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Evidence of a correlation between the non-linearity of chemical sensors and the asymmetry of their response and recovery curves
Sensors and Actuators B 106 (2005) 407–423
Evidence of a correlation between the non-linearity of chemical sensors and the asymmetry of their response and recovery curves Francis M´enil∗ , Marc Susbielles, H´el`ene Deb´eda, Claude Lucat, Pascal Tardy Laboratoire IXL, Universit´e de Bordeaux I, UMR 5818 du CNRS, 351 cours de la Lib´eration, 33405 Talence, France Received 22 July 2004; accepted 20 August 2004 Available online 7 October 2004
Abstract The experimental response/recovery time of a chemical sensor is considered as the sum of an intrinsic time, independent of experimental conditions and supposing an instantaneous change of the fluid surrounding the sensor, and of an extrinsic time, depending on the transient concentration of target species, linked to the fluid delivery system. The transient concentration in the test cell is assumed to follow the exponential variations described by the well-stirred tank model. The application of the law governing the sensor signal to the transient concentration yields the extrinsic response/recovery times and accounts well for the symmetry or asymmetry between the response and recovery curves, according to the linearity or non-linearity of the sensor. All major trends derived from this theoretical approach are verified experimentally with various types of sensors, and this is shown to have major implications in many practical cases, including microsensors and microfluidic or car exhaust sensors. © 2004 Elsevier B.V. All rights reserved. Keywords: Chemical sensors; Response times; Recovery times; Transient concentration; Transient signal
1. Introduction The performances of chemical sensors are often based on the so-called “3s” criteria: sensitivity, selectivity and stability. However, the response time to target species and the recovery time after suppressing or decreasing their concentration, cannot be overlooked when evaluating the overall performances. The response and recovery times reported in the literature or in technical notices of commercial sensors, automatically include the delays due to the fluid delivery system, specific to every experimental setup. Even if these delays can be reduced to a few milliseconds in some very special cases [1–4], they can never be totally suppressed. It is thus of primary importance to estimate, at least roughly, which part of the response/recovery time measured experimentally, arises from the fluid delivery system, before drawing conclusions about what is called hereafter the sensor “intrinsic” response/recovery time. To ∗
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our knowledge, this has been rarely done in the literature [3,5,6]. Most published works only deal with qualitative comparisons between the response time and the recovery time of the same sensor, or between the response or recovery times of various sensors. The validity of such comparisons is of course limited to a given experimental set-up. A very oftenquoted result is that recovery times are longer than response times. The traditional explanation supposes slower kinetics for the desorption of the target species than for its adsorption. If the explanation is certainly right in some cases, the aim of the present paper is to demonstrate that this is usually not the prevailing one. A very simple model to account for the asymmetry between the response and recovery curves was recently proposed by one of the author [7]. The model correlates the asymmetry to the non-linearity of chemical sensors. Linearity means a steady-state signal proportional to the concentration of target species. Amperometric sensors are known to be linear. Catalytic sensors, thermal conductivity sensors, can also be considered as approximately linear.
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Conversely, semi-conductor oxide sensors that follow a power law, potentiometric sensors that follow the logarithmic Nernst law or Pd–MOS structures that also exhibit a logarithmic dependency [8], are non-linear. The model will first be briefly recalled. Then experiments with both gas and liquid sensors operating on linear and non-linear principles, will be carried out to support the validity of the model. Since the model only uses an approximate estimation of the transient concentration of the target species at the sensor level, the experimental curves are not expected to fit the model perfectly, but only to follow its main trends and to yield the order of magnitude of response and recovery times. On the basis of the model and of its experimental validation, various practical cases will finally be discussed.
2. Simple theoretical approach The basis of the theoretical approach consists in considering the response/recovery times measured experimentally as the sum of an “intrinsic” response/recovery time, independent of experimental conditions, and of a delivery system response/recovery time, depending, in particular, on the volume of the test cell and on the fluid flow rates. The latter will hereafter be called “extrinsic” response/recovery time. As will be demonstrated below, this time depends not only on the delivery system but also on the operating principle of the sensor, which justifies the denomination “extrinsic”. 2.1. Experimental response and recovery times The classical definition is based on the sensor signal variation between two concentrations c1 and c2 of a given target
species, as a function of time (Fig. 1). A well-defined and stable signal is normally characteristic of each concentration after reaching the corresponding steady-state. The response time is supposed to represent the time required to switch from one value to the other. Let us call ‘∆’ the difference between these two values. A quantitative definition of the response time consists in measuring the time to reach a given percentage of ∆ after switching from one concentration to the other (90% in the case of Fig. 1). For the sake of clarity, we will suppose in the following the concentration of the target species c1 to be always smaller than c2 . The response time will only correspond to the switching from c1 to c2 . The time corresponding to the switching from c2 to c1 (often called return to the baseline in the literature) will hereafter be called recovery time. Obviously, these experimentally measured response and recovery times include both “intrinsic” and “extrinsic” components. 2.2. “Intrinsic” response and recovery times The intrinsic response/recovery time being independent of experimental conditions, its evaluation supposes an instantaneous step function change of the fluid concentration surrounding the chemical sensor. Some rare laboratory experiments have been carried out in this direction, with the aim to measure the response/recovery times of gas sensors in exhaust pipes of automotive engines. There are two ways to very quickly change the gas surrounding a chemical sensor. The first requires a small test cell with comparatively huge flow rates, alternated within a few milliseconds by the use of one or several fixed injectors [1,2]. Flow rates of several litres/sec will be required to exchange the atmosphere of a cell of a few cm3 within milliseconds. The second uses flow rates more traditional in laboratories, of
Fig. 1. Typical normalized transient signal of a chemical sensor as a function of time.
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system, which we called “extrinsic” can never be rigorously zero. 2.3. “Extrinsic” response and recovery times The renewing of the gaseous or liquid volume of the test cell requires a definite time. “Extrinsic” response and recovery times actually correspond to the progressive and not instantaneous concentration changes, effectively “seen” by the sensor, after the switching of the valves. The aim of the following subsections is to try to quantify these extrinsic times, on the basis of a model as simple as possible, but nevertheless realistic.
Fig. 2. The well-stirred tank model.
the order of 100 cm3 /min. Two injectors, which permanently inject both concentrations, are alternatively moved below the sensor within milliseconds [3,4]. Two types of sensors were studied with such devices allowing very fast exchange of atmospheres: semiconductor oxides [1] and silicon carbide GASFETs [3,4]. The measured response/recovery times, of the order of a few milliseconds or tens of milliseconds, appear much shorter than those usually reported in the literature with usual laboratory set-ups, often longer than a few seconds or even minutes (especially for recovery times). In spite of these drastic reductions, the possibility that the measured times include a delivery system component cannot be completely ruled out. In summary, whatever the quickness of the atmosphere change around the sensor, it cannot be instantaneous, and the part of the response/recovery time due to the delivery
2.3.1. Estimation of the transient concentration at the sensor level, on the basis of the so-called well-stirred tank model As suggested in [5–7], for many sensor test cells, the transient concentration of target species may approximately be described by the well-stirred tank model [9], much used for example in pharmacology and biology ([10] and references therein). This simple model (Fig. 2) assumes that the fluid is always perfectly and instantaneously homogeneous in the test cell and that the total pressure remains constant. This allows a first order analytical treatment. After switching the inlet valve from c1 to c2 , the number of target molecules entering the cell in a time dt is c2 × f × dt and that leaving the cell is c(t) × f × dt, where f is the flow rate. If V is the volume of the cell:(c + dc) × V = c × V + c2 × f × dt − c × f × dt This leads to an exponential rise of the transient concentration of target molecules between c1 and c2 : −t V c(t) = c2 − (c2 − c1 ) exp with τ = (1) τ f After reaching the steady-state corresponding to c2 , the switching of the inlet valve back to c1 leads to a symmet-
Fig. 3. Theoretical transient signal of sensors following a linear law with the concentration of target species.
