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[Kosuke Minami](https://orcid.org/0000-0003-4145-1118), [Kota Shiba](https://orcid.org/0000-0001-7775-0318), [Genki Yoshikawa](https://orcid.org/0000-0002-9136-8964)

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[Sorption-induced static mode nanomechanical sensing with viscoelastic receptor layers for multistep injection-purge cycles](https://mdr.nims.go.jp/datasets/f76d179b-0996-43a8-a69d-a180d6094a5e)

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Sorption-induced static mode nanomechanical sensing with viscoelastic receptor layers for multistep injection-purge cyclesJ. Appl. Phys. 129, 124503 (2021); https://doi.org/10.1063/5.0039045 129, 124503© 2021 Author(s).Sorption-induced static modenanomechanical sensing with viscoelasticreceptor layers for multistep injection-purgecyclesCite as: J. Appl. Phys. 129, 124503 (2021); https://doi.org/10.1063/5.0039045Submitted: 29 November 2020 . Accepted: 05 March 2021 . Published Online: 25 March 2021 Kosuke Minami,  Kota Shiba, and  Genki Yoshikawahttps://images.scitation.org/redirect.spark?MID=176720&plid=1401535&setID=379065&channelID=0&CID=496959&banID=520310235&PID=0&textadID=0&tc=1&type=tclick&mt=1&hc=71bf76294ba1eff3502a31fdb96fd8874112c042&location=https://doi.org/10.1063/5.0039045https://doi.org/10.1063/5.0039045http://orcid.org/0000-0003-4145-1118https://aip.scitation.org/author/Minami%2C+Kosukehttp://orcid.org/0000-0001-7775-0318https://aip.scitation.org/author/Shiba%2C+Kotahttp://orcid.org/0000-0002-9136-8964https://aip.scitation.org/author/Yoshikawa%2C+Genkihttps://doi.org/10.1063/5.0039045https://aip.scitation.org/action/showCitFormats?type=show&doi=10.1063/5.0039045http://crossmark.crossref.org/dialog/?doi=10.1063%2F5.0039045&domain=aip.scitation.org&date_stamp=2021-03-25Sorption-induced static mode nanomechanicalsensing with viscoelastic receptor layersfor multistep injection-purge cyclesCite as: J. Appl. Phys. 129, 124503 (2021); doi: 10.1063/5.0039045View Online Export Citation CrossMarkSubmitted: 29 November 2020 · Accepted: 5 March 2021 ·Published Online: 25 March 2021Kosuke Minami,1,2,a) Kota Shiba,2,3 and Genki Yoshikawa2,4AFFILIATIONS1International Center for Young Scientists (ICYS), National Institute for Materials Science (NIMS), 1-1 Namiki, Tsukuba,Ibaraki 305-0044, Japan2Center for Functional Sensor & Actuator (CFSN), Research Center for Functional Materials, National Institute for MaterialsScience (NIMS), 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan3John A. Paulson School of Engineering and Applied Sciences, Harvard University, 9 Oxford Street, Cambridge, Massachusetts 02138, USA4Materials Science and Engineering, Graduate School of Pure and Applied Science, University of Tsukuba, 1-1-1 Tennodai, Tsukuba,Ibaraki 305-8571, Japana)Author to whom correspondence should be addressed: MINAMI.Kosuke@nims.go.jpABSTRACTNanomechanical sensors and their arrays have been attracting significant attention for detecting, distinguishing, and identifying targetanalytes. In the static mode operation, sensing signals are obtained by a concentration-dependent sorption-induced mechanical strain/stress.The analytical models for the static mode nanomechanical sensing with viscoelastic receptor layers have been proposed, while they are notformulated for practical conditions, such as multistep injection-purge cycles. Here, we derive an analytical model of viscoelastic material-based nanomechanical sensing by extending the theoretical model via solving differential equations with recurrence relations. The presentedmodel is capable of reproducing the transient behaviors observed in the experimental signal responses with multistep injection-purge cycles,including drifts and/or changes in the baseline. Moreover, this model can be utilized for extracting viscoelastic properties of the receptormaterial/analyte pairs as well as the concentrations of analytes accurately by fitting a couple of injection-purge curves obtained from theexperimental data. The parameters of the model that best fit the data can be used for predicting the entire signal response.© 2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license(http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0039045I. INTRODUCTIONNanomechanical sensors have gained significant attention aspowerful tools for detecting target analytes,1–7 especially odors thatare composed of a complex mixture of gaseous molecules.8–11 Anarray of nanomechanical sensors can be potentially used as asensing unit for artificial olfaction. In the case of so-called staticmode operation, sensing signals are obtained by measuringmechanical stress/strain induced by the sorption of target mole-cules in a receptor layer designed to respond to a wide range ofchemical classes. Since such a multidimensional dataset obtainedby a nanomechanical sensor array contains a large amount ofinformation, multivariate analyses and machine learning can beeffectively applied to distinguish and identify each specimen. Toobtain highly accurate identification of target analytes by usingpattern recognition-based analyses, there are various methods toextract the effective features from the multidimensional dataset.12However, most of the feature extraction methods are generallybased on the empirical and/or mathematical interpretations and arenot necessarily related to the scientific interpretation, such as thephysical/chemical mechanism that is involved in the nanomechani-cal sensing.Nanomechanical sensors are coated with a receptor layer thatabsorbs the analyte. Sorption-induced bending of metal-coatedmicrocantilevers has been extensively studied.13,14 The responsesJournal ofApplied Physics ARTICLE scitation.org/journal/japJ. Appl. Phys. 129, 124503 (2021); doi: 10.1063/5.0039045 129, 124503-1© Author(s) 2021https://doi.org/10.1063/5.0039045https://doi.org/10.1063/5.0039045https://www.scitation.org/action/showCitFormats?type=show&doi=10.1063/5.0039045http://crossmark.crossref.org/dialog/?doi=10.1063/5.0039045&domain=pdf&date_stamp=2021-03-25http://orcid.org/0000-0003-4145-1118http://orcid.org/0000-0001-7775-0318http://orcid.org/0000-0002-9136-8964mailto:MINAMI.Kosuke@nims.go.jphttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/https://doi.org/10.1063/5.0039045https://aip.scitation.org/journal/japfrom metal-coated cantilever-type nanomechanical sensors exhibita first-order behavior representing an elastic behavior. In contrastto the elastic metal coating, polymers are often used as a coatingmaterial because polymers can be easily designed to respond to alarge variety of chemical classes. Polymers often show viscoelasticbehaviors, and hence the dynamic behaviors of nanomechanicalsensing reflect the sorption kinetics along with the viscoelasticstress relaxation. For theoretical investigations, Wenzel et al.15 pro-posed an analytical solution of dynamic responses of cantilever-type nanomechanical sensors coated with a viscoelastic material fora rectangular injection-purge system; it is possible to predict theviscoelastic signal responses using the parameters that describe thesteady-state elongation and the sorption time. Imamura et al.16recently