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[Kosuke Minami](https://orcid.org/0000-0003-4145-1118), [Genki Yoshikawa](https://orcid.org/0000-0002-9136-8964)

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[An Analytical Model of Sorption-Induced Static Mode Nanomechanical Sensing for Multicomponent Analytes](https://mdr.nims.go.jp/datasets/9d782388-4772-46b4-848c-54e646316381)

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An Analytical Model of Sorption-Induced Static Mode Nanomechanical Sensing for Multicomponent AnalytesAn Analytical Model of Sorption-Induced Static ModeNanomechanical Sensing for Multicomponent AnalytesKosuke Minami* and Genki YoshikawaCite This: Anal. Chem. 2025, 97, 19306−19312 Read OnlineACCESS Metrics & More Article Recommendations *sı Supporting InformationABSTRACT: Nanomechanical sensors and their arrays have attracted significant attentionfor detecting, distinguishing, and identifying target analytes, especially complex mixtures ofodors. In the static mode operation, sensing signals are obtained by a concentration-dependent sorption-induced mechanical strain/stress. Understanding of the dynamicresponses is crucial for developing practical artificial olfaction; however, the analyticalformulations are still limited to single-component analytes. Here, we derive an analyticalmodel of viscoelastic material-based static mode nanomechanical sensing for multi-component analytes by extending the theoretical model via solving differential equations.The present model can reduce the dynamic responses to the multicomponent target analytesobserved in the experimental signal responses. Moreover, the use of optimized fittingparameters extracted from pure vapors with viscoelastic parameters allows us to predict theconcentration of each analyte in the multicomponent system.Detecting odors composed of complex mixtures of gaseousmolecules is a fundamental requirement for a wide rangeof applications in electronic noses�models of the nose usingan array of chemical sensors. Since its proposal by Persaud andDodd,1 electronic noses and related technologies have beenextensively studied.2−8 Understanding the dynamic responsesto gases is critically important not only to analyze targetanalytes in the practical samples9,10 but also to extract effectivefeatures for subsequent multivariate analyses, includingmachine learning.11−14 However, analytical investigation ofthe dynamic responses of chemical sensors is still limited.Among the various sensors reported, nanomechanicalsensors have attracted much attention, partly because almostall materials can be used as a receptor layer.5,15,16 Thus, anarray of nanomechanical sensors can be potentially used as asensing unit for the detection of the complex mixtures of odorsin various fields,5 including food,11,13,17 agriculture,9,10,18,19environment,20−22 and medical and healthcare fields.23−29 Forthe so-called static mode operation of nanomechanicalsensors,5,30 the sensing signals are obtained via deformationinduced by the sorption of target molecules in a receptor layer(Figure 1a,b). Such sorption behavior in static modenanomechanical sensing is theoretically investigated for elasticand viscoelastic coatings.31−34 However, those theoreticalformulations are derived from the models of single analyteabsorption and are still unable to be applied to multi-component systems such as complex mixtures of odors.In this study, we extend the theoretical studies based onsingle-component systems32,33 to multicomponent systems andpropose a general expression that includes sorption kineticsalong with viscoelastic stress relaxation of receptor materialsfor nanomechanical sensing in static mode operation. Weformulated a new model by deriving an analytical solution ofoverall transient responses to the multicomponent analytes.This analytical model agrees well with sensor responses to thevapors of binary mixtures experimentally measured using oneof the nanomechanical sensors in static mode operation�Membrane-type Surface stress Sensor (MSS)35,36�coatedwith viscoelastic receptor materials (Figure 1b). The presentmodel can be utilized not only for reproducing entire dynamicresponses of nanomechanical sensors but also for analyzingmulticomponent vapors.