Patent Grant
8000903
This invention relates to use of carbon nanotube networks as sensors of chemical substances.
Chemical sensors have been developed for decades now to detect gases and vapors at various concentration levels for deployment in a wide range of industrial applications. The detection usually centers on a change of a particular property or status of the sensing material (such as temperature, electrical, optical characteristics, etc.) upon exposure to the chemical species of interest. The selection of sensing material itself has spanned across the periodic table with a range of inorganic, semiconducting elements and organic compounds either in bulk or in thin film form. Perhaps the most widely investigated class of sensors is the high temperature metal oxide sensor, due to its high sensitivity with tin oxide as an example of sensor material. The most common SnO2 sensor platform is a chemiresistor wherein the transport characteristics of a conducting channel of tin oxide are modulated by the adsorption of chemical species at elevated temperatures (T≧350° C.). Other types of sensors include electrochemical cells, conducting polymer sensors, surface acoustic wave sensors and catalytic bead sensors.
While commercial sensors based on some of the above approaches are available, research continues on new sensing materials and transducer platforms for improved performance. Desirable attributes of next generation sensors include high sensitivity, in the parts per million (ppm) to parts per billion (ppb) range, low power consumption, room temperature operation, rapid response time, high selectivity and long term stability. Sensors based on the emerging nanotechnology promise to provide improved performance on all of the above metrics compared to the current micro and macro sensors. Nanomaterials exhibit small size, light weight, very high surface to volume ratio, increased chemical reactivity compared to bulk materials, and mechanical stability so that a sensing material can be refreshed or regenerated many times. All these properties are ideal for developing extremely sensitive chemical sensors.
Among the numerous nanomaterials, carbon nanotubes (CNTs) have received significant attention due to their unique electronic and extraordinary mechanical properties. Single-wall carbon nanotubes (SWCNTs) have an enormous surface area, as high as about 1600 m2/gm, which leads to an increased adsorptive capacity for gases and vapors. With all the atoms on the surface, SWCNTs are expected to exhibit a change in properties sensitively upon exposure to the environment. Indeed, electrical conductivity of SWCNTs has been shown to change reproducibly in the presence of gases such as NO2 and NH3. This revelation has resulted in the fabrication of SWCNT-based chemical sensors by several groups.
The principal platform for such sensors has been a nanotube field effect transistor (“CNT-FET”) with a single SWCNT serving as the conducting channel. This platform faces some serious difficulties for commercialization. First, the CNT-FET requires semiconducting SWCNTs for its operation, and selective growth of metallic versus semiconducting nanotubes is not possible today. Second, if an in situ chemical vapor deposition (“CVD”) process is used in the device fabrication sequence, it is hard to make a single SWCNT grow horizontally in order to bridge a given distance between the source and the drain. Alternatively, one is forced to ‘pick and place’ a semiconducting SWNT from bulk samples. Finally, the chemical sensor market is too cost sensitive to rely on complex steps involved in CNT-FET fabrication resulting in low sensor yield and poor reproducibility.
What is needed is an approach using suitably modified nanomaterials, such as SWCNTs, that can detect presence of certain gas components whose presence cannot be detected by any simple means. Preferably, the method should provide high sensitivity (detection of parts per million or parts per billion of the target gas), high selectivity, room temperature operation, low power consumption, high throughput and low cost. Preferably, the method(s) should extend to detection of other gas components with at most modest changes in the nanomaterial modification procedures. Preferably, this method should allow estimation of the effects of change of local temperature, change of local relative humidity and temporal drift (change of baseline and sensitivity with elapsed time).
These needs are met by the invention, which provides a chemical sensor or sensor array for detecting presence, at or near room temperature, of one or more of N target gas components or molecules (N≧1) in a gas mixture contained in a chamber, by any ambient being considered. The sensor contains a network of SWCNTs that is connected to a controllably variable voltage difference or current source. The chamber may be closed, isolated and static; or, preferably, may allow gas flow-through and thus not be wholly isolated from the external environment. Alternatively, the chamber may be part or all of the external environment.
In a first embodiment, the SWCNTs in the network are partly or wholly coated with a selected polymer, such as chlorosulfonated polyethylene, hydroxypropyl cellulose, polystyrene or polyvinylalcohol, and the target molecule may be a hydrocarbon CmHn, (e.g., CH4 or C3H7 or C2H2), an alcohol CmHn,OH, a ketone (e.g., CH3(CO)CH3) or an aldehyde (e.g., C2H5(COH). An algorithm, applicable to any embodiment, provides an estimate of target molecule concentration.
