- Email: [email protected]
A spherical model of the clark electrode
J. Electroanal. Chem., 127 (1981) 59-66
59
Elsevier Sequoia S.A., L a u s a n n e - - P r i n t e d in The Netherlands
A SPHERICAL M O D E L OF THE CLARK ELECTRODE
J.S. U L T M A N , E. F I R O U Z T A L E and M.J. S K E R P O N
Department of Chemical Engineering and the Bioengineering Program, The Pennsylvania State University, University Park, PA 16802 (U.S.A.) (Received 1st December 1980; in revised form 12th March 1981)
ABSTRACT A spherical model that relates the sensitivity (S) of a Clark electrode to cathode radius (a), to electrolyte layer permeability ( Pe ) and thickness ( wc ), and to membrane permeability (Pm ) and thickness (win) has been developed. This model was used to correlate two different data sets, one obtained from polypropylene-covered (Pm - 1.7 × 10 t I cm 3 0 2 / c m . s . T o r r ) and the other obtained from Teflon-covered (Pro = 6 . 0 × 10 I I cm 3 O2/cm.s.Torr ) Clark electrodes. The two data sets spanned a range of cathode radii from 10 to 677.5 ~ m and a range of membrane thickness from 6.35 to 76.2/Lm. The spherical model fits the experimental sensitivity values with average relative errors of 13.3 and 23.9% for the polypropylene and Teflon data sets respectively; individual errors were randomly distributed over the range of membrane thicknesses, membrane permeabilities and cathode radii. The final regressed values of the free parameters (Pc~Pro and we) were found to be 69.22 and 27.7/~m, respectively, for the polypropylene data, and 6.13 and 154.6/~m, respectively, for the Teflon data. Although the Pc~Pro values are consistent with reported values in the literature, we values are larger than expected. For purposes of comparison, a one-dimensional planar model and a pseudo-two-dimensional model were also used to correlate the data sets. The planar model was incapable of fitting sensitivity values corresponding to small cathode radii, and the pseudo-two-dimensional model was incapable of fitting sensitivity values for the Clark electrodes covered with the more permeable (Teflon) membrane. Final regressed values of the two free parameters (Pe/Pm and We) were generally unrealistic for these two models.
INTRODUCTION
The sensitivity of a Clark electrode is the ratio of its current output to the undisturbed oxygen tension of the external medium in which it is immersed. The value of S is related to cathode radius, to electrolyte layer permeability and thickness, and to membrane permeability and thickness. In order to determine this relationship, one must integrate the two coupled partial differential equations which specify the diffusion of oxygen in the electrolyte and membrane layers. The exact analytical solution of this problem, if it were determined, would be quite complex; it is the objective of this work to find an approximate solution which is convenient for data correlation and electrode design. Most previous analyses have specific restrictions and cannot predict the full range of electrode characteristics. It was therefore our goal to develop a formulation which is uniformly valid for a wide range of
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''~'/
~'//
(a)
I'
' V' ,
,~ (b)
MEMB~A.E
"
Jr ~ I~ dr
LAYERELECTROLYTE CATHODE
(c)
Fig. I. Oxygen flux lines for: (a) general case, (b) planar model, (c) pseudo-t~o-dimensional model. Drawings of the axial plane of symmetry indicate the general (J), axial (Jz), and radial (Jr) flux lines respectively.
system parameter values. In particular, we sought a diffusion model with which to correlate the anomalously high current fluxes that have been observed for small cathode radii and for membranes of low oxygen permeability. In this work we considered the operation of a Clark electrode having a circular disk-shaped cathode. In most practical cases, oxygen diffusion occurs both perpendicular to (i.e. axially) and parallel to (i.e. radially) the cathode surface. Thus, the oxygen flux lines are generally curved (Fig. la). Previous analyses of the Clark electrode usually introduce simplifying assumptions regarding the shape of these flux lines. For example, the planar model, which has been utilized by several investigators including Schuler and Kreutzer [1], totally excludes edge effects. That is, this model excludes radial diffusion in the membrane and electrolyte layers, and consequently the oxygen flux lines are straight and perpendicular to the cathode surface at all points (Fig. lb). Since the importance of axial diffusion depends directly upon the available cross-sectional area, the one-dimensional model will be valid when the cathode radius becomes very large compared to the electrolyte and membrane thicknesses. The electrode sensitivity of the planar model can be expressed as S = "lra2KTPe/[Oae 4- ( P e / / P m ) ~ m ]
