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Expression of picogram sensitive bending modes in piezoelectric cantilever sensors with nonuniform electric fields generated by asymmetric electrodes

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Single-layer uniform cross-sectioned piezoelectric macro-cantilevers fabricated with an asymmet-ric electrode configuration enabled electrical measurement of picogram-sensitive resonant bending modes in liquids. Bending modes were otherwise not
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  REVIEW OF SCIENTIFIC INSTRUMENTS  81 , 125108 (2010) Expression of picogram sensitive bending modes in piezoelectric cantileversensors with nonuniform electric fields generated by asymmetric electrodes Blake N. Johnson and Raj Mutharasan a)  Department of Chemical and Biological Engineering, Drexel University, 3141 Chestnut Street, Philadelphia,Pennsylvania 19104, USA (Received 2 August 2010; accepted 30 October 2010; published online 27 December 2010)Single-layer uniform cross-sectioned piezoelectric macro-cantilevers fabricated with an asymmet-ric electrode configuration enabled electrical measurement of picogram-sensitive resonant bendingmodes in liquids. Bending modes were otherwise not electrically measurable without excitation bya nonuniform electric field created by the geometric asymmetry in electrode design used. Electrodemodification wasconfirmed byenergy-dispersive X-rayspectroscopy(EDS).Mass-change sensitivitywas tested using both bulk density changes and surface chemisorption experiments in a continuousflow apparatus. Significant response to density changes as small as 0.004 g/mL was measured. Asensitivity limit of   ∼ 1 picogram in liquid was determined from 1-dodecanethiol chemisorption ex-periments. The sensitivity decreased with chemisorbed mass and was log-linear over five orders of magnitude. The observed resonance responses were in agreement with previously reported modelsof resonating cantilever sensors. This work demonstrates experimentally for the first time that intro-ducing electrode asymmetry enables measurement of bending modes in cantilevers containing only asingle piezoelectric layer.  © 2010 American Institute of Physics . [doi:10.1063/1.3518925] I. INTRODUCTION Piezoelectric-excited millimeter-sized cantilever(PEMC) sensors containing both a piezoelectric layer(typically lead zirconate titanate: PZT) and a nonpiezoelec-tric layer (typically silica) enable excitation of mechanicalresonant modes. Such bi-layered cantilevers result in aneffective design for both exciting and sensing resonance. 1–5 The mechanical resonance of such a device is measured bythe phase angle (  ) between the excitation voltage and theresulting electric current in the piezoelectric layer. 6 From afrequency sweep, resonance is identified by a sharp change inphase angle due to altered electrical impedance of PZT due tohigher than normal strain. To improve mass-change sensitiv-ity of PEMC sensors, it is advantageous to examine designsthat eliminate the parasitic mass of the nonpiezoelectriclayer. However, when the parasitic nonpiezoelectric layer wasremoved from the bi-layer design, resulting in a single-layercantilever with geometrically symmetric electrodes, bendingmodes were no longer electrically measurable althoughthey are predicted in the investigated frequency range of 0–100 kHz. Hence the object of this report is to examinemethods that allow measurable expression of resonantbending modes in single-layer PZT cantilevers.When a PZT cantilever with geometrically symmetricelectrodes is excited periodically by an applied potentialalong the material polarization axis, the cantilever deformslongitudinally in a harmonic fashion due to the uniform elec-tric field. In such a configuration only longitudinal resonantmodes can be excited. On the other hand, if the electrode areaon one side of the cantilever is different than on the other, the a) Author to whom correspondence should be addressed. Electronic mail:mutharasan@drexel.edu. Tel.: (215) 895-2236. FAX: (215) 895-5837. cantilever experiences differential local longitudinal deforma-tion, thus inducing electrically measurable bending modesthat are absent without electrode asymmetry. The nonuni-form electric field creates measurable impedance change inthe PZT for bending modes, as well as longitudinal modes.WeshowthispropertyofPZT-onlycantileversexperimentallyby introducing asymmetry in electrode configuration.Resonant-mode cantilever sensors continue to be of great interest due to their high sensitivity and label-freebio-detection protocols. 