■ EXPERIMENTAL SECTIONMaterials. Polycaprolactone (PCL) and poly(vinylidenefluoride) (PVF) were purchased from Sigma-Aldrich Inc. N,N-Dimethylformamide (DMF) was purchased from Sigma-Aldrich Inc. and used for the preparation of receptor layers.n-Hexane, n-nonane, n-dodecane, methanol, ethanol, and 2-propanol were purchased from Sigma-Aldrich Inc., TokyoChemical Industry Co., Inc., or Nacalai Tesque Co., Ltd.Sensor Preparation. The construction of the MSS and itsworking principle have been previously reported (Figure1a,b).35,36 Briefly, MSS consists of a silicon membranesuspended by four sensing beams, in which piezoresistors areembedded. The membrane is coated with a receptor material,Received: June 11, 2025Revised: August 13, 2025Accepted: August 14, 2025Published: August 27, 2025Articlepubs.acs.org/ac© 2025 American Chemical Society19306https://doi.org/10.1021/acs.analchem.5c03397Anal. Chem. 2025, 97, 19306−19312This article is licensed under CC-BY 4.0Downloaded via 192.51.222.132 on September 11, 2025 at 00:36:56 (UTC).See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.https://pubs.acs.org/action/doSearch?field1=Contrib&text1="Kosuke+Minami"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Genki+Yoshikawa"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/showCitFormats?doi=10.1021/acs.analchem.5c03397&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?goto=articleMetrics&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?goto=recommendations&?ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?goto=supporting-info&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=tgr1&ref=pdfhttps://pubs.acs.org/toc/ancham/97/35?ref=pdfhttps://pubs.acs.org/toc/ancham/97/35?ref=pdfhttps://pubs.acs.org/toc/ancham/97/35?ref=pdfhttps://pubs.acs.org/toc/ancham/97/35?ref=pdfpubs.acs.org/ac?ref=pdfhttps://pubs.acs.org?ref=pdfhttps://pubs.acs.org?ref=pdfhttps://doi.org/10.1021/acs.analchem.5c03397?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://pubs.acs.org/ac?ref=pdfhttps://pubs.acs.org/ac?ref=pdfhttps://acsopenscience.org/researchers/open-access/https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/which generates surface stress caused by the sorption-inducedexpansion. PCL and PVF were deposited directly onto theMSS membrane by inkjet spotting using an inkjet spotter(LaboJet-500SP, MICROJET Co. Ltd.) equipped with anozzle (IJHBS-300, MICROJET Co. Ltd.). Each polymerwas dissolved in DMF at a concentration of 1 mg/mL, and theresulting solutions were deposited onto each surface of theMSS membrane. The inkjet conditions are used according tothe previous study.28Sensing System and Procedure. The MSS chip wassettled into a homemade Teflon chamber and placed in anincubator with a controlled temperature at 25.00 ± 0.02 °C.The chamber was connected to gas lines consisting of fourmass flow controllers (MFCs), three vials for target liquidsamples, and a mixing chamber (Figure S1). Three MFCs (i.e.,MFC-1, MFC-2, and MFC-3) were connected to each vial tointroduce the corresponding saturated vapor. In the injectionprocesses, the binary and ternary mixtures of homologousseries with different concentrations were prepared by varyingthe flow rates of the MFCs (Tables S1 and S2). MFC-4 wasused for diluting the vapors of binary and ternary mixtures aswell as for purging by changing the flow rates every 10 s (seealso Figure 2b). Total flow rate was maintained at 100 mL/minduring the experiments, that is, the concentration Pi/Pio of eachanalyte was varied in the range of 0−30%, where Pi and Pio arethe partial pressure and saturated vapor pressure of the i-thanalyte, respectively. Pure nitrogen gas was used as the carrierand purging gases. Data were measured with a bridge voltageof −0.5 V and recorded with a sampling rate of 10 or 20 Hz.The data collection program was designed using LabVIEW(Emerson Electric Co.).33Curve Fitting and Estimation of Parameters. Toextract the fitting parameters from pure vapors, we usedleast-squares methods with a trust region reflective algorithmusing Python 3 with SciPy module, according to the previousstudy.33 The amplitude constant γσsat., the diffusion timeconstant τi, the relaxation time constant τr, and the ratio ofunrelaxed and relaxed biaxial moduli M0/M∞, in addition tothe time t0 when the first injection starts were optimized usingthe derived formula in eq 7 (see Results and DiscussionSection). The initial fitting parameters were set as follows: γσsat.