (9) by using Eqs. (10)– (12) is " 1 X ½VO C1 fn2m þ ðhR 1Þ2 g1=2 ¼ 2hR C0 C1 n2m þ hRðhR 1Þ m¼1 # sin½nm ðr=RÞ 2 DV (13) exp nm 2 t nm ðr=RÞ R where h is kV/DV and vm is an eigenvalue with which tan n ¼ m=ð1 hRÞ is filled. As mentioned above, sensor output, which is s, is controlled by [VO ]. Therefore, response time of sensor (t90) was defined as the time when ð½VO C1 Þ=ðC0 C1 Þ is 0.1 at r ¼ 0. Since the numerical value for a term with the m value of 5 or more in the square brackets in Eq. (13) is negligibly small, we calculated the response time from a summation of terms with m value of 1–4 in Eq. (13).
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3. Experimental Cerium oxide powder was prepared by mist pyrolysis of Ce aqueous solution 1.0 mol/dm3 at 973 K. The paste, made by mixing the cerium oxide powders with organic binder, was screen-printed on an Al2O3 substrate. The obtained thick films were calcined at 773 K for 5 h in air, and fired at 1573 K for 2 h in air. Finally, the platinum electrodes were attached to both ends of the thick film. The microstructure of thick films was characterized by X-ray diffraction (XRD) and scanning electron microscopy (SEM). The electrical conductivity of thick films as an output of sensor was measured in the range of 1073– 1273 K and in oxygen partial pressure from 1:0 103 to 1:0 105 Pa.
4. Results 4.1. Calculation Fig. 1 shows the calculation results for Eq. (13) in a case that the value of kV was fixed at 1:0 1010 m/s and the value of DV was a variable. Fig. 2 shows results in a case that the value of DV was fixed at 1:0 1012 m2/s and the value of kV was variable. The diagrams of t90 versus R are shown in Figs. 1(a) and 2(a), and the diagrams of n in t90 / Rn versus R are shown in Figs. 1(b) and 2(b). The value of n was about 1 in the smaller particle size range (region [I]), while the value was about 2 in the larger particle size range (region
Fig. 1. Calculation results in case that the value of kV was fixed at 1:0 1010 m/s and that of DV was varied: (a) t90; (b) n in t90 / Rn .
Fig. 2. Calculation results in case that the value of DV was fixed at 1:0 1012 m2/s and that of kV was varied: (a) t90; (b) n in t90 / Rn .
[III]). Region [II] was defined as representing a particle size range of 1 < n < 2. The particle size where n was 1.01, that is, the boundary of regions [I] and [II] was defined as Rk , and the particle size where n was 1.99, that is, the boundary of regions [II] and [III] was defined as RD . The value of t90 increased as R increased. In the region [I], t90 did not change when DV was varied, but changed when kV was varied. The value of t90 was directly proportional to R/kV in this region. In the region [III], t90 did not change when kV was varied, but changed when DV was varied and the value of t90 was directly proportional to R2/DV. Therefore, it was became apparent that sensor kinetics in region [I] and [III] were controlled by surface reaction and by diffusion, respectively. In the region [II], sensor kinetics were controlled both by surface reaction and diffusion, since the value of t90 changed when DV and/or kV were varied. The diagram of n in t90 / Rn versus R is important in order to define region [II] because this region was largely undefined in the diagram of t90 versus R. The region [II] shifted to larger particle size in the diagram with increasing DV and/or decreasing kV. In any kV and DV, the region [I], where sensor kinetics is controlled by surface reaction, was located at a smaller particle size compared to the region [III], where the kinetics are controlled by diffusion. That is to say, the ratelimiting step is diffusion in the case of large R, and it is surface reaction in the case of small R in any materials. Figs. 3(a) and (b) show Rk and RD as a function of kV and DV, respectively. These figures give the critical particle size for the boundary between diffusion-limiting step and surface-reaction-limiting step. For example, when kV and DV are 1010 and 1012 m2/s respectively, Rk is approximately
102
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controlled by surface reaction at the area where the line of t90 was vertical, while the kinetics were controlled by diffusion in the area where the line was horizontal. It was reported that the diffusion coefficient for oxygen in CeO2 was 2:6 1014 m2/s at 1081 K [9,10] or 6:6 109 m2/s at 1244 K [11]. Generally, the following relational expression is known [12]: DO CO ¼ DV CV
(14)
where DO is the diffusion coefficient of oxygen, CO is the oxygen concentration and CV is the oxygen vacancy concentration. The CV/CO ratio was 103 or less when oxygen partial pressure was more than 105 Pa at 1073 K [13,14]. From Eq. (14), the DV/DO ratio was more than 103. Therefore, the diffusion coefficient for oxygen vacancy in CeO2 seemed to be in the range of 1011 to 105 m2/s. It was reported that the surface reaction coefficient for oxygen in CeO2 was 5:5 1010 m/s at 1081 K [9,10]. We assumed for the moment that the surface reaction coefficient for oxygen was equal to that for oxygen vacancy. By using these values of the diffusion coefficient and the surface reaction coefficient for oxygen vacancy, it became clear that the calculated value of t90 was in the range of about 103 to 104 s from Fig. 4. When kV was 5 1010 m/s and DV was 2 1011 m2/s, it became clear that Rk was 103 m from Fig. 3(a). However, an actual kV may be 1000 times larger than assumed (please refer to Section 5). Therefore, it is more certain that the kinetics of sensor, with a particle size of 1 mm or less, are dependent on the degree of surface reaction. Fig. 3. Particle size where the rate-limiting step changed as a function of kV and DV. Particle size where the rate-limiting step changed from (a) surface reaction to surface reaction and diffusion (Rk ) and (b) diffusion to surface reaction and diffusion (RD ).