reported the effective feature extraction method by usingWenzel’s theoretical model for analyte identification. In this model,it was assumed that the signal response reaches a steady-state.However, some of the signal responses do not reach a steady-stateeven after 1 h [Fig. 1(a)], probably due to the secondary and terti-ary viscoelastic behaviors.17 In such a case, it is difficult to extractthe accurate viscoelastic features, and also a long time durationlimits the applicability of nanomechanical sensors to practical arti-ficial olfaction.To measure a signal response with a relatively short duration ina practical condition,18–20 the injection-purge cycle is often repeated[Figs. 1(b) and 1(c)], which is similar to inhalation–exhalation cyclesof the human nose. By using the dataset obtained from the lattercycles, it is possible to identify various target analytes.4–6,8–10 Thismultistep injection-purge cycle with a relatively short duration canminimize the effects induced by the long-time injection process[Fig. 1(a)], such as secondary and tertiary viscoelastic behaviors.17However, it frequently suffers from a severe drift of signal responsesand/or change in their baseline [Fig. 1(c)], preventing the measure-ments of the reproducible signal responses. Although these phenom-ena must relate to the sorption of analytes into a receptor layer andsubsequent release, an analytical expression of comprehensivelydescribing the phenomena is still missing.In the present study, we derive a general analytical expres-sion that includes the sorption kinetics and the viscoelastic stressrelaxation. This model is applicable to the multistep injection-purge cycles without reaching a steady-state, allowing us to effec-tively analyze the sensor response even if it drifts and/or changesits baseline over time. On the basis of the theoretical model pro-posed by Wenzel et al.,15 we formulated a new model by derivinganalytical solutions of overall transient (and steady-state)responses with multistep injection-purge cycles. This analyticalmodel agrees well with sensor responses experimentally measuredusing a nanomechanical Membrane-type Surface stress Sensor(MSS)3 coated with viscoelastic materials because of the highrobustness and high sensitivity of MSS.4,21 The curve fitting pro-vides not only several physical parameters but also analyte con-centration that can be utilized as solid features for scientificallyreliable pattern recognition-based analyses.8–10II. EXPERIMENTAL SECTIONA. MaterialsPolystyrene (PS), polycaprolactone (PCL), andpoly(4-methylstyrene) (P4MS) were purchased from Sigma-Aldrich.Alkyl-functionalized inorganic nanoparticles (octadecyl-functionalizedsilica/titania-based hybrid nanoparticles; C18-STNPs) are fabricatedaccording to our previous papers.9,22 N,N-dimethylformamide(DMF), used as a solvent to prepare polymer solutions for inkjetspotting, was purchased from Wako Pure Chemical Industries.Ethanol, n-dodecane, and 1,2-dichlorobenzene (analytical or highergrade) used as solvent vapors were purchased from Wako PureChemical Industries, Nacalai tesque, and Sigma-Aldrich, respec-tively. All chemicals were used as purchased. MilliQ water (MerckMilliPore) was used to obtain water vapor.B. Fabrication of MSSThe construction of the MSS chips and its working principlehas been previously reported.3,21 Briefly, MSS consists of a silicon-based membrane suspended by four piezoresistive beams, compos-ing a full Wheatstone bridge [Fig. 2(a)]. The membrane is coatedwith a receptor material, which generates the surface stress causedby the sorption-induced expansion. The surface stress on the mem-brane is transduced to the four sensing beams as amplified uniaxialstress, resulting in the changes in the electrical resistance of the pie-zoresistors embedded in the beams. In contrast to simplecantilever-type nanomechanical sensors, in which a displacementof free-end Δz corresponds to a sensing signal, the signal output ofMSS (Vout) is provided by the total output resistance changeFIG. 1. Typical signal outputs measured by viscoelastic material-coated MSS.(a) Long time injection (90 min) of 10% n-dodecane to PS-coated MSS. (b) and(c) Multistep injection-purge cycle with duration time T ¼ 10 s. Water(10%)/PCL (b) and n-dodecane (10%)/PS are shown. Black arrows denote thebaseline drift.Journal ofApplied Physics ARTICLE scitation.org/journal/japJ. Appl. Phys. 129, 124503 (2021); doi: 10.1063/5.0039045 129, 124503-2© Author(s) 2021https://aip.scitation.org/journal/japobtained from the Wheatstone bridge circuit; it can be expressed asVout ¼ VB4ΔR1R1� ΔR2R2þ ΔR3R3� ΔR4R4� �, (1)where VB is the bridge voltage applied to the Wheatstone bridgecircuit and ΔRi=Ri is the relative resistance change in each sensingbeam [Fig. 2(b)].The receptor materials were coated directly onto the MSSmembrane using an inkjet spotting (LaboJet-500SP, Microjet Co.Ltd.) equipped with a nozzle (IJHBS-300, Microjet Corporation).Each receptor material was dissolved in DMF at a concentration of1 mgmL�1, and the resulting solutions were deposited onto eachchannel of the MSS. The injection speed, volume of a droplet, andnumbers of inkjet shots were fixed at ca. 5 m s�1, ca. 300 pL, and300 shots, respectively. A stage of the inkjet spotter was heated at80 �C to promote evaporation of DMF. In the present study, weused viscoelastic material-coated MSS with more than 6-monthaging to obtain reproducible signal responses.C. SensingThe coated MSS chips were placed in a Teflon chamber,which was placed in an incubator with a controlled temperatureof 25.00 ± 0.02 �C. The chamber was connected to a gas systemconsisting of two mass flow controllers (MFCs), a mixingchamber, a purging gas line, and a vial for a solvent liquid. Thevapor of each solvent was produced by bubbling carrier gas.Pure nitrogen gas was used as carrier and purging gases. Thetotal flow rate was maintained at 100 mL min�1 during theexperiments. The duration time was precisely controlled andthe concentrations of the four different solvent vapors were con-trolled using MFC-1 at Pa=Po of 0.1, 0.2, and 0.3, where Pa andPo denote the partial vapor pressure and saturated vapor pres-sure of the solvent, respectively. Before measuring MSS signals,pure nitrogen gas was introduced into the MSS chamber for1 min. Subsequently, MFC-1 (injection line) was switched on/offat each duration time T [s] with a controlled total flow rate of100 mL min�1 using MFC-2 for up to 20 injection-purge cycles.Data were measured with a bridge voltage of �0:5 V andrecorded with a sampling rate of 100 Hz. The data collectionprogram was designed using LabVIEW (NI Corporation).D. Curve fitting and estimation of parametersTo extract coating film properties from experimental data, weused least squares methods with trust region reflective algorithmusing Python 3 with SciPy module. The bounds for each parameterwere set at γ � σsat: . 0, EU=ER . 0, τs . 0, τr . 