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Voltage relaxation measurements of the electron and hole mobilities in yttria-doped zirconia
VOLTAGE RELAXATION MEASUREMENTS ELECTRON AND HOLE MOBILITIES YTTRIA-DOPED ZIRCONIA W. Stanford
University,
OF THE IN
WEPPNER
Stanford,
California 94305, U.S.A.
(Received 1 September 1976) Abstract-The voltage relaxation of galvanic cells with zirconia electrolytes polarized between an inert silver electrode and either a FY/air or a Fe/Fe0 electrode has been analysed to obtain the mobilities of both electronic minority charge carriers. At 900°C the mobility of the electrons is 2.4 x lO_’ cm’/Vs, that of the holes is 2 orders of magnitude lower, 1.6 x 10m4cmz/Vs.The activation enthalpy is 0.55 eV for the electrons and 1.4eV for the holes. From conductivity data, the concentrations of electrons and holes at 900°C and 1 atm oxygen partial pressure are calculated to be 3 x 10” and 6 x 10” cmm3, respectively. With the use of platinum inert electrodes the formation of intermetallic pt-Zr compounds appears at an oxygen partial pressure of 2.3 x lo-*’ atm at 900°C. From this the Gibbs formation energy of yttria-doped zirconia is calculated to be -9.1 eV at 900°C. 1. INTRODUCTION
At one side zirconia doped with 10 mole % yttria
Zirconia doped with several mole yO of yttria or some other lower valent oxides is of considerable interest as a solid oxygen ion conductor in many tecbnological and scientific applications. Its use is limited in some cases, however, because electronic species either electrons or holes can make an appreciable contribution to the total conductivity. In several investigations[l-181 the partial electronic conductivities of stabilized zirconia have been studied. P-type conduction occurs at high oxygen partial pressures, ey POz 2 IO-” atm at 9OO”C, whereas, n-type conduction occurs at low oxygen partial pressures. There is little information, however, about the relative contributions of the concentrations of both electronic charge carriers and their mobilities to the electronic portion of the total conductivity[5,12,14, 151. The work reported here involves measurements of the voltage relaxation in galvanic cells[i%24,151 in order to obtain diffusion coefficients (or mobilities) for the electrons and holes. From this information and the data for the conductivities of the electronic species, the concentrations of the electrons and holes can be calculated. This is the inverse of the charge transfer technique[23, 25-281 which allows the determination of the concentrations of the eIectronic species and the calculation of the mobilities from knowledge of the conductivity values.
is exposed to a defined reference oxygen partial pres-
sure, either air or a mixture of Fe and “Fee”, and an inert atmosphere, either nitrogen or argon, of very small volume, is present at the other side. Electrical contacts were made using thin sputtered silver layers. If a positive voltage is applied to the galvanic cell, no ionic current exists under steady state conditions, since the inert electrode cannot supply oxygen ions. Therefore, the only current that will pass through the cell will be that due to the transport of electronic species. Thus, by measuring the
current
the
electronic
conductivities
have
been
evaluated[S, 15-17,24,29-311. The chemical potential of the oxygen ions can be regarded as constant throughout the solid because of the high degree of disorder due to the presence of about 5% oxygen vacancies[32,33]. Any changes in the oxygen partial pressure have only a negligible influence upon the concentration of these defects. Since under steady state conditions no oxygen is transported across the cell, the gradient of the electrochemical potentia1 of the oxygen ions must vanish. Therefore, since the electrochemical potential q (per particle) of any species is equal to the sum of its chemical potential p (per particle) and the electrostatic potential c$ multiplied by the charge number 2 and the elementary charge q,
2. THEORY
we can conclude that there is no internal electric field within the electrolyte under steady state conditions.
The following galvanic cell has been employed:
EA
I
I
This leads to the conclusion that the transport of elecand holes can only be related to gradients in their concentrations and obeys Fick’s first law, in accordance with Wagner’s interpretation of the pofarization current[30,31]. The concentration gradient of the prevailing conductive electronic species is linear and the slope is inversely proportional to the diffusion coefficient if we can assume that the mobility is independent of
trons I
Pt, Air
Inert Ag Electrolyte: Zr9Y2021 Atmosphere’
Fe +%eO”
, x 0
L 721
W.
