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

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[Finite Element Analysis of Interface Dependence on Nanomechanical Sensing](https://mdr.nims.go.jp/datasets/9b159eec-9763-41c7-ade5-408e5d66db10)

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Finite Element Analysis of Interface Dependence on Nanomechanical SensingsensorsArticleFinite Element Analysis of Interface Dependence onNanomechanical SensingKosuke Minami 1,2,* and Genki Yoshikawa 2,3,41 International Center for Young Scientists (ICYS), National Institute for Materials Science (NIMS), 1-1 Namiki,Tsukuba, Ibaraki 305-0044, Japan2 Center for Functional Sensor & Actuator (CFSN), National Institute for Materials Science (NIMS),1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan; YOSHIKAWA.Genki@nims.go.jp3 International Research Center for Materials Nanoarchitectonics (MANA), National Institute for MaterialsScience (NIMS), 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan4 Materials Science and Engineering, Graduate School of Pure and Applied Science, University of Tsukuba,1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan* Correspondence: MINAMI.Kosuke@nims.go.jpReceived: 10 February 2020; Accepted: 4 March 2020; Published: 10 March 2020�����������������Abstract: Nanomechanical sensors and their arrays have been attracting significant attentionfor detecting, discriminating and identifying target analytes. The sensing responses can bepartially explained by the physical properties of the receptor layers coated on the sensing elements.Analytical solutions of nanomechanical sensing are available for a simple cantilever model includingthe physical parameters of both a cantilever and a receptor layer. These analytical solutions generallyrely on the simple structures, such that the sensing element and the receptor layer are fully attachedat their boundary. However, an actual interface in a real system is not always fully attached becauseof inhomogeneous coatings with low affinity to the sensor surface or partial detachments causedby the exposure to some analytes, especially with high concentration. Here, we study the effectsof such macroscopic interfacial structures, including partial attachments/detachments, for staticnanomechanical sensing, focusing on a Membrane-type Surface stress Sensor (MSS), through finiteelement analysis (FEA). We simulate various macroscopic interfacial structures by changing the sizes,numbers and positions of the attachments as well as the elastic properties of receptor layers (e.g.,Young’s modulus and Poisson’s ratio) and evaluate the effects on the sensitivity. It is found thatspecific interfacial structures lead to efficient sensing responses, providing a guideline for designingthe coating films as well as optimizing the interfacial structures for higher sensitivity includingsurface modification of the substrate.Keywords: Membrane-type Surface stress Sensor (MSS); nanomechanical sensors; static modeoperation; interface; finite element analysis (FEA)1. IntroductionNanomechanical sensors and their arrays have gained significant attention as a powerful tool fordetecting, discriminating and identifying target analytes [1–4], especially various odors composed ofa complex mixture of gaseous molecules [5–7]. The versatility of these sensors and their arrays arebased on physical and chemical properties of a receptor layer coated on a sensing element. In the caseof so-called static mode operation, sensing signals are given by mechanical stress/strain induced bysorption of target molecules in a receptor layer. To obtain high sensitivity in nanomechanical sensing,it is important to efficiently transduce the mechanical stress/strain derived from the deformation of thereceptor layers to the sensing elements. In the practical conditions, however, coating films and theSensors 2020, 20, 1518; doi:10.3390/s20051518 www.mdpi.com/journal/sensorshttp://www.mdpi.com/journal/sensorshttp://www.mdpi.comhttps://orcid.org/0000-0003-4145-1118http://www.mdpi.com/1424-8220/20/5/1518?type=check_update&version=1http://dx.doi.org/10.3390/s20051518http://www.mdpi.com/journal/sensorsSensors 2020, 20, 1518 2 of 11surface of sensing elements frequently have different affinities, reflecting the chemical properties ofeach material, such as organic polymers and an inorganic silicon substrate, leading to poor attachmentsat their interface and a reduction in the efficiency of the mechanical transduction.For theoretical investigation of the nanomechanical sensing, there are several analytical solutions,especially for a simple cantilever model. For example, the displacement of a free end of a cantilever(∆z) induced by isotropic internal strain in a receptor layer (εf) is given by the following equation [8]:∆z =3l2(t f + ts)(A + 4)t2f + (A−1 + 4)t2s + 6t f tsε f , (1)with:A =E f w f t f1− ν f/Eswsts1− νs, (2)where the subscripts “f ” and “s” denote the coating film and the cantilever substrate, respectively,and l, t, w, E, and ν correspond to length, thickness, width, Young’s modulus and Poisson’s ratio,respectively. The internal strain in a receptor layer (εf) can be replaced by other parameters, such asthree-dimensional internal stress in the coating film (σf; in the unit of [N m−2]) or two dimensionalsurface stress (σsurf.; in the unit of [N m−1]), via the relations ε f = σ f(1− ν f)/E f or σ f = σsurf./t f [8–10].When a cantilever is covered with a thin film having a same width (ts >> tf and ws = wf), Equation (1)reduces to the following equation, which is known as the Stoney’s equation [9]:∆z =3l2(1− νs)Est2sσsurf.. (3)However, these models are limited to simple analytical problems, and it is still difficult to expandthese models to complex problems, especially to the systems having various interfacial structures.On the other hand, it is possible to simulate numerical solutions by using finite element analysis (FEA)even for such complex problems of the nanomechanical sensors [3,8,11–16].In the present study, we investigate sensing responses of nanomechanical sensors, focusing on theinterfacial structures between a