In a second embodiment, the SWCNTs in the network are doped with a selected transition element, such as Pd, Pt, Rh, Ir, Ru, Os and/or Au. In either embodiment, a value of an electrical parameter, such as conductivity, resistivity, electrical current or voltage difference, is measured and compared with the parameter value for the network with no coating and no doping. SWCNTs are positioned between the electrodes using either a solution casting process in the form of a network or in situ growth using chemical vapor deposition (CVD) techniques. Polymer coating or transition element doping of SWCNTs allows selective sensing of certain gases, as demonstrated here for chlorine (Cl2), HCl, CH4 and COx vapor. The CNT sensors may be formed on a substrate, such as silicon, ceramic, glass and selected polymers. The sensor fabrication process is scalable for manufacturing products that include wafer scale interdigitated electrode fabrication and inkjet deposition of SWCNTs and polymers for coatings and metal nanoparticles for doping. Coating and doping may be collectively referred to as “loading.”
In a third embodiment, an array of substantially identical SWCNTs are (i) coated, as in the first embodiment, or (ii) doped, as in the second embodiment, a temperature gradient is imposed on the array, and a value of an electrical parameter is determined for each coating or doping tested, and a change in sensitivity is determined or estimated as a function of the local temperature value or gradient.
In a fourth embodiment, a sequence of substantially identical SWCNTs are (i) coated, as in the first embodiment, or (ii) doped, as in the second embodiment, different relative humidity values (e.g., 0, 15, 30, 50, 70 and 90 percent) or other environmental parameter values are imposed on each SWCNT sensor, a value of an electrical parameter is determined for each coating or doping tested, and a change in sensitivity is determined or estimated as a function of changes in the environmental parameter value.
In a fifth embodiment, an array of substantially identical SWCNTs are (i) coated, as in the first embodiment, or (ii) doped, as in the second embodiment, a value of an electrical parameter is determined for a selected local temperature and a selected local relative humidity, at the end of each of a sequence of selected time intervals, with temporal lengths from a few seconds to six months or more, if desired, and a change in baseline and sensitivity or “drift” is determined or estimated as a function of elapsed time. An algorithm is presented that can compensate for this drift in baseline and sensitivity, as a function of time.
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SWCNTs yield different signal responses when exposed to different gases and vapors and one must use pattern recognition or intelligent signal processing techniques for the identification of the gas constituent of interest. SWCNTs do not respond to exposure to certain gases and vapors, and in those cases, coating or doping of the nanotubes may elicit a signal. Hydroxypropyl cellulose, having a mass of 0.791 gm, was dissolved in a solvent of 25 ml chloroform for coating the nanotubes to detect presence of HCl. In each case, an aliquot of 5 ml polymer solution was drop-deposited onto the SWCNT network in
In one version of the system shown in
The electrical current through the sensors, at a constant voltage of 1 Volt, was monitored as different concentrations of chemicals, such as chlorine (Cl2) and of hydrochloric acid (HCl) vapor were sequentially introduced to the sensor's environment. A voltage difference of less than 1 Volt (or greater, if desired) can be used here. A computerized gas blending and dilution system, Environics 2040 (Environics, Inc. Tolland, Conn.), was used to create different concentration streams with a steady flow of 400 cc/min during both exposure and purge periods. A gas cylinder containing 98.3 ppm Cl2 gas balanced with nitrogen, and a gas cylinder containing 478 ppm HCl with nitrogen, were used as the source gases. Nitrogen was used both as the purge gas and as the balance gas for creating low concentration test samples. The test sample concentrations were 1, 2 and 5 ppm for Cl2 gas and 5, 10 and 40 ppm for HCl gas. The electrical signal (current) was collected using a semiconductor parameter analyzer HP4155B (Agilent, Palo Alto, Calif.). Other equivalent electrical parameters, such as conductance or resistance, can be used as a response value. In trials involving heating, a thermal controller, Micro-Infinity ICN77000 Series Controller (Newport Electronics, Inc., Santa Ana, Calif.) with a thermocouple, maintained a constant temperature for the sensor operation. Additionally, a vacuum pump and an ultraviolet lamp of wavelength 254 nm were employed on occasion to accelerate the recovery of the sensors between tests; other ultraviolet wavelengths, such as 300 nm and 360 nm, can also be used to accelerate recovery.
Carbon nanotubes do not sense presence of some gases and vapors due to the chemical and physical properties of CNTs as well as the nature of interaction between the gas molecules and nanotubes. SWCNTs have been found to detect presence of NH3 and/or NO2, based on the charge transfer between these gases and SWCNTs. Our early tests indicated that pristine SWNTs do not respond at all when exposed to some industrial chemicals, such as chlorine and hydrogen chloride. It is important to get some observable response before one can do signal processing or pattern recognition for selective identification.