122
WEPPNER
the oxygen partial pressure. In the case of air reference electrodes, a junction between p-type and n-type regions of the electrolyte appears at sufficiently high polarization voltages. If, after steady state conditions are established, the externally applied voltage is switched off, the concentration gradients will decay until the stoichiometry, as determined by the reference electrode, is uniformly established throughout the entire sample. The transport of electrons moving toward the right and of holes moving to the left is electrically compensated by an equivalent current of oxygen ions which are incorporated into the crystal at the reference electrode. As under steady state conditions, during the transient period, the movement of electrons and holes is also exclusively a concentration-driven diffusion process due to the lack of an internal field: If we insert the individual expressions for the electric cds 4 = c, 11,grad (A - ~8) ih =
&
C/I uk grad
(Pk +
9d)
= co>- u,,- grad&-
in the electroneutrality
- 2q4)
(2) (3)
(51
the electric field may be neglected with regard to the transport of both electrons and holes because the conductivity due to the motion of oxygen ions is always much larger than that due to the electronic species
Furthermore, since the oxygen ions have an essentially uniform chemical potential, their transport must be due to the presence of a very small electric field. In other words, the high ionic conductivity of the sample acts as an internal short circuit and the rapid motion of the ionic species annihilates any internal
EM
21
h
= ,&,-
- 211, = )+-
+ 2,&
(7)
and the local independence of hi-, the voltage is therefore also equal to the difference of the chemical potentials of the electrons and (with the reverse sign) the holes at both sides of the electrolyte divided by 4, If we may assume the behavior of ideal dilute solutions. we have the relation
(4)
condition
iozm + i, + ih = 0
electric field within the sample very rapidly relative to the transient period for the redistribution of electrons and holes. It might be pointed out that the major part of the voltage drop across the system occurs at the phase boundaries, especially at the side of the inert electrode. The change of the electron and hole concentrations at the inert electrode as a function of time is observed by the measurement of the voltage E across the galvanic cell. As long as the sample is a predominant oxygen ion conductor throughout, the voltage is given by the difference of the chemical potential of the neutral oxygen atom b at the left and right hand phase boundary divided by the charge transferred by one ion, 29. Due to the equilibrium condition
where C, k and T are the concentration, Boltzmann’s constant and the absolute temperature, respectively. The concentrations at x = L are fixed by equilibrium with the reference electrode. The relaxation of the concentration at x = 0 is determined by Fick’s second law. Beside the initial and boundary conditions already mentioned, we may assume that the exchange of oxygen and electrons at the inert electrode is suppressed at all times, ie, L&,/ax = 0. If the movement of both electrons and holes is involved in the relaxation process, eg using the air reference electrode and higher polarization voltages. a complication arises from the formation and annihilation of electrons and holes to establish thermodyna-
I
a4
Fig. 1. Theoretical dependence of the voltage as a function of time t (multiplied hy D/L’) for different polarizations under the assumption of equal electron and hole diffusion coefkients. U./a* (x = L) = 10-s; ?-= 900°C.
Voltage relaxation measurements of the electron and hole mobilities mic equilibrium, pe + pub= 0, which is superimposed upon the diffusion of these species and Fick’s second law is only valid in the form
For a ratio of lo-’ of the condu&vities of electrons and holes at x = L, which is approximately the case for an air reference electrode. the decay of the voltage has been calculated[24] and is shown in Fig. 1 under the assumption that the electrons and holes have equal values of the diffusion coefficient. It is seen that the voltage-time relation has a characteristic shoulder. The relaxation process is first limited by the diffusion of electrons, since the motion of the holes at the inert side is negligible by comparison, and their concentration increases only due to the electronic equilibrium. After a transitional period the holes will dominate and determine the reIaxation process. If the electrons and holes have different mobilities, for example, if we assume the case in which the diffusion coefficient for electrons is much greater than that for holes, the relaxation may be divided into two consecutive steps. First, a relaxation of electrons will occur while the concentration of the holes will hardly be affected. At longer times, a slow equilibration of the hole concentration will determine the time dependence of the voltage. The electron diffusion coefficient can be determined from the linear voltage decay dE/dt in the first part of the relaxation (at higher polarization voltages in the case of the air reference electrode), after there is a short initial transient, in which the voltage decay is proportional to the square root of time[24]
D = 4G2 e GS
dE dt’
00)
The hole diffusion coefficient can be obtained from the change in the length of the shoulder, z, as a function of the previously applied polarization voltage, ~~~241,
3. EXPERIMENTAL
CONSIUERATIONS
The bottom of flat ended tubes of ZrO, + 10 mole “/0Y,O, with about 1 cm outer dia and 0.16 cm wall thickness, commercially made by the Zirconium Corporation of America (Cleveland, Ohio) were used in the case of the application of air reference electrodes. Pellets of the same material were used when working with the Fe + “FeO” reference electrode. The thickness of the pellets was varied between 0.06 and 0.22 cm by abrading and polishing. Two galvanic cells with electrolytes of different thicknesses were put together in one experimental arrangement. Highly purified helium or nitrogen have been used as the inert gas. The silver electrodes were sputtered on the surface and had a thickness of _ 1 pm. The experimental details are reported elsewhere[24]. Platinum could not be used at the low oxygen partial pressure present at the inert electrode because of the formation
l8-
I
I
C-1 N,.Aq
EPI t
I
1PO,
0
I
I
Yz031
+iOm/o
Pt.Airt+l
900°C
-
_.-
723
I
200
400
600
I300 -t
IO00 [set]
Fig. 2. Experimental voltage relaxation of the galvanic cell with an air reference eletrode for higher polarization voltages between 1.4 and 2 V at T = 900°C.
of platinum-zirconium compounds which therrnodynamically influence the voltage across the cel1[34-36,241. In a series of experiments the formation of Pt-Zr compounds has been studied and thermodynamic data have been determined. Other otherwise suitable metals show similar effects. In the case of silver, however, compounds with zirconium do not become stable until zirconium activities greater than those used in this work are present[34].