7–9 Several innovative methods formeasuring resonance, both internal and external to thesensor, have been reported. 6,9 Within the class of macro-scale cantilever sensors, bi-layered PEMC sensors haveyielded successful label-free biosensing in liquid at picogram(10 –12 g, pg) sensitivity using low-order ( ∼ 0.1 MHz) resonantmodes, and femtogram (10 –15 g, fg) sensitivity using high-order ( ∼ 1 MHz) resonant modes. 1–5 Direct measurement inliquid was feasible because of high Reynolds number (Re =  ρ w 2 ω  /4 η  ∼ 10 4 –10 5 ;  ρ  =  fluid density,  w  =  cantileverwidth,  ω  =  frequency,  η  =  fluid viscosity). 10 The resonantmodes are not damped in liquids, but persist with reasonableQ-values (25–50) such that resonant frequency values canbe measured within  ± 1–3 Hz for low-order modes. 11 Sucha characteristic makes the macro-scale cantilever sensorsuseful for biosensing compared to micro-scale devices whichare damped in liquids due to viscous effects (Re ∼ 10 0 –10 2 ). II. THEORETICAL BACKGROUND The defining property of a piezoelectric material is thecoupling of mechanical and electric displacements. PEMCsensors are excited periodically by exploiting the conversepiezoelectric effect, which is the result of mechanical stress 0034-6748/2010/81(12)/125108/6/$30.00 © 2010 American Institute of Physics 81 , 125108-1  125108-2 B. N. Johnson and R. Mutharasan Rev. Sci. Instrum.  81 , 125108 (2010) caused by the application of an electric field. The constitutiverelations for a piezoelectric material are given by 12 T   = cS  − e   E  ,  (1)  D  = eS  + ε  E  ,  (2)where  T, c, e, S,  ε  , E  , and  D  are the stress, elasticity, coupling,strain, permittivity, electric field, and electric displacementmatrices. Since 0.1 sin(2 π ft) volts is applied in the PZT polar-ization axis, the piezoelectric material deforms periodically,where  t   is time, and  f   is the excitation frequency. The non-trivial components of strain, electric field, stress, and electricdisplacement are given by the following equations: 12 T  11  = T  22  = c 13 S  33 + e 31  E  3 ,  (3) T  33  = c 33 S  33 + e 33  E  3 ,  (4)where the subscripts 1, 2, and 3 stand for the  x  ,  y , and  z  di-rections, respectively. Note that if the electric field is nonuni-formly distributed over the material domain, the resulting lo-cal strain will likewise be nonuniformly distributed. Thus,cancellation effects which prohibit impedance coupling of resonant modes become significantly reduced. III. MATERIALS AND METHODSA. Fabrication Type 5A lead zirconate titanate (PZT-5A) (5  ×  1  × 0.127 mm 3 ; Piezo Systems, Inc., Woburn, MA) plates with100 nm thick Nickel (Ni) electrode deposited across thethickness served as the single piezoelectric cantilever layer.To provide electrical excitation of the cantilever, 30-gaugecopper (Cu) wires were attached to the Ni electrode at thecantilever anchor; schematics of symmetric and asymmet-ric electrode design are shown in Figs. 1(A) and 1(B), re- spectively. Subsequent to electrode connection, the sensorwas embedded in a 6 mm diameter glass tube filled withnon-conductive epoxy for providing anchoring. In Fig. 1(C),a photograph of the sensor at this point in fabrication isshown. Prior to etching, a polyurethane resist layer was spin-coated (30 sec at 1500 rpm) on all faces of the PZT except ∼ 1 mm 2 area at the tip of one side of the PZT. The exposedNi electrode was etched in freshly prepared ferric chloridesolution (FeCl 3 , 0.1 M in deionized water, Fisher Scientific,Waltham, MA) by immersing  ∼ 1 mm of the tip for 20 min.Such a treatment caused electrode Ni to react and dissolveby: 2FeCl 3  +  Ni  →  2FeCl 2  +  NiCl 2 , resulting in asym-metric electrode configuration. 