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[Meng-Qun Feng](https://orcid.org/0000-0002-3185-7555), [Kosuke Minami](https://orcid.org/0000-0003-4145-1118), [Yingcheng Zhou](https://orcid.org/0000-0002-6999-4897), [Genki Yoshikawa](https://orcid.org/0000-0002-9136-8964)

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[Analytical modeling and decoupling of humidity effects in nanomechanical sensing based on sorption kinetics and viscoelastic stress relaxation](https://mdr.nims.go.jp/datasets/78086aa9-f01b-4c3b-b6f4-90e0516a3776)

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Analytical modeling and decoupling of humidity effects in nanomechanical sensing based onsorption kinetics and viscoelastic stress relaxationMeng-Qun Feng,1, 2 Kosuke Minami,1, 3, ∗ Yingcheng Zhou,1, 2 and Genki Yoshikawa1, 2, †1Research Center for Macromolecules and Biomaterials,National Institute for Materials Science (NIMS), 1-1 Namiki, Tsukuba, Ibaraki 305-0044 Japan2Materials Science and Engineering, Graduate School of Pure and Applied Science,University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571 Japan3International Center for Young Scientists (ICYS), National Institute forMaterials Science (NIMS), 1-1 Namiki, Tsukuba, Ibaraki 305-0044 Japan(Dated: April 28, 2025)Nanomechanical sensors have gained significant attention as powerful tools for detecting target analytes;however, their signals are often influenced by environmental humidity. In static mode operation, these signalsare obtained by a concentration-dependent sorption-induced mechanical strain/stress. In this study, we derive ananalytical model to describe the response of viscoelastic material-coated nanomechanical sensors by incorpo-rating humidity effects based on sorption kinetics and viscoelastic stress relaxation of receptor materials. Thismodel is capable of reproducing the dynamic responses observed in the experimental signal responses undervarying humidity conditions. Moreover, it allows for the subtraction of humidity effects, facilitating the preciseisolation of analyte-specific signals. These results provide a theoretical framework for decoupling environmentalbackground factors, such as humidity effects, in nanomechanical sensors.I. INTRODUCTIONNanomechanical sensors and their arrays have emerged as apowerful platform for detecting a wide range of target gases,making them particularly valuable for practical applicationsin artificial olfaction, including odor detection, environmen-tal monitoring, agriculture, and healthcare and medical diag-nostics [1–11]. However, humidity is a critical issue in suchapplications, as it significantly influences and fluctuates theirperformance [12–15]. Variation in humidity often leads toa baseline shift, suppression of sensor performance, and dy-namic fluctuation in sensor response, complicating the accu-rate detection of target analytes [16–19]. For instance, Inadaet al. [9] recently proposed statistically reproducible protocolsfor breath analysis to mitigate this issue by selecting featuresrobust to the effects of humidity, yet the underlying mecha-nisms remain unclear. Therefore, understanding the humidityinfluence on sensing performance is crucial for achieving re-liable detection.In the so-called static mode operation, nanomechanical sen-sors are coated with a receptor material that absorbs the ana-lyte [1, 20]. The sensing signals are obtained through the de-formation induced by surface stress resulting from the sorp-tion of target analytes, including water molecules, into the re-ceptor material [Fig. 1(a)]. Based on theoretical models of thesurface stress [21–23], such sorption behaviors in the staticmode nanomechanical sensors have been extensively studiedtheoretically, including first-order sorption kinetics in elastic[24] and viscoelastic coatings [25–29] and two-step reactionkinetics [30]. However, these models do not cover humidityeffects as a background component and an analytical investi-gation into the influence of humidity is essential for improving∗ Contact author; MINAMI.Kosuke@nims.go.jp† Contact author; YOSHIKAWA.Genki@nims.go.jpthe reliability of nanomechanical sensors in practical applica-tions.In this study, we propose an analytical model that pre-dicts the dynamic responses of nanomechanical sensors un-der varying humidity conditions. By integrating sorption ki-netics and viscoelastic stress relaxation, the model providesa comprehensive description of the dynamic response dur-ing multistep injection-purge cycles. Numerical calculationsbased on this analytical model revealed the effects of humid-ity in sensor responses under different initial