4. RESULTS
Some experimental results at 900°C with polarization voltages of 1.4-2.OV and using an air reference electrode are shown in Fig. 2. At about 1.6V, it is seen that a shoulder is beginning to form. At higher initial voltages this shoulder becomes more marked and extends to longer times. The shape of these curves is similar to that calcuIated for the case of equal electron and hole diffusion coefficients. However, because of the faster initial voltage drop, the electron diffusion coefficient is obviously larger than that for the holes. The value of 2 x 10m3 cm’/s for the diffusion coefficient of electrons and 1.5 x lo-’ cm2/s as the diffusion coefficient for the holes results from the experimental data. Thus, we see that there is a difference of about 2 orders of magnitude between these values. Experiments using the iron and “FeO” reference electrode showed relaxation due only to electrons because of the low oxygen partial pressure established by that reference electrode. Experiments have been conducted on samples of varying thickness as the results of such experiments provide a very good indication of the applicability of this method. Results proportional to the square of the thickness should be observed if the process is diffusion controlled whereas a first power thickness dependence is to be expected if the charging of double layers is the rate determining process[23]. Experimental results are shown in Fig. 3 for a series of different initial polarization voltages applied across
724
w.
(-) N2 , Aq I ZrsYzOz,, T =
I Fe+FeO.
WEI’PNPR
Aq (+)
a25 ‘C
Experimental voltage relaxation of the galvanic cell with an Fe + “FeO” reference electrode for several polarization voltages and two different sample thicknesses (0.106 and 0.218 cm1 at 825°C. Fig. 3.
pellets of two different thicknesses; 0.106 and 0.218 cm, respectively. At higher polarization voltages, it is seen that the voltage changes linearly with time after an initial rapid decay which goes as the square root of time. The ratio of the slopes for the samples of different thickness is 3.9 which is close to the ratio of the squares of their thickness, 4.2, as expected, if bulk diffusion is the rate determining process. The agreement between the diffusion coefficients observed using both types of reference electrode is very good. The values of the electron diffusion coefficient that were observed are plotted in Fig. 4 as a function of 1/T. From the slope an activation enthalpy of 0.56 eV is obtained.
T
-3.5
Some experimental
I 8.8
I 9.2
I 96
I IO -
Fig. 4. DiBiision
coefficient of
and fundamental
errors may
a50 I
,
I 8.4
5. I)ISCUSSION
f-C]-
900
-2 ‘OP 4 [cm*/sec]
The diffusion coefficients for holes are plotted in Fig. 5 as a function of l/T. As mentioned earlier the values in this case are about 2 orders of magnitude less than for the electrons. For this reason, the stoichiometric point with equal electron and hole concentrations corresponds to an oxygen partial pressure about 8 orders of magnitude lower than that at equal conductivities. The activation enthalpy for the holes is 1.4 eV. The concentrations of electrons and holes can be calculated by making use of the partial conductivities of the respective species, which were determined from measurements of steady state current flux through the galvanic cell[lS, 241. These are shown in Fig. 6, in which the concentrations are plotted logarithmically against the logarithm of the oxygen partial pressure for three temperatures, 700, 800 and 900°C. Due to the higher activation enthalpy for the formation of electrons than for holes the intrinsic point of equal concentrations moves to higher oxygen partial pressures with increasing temperature. From the temperature dependence of the product of the electron and hole concentrations, one can estimate the electronic band gap to be about 4.1 eV. In the measurements using platinum instead of silver at the inert electrode, the voltage relaxation was characterized by special voltages which remained constant over long periods of time. In accordance with other thermodynamic measurements, this has been interpreted in terms of the formation of platinum-zirconium compounds which have unusually high stability[34-361. The voltages of the galvanic cell at which the formation of intermetallic zirconium-platinum compounds appears are plotted in Fig. 7 with respect to a reference electrode at 1 atm oxygen partial pressure. From these results the decomposition voltage of yttria-doped zirconia may be calculated, as shown in Fig. 8 as a function of temperature vs 1 atm oxygen partial pressure[24].
I(
IO4
the elctrons in Zr9Y2021 as a function between 700 and 900°C.
l/T
of the reciprocal
temperature
Vohage
relaxation
measurements
of
725
the electron and hole mobilities
Tbl--650
900
-4.5
750
600
700
I
-60 zsy2 -6.5
6.4
41 I 6.8
I 9.2
I 9.6
I IO -
Fig.
5. Diffusion
coefficient
as a function of the holes in ZrgYzO,, between 700 and 900°C.
Fig. 6. Concentrations data and the diffusion
lo*
Iw
be
10.4 l/T
of the reciprocal
temperature
[otml
of the electrons (e) and holes (h) in ZrpY20 21r calculated from conductivity coefficients of electrons and holes, as a function of the equilibrium oxygen partial pressure at 700, 800 and 900°C.