(1)
where K T = 16.06 A.s/cm 3 at 20°C and 1 atm total pressure. Unlike the planar model, the pseudo-two-dimensional model, developed by Jensen et al. [2], does attempt to account for edge effects. This model is based upon the assumptions that diffusion in the electrolyte layer is purely radial, while diffusion through the membrane is purely axial (Fig. lc). It is assumed, in addition, that the section of the electrolyte layer directly above the cathode is devoid of oxygen and that the electrolyte layer is so thin that it contains no axial concentration gradients. Physically, we expect that these model assumptions correspond to the physical restrictions of an electrolyte layer which is thin compared to the cathode radius, and a membrane permeability which is very small compared to the electrolyte permeability. The electrode sensitivity of the pseudo-two-dimensional model can be expressed
61 as
S=I,
1-4 a / R ,
Ko(a/R,)
(2)
where K 0 and K~ are modified Bessel functions of zero and first orders, respectively, and 11 = KTPm/LOm
(3)
and RI ~_ [(pe/Pm)(.,Je~rn ] 1/2
(4)
Based upon Siu and Cobbold's initial work [3], Ellis [4] recently presented a two-dimensional analysis which is more general than the pseudo-two-dimensional model of Jensen et al. [2]. This analysis, however, is accurate only for a small cathode. Moreover, it is inconvenient to use Ellis' analysis for data correlation since this would require repeated numerical integration over a semi-infinite domain. METHOD
Derivation of the one-dimensional spherical model
Numerical simulations [5] suggest that, in general, the oxygen isoconcentration surfaces surrounding the cathode can be approximated by hemispheres. Thus, the spherical model treats the cathode as a hemisphere surrounded by hemispherical shells of electrolyte and membrane (Fig. 2a). In the spherical model the flux is represented by radial lines, and the oxygen tension distribution depends upon radial position alone. As illustrated by Crank [6], the steady-state differential diffusion equations can be solved for the rate of oxygen diffusion. The sensitivity resulting from this diffusion rate is S = 2~rKTPer2[(re -- rc)[rc/re] -F [Pe/Pm][rc/re][rc/rm](rm -- re) ] -1
(5)
The parameters re, re and rm represent the radial distances from the origin to the cathode/electrolyte, electrolyte/membrane and membrane/external medium interMEDIUM MEMBRANE
.
ELECTROLYTE to#, LAYER CATHODE (Q)
p. ~ ) (b)
Fig. 2. Comparison of (a) surfaces of the spherical model to the (b) planar surfaces of an actual membrane-covered cathode.
62 faces, respectively. Since these interfaces are actually planar (Fig. 2b), the determination of effective values for rc, re and r m, in terms of the actual geometrical parameters a, ~oe and ~0m, is the critical factor in a complete specification of eqn. (5). This fact was overlooked by Buckles et al. [7], who proposed a similar spherical model. Effective values for rc, re and r m can be determined by considering two asymptotic limits: (1) the infinitely thick electrolyte layer; and (2) the infinitely large cathode radius. (1) If a circular disk-shaped cathode of radius a operates in a semi-infinite stagnant electrolyte medium, then the solution of Carslaw and Jaeger [8] can be employed to formulate the sensitivity: S = 4KTPe a
(6)
Physically, this situation corresponds to a Clark electrode, wherein the effective cathode radius becomes vanishingly small compared to the effective electrolyte layer thickness (i.e., rc << re and rc << rm). Applying this limit to eqn. (5) results in the definition: rc = 2a/~r
(7)
(2) If r~ becomes very large relative to (& - r e ) and (r m - r e ) , then the oxygen isoconcentration surfaces are planar and the sensitivity can be calculated from eqn. (1). Considering the corresponding limit of eqn. (5) results in the relationships: (r e - rc ) = 8toe/w 2
(8)
(rm -- re) = 8¢0m/~ 2
(9)
Equations (7)-(9) express re, re and rmin terms of the actual parameters of the Clark electrode a, ~oe and ~0n~ and thus, in terms of measurable parameters, the sensitivity is =
_
¢°e+C°m[Pe/Pm
1 +4
¢Oe+~4
--1
1
