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Physical description of the principle of an SO2 sensor
Sensors
and Actuators B 24-25 (1995) 400-402
Physical description of the principle of an SO2 sensor M. Horn, J. Pichlmaier, H.-R. Trgnkler Instihtt filr Me& und Aufomatisienmgstechn~
Universittitder Bundeweht Milnchen, D-85577 Neubiberg, Germany
Abstract The capacity of coated interdigital capacitors (IDCs) is related to the SO* concentration in the ambient air. The amount of gas adsorbed on the coating determines the change of its dielectric properties. SOz molecules are adsorbed in a two-step process in two states. The temperature dependence is completely determined by the equilibrium between adsorption and desorption in the first step of this two-step process. The shape of the sensor characteristic is strongly non-linear. It is determined by the quotient of the number of available sites for SO, molecules on the sensor surface and the number of initially existing nuclei, which cause a formation
of the molecules
necessary
for the transition
between
state 1 and state 2.
Keywordr: Gas sensors; Sulfur dioxide sensors; Modelling; Signal processing
1. Introduction SO, concentrations in the ambient air can be measured with interdigital capacitors (IDCs) (Fig. 1) coated with NNDE/?TMS (N,N-diethyl-3-aminopropyl-trimethoxysilane/n-propyltrimethoxysilane) [l]. NNDE mole&es serve as adsorption centres for SO, molecules. The cross-sensitivity to humidity was optimized by cocondensation with PTMS. Since the characteristics of these sensors are strongly non-linear and temperature dependent (see Fig. 3), the measured values have to be corrected by means of signal processing. The characteristics correspond qualitatively to the type V (BET) isotherm according to Ref. [2). Qaantitatively, however, they neither match the BET isotherms nor any other known adsorption isotherms [3,4]. For these reasons a physical description of the functional principle of the sensor has been developed, which
Fig. 1. IDC gas sensor. 0925~4005/95/$09.50 8 1995 Elsevier Science S.A. All rights reserved SSDI 0925-4005(95)01737-G
fundamentally results from the behaviour of the gas molecules that are adsorbed on the sensor surface.
2. CalculaEion of temperature dependence and activation energy Here, the adsorption of the SO, molecules at the sensor surface is described as a two-step process. The first step is adsorption in state Sl; the second step is adduction in state S2, as shown in Fig. 2. In the first step the gas is adsorbed reversibly on the surface in a state 1 (Sl) proportionally to its ambient concentration x. The number z of adsorbed molecules results from the equilibrium between the input stream ~iil and the output stream @,,1for state Sl.
Fig. 2. Potential energy in states Sl and S2.
M. Horn et al. I Sensors and Achxators b, Chemical 24-25 (1995) &MO2
According to Eq. (1) below, @ii is a weighted sum of the stream of molecules impacting on the surface from the gas space, according to the kinetic gas theory, and the thermally excited stream from S2 to Sl. @rl=ai $
+&n
exp( -E&T)
@0x=d,z exp( -E,lkl’J +d,,z exp( -E,JkT)
(2)
In the equilibrium state $,i and Gu have to be equal, and so Eq. (3) for z can be deduced: ‘=a’ 2
[d, exp(-E,lkQ+Id,,
exp(-E,JkT)]
n exp(-EJkT) +dzl [d, exp( -E,/kT) +dlz exp( -E,,/kT)]
(3)
As we shall see below, the activation energy EA is about 0.7 eV. Considering the.fact that the temperature during the measurements had a value of about 300 K, and making the assumption that El2 approximately equals E,, >E,+O.l eV, it can be stated that exp( -E,,lkT)=exp(
estimate the activation energy E, from k h&/x,) + 5 hQ’,IT,) EA=
1 --Tz
(1)
@..1consists of the stream of thermally excited molecules that are desorbed again to the gas space and the stream of molecules that moves from Sl to S2. It is given by
-E,,IkT)~eexp(
-E,IkT)
401
(6)
1 Tl
For the example shown in Fig. 3, EA was estimated as 0.716 eV.