Fig. 7. Voltages of the formation of Pt-Zr compounds between Ft and Zr,Y,O,,, related to an oxygen reference pressure of 1 atm. The dotted line is the corresponding voltage calculated from literature data[35,36], using the Gibbs formation energy of pure monoclinic ZrOz.
726
w. WEPPNER
-aGf(ZrO;U”
E ["I
_-9, t
ZrO* + IO IIll0YZO,
2.25-
1
WI
t -3
-89
22x-
215
- 8.8
-
-
E ~5.1 otm
----
E vs. a+r IO.21 ah)
I 900
..
. ..-86
I 1000
I it00 -TTC]
Fig. 8. Decomposition voltage and Gibbs formation energy of Zr,Y,O,, related to 1 atm. The dotted line shows the decomposition voltage for an air reference electrode.
limit the application of the voltage relaxation method for the determination of electron and hole mobilities in ionic conductors. First, the method will be influenced if the reference electrode has insufficient reversibility, and is not able to supply the oxygen necessary during the relaxation process. The maximum current that passes through the galvanic cell initially after the relaxation process is initiated is obviously equal to the steady state polarization current. From the measured values, and recognizing that the oxygen flux decreases with time, it is not expected that this will be a problem if refer-
ence electrodes with relatively high diffusion or permeability of oxygen are used. Also, if we assume a totally non-reversible electrode in place of the reference electrode, we see that the resulting diffusion coefficient would be changed by a factor of 4; this gives the maximum error. Another serious, but more or less avoidable, error is that the inert electrode may be a sink for oxygen, and therefore a source for electrons. But, with a small inert volume, and as long as the oxygen activity is low at the inert side, this effect may be neglected in comparison
with the change
of the electronic
concen-
trations by diffusion. A more fundamental possible limitation may be the amount of current necessary to charge the electrical double layers. If one assumes the value of IOOpF per cm*, it can be shown that for cell voltages that are not too low this double layer effect should occur at much shorter times than those experimentally observed. Therefore, it is not the rate determining process under these experimental conditons. However, for small values of cell voltage its influence might become more important. Compared to the charge transfer technique, this voltage relaxation method has the important advantage that it does not require the integration of low values of transient current with their unavoidable fluctuations over a long period of time. Even though such fluctuations may be small, they can become important for long integration times and introduce serious errors.
When making experiments upon systems with high diffusion coefficients for electrons and holes, the charge transfer technique may have some advantages.
However, in cases in which the diffusion coefficients of the electronic species are small, this voltage relaxation method is preferable.
Acknowledgements-The author is grateful to Professors R. A. Huggins and H. Rickert for helpful discussions, Special thanks are also due to Professor R. A. Huggins for very vahrable help in the preparation and presentation of the paper.
REFERENCES
1. H. Schmalzried, Z. ahvs. Chem. N.F. 38. 87 (19631. 2. B. C. H. Steele and-C: B. Alcock, Trans. M&l.‘Soc. AIME 233, 1359 (1965). 3. R. W. Vest and N. M. Tallan, J. appl. Phys. 36, 543 (1965). 4. H. H. Mobius and R. Hartung, Silikattechnik 16, 276 (1965). 5. A. W. Smith, F. W. Meszaros and C. D. Amata, J. Am. ceram. Sot. 49, 240 (1966). 6. H. H. Miibius, Silikattcchnik 17, 358 (1966). Y. D. Tretyakov, Jnorg. Marer. USSR 2, 432 (1966). :: J. W. Patterson, E. C. Bogren and R. A. Rapp, J. electrochem. Sm. 114, 752 (1967). 9. H. Ullmann, 2. phys. Chesn. Leipzig 237, 71 (1968). 10. W. A. Fischer and D. Janke, AI-C/L Eisenhiitlenwes. 39, 89 (1968). 11. Y. D. Tretyakov and A. Muan, J. electrochem. Sot. 116, 332 (1969). 12. R. Hartung and H. H. Miibius, Z. phys. Chem., Leipzig 243, 133 (1970). 13. W. A. Fischer and D. Janke, Z. phys. Chem. N.F. 69, 11 (1970). 14. L. Heyne and N. M. Beekmans, Proc. Br. ceram. Sm. 19, 229 (1970). 15. L. D. Burke, H. Rickert and R. Steiner, Z. phys. Chem. N.F. 74, 146 (1971). 16. L. M. Friedman, K. E. Oberg, W. M. Boorstein and RAIRapp, Metall. %UIS. 4, 69 (1973). 17. R. Hartung, Z. phys. Chew., Leipzig 254, 393 (1973).
