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 Gantzler†,∗, Adrian Henle‡,∗, Praveen Thallapally¶, Xiaoli Fern§, Cory M. Simon‡

†Department of Physics. Oregon State University. Corvallis, OR. ‡School of Chemical, Biological, and Environmental Engineering. Oregon State University. Corvallis, OR. ¶Paciﬁc Northwest National Laboratory. Richland, WA. §School 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 applications 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 ‡, 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 ‡ 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 ∈ 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. Let m ∈ RM in natural gas [45] (N2, CO2, and C2H6 in be the sensor array response vector, with mi CH4) using gravimetric, MOF-based sensors [g gas/g MOF] denoting the total mass of comprised of HKUST-1 and Co-MOF-74. We gas adsorbed in MOF i at thermodynamic use experimental gas adsorption data and ideal equilibrium, normalized by the mass of MOF adsorbed solution theory [46, 47] to model the in the sensor. We assume, as in practice, that mass of adsorbed gas in each MOF in response each MOF is cross-sensitive. As a result, mi to different gas compositions. Under several is comprised of contributions from each of the case studies, we visualize the unresponsive and G + 1 components of the gas phase. ranked responsive directions together with the underlying mixed gas adsorption model. The adsorption function. Mathematically, we Several computational and mathematical view gas sensor arrays as mapping from gas modeling approaches have been developed to composition space X := RG to sensor response design/analyze gas sensor arrays of MOFs space M := RM [36, 53]. The adsorption [35,36,48–52]. Our work focuses on identifying function a : x 7→ m maps each gas composition the limitations of non-injective, MOF-based vector x ∈ X to a sensor response vector gas sensor arrays by elucidating directions in m ∈ M. I.e., m = a(x) is a model composition space in which the gas sensor is for the total mass of gas adsorbed in each unresponsive to changes. MOF (at thermodynamic equilibrium) as a Unresponsive spaces in gas sensors 5 function of the gas composition. We assume unique solution, the inverse problem may a(x) is a continuously differentiable function, have a unique solution, no solution, or many disallowing adsorption/desorption hysteresis, solutions. gas-induced first-order structural transitions, chemical transformations, and condensation of 2.2. Non-injective gas sensor arrays with adsorbates in the pores. underdetermined inverse problems A model of the adsorption function a(x) In this work, we consider gas sensor arrays could be constructed from adsorption data comprised of fewer MOFs than degrees of D = {(x , m )}n , with n the number of i i i=1 freedom in the gas composition (M < G). The gas exposure experiments, through various mapping a : x 7→ m under a sensor array methods including: with M < G is generally non-injective (many- • interpolation of D. to-one). Consequently, the inverse problem • a statistical mechanical model of gas of determining the gas composition x from adsorption [54] whose parameters are the sensor response m is underdetermined; identified using D. typically, many gas compositions can produce • a statistical machine learning model the observed response. In the inverse problem, trained on D. observation of each MOF’s response, generally, provides one independent constraint, m = • ideal adsorption solution theory (IAST) i a (x), on the gas composition. For M < G, [55] applied to pure-component adsorp- i the response of the sensor array places an tion models constructed from D. insufficient number of constraints on the gas The source of D could be experimental composition to uniquely determine it. gas adsorption measurements and/or grand- As a result of their (view 1) non- canonical Monte Carlo molecular simulations injective mapping a : x 7→ m and (view of gas adsorption [56,57] in the MOFs. 2) underdetermined inverse problems, gas While we write the adsorption function sensor arrays with fewer MOFs than degrees a(x) as a vector-valued function, the mass of of freedom in the gas (M < G) suffer gas adsorbed in each MOF is independent. In from unresponsiveness to some changes in other words, we can independently build the gas composition: despite the change in adsorption model for each MOF i in the array, composition, the sensor response remains mi = ai(x). constant, and hence the composition change is not detected. Gas sensing as an inverse problem [35]. The forward problem is to predict the response of 3. Unresponsive and ranked responsive the sensor array m given the gas composition directions of non-injective gas sensors x. The solution is to evaluate the adsorption model m = a(x). The inverse problem Non-injective gas sensors contain fewer MOFs arises in gas sensing: we wish to predict the than degrees of freedom in the gas composi- gas composition x given the sensor response tion. They are vulnerable to blindness: the m using the adsorption model m = a(x). sensor is unresponsive to certain gas composi- While the forward problem always has a tion changes. Our work focuses on identifying Unresponsive spaces in gas sensors 6 these vulnerabilities in non-injective gas sen- 3.3. Linear approximation of a(x) near the sors. Moreover, the sensor is more sensitive to operating point xop certain gas composition changes than others Let us make a linear approximationa ˜(x) of (of the same magnitude). We show that the the adsorption function a(x) valid near the singular value decomposition of the Jacobian operating point x . A first-order Taylor matrix of the adsorption function a(x) is key op expansion of a(x) about x gives: to identifying directions in composition space op to which the sensor is most-, least-, and un- a(x) ≈ a(xop) + J(xop)(x − xop) =:a ˜(x), (2) responsive. with J = J(x) the M × G Jacobian matrix ∂mi 3.1. The operating point x as a reference of a(x) with Jij := . For a non-injective op ∂xj gas sensor array (M < G), J is fat. From a We will define unresponsive and ranked geometric viewpoint, the approximator m ≈ responsive directions in reference to some i a˜ (x) is a hyperplane tangent to the surface steady-state, reference gas composition, or i G ai(x) and passing through mi,op. operating point xop ∈ R . mop := a(xop) is the sensor response at this operating point. 3.4. The (local) unresponsive subspace of composition space 3.2. The unresponsive locus B We define the (locally valid) unresponsive The unresponsive locus B ⊂ X associated subspace B˜ ⊂ X of a sensor array associated with an operating point x is the set of gas op with operating point x as the directions in compositions such that the sensor response op composition space such that, for small changes remains constant at m : op in the gas composition in these directions, the sensor response remains approximately B := {x : a(x) = mop, x ∈ X }. (1) constant at mop. Thus, for a unit vector ˜ The unresponsive locus is an intersection of ∆x ∈ B, a(xop + ∆x) ≈ a(xop) with a small the level sets of the adsorption functions for number. Invoking the linear approximation M each MOF in the sensor array, B = ∩i=1{x : a˜(x) in eqn. 2, the local unresponsive subspace ai(x) = mi,op}. By definition, the sensor array is given by the null space of the Jacobian cannot distinguish between gas compositions matrix J(xop): in the unresponsive locus. A change in the ˜ 0 B := {∆x : J(xop)∆x = 0, ∆x ∈ X } (3) gas composition from xop ∈ B to x ∈ B is 0 ˜ undetectable despite x 6= xop. Fundamentally, =⇒ a˜(xop + ∆x) =a ˜(xop) = mop ∀∆x ∈ B. the response of a sensor array composed of (4) cross-sensitive MOFs remains constant under The null space of J will be non-trivial provided the composition change from x to x0 because op G > M. I.e, sensor arrays such that M < G the adsorption of one set of species balances [in possess non-trivial unresponsive subspaces due mass] the desorption of another set of species to the non-injective nature of the mapping (see Fig. 2). A trivial example of a non-trivial a˜ : x 7→ m. unresponsive locus is when all MOFs do not adsorb a particular component of the gas. Unresponsive spaces in gas sensors 7

