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Identifying Gas Composition Changes to Which Non-Injective Gas Sensor Arrays Are Unresponsive

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Identifying Gas Composition Changes to Which Non-Injective Gas Sensor Arrays Are Unresponsive Identifying gas composition changes to which non-injective gas sensor arrays are unresponsive Nickolas Gantzlery;∗, Adrian Henlez;∗, Praveen Thallapally{, Xiaoli Fernx, Cory M. Simonz yDepartment of Physics. Oregon State University. Corvallis, OR. zSchool of Chemical, Biological, and Environmental Engineering. Oregon State University. Corvallis, OR. {Pacific Northwest National Laboratory. Richland, WA. xSchool of Electrical Engineering and Computer Science. Oregon State University. Corvallis, OR, USA E-mail: [email protected] ∗equal contribution 17 June 2021 Abstract. Metal-organic frameworks (MOFs) have great prospects as recognition elements for gas sensors owing to their adsorptive sensitivity and selectivity. A gravimetric, MOF-based sensor functions by measuring the mass of gas adsorbed in a thin film of MOF. Changes in the gas composition are expected to produce detectable changes in the mass of gas adsorbed in the MOF. In practical settings, multiple components of the gas adsorb into the MOF and contribute to the sensor response. As a result, there are typically many distinct gas compositions that produce the same single-sensor response. The response vector of a gas sensor array places more constraints on the gas composition. Still, if the number of degrees of freedom in the gas composition is greater than the number of MOFs in the sensor array, the map from gas compositions to response vectors will be non-injective (many-to-one). Here, we outline a mathematical method to determine undetectable changes in gas composition to which non-injective gas sensor arrays are unresponsive. This is important for understanding their limitations and vulnerabilities. We focus on gravimetric, MOF-based gas sensor arrays. Our method relies on a mixed-gas adsorption model in the MOFs comprising the sensor array, which gives the mass of gas adsorbed in each MOF as a function of the gas composition. The singular value decomposition of the Jacobian matrix of the adsorption model uncovers (i) the unresponsive directions and (ii) the responsive directions, ranked by sensitivity, in gas composition space. We illustrate the identification of unresponsive subspaces and ranked responsive directions for gas sensor arrays based on Co-MOF-74 and HKUST-1 aimed at quantitative sensing of CH4/N2/CO2/C2H6 mixtures (relevant to the natural gas industry). Unresponsive spaces in gas sensors 2 1. Introduction Gas sensors [1] have a wide range of applica- tions in the chemical industry and in emerg- ing domains such as air quality monitoring [2], diagnosis of disease [3], food quality assess- ment [4], detection of chemical warfare agents and explosives [5,6], and crop monitoring [7]. Metal-organic frameworks (MOFs) [8] are Figure 1: A gravimetric sensor comprised nanoporous materials with high prospects as of a thin film of metal-organic framework recognition elements for enhanced gas sensors (MOF) [23] (the recognition element), in this [9{12] owing to their sensitivity and selectivity. case HKUST-1 [32], mounted on a quartz MOFs offer large internal surface areas and, crystal microbalance (QCM) [22,33] (the signal often, expose functional groups and open metal transducer). The QCM measures the (total) sites to the pore space. As a result, gas adsorbed mass of gas, m, in the thin film of molecules concentrate inside the pores of the MOF. The QCM-MOF functions as a sensor MOF by (selectively) adsorbing onto the pore because changes in the gas composition, x, are walls. The amount of gas adsorbed in a expected to cause changes in m. MOF at thermodynamic equilibrium depends upon the chemical potential of each component practical applications where the gas contains in the gas phase (and the temperature). multiple components that vary in concentra- Therefore, observation of the total mass of tion [34]. In the case of MOF-based gas sen- gas adsorbed in a MOF [13]|or some other sors, interfering gas species, in addition to the property that depends on the gas adsorbed analyte species, adsorb onto the MOF and con- in the MOF, such as electrical resistance tribute to the response (i.e. the total adsorbed [14, 15], color [16], luminescence [17, 18], or mass of gas). As a consequence, the inverse strain [19,20]|provides information about the problem [35] of inferring the concentration of composition of the gas. Fig. 1 illustrates a the analyte from a single-sensor response is un- miniaturized gravimetric, MOF-based sensor derdetermined [36]. To address the problem of composed of a quartz crystal microbalance cross-sensitivity, one might propose to design (QCM) coated with a thin film of MOF a MOF with a very high adsorptive selectivity [21,22]|a surface mounted MOF (SURMOF) for the analyte. However, this strategy may be [23]. Nanogram-scale changes in the adsorbed difficult or impossible, and it requires a differ- mass of gas in the MOF film