= 1 [mV], M0/M∞ = 3, τi = 30 [s], τr = 2 [s], and t0 = 0 [s].The bounds for each parameter were set at γσsat. > 0 [mV],M0/M∞ ≥ 1, τi > 0 [s], τr > 0 [s], and −1 < t0 < 2 [s].To demonstrate the prediction of each analyte concen-tration, we used least-squares methods with a trust regionreflective algorithm using Python 3 with SciPy module. Eachamplitude γσi and t0 was optimized using the derived formulain eq 7. The initial fitting parameter was σio, where σio was σiextracted from the responses to pure vapors in Table S1. Thebounds for each parameter were set at 0 ≤ σi ≤ 1.2σio.■ RESULTS AND DISCUSSIONGoverning Equations for Multicomponent Sensing.To derive theoretical formulations, we use a theory based onthe viscoelastic behavior derived from the three-parametersolid model as follows:32,37−39Figure 1. Sorption-induced model for nanomechanical sensors. (a) Working principle of nanomechanical sensors in static mode operation. (b)Typical geometries of one of the nanomechanical sensors, MSS. Color gradient represents the displacement in z-direction (i.e., perpendicular to themembrane surface) simulated by finite element analysis (COMSOL Multiphysics with the Structural Mechanics module). (c) Biaxial relaxationmodulus M(t) = σ(t)/ε for a constant strain in a viscoelastic material behaving as a three-parameter solid model. M0 = 100 [MPa]; M∞ = 40[MPa]; τr = 10 [s]. (d) Sorption-induced strain ε(t)/λiKiCi as a first-order sorption kinetics. τi = 30 [s]. (e) Typical responses to single analyte withτi = 5, 10, 20, 30, 60 [s]; M0/M∞ = 0.72/0.51; τr = 22 [s] calculated by the model in eq 7.Figure 2. Gas injection models. (a) A single rectangular injection-purge and corresponding typical response. (b) A rectangular pulsewave-like multistep injection-purge and corresponding typicalresponse.Analytical Chemistry pubs.acs.org/ac Articlehttps://doi.org/10.1021/acs.analchem.5c03397Anal. Chem. 2025, 97, 19306−1931219307https://pubs.acs.org/doi/suppl/10.1021/acs.analchem.5c03397/suppl_file/ac5c03397_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.analchem.5c03397/suppl_file/ac5c03397_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.analchem.5c03397/suppl_file/ac5c03397_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig1&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig1&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig1&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig1&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig2&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig2&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig2&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig2&ref=pdfpubs.acs.org/ac?ref=pdfhttps://doi.org/10.1021/acs.analchem.5c03397?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-as+ = +ttM t Mt( )dd( )ddr r 0 (1)where M0 and M∞ denote the unrelaxed (instantaneous) andthe relaxed (asymptotic) biaxial moduli, respectively, and τr isthe time constant of stress relaxation. The three-parametersolid model describes the stress (σ)/strain (ε) relationship in aviscoelastic solid exhibiting both viscous and elastic proper-ties.37,40 The behavior of the three-parameter solid modelgoverned by eq 1 is depicted in Figure 1c. In this theory, eq 1is directly applicable to the cantilever-type nanomechanicalsensors (Figure S2; SI Text) when the coating film issignificantly soft or thin , i.e., M0 ≪ Ms or hf ≪ hs, whereMs denotes the elastic biaxial modulus of the substrate; hf andhs are thicknesses of the coating film and the substrate,respectively.32 Notably, a signal response of the MSS used inthis study is directly proportional to the internal strain εf thecoating film according to numerical simulations,41 which issimilar to the cantilever-type sensors.42 Therefore, the modelrepresented in eq 1 can also be directly applied to the MSS.33In the sorption-induced nanomechanical sensing, theinternal strain εf in the receptor material at any time t ismodeled as a function of the concentration of analytes in thereceptor material as32,33=t tC( ) ( )f (2)for small volume expansion (i.e., εf ≪ 1). Here, C(t) denotesthe concentration matrix of analytes in the receptor material asa function of time and Λ is a proportional factor matrix of λicorresponding to the specific volume νi of the i-th analyte (i.e.,λi = νi/3).32,33In the case of typical gas sensing with nanomechanicalsensors, a single rectangular injection-purge or a rectangularpulse wave-like multistep injection-purge can be considered.33Here, we assign the odd and even