3.5. The sensitive direction in composition • responsive to gas composition changes in space the direction belonging to the subspace spanned by the first M right singular The sensor response is most sensitive to gas vectors v , v , ..., v . They are ranked composition changes in the direction of: 1 2 M by the magnitude of the sensor array response for composition changes of fixed ∆xs := arg max ||a˜(xop + ∆x) − a˜(xop)||. (5) ∆x:||∆x||= magnitude. • unresponsive to gas composition changes Of all gas composition changes ∆x from xop that are small ( > 0) in magnitude, those in in the direction belonging to the subspace spanned by the last G − M right singular the sensitive direction of ∆xs elicit the largest- magnitude change in the response of the sensor vectors {vM+1, vM+2, ..., vG}. This sub- array. space is the null space of J. Hence, 3.5.1. The singular value decomposition • the sensitive direction of the gas sensor (SVD) of J. The singular value decomposi- array, ∆xs in eqn. 5, is described by tion (SVD) [58,59] of the M × G Jacobian ma- the first right singular vector v1 of the trix J factorizes it as: Jacobian matrix J. The gas sensor array is responsive, but less sensitive, J = UΣV|, (6) to composition changes in the remaining where U is a M × M orthogonal matrix, Σ directions v2, v3, ..., vM . is an M × G diagonal matrix with singular • the local unresponsive subspace B˜ is

values σ1 ≥ σ2 ≥ · · · ≥ σM listed down the spanned by the last (unranked) G − M diagonal, and V is a G × G orthogonal matrix. right singular vectors {vM+1, vM+2, ..., vG}

The left singular vector ui ∈ M is column i of J. of U and lies in sensor response space. The We emphasize that the sensitive and unrespon- right singular vector vi ∈ X is column i of V sive directions are (i) depend on the operation

and lies in gas composition space. The M left gas composition xop and (ii) pertain to small (G right) singular vectors form an orthonormal changes in the gas composition since they rely basis for sensor response (gas composition) on the linear approximation of the generally space. Provided the MOFs are distinct, it is non-linear function a(x) in eqn. 2. extremely likely that J is full-rank (rank M)§, We remark on the uniqueness of the SVD so we assume σM > 0 in this work. in eqn. 6. The singular values are unique. Seen by JV = UΣ: The left and right singular vectors associated with the distinct, non-zero singular values σiui i ∈ {1, 2, ..., M} are unique, up to both corresponding vectors Jvi = (0 i ∈ {M + 1,M + 2, ..., G}. being multiplied by −1. The right singular (7) vectors spanning the null space are much more arbitrary in the sense that they can be any Therefore, under the linear approximation orthonormal basis for the null space. a˜(x) in eqn. 2, the sensor array is: Note, if the response vector is recorded § J could, however, be ill-conditioned [35]. in units g gas instead of g gas/g MOF, (i) Unresponsive spaces in gas sensors 8 the responsive and unresponsive subspaces do composition space, while the two orthonormal not change, but (ii) the ranked responsive left singular vectors {u1, u2} lie in M = 2- directions that span the responsive subspace dimensional response space. The Jacobian do change if diﬀerent masses of MOF are used matrix maps the right-singular vectors into in each sensor. response space as follows:

