Non-injective gas sensor arrays: identifying undetectable composition changes

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 CH 4 /N 2 /CO 2 /C 2 H 6 mixtures (relevant to the natural gas industry).

Computationally predicting the performance of gas sensor arrays for anomaly detection

“…Generally, the performance of a sensor array tends to increase with the number of sensing elements, as each additional sensor provides additional information about the gas [27,28], albeit with diminishing marginal returns [27]. If the number of sensors is fewer than the number of components in the gas phase, multiple distinct gas compositions produce an identical sensor array response and hence are indistinguishable by the array [28,29]. Deciding the number of sensors to comprise the array likely involves a tradeoff between performance, cost, and complexity.…”

“…17, m = Hp, is non-injective, meaning multiple gas compositions map to an identical sensor array response vector. Consequently, we cannot uniquely determine the ternary gas composition from the response of the two-sensor array [29]. While this non-injectivity could be resolved by adding an additional sensor, we elected to consider non-injectivity as a source of undetected anomalies.…”