3. Calculation of the characteristic In the second step some of the adsorbed molecules are adduced to the NNDEPTMS layer, which means that they are moving from state Sl to S2. The input and output streams for S2 are given by %=dnz
exp( -E&“)
(7)
%=&n
exp(-&JO
(8)
The parameters d12 and dZ1 describe the probability for a transition between Sl and S2. As a result of a closer inspection, the parameter d,, shows a dependence on n. For the calculation of n =n(z) the balance of the differential streams dQo,, and daii, can be considered. Using the simplified total differential of $2, the balance of differential streams leads to
(4)
(9)
and so Eq. (3) can be simplified to
(5) If we choose the number z of molecules in the state Sl as a new variable for the abscissa (Fig. 4), we obtain a single characteristic, which is independent of the temperature for the same measured points as those in Fig. 3. This means that two measured points at different temperatures but equal values of z lead to the same value of the resulting capacity! Using this fact we can
For small differences between Ezl and El2 the dependence of Eq. (9) on small changes in temperature is low. This is in accordance with the temperature independence of the sensor characteristic shown in Fig. 4. The transition from Sl to S2 is only probable if the molecules are in a correct orientation and position, i.e., that they are ‘formed’, and at the same time free sites for the molecules exist in state S2. 38
T
A ?$ 36 8 B 0 34 b : f 32 30
28
28
26
26 24
24 0
1000
2000
3000
4000 5000 voiume concentration
6000 7000 x [ppm]
Fig. 3. Sensor characteristics with the absolute temperature parameter.
as the
0
0.1
0.2
0.3
04
05
0.6
07
06 zN
0.9
IO
[WI. Units]
Fig. 4. Plot of measured capacity vs. the normalized number zN and characteristic according to Eq. (13) (solid line).
402
M. Horn et al. I Sensors and Actuarom b, Chemical 24-25 (1995) 400-#02
The progressive gradient for low values of z can be interpreted as an ‘exponential growth’ on nuclei or already adduced molecules that have the same effect as nuclei. ‘Exponential growth’ means that the number of nuclei, which cause the formation, increases exponentially. The number of adduced molecules n and nuclei n, is (nO+n). If N is the maximum number of free sites in state S2, the number of free sites is (N-n), which causes a saturation effect in accordance with the decreasing gradient for high z values in Fig. 4. The transition probability is given by the probability of finding both a nucleus and a free site. Therefore the parameter d,, can be written as a proportional function of (N-n)(n,+n), and so Eq. (9) results in the expression dn z
dV+no)
“P C=C,ck,N
exp(EAWx
_
1
JT
1
[
exp(Wkr)x rT
1no + _N
(14) The accuracy of this model is better than the relative hysteresis error of the IDC sensor of 1.5% which can be seen in Fig. 3.
4. Conclusions =b(N-n)(n,+n)
The proportional constant d,, and the nearly constant Boltzmann factor of Eq. (9) are combined in the constant b. The differential Eq. (10) as the main equation of the process describes the adduction of n molecules on maximum N free sites. Additionally, the process is influenced by the number n, of nuclei and the number n of already adduced molecules. The solution of the differential Eq. (10) is n=N
With Eq. (5) the complete model can be written as
ev[b(N+no)tl-l exp[b(N+n&]
hr
(11)
+ fl 4
Acknowledgement
The dielectric constant of the NNDEPTMS layer and with it the capacity of the IDC sensor changes as a linear combination of the number of adsorbed and adduced SO, molecules according to
C=C,+kc,z+kc,n(z)
(12)
The transfer coefficients k, and ka represent the polarizability of molecules in Sl and S2. Considering Eq. (ll), we find the characteristic described by C=C,,+k,,z+k,N
The parameters of the presented model, which are based on elementary physical assumptions, can be physically interpreted. By an analysis of experimental data on the basis of the described model, it is possible to estimate the maximum number of free sites for adducts N, the number of nuclei n, and the number of molecules adsorbed in state Sl when state S2 is saturated. The presented model is expected to be important both for future modelling of cross-sensitivities and in the field of investigations on reliability and aging of SO, sensors with NNDE/PTMS coating.
exp[b(N +n&] - 1 (13)
exp[W+d4 + % which is shown in Fig. 4 as the solid line. The abscissa is the normalized number of molecules zN in state Sl.
This work was supported by the Bundesministerium fiir Forschung und Technologie by contract numbers 13 AS 01089 and 13 MV 0156.
References 111J. Lm, S. MBller and E. Obermeier, Thin-film gas sensors with organically modified silicates for the measurement of SO2, Sensors and Acluafors B, 5 (1991) 219-221. PI S. Bnmauer, L.S. Deming, W.E. Deming and E. Teller, On a theory of the van der Waals adsorption of gases, 1. Am. Chem SK, 62 (1940) 1723-1732. [31 D.G. Kinniburgh, General purpose adsorption isotherms, Environ. Sci. Technol., 20 (1986) 895-904. [41 S.-L. Kuo and A.L. Hines, Adsorption of l,l,l-trichlorethane and tetrachlorethylene on silica gel, 1. Chem. Eng. Data, 37 (1992) l-3.