3.6. Special case: linear adsorption model Jv1 = σ1u1 (10) Jv2 = σ2u2 (11) Suppose gas adsorption in each MOF of the Jv = 0. (12) array is described by Henry’s law: 3

G+1 Provided the MOFs are distinct, the two rows

mi = Hi,jP xj, (8) of J are highly unlikely to point in exactly the j=1 X same direction; thus, we safely assume J is full- rank, i.e., σ > 0. Eqns. 10 and 11 indicate the with P the total pressure [bar] of the gas 2 sensor is responsive to gas composition changes phase and Hi,j the Henry coeﬃcient [g gas/(g in the direction of v and v but more sensitive MOF-bar)] of gas j in MOF i. Imposing the 1 2 to changes in the direction of v1 (σ1 ≥ σ2 constraint that the mole fractions {xj} sum to by deﬁnition, but likely σ > σ ). Eqn. 12 one, the adsorption model for the sensor array 1 2 indicates the null space of the Jacobian matrix is linear: is non-trivial and spanned by v3. Therefore, m = a(x) = Ax + b, (9) the linear mapa ˜ : x 7→ m is non-injective, as a consequence of M < G. with Aij := P (Hi,j −Hi,G+1), bi := Hi,G+1, and Consider small gas composition changes

x := [x1, x2, ..., xG]. Under this special case, (of magnitude > 0) from xop. The linear the Jacobian matrix J = A does not depend approximanta ˜(x) in eqn. 2 maps a sphere on a reference gas composition xop. Therefore, in 3D composition space centered at xop to the unresponsive and responsive subspaces are a solid ellipse in 2D sensor response space independent of xop, too. [58] centered at mop. See Fig. 3b. This follows from eqn. 2 and eqns. 10-12. The 4. Summarizing with a toy example composition change in the sensitive direction,

v1 (green in Fig. 3b), elicits the largest sensor We illustrate the identification of the unre- response—from mop to mop + σ1u1 on the sponsive and ranked responsive directions with endpoint of the major axis of the ellipse. a toy example: a sensor array of M = 2 dis- The sensor is responsive, but less sensitive, tinct MOFs aimed at sensing a quaternary to composition changes in the direction of v2 mixture at fixed temperature and pressure (yellow in Fig. 3b); the composition change (G = 3). See Fig. 3a. v2 moves the sensor response by the vector Fig. 3c illustrates the singular value σ2u2, placing it on the endpoint of the minor decomposition of a M = 2 × 3 = G Jacobian axis of the ellipse. The relative sensitivity of matrix J = J(xop) at some operating point the response to composition changes in the x . The three orthonormal right singular op direction of v1 vs. v2 depends on the singular vectors {v , v , v } lie in G = 3-dimensional 1 2 3 values, σ1 vs. σ2. On the other hand, v3 is Unresponsive spaces in gas sensors 9 the unresponsive direction; the composition 5.1. The adsorption model a(x), its Jacobian change v3 (red in Fig. 3b) does not elicit a matrix J, and its SVD. sensor response, and m remains at m despite op We construct an adsorption model for each the change in gas composition. Using this toy MOF on the basis of experimentally measured, sensor array, we therefore cannot detect small pure-component CH , CO ,C H , and N composition changes in the direction of v ! 4 2 2 6 2 3 adsorption data at or near 298 K. Fig. 4b displays the adsorption isotherms and Tab. 1 5. Demonstration: natural gas sensing provides the references. with gravimetric sensors based on HKUST-1 and Co-MOF-74 Table 1: References for experimentally measured, equilibrium gas adsorption isotherms at We now demonstrate the identification of the 298 K (except C H , at 296 K). unresponsive subspace and ranked responsive 2 6 directions in gas composition space for a MOF CH4 CO2 C2H6 N2 gravimetric, single-MOF sensor and two- HKUST-1 [61] [62] [63] [64] MOF sensor array for diluents and higher- Co-MOF-74 [65] [65] [63] [65] hydrocarbons in natural gas [45]. Specifically, suppose the sensor (array) is immersed in a We invoke ideal adsorbed solution theory variable-composition gas mixture involving N , 2 (IAST) [47, 55] and use pyIAST [66] to C H , CO , and CH at 1 bar and 298 K. 2 6 2 4 implement the adsorption function a(x). We consider two candidate MOFs as the Sec. 2.1 explains that a(x) gives the total mass recongition element(s), HKUST-1 [32] and Co- of gas adsorbed in each MOF [g/g] of the MOF-74 [60] (see Fig. 4a). array as a function of the gas composition In each case study below, the number described by x. IAST is a thermodynamic of degrees of freedom in the gas composition framework to predict mixed-gas adsorption (G, with G + 1 the number of components in from pure-component adsorption isotherms. the gas) is greater than the number of MOFs Fig. 4b shows the pure-component adsorption (M) comprising the sensor (array). Thus, isotherm models fit to the experimental (i) the sensor’s mapping under a from gas adsorption data and used as input for IAST composition space to sensor response space is calculations; the single-site Langmuir, dual- non-injective and (ii) the inverse problem to site (DS) Langmuir, and quadratic adsorption determine the gas composition from the sensor isotherm models fit the data well. N.b., though response is underdetermined. Accordingly, we write m = a(x), the adsorption model for the sensor (array) will contain a non-trivial each MOF in the array is independent; IAST unresponsive subspace. provides m = a (x) for each MOF in the array The data and Python code to reproduce i i independently. all plots in our article are available at github. We compute the Jacobian matrix J = com/SimonEnsemble/unresponsive_directions_ J(x ) of a(x) numerically via a second-order, in_sensors. op centered finite difference using numdifftools; we compute the SVD of J using numpy. Unresponsive spaces in gas sensors 10