coating film and a substrate of the sensors. Since the availableanalytical solutions of nanomechanical sensing so far are based on the model in which thecoating layer is fully attached on the surface of the sensing element, it is impossible to applythe solutions to complicated interfacial structures, such as partially attached models at their boundary.Thus, we perform numerical calculations of the various interfacial attachment models through FEA usingCOMSOL Multiphysics® 5.4 with the Structural Mechanics module. We focus on a nanomechanicalsensor, especially a Membrane-type Surface stress Sensors (MSS), which is one of the optimizednanomechanical sensors for static operation based on the integrated piezoresistive read-out with highsensitivity [3–7,11,12,17–20].It should be noted here that we will discuss a macroscopic model in this study, assuming the“ideal interface” at the attached parts as described in Section 2. Accordingly, the microscopic interfacephenomena, such as the lap shear or the interfacial slip, reported as the molecular level effects or thefinite size effects for surface stress-based signal responses [21–23] are not taken into account. The reasonfor this assumption is based on the experimental observation; we sometimes encounter a suddendecrease in sensing signals when a receptor layer-coated MSS is exposed to some target analytes,indicating a partial detachment of the receptor layer. This kind of partial detachment could be alsoconfirmed by optical microscope or scanning electron microscope observation. Even in the case withsuch a partially detached receptor layer, we still observe sensing signals and sometimes the signals areeven enhanced. Thus, we assume that the sensing signals are significantly affected by the interfacialstructures. To understand the effects of these partial interfacial attachments on the nanomechanicalsensing including the domain sizes of the partial attachments as well as the distributions of the partialSensors 2020, 20, 1518 3 of 11attachment points, we modeled various macroscopic structures at the interface between the membranesurface of MSS and the receptor layers.2. SimulationTo examine the effects of the interfacial structures, pillar-like structures with a fixed heighth (= 10 [nm]) and varying radius rpillar from 1 to 20 µm with the same materials of a coating film areplaced between the coating film and the surface of the MSS as the attached points at the interface(Figure 1). Note that these attached points are modeled as “ideal attachments” without assuming anylap shear or interfacial slip. The dimensions for the MSS were set according to the previous report(Figure 1a) [11]. The diameter and the thickness of the membrane were 300 µm and 3 µm, respectively.The membrane is suspended by the four sensing beams, in which piezoresistors are embedded (R1–R4).The dimensions of each beam in the directions x and y are as follows: sensing beams for R1 and R3,12 µm × 18 µm; sensing beams for R2 and R4, 28 µm × 13 µm. A receptor layer was set with radiusrf = 150 µm and thickness tf = 990 nm applying isotropic internal strain, εf = 1.0 × 10−5. Each geometrywas meshed over 20,000 elements, which give sufficient resolution for the present simulation. In thecase of an MSS, the surface stress on the membrane is transduced to the four sensing beams as anamplified uniaxial stress, resulting in the changes in electrical resistance of the piezoresistors embeddedin the beams. In contrast to simple cantilever-type nanomechanical sensors, in which a displacementof free end ∆z (see also Figure S1 in the Supplementary Material) corresponds to a sensing signal,we calculated the total output resistance change ∆R/R|total obtained from the Wheatstone bridge circuitcomposed of the four piezoresistors, providing the sensing signals of MSS. The p-type piezoresistors ofthe MSS are fabricated by doping boron onto a single crystal Si with (100) surface to take advantage ofits high piezocoefficient [3,11,24–26]. Assuming in-plane stress (i.e., σz = 0), relative resistance changecan be described as [26,27]:∆RiRi≈12π44(σx − σy), (4)where π44 (~138 × 10−11 [Pa−1]) is one of the fundamental piezoresistance coefficients of the siliconcrystal, and σx, σy and σz are stresses induced on the piezoresistors in [110], [1–10] and [001] directionsof the silicon crystal, respectively. The subscript of “i” indicates the position of the piezoresistors onthe MSS as illustrated in Figure 1a. The ∆R/R|total of all four resistors can be approximately given bythe following equation:∆RR∣∣∣∣∣total=(∆R1R1−∆R2R2+∆R3R3−∆R4R4), (5)where ∆Ri/Ri is the relative resistance change in Ri (i = 1–4) [3,11]. Due to the symmetric geometry,Equation (5) can be reduced to the following equation:∆RR∣∣∣∣∣total= 2(∆R1R1−∆R2R2). (6)Sensors 2020, 20, 1518 4 of 11Sensors 2020, 20, 4 of 11   Figure 1. Geometry of a partially attached interface model. (a) Configuration of an MSS. The piezoresistor-integrated sensing beams are magnified in the insets. All images are illustrated in top-view. Numbers indicate the dimensions in µm. (b) Cross-sectional image of a pillar as an interfacial attachment. The height of pillars is fixed at 10 nm with various radius, rpillar. (c,d) The results of the FEA simulations for the conventional fully attached model (c) and partially attached model (d), respectively. The distributions of relative resistance changes are plotted as a color gradient. (e) Typical signal output of the MSS measured in the experiments. Poly(methyl methacrylate) (PMMA) was coated on the surface of the membrane. Ef, f and tf are approximately 3 GPa, 0.4 and 1 µm, respectively. Signal outputs were measured at the bias voltage VB of –1.0 V. The concentrations of water vapor Pa/Po are set to 2%, 3% and 5%, where Pa and Po stand for the partial pressure and saturated vapor pressure, respectively. Black dashed line indicates the level of total resistance change (R/R|total = 5.82 × 10−4) simulated by FEA for the fully attached model (Ef = 3 [GPa], f = 0.4, tf = 1 [µm], f = 1 × 10−5). 