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Fig. 4. Theoretical transient signal of sensors following a power law (exponent < 1) with the concentration of target species.
rical exponential fall of the transient concentration: t c(t) = c1 + (c2 − c1 ) exp − τ V with the same time constant τ = f
(2)
One weak point of the model is certainly the assumption of a homogeneous transient concentration of target species in the test cell. Obviously, a concentration gradient occurs between the inlet and the outlet of the fluid flow. However, any other model, taking into account this concentration gradient as well as diffusion and convection processes in the cell, etc., would be far less simple, and in our opinion will not bring much more to our demonstration. 2.3.2. Application of the well-stirred tank model to linear sensors The theoretical curves reflecting the extrinsic response and recovery of linear sensors will be homothetic to those of the
transient concentration “seen” by the sensor, which follows Eqs. (1) and (2). They will be perfectly symmetrical (Fig. 3). In many practical cases where the intrinsic response/recovery times are negligible versus the extrinsic ones, the experimental response and recovery curves should also be symmetrical. 2.3.3. Application of the well-stirred model to non-linear sensors following a power law (exponent < 1) This category of sensors includes semi-conducting oxides, such as tin oxide, for which the steady-state conductance varies with the concentration of reducing gas, according to a more or less empirical power law with an exponent usually included between 0.2 and 0.9. Fig. 4 shows the theoretical extrinsic response and recovery of such sensors, obtained by applying a power law to the transient concentration of formula (1) and (2) with c1 = 0, and by normalizing the resulting values. The most striking feature of the curves is the asymmetry between the response and the recovery. The smaller the
Fig. 5. Theoretical transient signal of sensors following a logarithmic law with the concentration of target species.
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Table 1 Computed response and recovery times at 90% of the steady-state signal for various types of sensors Linear law
Power law
Logarithmic law
n = 0.8
n = 0.5
c2 /c1 = 102
n = 0.2
c2 /c1 = 104
c2 /c1 = 106
tresp (τ)
trec (τ)
tresp (τ)
trec (τ)
tresp (τ)
trec (τ)
tresp (τ)
trec (τ)
tresp (τ)
trec (τ)
tresp (τ)
trec (τ)
tresp (τ)
trec (τ)
2.30
2.30
2.09
2.88
1.66
4.60
0.89
11.50
0.99
5.13
0.51
8.80
0.29
12.72
The time unit is the time constant τ = V/f.
exponent, the faster the response and the slower the recovery. This observation seems of primary importance, since it enables to explain quite rationally numerous experimental curves reported in the literature for semi-conducting oxides. It may be worthwhile noting that for a hypothetical nonlinear sensor following a power law with an exponent larger than one, the extrinsic recovery time would be shorter than the response time! 2.3.4. Application of the well-stirred tank model to non-linear sensors following a logarithmic law This category of sensors is especially important, since it includes all sensors following the Nernst law: ISE, ISFET’s, potentiometric gas sensors using solid electrolytes. Pd–MOS structures for hydrogen sensing, which have been shown to display a logarithmic dependency over at least 10 decades [8] also belong to this category. Because of the characteristics of the logarithm function, the initial concentration c1 cannot be taken as zero any longer, conversely to the previous case. Three cases will be considered: c2 /c1 = 102 , c2 /c1 = 104 and c2 /c1 = 106 . Fig. 5 shows the theoretical extrinsic response and recovery of such sensors, obtained by applying the logarithm function to the transient concentration of formula (1) and (2) and by normalizing the resulting values. Again, the striking feature of the curves is the asymmetry between the response and the recovery. The larger the c2 /c1 ratio, the faster the response and the slower the recovery. This observation again enables to explain rationally many numerous experimental curves reported in the literature for these type of sensors. 2.3.5. Computation of extrinsic response/recovery times at 90% on the basis of the well-stirred tank model In order to get a more quantified view of the effect, the computation of some response and recovery times at 90% of the stationary state signals have been computed. Starting from formulas (1) and (2) with c1 = 0, the transient sensor signal was taken as c/c2 for linear sensors and as (c/c2 )n for sensors following a power law. For sensors governed by a logarithmic law and for which c1 cannot be zero, the signal was taken as 1 − ln[c/c2 ]/ln[c1 /c2 ]. These formulas ensure the normalization conditions, i.e. a signal equal to zero for c1 and 100% for c2 . The extrinsic response and recovery times at 90% are then given by: • for linear sensors: tresp = trec = −τ ln[0.1]
(3)
• for non-linear sensors following a power law with exponent n: 1 ln[0.1] tresp = −τ ln[1 − 0.91/n ] and trec = −τ n (4) • for non-linear sensors following a logarithmic law with k = c1 /c2 : 0.9 (1 − k0.1 ) k −k tresp = −τ ln and trec = −τ ln 1−k 1−k (5) The so calculated response and recovery times at 90% are reported in Table 1, as a function of the time constant τ = V/f. At this stage, it is interesting to note that τ corresponds to a virtual rinsing time of the cell, during which the inlet fluid would push away all the fluid inside without mixing with it. In the more realistic well-stirred tank model with an exponential evolution of the transient concentration, the rinsing process is achieved at 90% only after 2.3τ. This value is also that of the response and recovery times at 90% for linear sensors. For non-linear sensors governed either by a logarithmic law or by a power law with an exponent smaller than one, the response time is smaller than for linear sensors, whereas the recovery time is longer. As noticed previously, the asymmetry between response and recovery times is all the more marked that the exponent is smaller for power law sensors, or the concentration change c2 /c1 larger for sensors governed by a logarithmic law. Note the recovery time/response time ratio of 13 for a sensor following a power law with an exponent equal to 0.2, and of 44 for a sensor governed by a logarithmic law with c2 /c1 = 106 .
3. Experimental validation Various linear and non-linear sensors will be tested in order to validate the above model, both with gases and with liquids. The experimental validation supposes that the intrinsic response/recovery times remain as far as possible negligible versus the extrinsic ones. 3.1. Experimental set-up 3.1.1. Selection of sensors for the validation of the model Commercial sensors were preferred to laboratory-made sensors. One reason is a usually better reliability. Another is
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sponse/recovery times are required to be shorter than 100 ms • a zirconia lambda gauge also made by Bosch but working on a potentiometric principle. The application is identical, with similar requirements for response/recovery times • a tin-oxide-based sensor manufactured by Figaro (Japan). Response/recovery times are unfortunately not reported in the technical notices, but our own experience with tin oxide sensors make us believe that the intrinsic times are certainly no longer than 1 s. Two liquid sensors were selected: Fig. 6. Experimental set-up for validating the well-stirred tank model in the case of liquid sensors.
that the technical notices of commercial sensors, in principle give some indications about response and recovery times. As discussed previously, these experimentally measured times necessarily include some extrinsic part due to the delivery system, but the test conditions are more likely to be optimised to minimize it than in a laboratory. We considered these values as maximum intrinsic times. The selected gas sensors were the following: • a catalytic sensor (Pellistor) commercialized by Marconi Applied Technologies Ltd. (England), with a reported response time of 2 s • a thermal conductivity sensor made by Microsens (Switzerland). The reported gas exchange time constant is 100 ms • a zirconia proportional gauge made by Bosch GmbH (Germany), working on an amperometric principle. The application is engine control and re-
• a Clark-type electrode for dissolved oxygen operating on a amperometric principle; • a glass pH electrode operating on a potentiometric principle. Since we did not have any precise information about the response/recovery times of these electrodes, we determined a maximum intrinsic response/recovery time by plunging quickly by hand the electrodes from one solution to another and agitating vigorously at the same time. In the case of the pH electrode, the ratio c2 /c1 was chosen small (about 5) in order to minimize extrinsic effects due to the unavoidable liquid film or drop carried with the electrode when moving it from one solution to the other. Times required to get stable stationary state values were never longer than 3 s for both electrodes. 3.1.2. Test cells and delivery systems The experimental validation supposes that the values mentioned in the previous section, considered as maximum in-
Fig. 7. Stationary state signal values of the catalytic sensor as a function of methane concentration.
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Fig. 8. Experimental transient signals of the catalytic sensor exposed to 1% methane in dry air at various flow rates.
trinsic times, remain as far as possible negligible versus the extrinsic ones. Both tested liquid sensors are operated at room temperature and it was quite easy to design an experimental set-up resembling the well-stirred tank model (Fig. 6). The tank volume is 200 cm3 and the flow rate fixed by a peristaltic pump at 130 cm3 /min. The time constant V/f is thus equal to 92 s, i.e. about 30 times more than the maximum intrinsic time.