(a)

gas composition space sensor response space

mop xop

v3 2u2 a(x) 2

x m 3 v1 −−→

1u1

0 0 0 0 0 x1 x2 m1

(b)

J U Σ V| σ | ∂ m u u 1 v1 responsive (sensitive) xj i = 1 2 | σ2 v2 responsive (less sensitive) M G M M M G  |  × × × v3 unresponsive G G  ×  (c)

Figure 3: Elucidating the unresponsive and ranked responsive directions of a (toy) sensor array through the singular value decomposition (SVD) of the Jacobian matrix J of its adsorption function a(x). (a) A gravimetric, two-MOF sensor array immersed in a four-component gas mixture at constant temperature and pressure but varying composition. The gas composition x ∈ R3, while the sensor response m ∈ R2. (b) The adsorption function a(x) is a map a : x 7→ m from 3D gas composition space (left) to 2D sensor response space (right). The operating gas composition xop and associated response mop are shown as black points. Under the linear approximation of a(x) valid locally around xop: The image of the sphere of small radius (blue, left) maps to the solid ellipse (blue, right) with semi-major and semi-minor axes given by σ1u1 and σ2u2, respectively. Gas composition changes v1, v2, v3 form an orthogonal basis of composition space and are mapped to response changes σ1u1, σ2u2, 0, respectively. v3 describes the unresponsive direction, since small changes in composition in this direction do not

alter the response. Small composition changes in the direction of v1 elicit the largest-magnitude response. (c) The singular value decomposition J = UΣV| gives the vectors described in (b). Unresponsive spaces in gas sensors 11

(a)

HKUST-1 Co-MOF-74

14 CH4 14 CH4 N2 N2

12 CO2 12 CO2

C2H6 C2H6 10 Langmuir model 10 Quadratic model Langmuir model Langmuir model 8 Langmuir model 8 DSLangmuir model Langmuir model Quadratic model 6 6

4 4 gas uptake [g gas/g MOF] gas uptake [g gas/g MOF]

2 2

0 0 10 2 10 1 100 101 102 10 2 10 1 100 101 102 pressure [bar] pressure [bar]

(b)

Figure 4: The two MOFs as candidate recognition elements and their gas adsorption isotherms. (a) The crystal structures of HKUST-1 [32] (left) and Co-MOF-74 [60] (right). (b)

Experimentally measured, pure-component CH4 (298 K) [61,65], N2 (298 K) [64,65], CO2 (298 K) [62, 65], and C2H6 (296 K) [63], and adsorption isotherm data (◦, , +, and x, respectively) in HKUST-1 and Co-MOF-74. Lines show the pure-component adsorption models ﬁt to the data.

5.2. Case M = 1,G = 2: single-MOF sensor, v1 (green arrow) directions—given by the ternary gas mixture right singular vectors of the Jacobian matrix J(x )—at a reference gas composition x We first consider a single-MOF sensor, ei- op op (black dot). The rows correspond to the three ther IRMOF-1 or HKUST-1, immersed in different gas mixtures; the columns correspond a three-component mixture: CH /N /CO , 4 2 2 to the two different single-MOF sensors. Each CH /N /C H , or CH /C H /CO at 1 bar 4 2 2 6 4 2 6 2 panel displays the 2D gas composition space and 298 K. pertaining to the row. The colored lines are Fig. 5 visualizes, under these six scenarios, contours of the adsorption function a(x) and the adsorption function m = a(x) as well as are labeled by the total adsorbed mass of gas the unresponsive v2 (red arrow) and sensitive Unresponsive spaces in gas sensors 12

m = a(x) [g gas/g MOF] m = a(x) [g gas/g MOF] 0.30 0.30 xop xop

0.25 0.25 0.196 0.067 n n

o 0.20 o 0.20 i i

t t 0.171 c v1 0.053 c v1 a a r r f f

e 0.15 e 0.15 l l

o o 0.147 m m

2 0.040 2

O 0.10 v2 O 0.10 v2 0.122 x = [xN , xCO ] C C 2 2 0.098 0.027 0.05 0.05 0.073 0.049 0.00 0.013 0.00 0.024 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20 0.25 0.30