Voltage
relaxation
measurements
18. Y. Janke and W. A. Fischer, Arch. Eisenhiittenwes. 45, 477 [ 1975). 19. J. Yokota, J. ahvs. Sot. Javun 8. 595 (1953). 20. S. Miyatani, j. ihys. SOC. japan 10, 786 (l&X). 21. K. Weiss. Z. phvs. Chem. N.F. 59. 242 (1968). 22. K. Weiss. El&ochim. Acto 16, 201 (1971). 23. D. 0. Raleigh. Z. ph1.s. Chm~. N.F. 63. 319 (1969). 24. W. WEZDDIICX. to be nubl. in J. sdd staie Chem.: 2. Naturf&h. ‘31a, 13j6 (1976). 25. A. V. Joshi, Ph.D. Thesis, Northwestern University (1972). 26. A. V. Joshi, in Fast Ion Transport in Solids. Solid State Batteries and Deoices (Edited bv W. van Goal), p. 173. North Holland, Am&dam-Lbndon, (1973). _ 27. A. V. Joshi and J. B. Wagner. Jr., J. pkys. &em. Solids 3.3. 205 (I 972).
of the electron
and hole mobilities
727
28. A. V. Joshi and J. B. Wagner, Jr., J. elrctrochem. Sot. 122, 1071 (1975). 29. M. Hebb. J. them. Phvs. 20, 185 (1958). 30. C. Wagner. in Proc. ‘Int. Comm. Ele&ochem. Thermodyn. und Kinetics (CITCE), 7th Meeting, Lindnu 1955. p. 361. Butterworth, London (1957). 31. C. Wagner, Z. Ekktrockem. 60, 4 (1956). 32. F. Hund, 2. Elektrockem. 55, 363 (1951). 33. F. Hund. Z. nkus. Ckrw. 199. 142 (1952). 34. L. Brewer. P: R. Wengert, M&l. && 4, 83 (1973). 35. P. J. Meschter. Ph.D. Thesis. Universitv of Pennsvlvania, Philadelphia (1974). 36. P. J. Meschter, W. L. Worrell, preprint (to be publ. in Metal1 Trans.).
University,
OF THE IN
WEPPNER
Stanford,
California 94305, U.S.A.
(Received 1 September 1976) Abstract-The voltage relaxation of galvanic cells with zirconia electrolytes polarized between an inert silver electrode and either a FY/air or a Fe/Fe0 electrode has been analysed to obtain the mobilities of both electronic minority charge carriers. At 900°C the mobility of the electrons is 2.4 x lO_’ cm’/Vs, that of the holes is 2 orders of magnitude lower, 1.6 x 10m4cmz/Vs.The activation enthalpy is 0.55 eV for the electrons and 1.4eV for the holes. From conductivity data, the concentrations of electrons and holes at 900°C and 1 atm oxygen partial pressure are calculated to be 3 x 10” and 6 x 10” cmm3, respectively. With the use of platinum inert electrodes the formation of intermetallic pt-Zr compounds appears at an oxygen partial pressure of 2.3 x lo-*’ atm at 900°C. From this the Gibbs formation energy of yttria-doped zirconia is calculated to be -9.1 eV at 900°C. 1. INTRODUCTION
At one side zirconia doped with 10 mole % yttria
Zirconia doped with several mole yO of yttria or some other lower valent oxides is of considerable interest as a solid oxygen ion conductor in many tecbnological and scientific applications. Its use is limited in some cases, however, because electronic species either electrons or holes can make an appreciable contribution to the total conductivity. In several investigations[l-181 the partial electronic conductivities of stabilized zirconia have been studied. P-type conduction occurs at high oxygen partial pressures, ey POz 2 IO-” atm at 9OO”C, whereas, n-type conduction occurs at low oxygen partial pressures. There is little information, however, about the relative contributions of the concentrations of both electronic charge carriers and their mobilities to the electronic portion of the total conductivity[5,12,14, 151. The work reported here involves measurements of the voltage relaxation in galvanic cells[i%24,151 in order to obtain diffusion coefficients (or mobilities) for the electrons and holes. From this information and the data for the conductivities of the electronic species, the concentrations of the electrons and holes can be calculated. This is the inverse of the charge transfer technique[23, 25-281 which allows the determination of the concentrations of the eIectronic species and the calculation of the mobilities from knowledge of the conductivity values.
is exposed to a defined reference oxygen partial pres-
sure, either air or a mixture of Fe and “Fee”, and an inert atmosphere, either nitrogen or argon, of very small volume, is present at the other side. Electrical contacts were made using thin sputtered silver layers. If a positive voltage is applied to the galvanic cell, no ionic current exists under steady state conditions, since the inert electrode cannot supply oxygen ions. Therefore, the only current that will pass through the cell will be that due to the transport of electronic species. Thus, by measuring the
current
the
electronic
conductivities
have
been
evaluated[S, 15-17,24,29-311. The chemical potential of the oxygen ions can be regarded as constant throughout the solid because of the high degree of disorder due to the presence of about 5% oxygen vacancies[32,33]. Any changes in the oxygen partial pressure have only a negligible influence upon the concentration of these defects. Since under steady state conditions no oxygen is transported across the cell, the gradient of the electrochemical potentia1 of the oxygen ions must vanish. Therefore, since the electrochemical potential q (per particle) of any species is equal to the sum of its chemical potential p (per particle) and the electrostatic potential c$ multiplied by the charge number 2 and the elementary charge q,
2. THEORY
we can conclude that there is no internal electric field within the electrolyte under steady state conditions.