Equation (10) is the final equation for the one-dimensional spherical model. Note that the diffusion resistance of the medium is presumed to be zero in the three alternative models, eqns. (1), (2) and (10), that have been presented. Correlation of experimental data Using a least-squares non-linear regression program based on the method of Marquardt [9], and treating P e / P m and ~0e as free parameters, we fit eqn. (10), as well as eqns. (1) and (2), to two different sets of experimental sensitivity values, weighing all data points equally. The first data set obtained by Jensen et al. [2] consisted of seven sensitivity values corresponding to seven different cathode radii in the range from 10 to 350 /xm. In all cases the cathodes were covered with a 28 /~m thick polypropylene membrane. The second data set consisted of a total of 38 sensitivity values and was obtained
63
in our laboratory. We employed a Clark-type electrode consisting of a platinum wire embedded in a glass capillary which was, in turn, mounted axially in a 12.7 mm diameter polymethylmethacrylate rod. This plastic rod had an internal electrolyte reservoir containing an Ag/AgC1 reference anode constructed from silver foil. The polished cross-section of the platinum cathode was flush with the surface of the plastic rod. The entire surface was covered with a Teflon membrane of measured thickness varying from 6.35 to 76.2 #m; care was taken to apply the Teflon membrane in a reproducible manner such that variations in the thickness of the electrolyte layer trapped between the membrane and the cathode surface were minimized. The data set spanned a range of cathode radii of 64.75-677.5 ,~m. Each electrode assembly was alternatively exposed to N 2 / O 2 mixtures having oxygen partial pressures of 0, 159, 386, 611 and 760 Torr. The resulting current outputs were linearly correlated with the oxygen partial pressures and the electrode sensitivity was the slope of the resulting line. In keeping with. our goal of formulating a general model of the Clark electrode, these data sets were chosen so as to provide a wide range of cathode radii (10-677.5 /~m), a wide range of membrane thickness (6.35-76.2 /~m) and two significantly different membrane permeability values of 6 . 0 × 10 ~1 cm 3 O2/cm.s.Torr and 1.7 X 10 - l l cm 3 O2/cm.s.Torr. DISCUSSION OF THE RESULTS OF CORRELATION
Regarding the correlation of the polypropylene membrane data (Table 1), the fit to the data by the planar model is poor for cathode radii of 50/~m and less, but improves significantly as the cathode radius is increased. Moreover, this model progressively underpredicts the electrode sensitivity as the cathode radius decreases. This behavior is consistent with the fact that the planar model does not account for radial diffusion which, in fact, becomes more pronounced the smaller the cathode
TABLE 1 Correlation of the polypropylene membrane data (Jensen et al. [2])
a/l~m
S/A
Torr - 1
Predicted values as fraction of the experimental values Planar model
10 25 50 125 200 300 350
0.1065)< 0.2256)< 0.4386)< 0.9398 X 0.1598)< 0.3759)< 0.4699×
10 lo 10 lo 10 J0 10 - l0 10 9 10 -9 I0 9
Average error (%) relative to experimental values
0.037 0.110 0.227 0.662 0.997 0.953 1.038 43.6
Pseudotwo-dimensional model 0.813 0.688 0.664 0.966 1.158 0.963 1.005 15.3
Spherical model 0.913 0.719 0.673 0.968 1.159 0.963 1.005 13.3
64 TABLE 2
Final regressed parameter values for the polypropylene membrane data (Jensen et al. [2]) Model
Pe / P,~
we /
Average e r r o r / %
~m
Planar Pseudo-two-dimensional
30.32 68.72 (78.81) 69.22
Spherical
593,1 1.38 (2,55) 27,7
43.6 15.3 13.3
radius. The fit to the polypropylene data by the pseudo-two-dimensional and spherical models is almost identical, with the spherical model being slightly superior for the smallest cathode radii. Apparently, polypropylene is so impermeable that the edge effect is confined to the electrolyte layer as assumed in Jensen's model [2]. The final regresssed Pe/Pm parameter values (Table 2) for the spherical and pseudo-two-dimensional models are in reasonable agreement with the range of values 46-67 that have been previously reported [2,3,7]. The planar model predicts a value lying outside of the range. The actual value of the electrolyte layer thickness ~0e is unknown. However, the value of 593.1/~m predicted by the planar model appears to be too high, while the value of 1.38/~m predicted by the pseudo-two-dimensional model appears to be too low. The electrolyte layer thickness of 27.7/xm, predicted by the spherical model, though still probably too large, is possibly closer to the actual electrolyte layer thickness.