3. Results and Discussion To investigate the effects of the domain sizes of the interfacial attached points, we first simulated the interfacial structures with uniformly distributed pillars as a function of area. As presented in Figure 2a,b, 21 pillars were placed on the surface of the membrane of MSS as a model of the attached points. The changes in the relative resistance R/R|total obtained by Young’s modulus Ef and Poisson’s ratio f of a coating layer are calculated by FEA. To confirm the effects of the area of the interfacial attachments, the radii of the pillars rpillar are varied from 1 µm to 20 µm. The detailed parameters are listed in Table S1 in the Supplementary Material. As expected, the larger area of interfacial attachments gives the higher R/R|total in the wide range of the Young’s moduli of the coating films (Figure 2c,d, solid line with closed circles; see also Figure S2, in the Supplementary Material). As presented in Figure 2e, the Poisson’s ratio of the coating films considerably affected R/R|total; the higher Poisson’s ratio gives higher R/R|total (see also Figure S3, in the Supplementary Material). abc dxy e 0.2Signal output, Vout[mV]Time [s]0 20 40 60 800.150.10.050H2O N2 H2O N2 H2O N2 H2O N2 15DR/R |total (x10–4)1050300R 1 R 3R 2R 4R 1R 212181328P illars= Interfacial attachm entSilicon membraneCoating film990 nm10 nm3 µmFigure 1. Geometry of a partially attached interface model. (a) Configuration of an MSS.The piezoresistor-integrated sensing beams are magnified in the insets. All images are illustrated intop-view. Numbers indicate the dimensions in µm. (b) Cross-sectional image of a pillar as an interfacialattachment. The height of pillars is fixed at 10 nm with various radius, rpillar. (c,d) The results ofthe FEA simulations for the conventional fully attached model (c) and partially attached model (d),respectively. The distributions of relative resistance changes are plotted as a color gradient. (e) Typicalsignal output of the MSS measured in the experiments. Poly(methyl methacrylate) (PMMA) was coatedon the surface of the membrane. Ef, νf and tf are approximately 3 GPa, 0.4 and 1 µm, respectively.Signal outputs were measured at the bias voltage VB of –1.0 V. The concentrations of water vapor Pa/Poare set to 2%, 3% and 5%, where Pa and Po stand for the partial pressure and saturated vapor pressure,respectively. Black dashed line indicates the level of total resistance change (∆R/R|total = 5.82 × 10−4)simulated by FEA for the fully attached model (Ef = 3 [GPa], νf = 0.4, tf = 1 [µm], εf = 1 × 10−5).The signal output of the full Wheatstone bridge (Vout) is given by:Vout =VB4∆RR∣∣∣∣∣total, (7)where VB is bias voltage applied to the bridge [3,11]. Typical signal responses of poly(methylmethacrylate) (PMMA) for water vapors are shown in Figure 1e (see also the Supplementary Materialfor detailed sensing measurement). A fixed constraint was applied on the outer edges of the sensingbeams (Figure 1a).3. Results and DiscussionTo investigate the effects of the domain sizes of the interfacial attached points, we first simulatedthe interfacial structures with uniformly distributed pillars as a function of area. As presented inFigure 2a,b, 21 pillars were placed on the surface of the membrane of MSS as a model of the attachedpoints. The changes in the relative resistance ∆R/R|total obtained by Young’s modulus Ef and Poisson’sratio νf of a coating layer are calculated by FEA. To confirm the effects of the area of the interfacialattachments, the radii of the pillars rpillar are varied from 1 µm to 20 µm. The detailed parameters arelisted in Table S1 in the Supplementary Material. As expected, the larger area of interfacial attachmentsgives the higher ∆R/R|total in the wide range of the Young’s moduli of the coating films (Figure 2c,d,solid line with closed circles; see also Figure S2, in the Supplementary Material). As presented inFigure 2e, the Poisson’s ratio of the coating films considerably affected ∆R/R|total; the higher Poisson’sratio gives higher ∆R/R|total (see also Figure S3, in the Supplementary Material).Sensors 2020, 20, 1518 5 of 11Sensors 2020, 20, 5 of 11   Figure 2. Dependence of the relative resistance change R/R|total on the area of the interfacial attachments calculated by FEA. (a) Configuration of pillars as a model of interfacial attachment. (Left) Uniformly distributed model, number of pillars, N = 21; (center) Peripheral position model, N = 12; and (right) Center-distributed model, N = 9. (b) Magnified configuration of pillars shown in the rectangular in (a). Numbers indicate the dimensions in µm. The radii of pillars (rpillar) are varied from 1 µm to 20 µm. (c,d) Dependence of the total resistance change (R/R|total) on Young’s moduli as a function of the area of pillars. The Young’s moduli (Ef) are varied from 0.01 GPa to100 GPa (c) and from 1 GPa to 5 GPa (d). Poisson’s ratio of the coating film (f) is set at 0.4 and rpillar is varied as follows: 1, 2, 5, 10, 20 µm. (e) Dependence of the total resistance change (R/R|total) on Poisson’s ratios as a function of the area of pillars. The Young’s modulus of coating film (Ef) is set at 3.0 GPa, which is similar to that of PMMA. All dotted lines in (c), (d) and (e) are R/R|total of a conventional full attachment model without any pillars. (f) The comparison between the uniformly distributed model (N = 21) and the peripheral position model (N = 12). To investigate the effects of the distributions of the pillars, we then investigate the position dependence of pillars. When the pillars were placed at the position close to the center of the membrane surface (the right model in Figure 2a, N = 9), the changes in the relative resistance R/R|total significantly decreased (Figure 2c,d, dashed lines with open squares). On the other hand, interestingly, 12 pillars placed only at the peripheral position exhibited similar level of R/R|total (Figure 2a,c,d, dashed lines with open circles), especially for the pillars with larger area (rpillar ≥ 10 µm) (Figure 2f). To confirm this position dependence further, we simulated two different models as illustrated in Figure 3a: (i) pillars placed as a function of position, rpos., with a fixed number of pillars (Number-fixed