N2 mole fraction N2 mole fraction

m = a(x) [g gas/g MOF] m = a(x) [g gas/g MOF] 0.30 0.30 0.080 xop xop

0.25 0.25 0.122

n 0.067 n o o

i 0.20 i 0.20 t t c v c v a 1 a 1 r r f f

e 0.053 e 0.098 l 0.15 l 0.15 o o m m

6 6

H v2 0.040 H 2 0.10 2 0.10 0.073 v2 C C x = [xN2 , xC2H6 ] 0.05 0.027 0.05 0.049

0.024 0.00 0.013 0.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20 0.25 0.30

N2 mole fraction N2 mole fraction

m = a(x) [g gas/g MOF] m = a(x) [g gas/g MOF] 0.30 0.30 0.220 xop xop 0.093 0.107 0.120 0.25 0.25 0.196

0.067 n n

o 0.20 o 0.20 i i t t c c

a a 0.171 v1 r r f f v e 0.15 1 e 0.15 l l o o m m

2 0.040 0.080 2

O 0.10 O 0.10 C C 0.122 x = [xC2H6 , xCO2 ] 0.098 0.147 0.053 v2 0.05 0.05 0.027 v2 0.073 0.049 0.00 0.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20 0.25 0.30

C2H6 mole fraction C2H6 mole fraction

Figure 5: Case M = 1,G = 2: visualization of a(x) and the unresponsive- and sensitive directions for a single-MOF sensor immersed in a three-component gas mixture at 1 bar and 298 K under six distinct cases. The row indicates which three species compose the gas mixtures. The column indicates which MOF is used as the recognition element for the gravimetric sensor. In each panel, the plane represents gas composition space. Color-coded contours of the adsorption function a(x) are drawn to indicate unresponsive loci and are labeled with m = a(x) for all x on that contour. At the operating gas composition xop, the sensitive (green) and unresponsive (red) directions are shown as v1 and v2, respectively ( = 1/15). Unresponsive spaces in gas sensors 13

m in the respective single-MOF sensor. Each CH4/C2H6/CO2 mixture is most interesting, map a : x 7→ m is clearly non-injective, as an as the sensitive and unresponsive directions unresponsive locus of points elicits the same each have significant components in the sensor response m. Within each panel, (i) the direction of changes in CO2 and C2H6 mole non-linearity of the contours and the different fractions. Again reflected in the adsorption areas between successive contours reflect the isotherms in Fig. 4b, CO2-MOF and C2H6- non-linearity of a(x) and (ii) the slant of the MOF interactions are competitive. contours reflects the different affinity for the Comparing Co-MOF-74 and HKUST-1

MOF among the three diﬀerent components of for CH4/C2H6/CO2 sensing, the sensitive the gas. direction v1 for HKUST-1 has a larger

The sensitive direction v1 is orthogonal to component along x = [0.5, 0.5] (see Fig. 5). the contour {x : a(x) = mop}. Among all Therefore, HKUST-1 is a more suitable sensor fixed-magnitude changes in gas composition, for estimating the sum of the CO2 and C2H6 the change in the sensitive direction will mole fractions and, thus, the mole fraction of elicit the largest response (see eqn. 5). The CH4. In other words, the component of the unresponsive direction is tangential to the unresponsive direction for HKUST-1 along x = contour {x : a(x) = mop}. The unresponsive [0.5, 0.5] is smaller than for Co-MOF-74; so, subspace is one-dimensional owing to the single HKUST-1 is not as blind to changes in the CH4 constraint imposed by the response of the mole fraction in the CH4/C2H6/CO2 mixture sensor on the two degrees of freedom in the than Co-MOF-74. Fig. 6 clarifies: observation gas composition. For a small change in gas of the response mop of the HKUST-1 sensor composition in the unresponsive direction, the places tighter bounds on xCH4 than of the Co- sensor response remains approximately the MOF-74 sensor because xCH4 is approximately same and, hence, the composition change constant along its unresponsive direction. This is not detected by the sensor. Fig. 2 exemplifies how unresponsive directions and illustrates the underlying physical cause of loci could be used to juxtapose two non- the unresponsiveness to a composition change, injective sensors. However, a more rigorous using as an example the HKUST-1 sensor comparison of the two sensors should include immersed in a CH4/C2H6/CO2 mixture. The sensitivity to measurement noise [35,50]. gas changes from a dominantly CH4/C2H6 to a dominantly CH4/CO2 mixture, but the mass 5.3. Case M = 1,G = 3: single-MOF sensor, of adsorbed gas remains constant because the quaternary gas mixture desorption of C H approximately balances 2 6 Now consider a single-MOF sensor—either [in mass] the adsorption of CO (while CH 2 4 IRMOF-1 or HKUST-1— immersed in a adsorption changes only marginally). CH /N /CO /CH mixture at 1 bar and For the two gas mixtures involving N , the 4 2 2 4 2 298 K. sensitive direction is oriented approximately Fig. 7 visualizes the adsorption function in the direction of changes in the more m = a(x) by showing level sets of it in 3D gas readily adsorbing components, CO or C H . 2 2 6 composition space, colored by the associated N -MOF interactions are weaker than CO - 2 2 response m. The black point is the reference and C H -MOF interactions, reflected in 2 6 gas composition x . The sensitive direction the adsorption isotherms in Fig. 4b. The op Unresponsive spaces in gas sensors 14