humidity lev-els and varying humidity changes. Experimental validationusing a viscoelastic receptor material-coated nanomechani-cal Membrane-type Surface stress Sensor (MSS) [1, 31, 32]in static mode [Fig. 1(b)] under different humidity condi-tions demonstrates strong agreement between predicted andobserved sensing responses. Furthermore, our model enableseffective decoupling of the humidity effects from the sensingsignals, allowing analyte-induced components in the signalsto be accurately extracted in humid environments. Overcom-ing the long-standing challenges posed by humidity, this studyestablishes a robust predictive method for designing reliablenanomechanical sensor systems, advancing their practical ap-plications in various fields.II. THEORYTo derive the theoretical formulations, we employ a the-ory based on the viscoelastic behavior described by the three-parameter solid model. The model is given by [25, 26, 28, 33]σ(t)+ τrdσdt= M∞ε(t)+ τrM0dεdt, (1)where τr is the time constant of stress relaxation, M0 and M∞denote the unrelaxed (instantaneous) and the relaxed (asymp-totic) biaxial moduli, respectively. In this model, when thecoating film is significantly soft and/or thin, sorption induced2FIG. 1. Nanomechanical sensor and experimental setup. (a) Illustra-tion of the working principle of nanomechanical MSS in static modeoperation. (b) Schematic of an MSS sensing unit. (c) An experi-mental setup used for multistep gas injection and purge cycles undervaried humidity conditions. (d) The typical rectangle injection-purgesequence, where red indicates the analyte concentration (top) andblue indicates relative humidity (bottom). The initial RH is repre-sented as Cw.strain (and therefore the stress) in the coating film can be ap-proximated as uniform, thereby enabling Eq. (1) to be directlyapplied to cantilever-type nanomechanical sensors [26]. In thesorption-induced deformation in nanomechanical sensors, theinternal strain ε f in the coating film is modeled as a functionof the concentration of analytes Cgi (t) and water Cgw(t) in thecoating film as [26, 28, 29]ε f (t) = λiCgi (t)+λwCgw(t), (2)where λi and λw are the proportional factors corresponding tothe specific volume of analyte and water divided by three forsmall expansion [26].Gas sensing steps with nanomechanical sensors can be rep-resented by a single rectangular injection or a rectangularpulse waveform, such as a multistep injection-purge sequence[28]. In this setup [Fig 1(c)], odd steps correspond to theinjection process, while even steps correspond to the purgeprocess [Fig 1(d)]. Let Cg, Cw, and ∆Cw be a constant con-centration of analyte in the gas phase at the injection process(i.e., 2n–1 steps; n ∈ N), an initial constant relative humidity(RH), and constant humidity difference (i.e., humidity differ-ence between purge and injection processes), respectively [seeFig. 1(d)]. The concentrations of analyte Cgi,n and water Cgw,nat the gas phase at the n-th step can be described asCgi,n(t) =11A(n)Cg, (3a)Cgw,n(t) =Cw + 11A(n)∆Cw, (3b)where 11A(n) is the indicator function; 11A(n) = 1 if n is odd,and zero otherwize.In the derivation of the concentration equations of analyteswithin a viscoelastic receptor material coated on a nanome-chanical sensor during absorption and desorption, it is as-sumed that the absorption of analytes follows first-order kinet-ics and interactions among analytes are independent, as illus-trated in Fig. 1(a) [29]. Diffusion of analyte into the bulk of acoating film is generally a rate-limiting process in absorption.Under Fickian diffusion, the absorption rate becomes propor-tional to the difference between the equilibrium concentrationin the coating film and the concentration of analytes absorbedin the film. From Eq. (3), the reaction rate of the concentra-tion of analyte Ci,n and water Cw,n in the receptor material atthe n-th step can be modeled asdCi,ndt=1τs[KpCgi,n(t)−Ci,n(t)], (4a)dCw,ndt=1τw[KwCgw,n(t)−Cw,n(t)], (4b)where τs and τw denote the diffusion time constants of ana-lyte and water, respectively; Kp and Kw are the partition coef-ficients of analyte and water, respectively [28, 29]. AlthoughEq. (3) is a step function, the dynamic concentration must be acontinuous function. As for the boundary conditions at t = tn,Cw,n(tn) =Cw,n+1(tn) and at the beginning of the first injection(t = t0), the concentration should be Cw,n=1(t0) = KwCw. Bysolving differential equations in Eq. (4b) with Eq. (3b), the re-currence relation between the 2m-th and 2(m+1)-th purge pro-cesses and that between (2m+1)-th injection and 2m-th purgeprocesses for water can be found