As for gas sensors, they are operated above room temperature; and a new cell design, which also requires gas tightness, is not so easy. Consequently, we decided to re-use our regular test cells. The cell used for testing semiconductor oxide and catalytic sensors has a volume of 30 cm3 , and that used for testing potentiometric, amperometric and thermal conductivity sensors a volume of 165 cm3 . Initial experiments were carried out directly with these cells. Even if the main
Fig. 9. Experimental and theoretical transient signals of the catalytic sensor exposed to 1% methane in dry air at a flow rate of 500 cm3 /min.
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Fig. 10. Stationary state signal values of the thermal conductivity sensor as a function of carbon dioxide concentration.
Fig. 11. Experimental and theoretical transient signals of the thermal conductivity sensor exposed to a carbon dioxide concentration change from 0–100% in dry air.
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Fig. 12. Stationary state signal values of the amperometric zirconia sensor as a function of oxygen concentration in nitrogen.
Fig. 13. Experimental and theoretical transient signals of the amperometric zirconia sensor exposed to an oxygen concentration change from 200–200,000 ppm in nitrogen.
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trends observed in these conditions qualitatively agreed with the theoretical expectations, a set-up in which the test cell is located downstream of a larger “buffer” volume of 320 cm3 was finally preferred. Such a design yields a total volume of either 350 or 485 cm3 . This ensures larger time constants and practically negligible intrinsic times versus extrinsic ones. In the most unfavourable case, the flow rate was 900 cm3 /min, which yields a time constant of 23 s with the 30 cm3 cell, i.e. more than 10 times the maximum intrinsic time of 2 s. For all gas sensors studied in these conditions with flow rates ranging from 100–900 cm3 /min, the experimental response/recovery curves depend on the flow rate, in agreement with the wellstirred tank model. Moreover, for a given sensor, the curves can practically be superimposed by modifying the time scale proportionally to the flow rate. This demonstrates clearly that the intrinsic times are negligible versus the extrinsic ones. The counterpart of this experimental set-up with two tanks in series is that it does not stick so well to the simple one tank model of Fig. 2. A set-up similar to that of Fig. 6 for liquids, with only one large tank and a fan inside would certainly have been preferable. But as stated in the introduction, the aim of the paper is to demonstrate the reality of the trends evaluated in Section 2, and not to fit perfectly theoretical curves on the basis of a model, which is only approximate anyhow. 3.2. Model validation with gas sensors 3.2.1. Linear sensors Tests with a catalytic sensor, a thermal conductivity sensor and an amperometric sensor will be reported in this section. The linearity of each sensor will first be checked.
3.2.1.1. Catalytic sensor. Fig. 7 shows the relatively good linearity of the stationary state signal with methane concentration. In Fig. 8 are reported the normalized transient response/recovery curves to 1% methane in dry air, for flow rates ranging from 100–900 cm3 /min. The normalization suppresses the slight dependency of the stationary state values with the flow rate and allows better comparisons. As expected, the response/recovery time decreases with increasing flow rate. Moreover, recovery curves appear symmetrical to response curves, in agreement with Section 2.3.2. Fig. 9 shows the fit to the well-stirred one tank model with a total volume of 320 + 30 = 350 cm3 , in the case of a flow rate equal to 500 cm3 /min. For the sake of simplicity, only the curves corresponding to one flow rate are presented. As stated in Section 3.1.2, all curve shapes are quite similar when the time scale is changed proportionally to the inverse of the time constant and this is also true for the sensors discussed hereafter. For this reason and in order to avoid numerous unnecessary curves, only results corresponding to a flow rate of 500 cm3 /min, shown on the same time scale (0–1200 s), will be presented in the following. The comparison between experimental and theoretical curves shown in Fig. 9 indicates that the well-stirred one tank model is not a too bad approximation, even for an experimental set-up with a small tank in series with a larger buffer tank. Anyhow, the departure of the experimental curve from the theoretical one does not affect the excellent symmetry between the recovery and the response. 3.2.1.2. Thermal conductivity sensor. The thermal conductivity sensor was tested with carbon dioxide in dry air. Sta-
Fig. 14. Stationary state e.m.f of the potentiometric zirconia sensor as a function of oxygen concentration in nitrogen.
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Fig. 15. Experimental and theoretical transient signals of the potentiometric zirconia sensor exposed to an oxygen concentration change from 200–200,000 ppm in nitrogen.
Fig. 16. Comparison of the experimental transient signals of the potentiometric and amperometric zirconia sensors tested in exactly the same conditions.
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Fig. 17. Experimental transient signals of the potentiometric zirconia sensor exposed to three concentration changes: 20–200,000; 200–200,000 and 2000–200,000 ppm oxygen in nitrogen.
tionary state values are reported in Fig. 10. In agreement with the theoretical behaviour [11], the experimental points are not perfectly aligned. However, taking into account the approximations of the well-stirred tank model and of the experimental set-up versus the model, this slight non-linearity will be neglected. Fig. 11 reports the experimental transient
response/recovery curve to 100% carbon dioxide in dry air for a flow rate of 500 cm3 /min. The curve is compared to that of the well-stirred tank model for a total volume of 320 + 165 = 485 cm3 . The quality of the agreement is satisfying, and the recovery again appears basically symmetrical to the response.
Fig. 18. Stationary state signal values of the semiconductor oxide sensor as a function of methane concentration in dry air.
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Fig. 19. Experimental and theoretical transient signals of the semi-conductor oxide sensor exposed to 1% methane in dry air.
3.2.1.3. Amperometric sensor. The amperometric sensor was tested with oxygen in nitrogen in the same cell as that used for the thermal conductivity sensor, i.e. with a total volume of 485 cm3 . The linearity of the amperometric sensor is excellent in the range shown in Fig. 12. No measurement was carried out between 5000 and 200,000 ppm. The signal measured for 200,000 ppm was 6.5 V, which does not depart much
from the value of 6.1 V, extrapolated from Fig. 12. For our purpose, the amperometric sensor will be considered as linear. Fig. 13 reports the experimental transient response/recovery curve to concentrations changes from 200 to 200,000 ppm oxygen in nitrogen for a total flow rate of 500 cm3 /min. Even if the agreement with the well-stirred tank model is only approximate, the symmetry remains rather satisfying.
Fig. 20. Experimental transient signals of the amperometric electrode for various concentration changes of dissolved oxygen in water.
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3.2.2. Non-linear sensor following a logarithmic law The potentiometric oxygen gauge has an external size similar to the amperometric one, i.e. that of a sparking plug. Moreover, both sensors are operated at around 700 ◦ C. In both devices, the oxygen ion conductor is yttrium-stabilized zirconia. Comparisons between both sensors will be all the more valuable that all experimental conditions are identical: same test device, same oxygen concentration ratios, and same flow rates. Fig. 14 shows the well-known logarithmic dependency of the lambda gauge with a slope of about 46 mV per concentration decade, in perfect agreement with the theoretical value for an operating temperature of 700 ◦ C. Fig. 15 shows the transient response/recovery curve to concentrations changes from 200 to 200,000 ppm oxygen in nitrogen for a 500 cm3 /min flow rate. Even though the fit to the wellstirred model is not satisfying for the recovery, the asymmetry between the response and the recovery is not questionable. The plot on the same figure of the transient signals of the amperometric and the potentiometric sensors magnifies this result (Fig. 16). The observation of a response time shorter for the potentiometric sensor than for the amperometric one, also agrees well with Table 1, at least qualitatively. It is worth noting that a reduction of only 20% of the time constant, leads to a strong improvement of the fit of the experimental curve for the potentiometric sensor and to a lesser extent for the amperometric one. A decrease of the time constant corresponds to a reduction of the effec-
tive total volume in the same proportion. Even if such a reduction is not meaningless, a precise interpretation remains, however, hazardous and without any real interest for our purpose. Another interesting feature of the well-stirred tank model applied to potentiometric sensors is the theoretical increase of the asymmetry with the concentration ratio c2 /c1 . Fig. 17 shows the experimental comparison for three values of this ratio: 102 , 103 and 104 . Again, the results are in qualitative agreement with the model. Not only the recovery time increases when the ratio increases, but the response time appears slightly shorter for c2 /c1 = 103 and c2 /c1 = 104 than for c2 /c1 = 102 (the distinction between 103 and 104 is not visible). The experimental ratios trec /tresp at 90% are 4.0, 6.2 and 9.0 for c2 /c1 = 102 , 103 and 104 , respectively, whereas the values computed with formula (5) are 5.1, 9.9 and 17.0. The differences between experimental and theoretical values are certainly significant, but the trend and the orders of magnitude are there. 3.2.3. Non-linear sensor following a power law For this category of sensor, we used a classical tin oxide sensor. The exponent of the power law determined from the stationary state signals for various methane concentrations (Fig. 18) was unfortunately higher than expected (0.82). In spite of this value relatively close to unity, the asymmetry of the response/recovery curve is slightly visible (Fig. 19).