0, and10 � t0 � 13. The amplitude constant σsat:, the diffusion timeconstant τs, the relaxation time constant τr , and the ratio of unre-laxed and relaxed moduli EU=ER in addition to the time when thefirst injection starts (t0) were optimized using the derived formulain the present study. The initial fitting parameters are set as follow:γ � σsat: ¼ ymax � ymin, EU=ER ¼ 5, τs ¼ 100 [s], τr ¼ 6 [s], andt0 ¼ 11 [s], where ymax and ymin are the maximum and minimumvalues of each signal response.III. BACKGROUND THEORYFirst, we summarize the background theory used to calculatethe sorption-induced and concentration-dependent nanomechani-cal sensing models for elastic and viscoelastic materials, respec-tively, using a rectangular injection sequence as illustrated in Fig. 3.A more detailed discussion of the background theory is given inthe literature reported by Wenzel et al.15A. Sorption-induced nanomechanical sensingWhen a receptor material coated on a nanomechanical sensoris exposed to a chemical analyte, the receptor material expandsowing to the sorption-induced deformation. However, the receptormaterial is attached to a substrate and is not free to expand. In theFIG. 2. Structure of the membrane-type surface stress sensor (MSS). (a)Three-dimensional schematic image of MSS structure. (b) Circuit diagram of theMSS.FIG. 3. Rectangular injection. (a) Rectangular analyte injection. (b) Typicalsignal output.Journal ofApplied Physics ARTICLE scitation.org/journal/japJ. Appl. Phys. 129, 124503 (2021); doi: 10.1063/5.0039045 129, 124503-3© Author(s) 2021https://aip.scitation.org/journal/japcase of so-called static mode operation of a microcantilever plate,the sorption-induced expansion makes the cantilever beam bend.1Several analytical solutions have been proposed for the theoreticalformulation of the static mode operation of nanomechanicalsensing, especially for a microcantilever model. For example, thedisplacement of a free-end of a cantilever Δz induced by isotropicinternal strain in a receptor layer εf is given by23Δz ¼ 3l2 hf þ hs� �(Aþ 4)h2f þ (A�1 þ 4)h2s þ 6hf hsεf , (2)withA ¼ Ef wf hf1� νfEswshs1� νs�, (3)where the subscripts “f ” and “s” denote the coating film and thecantilever substrate, respectively; l, h, w, E, and ν represent thelength, height, width, Young’s modulus, and Poisson’s ratio,respectively. The internal strain in a coating film εf can bereplaced by other parameters, such as three-dimensional internalstress in the coating film σ f ([N m�2]) or two-dimensional surfacestress σsurf ([N m�1]) using the relations εf ¼ σ 1� νf� �=Ef orσ f ¼ σsurf=hf .23–25 When a cantilever is covered with a thinfilm having a same width (i.e., hf � hs and wf ¼ ws), Eq. (2)reduces to the following equation, which is known as Stoney’sequation:24Δz ¼ 3l2 1� νsð ÞEsh2sσsurf : (4)In the case of the sorption-induced nanomechanical sensing,there are several investigations using microcantilever sensors.13,15Wenzel et al. proposed the theoretical formulation of the dynamicbehavior for a cantilever plate, especially for that coated with a vis-coelastic material.15 In this model, the sorption-induced internalstrain in a coating film εf is approximated as follows: in the gasphase, the volume of a coating film V(C), which is not attached toa cantilever substrate, changes with the concentration of an analytein the coating film C (mol mL�1 of coating film) asV(C) ¼ V0 1þ Cvað Þ, (5)where V0 is the initial volume of the coating film and va is the spe-cific volume of the absorbed analyte. Assuming that the coatingfilm is isotropic and expands isotropically in all directions, theabsorption-induced strain varies with concentration asεf (C) ¼ 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ Cvap� 1, (6a)which has the linear approximation given byεf (C) ¼ 13Cva (6b)for small volume expansion (i.e., εf � 1). The absorption-inducedstrain can, therefore, be assumed directly proportional to theabsorbed analyte concentration. Let εf (t) ¼ λC(t), where λ ¼ 13 va;the deflection at the free-end of the cantilever Δz(t) for the casedescribed in Eq. (2) can be then given byΔz(t) ¼ 3l2 hf þ hs� �(Aþ 4)h2f þ (A�1 þ 4)h2s þ 6hf hsλC(t): (7)Stoney’s equation [Eq. (4)] can similarly be used to develop asimpler expression for hf � hs. Equation (7) relates the cantileverresponse to the sorbed analyte concentration C(t) in the coatingfilm. The expression indicates that the dynamic behavior of theabsorption causes deflection. Thus, it is necessary to know how theabsorbed analyte concentration in the coating film varies with time.It should be noted that a signal response of MSS generallycorrelates with the internal strain, which is similar to thecantilever-type sensors.26 In the present study, we used MSS forexperimental investigation. Although no analytical solution isreported for the signal response of MSS, we have revealed thatthe resistance change of piezoresistive sensing beams of MSSreflects the linear correlation with the internal strain εf throughexperiments4,6 and simulations using finite element analysis(FEA).6,27–29 Therefore, the signal responses of MSS can alsobe assumed to be directly proportional to the absorbed analyteconcentration as expressed in Eq. (7).B. Governing equations for single-stepabsorption/desorption processesFor the derivation of the equations governing the concentra-tion of an analyte A into a receptor material coated on a nanome-chanical sensor during absorption/desorption processes, we assumea first-order absorption of A with absorption rate constant ka anddesorption rate constant kd asA(g)kdOkaA(s), (8)where A(g) and A(s) denote an analyte in the gas phase and incoating film, respectively. The reaction rate of a concentration of ananalyte in the coating film C(t) is given byddtC(t) ¼ ka � Cg(t)� kd � C(t), (9a)where Cg(t) is the concentration of analyte in the gas phase as afunction of time. Although the absorption process of an analyteinto a bulk of the coating film is generally rate limited by the diffu-sion of the analyte across the surface barrier and into the coatingfilm,13 this reaction rate is equivalent to the diffusion time constantτs reported by Wenzel et al.15 asddtC(t) ¼ 1τsKpCg(t)� C(t)�  , (9b)where τs ¼ 1=kd ; Kp ¼ ka=kd is known as the partition coefficientin the case of a polymer coating film. We also use the time constantτs instead of the desorption rate constant kd .Journal ofApplied Physics ARTICLE scitation.org/journal/japJ. Appl. Phys. 129, 124503 (2021); doi: 10.1063/5.0039045 129, 124503-4© Author(s) 2021https://aip.scitation.org/journal/japIn the case of gas sensing using nanomechanical sensors, thesample gas is introduced by carrier gas (e.g., nitrogen); subse-quently, the gas line is switched to the purge gas line (e.g., nitrogen)to promote the desorption of the sample gases. Since the injectionof analyte is generally controlled by the continuous flow of head-space gas or bubbling liquid samples, it can be assumed to behomogeneous in time. Thus, a rectangular injection of analyte withconstant rate can be considered as depicted in Fig. 3(a). When theanalyte is injected at t ¼ t0 and maintained at a constant concentra-tion Cg until the time to start the purge t1, the concentration ofanalyte in the gas phase Cg(t) can be considered as a step function:Cg(t) ¼0, t , t0Cg , t0 � t , t1:0, t1 � t8<: (10)It is assumed that the initial concentration in the coating film C(t)is zero before injection starts (i.e., t , t0). Using Eq. (10), the reac-tion rate of A in the coating film [Eq. (9b)] can be described as astep function:ddtC(t) ¼0, t , t01τsKpCg � C(t)�  , t0 � t , t1:� 1τs� C(t) t1 � t8>>>><>>>>:(11)The general differential equations for the reaction rate during theabsorption process (i.e., the injection process; t0 � t , t1) and thedesorption process (i.e., the purge process; t � t1) can be solved asC(t) ¼0 t , t0KpCg þ C1 � e� tτs , t0 � t , t1,C2 � e� tτs , t1 � t8><>: (12)where C1 and C2 are the constants of integration.Although Eq. (12) is a step function, the dynamic concentra-tion in a coating film must be a continuous function. As for theboundary condition at t ¼ t0, the concentration of analyte in thecoating film should be zero. Consequently, the constant of integra-tion C1 ¼ �KpCg � et0=τs . Substituting this into Eq. (12), the con-centration of the analyte in the coating film can be obtained byC(t) ¼ KpCg 1� e�t�t0τs �, t0 � t , t1: (13)Subsequently, as a second boundary condition at the beginning ofthe desorption process (t ¼ t1), the concentration of analyte in thecoating film should be equal to that at the end of the absorptionprocess [Eq. (13)]. Thus, the constant C2 can be solved asC2 ¼ KpCg 1� e�t1�t0τs �� et1τs : (14)Substituting C2 into Eq. (12), the particular solution of the differen-tial equations can be solved. The concentration of the analyte inthe coating film at any time t is given byC(t) ¼0, t , t0KpCg 1� e�t�t0τs �, t0 � t , t1:KpCg 1� e�t1�t0τs �� e�t�t1τs , t1 � t8>><>>: (15a)Let T ¼ t1 � t0 be the duration time for injection; then the concen-tration of analyte A can be simplified asC(t) ¼0, t , t0KpCg 1� e�t�t0τs �, t0 � t , t1:KpCg 1� e�Tτs �� e�t�t1τs , t1 � t8>><>>: (15b)The analytical solution of the absorption process can be expressedas a typical first-order response. As described above, when thecoating film is an elastic material, the signal response of nanome-chanical sensing can be assumed to be proportional to the concen-tration of the analyte in the coating film, if the sorption-inducedexpansion is small. Indeed, the typical signal outputs of an MSSare transduced from the sorption-induced expansion with the esti-mated internal strain εf in the range from approximately 1� 10�6simulated by FEA.6,28C. Governing equations for viscoelastic materialsAmong a large variety of materials, viscoelastic propertiesarise from dynamic differences on molecular rearrangements.30Wenzel et al. proposed a theoretical model for a cantilever-typenanomechanical sensor coated with a viscoelastic material.15 Thetheoretical models are derived from the simplest three-parametersolid model:15,31τrEUddtε(t)þ ERε(t) ¼ τrddtσ(t)þ σ(t), (16)where EU and ER denote the unrelaxed (instantaneous) modulusand the relaxed (asymptotic) modulus, respectively, and τr is thetime constant of stress relaxation. The three-parameter solid modeldescribes the stress/strain relationship in a viscoelastic solid thatexhibits both viscous and elastic properties. As proposedby Wenzel et al.,15 the derived general differential equations fromEq. (16) can be greatly simplified asddtσ(t) ¼ �σ(t)τrþ ERλτsEUER� τsτr� �C(t)� EUKpλτsCg(t), (17)when the coating film is significantly soft (i.e., EU � Es) or thin(i.e., hf � hs). In the case of MSS, the membrane (substrate) ismade of silicon (approximately 170 GPa). In general, the substratematerial is considerably stiffer than the polymer materials, whoseunrelaxed moduli are usually in the range from 10 MPa to 5 GPa.Accordingly, the general differential equation in Eq. (17) can beutilized in the present study.Journal ofApplied Physics ARTICLE scitation.org/journal/japJ. Appl. Phys. 129, 124503 (2021); doi: 10.1063/5.0039045 129, 124503-5© Author(s) 2021https://aip.scitation.org/journal/japFor derivation of the equations governing the stress with a rec-tangular injection, we again consider the step function of the concen-tration of analyte in the gas phase Cg(t) in Eq. (10). SubstitutingEqs. (10) and (15a) into Eq. (17), the general differential equation ofstress can be described as a step function:ddtσ(t) ¼0, t , t0�σ(t)τr� σsat:τsEUER� τsτr� �e�t�t0τs � σsat:τr, t0 � t , t1,�σ(t)τrþ σsat:τsEUER� τsτr� �1� e�t1�t0τs �e�t�t1τs , t1 , t8>>>><>>>>:(18)withσsat: ¼ ERλKpCg : (19)σsat: denotes the stress at the saturated or the equilibrium state. Thegeneral differential equations of the stresses during the absorption anddesorption processes can be solved asσ(t) ¼σ0, t , t0�σsat: þ σsat:α � e�t�t0τs þC3 � e� tτr , t0 � t , t1,�σsat:α 1� e�t1�t0τs �� e�t�t1τs þ C4 � e� tτr , t1 � t8><>:(20)withα ¼ 1τsEUER� τsτr� �1τs� 1τr� ��1, (21)where C3 and C4 are the constants of integration.During rectangular injection-purge, again the stresses must bea continuous function. As for the boundary condition at the begin-ning of the absorption process, the stress must be σ0. Then, C3 canbe solved asC3 ¼ σsat:(1� α) � et0τr þ σ0 � et0τr : (22)The substitution of C3 into Eq. (20) yields the stress of the absorp-tion process asσ(t) ¼ �σsat: þ σsat:α � e�t�t0τsþ σsat:(1� α) � e�t�t0τr þ σ0 � e�t�t0τr : (23)From Eq. (23), the second boundary condition of the stress at thebeginning of purge can be estimated. The constant C4 is, therefore,given byC4 ¼ σ0et0τr � σsat:(1� α) 1� e�t1�t0τr �� et1τr : (24)Substituting C4 into Eq. (20), the particular solutions of the differ-ential equations can be solved and the stress at any time t can beobtained as follows:σ(t) ¼σ0, t , t0�σsat: þ σsat:α � e�t�t0τs þ σ0 � e�t�t0τr þ σsat:(1� α) � e�t�t0τr , t0 � t , t1:�σsat:α 1� e�t1�t0τs �� e�t�t1τs þ σ0 � e�t�t0τr � σsat:(1� α) 1� e�t1�t0τr �� e�t�t1τr , t1 � t8><>: (25)Although the resulting stresses in Eq. (25) are given in negative values,the original definition proposed by Wenzel et al.15 states that the posi-tive z is directed downward of a cantilever plate; thus, the correspond-ing analytical solution of the deflection at the free-end of thecantilever is also multiplied by negative one, as can be seen in Eq.(20b) in Ref. 15. It should be noted that we use the equations multi-plied by negative one to apply the analytical solutions of viscoelasticmodels to the MSS with the same directions of surface stress; a posi-tive sign corresponds to the initial compressive stress and vice versa.It should also be noted that when the signal reaches the steady-state or equilibrium state, the stresses in Eq. (25) can be further sim-plified and be equivalent to Eqs. (11b) and (12b) in Ref. 16, ifσ0 ¼ 0. In the simplified equations for the steady-state, the risingand decay curves become symmetric.IV. ANALYTICAL SOLUTIONSIn nanomechanical sensing, the multistep injection-purgecycles are often used to obtain the repetitive signal patterns toextract multistep sets of features for the pattern recognition(Fig. 1).4–6,8–10,32 However, the signal output is sometimes obtainedwith monotonous increase in the baseline as depicted in Figs. 1(b)and 1(c). This can be due to the asymmetric nature between theabsorption and desorption processes, even the desorption rate constantis the same as kd or equivalently 1=τs. In the present study, using theabove theoretical framework, we now derive the analytical solutions ofthe