asCw,2m+1(t)−Cw,2m(t) =Kw∆Cw(1− e−t−t2mτw), (5a)Cw,2(m+1)(t)−Cw,2m(t) =Kw∆Cw(e−t−t2m+1τw − e−t−t2mτw),(5b)with the concentrations at the first injection and purge pro-cesses given byCw,n=1(t) =Kw[Cw +∆Cw(1− e−t−t0τw)], (6a)Cw,n=2(t) =Kw[Cw +∆Cw(e−t−t1τw − e−t−t0τr)]. (6b)For analyte, see [28]. Then, the recurrence relations in Eq. (5)with Eq. (6) can be solved to obtain the concentrations of wa-ter in the receptor material at the (2m–1)-th and 2m-th steps aswell as the concentrations of analyte [28, 29], and hence thedynamic concentration Ci,n and Cw,n at any time t of the n-thstep can be simplified asCi,n(t) =KpCg (11A −as,n) , (7a)Cw,n(t) =Kw [Cw +∆Cw (11A −aw,n)] , (7b)withas,n(t) =n−1∑j=0(−1) je−t−t jτs , aw,n(t) =n−1∑j=0(−1) je−t−t jτw . (8)The substitution of Eqs. (2) and (7) into Eq. (1) yields thedifferential equation of the dynamic stress change in a vis-coelastic receptor of nanomechanical sensors at the n-th step3FIG. 2. Numerical calculations of the signal responses for multistep injection-purge cycles using the derived model in Eq. (9). (a) Responses toanalytes with different τs with the fixed humidity difference (∆σw = 1) under varied initial humidity. (b) Offset responses σ(t)−σw to analyteswith different τs with varied humidity differences ∆σw. (c,d) Model responses to highly humidified analytes, i.e., fixed (σw +∆σw = 5) undervaried initial humidity σw. Responses σ(t) (c) and normalized responses [σ(t)−σw]/σsat. (d) are shown. Colors indicate the different τs:blue, τs = 5 [s]; sky blue, τs = 10 [s]; green, τs = 20 [s]; yellow, τs = 30 [s]; red, τs = 60 [s]. Other parameters are fixed: σi = 1; τw = 15 [s];M0/M∞ = 0.72/0.51; τr = 22 [s].σn(t). In the case of the diffusion time constants do not equalto the relaxation time constant, i.e., τs ̸= τr and τw ̸= τr, thedifferential equation can be solved and found the recurrencerelation between the 2m-th and 2(m+1)-th purge processes andthat between (2m+1)-th injection and 2m-th purge processeswith the stresses at the first injection and purge processes [Ap-pendix A]. The recurrence relations in Eq. (A1) with Eq. (A2)can be solved to obtain the dynamic stress change at the (2m–1)-th and 2m-th steps, and hence the dynamic stress changeσn(t) at any time t of the n-th step can be simplified asσn(t) =σi [11A −αias,n − (1−αi)ar,n]+σw+∆σw [11A −αwaw,n − (1−αw)ar,n] , (9)with stress components in sorption kinetics as,n and aw,n inEq. (8) and in viscoelastic stress relaxation ar,n at any time tof the n-th step, which is given byar,n(t) =n−1∑j=0(−1) je−t−t jτr , (10)where σi = M∞λiKpCg, σw = M∞λwKwCw, and ∆σw =M∞λwKw∆Cw; and αi and αw are respectivelyαi =1τs(M0M∞− τsτr)(1τs− 1τr)−1, (11a)αw =1τw(M0M∞− τwτr)(1τw− 1τr)−1, (11b)if τs ̸= τr and τw ̸= τr.If the relaxation time constant equals one of the diffusiontime constant (i.e., τs = τr or τw = τr), the recurrence relationsbetween the 2m-th and 2(m+1)-th purge processes and that be-tween (2m+1)-th injection and 2m-th purge processes can befound similarly by substituting Eqs. (2) and (7) into Eq. (1)with the stresses at the first injection and purge processes [Ap-pendix A]. The recurrence relations in Eq. (A3) with Eq. (A4)if τs = τr and in Eq. (A5) with Eq. (A6) if τs ̸= τr = τw canbe solved to obtain the dynamic stress change at the (2m–1)-th and 2m-th steps. Then, the dynamic stress change σn(t) inthe cases of τs = τr ̸= τw and τs ̸= τr = τw respectively can besimplified asσn(t) =σi [11A −ar,n +βrbr,n]+σw+∆σw [11A −αwaw,n − (1−αw)ar,n] , (12a)σn(t) =σi [11A −αias,n − (1−αi)ar,n]+σw+∆σw [11A −ar,n +βrbr,n] , (12b)whereβr =1τr(M0M∞−1), (13)br,n =n−1∑j=0(−1) j(t − t j)e−t−t jτr . (14)The stress σn(t) given in Eqs. (9) and (12) is directly pro-portional to the signal responses of nanomechanical sensors[26, 28]. It should be noted that the amplitude (intensity) σsat.at the equilibrium or steady state of the injection process canbe described asσsat. =[limt→∞σn(t)]−σn(t0) = σi +∆σw, (15)where σn(t0) = σw. The stress σw derived from initial humid-ity Cw only influences the baseline.4FIG. 3. Numerical calculations of the signal responses for multistep injection-purge cycles using the derived model in Eq. (9). (a) Offsetresponse σ(t)−σw to analytes with different τs with varied humidity differences ∆σw. (b) Model responses to highly humidified analytes, i.e.,fixed (σw +∆σw = 5) under varied initial humidity σw. Colors indicate the different τs: blue, τs = 5 [s]; sky blue, τs = 10 [s]; green, τs = 20[s]; yellow, τs = 30 [s]; red, τs = 60 [s]. Other parameters are fixed: σi = 1; τw = 15 [s]; M0/M∞ = 0.72/0.51; τr = 22 [s]; T = tn − tn−1 = 10[s], where T is the duration.III. NUMERICAL CALCULATIONSThe