TABLE 3
Correlation of the Teflon membrane data
a/pom
Wm//.~m
No. of
10SSavJ
averaged data points
A Torr
Predicted values as fraction of the exper, values 1
Planar model
Pseudotwo-dimensional model
Spherical model
677.5 677.5 677.5 677.5
6.35 19.05 25.40 76.20
7 4 5 2
1.501 1.309 0.8321 0.7770
0.984 0.862 1.213 0.704
1.105 0.668 0.908 0.600
0,984 0.858 1.209 0.730
179.35 179.35 179.35
6.35 12.70 25.40
6 1 1
0.2348 0.1910 0.1064
0.441 0.469 0.664
1.857 1.796 2.625
0.765 0.855 1.308
64.75 64.75 64.75
6.35 12.70 25.40
8 1 3
0.06642 0.04736 0.03152
0.203 0.247 0.292
3.556 4.267 5.590
0.716 0.953 1.309
Average error (%) relative to experimental values
41.6
119.2
23.9
65
The values of Pe/Pm and we within the parentheses in Table 2 are those reported by Jensen et al. [2], while the corresponding values are the result of our own least-squares regression analysis. Jensen et al. [2] do not discuss their method of non-linear regression and the discrepancy between the two sets of values may be due to differences in our regression techniques and the sensitivity of the ~% free parameter. The fit to the experimental Teflon membrane data (Table 3) indicates that the planar and pseudo-two-dimensional models both give reasonable results for the cathode radius of 677.5/~m. However, as the cathode radius decreases to 179.35/~m, the planar model underpredicts the sensitivity values by a factor of 1.5 to 2.25, and the pseudo-two-dimensional model overpredicts them by a factor of 1.80 to 2.60. As the cathode radius drops to a still lower value of 64.75 tzm, the lack of fit by both models becomes more pronounced. For this cathode radius, the planar model underpredicts and the pseudo-two-dimensional model overpredicts the sensitivity values by a factor of 3.5-5.0. As was the case with polypropylene, the underprediction of cathode sensitivity by the planar model for small cathodes is due to the theoretical exclusion of radial diffusion. The pseudo-two-dimensional model, on the other hand, progressively overpredicts the sensitivity values as the cathode radius is reduced. In the derivation of the pseudo-two-dimensional model, it is assumed that diffusion in the electrolyte layer is only radial and thus this model is valid only for relatively impermeable membranes such as polypropylene. For permeable membranes, such as Teflon, the contribution of radial diffusion in the electrolyte layer is reduced since the flux lines tend to bend more toward the axial direction. This results in less actual electrode sensitivity than would be predicted by the pseudo-two-dimensional model. (Note that the average relative error of 119% for the pseudo-two-dimensional model is deceptive. The average errors reported here are relative to the experimental values, and since the pseudo-two-dimensional model greatly overpredicts the sensitivity values, the result is a large relative error. In terms of absolute error values, the planar and pseudo-two-dimensional models are comparable.) Table 3 demonstrates that the spherical model, in contrast to the planar and pseudo-two-dimensional models, fits all the experimental sensitivity values equally well for the range of cathode radii from 64.75 to 677.5 /~m and the membrane thickness of 6.35-76.2/~m. Moreover, the residuals are randomly scattered about the regression line. The average relative error associated with the fit of the spherical
TABLE 4 Final regressed parameter values for Teflon membrane data Model
Pe~Pro
~e/~ m
Average error/%
Planar Pseudo-two-dimensional Spherical
3.48 53.83 6.13
120.9 471.9 154.6
41.9 119.2 23.9
66
model to the Teflon data is 23.9% which, compared to the actual reproducibility of the experimental data of 18.3%, is an excellent value. The final regressed values of the Pe/Pm parameter resulting from least-squares fit to the Teflon membrane data (Table 4), for the planar and pseudo-two-dimensional models, are far outside the range of 7.0-18.0 previously reported in the literature [2,3,7]. However, the Pe~Provalue for the spherical model is very close to the low end of Pe/P~nvalues previously reported. The regressed values of %, for all three models, seem to be large. CONCLUSIONS
The spherical model of the Clark electrode fits the experimental sensitivity values quite precisely for a wide range of cathode radii (10 677.5 ~tLm), a wide range of membrane thickness (6.35 76.2 btm) and significantly different membrane permeabilities (1.7 X 10 -11 cm3 O2/cm-s-Torr and 6.0 X 10 11 cm ~ O2/cm_s_Torr). By comparison, the planar and pseudo-two-dimensional models previously reported in the literature are valid only for large cathodes and impermeable membranes respectively. The spherical model predicts a value for P~/P~n, for the data of both polypropylene and Teflon membranes, that is consistent with reported values in the literature; the planar and pseudo-two-dimensional models generally predict Pe/Pm values that are outside the reported range of values. All three models predict 0oe values that seem to be either too low or too high. Although the spherical model may not be predictive, it is an excellent tool for correlating the sensitivity values for the Clark electrodes with a wide range of geometrical parameter values. ACKNOWLEDGEMENT
This work was supported in part by National Institute of Health Grant HL19190. REFERENCES l R. Schuler and F. Kreuzer, Respir. Physiol., 3 (1967) 90. 2 0 . J . Jensen, I. Jacobsen and K. Thomsem J. Electroanal. Chem., 87 (1978) 203. 3 W. Siu and R.S.C. Cobbold, Med. Biol. Eng., (1976) 109. 4 C.G. Ellis, Ph.D. Dissertation, Northwestern University, 1980. 5 C.G. Ellis, H.E. G u t h e r m a n and T.K. Goldstick, Proc. Can. Med. Biol. Eng. Conf., 7 (1978) 163. 6 J. Crank, The Mathematics of Diffusion, Oxford University Press. London, 1957. p. 84. 7 R.G. Buckles, H. Heitmann and M.B. Laver. N.B.S. Spec. Pub., 450 (1975) 207. 8 .S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, 2nd ed.. Oxford Univcrsity Press. London, 1960. 9 D.W. Marquardk J. Soc. Indust. Appl. Math., I I (1963) 431.