and Actuators B 24-25 (1995) 400-402
Physical description of the principle of an SO2 sensor M. Horn, J. Pichlmaier, H.-R. Trgnkler Instihtt filr Me& und Aufomatisienmgstechn~
Universittitder Bundeweht Milnchen, D-85577 Neubiberg, Germany
Abstract The capacity of coated interdigital capacitors (IDCs) is related to the SO* concentration in the ambient air. The amount of gas adsorbed on the coating determines the change of its dielectric properties. SOz molecules are adsorbed in a two-step process in two states. The temperature dependence is completely determined by the equilibrium between adsorption and desorption in the first step of this two-step process. The shape of the sensor characteristic is strongly non-linear. It is determined by the quotient of the number of available sites for SO, molecules on the sensor surface and the number of initially existing nuclei, which cause a formation
of the molecules
necessary
for the transition
between
state 1 and state 2.
Keywordr: Gas sensors; Sulfur dioxide sensors; Modelling; Signal processing
1. Introduction SO, concentrations in the ambient air can be measured with interdigital capacitors (IDCs) (Fig. 1) coated with NNDE/?TMS (N,N-diethyl-3-aminopropyl-trimethoxysilane/n-propyltrimethoxysilane) [l]. NNDE mole&es serve as adsorption centres for SO, molecules. The cross-sensitivity to humidity was optimized by cocondensation with PTMS. Since the characteristics of these sensors are strongly non-linear and temperature dependent (see Fig. 3), the measured values have to be corrected by means of signal processing. The characteristics correspond qualitatively to the type V (BET) isotherm according to Ref. [2). Qaantitatively, however, they neither match the BET isotherms nor any other known adsorption isotherms [3,4]. For these reasons a physical description of the functional principle of the sensor has been developed, which
Fig. 1. IDC gas sensor. 0925~4005/95/$09.50 8 1995 Elsevier Science S.A. All rights reserved SSDI 0925-4005(95)01737-G
fundamentally results from the behaviour of the gas molecules that are adsorbed on the sensor surface.
2. CalculaEion of temperature dependence and activation energy Here, the adsorption of the SO, molecules at the sensor surface is described as a two-step process. The first step is adsorption in state Sl; the second step is adduction in state S2, as shown in Fig. 2. In the first step the gas is adsorbed reversibly on the surface in a state 1 (Sl) proportionally to its ambient concentration x. The number z of adsorbed molecules results from the equilibrium between the input stream ~iil and the output stream @,,1for state Sl.
Fig. 2. Potential energy in states Sl and S2.
M. Horn et al. I Sensors and Achxators b, Chemical 24-25 (1995) &MO2
According to Eq. (1) below, @ii is a weighted sum of the stream of molecules impacting on the surface from the gas space, according to the kinetic gas theory, and the thermally excited stream from S2 to Sl. @rl=ai $
+&n
exp( -E&T)
@0x=d,z exp( -E,lkl’J +d,,z exp( -E,JkT)
(2)
In the equilibrium state $,i and Gu have to be equal, and so Eq. (3) for z can be deduced: ‘=a’ 2
[d, exp(-E,lkQ+Id,,
exp(-E,JkT)]
n exp(-EJkT) +dzl [d, exp( -E,/kT) +dlz exp( -E,,/kT)]
(3)
As we shall see below, the activation energy EA is about 0.7 eV. Considering the.fact that the temperature during the measurements had a value of about 300 K, and making the assumption that El2 approximately equals E,, >E,+O.l eV, it can be stated that exp( -E,,lkT)=exp(
estimate the activation energy E, from k h&/x,) + 5 hQ’,IT,) EA=
1 --Tz
(1)
@..1consists of the stream of thermally excited molecules that are desorbed again to the gas space and the stream of molecules that moves from Sl to S2. It is given by
-E,,IkT)~eexp(
-E,IkT)
401
(6)
1 Tl
For the example shown in Fig. 3, EA was estimated as 0.716 eV.