The following galvanic cell has been employed:
EA
I
I
This leads to the conclusion that the transport of elecand holes can only be related to gradients in their concentrations and obeys Fick’s first law, in accordance with Wagner’s interpretation of the pofarization current[30,31]. The concentration gradient of the prevailing conductive electronic species is linear and the slope is inversely proportional to the diffusion coefficient if we can assume that the mobility is independent of
trons I
Pt, Air
Inert Ag Electrolyte: Zr9Y2021 Atmosphere’
Fe +%eO”
, x 0
L 721
W.
122
WEPPNER
the oxygen partial pressure. In the case of air reference electrodes, a junction between p-type and n-type regions of the electrolyte appears at sufficiently high polarization voltages. If, after steady state conditions are established, the externally applied voltage is switched off, the concentration gradients will decay until the stoichiometry, as determined by the reference electrode, is uniformly established throughout the entire sample. The transport of electrons moving toward the right and of holes moving to the left is electrically compensated by an equivalent current of oxygen ions which are incorporated into the crystal at the reference electrode. As under steady state conditions, during the transient period, the movement of electrons and holes is also exclusively a concentration-driven diffusion process due to the lack of an internal field: If we insert the individual expressions for the electric cds 4 = c, 11,grad (A - ~8) ih =
&
C/I uk grad
(Pk +
9d)
= co>- u,,- grad&-
in the electroneutrality
- 2q4)
(2) (3)
(51
the electric field may be neglected with regard to the transport of both electrons and holes because the conductivity due to the motion of oxygen ions is always much larger than that due to the electronic species
Furthermore, since the oxygen ions have an essentially uniform chemical potential, their transport must be due to the presence of a very small electric field. In other words, the high ionic conductivity of the sample acts as an internal short circuit and the rapid motion of the ionic species annihilates any internal
EM
21
h
= ,&,-
- 211, = )+-
+ 2,&
(7)
and the local independence of hi-, the voltage is therefore also equal to the difference of the chemical potentials of the electrons and (with the reverse sign) the holes at both sides of the electrolyte divided by 4, If we may assume the behavior of ideal dilute solutions. we have the relation
(4)
condition
iozm + i, + ih = 0
electric field within the sample very rapidly relative to the transient period for the redistribution of electrons and holes. It might be pointed out that the major part of the voltage drop across the system occurs at the phase boundaries, especially at the side of the inert electrode. The change of the electron and hole concentrations at the inert electrode as a function of time is observed by the measurement of the voltage E across the galvanic cell. As long as the sample is a predominant oxygen ion conductor throughout, the voltage is given by the difference of the chemical potential of the neutral oxygen atom b at the left and right hand phase boundary divided by the charge transferred by one ion, 29. Due to the equilibrium condition
where C, k and T are the concentration, Boltzmann’s constant and the absolute temperature, respectively. The concentrations at x = L are fixed by equilibrium with the reference electrode. The relaxation of the concentration at x = 0 is determined by Fick’s second law. Beside the initial and boundary conditions already mentioned, we may assume that the exchange of oxygen and electrons at the inert electrode is suppressed at all times, ie, L&,/ax = 0. If the movement of both electrons and holes is involved in the relaxation process, eg using the air reference electrode and higher polarization voltages. a complication arises from the formation and annihilation of electrons and holes to establish thermodyna-
I
a4
Fig. 1. Theoretical dependence of the voltage as a function of time t (multiplied hy D/L’) for different polarizations under the assumption of equal electron and hole diffusion coefkients. U./a* (x = L) = 10-s; ?-= 900°C.
Voltage relaxation measurements of the electron and hole mobilities mic equilibrium, pe + pub= 0, which is superimposed upon the diffusion of these species and Fick’s second law is only valid in the form
For a ratio of lo-’ of the condu&vities of electrons and holes at x = L, which is approximately the case for an air reference electrode. the decay of the voltage has been calculated[24] and is shown in Fig. 1 under the assumption that the electrons and holes have equal values of the diffusion coefficient. It is seen that the voltage-time relation has a characteristic shoulder. The relaxation process is first limited by the diffusion of electrons, since the motion of the holes at the inert side is negligible by comparison, and their concentration increases only due to the electronic equilibrium. After a transitional period the holes will dominate and determine the reIaxation process. If the electrons and holes have different mobilities, for example, if we assume the case in which the diffusion coefficient for electrons is much greater than that for holes, the relaxation may be divided into two consecutive steps. First, a relaxation of electrons will occur while the concentration of the holes will hardly be affected. At longer times, a slow equilibration of the hole concentration will determine the time dependence of the voltage. The electron diffusion coefficient can be determined from the linear voltage decay dE/dt in the first part of the relaxation (at higher polarization voltages in the case of the air reference electrode), after there is a short initial transient, in which the voltage decay is proportional to the square root of time[24]
D = 4G2 e GS
dE dt’
00)
The hole diffusion coefficient can be obtained from the change in the length of the shoulder, z, as a function of the previously applied polarization voltage, ~~~241,
3. EXPERIMENTAL
CONSIUERATIONS
The bottom of flat ended tubes of ZrO, + 10 mole “/0Y,O, with about 1 cm outer dia and 0.16 cm wall thickness, commercially made by the Zirconium Corporation of America (Cleveland, Ohio) were used in the case of the application of air reference electrodes. Pellets of the same material were used when working with the Fe + “FeO” reference electrode. The thickness of the pellets was varied between 0.06 and 0.22 cm by abrading and polishing. Two galvanic cells with electrolytes of different thicknesses were put together in one experimental arrangement. Highly purified helium or nitrogen have been used as the inert gas. The silver electrodes were sputtered on the surface and had a thickness of _ 1 pm. The experimental details are reported elsewhere[24]. Platinum could not be used at the low oxygen partial pressure present at the inert electrode because of the formation
l8-
I
I
C-1 N,.Aq
EPI t
I
1PO,
0
I
I
Yz031
+iOm/o
Pt.Airt+l
900°C
-
_.-
723
I
200
400
600
I300 -t
IO00 [set]
Fig. 2. Experimental voltage relaxation of the galvanic cell with an air reference eletrode for higher polarization voltages between 1.4 and 2 V at T = 900°C.