nanomechanical sensing responses were numericallycalculated under the influence of humidity using the derivedequation in Eq. (9), as illustrated in Fig. 2. In our previousstudy, we demonstrated that the diffusion time constants τscan be robustly extracted even under varying relative humid-ity conditions [11]. Therefore, we first numerically calculatedthe influence of the initial humidity Cw. Figure 2(a) shows thestress σn(t) with a fixed humidity difference, ∆Cw ∝ ∆σw = 1,under varied initial humidity, Cw ∝ σw. Since the sorption be-haviors of analyte and water are assumed to be independentin this study, as expected, increasing the initial humidity Cwonly increases the baseline, i.e., σw in Eq. (9), upward whilemaintaining their dynamic behavior. In contrast, the humiditychange ∆Cw—the humidity difference between purge and in-jection processes—greatly influences the dynamic responsesof nanomechanical sensors as can be seen in Fig. 2(b) andFig. S1 in the Supplemental Material [34].It should be noted that the present model in Eq. (9) alsoallows numerical calculations in the cases of negative humid-ity differences (∆σw < 0), such as when the humidity of targetgas is lower than that of the purge gas. In these cases, not onlythe undershoot trends (i.e., the desorption of water molecules)but also the overshoot trends are calculated. For example, inthe case of τs = 5 [s] and τw = 15 [s] with ∆σw = –1 or –2 inFig. 2(b) and Fig. S1 in the Supplemental Material [34], theovershoot trends are observed, originating from the rapid ab-sorption of analytes, and subsequently, the response drops dueto the slower desorption of water as well as viscoelastic stressrelaxation.The present model in Eq. (9) is also applicable to the multi-step injection-purge cycles, i.e., rectangular pulse-wave likeinjection purge cycles [Fig. 1(d)] [28]. Figure 3(a) showsthe numerically calculated responses to the various humid-ity changes, including negative humidity difference, ∆σw < 0(see also Fig. S2 in the Supplemental Material [34]). Over-all trends, including baseline drifts for the multistep injection-purge cycles, align with the trends observed in single injectioncurves, as shown by dashed lines in Fig. 3(a). These char-acteristic baseline transitions clearly indicate the viscoelasticresponse by sorption-induced nanomechanical sensing.More importantly, the present model is capable of numeri-cally calculating responses to fixed humidity in a target sam-ple, particularly, for highly humidified samples such as bever-ages and exhaled breath [2, 5–10]. In practical applications ofolfactory sensors, e.g., in breath diagnosis [7–10], the exhaledbreath typically contains almost 100%RH, while the ambi-ent air used as purge gas depends on the surrounding rela-tive humidity. In our previous study towards breath diagno-sis, measurements of exhaled breath samples with the well-controlled humidity in purge gas yield high reproducibility,while the normalization of the signal can reduce the humidityinfluence if appropriate features are selected [9]. Figures 2(c)and 2(d) show the numerically calculated signal responses ofthe model for a highly humidified sample, where the totalhumidity is fixed, i.e., resulting stress σw +∆σw = 5, whilethe initial humidity varies σw ranging from 0 to 5. Whenthe total humidity in the sample is fixed, the humidity dif-ference ∆Cw depends on the initial humidity, resulting in achange in the dynamic response σ , even when the concentra-tion of analyte Cg remains fixed, as shown in Fig. 2(c). Sincelimt→∞ σ(t) = σi +σw +∆σw, the stress at equilibrium stateis the same as shown in Fig. 2(c). The initial humidity has a5FIG. 4. Prediction of dynamic responses to humidity environments measured by polymer-coated MSS. (a) Responses to pure water withdifferent initial humidity at Cw = 0%RH, 20%RH, and 40%RH. (b–d) Responses to a fixed humidity change of ∆Cw = 20%RH with tolueneconcentration Cg of 5% (b), 10% (c), and 15% (d) under varied initial humidity conditions at Cw = 0%RH, 20%RH, and 40%RH. (e–h)Responses to BETX at different concentrations under ∆Cw = 20%RH: benzene (e), toluene (f), ethylbenzene (g), and m-xylene (h). Initialhumidity is fixed at Cw = 20%RH. Colors indicate the analyte concentrations Cg: blue, 5%; orange, 10%; and green, 15%. (i–l) Responses toBETX at concentration Cg = 10% under varied humidity differences ∆Cw: benzene (i), toluene (j), ethylbenzene (k), and m-xylene (l). Initialhumidity is fixed at Cw = 20%RH. Humidity differences ∆Cw are 20%RH, 30%RH, and 40%RH. Dashed lines indicate the predicted responsesbased on the extracted fitting parameters using Eq. (9).greater influence