59
Elsevier Sequoia S.A., L a u s a n n e - - P r i n t e d in The Netherlands
A SPHERICAL M O D E L OF THE CLARK ELECTRODE
J.S. U L T M A N , E. F I R O U Z T A L E and M.J. S K E R P O N
Department of Chemical Engineering and the Bioengineering Program, The Pennsylvania State University, University Park, PA 16802 (U.S.A.) (Received 1st December 1980; in revised form 12th March 1981)
ABSTRACT A spherical model that relates the sensitivity (S) of a Clark electrode to cathode radius (a), to electrolyte layer permeability ( Pe ) and thickness ( wc ), and to membrane permeability (Pm ) and thickness (win) has been developed. This model was used to correlate two different data sets, one obtained from polypropylene-covered (Pm - 1.7 × 10 t I cm 3 0 2 / c m . s . T o r r ) and the other obtained from Teflon-covered (Pro = 6 . 0 × 10 I I cm 3 O2/cm.s.Torr ) Clark electrodes. The two data sets spanned a range of cathode radii from 10 to 677.5 ~ m and a range of membrane thickness from 6.35 to 76.2/Lm. The spherical model fits the experimental sensitivity values with average relative errors of 13.3 and 23.9% for the polypropylene and Teflon data sets respectively; individual errors were randomly distributed over the range of membrane thicknesses, membrane permeabilities and cathode radii. The final regressed values of the free parameters (Pc~Pro and we) were found to be 69.22 and 27.7/~m, respectively, for the polypropylene data, and 6.13 and 154.6/~m, respectively, for the Teflon data. Although the Pc~Pro values are consistent with reported values in the literature, we values are larger than expected. For purposes of comparison, a one-dimensional planar model and a pseudo-two-dimensional model were also used to correlate the data sets. The planar model was incapable of fitting sensitivity values corresponding to small cathode radii, and the pseudo-two-dimensional model was incapable of fitting sensitivity values for the Clark electrodes covered with the more permeable (Teflon) membrane. Final regressed values of the two free parameters (Pe/Pm and We) were generally unrealistic for these two models.
INTRODUCTION
The sensitivity of a Clark electrode is the ratio of its current output to the undisturbed oxygen tension of the external medium in which it is immersed. The value of S is related to cathode radius, to electrolyte layer permeability and thickness, and to membrane permeability and thickness. In order to determine this relationship, one must integrate the two coupled partial differential equations which specify the diffusion of oxygen in the electrolyte and membrane layers. The exact analytical solution of this problem, if it were determined, would be quite complex; it is the objective of this work to find an approximate solution which is convenient for data correlation and electrode design. Most previous analyses have specific restrictions and cannot predict the full range of electrode characteristics. It was therefore our goal to develop a formulation which is uniformly valid for a wide range of
60 d
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/:.".1!
dz
MEDIUM
i
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:: •
,
,/
....
....
,
:
,
:
:
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: ~ :
I"
''~'/
~'//
(a)
I'
' V' ,
,~ (b)
MEMB~A.E
"
Jr ~ I~ dr
LAYERELECTROLYTE CATHODE
(c)
Fig. I. Oxygen flux lines for: (a) general case, (b) planar model, (c) pseudo-t~o-dimensional model. Drawings of the axial plane of symmetry indicate the general (J), axial (Jz), and radial (Jr) flux lines respectively.
system parameter values. In particular, we sought a diffusion model with which to correlate the anomalously high current fluxes that have been observed for small cathode radii and for membranes of low oxygen permeability. In this work we considered the operation of a Clark electrode having a circular disk-shaped cathode. In most practical cases, oxygen diffusion occurs both perpendicular to (i.e. axially) and parallel to (i.e. radially) the cathode surface. Thus, the oxygen flux lines are generally curved (Fig. la). Previous analyses of the Clark electrode usually introduce simplifying assumptions regarding the shape of these flux lines. For example, the planar model, which has been utilized by several investigators including Schuler and Kreutzer [1], totally excludes edge effects. That is, this model excludes radial diffusion in the membrane and electrolyte layers, and consequently the oxygen flux lines are straight and perpendicular to the cathode surface at all points (Fig. lb). Since the importance of axial diffusion depends directly upon the available cross-sectional area, the one-dimensional model will be valid when the cathode radius becomes very large compared to the electrolyte and membrane thicknesses. The electrode sensitivity of the planar model can be expressed as S = "lra2KTPe/[Oae 4- ( P e / / P m ) ~ m ]