steps to the injection andpurge processes, respectively (Figure 2). Let Cg be a matrix ofthe constant concentration Ci of the i-th analyte in the gasphase during injection processes (i.e., 2n−1 steps; n ∈ ). Theanalyte concentrations Cg in the gas phase at the n-th step canbe considered as= <t n t t tC 1 C( ) ( ) , ,g A g n n1 (3)where 1A(n) is the indicator function; 1A(n) = 1 if n is odd, andzero otherwise.For the derivation of the equations governing theconcentration of each analyte in a receptor material coatedon a nanomechanical sensor during absorption/desorptionprocesses, we assume a first-order absorption of each analytealong with an independent sorption among analytes asillustrated in Figure 1a. Diffusion of analytes into the bulk ofa coating film is generally the rate-limiting process inabsorption.31 If the diffusion of each analyte is assumed tobe Fickian, the absorption rate can be approximated to beproportional to the difference between the equilibriumconcentration in the receptor material and the concentrationof each analyte absorbed in the receptor material C(t).32,33From eq 3, the reaction rate of the concentration of eachanalyte in the receptor material C(t) is given by= [ ]tt tCK C Cdd( ) ( )s p g1(4)where Ts is a diagonal matrix of the diffusion time constant τiof the i-th analyte; Kp is a diagonal matrix of the partitioncoefficient Ki of the i-th analyte (see Figure 1d).32 Then, fromeq 4 with eq 3, the recurrence relation can be found (see SIText in Supporting Information). The recurrence formula canbe solved, and hence the dynamic concentration Cn(t) at the n-th step can be obtained as= [ ]tC K 1 I A C( )n p A i n g (5)where Ii is an identity matrix and a diagonal matrix An at anytime t of the n-th step is given by==t eA ( ) ( 1)njnj t t01( )j s1(6)Since the internal strain εf can be assumed to be directlyproportional to the concentrations of each analyte in thereceptor material in eq 2 in the case of small expansion (i.e., εf≪ 1),32,33 the dynamic stress change of nanomechanicalsensors at the n-th step σn(t) can be solved by substituting eqs2 and 5 into eq 1 (see also SI Text) and is obtained as= [ ]t M aK 1 I A B I B C( ) ( )n p A i n n i g (7)with stress components in sorption kinetics An in eq 6 and inviscoelastic stress relaxation an at any time t of the n-th step,which is given by==a t e( ) ( 1)njnj01 t tjr(8)where a diagonal matrix B is=ikjjjjjy{zzzzzikjjjjjy{zzzzzMMB I I1 1s irs sri1 0 11(9)if τi ≠ τr. In eq 7, the analytical solution clearly expresses thestress in terms of the elastic properties and stress relaxationprofiles in different forms with the viscoelastic relation in eq 9.The stress σn(t) given in eq 7, which is assumed to be directlyproportional to the concentrations of each analyte in the gasphase, is directly proportional to the signal responses ofnanomechanical sensors.32,33 It should be noted that the stressσsat. at the equilibrium state or a steady state of the injectionprocess can be described as= = =t M K Clim ( )tn p giisat.(10)where σi is the stress at the equilibrium state derived from thesorption of the i-th analyte, which is given by σi = M∞λiKiCi.33Numerical Calculations of Nanomechanical Sensing.In this section, viscoelastic material-coated nanomechanicalsensing responses are numerically calculated using eq 7. One ofthe important features of viscoelastic behaviors in nano-mechanical sensing is an overshoot as can be seen in Figure1e.32 If absorption of analytes occurs faster than the stressrelaxation of a receptor material (Figure 1c,d), the responseresulting from the absorption-induced deformation will be asignal output higher than the level of amplitude σsat. because ofthe accumulation of unrelaxed stress and, subsequently, therelaxed stress (Figure 1e). In single analyte sensing,32 theresponse exhibits an overshoot only if M0 > M∞ and if< MMi r0(11)Analytical Chemistry pubs.acs.org/ac Articlehttps://doi.org/10.1021/acs.analchem.5c03397Anal. Chem. 2025, 97, 19306−1931219308https://pubs.acs.org/doi/suppl/10.1021/acs.analchem.5c03397/suppl_file/ac5c03397_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.analchem.5c03397/suppl_file/ac5c03397_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.analchem.5c03397/suppl_file/ac5c03397_si_001.pdfpubs.acs.org/ac?ref=pdfhttps://doi.org/10.1021/acs.analchem.5c03397?