on the responses of the multistep injection-purge cycles for the highly humidified model in Fig. 3(b)and Fig. S3 in the Supplemental Material [34]. Figure 2(d)shows the normalized offset responses, [σ(t)−σw]/σsat., forthe highly humidified model. The initial humidity influencesthe part of the dynamic responses σ depending on the timet. These results support the influence of humidity on breathanalysis [9].IV. EXPERIMENTSPolycaprolactone (PCL) was purchased from Sigma-Aldrich and employed as a receptor material for MSS(Schematic of the MSS structure is shown in [Fig. 1(b)].N,N-Dimethylformamide (DMF; 99.5%; Wako Pure Chem-ical Industries, Ltd.) was used as the solvent for prepar-ing receptor material solutions. For sensing measurements,toluene (99.8%; Sigma-Aldrich), benzene (>99.5%; KantoChemical Co., Inc.), ethylbenzene (99.8%; Sigma-Aldrich),m-xylene (>98.0%; Kanto Chemical Co., Inc), and ultrapurewater (MilliQ, Merck MilliPore) were used as target samplegases. All chemicals and materials were used as received.To prepare PCL-coated MSS, PCL was dissolved in DMFat a concentration of 1.0 g/L. The resulting solution was de-posited onto the surface of an MSS chip using an inkjet spotter(LaboJet-600 Bio, MICROJET Corporation) equipped with anozzle (IJHBS-300, MICROJET Corporation). To acceleratethe evaporation of DMF, the inkjet stage was maintained at80 °C during the deposition process. For inkjet spotting, 300shots, delivering approximately 300 pL per shot, were appliedto each membrane. The resulting MSS chip was subsequentlyemployed in sensing measurements.The MSS chip was mounted in a custom-made Teflon-6TABLE I. Extracted fitting parameters of pure water at different concentrations.∆Cw [%RH] Cw [%RH] γ∆σwa [mV] τwa [s] τra [s] M0/M∞a20 0 0.392 ± 0.002 6.596 ± 0.193 0.608 ± 0.004 7.833 ± 0.16120 0.402 ± 0.002 4.805 ± 0.028 0.518 ± 0.004 6.764 ± 0.00540 0.433 ± 0.004 4.582 ± 0.260 0.536 ± 0.006 6.280 ± 0.27130 0 0.600 ± 0.003 5.889 ± 0.109 0.561 ± 0.002 7.552 ± 0.11420 0.633 ± 0.003 4.671 ± 0.068 0.527 ± 0.002 6.478 ± 0.06840 0.706 ± 0.003 4.654 ± 0.069 0.547 ± 0.005 6.241 ± 0.05540 0 0.797 ± 0.001 5.250 ± 0.035 0.545 ± 0.003 6.973 ± 0.06720 0.840 ± 0.009 4.343 ± 0.044 0.525 ± 0.001 6.076 ± 0.04940 0.926 ± 0.004 4.335 ± 0.063 0.550 ± 0.003 5.820 ± 0.054a An average value ± standard deviation from six independent measurementsTABLE II. Extracted fitting parameters of pure BTEX at different concentrations.Sample Cga [%] γσia [mV] τsa [s] τra [s] M0/M∞aBenzene 5 2.235 ± 0.017 22.410 ± 1.052 1.614 ± 0.036 6.901 ± 0.14910 4.380 ± 0.014 18.265 ± 0.189 1.364 ± 0.015 6.612 ± 0.01915 6.595 ± 0.032 18.088 ± 0.419 1.303 ± 0.010 6.813 ± 0.097Toluene 5 1.918 ± 0.012 29.587 ± 2.120 1.804 ± 0.020 7.708 ± 0.30410 3.835 ± 0.018 23.000 ± 1.252 1.609 ± 0.032 6.956 ± 0.18515 5.723 ± 0.011 20.155 ± 0.429 1.457 ± 0.017 6.722 ± 0.056Ethylbenzene 5 1.562 ± 0.012 34.099 ± 3.443 1.902 ± 0.025 8.044 ± 0.47310 3.136 ± 0.004 25.011 ± 1.361 1.731 ± 0.039 6.842 ± 0.19715 4.709 ± 0.017 22.240 ± 0.734 1.602 ± 0.021 6.537 ± 0.116m-Xylene 5 1.679 ± 0.004 30.862 ± 0.971 1.919 ± 0.011 7.086 ± 0.12610 3.337 ± 0.011 24.460 ± 0.516 1.741 ± 0.011 6.397 ± 0.06515 4.977 ± 0.016 22.292 ± 0.374 1.622 ± 0.010 6.250 ± 0.050a An average value ± standard deviation from six independent measurementsbased chamber placed inside an incubator maintained at 25.00± 0.02 °C. The chamber was connected to a gas control systemwith three mass flow controllers (MFCs), two vials contain-ing the target liquid sample and water, and a mixing chamber[Fig. 1(c)]. Two MFCs (MFC-1 and MFC-2) were used to de-liver vapors from the vials to the chamber. During the vaporinjection process, humidified samples with varying concentra-tions were generated by adjusting the flow rates of the MFCs.The target analyte flow rate was controlled between 0 and 15mL/min, while the humidity flow rate ranged from 0 to 80mL/min. MFC-3 was employed to dilute the vapor mixtureand purge the chamber. The vapor injection and purge wereswitched at every 10 s [Fig. 1(d)]. The total gas flow ratewas kept constant at 100 mL/min throughout the experiments.This setup allowed the analyte concentration ratio (Pi/P0) tobe varied between 0% and 15%, where Pi and P0 representthe partial pressure and saturated vapor pressure of the an-alyte, and the relative humidity to be adjusted from 0%RHto 80%RH. Nitrogen gas was used as both the carrier andthe purging gases. The measurements were conducted witha bridge voltage of –0.5 V, and data were recorded at a sam-pling frequency of 20 Hz. The data acquisition program, de-signed and developed in LabVIEW (Emerson Electric Co.),was