m = a(x) [g gas/g MOF] m = a(x) [g gas/g MOF] 0.30 0.30 = {x : a(x) = mop} = {x : a(x) = mop} 0.84 < x < 0.86 0.71 < x < 0.88 0.25 CH4 0.25 CH4 xop xop n n

o 0.20 o 0.20 x i i C t t O c c 2 + a a

r r x f f C

2 H e 0.15 e 0.15 l l 6 = o x o 0 C . m O m 2

2 9 2 x + 2

O C x O x 0.10 O C 0.10

C C C 2 + 2 H O x = [xC2H6 , xCO2 ] x 6 = 2 + C 0 x 2 H .1 C 0.05 6 = 6 0.05 2 H 0 6 = .1 0 4 .1 2 0.00 0.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20 0.25 0.30

C2H6 mole fraction C2H6 mole fraction

Figure 6: Case M = 1,G = 2: comparing the utility of two non-injective gas sensors for determining xCH4 in a CH4/C2H6/CO2 mixture. Along the unresponsive locus B (black curve) associated with xop (black dot), the response of the sensor remains ﬁxed at mop. Observation of the response mop in each sensor places a lower and upper bound on the mole fraction of CH4, xCH4 . The HKUST-1 sensor gives a more useful constraint on xCH4 than the Co-MOF-74 sensor, since xCH4 is constrained to lie in a smaller interval by the response of the HKUST-1 sensor.

v1 (green vector) is orthogonal to the level two sensors at the operating point set {x : a(x) = mop}, and the unresponsive xop = [0.25, 0.2, 0.2] (13) subspace, a plane spanned by v2 and v3 (red vectors), is tangent to it. Note the vectors we ﬁnd: v2 and v3 are not ranked, i.e. one is not v1 ≈ −0.02 0.67 0.74 HKUST-1 (14) more or less unresponsive than the other, and h i are not unique, in that any two orthonormal v1 ≈ 0.02 0.96 0.27 Co-MOF-74. (15) vectors that lie in the unresponsive plane can Both sensorsh are muchi more sensitive to be chosen. The unresponsive subspace is changes in CO2 and C2H6 than to changes two-dimensional due to the single constraint in N2. Note the three faces of each cube in imposed by the response of the sensor on Fig. 7 when one mole fraction is set to zero are the three degrees of freedom in the gas equivalent to Fig. 5. composition. Small changes in gas composition on the unresponsive plane are undetectable 5.4. Case M = 2,G = 3: two-MOF sensor since the sensor response remains constant. array, quaternary gas mixture The sensitive direction v1 is orthogonal to the unresponsive plane. Now consider a two-MOF sensor array, com- Comparing the sensitive directions of the prised of IRMOF-1 and HKUST-1, immersed in a CH4/N2/CO2/CH4 mixture. See Fig. 8a. Unresponsive spaces in gas sensors 15

x = [xN2 , xCO2 , xC2H6 ]

Figure 7: Case M = 1,G = 3: visualization of m = a(x) and the unresponsive plane and sensitive direction for a single-MOF sensor immersed in a CH4/N2/CO2/C2H6 gas mixture at 1 bar and 298 K. The column indicates which MOF is used as the recognition element for the gravimetric sensor. Each panel shows gas composition space. Level surfaces of the adsorption function a(x) are color-coded according to the associated m. At the operating gas composition xop (black point), (i) the sensitive direction v1 is shown by the green vector and (ii) the unresponsive plane is spanned by the two unresponsive directions v2, v3 shown by the red vectors.

Fig. 8b is a visualization to understand the image of the sphere is not exactly a solid the mapping a : x 7→ m, with x in a ellipse owing to the nonlinearity of a(x). 3D gas composition space and m in a 2D Magnitude- changes in the gas composi- sensor response space. The image of the tion in the sensitive and intermediate direc- sphere of radius = 0.04 in gas composition tions in gas composition space—changes of v1 space (blue, left) is the solid ellipse-like and v2, green and yellow vectors in Fig. 8b— shape (blue, right) in sensor response space; are mapped to the endpoints of the semimajor- they are (approximately, for the case of the like and semiminor-like axes of the ellipse-like ellipse-like shape) centered, respectively, at shape— σ1u1 and σ2u2, respectively. Among the reference gas composition xop and its all magnitude- changes in the gas composition associated response mop. Unlike the M = on the sphere (left), a change ∼ v1 produces 2,G = 3 case in the toy sensor array Fig. 3b, the largest-magnitude sensor array response. Unresponsive spaces in gas sensors 16