of platinum-zirconium compounds which therrnodynamically influence the voltage across the cel1[34-36,241. In a series of experiments the formation of Pt-Zr compounds has been studied and thermodynamic data have been determined. Other otherwise suitable metals show similar effects. In the case of silver, however, compounds with zirconium do not become stable until zirconium activities greater than those used in this work are present[34].
4. RESULTS
Some experimental results at 900°C with polarization voltages of 1.4-2.OV and using an air reference electrode are shown in Fig. 2. At about 1.6V, it is seen that a shoulder is beginning to form. At higher initial voltages this shoulder becomes more marked and extends to longer times. The shape of these curves is similar to that calcuIated for the case of equal electron and hole diffusion coefficients. However, because of the faster initial voltage drop, the electron diffusion coefficient is obviously larger than that for the holes. The value of 2 x 10m3 cm’/s for the diffusion coefficient of electrons and 1.5 x lo-’ cm2/s as the diffusion coefficient for the holes results from the experimental data. Thus, we see that there is a difference of about 2 orders of magnitude between these values. Experiments using the iron and “FeO” reference electrode showed relaxation due only to electrons because of the low oxygen partial pressure established by that reference electrode. Experiments have been conducted on samples of varying thickness as the results of such experiments provide a very good indication of the applicability of this method. Results proportional to the square of the thickness should be observed if the process is diffusion controlled whereas a first power thickness dependence is to be expected if the charging of double layers is the rate determining process[23]. Experimental results are shown in Fig. 3 for a series of different initial polarization voltages applied across
724
w.
(-) N2 , Aq I ZrsYzOz,, T =
I Fe+FeO.
WEI’PNPR
Aq (+)
a25 ‘C
Experimental voltage relaxation of the galvanic cell with an Fe + “FeO” reference electrode for several polarization voltages and two different sample thicknesses (0.106 and 0.218 cm1 at 825°C. Fig. 3.
pellets of two different thicknesses; 0.106 and 0.218 cm, respectively. At higher polarization voltages, it is seen that the voltage changes linearly with time after an initial rapid decay which goes as the square root of time. The ratio of the slopes for the samples of different thickness is 3.9 which is close to the ratio of the squares of their thickness, 4.2, as expected, if bulk diffusion is the rate determining process. The agreement between the diffusion coefficients observed using both types of reference electrode is very good. The values of the electron diffusion coefficient that were observed are plotted in Fig. 4 as a function of 1/T. From the slope an activation enthalpy of 0.56 eV is obtained.
T
-3.5
Some experimental
I 8.8
I 9.2
I 96
I IO -
Fig. 4. DiBiision
coefficient of
and fundamental
errors may
a50 I
,
I 8.4
5. I)ISCUSSION
f-C]-
900
-2 ‘OP 4 [cm*/sec]
The diffusion coefficients for holes are plotted in Fig. 5 as a function of l/T. As mentioned earlier the values in this case are about 2 orders of magnitude less than for the electrons. For this reason, the stoichiometric point with equal electron and hole concentrations corresponds to an oxygen partial pressure about 8 orders of magnitude lower than that at equal conductivities. The activation enthalpy for the holes is 1.4 eV. The concentrations of electrons and holes can be calculated by making use of the partial conductivities of the respective species, which were determined from measurements of steady state current flux through the galvanic cell[lS, 241. These are shown in Fig. 6, in which the concentrations are plotted logarithmically against the logarithm of the oxygen partial pressure for three temperatures, 700, 800 and 900°C. Due to the higher activation enthalpy for the formation of electrons than for holes the intrinsic point of equal concentrations moves to higher oxygen partial pressures with increasing temperature. From the temperature dependence of the product of the electron and hole concentrations, one can estimate the electronic band gap to be about 4.1 eV. In the measurements using platinum instead of silver at the inert electrode, the voltage relaxation was characterized by special voltages which remained constant over long periods of time. In accordance with other thermodynamic measurements, this has been interpreted in terms of the formation of platinum-zirconium compounds which have unusually high stability[34-361. The voltages of the galvanic cell at which the formation of intermetallic zirconium-platinum compounds appears are plotted in Fig. 7 with respect to a reference electrode at 1 atm oxygen partial pressure. From these results the decomposition voltage of yttria-doped zirconia may be calculated, as shown in Fig. 8 as a function of temperature vs 1 atm oxygen partial pressure[24].