can be inferred ent MOF highly tailored to each sensing task. from changes in the resonance frequency of Gas sensor arrays tackle the issue of cross- the piezoelectric quartz crystal, forced by an sensitivity by monitoring the response of a alternating voltage [9,10]. Several gravimetric, diverse set of (likely, cross-sensitive) MOFs MOF-based sensors have been experimentally [37, 38], analogous to mammalian olfactory demonstrated using QCMs [13, 24{28] and receptors [39]. Each response of a cross- surface acoustic wave devices (SAWs) [29{31] sensitive MOF, though having contributions as signal transducers. from the analyte and interferents, places one Cross-sensitivity plagues gas sensors in Unresponsive spaces in gas sensors 3 Figure 2: Illustrating a gas composition change to which a non-injective, gravimetric, MOF- based gas sensor is unresponsive. A single QCM-HKUST-1 sensor is immersed in a gas mixture of two different compositions: on the left, a predominantly CH4/C2H6 mixture; on the right, a predominantly CH4/CO2 mixture. The response of the sensor to the two distinct gas compositions is identical; hence, the sensor cannot be used to distinguish between these two compositions. The total mass of gas adsorbed in the HKUST-1 film remains the same upon changing from the CH4/C2H6 to CH4/CO2 mixture because the desorption of C2H6 balances [in mass] the adsorption of CO2, while the adsorption of CH4 remains approximately constant. constraint on the gas composition. As a whole, component is introduced. the response vector of the gas sensor array In this article, we outline a mathematical may provide sufficient information to uniquely method to identify changes in the gas compo- determine the gas composition. Typically, a sition to which reversible z, quantitative gas machine learning algorithm is used to parse sensors /sensor arrays are unresponsive. Gas the high-dimensional response of a sensor sensing can be viewed as the inverse prob- array [40]. A few MOF-based gas sensor lem of finding the gas composition consistent arrays have been experimentally demonstrated with the sensor response governed by a mixed- [14, 27, 41{43]. Still, even gas sensor arrays gas adsorption model [35]. Owing to cross- are susceptible to cross-sensitivity; the inverse sensitivity, the mapping, under the adsorption problem (determine the gas composition from model, from gas composition space to sensor the sensor array response vector) could still be array response space could be non-injective underdetermined|particularly, if the number (many-to-one). Consequently, some gas com- of MOFs comprising the array is less than position changes are undetectable. By our defi- the number of degrees of freedom in the gas z Gas sensors based on physical adsorption in stable composition. This circumstance can arise MOFs lacking adsorption/desorption hysteresis are (i) in an environment with highly complex reversible. Suppose the gas composition changes gas mixtures and/or (ii) a sensor array is from (i) a reference composition to (ii) some calibrated for gas mixtures with a specific set new composition, then (iii) back to the reference composition. The equilibrium mass of gas adsorbed of components, but an unexpected, interfering in the MOF will be the same in (i) and (iii). Unresponsive spaces in gas sensors 4 nition, when the gas composition changes in an 2. A mathematical model of gas sensor unresponsive direction in composition space, arrays the sensor response remains constant. When the gas composition changes in an unrespon- 2.1. Problem setup sive direction, the adsorption of one set of Suppose a gravimetric sensor array of M species balances (in mass) the desorption of an- MOFs aims to determine the composition of other set of species, thus, keeping the sensor re- a gas (temperature, pressure fixed) with G + 1 sponse fixed. These undetectable gas composi- components. tion changes generally depend upon the initial composition. The identification of unrespon- G The gas composition vector. Let x 2 R be sive directions of gas sensors/sensor arrays is the gas composition vector, with xi the mole important for understanding their limitations fraction of component i in the gas phase. We and vulnerabilities to adversarial attacks [44]. omit component G + 1 of the gas from x Specifically, we show that the singular since its mole fraction follows from xG+1 = value decomposition of the Jacobian matrix 1 − (x1 + x2 + ··· + xG). Thus, the only of the mapping from gas composition vectors G constraints on x are i=1 xi ≤ 1 and x ≥ 0. to sensor response vectors is key to determin- We assume the gas is held at constant pressure P ing unresponsive directions and responsive di- and temperature, implying (i) x fully specifies rections, ranked by sensitivity, in composition the thermodynamic state of the gas phase and space. In particular, unresponsive directions (ii) G is the number of degrees of freedom in follow from the null space of the Jacobian. We the gas phase. illustrate this under the context of quantitative sensing for diluents and higher-hydrocarbons The sensor response vector.
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