dynamic concentrations of the analyte in the receptor materialswith viscoelastic properties at the n-th injection and purge processes.A. Concentrations of analyte at n-th injection and purgeFor the multistep injection-purge cycles, we consider a rectan-gular pulse wave-like sequence (Fig. 4). The concentration of ananalyte in the gas phase Cg(t) can be described as a step function:Cg(t) ¼0, t , t0Cg , t2(n�1) � t , t2n�1,0, t2n�1 � t , t2n8><>:n ¼ 1, 2, . . . :(26)Journal ofApplied Physics ARTICLE scitation.org/journal/japJ. Appl. Phys. 129, 124503 (2021); doi: 10.1063/5.0039045 129, 124503-6© Author(s) 2021https://aip.scitation.org/journal/japLet inj:Cn(t) and purg:Cn(t) be the concentrations of analyte in acoating film at the n-th injection and at n-th purge, respectively.Then, the reaction rate can be described as a step function:ddtC(t) ¼ 0, t , t0ddtinj: Cn(t) ¼ 1τsKpCg �inj: Cn(t)�  , t2(n�1) � t , t2n�1,ddtpurg: Cn(t) ¼ � 1τspurg: Cn(t), t2n�1 � t , t2n8>>>>>><>>>>>>:n ¼ 1, 2, . . . :(27)In the n-th purge process (i.e., 2n-th step), the general differ-ential equation can be solved asinj:Cn(t) ¼ KpCg þinj: Cn � e� tτs , t2(n�1) � t , t2n�1, (28a)purg:Cn(t) ¼ purg: Cn � e� tτs , t2n�1 � t , t2n, (28b)where inj:Cn and purg:Cn are the arbitrary constants of the n-thinjection and purge, respectively. The concentration at the begin-ning of the n-th purge can be estimated using the concentration ofthe n-th injection process [i.e., (2n� 1)-th step] as inj:Cn(t2n�1).From Eq. (28b), the concentration of n-th purge can be obtained aspurg:Cn(t) ¼inj: Cn(t2n�1) � e�t�t2n�1τs : (29)Similarly, the concentration at the beginning of the (n+1)-th injec-tion process (i.e., t ¼ t2n) should be the concentration at the end ofthe n-th purge process. From Eqs. (28a) and (29), the concentrationof the (n+1)-th injection, therefore, can be found asinj:Cnþ1(t) ¼ KpCg 1� e�t�t2nτs �þinj: Cn(t2n�1) � e�t2n�1τs : (30)Again, owing to the connectivity of the dynamic concentrationbehavior between at the end of the purge and at the beginning ofthe injection, the concentration of the (n+1)-th purge process canbe obtained from Eqs. (28b) and (30) as follows:purg:Cnþ1(t) ¼ KpCg e�t�t2nþ1τs � e�t�t2nτs �þinj: Cn(t2n�1) � e�t�t2n�1τs :(31)From Eqs. (29) and (31), the recurrence relation of the con-centration between the n-th and (n+1)-th purges and that between(n+1)-th injection and n-th purge can be found bypurg:Cnþ1(t)�purg: Cn(t) ¼ KpCg e�t�t2nþ1τs � e�t�t2nτs �, (32a)inj:Cnþ1(t)�purg: Cn(t) ¼ KpCg 1� e�t�t2nτs �, (32b)respectively, with the concentration at the first purge is given inEq. (15a) aspurg:C1(t) ¼ KpCg e�t�t1τs � e�t�t0τs �: (32c)Thus, the recurrence formula can be solved, and hence theconcentration at the n-th injection and purge processes can beobtained asinj:Cn(t) ¼ KpCg � KpCg � e� tτsX2(n�1)i¼0(� 1)ietiτs , (33a)purg:Cn(t) ¼ �KpCg � e� tτsX2n�1i¼0(� 1)ietiτs : (33b)It should be noted that the derived n-th injection and purgeconcentrations in Eq. (33) can be further simplified when theduration of injection and purge are fixed at T (i.e.,T ¼ t2n � t2n�1 ¼ t2n�1 � t2(n�1)). Because tn � t0 ¼ n � T , multi-plying the right-hand side of Eq. (33) by (e�t0=τs � et0=τs ), the con-centrations can be modified asinj:Cn(t) ¼ KpCg � KpCg � e�t�t0τsX2(n�1)i¼0�eTτs �i(34a)¼ KpCg � KpCg � e�t�t2(n�1)τsX2(n�1)i¼0�e�Tτs �i, (34b)t2(n�1) � t , t2n�1,purg:Cn(t) ¼ �KpCg � e�t�t0τsX2n�1i¼0�eTτs �i(35a)¼ KpCg � e�t�t2n�1τsX2n�1i¼0�e�Tτs �i, (35b)t2n�1 � t , t2n:The models given in Eqs. (34) and (35) are assumed to be propor-tional to the concentration of analyte in gas phase Cg ; additionally,FIG. 4. Rectangular pulse wave-like gas injection.Journal ofApplied Physics ARTICLE scitation.org/journal/japJ. Appl. Phys. 129, 124503 (2021); doi: 10.1063/5.0039045 129, 124503-7© Author(s) 2021https://aip.scitation.org/journal/japKpCg is related to the amplitude of the signal. In Eqs. (34b)and (35b), the terms t2n�1 and t2(n�1) denote the time when thesignal starts to rise and to decay for each step in the rectangularpulse wave-like injection-purge cycles, respectively. Since the signaloutput of the nanomechanical sensing is directly proportional tothe concentration C(t) as described in Sec. III A [Eq. (7)], thesignal output is analytically derived for the rectangular injection-purge cycles.In Eqs. (34) and (35), if the duration T is long enough (i.e.,the signal reaches equilibrium state), the term e�n�T=τs is negligible;then, the models given in Eqs. (34) and (35) are symmetricinjection-purge curves as follows:inj:Cn(t) ¼ KpCg � KpCg � e�t�t2(n�1)τs , t2(n�1) � t , t2n�1, (36a)purg:Cn(t) ¼ KpCg � e�t�t2n�1τs , t2n�1 � t , t2n: (36b)In contrast, when the duration T is short or the desorption timeconstant τs is slow, the signal gives asymmetric curves, leading toan increase in the baseline as shown in Fig. 5. Note that the signalreaches symmetric patterns upon increasing the injection-purgecycles due to limn!1 e�n�T=τs ¼ 0 [Fig. 5(b)]. The model clearlyindicates that the analyte with fast τs desorbs from the coating filmwhile the analyte with slow τs accumulates in the coating film whenduration time T is relatively short [Fig. 5(a); e.g., τs ¼ 120].B. Stress with viscoelastic materials at n-th injectionand purgeWe derived the concentration of analyte A in the coating filmat the n-th injection and purge in Eq. (33). According to the for-mulation derived by Wenzel et al.,15 the derived general differentialequations of stress in Eq. (17) can be extended to the rectangularpulse wave-like injection models. Substituting Eq. (27) into Eq. (17)with Eq. (26), the general differential equations of stress can berewritten at the n-th injection and purge as a step function, andcan be solved asinj:σn(t) ¼ �σsat: þ σsat:α � e� tτsX2(n�1)i¼0(�1)ietiτs þinj: Cσn � e� tτr ,t2(n�1) � t , t2n�1, (37a)FIG. 5. Dynamic variation of an analyte concentration in a coating film esti-mated by Eqs. (34) and (35). The numbers of injection-purge cycles is 5 (a)and 30 (b). τs is varied with 120, 60, 30, 20, 10, 5, 1, and 0.1 from the top tobottom, and other parameters are fixed; Kp ¼ 1; Cg ¼ 1; T ¼ 10; and t0 ¼ 0.All black dashed lines are the level of the amplitude KpCg , while the blackdashed lines in the top five graph of (a) are not shown because of out of range.FIG. 6. Dynamic stress change derived from sorption-induced deformation esti-mated by Eqs. (44) and (45). The numbers of injection-purge cycles are 5 (a)and 30 (b). τs is varied with 120, 60, 30, 20, 10, 5, 1, and 0.1 from top tobottom, and other parameters are fixed; EU=ER ¼ 1:75; τ r ¼ 25 [s]; σsat: ¼ 1;and t0 ¼ 0. Solid lines and dashed lines for the stress change denote the dura-tion time T ¼ 10 and .600 [s], respectively. All black dashed horizontal linesare the level of the amplitude σsat: . Black vertical dotted lines are the time whenan overshoot starts to decay tmax:.Journal ofApplied Physics ARTICLE scitation.org/journal/japJ. Appl. Phys. 129, 124503 (2021); doi: 10.1063/5.0039045 129, 124503-8© Author(s) 2021https://aip.scitation.org/journal/jappurg:σn(t) ¼ σsat:α � e� tτsX2n�1i¼0(�1)ietiτs þpurg: Cσn � e� tτr ,t2n�1 � t , t2n, (37b)where inj:σn(t) and purg:σn(t) denote the