Carbon nanotubes coated with different polymers, such as chlorosulfonated polyethylene, hydroxypropyl cellulose, polystyrene, polyvinylalcohol, etc. used in commercial polymer based chemical sensors available for organic vapor detection, can provide specific interactions with a chemical species of interest. As this chemical treatment aims to provide a specific interaction between the carbon nanotube matrix and specific gas molecules, the treatment can improve the selectivity while maintaining the high sensitivity expected of a nanosensor.
Several polymer-coated carbon nanotube sensors have been tested for different toxic gases, such as chlorine and hydrogen chloride; for comparison, other gases, including oxides of nitrogen (NOp; p=1-3), ammonia (NH3), benzene, nitrotoulene and acetone, have also been tested.
Sensors with SWCNTs coated with hydroxypropyl cellulose have been tested for HCl detection.
A comparison experiment was conducted on sensors using pure, uncoated SWCNTs exposed to different gas and vapor analytes, with results shown in FIG. 4. The pure or uncoated SWCNT sensors showed no observable response when exposed to Cl2 or to HCl, but displayed positive response signals, varying with concentration, for NOp, nitrotoulene, and benzene, and showed a negative response signal (opposite polarity) for NH3. Thus, the SWCNT sensor has some low level of discriminating power by itself to some, but not all, gases and vapors. It is clear from
A similar comparison experiment was carried out on the polymer-coated SWCNT sensors exposed to different analytes, with results shown in
These studies also show that both pure SWCNTs and chlorosulfonated polyethylene-coated SWCNTs do not respond to 100 ppm concentration of HCl gas in nitrogen. Higher concentrations of HCl were not tested as these high levels are not of interest for a nanosensor. In contrast, hydroxypropyl cellulose-coated SWCNTs respond to presence of HCl, but this sensor is also sensitive to NO2. Presence of the OH groups in the polymer coating may be responsible for the response signal differences in interaction with acidic gases. Because this sensor gives a significant response to HCl that other SWCNT sensors do not, the sensor can be used in a sensor array to provide a chemical signature that differentiates the HCl gas from other chemicals.
We have demonstrated a simple nano-chemical sensor using polymer-coated SWCNTs as the sensing medium. Because pristine (uncoated or unmodified) nanotubes do not respond observably to some chemicals of interest, it is important to explore coating or doping techniques to promote observable responses so that a broad application coverage can be ensured. We have found that the polymer coating enables selective sensing of chlorine and hydrochloric acid vapor at a sensitivity level of 5 ppm and above. It is important to recognize that coating or doping alone is unlikely to provide absolute discrimination. As with most sensors (of any size or exploiting any property change), pattern recognition techniques would be a valuable and necessary complement to provide discrimination. In that regard, the use of sensor arrays with multiple elements is an effective approach to chemical sensing, wherein the data from multiple sensors can be routed to a signal processing chip, integrated into the system, for data fusion and analysis. Advanced signal processing and pattern recognition techniques can be used to confirm (or refute) the assumed presence of a given species; in addition to the help from the selective coatings. Multiple sensing element arrays offer additional operational freedom when sensor recovery is slow and is a rate limiting process. Under such circumstances, a sensor would always be available while other sensors are in recovery mode.
Using the results shown in
It is assumed initially in Appendix A that (i) the response value difference varies linearly with concentration difference of a single constituent that is present and (ii) the response value difference, in the presence of two or more gas constituents in the gas G, is the sum of the response value differences of the single constituent gases. Linear response coefficients aij for the response value differences are assumed to be determined experimentally or otherwise provided. As an example, assume that one reference gas (e.g., NO2 or NH3) plus first and second target gases (e.g., Cl2 and HCl), are suspected to be present in the gas G. Estimates of each of the concentration values cm0 for the initial (unaugmented) gas G are obtained from inversion of an M×M matrix equation relating these concentration values to response value differences for N coatings, where M (≧2) is the number of gas components (reference and target) believed to be present and N (≧1) is the number of coatings (or dopings) used for the measurements.
The approach discussed in Appendix A allows separate weights, wn and w′n (or the same weight), to be assigned to the measurements of the initial gas and augmented gas. Preferably, at least two of the weight values in Eq. (3) are positive (e.g., (w1, w2) or (w′1, w′2) or (w1, w′2) or (w′1, w2)) for the example with N=2, and the relative sizes of the non-zero weights reflect the relative importance of the response measurements. If, as is likely, the four response measurements are believed to be equally important, one can choose w1=w2=w′1=w′2=1. One can ignore one or two of the four measurements, in which event the corresponding weight value(s) is set equal to 0.
The response coefficients aij used in Eqs. (1) and (2) are not necessarily positive. For example, the response coefficient aij for the gas constituent NO2 is positive for several of the SWCNT coatings used, while the response coefficient aij for NH3 is observed to be negative for some of these coatings.
Appendix B, and the corresponding flow chart in
Exposure of the coated SWCNT network to ultraviolet light can reduce the recovery time (normally ten hours or more) required to return the network to a substantially uncoated condition, by promoting accelerated detachment of the coating material from the SWCNT network.