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asFigure 3a shows the numerically calculated responses to thebinary mixtures with varied stresses σi derived from the i-thanalyte. The diffusion time constants τ1 and τ2 are 5 and 60 s,respectively, with the response of an analyte 1 exhibiting anovershoot and the other not (i.e., an analyte 2), as shown inFigure 1e (see also Figure S3 in Supporting Information formore details of other combinations of τi). By increasing thecontributions of σ1 (∝C1), a shoulder appears in the response(e.g., σ1 = 0.3−0.4), and then, an overshoot is observed(Figure 3a). Notably, in a binary system, even if an overshootis observed, the response may subsequently reach a higher levelof equilibrium amplitude σsat. than that of the local maximumof the overshoot in the case of a specific balance between σ1and σ2, e.g., σ1:σ2 = 0.7:0.3 as an example shown in Figure 3b.These trends can also be seen in the numerical calculations formixtures of three or more analytes as shown in Figure S4 (seealso supporting Python code in Supporting Information). It istheoretically confirmed that these trends never occur in thecase of a single analyte according to the previous model.32,33The present model in eq 7 is also applicable to the multistepinjection-purge cycles (Figure 2b). Figure 3c shows thenumerically calculated responses to various binary mixtures.If τi is large, analytes diffused into a receptor material duringthe injection process are not desorbed completely during thepurge process within a limited duration, resulting in an increasein the baseline. The responses to these binary mixtures alsoexhibit baseline drift as shown in Figure 3c, because of thelarge enough values of τ1 and τ2 (see also Figure S5 for moredetails on other combinations of τi; the supporting Pythoncode for mixtures of three or more analytes).It should be noted here that a similar dynamic response ofthe nanomechanical sensor is often experimentally reproducedin each injection-purge cycle by repeating the cycles when theduration of each injection and purge is fixed.11,25,41 In the caseof the fixed duration τ, i.e., τ = tn − tn−1, eqs 6 and 8 can besimplified (see SI Text). If the number of injection-purgecycles is sufficiently large (e.g., n → ∞), An and an can befurther simplified as= +t e eA Ilim ( ) ( 1) ( )nnnit t1 1 ( )s n s111(12)andFigure 3. Numerical calculations of the signal responses using thederived model in eq 7. (a) Numerically calculated responses to binarymixtures for the first injection (i.e., n = 1). σsat. = σ1 + σ2, while σ1 isvaried from 0 to 1. τ1 = 5 [s], τ2 = 60 [s]. (b) Response to the binarymixture with σ1/σsat. = 0.7 and σ2/σsat. = 0.3. Dashed lines correspondto the stresses derived from analytes 1 and 2. (c) Responses to thebinary mixture for multistep injection-purge processes. Dashed linescorrespond to the first step injection shown in (a). Other parametersare fixed: M0/M∞ = 0.72/0.51; τr = 22 [s].Figure 4. Numerically calculated dynamic responses to the binary mixtures measured experimentally by polymer-coated MSS. (a) Responses ofPCL-coated MSS to pure alkanes: n-hexane (C6; green); n-nonane (C9; red); and n-dodecane (C12; blue) with fitting results (black dashed lines).(b) Responses of PVF-coated MSS to alcohols: methanol (MeOH; green); ethanol (EtOH; red); and 2-propanol (IPA; blue) with fitting results(black dashed lines). (c, d) Responses to binary mixtures of homologous series of alkanes (c) and alcohols (d) with predicted responses based onthe extracted fitting parameters (dashed lines). The color gradient from purple to red indicates the conditions from entries 1 to 7 in Table S1.Analytical Chemistry pubs.acs.org/ac Articlehttps://doi.org/10.1021/acs.analchem.5c03397Anal. Chem. 2025, 97, 19306−1931219309https://pubs.acs.org/doi/suppl/10.1021/acs.analchem.5c03397/suppl_file/ac5c03397_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.analchem.5c03397/suppl_file/ac5c03397_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.analchem.5c03397/suppl_file/ac5c03397_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.analchem.5c03397/suppl_file/ac5c03397_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.analchem.5c03397/suppl_file/ac5c03397_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig3&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig3&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig3&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig3&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig4&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig4&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig4&ref=pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.analchem.5c03397/suppl_file/ac5c03397_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig4&ref=pdfpubs.acs.org/ac?ref=pdfhttps://doi.org/10.1021/acs.analchem.5c03397?