stress at the n-th injectionand purge, respectively; inj:Cσn and purg:Cσn are the arbitrary con-stants of the n-th injection and purge, respectively.As the boundary condition, the stress at the beginning of then-th purge process should be inj:σn(t2n�1). From Eq. (37) withinj:σn(t2n�1), the stress of the n-th purge process can be obtained bypurg:σn(t) ¼ σsat:α � e� tτsX2n�1i¼0(�1)ietiτs þ an, (38)withan ¼ inj:σn(t2n�1)� σsat:α � e�t2n�1τrX2n�1i¼0(�1)ietiτs" #e�t�t2n�1τr :Similarly, from the boundary conditions at the beginning of the(n+1)-th injection and purge processes, the stresses can also bewritten using inj:σn(t2n�1) asinj:σnþ1(t) ¼ �σsat: þ σsat:α � e� tτsX2ni¼0(�1)ietiτsþ σsat:(1� α) � e�t�t2nτr þ an, (39)purg:σnþ1(t) ¼ σsat:α � e� tτsX2nþ1i¼0(�1)ietiτsþ σsat:(1� α) et2nτr � et2nþ1τr �� e� tτr þ an: (40)From the recurrence relations of the stress between the n-thand (n+1)-th purge and that between the (n+1)-th injection andn-th purge processes in Eqs. (38)–(40) with Eq. (25), the recurrenceformula of the stress can be found aspurg:σnþ1(t)�purg: σn(t) ¼ σsat:α � e� tτs et2nτs � et2nþ1τs �þ σsat:(1� α) � e� tτr et2nτr � et2nþ1τr �, (41a)inj:σnþ1(t)�purg: σn(t) ¼ �σsat: þ σsat:α � e�t�t2nτsþ σsat:(1� α)e�t�t2nτr : (41b)The recurrence formula can be solved, and hence the stress at then-th injection and purge processes can be obtained asinj:σn(t) ¼ �σsat: þ σsat:α � e� tτsX2(n�1)i¼0(�1)ietiτsþ σsat:(1� α) � e� tτrX2(n�1)i¼0(�1)ietiτr þ σ0 � e�t�t0τr ,t2(n�1) � t , t2n�1, (42)purg:σn(t) ¼ σsat:α � e� tτsX2n�1i¼0(�1)ietiτsþ σsat:(1� α) � e� tτrX2n�1i¼0(�1)ietiτr þ σ0 � e�t�t0τr ,t2n�1 � t , t2n:(43)Note that the stress at the n-th injection and purge processesin Eqs. (42) and (43) can also be simplified when the duration timeof injection and purge are fixed at T . From Eqs. (42) and (43), thestresses can be modified asinj:σn(t) ¼ �σsat: þ σsat:α � e�t�t0τsX2(n�1)i¼0�eTτs �iþ σsat:(1� α) � e�t�t0τrX2(n�1)i¼0�eTτr �iþ σ0 � e�t�t0τr (44a)¼ �σsat: þ σsat:α � e�t�t2(n�1)τsX2(n�1)i¼0�e�Tτs �iþ σsat:(1� α) � e�t�t2(n�1)τrX2(n�1)i¼0�e�Tτr �iþ σ0 � e�t�t0τr , t2(n�1) � t , t2n�1, (44b)purg:σn(t) ¼ σsat:α � e�t�t0τsX2n�1i¼0�eTτs �iþ σsat:(1� α) � e�t�t0τrX2n�1i¼0�eTτr �iþ σ0 � e�t�t0τr (45a)¼ �σsat:α � e�t�t2n�1τsX2n�1i¼0�e�Tτs �i� σsat:(1� α) � e�t�t2n�1τrX2n�1i¼0�e�Tτr �iþ σ0 � e�t�t0τr , t2n�1 � t , t2n: (45b)Journal ofApplied Physics ARTICLE scitation.org/journal/japJ. Appl. Phys. 129, 124503 (2021); doi: 10.1063/5.0039045 129, 124503-9© Author(s) 2021https://aip.scitation.org/journal/japIn Eqs. (44) and (45), the analytical solutions clearly express thestress in terms of the elastic properties and relaxation properties indifferent forms with the viscoelastic relation in Eq. (21). Themodels given in Eqs. (44) and (45) are assumed to be proportionalto the concentration of analyte in the gas phase Cg , and σsat: isrelated to the amplitude of the signal. Similarly, with the analyticalsolutions of the elastic models in Eqs. (34b) and (35b), the termst2n�1 and t2(n�1) in Eqs. (44b) and (45b) denote the time when thesignal starts to rise and to decay for the corresponding steps of therectangular pulse wave-like injection-purge cycles, respectively.Again, since the signal output of the nanomechanical sensing isdirectly proportional to the concentration C(t) as described inSec. III A [Eq. (7)], the signal output is analytically derived for therectangular multistep injection-purge cycles with viscoelastic mate-rials. It should be noted that the analytical solutions assume thefirst-order kinetics to the single analyte as shown in Eq. (8). Thus,to describe a multicomponent system, such as complex mixtures,some additional assumptions or some other equations are required.V. RESULTS AND DISCUSSIONA. Numerical calculations of nanomechanical sensingcoated with viscoelastic materialIn this section, the responses of viscoelastic material-coatednanomechanical sensing are numerically calculated using theFIG. 7. Experimentally measured signal responses of viscoelastic material-coated MSS to four different analytes with different concentrations. The signal responses for polystyrene(a)–(d), polycaprolactone (e)–(h), poly(4-methylstyrene) (i)–(l), and alkyl-functionalized silica-titania hybrid nanoparticles (C18-STNPs) (m–p) are shown. Red and blue lines are theoriginal signal outputs and black dashed lines show the numerically calculated responses using Eqs. (44) and (45). The overshoot tendency in Eq. (46) is represented by red lines[observed only in (a)], and the conditions with no overshoot tendency are denoted by blue lines. The exposed gases used in this study are water [(a), (e), (i), and (m)], ethanol [(b),(f ), ( j), and (n)], n-dodecane [(c), (g), (k), and (o)], and 1,2-dichlorobenzene [(d), (h), (i), and (p)]. The concentration is varied with 10%, 20%, and 30% of the saturated vapors.Journal ofApplied Physics ARTICLE scitation.org/journal/japJ. Appl. Phys. 129, 124503 (2021); doi: 10.1063/5.0039045 129, 124503-10© Author(s) 2021https://aip.scitation.org/journal/japmodels derived in Sec. IV B. As presented in Fig. 5, the signaloutput of elastic material-coated nanomechanical sensors onlydepends on the desorption time constant τs. When the desorptionof the analyte is slow (e.g., τs ¼ 120 [s]), the analytes adsorbedand diffused into the coating film during the injection process donot completely desorb during the purge process, resulting in anincrease in the baseline. In contrast, if the desorption time cons-tant is fast enough such that the signal reaches a steady-state oran equilibrium state within the injection and purge processes, thesignal obtains a symmetric response. Therefore, no baselineincrease is observed.When a viscoelastic material is used for a receptor layer of ananomechanical sensor, the corresponding signal drastically changes.In the case of a single rectangular injection model with viscoe-lastic coating, Wenzel et al. widely discussed the detailed ten-dencies of signal responses.15 For example, if sorption occursfaster than the relaxation process (i.e., τs � τr), the responsederived from absorption-induced deformation yields highersignal output than the level of amplitude σsat: because of abuilt-up of unrelaxed stress, followed by a decrease due to relax-ation. Then, the signal response exhibits an overshoot, depictedas red signals in Fig. 6, with long enough duration (dashed lines).Wenzel et al. derived the condition for which the response exhibitsan overshoot asEUER� τsτr. 0, (46)only if EU . ER with long duration.TABLE I. The optimized parameters for each VOC/polymer combination.Receptor VOCCga γ ⋅ σsat. τs τrEU/ER Receptor VOCCga γ ⋅ σsat. τs τrEU/ER(%) (mV) (s) (s) (%) (mV) (s) (s)PCL Water 10 0.28 11.03 0.94 7.18 P4MS Water 10 0.82 11.49 1.14 6.4320 0.52 7.89 0.70 6.90 20 1.36 10.04 0.97 7.2630 0.78 6.96 0.67 6.38 30 1.90 11.14 0.98 8.47Ethanol 10 1.29 23.82 1.72 6.47 Ethanol 10 1.60 26.92 2.25 4.1120 2.85 26.13 1.68 6.98 20 3.74 39.27 2.33 4.9730 4.56 25.28 1.60 7.00 30 5.48 38.96 2.26 5.12n-dodecane 10 0.95 31.21 1.65 5.49 n-dodecane 10 0.99 40.23 2.47 4.5620 2.06 32.19 1.58 5.74 20 2.23 47.42 2.36 5.0130 2.93 26.54 1.47 5.51 30 2.75 35.41 2.22 4.71ODCBb 10 3.27 