model, N = 12), and (ii) pillars placed as a function of position, rpos., with a fixed distance between the center to center of pillars (Distance-fixed model, dpillar = 3 µm). We fixed the radius of pillars, rpillar, at 1 µm in both models. Detailed positions and their parameters are listed in Tables 1 and 2. As observed in the result of FEA as a function of position, rpos., the positions strongly affected R/R|total in both Number-fixed and Distance-fixed models (Figure 3b–d; see also Supplementary Material for detailed data). Comparing the models with pillars placed near the center and the peripheral positions, R/R|total values of the latter cases reached approximately 300 times and 600 times higher than those of the former cases in the cases of Number-fixed and Distance-fixed models (rpos. = 6.8 µm), respectively. The obtained values are in good agreement with the conventional fully attached models, which have a receptor layer with a radius rf = rpos. (Figures S4 and S5 in the Supplementary Material). These results indicate that rpos. significantly affects the effective coverage of each receptor layer, sometimes resulting in the significant loss of the deformation-induced surface stress applied to the membrane surface of MSS. 101 102 10310–710–110–510–410–610–210–3∆R/RArea of pillars [µm2]1101000.10.01[GPa]104 105 101 102 1030104628∆R/R (x10–4)Area of pillars [µm2]34521[GPa]104 105 101 102 103062314∆R/R (x10–4)Area of pillars [µm2]104 105nf = 0nf = 0.25nf = 0.45c d eFully attachedf0 5 100100204060∆R/R12pillar∆R/R21pillarRadius of pillars, rpillar [µm]15 20801101000.10.01[GPa][%]rpillarr= 150705Beam MembraneU niformN = 21 (12 + 8 + 1)P eripheralN = 12C enterN = 9 (8 + 1)a bN = 21 (Uniform) N = 12 (Peripheral) N = 9 (Center)Figure 2. Dependence of the relative resistance change ∆R/R|total on the area of the interfacialattachments calculated by FEA. (a) Configuration of pillars as a model of interfacial attachment. (Left)Uniformly distributed model, number of pillars, N = 21; (center) Peripheral position model, N = 12; and(right) Center-distributed model, N = 9. (b) Magnified configuration of pillars shown in the rectangularin (a). Numbers indicate the dimensions in µm. The radii of pillars (rpillar) are varied from 1 µm to20 µm. (c,d) Dependence of the total resistance change (∆R/R|total) on Young’s moduli as a function ofthe area of pillars. The Young’s moduli (Ef) are varied from 0.01 GPa to100 GPa (c) and from 1 GPato 5 GPa (d). Poisson’s ratio of the coating film (νf) is set at 0.4 and rpillar is varied as follows: 1, 2, 5,10, 20 µm. (e) Dependence of the total resistance change (∆R/R|total) on Poisson’s ratios as a functionof the area of pillars. The Young’s modulus of coating film (Ef) is set at 3.0 GPa, which is similar tothat of PMMA. All dotted lines in (c), (d) and (e) are ∆R/R|total of a conventional full attachment modelwithout any pillars. (f) The comparison between the uniformly distributed model (N = 21) and theperipheral position model (N = 12).To investigate the effects of the distributions of the pillars, we then investigate the positiondependence of pillars. When the pillars were placed at the position close to the center of the membranesurface (the right model in Figure 2a, N = 9), the changes in the relative resistance ∆R/R|total significantlydecreased (Figure 2c,d, dashed lines with open squares). On the other hand, interestingly, 12 pillarsplaced only at the peripheral position exhibited similar level of ∆R/R|total (Figure 2a,c,d, dashed lineswith open circles), especially for the pillars with larger area (rpillar ≥ 10 µm) (Figure 2f). To confirmthis position dependence further, we simulated two different models as illustrated in Figure 3a: (i)pillars placed as a function of position, rpos., with a fixed number of pillars (Number-fixed model,N = 12), and (ii) pillars placed as a function of position, rpos., with a fixed distance between the centerto center of pillars (Distance-fixed model, dpillar = 3 µm). We fixed the radius of pillars, rpillar, at 1 µmin both models. Detailed positions and their parameters are listed in Tables 1 and 2. As observedin the result of FEA as a function of position, rpos., the positions strongly affected ∆R/R|total in bothNumber-fixed and Distance-fixed models (Figure 3b–d; see also Supplementary Material for detaileddata). Comparing the models with pillars placed near the center and the peripheral positions, ∆R/R|totalvalues of the latter cases reached approximately 300 times and 600 times higher than those of theformer cases in the cases of Number-fixed and Distance-fixed models (rpos. = 6.8 µm), respectively.The obtained values are in good agreement with the conventional fully attached models, which havea receptor layer with a radius rf = rpos. (Figures S4 and S5 in the Supplementary Material). These resultsindicate that rpos. significantly affects the effective coverage of each receptor layer, sometimes resultingin the significant loss of the deformation-induced surface stress applied to the membrane surfaceof MSS.Sensors 2020, 20, 1518 6 of 11Sensors 2020, 20, 6 of 11   Figure 3. Dependence of the relative resistance change (R/R|total) on the position of the interfacial attachments (rpos.) calculated by FEA. (a) Configuration of pillars as a model of interfacial attachment. The radius of pillars (rpillar) is fixed at 1 µm. (Left) Number-fixed model. The number of the pillars (N) is fixed at 12. (Right) Distance-fixed model. Center-to-center distance between pillars is fixed at ca. 3 µm, indicating that the shortest distance between pillars is 1 µm. Detailed distances are listed in Tables 1 and 2. (b–c) Dependence of the total resistance change R/R|total on Young’s modulus as a function of the position of pillars (rpos.). The Young’s moduli (Ef) are varied from 0.01 GPa to 100 GPa (b) and from 1 GPa to 5 GPa (c). Poisson’s ratio of a coating film (f) is fixed at 0.4. (d) Dependence of the total resistance change R/R|total on Poisson’s ratio as a function of the position of pillars (rpos.). The Young’s modulus of coating film is fixed at 3.0 GPa, which is similar to that of