(1)
where K T = 16.06 A.s/cm 3 at 20°C and 1 atm total pressure. Unlike the planar model, the pseudo-two-dimensional model, developed by Jensen et al. [2], does attempt to account for edge effects. This model is based upon the assumptions that diffusion in the electrolyte layer is purely radial, while diffusion through the membrane is purely axial (Fig. lc). It is assumed, in addition, that the section of the electrolyte layer directly above the cathode is devoid of oxygen and that the electrolyte layer is so thin that it contains no axial concentration gradients. Physically, we expect that these model assumptions correspond to the physical restrictions of an electrolyte layer which is thin compared to the cathode radius, and a membrane permeability which is very small compared to the electrolyte permeability. The electrode sensitivity of the pseudo-two-dimensional model can be expressed
61 as
S=I,
1-4 a / R ,
Ko(a/R,)
(2)
where K 0 and K~ are modified Bessel functions of zero and first orders, respectively, and 11 = KTPm/LOm
(3)
and RI ~_ [(pe/Pm)(.,Je~rn ] 1/2
(4)
Based upon Siu and Cobbold's initial work [3], Ellis [4] recently presented a two-dimensional analysis which is more general than the pseudo-two-dimensional model of Jensen et al. [2]. This analysis, however, is accurate only for a small cathode. Moreover, it is inconvenient to use Ellis' analysis for data correlation since this would require repeated numerical integration over a semi-infinite domain. METHOD
Derivation of the one-dimensional spherical model
Numerical simulations [5] suggest that, in general, the oxygen isoconcentration surfaces surrounding the cathode can be approximated by hemispheres. Thus, the spherical model treats the cathode as a hemisphere surrounded by hemispherical shells of electrolyte and membrane (Fig. 2a). In the spherical model the flux is represented by radial lines, and the oxygen tension distribution depends upon radial position alone. As illustrated by Crank [6], the steady-state differential diffusion equations can be solved for the rate of oxygen diffusion. The sensitivity resulting from this diffusion rate is S = 2~rKTPer2[(re -- rc)[rc/re] -F [Pe/Pm][rc/re][rc/rm](rm -- re) ] -1
(5)
The parameters re, re and rm represent the radial distances from the origin to the cathode/electrolyte, electrolyte/membrane and membrane/external medium interMEDIUM MEMBRANE
.
ELECTROLYTE to#, LAYER CATHODE (Q)
p. ~ ) (b)
Fig. 2. Comparison of (a) surfaces of the spherical model to the (b) planar surfaces of an actual membrane-covered cathode.
62 faces, respectively. Since these interfaces are actually planar (Fig. 2b), the determination of effective values for rc, re and r m, in terms of the actual geometrical parameters a, ~oe and ~0m, is the critical factor in a complete specification of eqn. (5). This fact was overlooked by Buckles et al. [7], who proposed a similar spherical model. Effective values for rc, re and r m can be determined by considering two asymptotic limits: (1) the infinitely thick electrolyte layer; and (2) the infinitely large cathode radius. (1) If a circular disk-shaped cathode of radius a operates in a semi-infinite stagnant electrolyte medium, then the solution of Carslaw and Jaeger [8] can be employed to formulate the sensitivity: S = 4KTPe a
(6)
Physically, this situation corresponds to a Clark electrode, wherein the effective cathode radius becomes vanishingly small compared to the effective electrolyte layer thickness (i.e., rc << re and rc << rm). Applying this limit to eqn. (5) results in the definition: rc = 2a/~r
(7)
(2) If r~ becomes very large relative to (& - r e ) and (r m - r e ) , then the oxygen isoconcentration surfaces are planar and the sensitivity can be calculated from eqn. (1). Considering the corresponding limit of eqn. (5) results in the relationships: (r e - rc ) = 8toe/w 2
(8)
(rm -- re) = 8¢0m/~ 2
(9)
Equations (7)-(9) express re, re and rmin terms of the actual parameters of the Clark electrode a, ~oe and ~0n~ and thus, in terms of measurable parameters, the sensitivity is =
_
¢°e+C°m[Pe/Pm
1 +4
¢Oe+~4
--1
1