103 m. In this case, the kinetics of sensors with a particle size of 103 m or less are controlled by surface reaction. The response time, t90, as a function of DV and kV for R ¼ 1 mm is shown in Fig. 4. The sensor kinetics were
Fig. 4. The response time t90 of resistive-type sensors as a function of DV and kV with R fixed at 1 mm.
4.2. Experiment results All of the XRD peaks were identified as CeO2. The SEM observation for the sensors obtained after firing at 1573 K showed there was neck growth and it revealed the threedimensional network structure of cerium oxide. An average particle size was 1 mm. Fig. 5 shows typical dynamic change
Fig. 5. Dynamic change of normalized output of sensors with an average particle size of 1 mm after a sudden change from 1:0 105 to 1:0 103 Pa at 1081 K. The variable s(t) is the electrical conductivity at the arbitrary selected time t, and s1 and s100 denotes the stable electrical conductivity in oxygen particle pressure of 1:0 103 and 1:0 105 Pa, respectively.
N. Izu et al. / Sensors and Actuators B 87 (2002) 99–104 Table 1 Response time of the sensors using cerium oxide powder having an average particle size of 1 mm Temperature (T, K)
Response time (t90, s)
1081 1130 1178 1226 1275
35 22 15 11 8
in normalized output of sensors after a sudden change of oxygen partial pressure from 1:0 105 to 1:0 103 Pa at 1081 K, where s(t) is the electrical conductivity at an arbitrary point in time t, and s1 and s100 are the stable electrical conductivity in oxygen partial pressure of 1:0 103 and 1:0 105 Pa, respectively. An output signal was very stable in state of the equilibrium state and the measurement results had reproducibility. The t90 value obtained from the experimental results was 35 s. The response times measured at various temperatures are summarized in Table 1.
5. Discussion We firstly discuss the difference between calculations and the experimental results of the response time. There are two possibilities as this reason. One possibility is that the assumption that the surface reaction coefficient of oxygen, kO, is equal to that of oxygen vacancy, kV, is false. Like the diffusion coefficient, a difference between the surface reaction coefficient of oxygen and that of oxygen vacancy may exist. If kV/kO ratio is equal to about 103 like DV/DO ratio, the t90 value of calculation is in the range of about 1–10 s and similar to the t90 value of the experiment. Another possibility is the difference in kV due to different sample shape. The sample in the present study was a porous thick film consisting of small particles, while a bulk sample was used in previous study [9,10]. The radius of curvature on the surface of cerium oxide particles used in this study was very small, while the radius of curvature on the surface of the bulk cerium oxide was very large [9,10]. The surface energy of small particles must be larger than that of bulk. The activated surface may increase the surface reaction coefficient. The t90 for the sensors with the particle radius of 100 and 1000 nm at 888 K were about 32 and 330 s [15]. As described in Section 4, the kinetics of sensor based on cerium oxide with the particles size of 1 mm or less are dependent on the degree of surface reaction and t90 is directly proportional to R. This calculated relation is consistent with the experiment result that reducing R by 1/10 reduced t90 by 1/10. In the future, it will be necessary to develop a processing method capable of making nano-particulate porous thick films in order to decrease R. The value of k will be increased
103
by the surface modification through a catalyst. In addition, it may be increased depending on the shape when R decreases.
6. Conclusion The response time of resistive-type oxygen sensors based on cerium oxide can be calculated as a function of diffusion coefficient, surface reaction coefficient and particle size. The kinetics of resistive-type oxygen gas sensors can be controlled by diffusion in the case of large particle size and by surface reaction in the case of small size. The rate-limiting step was surface reaction for oxygen gas sensors based on cerium oxide thick film with a particle size of 1 mm or less. Therefore, in order to shorten the response time of these sensors it is necessary to increase k and/or decrease R.
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Biographies Noriya Izu received his BS, MS and Dr. Eng. degrees in materials science and processing from Osaka University in 1995, 1997, and 2001, respectively. He is currently a research scientist in the environmental sensors team at Synergy Materials Research Center, National Institute of Advanced Industrial Science and Technology (AIST) in Nagoya, Japan. His research interests include environmental sensors and defect chemistry. Woosuck Shin studied material science and engineering and finished his master’s course in 1994 at KAIST in Korea. After receiving doctorate 1998 in applied chemistry from the Nagoya University in Japan, he has been
employed at AIST, in Nagoya, Japan. His research activity includes optical properties of nano-structured materials, electronic structures of wide-bandgap materials, and oxide thermoelectrics. His scientific interests are particularly in the gas sensor technology with microsystems. Norimitsu Murayama is a leader in the environmental sensors team at Synergy Materials Research Center, AIST. He received his BS, and MS degrees in electronics from Kyoto University in 1982, and 1984, respectively, and his Dr. Eng. degree in materials science and engineering from Tokyo Institute of Technology in 1992. His research interests focus on novel processing for conductive ceramics including oxide superconductors, oxide thermoelectrics, and gas sensors.