PMMA. Dotted lines in (b), (c) and (d) are R/R|total of the full attachment model. It should be noted that pillars placed at the peripheral position in the case of Distance-fixed model (Figure 3) provide higher R/R|total than that of Number-fixed model with rpillar = 20 µm (Figure 2), even though the area of Distance-fixed model with rpillar = 1 µm (942 µm2; Table 2) is only 6% of Number-fixed model with rpillar = 20 µm (15,080 µm2; Table S1 in the Supplementary Materials). This result indicates that the distance between interfacial attachments (dpillar) affects R/R|total more significantly than the area of interfacial attachments (rpillar). To confirm this aspect, we constructed the modified model as illustrated in Figure 4a. The pillars were placed at the peripheral position rpos. = 145 [µm] with the radius of pillars rpillar = 1 [µm] by varying the number of pillars ranging from 4 to 768. In addition, we also constructed the fully-connected-pillars model (Figure 4a). Detailed positions and their parameters are listed in Table 3. Figure 4b–d show the effects on R/R|total as a function of distance (dpillar), angle (), and number of pillars (N), respectively. It has been found that the higher signal response can be obtained with shorter distances, and R/R|total with the shortest distance of pillars (dpillar – 2 × rpillar) less than 5 µm reached the similar level of the signal response to the fully-connected-pillars model (Figure 4d–g). Notably, the pillars placed nearby the four sensing beams of the MSS yield largest signal response, for example, in the case of the number of pillars N = 4, and the distance of the pillars from the sensing beams significantly affects the signal response because of the geometry of the MSS (see also Appendix A).  rpillar = 1 µmrpos. rpos.dpillarN  = 12ffd pillar = 3.0 µma b0 50 1000104628∆R/R (x10–4)rpos. [µm]34521[GPa]150 200 0 50 100062314∆R/R (x10–4)rpos. [µm]150 200nf = 0nf = 0.25nf = 0.45c d0 50 10010–910–110–510–410–610–210–3∆R/Rrpos. [µm]1101000.10.01[GPa]150 20010–710–8Figure 3. Dependence of the relative resistance change (∆R/R|total) on the position of the interfacialattachments (rpos.) calculated by FEA. (a) Configuration of pillars as a model of interfacial attachment.The radius of pillars (rpillar) is fixed at 1 µm. (Left) Number-fixed model. The number of the pillars(N) is fixed at 12. (Right) Distance-fixed model. Center-to-center distance between pillars is fixed atca. 3 µm, indicating that the shortest distance between pillars is 1 µm. Detailed distances are listedin Tables 1 and 2. (b–c) Dependence of the total resistance change ∆R/R|total on Young’s modulus asa function of the position of pillars (rpos.). The Young’s moduli (Ef) are varied from 0.01 GPa to 100 GPa(b) and from 1 GPa to 5 GPa (c). Poisson’s ratio of a coating film (νf) is fixed at 0.4. (d) Dependence ofthe total resistance change ∆R/R|total on Poisson’s ratio as a function of the position of pillars (rpos.).The Young’s modulus of coating film is fixed at 3.0 GPa, which is similar to that of PMMA. Dotted linesin (b), (c) and (d) are ∆R/R|total of the full attachment model.Table 1. Detailed parameters of the number-fixed model.Number-Fixed Model, N = 12rpos. [µm] a φ [◦] a dpillar [µm] a,b No. of Pillars Area of Pillars [µm2]145 30 74.5 (72.5) 12 38135 30 69.4 (67.4) 12 38125 30 64.2 (62.2) 12 38115 30 59.0 (57.0) 12 38105 30 53.8 (53.8) 12 3895 30 48.7 (46.7) 12 3885 30 43.5 (41.5) 12 3875 30 38.3 (36.3) 12 3865 30 33.1 (31.1) 12 3855 30 28.0 (26.0) 12 3845 30 22.8 (20.8) 12 3835 30 17.6 (15.6) 12 3825 30 12.4 (10.4) 12 3815 30 7.25 (5.25) 12 386.8 30 3.00 (1.00) 12 38a rpos., φ, and dpillar are denoted in Figure 3a. b Values in parentheses are the shortest distance between theneighboring pillars (dpillar − 2 × rpillar).Sensors 2020, 20, 1518 7 of 11Table 2. Detailed parameters of the distance-fixed model.Distance-Fixed Model, dpillar = 3 µmrpos. [µm] a φ [◦] a dpillar [µm] a,b No. of pillars Area of pillars [µm2]145 1.2 3.02 (1.02) 300 942115.7 1.5 3.00 (1.00) 240 75487 2 3.00 (1.00) 180 56569.8 2.5 3.00 (1.00) 144 45258.3 3 3.00 (1.00) 120 37746.8 3.75 3.00 (1.00) 96 30235.4 5 3.00 (1.00) 72 22629.7 6 3.00 (1.00) 60 18823.9 7.5 3.00 (1.00) 48 15118.2 10 3.00 (1.00) 36 11312.5 15 3.00 (1.00) 24 756.8 30 3.00 (1.00) 12 38a rpos., φ, and dpillar are denoted in Figure 3a. b Values in parentheses are the shortest distance between theneighboring pillars (dpillar − 2 × rpillar).It should be noted that pillars placed at the peripheral position in the case of Distance-fixedmodel (Figure 3) provide higher ∆R/R|total than that of Number-fixed model with rpillar = 20 µm(Figure 2), even though the area of Distance-fixed model with rpillar = 1 µm (942 µm2; Table 2) isonly 6% of Number-fixed model with rpillar = 20 µm (15,080 µm2; Table S1 in the SupplementaryMaterials). This result indicates that the distance between interfacial attachments (dpillar) affects∆R/R|total more significantly than the area of interfacial attachments (rpillar). To confirm this aspect,we constructed the modified model as illustrated in Figure 4a. The pillars were placed at the peripheralposition rpos. = 145 [µm] with the radius of pillars rpillar = 1 [µm] by varying the number of pillarsranging from 4 to 768. In addition, we also constructed the fully-connected-pillars model (Figure 4a).Detailed positions and their parameters are listed in Table 3. Figure 4b–d show the effects on ∆R/R|totalas a function of distance (dpillar), angle (φ), and number of pillars (N), respectively. It has been foundthat the higher signal response can be obtained with shorter distances, and ∆R/R|total with the shortestdistance of pillars (dpillar − 2 × rpillar) less than 5 µm reached the similar level of the signal response tothe fully-connected-pillars model (Figure 4d–g). Notably, the pillars placed nearby the four sensingbeams of the MSS yield largest signal response, for example, in the case of the number of pillars N = 4,and the distance of the pillars from the sensing beams significantly affects the signal response becauseof the geometry of the MSS (see also Appendix A).Sensors 