For some relatively small molecules, such as methane (CH4), other hydrocarbons, and oxides of carbon (COx; x=1, 2), an SWCNT network, doped with a transition element (“TE”,) such as Pd, Pt, Rh, Ir, Ru, Os and Au, can be used to detect presence of these molecules by detecting a change in an electrical parameter (conductance, resistance, current or voltage difference) or response value associated with a path defined by an SWCNT network that extends between two electrodes having a controllable voltage difference or current. Some molecules, including nitrotoluene and phenol, are relatively strong electron donors and/or electron acceptors, and these molecules' presence can be readily detected using “bare” or unmodified SWCNTs. Other molecules, including but not limited to methane, hydrocarbons and carbon oxides, manifest little or no electron donor or electron acceptor action so that monitoring an electric parameter value V of an unmodified or “bare” SWCNT network will, by itself, not indicate presence or absence of these molecules.
Where CH4 is adsorbed in a SWCNT/Pd matrix, the combination forms a weakly bound complex, such as Pdδ(CH4)−δ, where δ is a relatively small positive number that need not be an integer. Methane, hydrocarbons and carbon oxides are “greenhouse” gases and require detection capabilities in the ppb—ppm range to have much utility in environmental monitoring. The sensing platform is similar to that illustrated in
Fabrication of a sensing platform for the SWCNT/TE network begins with sputter coating of about 10 nm thick Pd onto a pile of SWCNT powder. The TE-loaded or TE-doped SWCNTs are then dispersed into distilled, deionized water (e.g., 0.1 mg of SWCNT/Pd in 10 ml of the water). This solution is then sonicated and drop deposited onto interdigitated electrode fingers to create an electrical sensor with an initial resistance in a range of about 0.2-1 kilo-Ohm. Current through the network, with a 1 Volt difference, was monitored where 6, 15, 30 and 100 ppm of CH4 was present, using a gas stream flow of about 400 cc/min during exposure and during purge. Apart from preparation of the doped or loaded SWCNT network (as distinguished from coating a SWCNT network), the procedure for estimation of constituents present in a gas using a doped SWCNT network, is parallel to the procedure using a coated SWCNT network set forth in the
Vacuum pumping and exposure of the network to ultraviolet light (δ=254 nm) are used to reduce the recovery time of the SWCNT/Pd (i.e., removal of the CH4) between tests (no recovery if these recovery acceleration procedures are not implemented).
V/V0≈a ln(C)−b=ln{Ca/exp(b)}, (1)
to relative response V/V0 versus CH4 concentration c, for different sensor networks. Resistance, as the electrical parameter, normally decreases with increasing concentration C of the constituent, while conductance, electrical current and voltage difference normally increase with increasing concentration c. The algorithm set forth in Appendix 1 and illustrated in Eqs. (A3)-(A6) can be applied to estimate concentration of one or more of the gases CH4, CmHn and/or COx, by replacing the concentration C1 or C2 or C3=C by the quantity
x=ln{Ca/exp(b)}, (2)
where the parameters a and b will vary with the particular gas constituent of interest.
Methane, in the presence of the SWCNT network, may form a complex such as H[Pd]·CH3. The H atoms in CH4 tend to attract electrons from Pd, which in turn can obtain electrons from the SWCNTs to facilitate formation of the complex. This behavior should also be manifest for some or all of similar transition metals, such as Pt, Ru, Rh, Ir, Os and Au. The detection lower limit for CH4 at room temperature, using a Pd-doped SWCNT network, is estimated to be a few hundred ppb to a few ppm. This compares with a CH4 detection lower limit Of 0.5-1 percent for conventional sensors, at temperatures T(min)≧450° C.
The SWCNT/TE sensor, with TE=Pd, has been tested at 15 ppm and 30 ppm concentrations of CH4 at gas temperatures of T=40° C., 80° C. and 150° C. The response parameter value (e.g., conductance or current) increases with increasing temperature, perhaps due to an enhanced catalytic effect of SWCNT/Pd binding with increasing temperature.
Experimental results for other hydrocarbons, for COx, for ketones and for aldehydes are qualitatively similar to those for CH4.