3. Calculation of the characteristic In the second step some of the adsorbed molecules are adduced to the NNDEPTMS layer, which means that they are moving from state Sl to S2. The input and output streams for S2 are given by %=dnz
exp( -E&“)
(7)
%=&n
exp(-&JO
(8)
The parameters d12 and dZ1 describe the probability for a transition between Sl and S2. As a result of a closer inspection, the parameter d,, shows a dependence on n. For the calculation of n =n(z) the balance of the differential streams dQo,, and daii, can be considered. Using the simplified total differential of $2, the balance of differential streams leads to
(4)
(9)
and so Eq. (3) can be simplified to
(5) If we choose the number z of molecules in the state Sl as a new variable for the abscissa (Fig. 4), we obtain a single characteristic, which is independent of the temperature for the same measured points as those in Fig. 3. This means that two measured points at different temperatures but equal values of z lead to the same value of the resulting capacity! Using this fact we can
For small differences between Ezl and El2 the dependence of Eq. (9) on small changes in temperature is low. This is in accordance with the temperature independence of the sensor characteristic shown in Fig. 4. The transition from Sl to S2 is only probable if the molecules are in a correct orientation and position, i.e., that they are ‘formed’, and at the same time free sites for the molecules exist in state S2. 38
T
A ?$ 36 8 B 0 34 b : f 32 30
28
28
26
26 24
24 0
1000
2000
3000
4000 5000 voiume concentration
6000 7000 x [ppm]
Fig. 3. Sensor characteristics with the absolute temperature parameter.
as the
0
0.1
0.2
0.3
04
05
0.6
07
06 zN
0.9
IO
[WI. Units]
Fig. 4. Plot of measured capacity vs. the normalized number zN and characteristic according to Eq. (13) (solid line).
402
M. Horn et al. I Sensors and Actuarom b, Chemical 24-25 (1995) 400-#02
The progressive gradient for low values of z can be interpreted as an ‘exponential growth’ on nuclei or already adduced molecules that have the same effect as nuclei. ‘Exponential growth’ means that the number of nuclei, which cause the formation, increases exponentially. The number of adduced molecules n and nuclei n, is (nO+n). If N is the maximum number of free sites in state S2, the number of free sites is (N-n), which causes a saturation effect in accordance with the decreasing gradient for high z values in Fig. 4. The transition probability is given by the probability of finding both a nucleus and a free site. Therefore the parameter d,, can be written as a proportional function of (N-n)(n,+n), and so Eq. (9) results in the expression dn z
dV+no)
“P C=C,ck,N
exp(EAWx
_
1
JT
1
[
exp(Wkr)x rT
1no + _N
(14) The accuracy of this model is better than the relative hysteresis error of the IDC sensor of 1.5% which can be seen in Fig. 3.
4. Conclusions =b(N-n)(n,+n)
The proportional constant d,, and the nearly constant Boltzmann factor of Eq. (9) are combined in the constant b. The differential Eq. (10) as the main equation of the process describes the adduction of n molecules on maximum N free sites. Additionally, the process is influenced by the number n, of nuclei and the number n of already adduced molecules. The solution of the differential Eq. (10) is n=N
With Eq. (5) the complete model can be written as
ev[b(N+no)tl-l exp[b(N+n&]
hr
(11)
+ fl 4
Acknowledgement
The dielectric constant of the NNDEPTMS layer and with it the capacity of the IDC sensor changes as a linear combination of the number of adsorbed and adduced SO, molecules according to
C=C,+kc,z+kc,n(z)
(12)
The transfer coefficients k, and ka represent the polarizability of molecules in Sl and S2. Considering Eq. (ll), we find the characteristic described by C=C,,+k,,z+k,N
The parameters of the presented model, which are based on elementary physical assumptions, can be physically interpreted. By an analysis of experimental data on the basis of the described model, it is possible to estimate the maximum number of free sites for adducts N, the number of nuclei n, and the number of molecules adsorbed in state Sl when state S2 is saturated. The presented model is expected to be important both for future modelling of cross-sensitivities and in the field of investigations on reliability and aging of SO, sensors with NNDE/PTMS coating.
exp[b(N +n&] - 1 (13)
exp[W+d4 + % which is shown in Fig. 4 as the solid line. The abscissa is the normalized number of molecules zN in state Sl.
This work was supported by the Bundesministerium fiir Forschung und Technologie by contract numbers 13 AS 01089 and 13 MV 0156.
References 111J. Lm, S. MBller and E. Obermeier, Thin-film gas sensors with organically modified silicates for the measurement of SO2, Sensors and Acluafors B, 5 (1991) 219-221. PI S. Bnmauer, L.S. Deming, W.E. Deming and E. Teller, On a theory of the van der Waals adsorption of gases, 1. Am. Chem SK, 62 (1940) 1723-1732. [31 D.G. Kinniburgh, General purpose adsorption isotherms, Environ. Sci. Technol., 20 (1986) 895-904. [41 S.-L. Kuo and A.L. Hines, Adsorption of l,l,l-trichlorethane and tetrachlorethylene on silica gel, 1. Chem. Eng. Data, 37 (1992) l-3.