implemented for collecting, processing, and real-time vi-sualization of sensor responses.The fitting parameters for pure vapors were extracted usingthe least squares method with a trust region reflective algo-rithm implemented in Python 3 with the SciPy module, as de-scribed in prior work [28]. The optimized parameters in thisprocess included the amplitude constants (γσi and γ∆σw), dif-fusion time constant (τs), relaxation time constant (τr), theratio of unrelaxed to relaxed biaxial moduli (M0/M∞), andthe initial injection time (t0). These parameters were esti-mated using the formula shown in Eq. (9) (see Sec. II). Ini-tial guesses for these parameters were set as follows: γσi =max(Vout)−min(Vout), M0/M∞ = 5, τs = 50 [s], τr = 6 [s], andt0 = 0 [s], where γ is the proportionality factor (see Sec. II) andVout represents the signal responses. Lower and upper boundsapplied to these parameters were: for analyte, γσi > 0 [mV],M0/M∞ ≥ 1, τw > 0 [s], τr > 0 [s], and –1 < t0 < 2 [s]; forwater, γ∆σw > 0 [mV] with increasing humidity or γ∆σw < 0[mV] with decreasing humidity, M0/M∞ ≥ 1, τw > 0 [s], τr >0 [s], and –1 < t0 < 2 [s].V. EXPERIMENTAL VALIDATIONSThe validation of the proposed model, as described inEq. (9), was achieved through experimentally measured signalresponses from viscoelastic material-coated MSS [1, 31, 32]exposed to humified target analytes. The target analytes inthis study are BTEX (benzene, toluene, ethylbenzene, and7FIG. 5. Responses to toluene under positive and negative humiditychanges (∆Cw = ±20%RH). (a,b) The responses to toluene vapor atCg = 10% under positive (red; ∆Cw = 20%RH) and negative humid-ity changes (blue; ∆Cw = –20%RH). The initial humidity conditionsare Cw = 20%RH (a) and Cw = 40%RH (b). The output signals topure toluene (i.e., Cw = ∆Cw = 0%RH) are also shown as gray lines.Dashed lines are the predicted responses based on the extracted fit-ting parameters, respectively. (c,d) Comparison of signal responsesto pure toluene and the responses after humidity change subtraction.Gray, pure toluene; red dashed lines, ∆Cw = 20%RH; blue dashedlines, ∆Cw = −20%RH. See also Fig. S11 in the Supplemental Ma-terial [34] for comparison between the pure signal and subtractedresponses.xylenes), which are widely recognized as common toxicgases due to their carcinogenicity, neurotoxicity, and ubiqui-tous emission contributions to environmental pollution [35].BTEX are also known for their similar chemical propertiesand limited solubility in water, allowing for the assumptionof independent sorption behavior for each component. Thisindependence facilitated the extraction of sorption kinetic pa-rameters and viscoelastic properties from the responses to sin-gle analyte systems. The MSS signal output was observed tofollow the relationship Vout(t) = γσ(t) +V0, where γ repre-sents a proportionality factor and V0 denotes the baseline out-put. The measured responses highlighted the strong correla-tion between MSS signals and internal strain, consistent withprevious findings in cantilever-type sensors [31, 32, 36].Since the responses of three to four injection-purge cyclesyielded more reliable parameter extractions compared to asingle injection [28], we measured signal responses of PCL-coated MSS to pure water over four injection-purge cycles asshown in Fig. 4(a) and Fig. S4 in the Supplemental Mate-rial [34]. As mentioned in Fig. 2(a), the initial humidity Cwonly affected the baseline as shown in Fig. 4(a) and Fig. S4 inthe Supplemental Material [34]. Increasing the initial humid-ity from 0%RH to 40%RH increases the baseline (i.e., base-line V0 + γσw) from –10.871 ± 0.003 mV to –10.468 ± 0.002and –10.095 ± 0.003 mV for 20%RH and 40%RH, respec-tively [Fig. 4(a)]. This corresponds to γσw = 0.40 [mV] for20%RH and γσw = 0.78 [mV] for 40%RH. The responses toeach pure BTEX with varied analyte concentrations were alsomeasured (see also Fig. S4 in the Supplemental Material [34]).The sorption kinetic parameters of pure water (γ∆σw and τw)and pure BETX (γσi and τs) as well as the viscoelastic pa-rameters (τr and M0/M∞) were extracted as listed in TablesI and II. It was found that the sorption kinetic parameters(γσi, τs, γ∆σw, and τw) vary depending on the chemical prop-erties of individual analytes or water. In contrast, althoughthe viscoelastic parameters (τr and M0/M∞) also vary to someextent possibly because of their different mechanical behav-iors based on their chemical propoerties, the variation of themremained relatively small across different species, especiallyamong BTEX.It should be noted that the amplitude (intensity) to waterγ∆σw extracted from the signals to pure water