Fig. 21. Experimental and theoretical transient signals of the amperometric electrode exposed to a concentration change from 0.4–8 mg/l of dissolved oxygen in water.
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Fig. 22. Stationary state emf values of the potentiometric electrode as a function of the pH of a chloric acid solution.
Fig. 23. Experimental transient signals of the potentiometric glass electrode exposed to three pH concentration changes: from 5 to 1, 5 to 2 and 5 to 3.
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3.3. Model validation with liquid sensors 3.3.1. Linear amperometric sensor A Clark-type electrode for oxygen dissolved in water was used for this type of sensor. Fig. 20 shows the transient responses for various concentration changes. The stationary state values indicate an approximately linear behaviour at least in the studied range of 0.2–8 mg/l of dissolved oxygen. Moreover, the three curves appear roughly symmetrical. The fit to the well-stirred tank model for a concentration change from 0.4 to 8 mg/l is not too bad (Fig. 21). 3.3.2. Non-linear sensor following a logarithmic law A glass pH electrode was used as representative of potentiometric sensors. Fig. 22 shows the characteristic linear dependence of the emf on the pH of the HCl solution. Three concentration ratios c2 /c1 were studied: 102 , 103 and 104 corresponding to pH changes from 3–5, 2–5 and 1–5, respectively. The corresponding transient normalized signals are shown in Fig. 23. The asymmetry of the curves is not questionable, even if the fits (not shown) to the well-stirred model are not excellent, especially for the recovery. As expected from the model and as observed for the potentiometric gaseous oxygen sensor, the larger the c2 /c1 ratio, the shorter the response time and the longer the recovery time. It would have been interesting to compare directly on the same figure the normalized responses of both oxygen and pH electrodes, as was done in Fig. 16 for the amperometric and potentiometric gaseous oxygen sensors. However, the target species are different and, moreover, we were unable to get a concentration ratio c2 /c1 larger than about 20 for dissolved oxygen. An extrapolation of the curves in Fig. 20 may let suppose that a ratio of 100 instead of 20, would not modify the basic symmetry of the transient signal, in which case the comparison with the curve of Fig. 23 corresponding to a pH change from 5 to 3, would be meaningful.
4. Application of the model to practical cases The experimental validation of the model was on purpose carried out with large volumes, in order that the intrinsic times remain negligible versus the extrinsic ones. The aim of the present section is to discuss in the light of the above results, to which extent the model can be applied to practical cases. After examining the real meaning of experimentally recorded response/recovery times, the case of microsensors with delivery systems based on microfluidic and that of car exhaust sensors requiring response/recovery times of the order of a few millisecond will be briefly dealt with.
on the basis of intrinsic properties of the sensor, especially interaction kinetics with the target species. In the light of the above results, such an assertion is obviously meaningless if the extrinsic response/recovery times are not taken into account. In fitting experimentally measured response/recovery times of semiconductor oxide sensors to a theoretical model, Pilling et al. [6] incorporated formula (1) and (2), characteristics of the delivery system (what is called here extrinsic) to surface reaction kinetics, characteristic of the sensor (what is called here intrinsic). The theoretical treatment of such a global approach, is however, not straightforward. A simpler but more restricted approach can be carried out as follows. The knowledge of the dependency of the sensor stationary state signal with the concentration of target species and the knowledge of the experimental test conditions (cell volume and flow rate), allow a very easy estimation of the extrinsic response and recovery times, just by looking at Table 1 or by applying formula (3), (4) or (5). If the estimated values are significantly shorter than the experimentally recorded times, then one may safely conclude that the latter effectively correspond to the intrinsic times of the sensor. This type of approach, rarely found in the literature, was done by Herne et al. [5] in the study of the response time of optical sensors based on luminescence quenching. As suggested by Professor Ingemar Lundstr¨om (Link¨oping University, Sweden), another simple approach to know whether the experimental response and recovery times correspond to extrinsic or intrinsic times, is to study the dependency on the sensor operating temperature. If the response/recovery times are temperature dependent, it is clear that they correspond to intrinsic times, since the model proposed in the present paper for extrinsic times is basically independent of the sensor operating temperature. 4.2. Microsensors and microfluidic At a time where sophisticated technologies allow drastic miniaturizations of sensing devices, one might wonder whether the model also applies to microsensors with delivery systems based on microfluidics. A decrease of test cell volumes is often accompanied by a decrease in similar proportions of the section of inlet and outlet tubes. The flow rates are smaller and the time constants may not differ so much from those of more traditional experimental set-up. Even in cases for which the time constant is strongly reduced, Fig. 8 suggests that the shape of the curves will remain basically unchanged by dilating the time scale proportionally to the inverse of the time constant. 4.3. Car exhaust sensors
4.1. Meaning of the experimental response/recovery times As stated in the Section 1, the asymmetry between the response and recovery curves is often explained in the literature
Modern engines already require closed loop lambda control within a few tens of milliseconds, and future engines will require control times shorter than 10 ms. Flow rates in the exhaust pipe are of the order of 20 l/sec. Assuming a section of
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the exhaust pipe of about 30 cm2 and a step function change of the oxygen concentration after crossing the lambda point, the time required to rinse a length of pipe of 1 cm (which is the order of magnitude of the size of a sensing element) is 1.5 ms. Thus, a first limitation to response/recovery times of the order of the milliseconds arises from the size of the sensing element. In our opinion, there should be a second limitation, independent of the sensor size. This limitation is related to the fact that the oxygen concentration after crossing the lambda point is unlikely to follow a perfect step function change in the exhaust pipe. The concentration probably includes a time-dependent component, more or less exponential. Since the oxygen concentration decreases by several orders of magnitude when crossing the lambda point from the lean to the rich-burn region, the requirement of very short recovery times should be more difficult to meet for non-linear devices (especially those following logarithmic laws) than for linear ones.
5. Conclusion The starting point of the mechanisms presented here to account for the response and recovery curves of chemical sensors is to distinguish an intrinsic response/recovery time, independent of experimental conditions and supposing a step function change of the target species concentration surrounding the sensor, from an extrinsic time, depending especially on the fluid flow and on the test cell volume. The extrinsic time is linked to the transient concentration at the sensor level. The time dependency of the transient concentration is assumed to be exponential, according to the simple well-stirred tank model. Then the application of the law governing the sensor signal to the transient concentration, yields either symmetrical or asymmetrical response/recovery curves, whether the sensor is linear or not. All major trends of the model have been verified experimentally, with sensors based on various linear and not linear principles and with target species of various types. Finally,
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the model was shown to have major implications in many practical cases, including microsensors and microfluidic or car exhaust sensors.
Acknowledgements We are especially grateful to Professor Ingemar Lundstr¨om (University of Link¨oping, Sweden), for his interest in the original model developed in the present manuscript and for highly relevant suggestions.