2020, 20, 1518 8 of 11Sensors 2020, 20, 8 of 11   Figure 4. Dependence of the relative resistance change (R/R|total) on the distance of the interfacial attachments (dpillar) calculated by FEA. (a) Configuration of pillars as a model of interfacial attachment. The radius of pillars (rpillar) is fixed at 1 µm. Each piezoresistor-integrated sensing beam is magnified in the bottom insets. (b–d) The Young’s modulus-dependent total resistance change R/R|total on the distance (b), area (c) and number (d) of the pillars. Details are listed in Table 3. The Young’s moduli are varied in the range of 0.01 GPa to 100 GPa. (e) Dependence of R/R|total as a function of the number of pillars. Ef = 1–5 [GPa]. Poisson’s ratio of a coating film (f) is 0.4. (f) The Poisson’s ratio-dependent total resistance change (R/R|total) as a function of the position of pillars (rpos.). The Young’s modulus of coating film is 3.0 GPa, which is similar to that of PMMA. Closed and open circles are R/R|total on dpillar and fully attached model, respectively. Dotted lines in (b)–(f) are R/R|total of the fully attached model. (g) Relative sensing responses compared to R/R|total of the fully attached model. 4. Conclusions In summary, we have demonstrated the FEA simulations for investigating the effects of interfacial structures on the signal responses of nanomechanical sensors, especially MSS. Despite the various advantages of the nanomechanical sensors, it is often difficult to achieve high sensitivity and selectivity because of the low affinity between a coating film and a substrate of the sensing element, including the mechanical detachments between them. To properly investigate the actual contribution of a coating film to a sensing signal, it is important to understand the effects of interfacial attachments on the sensing responses. In the present study, we have shown the effects of the interfacial structures using the several models analyzed by FEA simulations. The effects of the physical parameters of coating films, such as Young’s modulus and Poisson’s ratio are also discussed. It was demonstrated that the attachments at the peripheral positions give the signal responses as high as uniformly 100 101 1020104628∆R/R (x10–4)Number of pillars34521[GPa]103 104ab c de f g0 30 6010–710–110–510–410–610–210–3∆R/RAngle, f [°]1101000.10.01[GPa]90 120 100 101 10210–710–110–510–410–610–210–3∆R/RNumber of pillars1101000.10.01[GPa]103 1040 50 10010–710–110–510–410–610–210–3∆R/RDistance, dpillar – 2 rpillar [µm]1101000.10.01[GPa]150 250200100 101 102062314∆R/R (x10–4)Number of pillars103 104nf = 0nf = 0.25nf = 0.450.01 0.1 10100204060∆R/Rperipheral∆R/RFullYoung’s modulus, E f [GPa]10 10080[%]N = 4N ≥192N = 96N = 48N = 24N = 12N = 8N = 4 N = 12 N = 384 N = 768 N = ∞•••Overlapped Fully connected5 µm90° 30°Pillarsrpillar = 1 µmFigure 4. Dependence of the relative resistance change (∆R/R|total) on the distance of the interfacialattachments (dpillar) calculated by FEA. (a) Configuration of pillars as a model of interfacial attachment.The radius of pillars (rpillar) is fixed at 1 µm. Each piezoresistor-integrated sensing beam is magnifiedin the bottom insets. (b–d) The Young’s modulus-dependent total resistance change ∆R/R|total on thedistance (b), area (c) and number (d) of the pillars. Details are listed in Table 3. The Young’s moduli arevaried in the range of 0.01 GPa to 100 GPa. (e) Dependence of ∆R/R|total as a function of the number ofpillars. Ef = 1–5 [GPa]. Poisson’s ratio of a coating film (νf) is 0.4. (f) The Poisson’s ratio-dependenttotal resistance change (∆R/R|total) as a function of the position of pillars (rpos.). The Young’s modulusof coating film is 3.0 GPa, which is similar to that of PMMA. Closed and open circles are ∆R/R|total ondpillar and fully attached model, respectively. Dotted lines in (b)–(f) are ∆R/R|total of the fully attachedmodel. (g) Relative sensing responses compared to ∆R/R|total of the fully attached model.Table 3. Detailed parameters of the model in Figure 4.No. of Pillars Angle, φ [◦] a Distance, dpillar [µm] a,b4 90 204 (202)12 30 74.5 (72.5)24 15 37.6 (35.6)48 7.5 18.8 (16.8)96 3.8 9.42 (7.42)192 1.9 4.71 (2.71)300 1.2 3.02 (1.02)384 0.94 2.36 (0.36)768 0.47 1.18 (−0.82)a φ and dpillar are denoted in Figure 3a; b Values in parentheses are the shortest distance between the neighboringpillars (dpillar − 2 × rpillar).4. ConclusionsIn summary, we have demonstrated the FEA simulations for investigating the effects of interfacialstructures on the signal responses of nanomechanical sensors, especially MSS. Despite the variousSensors 2020, 20, 1518 9 of 11advantages of the nanomechanical sensors, it is often difficult to achieve high sensitivity and selectivitybecause of the low affinity between a coating film and a substrate of the sensing element, including themechanical detachments between them. To properly investigate the actual contribution of a coatingfilm to a sensing signal, it is important to understand the effects of interfacial attachments on thesensing responses. In the present study, we have shown the effects of the interfacial structures usingthe several models analyzed by FEA simulations. The effects of the physical parameters of coatingfilms, such as Young’s modulus and Poisson’s ratio are also discussed. It was demonstrated that theattachments at the peripheral positions give the signal responses as high as uniformly distributed orfully attached models, while the attachments at the inner positions of the membrane cannot efficientlytransduce the mechanical response of a coating film to the membrane, leading to significant lossesof sensing signals. The presented study will provide a strategy for designing the coating films aswell as optimizing the interfacial structures for higher sensitivity including surface modification ofa substrate [2,28].Supplementary Materials: Supplementary Materials: The following are available online at http://www.mdpi.com/1424-8220/20/5/1518/s1, Supplementary text: Detailed experimental procedure, Figure S1: Distribution ofdisplacements and relative resistance change on the interfacial structure, Figures S2, S3, and