are consis-tent with the baseline shifts γσw. γ∆σw for ∆Cw = 20%RH,30%RH, and 40%RH are 0.41, 0.65, and 0.85 mV, respec-tively [Table I], while γσw observed in the baseline shifts forCw = 20%RH and 40%RH are 0.40 and 0.78 mV, respec-tively, as noted above. The concentration-dependent sorption-induced strain ε f in Eq. (2) is estimated by finite element anal-ysis (FEA) through COMSOL multiphysics according to theliteratures [37, 38]. The p-type piezoresistors embedded in thesensing beams of MSS are created by boron diffusion ontoa single-crystal silicon with a (100) surface [31]. Assumingplain stress (i.e., σz = 0), each relative resistance change canbe approximated as [31, 32]∆RR≈ 12π44 (σx −σy) , (16)where π44 is one of the fundamental piezoresistance coeffi-cients of the silicon crystal; σx, σy, and σz are stresses in-duced on the piezoresistor in [111], [11̄0], and [001] direc-tions of the crystal, respectively [Fig. 1(b)] [32]. The signaloutput of MSS Vout is provided by the total output resistancechange obtained from the Wheatstone bridge circuit and givenby [31, 32]Vout =VB4(∆R1R1− ∆R2R2+∆R3R3− ∆R4R4), (17)where VB is the bridge voltage applied to the Wheatstonebridge circuit (i.e., VB = −0.5 [V] in this study) and ∆Ri/Riare the relative resistance changes in each sensing beam[Fig. 1(b)]. Using representative values for M∞ and Pois-son’s ratio of PCL [39, 40], total relative resistance change∆R/R|total per stain is estimated as 1.13 by FEA. FromEq. (17), this corresponds to 1.52 × 10−4 strain per %RH.These results confirm the validity of the present model for hu-midity effects as sorption-induced strain corresponding to theinitial humidity Cw and the humidity difference ∆Cw.Using the derived equation in Eq. (9) with extracted pa-rameters obtained from pure water and pure BTEX [TablesI and II], we further verified the model for predicting thesignal responses to BTEX under varied humidity conditions.The observed baseline shifts in Figs. 4(b)–4(d) correspond to8the initial humidity ranging from 0%RH to 40%RH. Thesebaseline shifts are consistent with those observed in the re-sponses to pure water [Fig. 4(a)]. Meanwhile, the changes inthe initial humidity have no significant influence on the sig-nal responses, which aligns with the numerical results pre-sented in Fig. 2(a). Figure 4(e)–4(h) show the responses toeach BTEX under varied humidity differences ∆Cw (see alsofor the different initial humidity Cw in Fig. S5 in the Supple-mental Material [34]). By applying the theoretical concentra-tion profiles controlled by MFCs to calculate each responseto humidified analyte mixtures, the predicted responses usingEq. (9) were compared with experimental data in Figs. 4(b)–4(l) (see also Figs. S5–S7 in the Supplemental Material [34]).The predicted responses agree well with the experimentallymeasured responses to BTEX with concentrations Cg variedat 5%, 10%, and 15% under varied humidity differences ∆Cwat 20%RH, 30%RH, and 40%RH, demonstrating the potentialof the present model in capturing complex interactions underhumid environments.We also demonstrated the cases of negative humiditychanges, as discussed in Fig. 2(b). Figures 5(a) and 5(b) showthe dynamic responses of PCL-coated MSS to 5% toluene va-por with both positive and negative humidity changes (∆Cw =±20%RH) under different initial humidity conditions (Cw =20%RH and 40%RH). Using Eq. (9) along with the extractedparameters obtained from pure water and pure toluene [TablesI and II], the signal responses were predicted. Although thehumidity influence on the responses to toluene vapors is smallbecause of the lower sensitivity of PCL-coated MSS to wa-ter compared to toluene, the predicted responses show goodagreement with the experimentally measured signal responsesas shown in Figs. 5(a) and 5(b) (see also Figs. S8–S10 in theSupplemental Material [34]). These results indicate the po-tential of the present model for predicting the signal responsesunder various humidity conditions.Since we achieved the prediction of signal responses undervarious humidity conditions, the present model suggests thepossibility of subtracting the humidity influences. To demon-strate this in practical terms, we subtracted humidity influ-ences using the humidity responses predicted by Eq. (9) fromthe responses under the positive and negative humidity dif-ferences. The subtracted responses showed excellent over-lap with the responses to pure toluene vapor, as shown inFigs. 