0 The sensor array is responsive to, but with re- operating gas composition xop, to m . Invoking duced sensitivity, changes in the intermediate the linear approximation in eqn. 2, the inverse 0 direction v2. On the other hand, the sensor problem is to ﬁnd x that satisﬁes: array is unresponsive to small changes in the 0 0 gas composition in the unresponsive direction J(xop)(x − xop) = m − mop =: ∆m. (16)

of v3. The unresponsive subspace is one di- Because J(xop) is fat for a non-injective mensional here because the response of the two (M < G) gas sensor array, eqn. 16 MOFs places two constraints on the three de- has infinite solutions (fewer equations than grees of freedom in the gas composition. unknowns). However, if we impose an additional assumption that the minimal-L2- 6. Non-injective gas sensor arrays are 0 norm gas composition change ∆x := x − xop not useless from xop is the culprit of the change in the sensor response ∆m, the inverse problem is The inverse problem [35] of predicting the gas granted a unique solution. The altered inverse composition x from the response vector m problem: of a non-injective gas sensor array typically has infinite solutions, meaning that infinitely min ||∆x|| (17) many gas compositions are consistent with the ∆x: J(xop)∆x=∆m response. We therefore cannot make a unique has a unique solution given by ∆x = J†∆m, prediction of the gas composition without with J† the Moore-Penrose psuedo-inverse [59] imposing further assumptions. Despite this, of J. Again, the SVD in eqn. 6 plays the non-injective gas sensor arrays could still be central role, as the pseudo-inverse can be practical. constructed from it [59]. First, the response of a non-injective sensor arrays places potentially useful constraints 7. Conclusions and Discussion on the gas composition. For example, consider the single HKUST-1 sensor in Fig. 5. The Mathematically, gas sensor arrays map each response under CH4/CO2/N2 mixtures places gas composition vector x to an equilibrium tight bounds on the mole fraction of CO2 in the response vector m. For gravimetric, MOF- mixture. On the other hand, the response un- based gas sensor arrays, this mapping is der CH4/CO2/C2H6 mixtures cannot delineate characterized by a mixed-gas adsorption model between dominantly CH4/CO2 and CH4/C2H6 m = a(x). An observation of the [total] mass mixtures (see Fig. 2). Nevertheless, the re- of gas adsorbed in a MOF generally places sponse does place tight bounds on the sum of one independent constraint mi = ai(x) on the CO2 and C2H6 mole fractions and, hence, the gas composition. We considered when the CH4 mole fraction. See Fig. 6. This tight there are fewer MOFs in the array than bound on xCH4 could be very useful and suffi- degrees of freedom in the gas composition (the cient for certain applications. number of components minus one, given fixed Second, imposing additional assumptions temperature and pressure). As a result, the about the gas composition can grant the mapping a : x 7→ m is non-injective (many- inverse problem a unique solution. Suppose to-one). Then, some gas composition changes 0 the sensor response changed from mop, at the from xop to x do not cause changes in the Unresponsive spaces in gas sensors 17

x = [xN2 , xCO2 , xC2H6 ]

(a)

gas composition space sensor response space 1u1 x mop op 0.175

0.170 4 7 -

F 0.165 O

mole fraction 2u2 M - 0.160

0.18 o C

C a(x)

2 n i H 0.16

0.155 ] 6 v1 −−→ g 0.14 / v3 g

[ 0.150

0.12 v2 2 m 0.145

0.140 0.06 0.08 0.02 0.04 0.10

C 0.06 mole fractionO 0.12 0.08 2 0.14 N2 0.135 0.06 0.07 0.08 mole fraction m1 [g/g] in HKUST-1

(b)

Figure 8: Case M = 2,G = 3: visualization of a(x) and unresponsive directions for a two-MOF sensor array immersed in a CH4/N2/CO2/C2H6 gas mixture at 1 bar and 298 K. (a) The two- MOF sensor array uses HKUST-1 and Co-MOF-74 as recognition elements. (b) The mapping from gas composition space (left) to sensor response space (right) under a(x) for the sensor array in (a). The small sphere of radius = 0.04 in composition space (blue) is mapped to the solid ellipse-like shape (not an exact ellipse like in Fig. 3b because a(x) is non-linear) in response space (blue). Under the linear approximant of a(x),a ˜(x), the vectors v2, v2, v3 are mapped to σ1u1, σ2u2, 0, respectively. The unresponsive direction is v3 (red) and the sensitive direction is v1 (green).

0 sensor response mop despite x 6= xop and determine the local unresponsive subspace of thus are undetectable. We showed how to gas composition space from the null space Unresponsive spaces in gas sensors 18