I(
IO4
the elctrons in Zr9Y2021 as a function between 700 and 900°C.
l/T
of the reciprocal
temperature
Vohage
relaxation
measurements
of
725
the electron and hole mobilities
Tbl--650
900
-4.5
750
600
700
I
-60 zsy2 -6.5
6.4
41 I 6.8
I 9.2
I 9.6
I IO -
Fig.
5. Diffusion
coefficient
as a function of the holes in ZrgYzO,, between 700 and 900°C.
Fig. 6. Concentrations data and the diffusion
lo*
Iw
be
10.4 l/T
of the reciprocal
temperature
[otml
of the electrons (e) and holes (h) in ZrpY20 21r calculated from conductivity coefficients of electrons and holes, as a function of the equilibrium oxygen partial pressure at 700, 800 and 900°C.
Fig. 7. Voltages of the formation of Pt-Zr compounds between Ft and Zr,Y,O,,, related to an oxygen reference pressure of 1 atm. The dotted line is the corresponding voltage calculated from literature data[35,36], using the Gibbs formation energy of pure monoclinic ZrOz.
726
w. WEPPNER
-aGf(ZrO;U”
E ["I
_-9, t
ZrO* + IO IIll0YZO,
2.25-
1
WI
t -3
-89
22x-
215
- 8.8
-
-
E ~5.1 otm
----
E vs. a+r IO.21 ah)
I 900
..
. ..-86
I 1000
I it00 -TTC]
Fig. 8. Decomposition voltage and Gibbs formation energy of Zr,Y,O,, related to 1 atm. The dotted line shows the decomposition voltage for an air reference electrode.
limit the application of the voltage relaxation method for the determination of electron and hole mobilities in ionic conductors. First, the method will be influenced if the reference electrode has insufficient reversibility, and is not able to supply the oxygen necessary during the relaxation process. The maximum current that passes through the galvanic cell initially after the relaxation process is initiated is obviously equal to the steady state polarization current. From the measured values, and recognizing that the oxygen flux decreases with time, it is not expected that this will be a problem if refer-
ence electrodes with relatively high diffusion or permeability of oxygen are used. Also, if we assume a totally non-reversible electrode in place of the reference electrode, we see that the resulting diffusion coefficient would be changed by a factor of 4; this gives the maximum error. Another serious, but more or less avoidable, error is that the inert electrode may be a sink for oxygen, and therefore a source for electrons. But, with a small inert volume, and as long as the oxygen activity is low at the inert side, this effect may be neglected in comparison
with the change
of the electronic
concen-
trations by diffusion. A more fundamental possible limitation may be the amount of current necessary to charge the electrical double layers. If one assumes the value of IOOpF per cm*, it can be shown that for cell voltages that are not too low this double layer effect should occur at much shorter times than those experimentally observed. Therefore, it is not the rate determining process under these experimental conditons. However, for small values of cell voltage its influence might become more important. Compared to the charge transfer technique, this voltage relaxation method has the important advantage that it does not require the integration of low values of transient current with their unavoidable fluctuations over a long period of time. Even though such fluctuations may be small, they can become important for long integration times and introduce serious errors.
When making experiments upon systems with high diffusion coefficients for electrons and holes, the charge transfer technique may have some advantages.
However, in cases in which the diffusion coefficients of the electronic species are small, this voltage relaxation method is preferable.
Acknowledgements-The author is grateful to Professors R. A. Huggins and H. Rickert for helpful discussions, Special thanks are also due to Professor R. A. Huggins for very vahrable help in the preparation and presentation of the paper.
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Voltage
relaxation
measurements
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of the electron
and hole mobilities
727
28. A. V. Joshi and J. B. Wagner, Jr., J. elrctrochem. Sot. 122, 1071 (1975). 29. M. Hebb. J. them. Phvs. 20, 185 (1958). 30. C. Wagner. in Proc. ‘Int. Comm. Ele&ochem. Thermodyn. und Kinetics (CITCE), 7th Meeting, Lindnu 1955. p. 361. Butterworth, London (1957). 31. C. Wagner, Z. Ekktrockem. 60, 4 (1956). 32. F. Hund, 2. Elektrockem. 55, 363 (1951). 33. F. Hund. Z. nkus. Ckrw. 199. 142 (1952). 34. L. Brewer. P: R. Wengert, M&l. && 4, 83 (1973). 35. P. J. Meschter. Ph.D. Thesis. Universitv of Pennsvlvania, Philadelphia (1974). 36. P. J. Meschter, W. L. Worrell, preprint (to be publ. in Metal1 Trans.).