Effects of varying relative humidity (RH) on relative response, referenced to the response (V0) at RH=0 percent, have been measured for several coating materials and doping materials, for a sequence of RH values. It is expected that relative response V/V0 will decrease monotonically as the RH value increases, in part because the presence of a polar substance such as water would interfere with, and partly mask, the change ΔW in an electrical parameter, for substantially all coating and dopant materials of interest. This expectation is borne out in measurements of V/V0, at T=T0=40° C., presented graphically in
V/V0(RH;T0)=Fe((RH/RH0)m;a)≈a·sech{(RH/RH0)m}+(1−a) (3).
where RH0 is a reference RH value, m is a positive number and 0<a≦1, each value being chosen for the particular coating material or dopant material of interest. For sufficiently small values of the quantity RH/RH0, the parametrized curve in Eq. (3) is further approximated as
which is linear and decreasing in the quantity (RH/RH0)2m, and thus linear in the variable RH if 2m=1. More generally, the measured relative response value V/V0 (RH;T0) is substantially monotonically decreasing in the value RH and resembles a trapezoid with a non-zero tail value, as illustrated in the generalized curve shown in
Introducing the dimensionless variable
u=(RH/RH0)m, (5)
several interesting values of u can be identified on
(∂2Fe/∂u2)wK=a·sech3u{sinh2u−1}K=0, (6A)
(∂3Fe/∂u3)wMS=a·sech4u{cosh2u−4}MS=0, (6B)
The value of the parameter a in the interval 0<a≦1 is irrelevant here. The solutions of Eqs. (6A)— and (6B) are verified to be
uK=sinh−1{1)=ln{√2±1}, (7A)
uK\MS=ln{2±√3}, (7B)
With appropriate choices of the parameter values RH0 and m in Eq. (5), the experimentally observed locations of the values uMS and uK1 can be matched.
An appropriate value of the parameter a can be determined as follows. Let u =uf correspond to the value of u for which Fe(uf)=f(0<f<1). From Eq. (3), this requires that
a·sech(uf)+1−a=f, (8A)
sech(uf)=1−(1−f)/a), (8B)
which requires that a lie in a reduced range, 1-f<a≦1. Equations (6A), (6B) and either (7A) or (7B) can be used to estimate RH0, m and a.
If the approximation in Eq. (3) is adopted, the zero point relative response V/V0(RH=0) for a particular coating or dopant material can be compensated for the presence of moisture (RH>0) by a compensation factor such as
V0(RH=0)=V(RH>0)/{a·sech{(RH/RH0)m}+(1−a)}. (9)
Other approximations, replacing the sech(u) function in Eq. (3) by sec{u}, or by exp{−u2/u02}, for example, can also be used here for compensation.
For a given coating or doping material and fixed temperature, the measured relative response V(RH;meas)/V0, as relative humidity RH is increased over a sequence of values, can be compared with corresponding reference values V(RH;ref;h)/V0 for each of a plurality of candidate gases (h=1, . . . , H) to determine if a particular candidate gas is present. Appendix C sets forth an analytical procedure for determining if a target gas is likely to be present, from a comparison of measured relative response values V/V0 for a variable environmental parameter, such as relative humidity, temperature or pressure.
In each of the graphs in
For any one of the graphs in
where the NO2 concentration is assumed to change from c(NO2)=0 at t<t1 to c(NO2;1)>0 at t=t1, and to change from c(NO2;1)=0 to c(NO2)=0 at t=t2 (>t1). The exponent coefficients b1, b2, b3 and b4 are monotonically increasing with increasing values of the relative humidity parameter RH. The coefficient value W1(t1) is the initial value of the parameter W at t=t1, when the concentration changes from c(NO2)=0 to c(NO2)=c(NO2;1); the coefficient W2(t1) is the asymptotic value of the parameter W, where the concentration c(NO2)=c(NO2) is maintained indefinitely; and the coefficient W4(t2) is the asymptotic value of the parameter W where the concentration c(NO2)=0 is maintained indefinitely.
The measured relative response V/V0 or parameter value W may also drift with elapsed time Δt, measured relative to a time at which the coated or doped SWCNT sensor was initially prepared. If it is assumed here that the drift is a single first order process, the relative response, as a function of elapsed time Δt=t −t0, may be approximated as
V(Δt)/V(t0)≈b·exp{(−ΔtΔt0)}+(1−b), (12)
where b is a non-zero value (positive or negative) and Δt0 is a positive time increment. The quantity Δt0 can be determined from the combination
The quantity b is then estimated from a relation
b≈{1−V(Δt)/V(t0)}/{1−exp{(−Δt/Δt0)}. (15)
Drift with elapsed time Δt of the measured relative response value V/V0 can then be compensated by computing a value V(Δt2) in terms of a value at another time
V(Δt2)=V(Δt1)+b exp{(−Δt2/Δt0)−exp{(−Δt1/Δt0)}, (16)
V(t0)≈V(Δt)/{b·exp{(−Δt/Δt0)}+(1−b)}, (17)
where the quantities b and Δt0 are assumed known with reasonable accuracy. The correctness of the representation in Eq. (12) can be evaluated by forming
R(Δt)=ln|V(2Δt)/V(t0)−V(Δt)/V(t0)−ln|V(Δt)/V(t0−V(0)/V(t0)|, (18)
for two or more selected distinct positive values, Δt=Δt1 and Δt=Δt2, If the representation in Eq. (12) is substantially correct, the ratio identity
R(Δt1)/R(Δt2)=Δt1/Δt2 (19)
will be substantially satisfied for any distinct, positive values of Δt1 and Δt2.