References [1] J. Gerblinger, H. Meixner, Fast oxygen sensors based on sputtered strontium titanate, Sens. Actuators B 4 (1991) 99–102. [2] P.P. Tobias, A. Branzahi, A. Lloyd Spetz, O. Kordina, E. Jansz´en, I. Lundstr¨om, IEEE Electron. Dev. Lett. 18 (1997) 287–289. [3] P.P. Tobias, P. Martensson, A. G¨oras, I. Lundstr¨om, A. Lloyd Spetz, Moving gas outlets for the evaluation of fast gas sensors, Sens. Actuators B 58 (1999) 389–393. [4] H. Wingbrant, I. Lundstr¨om, A. Llyod Spetz, The speed of response of MISiCFET devices, Sens. Actuators B 93 (2003) 286–294. [5] R. Herne, J. Brocas, E. Vander Donckt, The response time of optical sensors based on luminescence quenching, Anal. Chim. Acta 364 (1998) 131–141. [6] R.S. Pilling, G. Bernhardt, C.S. Kim, J. Duncan, C.B.H. Crothers, D. Kleinschmidt, D.J. Frankel, R.J. Lad, B.G. Frederick, Quantifying gas sensor and delivery system response time using GC/MS, Sens. Actuators B 96 (2003) 200–214. [7] F. M´enil, Mod´elisation des temps de r´eponse des capteurs chimiques, C. R. Acad. Sc. Paris, Chimie/Chemistry 4 (2001) 899–904. [8] L.-G. Ekedahl, M. Eriksson, I. Lunstr¨om, Hydrogen sensing mechanisms of metal–insulator interfaces, Acc. Chem. Res. 31 (1998) 249–256. [9] W.E. Boyce, R.C. Di Prima, Elementary Differential Equations and Boundary Value Problems, Wiley, New York, 1977. [10] Ring C., Beitr¨age zur mathematischen Modellierung der hepatischen Pharmakokinetik, Thesis, July 27th 2000. Martin Luther University, Wittenberg (Germany). [11] P. Tardy, J.-R. Coulon, C. Lucat, F. M´enil, Dynamic thermal conductivity sensor for gas detection, Sens. Actuators B 98 (2004) 63– 68.
Evidence of a correlation between the non-linearity of chemical sensors and the asymmetry of their response and recovery curves Francis M´enil∗ , Marc Susbielles, H´el`ene Deb´eda, Claude Lucat, Pascal Tardy Laboratoire IXL, Universit´e de Bordeaux I, UMR 5818 du CNRS, 351 cours de la Lib´eration, 33405 Talence, France Received 22 July 2004; accepted 20 August 2004 Available online 7 October 2004
Abstract The experimental response/recovery time of a chemical sensor is considered as the sum of an intrinsic time, independent of experimental conditions and supposing an instantaneous change of the fluid surrounding the sensor, and of an extrinsic time, depending on the transient concentration of target species, linked to the fluid delivery system. The transient concentration in the test cell is assumed to follow the exponential variations described by the well-stirred tank model. The application of the law governing the sensor signal to the transient concentration yields the extrinsic response/recovery times and accounts well for the symmetry or asymmetry between the response and recovery curves, according to the linearity or non-linearity of the sensor. All major trends derived from this theoretical approach are verified experimentally with various types of sensors, and this is shown to have major implications in many practical cases, including microsensors and microfluidic or car exhaust sensors. © 2004 Elsevier B.V. All rights reserved. Keywords: Chemical sensors; Response times; Recovery times; Transient concentration; Transient signal
1. Introduction The performances of chemical sensors are often based on the so-called “3s” criteria: sensitivity, selectivity and stability. However, the response time to target species and the recovery time after suppressing or decreasing their concentration, cannot be overlooked when evaluating the overall performances. The response and recovery times reported in the literature or in technical notices of commercial sensors, automatically include the delays due to the fluid delivery system, specific to every experimental setup. Even if these delays can be reduced to a few milliseconds in some very special cases [1–4], they can never be totally suppressed. It is thus of primary importance to estimate, at least roughly, which part of the response/recovery time measured experimentally, arises from the fluid delivery system, before drawing conclusions about what is called hereafter the sensor “intrinsic” response/recovery time. To ∗
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our knowledge, this has been rarely done in the literature [3,5,6]. Most published works only deal with qualitative comparisons between the response time and the recovery time of the same sensor, or between the response or recovery times of various sensors. The validity of such comparisons is of course limited to a given experimental set-up. A very oftenquoted result is that recovery times are longer than response times. The traditional explanation supposes slower kinetics for the desorption of the target species than for its adsorption. If the explanation is certainly right in some cases, the aim of the present paper is to demonstrate that this is usually not the prevailing one. A very simple model to account for the asymmetry between the response and recovery curves was recently proposed by one of the author [7]. The model correlates the asymmetry to the non-linearity of chemical sensors. Linearity means a steady-state signal proportional to the concentration of target species. Amperometric sensors are known to be linear. Catalytic sensors, thermal conductivity sensors, can also be considered as approximately linear.
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Conversely, semi-conductor oxide sensors that follow a power law, potentiometric sensors that follow the logarithmic Nernst law or Pd–MOS structures that also exhibit a logarithmic dependency [8], are non-linear. The model will first be briefly recalled. Then experiments with both gas and liquid sensors operating on linear and non-linear principles, will be carried out to support the validity of the model. Since the model only uses an approximate estimation of the transient concentration of the target species at the sensor level, the experimental curves are not expected to fit the model perfectly, but only to follow its main trends and to yield the order of magnitude of response and recovery times. On the basis of the model and of its experimental validation, various practical cases will finally be discussed.
2. Simple theoretical approach The basis of the theoretical approach consists in considering the response/recovery times measured experimentally as the sum of an “intrinsic” response/recovery time, independent of experimental conditions, and of a delivery system response/recovery time, depending, in particular, on the volume of the test cell and on the fluid flow rates. The latter will hereafter be called “extrinsic” response/recovery time. As will be demonstrated below, this time depends not only on the delivery system but also on the operating principle of the sensor, which justifies the denomination “extrinsic”. 2.1. Experimental response and recovery times The classical definition is based on the sensor signal variation between two concentrations c1 and c2 of a given target
species, as a function of time (Fig. 1). A well-defined and stable signal is normally characteristic of each concentration after reaching the corresponding steady-state. The response time is supposed to represent the time required to switch from one value to the other. Let us call ‘∆’ the difference between these two values. A quantitative definition of the response time consists in measuring the time to reach a given percentage of ∆ after switching from one concentration to the other (90% in the case of Fig. 1). For the sake of clarity, we will suppose in the following the concentration of the target species c1 to be always smaller than c2 . The response time will only correspond to the switching from c1 to c2 . The time corresponding to the switching from c2 to c1 (often called return to the baseline in the literature) will hereafter be called recovery time. Obviously, these experimentally measured response and recovery times include both “intrinsic” and “extrinsic” components. 2.2. “Intrinsic” response and recovery times The intrinsic response/recovery time being independent of experimental conditions, its evaluation supposes an instantaneous step function change of the fluid concentration surrounding the chemical sensor. Some rare laboratory experiments have been carried out in this direction, with the aim to measure the response/recovery times of gas sensors in exhaust pipes of automotive engines. There are two ways to very quickly change the gas surrounding a chemical sensor. The first requires a small test cell with comparatively huge flow rates, alternated within a few milliseconds by the use of one or several fixed injectors [1,2]. Flow rates of several litres/sec will be required to exchange the atmosphere of a cell of a few cm3 within milliseconds. The second uses flow rates more traditional in laboratories, of
Fig. 1. Typical normalized transient signal of a chemical sensor as a function of time.
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system, which we called “extrinsic” can never be rigorously zero. 2.3. “Extrinsic” response and recovery times The renewing of the gaseous or liquid volume of the test cell requires a definite time. “Extrinsic” response and recovery times actually correspond to the progressive and not instantaneous concentration changes, effectively “seen” by the sensor, after the switching of the valves. The aim of the following subsections is to try to quantify these extrinsic times, on the basis of a model as simple as possible, but nevertheless realistic.
Fig. 2. The well-stirred tank model.
the order of 100 cm3 /min. Two injectors, which permanently inject both concentrations, are alternatively moved below the sensor within milliseconds [3,4]. Two types of sensors were studied with such devices allowing very fast exchange of atmospheres: semiconductor oxides [1] and silicon carbide GASFETs [3,4]. The measured response/recovery times, of the order of a few milliseconds or tens of milliseconds, appear much shorter than those usually reported in the literature with usual laboratory set-ups, often longer than a few seconds or even minutes (especially for recovery times). In spite of these drastic reductions, the possibility that the measured times include a delivery system component cannot be completely ruled out. In summary, whatever the quickness of the atmosphere change around the sensor, it cannot be instantaneous, and the part of the response/recovery time due to the delivery
2.3.1. Estimation of the transient concentration at the sensor level, on the basis of the so-called well-stirred tank model As suggested in [5–7], for many sensor test cells, the transient concentration of target species may approximately be described by the well-stirred tank model [9], much used for example in pharmacology and biology ([10] and references therein). This simple model (Fig. 2) assumes that the fluid is always perfectly and instantaneously homogeneous in the test cell and that the total pressure remains constant. This allows a first order analytical treatment. After switching the inlet valve from c1 to c2 , the number of target molecules entering the cell in a time dt is c2 × f × dt and that leaving the cell is c(t) × f × dt, where f is the flow rate. If V is the volume of the cell:(c + dc) × V = c × V + c2 × f × dt − c × f × dt This leads to an exponential rise of the transient concentration of target molecules between c1 and c2 : −t V c(t) = c2 − (c2 − c1 ) exp with τ = (1) τ f After reaching the steady-state corresponding to c2 , the switching of the inlet valve back to c1 leads to a symmet-
Fig. 3. Theoretical transient signal of sensors following a linear law with the concentration of target species.