Equation (10) is the final equation for the one-dimensional spherical model. Note that the diffusion resistance of the medium is presumed to be zero in the three alternative models, eqns. (1), (2) and (10), that have been presented. Correlation of experimental data Using a least-squares non-linear regression program based on the method of Marquardt [9], and treating P e / P m and ~0e as free parameters, we fit eqn. (10), as well as eqns. (1) and (2), to two different sets of experimental sensitivity values, weighing all data points equally. The first data set obtained by Jensen et al. [2] consisted of seven sensitivity values corresponding to seven different cathode radii in the range from 10 to 350 /xm. In all cases the cathodes were covered with a 28 /~m thick polypropylene membrane. The second data set consisted of a total of 38 sensitivity values and was obtained
63
in our laboratory. We employed a Clark-type electrode consisting of a platinum wire embedded in a glass capillary which was, in turn, mounted axially in a 12.7 mm diameter polymethylmethacrylate rod. This plastic rod had an internal electrolyte reservoir containing an Ag/AgC1 reference anode constructed from silver foil. The polished cross-section of the platinum cathode was flush with the surface of the plastic rod. The entire surface was covered with a Teflon membrane of measured thickness varying from 6.35 to 76.2 #m; care was taken to apply the Teflon membrane in a reproducible manner such that variations in the thickness of the electrolyte layer trapped between the membrane and the cathode surface were minimized. The data set spanned a range of cathode radii of 64.75-677.5 ,~m. Each electrode assembly was alternatively exposed to N 2 / O 2 mixtures having oxygen partial pressures of 0, 159, 386, 611 and 760 Torr. The resulting current outputs were linearly correlated with the oxygen partial pressures and the electrode sensitivity was the slope of the resulting line. In keeping with. our goal of formulating a general model of the Clark electrode, these data sets were chosen so as to provide a wide range of cathode radii (10-677.5 /~m), a wide range of membrane thickness (6.35-76.2 /~m) and two significantly different membrane permeability values of 6 . 0 × 10 ~1 cm 3 O2/cm.s.Torr and 1.7 X 10 - l l cm 3 O2/cm.s.Torr. DISCUSSION OF THE RESULTS OF CORRELATION
Regarding the correlation of the polypropylene membrane data (Table 1), the fit to the data by the planar model is poor for cathode radii of 50/~m and less, but improves significantly as the cathode radius is increased. Moreover, this model progressively underpredicts the electrode sensitivity as the cathode radius decreases. This behavior is consistent with the fact that the planar model does not account for radial diffusion which, in fact, becomes more pronounced the smaller the cathode
TABLE 1 Correlation of the polypropylene membrane data (Jensen et al. [2])
a/l~m
S/A
Torr - 1
Predicted values as fraction of the experimental values Planar model
10 25 50 125 200 300 350
0.1065)< 0.2256)< 0.4386)< 0.9398 X 0.1598)< 0.3759)< 0.4699×
10 lo 10 lo 10 J0 10 - l0 10 9 10 -9 I0 9
Average error (%) relative to experimental values
0.037 0.110 0.227 0.662 0.997 0.953 1.038 43.6
Pseudotwo-dimensional model 0.813 0.688 0.664 0.966 1.158 0.963 1.005 15.3
Spherical model 0.913 0.719 0.673 0.968 1.159 0.963 1.005 13.3
64 TABLE 2
Final regressed parameter values for the polypropylene membrane data (Jensen et al. [2]) Model
Pe / P,~
we /
Average e r r o r / %
~m
Planar Pseudo-two-dimensional
30.32 68.72 (78.81) 69.22
Spherical
593,1 1.38 (2,55) 27,7
43.6 15.3 13.3
radius. The fit to the polypropylene data by the pseudo-two-dimensional and spherical models is almost identical, with the spherical model being slightly superior for the smallest cathode radii. Apparently, polypropylene is so impermeable that the edge effect is confined to the electrolyte layer as assumed in Jensen's model [2]. The final regresssed Pe/Pm parameter values (Table 2) for the spherical and pseudo-two-dimensional models are in reasonable agreement with the range of values 46-67 that have been previously reported [2,3,7]. The planar model predicts a value lying outside of the range. The actual value of the electrolyte layer thickness ~0e is unknown. However, the value of 593.1/~m predicted by the planar model appears to be too high, while the value of 1.38/~m predicted by the pseudo-two-dimensional model appears to be too low. The electrolyte layer thickness of 27.7/xm, predicted by the spherical model, though still probably too large, is possibly closer to the actual electrolyte layer thickness.