13 After etching, the sensorwas rinsed with copious amounts of de-ionized water (DIW,18 M  , Milli-Q system, Millipore, Billerica, MA) to removeresidual ferric chloride etchant solution and stored in freshDIW for 30 min to allow mobile solutes that entered PZTduring etching to be leached. Subsequently, the sensors weredried 48 h in a vacuum oven at 80  ◦ C. For electrical insulationand in-liquid use the sensors were coated with 5 micron thickparylene-c following vendor supplied protocol (PDS 2010Labcoter R  2, Specialty Coating Systems, Indianapolis, IN).For 1-dodecanethiol (DDT, Fisher Scientific, Waltham, MA)chemisorption experiments in liquid, gold (100 nm thick) was ABC  Ni ElectrodePZT Ni ElectrodeEtched Ni Electrode twLE 3 E 3 E 3 E 3 E 3 E 3SEM/EDS Sample Area FIG. 1. (Color online) Schematic diagram of PEMC sensor before electrodeetching (A) and after Ni electrode etching using 0.1 M FeCl 3  solution (B).Excitation at 0.1 sin (2 π ft) volts applied across the Ni electrodes createsan electric field distribution (E 3 ) along the polarization axis. Hashed-edgesrepresent boundaries of zero-displacement. Diagram not drawn to scale. Forelectric field calculation results see Figs. 3 and 4. (C) Photograph of a PZTcantilever prior to polyurethane resist layer application. sputtered (Denton Desk IV sputtering unit, Denton VacuumLLC, Moorestown, NJ) over 0.75 mm 2 area on each side atthe cantilever tip. B. Experimental Apparatus Frequency response of the cantilever sensors was mea-sured using an impedance analyzer (Hewlett-Packard 4192Aor Agilent 4294A). The sensor was excited by applying0.1 sin(2 π ft) volts across the PZT thickness dimension;  t   istime and  f   is excitation frequency. The two Ni electrodesprovided for simultaneous actuation and measurement of PZT electrical impedance. Resonance is indicated by sharpchanges in electrical impedance due to higher than normalstrain that occurs at resonance condition.The experimental apparatus (Fig. 2) consists of several reagent reservoirs for DIW, ethanol (0.789 g/mL, EtOH, ana-lytical grade, Sigma-Aldrich, St. Louis, MO), and phosphatebuffered saline (PBS, 2.5–10 mM, pH 7.4, Sigma-Aldrich, St.Louis, MO) solutions connected via a 4-port valve. The in-let valve feeds the flow cell in which the sensor is housed. Asingle outlet stream containing a peristaltic pump for main-taining the desired flow rate is connected to either recycleor waste. The flow cell is housed within an incubator main-tained at 23  ±  0.1 ◦ C (density experiments) or 30  ±  0.1 ◦ C(chemisorptions experiments). IV. RESULTSA. Finite Element Model (FEM) Calculations The structural mechanics and piezoelectric module withplane-stress approximation in the finite element modeling  125108-3 B. N. Johnson and R. Mutharasan Rev. Sci. Instrum.  81 , 125108 (2010) µµµµ A FIG. 2. (Color online) Schematic block diagram (A) and picture of exper-imental apparatus (B). Agilent impedance analyzer (4294A) interfaced to acomputer both excites the PZT and measures resonant frequency and electri-cal impedance of a PEMC sensor installed in a custom flow cell. Data acqui-sition is carried out by a custom LABVIEW R  program. Isothermal conditionis maintained by the incubator. Various reservoirs are used to load runningfluids and valves (V1–V4) are used to manipulate the desired fluid flow intothe flow cell facilitated by the peristaltic pump. Insert shows flow cell andsensor assembly. platform COMSOL Multiphysics R  3.4 (COMSOL Group,Burlington, MA) was used to calculate the coupling of elec-trical and mechanical effects in the PZT cantilever sensor.Convergence in resonant frequency and electric current val-ues was attained within 0.1% with  ∼ 300 elements. A 2DFEM neglecting fluid-structure interaction of a 3.2  ×  0.127mm PZT cantilever conducted in vacuum yielded first, sec-ond, and third bending modes at 5.8, 36.2, and 100.3 kHz. Inthe section where Ni electrode was absent, electrical insula-tion boundary condition was applied. At the