*Where the temporal drift involves two or more different first order processes so that Eq. (19) is not substantially satisfied, for example,
v(Δt) ·=V(Δt)/V(t0)≈b1 exp{(−Δt/Δt01)}+b2 exp{(−Δt/Δt02)}(1−b1−b2), (20)
the computations are somewhat more complex. For simplicity, it is assumed that Δt02>>Δt01 and that Δt≧Δt01 so that
exp{(−Δt/Δt02)}>>exp{(−Δt/Δt01)} (21)
and exp{(−Δt/Δt02)} can be ignored relative to exp{(−Δt/Δt01)}. This assumption should be verified a posteriori. One how forms the differences
v(Δt)−v(0)≈b1(x1−1)+b2, (22A)
v(2Δt)−v(Δt)≈b1·x1 (x1−1), (22B)
v(3Δt)−v(2Δt)≈b1·x12(x1−1) (22C)
x1=exp(−Δt/Δt01), (22D)
and the ratio
Eq. (23) is a cubic polynomial in the unknown x1 and can be factored into
x1(x1−1)(x1−R′)=0 (24)
for which the realistic solution is
x1=R, (25A)
Δt01=Δt/ln{1/R}. (25B)
From the preceding Eqs. (20A), (20B) and (23), one finds that
b1≈{v(2Δt)−v(Δt)}/R′(R′−1), (26)
b2={v(Δt)−v(0)}−{v(2Δt)−v(Δt)}/R, (27)
Δt2=−Δt/{ln{v(Δt)−b1exp(−Δt/Δt01)+b1+b2−1}−ln(b2)}, (28)
V(t0)=V(Δt)/{b1 exp{(−Δt/Δt01)}+b2 exp{(−Δt/Δt02)} (1−b1−b2)}, (29)
Recall that v(0), v(Δt), v(2Δt) and v(3Δt) are measured quantities.
If the answer to the query in step 172 is “yes,” the system determines if the response value V(t) varies according to a single first order process, in step 174, for example, using the analysis developed in connection with Eqs. (12), (18) and (19), or another suitable analysis. If the answer to the query in step 174 is “yes” so that the relative response set forth in Eq. (12) is substantially correct, the system determines the parameters b and Δt0, as developed in Eqs. (12)-(15), in step 175. The system optionally compensates for temporal drift, for this single first order process, for example, as indicated in Eq. (17)), in step 176.
If the answer to the query in step 174 is “no,” the system optionally assumes that Eq. (20) is substantially correct, in step 177, where Δt02<<Δt01 is assumed, and estimates the parameters b1, b2, Δt01 and Δt02, for example, as discussed in connection with Eqs. (20)-(28), in step 178. The system optionally compensates for temporal drift for this multiple first order process, for example, as indicated in Eq. (29), in step 179.
Appendix A. Estimation of Concentration of a Gas Component.
For an SWCNT coated with coating number n,
Vn(G;meas)−V0(G;meas)=an,1·cNO2+an,2·cCl2+an,3·cHCl, (A1)
where, for example, cNO2 represents the NO2 concentration (e.g., expressed in ppm or in ppb). The set of response coefficients {an,m}m for different coatings, n=n1 and for n=n2 (≠n1), will differ from each other, but each set is determined or estimated by measurement of the response value difference, Vn(G;meas)−V0(G;meas), of coated (n) versus uncoated (n=0), for each of the two (or, more generally, M≧2) single constituent gases present in a known concentration. For a single constituent gas NO2 and no coating (n=0), for example, a0.1=0.034±0.002.
In step 82, a known increment of one (or more) of the (suspected) constituent (e.g., NO2 or Cl2 or HCl), is added to the gas unknown gas G to provide an augmented gas G′. In step 83, the response values, V0(G;meas) and V0(G′;meas), for the uncoated SWCNT network (n=0), in the presence of the gases G and G′, are measured or otherwise provided. In step 84, the response values, Vn(G;meas) and Vn(G′;meas), for the SWCNT network coated with the (single) coating number n, in the presence of the gases G and G′, respectively, are measured or otherwise provided.
In step 85, an error function ε, defined by
2ε(x,y,z)=Σnwn·{Vn(G;meas)−V0(G;meas)−an,1c1−an,c2+an,3c3}2+Σnw′n·{Vn(G′;meas)−V0(G′;meas)−an,1(c1+Δc1)−an,2c2+an,3c3}2, (A2)
is provided, where c1, c2 and c3 refer to the concentrations of the reference molecule, the first gas molecule and the second gas molecule, Δc1 is a known concentration increment of a selected one (c1) of the reference molecule, the first gas molecule or the second gas molecule, added to the gas G to provide the gas G′, and wn, and w′n are selected non-negative weight values. The two sums in Eq. (A2) represent the contributions of the initial composition and the augmented composition, respectively. These sums over n may include one, two, three or more coatings for which the response coefficients are known. In this example, n=1, 2.