30.28 4.16 3.18 ODCBb 10 0.50 26.85 4.88 1.9620 6.60 31.27 4.43 3.25 20 1.25 45.85 5.95 2.4830 10.16 33.30 4.68 3.36 30 2.11 60.60 6.83 2.71PS Water 10 0.01 15.92 0.21 348.46 C18-STNPs Water 10 12.00 24.50 1.62 6.2820 0.06 10.95 0.15 124.51 20 18.11 20.82 1.41 7.5030 0.11 8.91 0.18 69.35 30 22.84 17.74 1.06 9.11Ethanol 10 1.43 33.97 2.22 4.51 Ethanol 10 4.17 34.68 1.20 7.4120 3.18 42.61 2.25 4.87 20 6.53 29.55 1.18 7.1030 4.40 39.06 2.18 4.77 30 8.28 25.38 1.22 6.66n-dodecane 10 1.00 36.95 2.93 3.42 n-dodecane 10 1.37 61.93 61.95 2.7120 2.07 40.67 2.64 3.64 20 2.40 57.37 57.38 2.7530 2.55 30.02 1.96 3.63 30 3.30 48.95 48.96 2.45ODCBb 10 0.50 32.58 1.98 6.76 ODCBb 10 1.88 18.23 3.36 2.8820 0.86 34.35 1.99 6.90 20 3.39 16.46 3.39 2.7230 1.10 32.16 2.04 6.09 30 4.64 14.34 3.12 2.51aConcentration of VOCs at the injection process [see also Eq. (26)]. Values correspond to Pa/Po, where Pa and Po denote the partial pressure and saturatedvapor pressure, respectively.b1,2-dichlrorobenzene.FIG. 8. Experimentally measured signal responses of PCL-coated MSS to1,2-dichlorobenzene. (a) Result of Fig. 7(h). (b) Fitting curves with optimizedparameters calculated from 10% signal response. Corresponding γ � σsat: ismultiplied by 20%/10% and 30%/10%, respectively.Journal ofApplied Physics ARTICLE scitation.org/journal/japJ. Appl. Phys. 129, 124503 (2021); doi: 10.1063/5.0039045 129, 124503-11© Author(s) 2021https://aip.scitation.org/journal/japIf the duration is shorter than the time tmax: when the relaxa-tion starts to decay (i.e., T � tmax:), it is expected that the signaldoes not yield the overshoot characteristics (Fig. 6). Importantly,the baseline at time t2n, when the injection starts, exhibits a clearcorrelation with the signal responses of long duration (dottedlines). These characteristic baseline transitions clearly indicate theviscoelastic response by a sorption-induced nanomechanicalsensing. If the desorption time constant is small, the overshoot canbe observed even in the short duration (10 s). In such a case, thebaseline monotonously decreases and reaches the steady-state.Conversely, when the desorption time constant is relatively large,the baseline shows an overshoot-like tendency, i.e., the baselinefirst increases, then decreases after tmax: (Fig. 6; see also Figs. S1and S2 in the supplementary material). In the case that the condi-tion given in Eq. (46) does not hold, the overshoot does not occuras shown in blue lines in Fig. 6.The nanomechanical sensing responses with different dura-tions of injection-purge cycles are also numerically simulated usingthe model derived in Sec. IV B as described in the Appendix. UsingEqs. (42) and (43), it is possible to design the effective injection-purge sequence depending on the target features to be extracted.FIG. 9. Fitting accuracy between a single injection signal response and multistep injection-purge cycle. (a) and (b) 1,2-Dichlorobenzene/PCL and (c) and (d) n-dodecane/PS are shown (Pa=Po ¼ 30%). Red colored signal responses are used for optimizing each fitting curve. Black dashed lines are the corresponding fitting curves. Blackarrow indicates the point where the secondary or tertiary viscoelastic response may start taking place.FIG. 10. Dynamic stress change derived from sorption-induced deformationestimated by Eqs. (42) and (43) with different durations. (a) The injection andpurge durations are set at 10 and 50 s, respectively. EU=ER ¼ 1:75;τ r ¼ 25 [s]; τs ¼ 30 [s]; σsat: ¼ 1; and t0 ¼ 0. (b) The injection and purgedurations are set at 50 and 10 s, respectively. EU=ER ¼ 1:75; τ r ¼ 25 [s];τs ¼ 10 [s]; σsat: ¼ 1; t0 ¼ 0. Blue and red lines indicate the injection andpurge responses, respectively. (c) The numbers of injection-purge cycles are 15with different durations T ; 1–5 cycles, 5 s; 6–10 cycles, 10 s; and 11–15 cycles,30 [s]. EU=ER ¼ 10; τ r ¼ 15 [s]; τs ¼ 10 [s]; σsat: ¼ 1; and t0 ¼ 0.Journal ofApplied Physics ARTICLE scitation.org/journal/japJ. Appl. Phys. 129, 124503 (2021); doi: 10.1063/5.0039045 129, 124503-12© Author(s) 2021https://www.scitation.org/doi/suppl/10.1063/5.0039045https://aip.scitation.org/journal/japB. Estimation of parametersThe experimentally obtained signal output of MSS was fittedusing Eqs. (44) and (45) and the parameters are optimized. Wefabricated viscoelastic material-coated MSS with three different vis-coelastic polymers, PS, PCL, and P4MS, which are a series ofhydrophobic polymers. We also fabricated viscoelastic alkyl-functionalized inorganic nanoparticles, i.e., octadecyl-functionalized silica-titania hybrid nanoparticles (C18-STNPs).9,22Their signal responses to a varieties of volatile organic compounds(VOCs), i.e., ethanol, n-dodecane, and 1,2-dichlorobenzene, as wellas water were measured.Since the signal output is directly proportional to the concen-tration of an analyte in a coating film as described in Sec. III A, thesignal output of MSS Vout can be given byVout(t) ¼ γ � C(t)þ Vout(t0) (47a)for the elastic model, andVout(t) ¼ γ � σ(t)þ Vout(t0) (47b)for the viscoelastic model, where γ and Vout(t0) are the proportional-ity factor and the signal output at t ¼ t0 (i.e., baseline), respectively.When γ ¼ 1, the analytical models in Eqs. (34)–(35) and (42)–(43)are directly applicable to the sensing signal of MSS. We fit the signaloutput with Eqs. (44) and (45) for five cycles. The parameters areoptimized through non-linear least squares curve fitting using Pythonwith SciPy module. The details are presented in Sec. II.Figures 7(a)–7(h) show the fitting results for the signal outputsof viscoelastic polymer-coated MSS. The optimized parameters aresummarized in Table I. The signal responses are well fitted with thederived equations for five injection-purge cycles. The optimizedstress relaxation parameters (EU=ER and τr) of each analyte/polymer pair show a good agreement in the different concentra-tions, except water/PS pair. In the case of the signal response ofPS-coated MSS to water vapor, the strong overshoot was observed.According to the result in Fig. 6, the tendency of a strong overshootindicates that the diffusion time constant τs is significantly largerthan the relaxation time constant τr (τs � 0:1). It is probablybecause the diffusion speed is extremely slow, due to the hydropho-bic nature of PS, and the signal response only reflects the adsorptionprocess, which is generally much faster than the absorptionprocess.13 Such an extremely small time constant cannot be numeri-cally calculated (i.e., e�1000 ! e�1 ¼ 0), resulting in the unopti-mized parameters. Importantly, each baseline transition clearly fitswith the results in the numerical solutions (Fig. 6). Water/PS showsthe overshoot response; therefore, the baseline rapidly decreases,while the other cases do not exhibit overshoot tendency with rela-tively large τs, leading to the increase in the baseline.We also demonstrated the fitting with alkyl-functionalized inor-ganic nanoparticles [Figs. 7(i)–7(l)]. Generally, inorganic bulk materi-als tend to have elastic properties, while the nanoparticlesfunctionalized with organic groups often exhibit viscoelastic behavior.The signal responses of C18-STNPs are well fitted with thederived equations