5(c) and 5(d). Additional details regarding the subtrac-tion for other analyte concentrations are provided in Figs. S8–S11 in the Supplemental Material [34]. The results supportthat the pure analyte signals in humidified samples can be ac-curately extracted, enabling reliable quantification of analyteconcentrations.VI. CONCLUSIONWe propose an analytical model for humidity effects innanomechanical sensing based on sorption kinetics and vis-coelastic behaviors of receptor materials. By this analyticalmodel, the humidity influences observed in previous studies[9, 11] were theoretically revealed, i.e., the existence of fea-tures robust to the effects of humidity [9] and the robust ex-traction of the diffusion time constant τs under varying initialhumidity [11]. The model was experimentally validated us-ing BTEX as target analytes on MSS under various humidityconditions, demonstrating excellent agreement between theo-retical predictions and output responses. It should be notedthat the present model can be applicable in the practical casesof olfactory sensors by subtracting the humidity influences.This capability enables the elimination of humidity interfer-ence, isolating the analyte-specific sensing signals. Since thepresent model can be extended to multi-component analytes[29], the proposed model provides a robust approach to ana-lyzing analytes in various humid environments, offering sig-nificant potential for applications of nanomechanical sensorsas olfactory sensors in odor detection, environmental monitor-ing, and healthcare and medical diagnosis.ACKNOWLEDGEMENTSM.-Q.F. and Y.Z. thanks NIMS Junior program, NIMS.K.M. acknowledges the International Center for Young Sci-entists (ICYS) program, NIMS, Japan. Y.Z. thanks the Sup-port by Pioneering Research Initiated by the Next Generation(SPRING) program, JST, MEXT, Japan.This 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); a Grant-in-Aid for Challenging Re-search (Pioneering), JSPS, MEXT, Japan (no. 20K20554); thePublic/Private R&D Investment Strategic Expansion Program(PRISM), Cabinet Office, Japan; the Support by PioneeringResearch Initiated by the Next Generation (SPRING), JST,MEXT, Japan (no. JPMJSP2124); and ICYS, NIMS.DATA AVAILABILITYThe data are not publicly available. The data are availablefrom the authors upon reasonable request.Appendix A: Recurrence relations of the dynamic stress changeThe substitution of Eqs. (2) and (7) into Eq. (1) yields thedifferential equation of the dynamic stress change in a vis-coelastic receptor of nanomechanical sensors at the n-th stepσn(t). Although Eq. (3) is a step function, the dynamic stresschange must be a continuous function. As for the bound-ary conditions at t = tn, the stress σn(tn) should be equal toσn+1(tn) and the stress at the beginning of the first injection(t = t0) can be assumed as zero, i.e., σn=1(t0) = 0. The dif-ferential equation can be solved and found the recurrence re-lation between the 2m-th and 2(m+1)-th purge processes andthat between (2m+1)-th injection and 2m-th purge processesas9σ2m+1(t)−σ2m(t) =σiαi(1− e−t−t2mτs)+∆σwαw(1− e−t−t2mτw)+[σi (1−αi)+∆σw (1−αw)](1− e−t−t2mτr), (A1a)σ2(m+1)(t)−σ2m(t) =σiαi(e−t−t2m+1τs − e−t−t2mτs)+∆σwαw(e−t−t2m+1τw − e−t−t2mτw)+[σi(1−αi)+∆σw(1−αw)](e−t−t2m+1τr − e−t−t2mτr), (A1b)with the stresses at the first injection and purge processes given byσn=1(t) =σw +σi(1−αie− t−t0τs)+∆σw(1−αwe−t−t0τw)− [σi(1−αi)+∆σw(1−αw)]e− t−t0τr , (A2a)σn=2(t) =σw +σiαi(e−t−t1τs − e−t−t0τs)+∆σwαw(e−t−t1τw − e−t−t0τw)+[σi(1−αi)+∆σw(1−αw)](e−t−t1τr − e−t−t0τr), (A2b)for the case of τs ̸= τr ̸= τw,σ2m+1(t)−σ2m(t) =σi(1− e−t−t2mτr)+∆σw(1−αwet−t2mτw)+σiβr (t − t2m)e−t−t2mτr −∆σw(1−αw)e− t−t2mτr , (A3a)σ2(m+1)(t)−σ2m(t) =[σi +∆σw(1−αw)](e−t−t2m+1τr − e−t−t2mτr)+∆σwαw(e−t−t2m+1τw − e−t−t2mτw)−σiβr[(t − t2m+1)e− t−t2m+1τr − (t − t2m)e− t−t2mτr], (A3b)with the stresses at the first injection and purge processes given byσn=1(t) =σw +σi(1− e−t−t0τr)+∆σw(1−αwe−t−t0τw)+σiβr(t − t0)e− t−t0τr −∆σw(1−αw)e− t−t0τr , (A4a)σn=2(t) =σw +[σi +∆σw(1−αw)](e−t−t1τr − e−t−t0τr)+∆σwαw(e−t−t1τw − e−t−t0τw)−σiβr[(t − t1)e− t−t1τr − (t − t0)e− t−t0τr], (A4b)for the case of τs = τr ̸= τw, andσ2m+1(t)−σ2m(t) =σi(1−αie− t−t2mτs)+∆σw(1− et−t2mτr)+∆σwβr (t − t2m)e−t−t2mτr −σi(1−αi)e− t−t2mτr , (A5a)σ2(m+1)(t)−σ2m(t) =[σi(1−αi)+∆σw](e−t−t2m+1τr − e−t−t2mτr)+σiαi(e−t−t2m+1τs − e−t−t2mτs)−∆σwβr[(t − t2m+1)e− t−t2m+1τr − (t − t2m)e− t−t2mτr], (A5b)with the stresses at the first injection and purge processes given byσn=1(t) =σw +σi(1−αie− t−t0τs)+∆σw(1− e−t−t0τr)+∆σwβr(t − t0)e− t−t0τr −σi(1−αi)e− t−t0τr , (A6a)σn=2(t) =σw +[σi(1−αi)+∆σw](e−t−t1τr − e−t−t0τr)+σiαi(e−t−t1τs − e−t−t0τs)−∆σwβr[(t − t1)e− t−t1τr − (t − t0)e− t−t0τr], (A6b)for the case of τs ̸= τr = τw, where αi, αw, and βr are given in Eqs. 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