anchor boundary,zero displacement and zero slope boundary condition werealso applied. In Fig. 3 the effects of the asymmetric electrodeon electric field across the PZT thickness is illustrated. Sur-face plots of the electric potential over the thickness dimen-sion of the cantilever excited at the fundamental frequency( ∼ 5.9 kHz) are shown in Fig. 3. Although both cantilevers are of same size and material properties, only the cantileverexcited by asymmetric electrodes deflects in the transversedirection. For the case of symmetric electrodes [Fig. 3(A)], the electric field is distributed equally across the entire thick-ness of the cantilever. The uniform electric field resulted inonly longitudinal motion deformation, even though the can-tilever was excited at the various bending resonant frequen-cies. For the case of asymmetric electrodes [Fig. 3(B)], theelectric field profile is nonuniform due to the asymmetry in 0 31 102030405060708090100    E   l  e  c   t  r   i  c  p  o   t  e  n   t   i  a   l ,     ϕ    [  m   V   ] ABCDC’D’A’B’ BA Symmetric electrode:Axial tip displacement = 43 nmTransverse tip displacement = 0.02 nmAsymmetric electrode:Axial tip displacement = 28 nmTransverse tip displacement = 0.59 nmAnchor cross-sectionElectrode edge cross-section FIG. 3. (Color online) (A) FEM surface plot of electrical potential of the firstmode at 5.89 kHz for the 3.2  ×  1  ×  0.127 mm cantilever with symmetricelectrode. (B) FEM surface plot of electrical potential for the first mode at5.89 kHz for the 3.2 × 1 × 0.127 mm cantilever with asymmetric electrode.Electrode length on one side of the PZT layer is 3.2 mm, while on the otheris 1.5 mm. Solid black lines show an unexcited sensor. Anchor at A–B andA  –B  . Electrode edge at C–D and C  –D  . the applied electric field. Although the net current througheach electrode is equal due to conservation of charge, the cur-rent flux is different because electrode areas are not equal.The resulting asymmetry in electric field causes asymmetricstresses in PZT and enables coupling of bending modes withelectrical impedance.In Fig. 4, the electric potential at various cross-sections along the cantilever length is compared for the symmetric andasymmetric electrode designs shown in Fig. 3. For the case of symmetric electrodes, the electric potential changes lin-early across the thickness at each of the two cross-sectionsshown (anchor cross-section, A–B, at  X   =  0 and electrodeedge cross-section, C–D, at  X   =  1.5 mm). However, for thecase of asymmetric electrodes, the electric potential acrossthe thickness at the same two cross-sections exhibits inter-esting nonlinear behavior. Since the electric field with sym-metric electrodes is uniform (i.e., symmetric), the materialconstitutive relations [Eqs. (3) and (4)] indicate that sym- metric stresses occur resulting only in longitudinal motioneven though bending mode resonant frequencies are excited[Fig. 3(A)]. On the other hand, with asymmetric electrodes, the electric field is nonuniform and the coupled mechani-cal stresses in the material are no longer symmetric, thuscauses bending motion when excited at the same frequencies[Fig. 3(B)]. B. Experimental manifestation of bending modesby asymmetric electrode design Removal of Ni was confirmed by Energy-dispersiveX-ray spectroscopy (EDS) (Fig. 5). Measurements revealedhigh intensity of Ni bands in untreated region (indicated byred in EDS image) and near zero Ni band intensity in regiontreated by the etch solution (indicated by black in EDS im-age). An EDS line scan over a 160 micron segment that tra-verses the Ni-PZT etch interface revealed  < 4 % Ni remainedin the etched region relative to the untreated electrode. Such  125108-4 B. N. Johnson and R. Mutharasan Rev. Sci. Instrum.  