The error function ε(c1, c2, c3) is to be minimized with respect to choices of the (unknown) concentration values c1, c2 and c3. Differentiating ε with respect to each of the variables c1, c2 and c3, in step 86, one obtains three coupled linear equations in these variables:
In step 86, Eqs. (A1)-(A5) in the unknowns c1, c2 and c3 are determined, using standard matrix inversion techniques, after verification that a 3×3 (more generally, M×M) coefficient matrix for the vector [c1 c2 c3]tr has a non-zero determinant. These solutions, [c1 c2 c3]tr, provide estimates of the concentration values of the corresponding chemicals in the gas G (or in the gas G′) in step 66 of
Preferably, at least two of the weight values in Eq. (A2) are positive (e.g., (w1, w2) or (w′1, w′2) or (w1, w′2) or (w′1, w2)), and the relative sizes of the non-zero weights reflect the relative importance of the response measurements. If, as is likely, the four response measurements are believed to be equally important, one can choose w1=w2=w′1=w′2=1. One can ignore one or two of the four measurements, in which event the corresponding weight value(s) is set equal to 0.
The response coefficients an,m used in Eqs. (1) and (2) are not necessarily positive. For example, the response coefficient aij for the gas constituent NO2 is positive for several of the SWCNT coatings used, while the response coefficient aij for NH3 is observed to be negative for at least one of these coatings (
More generally, where M reference gas components (numbered m=1, . . . , M1) and target gas components (numbered m=M1+1, . . . , M1+M2=M) with unknown concentrations are believed to be present and N coatings (numbered n=1, . . . , N), the error function ε (analogous to Eq. (A2)) is defined by
2ε(c1, . . . ,cM))=Σnwn·{Vn(G;meas)−V0(G;meas)−Σman,mcm}2+Σnw′n·{Vn(G′;meas)−V0(G′;meas)−Σman,1(cm+Δcm)}2, (A6)
where one, or more then one, concentration value cm is augmented by a known amount Δcm. The error function ε is minimized by differentiation with respect to each of the unknown concentration values cm. This yields M coupled equations
Σnwn·{Vn(G;meas)−V0(G;meas)−Σman,mcm0}an,m0+Σnw′n·{Vn(G′;meas)−V0(G′;meas)−Σman,m(cm0+Δcm0)}an,m0=0, (A7)
for index values m0=1, 2, . . . , M. These can be restated in a matrix format as
Σn(wn·+w′n){Σman,mcm0}an,m0}=Σnwn·{Vn(G;meas)−V0(G;meas)}+Σnw′n·{Σman,mΔcm0)}an,m0, +Σnw′n·{Vn(G′;meas)−V0(G′;meas)}. (A8)
After verifying that the determinant of the M×M matrix of coefficients for the quantities cmo in Eq. (A8) is non-zero, this M×M matrix can be inverted to determine estimates for the concentration values cm0 (m0=1, . . . , M). These concentration value estimates will depend, in part, upon the relative values chosen for the weight values wn and w′n for the coatings. Where one or more of the reference molecule concentration values cm0 (m0=1, . . . , M1) are known in advance, the estimates for these reference concentration values can be compared with the corresponding known values to evaluate the likely accuracy of the remaining estimated values.
The approach set forth in this Appendix A can also be used to estimate an initial concentration value cm0 where the CNT network is doped or otherwise loaded, rather than being coated.
Appendix B. Determination of Bound on Gas Component Concentration.