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-as=+a teelim ( )( 1)1nnn 1/ rt tnr1(13)respectively. From eq 7 with eqs 12 and 13, it was theoreticallyconfirmed that each injection-purge cycle would exhibit thesame dynamic response if the number of injection-purge cyclesis sufficiently large, as can be seen in Figure 3c.Experimental Validation of the Model. To validate thepresent model in eq 7, we experimentally measured signalresponses of viscoelastic material-coated MSS to binarymixtures of homologous series. Since homologous series oflinear alkanes and short-chain alcohols (e.g., methanol,ethanol, and propanol) tend to have similar chemicalproperties, each homologous series can be assumed to exhibitindependent sorption behaviors.43 It should be noted that asignal response of MSS generally correlates with the internalstrain similar to the cantilever-type sensors.33,41,42,44,45 Thesignal output of MSS is given by Vout(t) = γσn(t) + Vout(t0),where γ is a proportionality factor that converts stress σn intoan output signal of MSS and Vout(t0) is the signal output at t =t0, i.e., baseline output (see Figure 2).33 As viscoelastic receptormaterials, polycaprolactone (PCL) and poly(vinylidene fluo-ride) (PVF) were coated on MSS because of their sensitivity toalkanes and alcohols, respectively. To measure the signalresponses to the binary mixtures, we constructed themeasurement setup as shown in Figure S1. Three MFCs(MFC-1, MFC-2, and MFC-3) were connected to each vial tointroduce the corresponding saturated vapor. By varying theflow rates of MFCs in the injection processes with purenitrogen by MFC-4, the binary mixtures of homologous serieswith different concentrations were prepared (Table S1).According to our previous study,33 parameters extractedfrom signal responses of three to four injection-purge cyclescan yield more accurate values than those extracted from asingle injection. Therefore, four injection-purge responsecycles of PCL-coated MSS to pure n-alkanes and PVF-coatedMSS to pure alcohols were measured, as shown in Figure 4a,b.First, we extracted the sorption kinetic parameters andviscoelastic parameters from the responses to pure vapors,i.e., a single analyte system.33 The fitting results are shown asdotted lines in Figure 4a,b, and the extracted fitting parametersare summarized in Table 1. As expected from eqs 6−8, thesorption kinetic parameters (i.e., γσi and τi) depend on thechemical properties of the i-th vapor, while the viscoelasticparameters (i.e., τr and M0/M∞) are reasonably consistent withthe vapors for each viscoelastic receptor material.To demonstrate the predictability of the responses tomulticomponent vapors, the fitting parameters extracted fromthe responses to each pure vapor in Table 1 were used tocalculate the signal responses to each binary and ternarymixture using eq 7 with theoretical concentrations controlledby MFCs (Tables S1 and S2). The comparisons between thepredicted results and the experimental responses of the binaryand ternary mixtures are shown in Figures 4c,d and 5. Thepredicted responses agree well with the experimentallymeasured responses to both the binary mixtures and theternary mixtures, demonstrating the potential of the presentmodel for predictive capability.We also demonstrated the ability of the present model in eq7 to predict the concentrations of each analyte (i.e., γσi ∝ Ci)in the binary mixtures using known sorption kinetic parameters(i.e., τi) extracted from each pure vapor along with viscoelasticparameters of each material (i.e., τr and M0/M∞). Figure 6shows the fitting results and plots of predicted γσi from threeindependent measurements for each binary mixture. In thecase of the binary mixtures of MeOH and EtOH, an explicitconcentration dependency was observed (Figure 6d), althoughthe predicted γσi, especially for EtOH, do not linearly correlatewith the theoretical vapor concentration Ci because thesorption processes of each analyte in the binary mixtures arenot perfectly independent. However, when the concentrationof IPA in the