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Fig. 4. Theoretical transient signal of sensors following a power law (exponent < 1) with the concentration of target species.
rical exponential fall of the transient concentration: t c(t) = c1 + (c2 − c1 ) exp − τ V with the same time constant τ = f
(2)
One weak point of the model is certainly the assumption of a homogeneous transient concentration of target species in the test cell. Obviously, a concentration gradient occurs between the inlet and the outlet of the fluid flow. However, any other model, taking into account this concentration gradient as well as diffusion and convection processes in the cell, etc., would be far less simple, and in our opinion will not bring much more to our demonstration. 2.3.2. Application of the well-stirred tank model to linear sensors The theoretical curves reflecting the extrinsic response and recovery of linear sensors will be homothetic to those of the
transient concentration “seen” by the sensor, which follows Eqs. (1) and (2). They will be perfectly symmetrical (Fig. 3). In many practical cases where the intrinsic response/recovery times are negligible versus the extrinsic ones, the experimental response and recovery curves should also be symmetrical. 2.3.3. Application of the well-stirred model to non-linear sensors following a power law (exponent < 1) This category of sensors includes semi-conducting oxides, such as tin oxide, for which the steady-state conductance varies with the concentration of reducing gas, according to a more or less empirical power law with an exponent usually included between 0.2 and 0.9. Fig. 4 shows the theoretical extrinsic response and recovery of such sensors, obtained by applying a power law to the transient concentration of formula (1) and (2) with c1 = 0, and by normalizing the resulting values. The most striking feature of the curves is the asymmetry between the response and the recovery. The smaller the
Fig. 5. Theoretical transient signal of sensors following a logarithmic law with the concentration of target species.
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Table 1 Computed response and recovery times at 90% of the steady-state signal for various types of sensors Linear law
Power law
Logarithmic law
n = 0.8
n = 0.5
c2 /c1 = 102
n = 0.2
c2 /c1 = 104
c2 /c1 = 106
tresp (τ)
trec (τ)
tresp (τ)
trec (τ)
tresp (τ)
trec (τ)
tresp (τ)
trec (τ)
tresp (τ)
trec (τ)
tresp (τ)
trec (τ)
tresp (τ)
trec (τ)
2.30
2.30
2.09
2.88
1.66
4.60
0.89
11.50
0.99
5.13
0.51
8.80
0.29
12.72
The time unit is the time constant τ = V/f.
exponent, the faster the response and the slower the recovery. This observation seems of primary importance, since it enables to explain quite rationally numerous experimental curves reported in the literature for semi-conducting oxides. It may be worthwhile noting that for a hypothetical nonlinear sensor following a power law with an exponent larger than one, the extrinsic recovery time would be shorter than the response time! 2.3.4. Application of the well-stirred tank model to non-linear sensors following a logarithmic law This category of sensors is especially important, since it includes all sensors following the Nernst law: ISE, ISFET’s, potentiometric gas sensors using solid electrolytes. Pd–MOS structures for hydrogen sensing, which have been shown to display a logarithmic dependency over at least 10 decades [8] also belong to this category. Because of the characteristics of the logarithm function, the initial concentration c1 cannot be taken as zero any longer, conversely to the previous case. Three cases will be considered: c2 /c1 = 102 , c2 /c1 = 104 and c2 /c1 = 106 . Fig. 5 shows the theoretical extrinsic response and recovery of such sensors, obtained by applying the logarithm function to the transient concentration of formula (1) and (2) and by normalizing the resulting values. Again, the striking feature of the curves is the asymmetry between the response and the recovery. The larger the c2 /c1 ratio, the faster the response and the slower the recovery. This observation again enables to explain rationally many numerous experimental curves reported in the literature for these type of sensors. 2.3.5. Computation of extrinsic response/recovery times at 90% on the basis of the well-stirred tank model In order to get a more quantified view of the effect, the computation of some response and recovery times at 90% of the stationary state signals have been computed. Starting from formulas (1) and (2) with c1 = 0, the transient sensor signal was taken as c/c2 for linear sensors and as (c/c2 )n for sensors following a power law. For sensors governed by a logarithmic law and for which c1 cannot be zero, the signal was taken as 1 − ln[c/c2 ]/ln[c1 /c2 ]. These formulas ensure the normalization conditions, i.e. a signal equal to zero for c1 and 100% for c2 . The extrinsic response and recovery times at 90% are then given by: • for linear sensors: tresp = trec = −τ ln[0.1]
(3)
• for non-linear sensors following a power law with exponent n: 1 ln[0.1] tresp = −τ ln[1 − 0.91/n ] and trec = −τ n (4) • for non-linear sensors following a logarithmic law with k = c1 /c2 : 0.9 (1 − k0.1 ) k −k tresp = −τ ln and trec = −τ ln 1−k 1−k (5) The so calculated response and recovery times at 90% are reported in Table 1, as a function of the time constant τ = V/f. At this stage, it is interesting to note that τ corresponds to a virtual rinsing time of the cell, during which the inlet fluid would push away all the fluid inside without mixing with it. In the more realistic well-stirred tank model with an exponential evolution of the transient concentration, the rinsing process is achieved at 90% only after 2.3τ. This value is also that of the response and recovery times at 90% for linear sensors. For non-linear sensors governed either by a logarithmic law or by a power law with an exponent smaller than one, the response time is smaller than for linear sensors, whereas the recovery time is longer. As noticed previously, the asymmetry between response and recovery times is all the more marked that the exponent is smaller for power law sensors, or the concentration change c2 /c1 larger for sensors governed by a logarithmic law. Note the recovery time/response time ratio of 13 for a sensor following a power law with an exponent equal to 0.2, and of 44 for a sensor governed by a logarithmic law with c2 /c1 = 106 .
3. Experimental validation Various linear and non-linear sensors will be tested in order to validate the above model, both with gases and with liquids. The experimental validation supposes that the intrinsic response/recovery times remain as far as possible negligible versus the extrinsic ones. 3.1. Experimental set-up 3.1.1. Selection of sensors for the validation of the model Commercial sensors were preferred to laboratory-made sensors. One reason is a usually better reliability. Another is
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sponse/recovery times are required to be shorter than 100 ms • a zirconia lambda gauge also made by Bosch but working on a potentiometric principle. The application is identical, with similar requirements for response/recovery times • a tin-oxide-based sensor manufactured by Figaro (Japan). Response/recovery times are unfortunately not reported in the technical notices, but our own experience with tin oxide sensors make us believe that the intrinsic times are certainly no longer than 1 s. Two liquid sensors were selected: Fig. 6. Experimental set-up for validating the well-stirred tank model in the case of liquid sensors.
that the technical notices of commercial sensors, in principle give some indications about response and recovery times. As discussed previously, these experimentally measured times necessarily include some extrinsic part due to the delivery system, but the test conditions are more likely to be optimised to minimize it than in a laboratory. We considered these values as maximum intrinsic times. The selected gas sensors were the following: • a catalytic sensor (Pellistor) commercialized by Marconi Applied Technologies Ltd. (England), with a reported response time of 2 s • a thermal conductivity sensor made by Microsens (Switzerland). The reported gas exchange time constant is 100 ms • a zirconia proportional gauge made by Bosch GmbH (Germany), working on an amperometric principle. The application is engine control and re-
• a Clark-type electrode for dissolved oxygen operating on a amperometric principle; • a glass pH electrode operating on a potentiometric principle. Since we did not have any precise information about the response/recovery times of these electrodes, we determined a maximum intrinsic response/recovery time by plunging quickly by hand the electrodes from one solution to another and agitating vigorously at the same time. In the case of the pH electrode, the ratio c2 /c1 was chosen small (about 5) in order to minimize extrinsic effects due to the unavoidable liquid film or drop carried with the electrode when moving it from one solution to the other. Times required to get stable stationary state values were never longer than 3 s for both electrodes. 3.1.2. Test cells and delivery systems The experimental validation supposes that the values mentioned in the previous section, considered as maximum in-
Fig. 7. Stationary state signal values of the catalytic sensor as a function of methane concentration.