of the Jacobian matrix J of the adsorption Does the notion and identification of unrespon- model a(x). Whereupon small changes in gas sive directions generalize beyond gravimetric, composition in an unresponsive direction, the MOF-based sensors? The notion and process sensor response remains constant. More, the of identifying unresponsive directions herein right singular vectors of J associated with its generalizes beyond the specific sensing mech- non-zero singular values give directions in gas anism of observing the total mass of adsorbed compositions to which the sensor is responsive, gas in a MOF. To adapt to a generic array of ranked by sensitivity. We demonstrated sensing elements, each exhibiting some observ- the identification of unresponsive and ranked able property dependent on the gas composi- responsive directions in non-injective, single- tion, redefine: and double-MOF sensor arrays based on • m as the vector containing the observed HKUST-1 and Co-MOF-74 immersed in a properties of the M sensing elements in mixture involving CH4, CO2,C2H6, and response to the gas composition x. N relevant to natural gas sensing [45]. 2 • a(x) as a function that models the Our case studies involved ≤two MOFs and response m to each gas composition x. ≤four components in the gas, instructively allowing us to visualize a(x) and clarify Even without much understanding of the the meaning of the unresponsive and ranked underlying physics linking the response to the responsive directions in gas composition space. gas composition, we could build the model The SVD in eqn. 6 is particularly powerful a(x) as follows. Construct the sensor array. for large sensor arrays and high-dimensional Conduct gas exposure experiments: observe gas composition spaces where, unfortunately, the response over a grid of gas compositions

visualization of a(x) is extremely difficult. to generate data D = {(xi, mi)}. Build a Unsurprisingly, the identification of unre- regression model of a(x) using the training sponsive and ranked responsive directions have data D. analogies in multiple-input, multiple-output Gravimetric MOF sensors are attractive control systems: observability [67–69] and di- from a modeling perspective because we can rectionality [70,71], respectively. construct a reasonable model a(x) without ac- tually constructing the sensor array and con- Why is it useful to identify unresponsive ducting laborious experiments. Gas adsorp- directions in gas sensors? A gas sensor tion data in MOFs is abundant; thermody- is unresponsive to and thus cannot detect namic theories of gas adsorption are well- gas composition changes in the unresponsive developed; and molecular models and simula- direction. Determining the unresponsive tions can predict adsorption in MOFs. The lat- directions of a gas sensor is important ter enables the computational design of MOF- to quantify its limitations and identify its based gas sensor arrays, as exemplified by vulnerabilities to adversarial attacks [44]. In Wilmer and coworkers [49,50], who use molec- addition, given a set of candidate non-injective ular simulations of adsorption to predict the gas sensors for a sensing task, we may be response of gravimetric, MOF-based sensor ar- able rank them according to their unresponsive rays and then evaluate their fitness for gas directions e.g. having a minimal component sensing. along a pertinent direction. Unresponsive spaces in gas sensors 19

How can we remedy non-injective gas sensors response times, (ii) posing inverse problems in to give a determined inverse problem? The terms of the dynamic response of the sensor, observed response of a non-injective gas sen- as in Rajagopalan and Petit [52], and (ii) sor generally places an insufficient number of extracting additional information about the constraints on the gas composition to give the gas composition from the dynamics of the inverse problem a unique solution. We can response. make a non-injective sensor array injective by Second, we assumed isothermal condi- (i) retrofitting it with additional sensing ele- tions. We may wish to use a gas sensor to ments (MOFs), (ii) designing sensing elements robustly predict the gas composition in a range (MOFs) that do not respond appreciably to in- of temperatures. Future work entails building terferents in the gas that we are not concerned an adsorption model a(x; T ) that applies under about, or (iii) imposing additional assumptions different temperatures T , then (a) inputting a on the gas composition as in eqn. 17. separate measurement of T or (b) determining T as part of the inverse problem. Is an injective sensor array the end goal? The Third, imposing additional assumptions inverse problem associated with an injective on the gas composition can endow the sensor array will theoretically always have a underdetermined inverse problem associated unique solution. However, in practice, the with a non-injective gas sensor with a unique measurements of the masses of gas adsorbed solution. This idea of imposing additional in each MOF are corrupted by measurement assumptions, consistent with but not imposed error. As a result, the applied inverse problem by the physics underlying the problem, is [72] for an injective sensor array could still a common approach to, given an inverse (i) have no solution or (ii) be ill-conditioned, problems with nonunique solutions, grant it a where small measurement errors lead to large unique solution [72]. We provided one example changes in the predicted gas composition in eqn. 17, where we sought the minimum- [35]. The conditioning of the inverse problem, norm change in the gas composition consistent colloquially speaking, depends on the diversity with the change in the response. of the MOFs in the array and their interactions with the components of the gas. One method Acknowledgements. to evaluate the fitness of different injective sensor arrays is to analyze the conditioning of We acknowledge support from the U.S. De- their inverse problem [35]. partment of Defense (DoD) Defense Threat Reduction Agency (HDTRA-19-31270) (NG, Future work remains. First, we posed the gas CMS, PT) and the National Science Founda- sensing problem in terms of the equilibrium tion (NSF) (award 1920945) (AH, CMS, XF). response of the gas sensor. However, the mass of gas adsorbed in a MOF is generally 8. References dynamic; gas must enter into and diffuse [1] Wenwen Hu, Liangtian Wan, Yingying Jian, Cong in the thin film of the MOF attached to Ren, Ke Jin, Xinghua Su, Xiaoxia Bai, Hossam the QCM [73–75]. Future work entails Haick, Mingshui Yao, and Weiwei Wu. Elec- (i) considering the diffusion kinetics when tronic noses: From advanced materials to sen- evaluating MOF-based sensors to avoid slow sors aided with data processing. Advanced Ma- Unresponsive spaces in gas sensors 20

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