binary mixture is low, the predicted γσi of IPAyields almost zero (Figure 6e,f). This tendency is particularlypronounced when the difference in τi is small, e.g., the binarymixtures of EtOH and IPA (Figure 6f). This may be attributedTable 1. Extracted Fitting Parameters of Pure Alkanes and Alcoholssample γσi [mV]a,b τi [s]a,b τr [s]a,b M0/M∞a,bPCL/Alkanesn-hexane 3.05 ± 0.01 10.11 ± 0.20 1.00 ± 0.02 5.04 ± 0.04n-nonane 2.50 ± 0.20 21.39 ± 3.30 1.63 ± 0.06 6.44 ± 0.58n-dodecane 1.76 ± 0.13 22.86 ± 2.02 2.28 ± 0.19 4.76 ± 0.05PVF/Alcoholsmethanol 4.65 ± 0.15 28.21 ± 0.93 1.98 ± 0.07 3.74 ± 0.07ethanol 2.14 ± 0.23 45.81 ± 6.43 2.30 ± 0.04 3.74 ± 0.182-propanol 1.22 ± 0.12 53.08 ± 5.83 1.98 ± 0.40 3.50 ± 0.21aAn average value ± standard deviation from six independent measurements. bγσi, signal amplitude of the i-th analyte at the steady-state; τi,diffusion time constant of the i-th analyte; τr, relaxation time constant of coating film; M0/M∞, the ratio of unrelaxed and relaxed biaxial moduli ofcoating film.Figure 5. Numerically calculated dynamic responses to the ternarymixtures of alcohols experimentally measured by PVF-coated MSS. (a,b) Responses of PVF-coated MSS to ternary mixtures of MeOH (C1),EtOH (C2), and IPA (C3) with fitting results (dashed lines) based onthe extracted fitting parameters (τi, τr, and M0/M∞). MeOH andEtOH concentrations are varied from 0% to 10% with fixed IPAconcentrations at 2% (a) and 10% (b) in Table S2. The color gradientfrom purple to red indicates the conditions of entries 1−6 (a) andentries 7−12 (b) in Table S2.Analytical Chemistry pubs.acs.org/ac Articlehttps://doi.org/10.1021/acs.analchem.5c03397Anal. Chem. 2025, 97, 19306−1931219310https://pubs.acs.org/doi/suppl/10.1021/acs.analchem.5c03397/suppl_file/ac5c03397_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.analchem.5c03397/suppl_file/ac5c03397_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.analchem.5c03397/suppl_file/ac5c03397_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig5&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig5&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig5&ref=pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.analchem.5c03397/suppl_file/ac5c03397_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.analchem.5c03397/suppl_file/ac5c03397_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig5&ref=pdfpubs.acs.org/ac?ref=pdfhttps://doi.org/10.1021/acs.analchem.5c03397?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asto the small contribution of IPA in the signal responsescompared to MeOH or EtOH. Although there are somelimitations, these results demonstrate the potential of thederived equation for predicting the concentrations of eachanalyte in the mixture.It is noteworthy that the logarithm of the partitioncoefficient of each analyte log Ki generally yields an inverserelationship with the logarithm of the saturated vapor pressurelog Pio, particularly within homologous series (i.e., Ki ∝ 1/Pio).43,46 This trend implies that the partition coefficient forethanol is higher than that for methanol. Consequently,ethanol molecules may tend to be retained more in the solidphase, while methanol molecules are likely to be in the gasphase, resulting in a convex concentration profile for ethanoland a slightly concave profile for methanol (Figure 6d). Thisbehavior is analogous to derivations of the vapor pressure fromRaoult’s law, often observed in gas−liquid equilibrium.46■ CONCLUSIONWe derived a general analytical expression that describes thedynamic responses of viscoelastic material-based static modenanomechanical sensors to multicomponent analytes. Thetheory includes the viscoelastic stress relaxation and sorption-induced responses with multiple analytes. Although the presentmodel assumes that the sorption behaviors of each analyte areindependent, the model is in good agreement withexperimental results measured by MSS coated with viscoelasticreceptor materials. The present model has the predictivecapability of the dynamic responses of nanomechanicalsensing, including the trends of overshoot. Moreover, themodel can be utilized for predicting each analyte concentrationin the mixed vapors using the known sorption parametersextracted from pure vapors, along with the viscoelasticparameters. The present model has significant potential toanalyze the complex mixtures of odors as well as the analytes inthe presence of interfering gases such as humidity, contributingto the development of practical artificial olfaction.