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Fig. 8. Experimental transient signals of the catalytic sensor exposed to 1% methane in dry air at various flow rates.
trinsic times, remain as far as possible negligible versus the extrinsic ones. Both tested liquid sensors are operated at room temperature and it was quite easy to design an experimental set-up resembling the well-stirred tank model (Fig. 6). The tank volume is 200 cm3 and the flow rate fixed by a peristaltic pump at 130 cm3 /min. The time constant V/f is thus equal to 92 s, i.e. about 30 times more than the maximum intrinsic time.
As for gas sensors, they are operated above room temperature; and a new cell design, which also requires gas tightness, is not so easy. Consequently, we decided to re-use our regular test cells. The cell used for testing semiconductor oxide and catalytic sensors has a volume of 30 cm3 , and that used for testing potentiometric, amperometric and thermal conductivity sensors a volume of 165 cm3 . Initial experiments were carried out directly with these cells. Even if the main
Fig. 9. Experimental and theoretical transient signals of the catalytic sensor exposed to 1% methane in dry air at a flow rate of 500 cm3 /min.
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Fig. 10. Stationary state signal values of the thermal conductivity sensor as a function of carbon dioxide concentration.
Fig. 11. Experimental and theoretical transient signals of the thermal conductivity sensor exposed to a carbon dioxide concentration change from 0–100% in dry air.
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Fig. 12. Stationary state signal values of the amperometric zirconia sensor as a function of oxygen concentration in nitrogen.
Fig. 13. Experimental and theoretical transient signals of the amperometric zirconia sensor exposed to an oxygen concentration change from 200–200,000 ppm in nitrogen.
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trends observed in these conditions qualitatively agreed with the theoretical expectations, a set-up in which the test cell is located downstream of a larger “buffer” volume of 320 cm3 was finally preferred. Such a design yields a total volume of either 350 or 485 cm3 . This ensures larger time constants and practically negligible intrinsic times versus extrinsic ones. In the most unfavourable case, the flow rate was 900 cm3 /min, which yields a time constant of 23 s with the 30 cm3 cell, i.e. more than 10 times the maximum intrinsic time of 2 s. For all gas sensors studied in these conditions with flow rates ranging from 100–900 cm3 /min, the experimental response/recovery curves depend on the flow rate, in agreement with the wellstirred tank model. Moreover, for a given sensor, the curves can practically be superimposed by modifying the time scale proportionally to the flow rate. This demonstrates clearly that the intrinsic times are negligible versus the extrinsic ones. The counterpart of this experimental set-up with two tanks in series is that it does not stick so well to the simple one tank model of Fig. 2. A set-up similar to that of Fig. 6 for liquids, with only one large tank and a fan inside would certainly have been preferable. But as stated in the introduction, the aim of the paper is to demonstrate the reality of the trends evaluated in Section 2, and not to fit perfectly theoretical curves on the basis of a model, which is only approximate anyhow. 3.2. Model validation with gas sensors 3.2.1. Linear sensors Tests with a catalytic sensor, a thermal conductivity sensor and an amperometric sensor will be reported in this section. The linearity of each sensor will first be checked.
3.2.1.1. Catalytic sensor. Fig. 7 shows the relatively good linearity of the stationary state signal with methane concentration. In Fig. 8 are reported the normalized transient response/recovery curves to 1% methane in dry air, for flow rates ranging from 100–900 cm3 /min. The normalization suppresses the slight dependency of the stationary state values with the flow rate and allows better comparisons. As expected, the response/recovery time decreases with increasing flow rate. Moreover, recovery curves appear symmetrical to response curves, in agreement with Section 2.3.2. Fig. 9 shows the fit to the well-stirred one tank model with a total volume of 320 + 30 = 350 cm3 , in the case of a flow rate equal to 500 cm3 /min. For the sake of simplicity, only the curves corresponding to one flow rate are presented. As stated in Section 3.1.2, all curve shapes are quite similar when the time scale is changed proportionally to the inverse of the time constant and this is also true for the sensors discussed hereafter. For this reason and in order to avoid numerous unnecessary curves, only results corresponding to a flow rate of 500 cm3 /min, shown on the same time scale (0–1200 s), will be presented in the following. The comparison between experimental and theoretical curves shown in Fig. 9 indicates that the well-stirred one tank model is not a too bad approximation, even for an experimental set-up with a small tank in series with a larger buffer tank. Anyhow, the departure of the experimental curve from the theoretical one does not affect the excellent symmetry between the recovery and the response. 3.2.1.2. Thermal conductivity sensor. The thermal conductivity sensor was tested with carbon dioxide in dry air. Sta-
Fig. 14. Stationary state e.m.f of the potentiometric zirconia sensor as a function of oxygen concentration in nitrogen.
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Fig. 15. Experimental and theoretical transient signals of the potentiometric zirconia sensor exposed to an oxygen concentration change from 200–200,000 ppm in nitrogen.
Fig. 16. Comparison of the experimental transient signals of the potentiometric and amperometric zirconia sensors tested in exactly the same conditions.
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Fig. 17. Experimental transient signals of the potentiometric zirconia sensor exposed to three concentration changes: 20–200,000; 200–200,000 and 2000–200,000 ppm oxygen in nitrogen.
tionary state values are reported in Fig. 10. In agreement with the theoretical behaviour [11], the experimental points are not perfectly aligned. However, taking into account the approximations of the well-stirred tank model and of the experimental set-up versus the model, this slight non-linearity will be neglected. Fig. 11 reports the experimental transient
response/recovery curve to 100% carbon dioxide in dry air for a flow rate of 500 cm3 /min. The curve is compared to that of the well-stirred tank model for a total volume of 320 + 165 = 485 cm3 . The quality of the agreement is satisfying, and the recovery again appears basically symmetrical to the response.
Fig. 18. Stationary state signal values of the semiconductor oxide sensor as a function of methane concentration in dry air.
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Fig. 19. Experimental and theoretical transient signals of the semi-conductor oxide sensor exposed to 1% methane in dry air.
3.2.1.3. Amperometric sensor. The amperometric sensor was tested with oxygen in nitrogen in the same cell as that used for the thermal conductivity sensor, i.e. with a total volume of 485 cm3 . The linearity of the amperometric sensor is excellent in the range shown in Fig. 12. No measurement was carried out between 5000 and 200,000 ppm. The signal measured for 200,000 ppm was 6.5 V, which does not depart much
from the value of 6.1 V, extrapolated from Fig. 12. For our purpose, the amperometric sensor will be considered as linear. Fig. 13 reports the experimental transient response/recovery curve to concentrations changes from 200 to 200,000 ppm oxygen in nitrogen for a total flow rate of 500 cm3 /min. Even if the agreement with the well-stirred tank model is only approximate, the symmetry remains rather satisfying.
Fig. 20. Experimental transient signals of the amperometric electrode for various concentration changes of dissolved oxygen in water.