S5: the distributionsof relative resistance changes, Figure S4: Comparison between the distance-fixed model and fully attached modelwith different radius of a receptor layer, Table S1; The list of detailed parameters of the interfacial structures.Author Contributions: K.M. and G.Y. conceived the present idea. K.M. designed the present research. K.M.performed and analyzed simulations. K.M. and G.Y. wrote the manuscript. All authors have read and agreed tothe published version of the manuscript.Funding: This work was supported by the MSS alliance; JST CREST (JPMJCR1665); a Grant-in-Aid for ScientificResearch (A), 18H04168, MEXT, Japan; the Public/Private R&D Investment Strategic Expansion Program (PRISM),Cabinet Office, Japan; the International Center for Young Scientists (ICYS), NIMS; the Center for Functional Sensor& Actuator (CFSN), NIMS; and the World Premier International Research Center Initiative (WPI) on MaterialsNanoarchitectonics (MANA), NIMS.Acknowledgments: K.M. thanks International Center for Young Scientists (ICYS), National Institute for MaterialsScience (NIMS).Conflicts of Interest: The authors declare no conflict of interest.Appendix AOwing to the geometry of the MSS, the distance between pillars and the four sensing beamssignificantly affects the signal response. As illustrated in Figure A1a, four pillars are placed at theperipheral position (rpos = 145) with varied angle θ [◦] from the y axis (0◦ ≤ θ < 90◦). As presented inFigure A1b,c, the relative resistance change (∆R/R|total) dramatically decreases and reaches ca. 30%at θ = 45◦ compared to ∆R/R|total (θ = 0◦). While the Young’s modulus of the coating films (Ef) areslightly affects the signal responses, the Poisson’s ratio of the coating films (νf) has almost effects on∆R/R|total (Figure A1c)http://www.mdpi.com/1424-8220/20/5/1518/s1http://www.mdpi.com/1424-8220/20/5/1518/s1Sensors 2020, 20, 1518 10 of 11Sensors 2020, 20, 9 of 11  distributed or fully attached models, while the attachments at the inner positions of the membrane cannot efficiently transduce the mechanical response of a coating film to the membrane, leading to significant losses of sensing signals. The presented study will provide a strategy for designing the coating films as well as optimizing the interfacial structures for higher sensitivity including surface modification of a substrate [2,28]. Table 3. Detailed parameters of the model in Figure 4. No. of Pillars Angle,  [°]a Distance, dpillar [µm]a,b 4 90 204 (202) 12 30 74.5 (72.5) 24 15 37.6 (35.6) 48 7.5 18.8 (16.8) 96 3.8 9.42 (7.42) 192 1.9 4.71 (2.71) 300 1.2 3.02 (1.02) 384 0.94 2.36 (0.36) 768 0.47 1.18 (–0.82) a  and dpillar are denoted in Figure 3a; b Values in parentheses are the shortest distance between the neighboring pillars (dpillar – 2 × rpillar). Appendix A Owing to the geometry of the MSS, the distance between pillars and the four sensing beams significantly affects the signal response. As illustrated in Figure A1a, four pillars are placed at the peripheral position (rpos = 145) with varied angle  [°] from the y axis (0° ≤  < 90°). As presented in Figure A1b,A1c, the relative resistance change (R/R|total) dramatically decreases and reaches ca. 30% at = 45° compared to R/R|total ( = 0°). While the Young’s modulus of the coating films (Ef) are slightly affects the signal responses, the Poisson’s ratio of the coating films (f) has almost effects on R/R|total (Figure A1c)  Figure A1. Dependence of the relative resistance change (R/R|total) on the position of the interfacial attachments from the four sensing beams of the MSS calculated by FEA. (a) Configuration of pillars as a model of interfacial attachment. The radius of pillars (rpillar) and the number of pillars (N) are fixed at 1 µm and 4, respectively, with the angle  = 90°. (b) Dependence of the total resistance change R/R|total on Young’s modulus as a function of the angle of pillars from y axis. The Young’s moduli (Ef) are varied from 0.01 GPa to 100 GPa. Poisson’s ratio of a coating film (f) is fixed at 0.4. Dotted lines are R/R|total of the fully attached models. (c) Dependence of the relative resistance change R/R|total on Young’s modulus and Poisson’s ratio. Supplementary Materials: Supplementary Materials: The following are available online at www.mdpi.com/xxx/s1, Supplementary text: Detailed experimental procedure, Figure S1: Distribution of displacements and relative resistance change on the interfacial structure, Figures S2, S3, and S5: the distributions of relative resistance changes, Figure S4: Comparison between the distance-fixed model and fully attached model with different radius of a receptor layer, Table S1; The list of detailed parameters of the interfacial structures. qf = 90°a b0 15 4510–510–410–610–210–3∆R/Rq [°]1101000.10.01[GPa]75 9010–730 60∆R/R| q∆R/R| q= 0°[%]100806040200q = 0, 90°nf = 0nf = 0.25nf = 0.4q = 15, 75°q = 30, 60°q = 45°0.01 0.1 1 10 100Young’s modulus, Ef [GPa]cFigure A1. Dependence of the relative resistance change (∆R/R|total) on the position of the interfacialattachments from the four sensing beams of the MSS calculated by FEA. (a) Configuration of pillarsas a model of interfacial attachment. The radius of pillars (rpillar) and the number of pillars (N) arefixed at 1 µm and 4, respectively, with the angle φ = 90◦. (b) Dependence of the total resistance change∆R/R|total on Young’s modulus as a function of the angle of pillars from y axis. The Young’s moduli (Ef)are varied from 0.01 GPa to 100 GPa. Poisson’s ratio of a coating film (νf) is fixed at 0.4. Dotted linesare ∆R/R|total of the fully attached models. (c) Dependence of the relative resistance change ∆R/R|totalon Young’s modulus and Poisson’s ratio.References1. Gimzewski, J.K.; Gerber, C.; Meyer, E.; Schlittler, R.R. Observation of a Chemical-Reaction Usinga Micromechanical Sensor. Chem. Phys. Lett. 1994, 217, 589–594. [CrossRef]2. Goeders, K.M.; Colton, J.S.; Bottomley, L.A. Microcantilevers: Sensing chemical interactions via mechanicalmotion. Chem. Rev. 2008, 108, 522–542. [CrossRef] [PubMed]3. Yoshikawa, G.; Akiyama, T.; Gautsch, S.; Vettiger, P.; Rohrer, H. Nanomechanical membrane-type surfacestress sensor. Nano Lett. 2011, 11, 1044–1048. [CrossRef] [PubMed]4. Minami, K.; Imamura, G.; Nemoto, T.; Shiba, K.; Yoshikawa, G. Pattern recognition of solid materials bymultiple probe gases. Mater. Horiz. 