■ ASSOCIATED CONTENT*sı Supporting InformationThe Supporting Information is available free of charge athttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397.Detailed derivation of eqs 7, 12, and 13, and detailednumerical calculations (PDF)Python code for visualizing numerical calculations (ZIP)■ AUTHOR INFORMATIONCorresponding AuthorKosuke Minami − Research Center for Macromolecules andBiomaterials, National Institute for Materials Science(NIMS), Tsukuba, Ibaraki 305-0044, Japan; InternationalCenter for Young Scientists (ICYS), National Institute forMaterials Science (NIMS), Tsukuba, Ibaraki 305-0044,Japan; orcid.org/0000-0003-4145-1118;Email: MINAMI.Kosuke@nims.go.jpAuthorGenki Yoshikawa − Research Center for Macromolecules andBiomaterials, National Institute for Materials Science(NIMS), Tsukuba, Ibaraki 305-0044, Japan; MaterialsScience and Engineering, Graduate School of Pure andApplied Science, University of Tsukuba, Tsukuba, Ibaraki305-8571, Japan; orcid.org/0000-0002-9136-8964Complete contact information is available at:https://pubs.acs.org/10.1021/acs.analchem.5c03397FundingThis study was financially supported by a Grant-in-Aid forScientific Research (A), JSPS, MEXT, Japan (no. 18H04168);a Grant-in-Aid for Scientific Research (C), JSPS, MEXT, Japan(no. 22K05324 and no. 25K08830); a Grant-in-Aid forFigure 6. Prediction of vapor concentrations in the binary mixtures. (a−c) Responses of PVF-coated MSS to binary mixtures of MeOH−EtOH(a), MeOH−IPA (b), and EtOH−IPA (c) with fitting results (dashed lines) based on the extracted fitting parameters (τi, τr, and M0/M∞). Colorgradient from purple to red indicates the conditions from entries 1 to 7 in Table S1. (d−f) Plots of predicted γσi as a function of theoretical vaporconcentrations Ci. Green, MeOH; red, EtOH; blue, IPA. Error bars are ± standard deviations.Analytical Chemistry pubs.acs.org/ac Articlehttps://doi.org/10.1021/acs.analchem.5c03397Anal. Chem. 2025, 97, 19306−1931219311https://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?goto=supporting-infohttps://pubs.acs.org/doi/suppl/10.1021/acs.analchem.5c03397/suppl_file/ac5c03397_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.analchem.5c03397/suppl_file/ac5c03397_si_002.ziphttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Kosuke+Minami"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://orcid.org/0000-0003-4145-1118mailto:MINAMI.Kosuke@nims.go.jphttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Genki+Yoshikawa"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://orcid.org/0000-0002-9136-8964https://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig6&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig6&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig6&ref=pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.analchem.5c03397/suppl_file/ac5c03397_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acs.analchem.5c03397?fig=fig6&ref=pdfpubs.acs.org/ac?ref=pdfhttps://doi.org/10.1021/acs.analchem.5c03397?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asChallenging Research (Pioneering), JSPS, MEXT, Japan (no.20K20554); the Public/Private R&D Investment StrategicExpansion Program (PRISM), Cabinet Office, Japan; andICYS, NIMS.NotesThe authors declare no competing financial interest.■ ACKNOWLEDGMENTSWe thank Yukiko Nakayama, NIMS, Japan for her technicalassistance. K.M. acknowledges the International Center forYoung Scientists (ICYS) program, NIMS, Japan.■ REFERENCES(1) Persaud, K.; Dodd, G. Nature 1982, 299, 352−355.(2) Potyrailo, R. A. Chem. Rev. 2016, 116, 11877−11923.(3) Covington, J. A.; Marco, S.; Persaud, K. C.; Schiffman, S. S.;Nagle, H. T. IEEE Sens. J. 2021, 21, 12969−12990.(4) Manzini, I.; Schild, D.; Di Natale, C. Physiol. Rev. 2022, 102, 61−154.(5) Minami, K.; Imamura, G.; Tamura, R.; Shiba, K.; Yoshikawa, G.Biosensors 2022, 12, 762.(6) Milone, A.; Monteduro, A. G.; Rizzato, S.; Leo, A.; Di Natale, C.;Kim, S. S.; Maruccio, G. Adv. Sustainable Syst. 2023, 7, 2200083.(7) Peveler, W. J. ACS Sens. 2024, 9, 1656−1665.(8) Wesoły, M.; Przewodowski, W.; Ciosek-Skibinśka, P. TrACTrends Anal. Chem. 2023, 164, 117082.(9) Minami, K.; Kobayashi, H.; Matoba, M.; Kamiya, Y.; Maji, S.;Nemoto, T.; Tohno, M.; Nakakubo, R.; Yoshikawa, G. 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