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3.2.2. Non-linear sensor following a logarithmic law The potentiometric oxygen gauge has an external size similar to the amperometric one, i.e. that of a sparking plug. Moreover, both sensors are operated at around 700 ◦ C. In both devices, the oxygen ion conductor is yttrium-stabilized zirconia. Comparisons between both sensors will be all the more valuable that all experimental conditions are identical: same test device, same oxygen concentration ratios, and same flow rates. Fig. 14 shows the well-known logarithmic dependency of the lambda gauge with a slope of about 46 mV per concentration decade, in perfect agreement with the theoretical value for an operating temperature of 700 ◦ C. Fig. 15 shows the transient response/recovery curve to concentrations changes from 200 to 200,000 ppm oxygen in nitrogen for a 500 cm3 /min flow rate. Even though the fit to the wellstirred model is not satisfying for the recovery, the asymmetry between the response and the recovery is not questionable. The plot on the same figure of the transient signals of the amperometric and the potentiometric sensors magnifies this result (Fig. 16). The observation of a response time shorter for the potentiometric sensor than for the amperometric one, also agrees well with Table 1, at least qualitatively. It is worth noting that a reduction of only 20% of the time constant, leads to a strong improvement of the fit of the experimental curve for the potentiometric sensor and to a lesser extent for the amperometric one. A decrease of the time constant corresponds to a reduction of the effec-
tive total volume in the same proportion. Even if such a reduction is not meaningless, a precise interpretation remains, however, hazardous and without any real interest for our purpose. Another interesting feature of the well-stirred tank model applied to potentiometric sensors is the theoretical increase of the asymmetry with the concentration ratio c2 /c1 . Fig. 17 shows the experimental comparison for three values of this ratio: 102 , 103 and 104 . Again, the results are in qualitative agreement with the model. Not only the recovery time increases when the ratio increases, but the response time appears slightly shorter for c2 /c1 = 103 and c2 /c1 = 104 than for c2 /c1 = 102 (the distinction between 103 and 104 is not visible). The experimental ratios trec /tresp at 90% are 4.0, 6.2 and 9.0 for c2 /c1 = 102 , 103 and 104 , respectively, whereas the values computed with formula (5) are 5.1, 9.9 and 17.0. The differences between experimental and theoretical values are certainly significant, but the trend and the orders of magnitude are there. 3.2.3. Non-linear sensor following a power law For this category of sensor, we used a classical tin oxide sensor. The exponent of the power law determined from the stationary state signals for various methane concentrations (Fig. 18) was unfortunately higher than expected (0.82). In spite of this value relatively close to unity, the asymmetry of the response/recovery curve is slightly visible (Fig. 19).
Fig. 21. Experimental and theoretical transient signals of the amperometric electrode exposed to a concentration change from 0.4–8 mg/l of dissolved oxygen in water.
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Fig. 22. Stationary state emf values of the potentiometric electrode as a function of the pH of a chloric acid solution.
Fig. 23. Experimental transient signals of the potentiometric glass electrode exposed to three pH concentration changes: from 5 to 1, 5 to 2 and 5 to 3.
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3.3. Model validation with liquid sensors 3.3.1. Linear amperometric sensor A Clark-type electrode for oxygen dissolved in water was used for this type of sensor. Fig. 20 shows the transient responses for various concentration changes. The stationary state values indicate an approximately linear behaviour at least in the studied range of 0.2–8 mg/l of dissolved oxygen. Moreover, the three curves appear roughly symmetrical. The fit to the well-stirred tank model for a concentration change from 0.4 to 8 mg/l is not too bad (Fig. 21). 3.3.2. Non-linear sensor following a logarithmic law A glass pH electrode was used as representative of potentiometric sensors. Fig. 22 shows the characteristic linear dependence of the emf on the pH of the HCl solution. Three concentration ratios c2 /c1 were studied: 102 , 103 and 104 corresponding to pH changes from 3–5, 2–5 and 1–5, respectively. The corresponding transient normalized signals are shown in Fig. 23. The asymmetry of the curves is not questionable, even if the fits (not shown) to the well-stirred model are not excellent, especially for the recovery. As expected from the model and as observed for the potentiometric gaseous oxygen sensor, the larger the c2 /c1 ratio, the shorter the response time and the longer the recovery time. It would have been interesting to compare directly on the same figure the normalized responses of both oxygen and pH electrodes, as was done in Fig. 16 for the amperometric and potentiometric gaseous oxygen sensors. However, the target species are different and, moreover, we were unable to get a concentration ratio c2 /c1 larger than about 20 for dissolved oxygen. An extrapolation of the curves in Fig. 20 may let suppose that a ratio of 100 instead of 20, would not modify the basic symmetry of the transient signal, in which case the comparison with the curve of Fig. 23 corresponding to a pH change from 5 to 3, would be meaningful.
4. Application of the model to practical cases The experimental validation of the model was on purpose carried out with large volumes, in order that the intrinsic times remain negligible versus the extrinsic ones. The aim of the present section is to discuss in the light of the above results, to which extent the model can be applied to practical cases. After examining the real meaning of experimentally recorded response/recovery times, the case of microsensors with delivery systems based on microfluidic and that of car exhaust sensors requiring response/recovery times of the order of a few millisecond will be briefly dealt with.
on the basis of intrinsic properties of the sensor, especially interaction kinetics with the target species. In the light of the above results, such an assertion is obviously meaningless if the extrinsic response/recovery times are not taken into account. In fitting experimentally measured response/recovery times of semiconductor oxide sensors to a theoretical model, Pilling et al. [6] incorporated formula (1) and (2), characteristics of the delivery system (what is called here extrinsic) to surface reaction kinetics, characteristic of the sensor (what is called here intrinsic). The theoretical treatment of such a global approach, is however, not straightforward. A simpler but more restricted approach can be carried out as follows. The knowledge of the dependency of the sensor stationary state signal with the concentration of target species and the knowledge of the experimental test conditions (cell volume and flow rate), allow a very easy estimation of the extrinsic response and recovery times, just by looking at Table 1 or by applying formula (3), (4) or (5). If the estimated values are significantly shorter than the experimentally recorded times, then one may safely conclude that the latter effectively correspond to the intrinsic times of the sensor. This type of approach, rarely found in the literature, was done by Herne et al. [5] in the study of the response time of optical sensors based on luminescence quenching. As suggested by Professor Ingemar Lundstr¨om (Link¨oping University, Sweden), another simple approach to know whether the experimental response and recovery times correspond to extrinsic or intrinsic times, is to study the dependency on the sensor operating temperature. If the response/recovery times are temperature dependent, it is clear that they correspond to intrinsic times, since the model proposed in the present paper for extrinsic times is basically independent of the sensor operating temperature. 4.2. Microsensors and microfluidic At a time where sophisticated technologies allow drastic miniaturizations of sensing devices, one might wonder whether the model also applies to microsensors with delivery systems based on microfluidics. A decrease of test cell volumes is often accompanied by a decrease in similar proportions of the section of inlet and outlet tubes. The flow rates are smaller and the time constants may not differ so much from those of more traditional experimental set-up. Even in cases for which the time constant is strongly reduced, Fig. 8 suggests that the shape of the curves will remain basically unchanged by dilating the time scale proportionally to the inverse of the time constant. 4.3. Car exhaust sensors
4.1. Meaning of the experimental response/recovery times As stated in the Section 1, the asymmetry between the response and recovery curves is often explained in the literature
Modern engines already require closed loop lambda control within a few tens of milliseconds, and future engines will require control times shorter than 10 ms. Flow rates in the exhaust pipe are of the order of 20 l/sec. Assuming a section of
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the exhaust pipe of about 30 cm2 and a step function change of the oxygen concentration after crossing the lambda point, the time required to rinse a length of pipe of 1 cm (which is the order of magnitude of the size of a sensing element) is 1.5 ms. Thus, a first limitation to response/recovery times of the order of the milliseconds arises from the size of the sensing element. In our opinion, there should be a second limitation, independent of the sensor size. This limitation is related to the fact that the oxygen concentration after crossing the lambda point is unlikely to follow a perfect step function change in the exhaust pipe. The concentration probably includes a time-dependent component, more or less exponential. Since the oxygen concentration decreases by several orders of magnitude when crossing the lambda point from the lean to the rich-burn region, the requirement of very short recovery times should be more difficult to meet for non-linear devices (especially those following logarithmic laws) than for linear ones.
5. Conclusion The starting point of the mechanisms presented here to account for the response and recovery curves of chemical sensors is to distinguish an intrinsic response/recovery time, independent of experimental conditions and supposing a step function change of the target species concentration surrounding the sensor, from an extrinsic time, depending especially on the fluid flow and on the test cell volume. The extrinsic time is linked to the transient concentration at the sensor level. The time dependency of the transient concentration is assumed to be exponential, according to the simple well-stirred tank model. Then the application of the law governing the sensor signal to the transient concentration, yields either symmetrical or asymmetrical response/recovery curves, whether the sensor is linear or not. All major trends of the model have been verified experimentally, with sensors based on various linear and not linear principles and with target species of various types. Finally,
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the model was shown to have major implications in many practical cases, including microsensors and microfluidic or car exhaust sensors.
Acknowledgements We are especially grateful to Professor Ingemar Lundstr¨om (University of Link¨oping, Sweden), for his interest in the original model developed in the present manuscript and for highly relevant suggestions.
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