2019, 6, 580–586. [CrossRef]5. Imamura, G.; Shiba, K.; Yoshikawa, G. Smell identification of spices using nanomechanical membrane-typesurface stress sensors. Jpn. J. Appl. Phys. 2016, 55, 1102B3. [CrossRef]6. Shiba, K.; Tamura, R.; Imamura, G.; Yoshikawa, G. Data-driven nanomechanical sensing: Specific informationextraction from a complex system. Sci. Rep. 2017, 7, 3661. [CrossRef]7. Shiba, K.; Tamura, R.; Sugiyama, T.; Kameyama, Y.; Koda, K.; Sakon, E.; Minami, K.; Ngo, H.T.; Imamura, G.;Tsuda, K.; et al. Functional Nanoparticles-Coated Nanomechanical Sensor Arrays for Machine Learning-BasedQuantitative Odor Analysis. ACS Sens. 2018, 3, 1592–1600. [CrossRef]8. Yoshikawa, G. Mechanical analysis and optimization of a microcantilever sensor coated with a solid receptorfilm. Appl. Phys. Lett. 2011, 98, 173502. [CrossRef]9. Stoney, G.G. The Tension of Metallic Films Deposited by Electrolysis. Proc. R. Soc. London Ser. A 1909, 82, 172–175.10. Chu, W.-H.; Mehregany, M.; Mullen, R.L. Analysis of tip deflection and force of a bimetallic cantilevermicroactuator. J. Micromech. Microeng. 1993, 3, 4–7. [CrossRef]11. Yoshikawa, G.; Akiyama, T.; Loizeau, F.; Shiba, K.; Gautsch, S.; Nakayama, T.; Vettiger, P.; de Rooij, N.F.;Aono, M. Two dimensional array of piezoresistive nanomechanical Membrane-type Surface Stress Sensor(MSS) with improved sensitivity. Sensors 2012, 12, 15873–15887. [CrossRef] [PubMed]12. Yoshikawa, G.; Lee, C.J.; Shiba, K. Effects of coating materials on two dimensional stress-induced deflectionof nanomechanical sensors. J. Nanosci. Nanotechnol. 2014, 14, 2908–2912. [CrossRef] [PubMed]13. Imamura, G.; Shiba, K.; Yoshikawa, G. Finite Element Analysis on Nanomechanical Detection of SmallParticles: Toward Virus Detection. Front. Microbiol. 2016, 7, 488. [CrossRef] [PubMed]14. Sader, J.E. Surface stress induced deflections of cantilever plates with applications to the atomic forcemicroscope: Rectangular plates. J. Appl. Phys. 2001, 89, 2911–2921. [CrossRef]15. Lachut, M.J.; Sader, J.E. Effect of surface stress on the stiffness of cantilever plates. Phys. Rev. Lett. 2007, 99, 206102.[CrossRef]16. Lachut, M.J.; Sader, J.E. Effect of surface stress on the stiffness of thin elastic plates and beams. Phys. Rev. B2012, 85, 085440. [CrossRef]17. Shiba, K.; Sugiyama, T.; Takei, T.; Yoshikawa, G. Controlled growth of silica-titania hybrid functionalnanoparticles through a multistep microfluidic approach. Chem. Commun. 2015, 51, 15854–15857. [CrossRef]http://dx.doi.org/10.1016/0009-2614(93)E1419-Hhttp://dx.doi.org/10.1021/cr0681041http://www.ncbi.nlm.nih.gov/pubmed/18229951http://dx.doi.org/10.1021/nl103901ahttp://www.ncbi.nlm.nih.gov/pubmed/21314159http://dx.doi.org/10.1039/C8MH01169Ahttp://dx.doi.org/10.7567/JJAP.55.1102B3http://dx.doi.org/10.1038/s41598-017-03875-7http://dx.doi.org/10.1021/acssensors.8b00450http://dx.doi.org/10.1063/1.3583451http://dx.doi.org/10.1088/0960-1317/3/1/002http://dx.doi.org/10.3390/s121115873http://www.ncbi.nlm.nih.gov/pubmed/23202237http://dx.doi.org/10.1166/jnn.2014.8604http://www.ncbi.nlm.nih.gov/pubmed/24734709http://dx.doi.org/10.3389/fmicb.2016.00488http://www.ncbi.nlm.nih.gov/pubmed/27148181http://dx.doi.org/10.1063/1.1342018http://dx.doi.org/10.1103/PhysRevLett.99.206102http://dx.doi.org/10.1103/PhysRevB.85.085440http://dx.doi.org/10.1039/C5CC07230ASensors 2020, 20, 1518 11 of 1118. Osica, I.; Imamura, G.; Shiba, K.; Ji, Q.; Shrestha, L.K.; Hill, J.P.; Kurzydlowski, K.J.; Yoshikawa, G.; Ariga, K.Highly Networked Capsular Silica-Porphyrin Hybrid Nanostructures as Efficient Materials for AcetoneVapor Sensing. ACS Appl. Mater. Interfaces 2017, 9, 9945–9954. [CrossRef]19. Ngo, H.T.; Minami, K.; Imamura, G.; Shiba, K.; Yoshikawa, G. Effects of Center Metals in Porphines onNanomechanical Gas Sensing. Sensors 2018, 18, 1640. [CrossRef]20. Minami, K.; Shiba, K.; Yoshikawa, G. Discrimination of structurally similar odorous molecules with variousconcentrations by using a nanomechanical sensor. Anal. Methods 2018, 10, 3720–3726. [CrossRef]21. Suhir, E. Interfacial Stresses in Bi-Metal Thermostats. J. Appl. Mech. -Trans. ASME 1989, 56, 595–600.[CrossRef]22. Murray, C.E.; Noyan, I.C. Finite-size effects in thin-film composites. Philos. Mag. A 2002, 82, 3087–3117.[CrossRef]23. Zhang, Y. Extended Stoney’s Formula for a Film-Substrate Bilayer with the Effect of Interfacial Slip.J. Appl. Mech. 2008, 75, 011008. [CrossRef]24. Pfann, W.G.; Thurston, R.N. Semiconducting Stress Transducers Utilizing the Transverse and ShearPiezoresistance Effects. J. Appl. Phys. 1961, 32, 2008–2019. [CrossRef]25. Kanda, Y. A graphical representation of the piezoresistance coefficients in silicon. IEEE Trans. Electron Devices1982, 29, 64–70. [CrossRef]26. Kanda, Y. Piezoresistance effect of silicon. Sens. Actuators A Phys. 1991, 28, 83–91. [CrossRef]27. Rasmussen, P.A.; Hansen, O.; Boisen, A. Cantilever surface stress sensors with single-crystalline siliconpiezoresistors. Appl. Phys. Lett. 2005, 86, 203502. [CrossRef]28. Maboudian, R.; Carraro, C. Surface chemistry and tribology of MEMS. Annu. Rev. Phys. Chem. 2004, 55, 35–54.[CrossRef]© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open accessarticle distributed under the terms and conditions of the Creative Commons Attribution(CC BY) license (http://creativecommons.org/licenses/by/4.0/).http://dx.doi.org/10.1021/acsami.6b15680http://dx.doi.org/10.3390/s18051640http://dx.doi.org/10.1039/C8AY01224Ehttp://dx.doi.org/10.1115/1.3176133http://dx.doi.org/10.1080/01418610208239635http://dx.doi.org/10.1115/1.2745387http://dx.doi.org/10.1063/1.1728280http://dx.doi.org/10.1109/T-ED.1982.20659http://dx.doi.org/10.1016/0924-4247(91)85017-Ihttp://dx.doi.org/10.1063/1.1900299http://dx.doi.org/10.1146/annurev.physchem.55.091602.094445http://creativecommons.org/http://creativecommons.org/licenses/by/4.0/. Introduction  Simulation  Results and Discussion  Conclusions   References