TABLE 3
Correlation of the Teflon membrane data
a/pom
Wm//.~m
No. of
10SSavJ
averaged data points
A Torr
Predicted values as fraction of the exper, values 1
Planar model
Pseudotwo-dimensional model
Spherical model
677.5 677.5 677.5 677.5
6.35 19.05 25.40 76.20
7 4 5 2
1.501 1.309 0.8321 0.7770
0.984 0.862 1.213 0.704
1.105 0.668 0.908 0.600
0,984 0.858 1.209 0.730
179.35 179.35 179.35
6.35 12.70 25.40
6 1 1
0.2348 0.1910 0.1064
0.441 0.469 0.664
1.857 1.796 2.625
0.765 0.855 1.308
64.75 64.75 64.75
6.35 12.70 25.40
8 1 3
0.06642 0.04736 0.03152
0.203 0.247 0.292
3.556 4.267 5.590
0.716 0.953 1.309
Average error (%) relative to experimental values
41.6
119.2
23.9
65
The values of Pe/Pm and we within the parentheses in Table 2 are those reported by Jensen et al. [2], while the corresponding values are the result of our own least-squares regression analysis. Jensen et al. [2] do not discuss their method of non-linear regression and the discrepancy between the two sets of values may be due to differences in our regression techniques and the sensitivity of the ~% free parameter. The fit to the experimental Teflon membrane data (Table 3) indicates that the planar and pseudo-two-dimensional models both give reasonable results for the cathode radius of 677.5/~m. However, as the cathode radius decreases to 179.35/~m, the planar model underpredicts the sensitivity values by a factor of 1.5 to 2.25, and the pseudo-two-dimensional model overpredicts them by a factor of 1.80 to 2.60. As the cathode radius drops to a still lower value of 64.75 tzm, the lack of fit by both models becomes more pronounced. For this cathode radius, the planar model underpredicts and the pseudo-two-dimensional model overpredicts the sensitivity values by a factor of 3.5-5.0. As was the case with polypropylene, the underprediction of cathode sensitivity by the planar model for small cathodes is due to the theoretical exclusion of radial diffusion. The pseudo-two-dimensional model, on the other hand, progressively overpredicts the sensitivity values as the cathode radius is reduced. In the derivation of the pseudo-two-dimensional model, it is assumed that diffusion in the electrolyte layer is only radial and thus this model is valid only for relatively impermeable membranes such as polypropylene. For permeable membranes, such as Teflon, the contribution of radial diffusion in the electrolyte layer is reduced since the flux lines tend to bend more toward the axial direction. This results in less actual electrode sensitivity than would be predicted by the pseudo-two-dimensional model. (Note that the average relative error of 119% for the pseudo-two-dimensional model is deceptive. The average errors reported here are relative to the experimental values, and since the pseudo-two-dimensional model greatly overpredicts the sensitivity values, the result is a large relative error. In terms of absolute error values, the planar and pseudo-two-dimensional models are comparable.) Table 3 demonstrates that the spherical model, in contrast to the planar and pseudo-two-dimensional models, fits all the experimental sensitivity values equally well for the range of cathode radii from 64.75 to 677.5 /~m and the membrane thickness of 6.35-76.2/~m. Moreover, the residuals are randomly scattered about the regression line. The average relative error associated with the fit of the spherical
TABLE 4 Final regressed parameter values for Teflon membrane data Model
Pe~Pro
~e/~ m
Average error/%
Planar Pseudo-two-dimensional Spherical
3.48 53.83 6.13
120.9 471.9 154.6
41.9 119.2 23.9
66
model to the Teflon data is 23.9% which, compared to the actual reproducibility of the experimental data of 18.3%, is an excellent value. The final regressed values of the Pe/Pm parameter resulting from least-squares fit to the Teflon membrane data (Table 4), for the planar and pseudo-two-dimensional models, are far outside the range of 7.0-18.0 previously reported in the literature [2,3,7]. However, the Pe~Provalue for the spherical model is very close to the low end of Pe/P~nvalues previously reported. The regressed values of %, for all three models, seem to be large. CONCLUSIONS
The spherical model of the Clark electrode fits the experimental sensitivity values quite precisely for a wide range of cathode radii (10 677.5 ~tLm), a wide range of membrane thickness (6.35 76.2 btm) and significantly different membrane permeabilities (1.7 X 10 -11 cm3 O2/cm-s-Torr and 6.0 X 10 11 cm ~ O2/cm_s_Torr). By comparison, the planar and pseudo-two-dimensional models previously reported in the literature are valid only for large cathodes and impermeable membranes respectively. The spherical model predicts a value for P~/P~n, for the data of both polypropylene and Teflon membranes, that is consistent with reported values in the literature; the planar and pseudo-two-dimensional models generally predict Pe/Pm values that are outside the reported range of values. All three models predict 0oe values that seem to be either too low or too high. Although the spherical model may not be predictive, it is an excellent tool for correlating the sensitivity values for the Clark electrodes with a wide range of geometrical parameter values. ACKNOWLEDGEMENT
This work was supported in part by National Institute of Health Grant HL19190. REFERENCES l R. Schuler and F. Kreuzer, Respir. Physiol., 3 (1967) 90. 2 0 . J . Jensen, I. Jacobsen and K. Thomsem J. Electroanal. Chem., 87 (1978) 203. 3 W. Siu and R.S.C. Cobbold, Med. Biol. Eng., (1976) 109. 4 C.G. Ellis, Ph.D. Dissertation, Northwestern University, 1980. 5 C.G. Ellis, H.E. G u t h e r m a n and T.K. Goldstick, Proc. Can. Med. Biol. Eng. Conf., 7 (1978) 163. 6 J. Crank, The Mathematics of Diffusion, Oxford University Press. London, 1957. p. 84. 7 R.G. Buckles, H. Heitmann and M.B. Laver. N.B.S. Spec. Pub., 450 (1975) 207. 8 .S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, 2nd ed.. Oxford Univcrsity Press. London, 1960. 9 D.W. Marquardk J. Soc. Indust. Appl. Math., I I (1963) 431.