for viscoelastic material in Eqs. (44) and (45);however, they do not fit well with the elastic equations inEqs. (34) and (35) (data not shown). This result clearly indicatesthat the alkyl-functionalized nanoparticles also exhibit the vis-coelastic behavior.It should be noted that most of the analyte/coating pairs showthe amplitude parameter γ � σsat: proportional to the concentrationsof analytes. As shown in Fig. 8, the signal responses of differentconcentrations (20% and 30%) can be entirely predicted with thefitting parameters extracted from the 10% signal response bysimply multiplying the amplitude parameter γ � σsat:. The goodagreement of the concentration dependency demonstrates that thepredictive capabilities of the theoretical models.Importantly, the optimized parameters generally depend onthe least squares techniques and the initial fitting parameters (seealso Fig. S3 and Table S1). When the measured signal responsesare short (e.g., 30 s), then the parameters extracted from theexperimental results cannot predict the entire signal responses(Fig. 9; see also Fig. S4 for the coefficient of determination, R2).If the signal response seems not to reach the steady state, theparameters extracted from a single injection procedure do not fitwell to the experimental results (i.e., 1,2-dichlorobenzene/PCL)[Fig. 9(a); Fig. S4a]. Conversely, in the case of the multistepinjection-purge cycle, the curves predicted by the extractedparameters fit well with the experimental results [Fig. 9(b);Fig. S4b]. Furthermore, polymers sometimes exhibit the secon-dary and tertiary viscoelastic relaxation properties.17 As can beseen in Fig. 9(d), PS changes the signal response at approximately180 s, followed by a decrease in the signal output. In such cases, itis difficult to measure the response of the steady state, therebyresulting in the extraction of less accurate parameters. Conversely,in the case of the multistep injection-purge cycles, the analytedesorbs from a receptor during the purge step, thereby minimiz-ing such changes in the elastic property [Fig. 9(d); Fig. S4d].Therefore, the optimized parameters extracted from a couple ofinjection-purge cycles can be clearly predicted by the experimen-tal responses.VI. CONCLUSIONThe theory of the nanomechanical sensors coated with a vis-coelastic material proposed by Wenzel et al.15 was extended to themultistep injection-purge cycle model. This theory includes thestress relaxation effects of a viscoelastic coating film and representsthe accurate signal responses. The numerical calculations show verygood agreement with the trends observed in the experimentalresults measured by MSS coated with different viscoelastic materi-als, including polymers and inorganic nanoparticles. Therefore, thetheoretical model can be utilized to extract values of the variouscoating parameters as well as the coating film/analyte parameters.More importantly, the present derived model can be used topredict and/or analyze the signal responses of a nanomechanicalsensor during the absorption and desorption processes withoutmeasuring until the signal reaches the steady-state. By measuring acouple of injection-purge cycles with relatively short duration, theactual values of the coating properties can be extracted.Journal ofApplied Physics ARTICLE scitation.org/journal/japJ. Appl. Phys. 129, 124503 (2021); doi: 10.1063/5.0039045 129, 124503-13© Author(s) 2021https://aip.scitation.org/journal/japAnalysis of the transient response using the presentedmodels will be beneficial for the improvement in recognitionaccuracy of analytes based on scientific interpretation. The opti-mized parameters can be extracted from the multistep injection-purge cycles more accurately than that extracted from a singleinjection or purge curve and, hence, the optimized parameters(e.g., τs) can be directly used as an effective index for the identifi-cation of gas species as proposed in the previous literature.16 Thepresented model can be utilized for the analyses of repeatedinjection-purge cycles with a short duration, contributing to thedevelopment of the practical artificial olfaction.SUPPLEMENTARY MATERIALSee the supplementary material for the detailed numerical cal-culations of viscoelastic responses.ACKNOWLEDGMENTSWe thank Mr. Takahiro Nemoto, Center for FunctionalSensor & Actuator (CFSC), National Institute for Materials Science(NIMS), Japan for fruitful discussion on this present study. K.M.acknowledges the International Center for Young Scientists (ICYS)program, NIMS. This study was financially supported by JSTCREST (No. JPMJCR1665); a Grant-in-Aid for Scientific Research(A), MEXT, Japan (No. 18H04168); Grant-in-Aid for ChallengingResearch (Pioneering) (No. 20K20554); the Public/Private R&DInvestment Strategic Expansion Program (PRISM), Cabinet Office,Japan; the Izumi Science and Technology Foundation (No.2020-J-070); ICYS, NIMS; CFSN, NIMS; and the World PremierInternational Research Center Initiative (WPI) on MaterialsNanoarchitectonics (MANA), NIMS.APPENDIX: PREDICTION OF INJECTION-PURGECYCLES WITH DIFFERENT DURATIONThe viscoelastic model derived in Sec. IV B can be utilized forpredicting and simulating the signal response of injection-purgecycles with different durations. For example, in the case of follow-ing parameters: EU=ER ¼ 1:75; τr ¼ 25 [s]; τs ¼ 10 [s], the signalresponse yields significant baseline drift (see Fig. 6), while each ofinjection and purge set at 10 and 50 s yields the signal responseswithout baseline drift [Fig. 10(a)]. Similarly, the small diffusiontime constant τs ¼ 10 [s] shows a negative baseline shift due to theviscoelastic behavior, while the baseline drifts can be minimized bytuning each duration time of the injection and purge as 50 and 10 s,respectively [Fig. 10(b)]. Additionally, when we change the durationtime, for example, from 10 s to 30 s, the first and second injectionsyield the different intensities [Fig. 10(c), black arrow]. After optimiz-ing the parameters from the experimental results, it is possible topredict and simulate the signal responses with any kind of injection-purge sequence using the presented model.DATA AVAILABILITYThe data that support the findings of this study are availablefrom the corresponding author upon reasonable request.REFERENCES1J. K. Gimzewski, Ch. Gerber, E. Meyer, and R. R. Schlittler, “Observation of achemical-reaction using a micromechanical sensor,” Chem. Phys. Lett. 217,589–594 (1994).2K. M. Goeders, J. S. Colton, and L. A. 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EXPERIMENTAL SECTION A. Materials B. Fabrication of MSS C. Sensing D. Curve fitting and estimation of parameters III. BACKGROUND THEORY A. Sorption-induced nanomechanical sensing B. Governing equations for single-step absorption/desorption processes C. Governing equations for viscoelastic materials IV. ANALYTICAL SOLUTIONS A. Concentrations of analyte at n-th injection and purge B. Stress with viscoelastic materials at n-th injection and purge V. RESULTS AND DISCUSSION A. Numerical calculations of nanomechanical sensing coated with viscoelastic material B. Estimation of parameters VI. CONCLUSION SUPPLEMENTARY MATERIAL DATA AVAILABILITY References