A second algorithm does not require provision of a large number of response coefficients aij but only seeks to determine if a particular target molecule is present in at least a selected concentration. For a selected coating, such as chlorosulfonated polyethylene or hydroxypropyl cellulose, on the CNT, a measurement of the response value difference ΔV=V(coated)−V(uncoated) is taken for modified gases, G′(1) and G′(2), where each of two distinct supplemental concentration values, Δ1c(m0) and Δ2c(m0), respectively, for a selected molecule no. m0 (e.g., NOx or Cl2 or HCl) is added to the original gas G. The concentration value c0(m0) of the selected molecule present in the original gas G is unknown, and the configuration of the CNT network is unknown. It is assumed that the response value difference ΔV increases approximately linearly with the concentration difference Δc(m0) of the selected molecule so that
ΔV1(m0)=v0+v1·(c0(m0)+Δ1c(m0)), (B1)
ΔV2(m0)=v0+v1·(c0(m0)+Δ2c(m0)). (B2)
A molecule m0 should be chosen for which |Δ2c(m0)−Δ1c(m0)| is at least equal to a selected positive threshold. The quantities v0, v1 and c0(m0) are then related by the equations
and v0 and v1 are determined, in part, by the CNT network configuration (assumed fixed and reusable) that is present. Where, as is likely, v0≧0, one infers that the initial concentration value c0(m0) for the molecule m0 is limited by
c0(m0)={ΔV1(m0)−v0}/v1−Δ1c(m0)≦{ΔV1(m0)/v1}−Δ1c(m0) (B5-1)
or
c0(m0)={ΔV2(m0)−v0}/v1−Δ2c(m0) ≦{ΔV2(m0)/v1}−Δ2c(m0) (B5-2)
Equations (B5-1) and (B5-2) provide[[s]] an upper bound for the quantity c0(m0). Where it is known that the coefficient v0 is non-positive, Eqs._(B5-1) and (B5-2) can be inverted to provide lower bounds for the concentration:
c0(m0)={ΔV1(m0)−v0}/v1−Δ1c(m0) ≧{ΔV1(m0)/v1}−Δ1c(m0) (B6-1)
or
c0(m0)={ΔV2(m0)−v0}/v1−Δ2c(m0) ≧{ΔV2(m0)/v1}−Δ2c(m0) (B6-2)
This approach does not provide a direct estimate for the quantity C0(m0), only an indication of whether the molecule m0 is or is not present in a concentration of no more than the right hand quantity in Eqs. (B5-1) or (B5-2). However, this approach does not require determination and use of the response coefficients aij that are required for the putatively more accurate method set forth in Appendix A. The method of Appendix B can be used to estimate upper (or lower) bounds for concentration values C of one, two or more selected molecules.
ΔV1(m0))=V(G1;meas)−V(G;meas), (B7)
ΔV2(m0))=V(G2;meas)−V(G;meas), (B8)
are measured or otherwise provided. In step 93, the coefficient v1 in an approximation for response value differences
ΔV1(m0)=v0+v1·{c0(m0)+Δ1c(m0)}, (B9)
ΔV2(m0)=v0+v1·{c0(m0)+Δ2c(m0)}, (B10)
is determined according to
v1=(ΔV2(m0)−ΔV1(m0))/{Δ2c(m0)−Δ1c(m0)}. (B11)
In step 94, the system queries whether the coefficient v0 is likely non-negative. If the answer to the query in step 94 is “yes,” the system estimates an upper bound for the initial concentration value c0(m0), in step 95:
c0(m0)≦{ΔV1(m0)/v1−Δ1c(m0) =ΔV2(m0)/v1−Δ2c(m0) (v0≧0). (B12)
If the answer to the query in step 94 is “no,” the system estimates a lower bound for the initial concentration value c0(m0), in step 96:
c0(m0)≧{ΔV1(m0)/v1−Δ1c(m0) =ΔV2(m0)/v1−Δ2c(m0) (v0≧0). (B13)
Appendix C. Effect of Varying Environmental Parameter.
In step 181, measurements of the response V(EPp;meas) (p=0, 1, . . . , P;P≧1) are provided, where p=0 may correspond to EP equal to a reference value, such as RH=0 or 0.1, for example. In step 182, a sequence of reference values V(EPp;ref;h) (h=1, . . . , H:H≧1) is provided, for each of H candidate gas components and for each of P reference values of the environmental parameter EP. In step 183, the system determines an error value
where wp is a non-negative weight associated with the environmental parameter value EPp and q is a selected positive number (e.g., q=1 or 2). If desired, the approximation Fe((EP/EP0)m;a) in Eq. (3) may be substituted for the reference quantity V(EPp;ref;h).
In steps 184 and 185, a minimum error value (or values)
ε(min)=min{ε(h=1), . . . , ε(h=H)} (C2)
is/are determined (e.g., ε(min)=ε(h=h0)), and ε(min) is compared with a selected threshold value ε(min;thr). If
ε(min)=ε(h=h0)≦ε(min;thr), (C3)
the system interprets these conditions as indicating that the candidate gas component(s) corresponding to h=h0 is/are likely present in the target gas, in step 186. If the condition (C3) is not satisfied, the system interprets this result as indicating that none of the gas component(s) h0 is likely to be present in the target gas, in step 187.
This invention is a Continuation-In-Part of U.S. Ser. No. 11/178,079, filed 8 Jul. 2005, and was made, in part or whole, by one or more employees of the U.S. government.
The U.S. government has the right to make, use and/or sell the invention described herein without payment of compensation, including but not limited to payment of royalties.
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| Number | Date | Country | |
|---|---|---|---|
| Parent | 11178079 | Jul 2005 | US |
| Child | 11416505 | US |