81 , 125108 (2010) 020406080100020406080100120    V  o   l   t  a  g  e ,     ϕ    [  m   V   ] Thickness, t [microns]C’ –D’A’ –B’A –BC –D FIG. 4. (Color online) FEM results of electric potential at various thicknesscross-sections along the PZT cantilever length for the symmetric and asym-metric electrode configurations. Symmetric electrodes yield linear electricpotential profile at all cross-sections, while asymmetric electrodes give riseto nonlinear electric potential profiles at both the anchor cross-section and atthe electrode edge cross-section. a value is within measurement uncertainty. Analysis of lead(Pb) and titanium (Ti) surface energy density in both treatedand untreated regions confirmed FeCl 3  had negligible effecton the underlying PZT layer. The measurements confirmedNi was absent in the etched region, thus ensuring applicationof a nonuniform electric field.We compare the spectra of PZT-only cantilevers (3.4 ×  1  ×  0.127 mm 3 ) prepared with symmetric and asymmet- -88.6-88.4-88.2-88.0-87.8-87.6-87.4-87.2-87.038394041424344    P   h  a  s  e  a  n  g   l  e ,      Φ     [   d  e  g  r  e  e  s   ] Frequency, f [kHz]Post-etchf  res = 40.95 kHzQ = 69Pre-etchf  res = N.A.Q= N.A EDSSEMNi RegionNi RegionEtch RegionEtch Region FIG. 5. (Color online) Resonance spectra of second bending mode showspartial etching of Ni electrode on one side of PZT leads to expression of resonant modes not observed in the sensor with symmetric electrode config-uration. Scanning electron microscope (SEM) image of PZT surface show-ing both the etched (light gray) and not etched (dark gray) portions of theelectrode. Energy-dispersive X-ray Spectroscopy (EDS) shows high energydensity of Ni (red) where no etching occurred and very low energy density of Ni over etched region. For location of SEM/DES images, see Fig. 1. -88.5-88.0-87.5-87.0-86.57075808590    P   h  a  s  e  a  n  g   l  e ,      Φ    [   d  e  g  r  e  e  s   ] Frequency, f [kHz]In Ethanolf  res = 72.6 kHzQ= 40In Airf  res = 86.9 kHzQ= 61 FIG. 6. (Color online) Resonant mode used for measuring 1-dodecanethiol(DDT) chemisorption in air (  f  = 86.9 kHz,  Q = 61) and in ethanol (  f  = 72.6kHz, Q = 40).The33%decreasein Q -valueandresonantfrequencydecreaseof 14.3 kHz was observed due to immersion in higher density ethanol. ric electrodes in Fig. 5 for verification of the electrode asym-metry effects obtained from FEM (see Sec. IV A). In Fig. 5, the second bending mode is not observable in the frequencyrange of 38–44 kHz for the case of symmetric electrodes(labeled as pre-etch) although it is predicted to be located at ∼ 47 kHz using analytical cantilever model. 14 With asymmet-ric electrode, the second bending mode became observable at40.99 kHz (   =  1.1 o ;  Q air  =  69). In the example shown,1 mm of the Ni electrode on the distal end of PZT was etched.Based on fabrication of over 20 sensors, we found symmetricelectrode sensors did not express the bending resonant modes,while asymmetric electrode sensors always yielded electricalexpression of mechanical bending resonance with  Q -valuestypically 50–75 in air.The  n th mode resonant frequency of a uniform cross-section cantilever subjected to small deformations (“planestress” model) 14 is given by  f  n  = 12 π   k m eff  , n  1 / 2 ,  (5)where  m eff, n  =  3 m  /  λ n 4 is the effective mass,  m  is can-tilever mass,  k   =  3  EI   z  /   L 3 is effective spring constant,  E   isYoung’s modulus (  E  1  =  66 GPa),  I   z  is moment of inertia( = 1.71 × 10 –4 mm 4 ),  L  is cantilever length ( = 3.4 mm), and λ n  is  n th positive root 14 of 1  +  cosh( λ )cos( λ )  =  0. Ana-lytically, the first three bending modes of the 3.4 mm PZT-only cantilever are at 5.2, 32.4, and 90.6 kHz. Experimen-tally, the modes were measured at comparable values of 5.5  ±  0.4, 30.8  ±  1.4, and 89.1  ±  3.7 kHz ( n  =  three de-vices of same length). The corresponding  Q -values in airwere 47  ±  8, 48  ±  7, and 37  ±  3. In DIW, they were 33 ±  5, 43  ±  6, and 29  ±  2, respectively. The minimal changein  Q -value upon immersion in a higher density fluid is dueto high Re ( ∼ 10 4 ) of the vibrating cantilever. For example,the persistence of bending resonant modes under completeimmersion in ethanol is illustrated in Fig. 6. The bending  125108-5 B. N. Johnson and R. Mutharasan Rev. Sci. Instrum.  81 , 125108 (2010) mode shown at 86.9 kHz in air and 72.6 kHz in ethanol(EtOH) corresponds to a Reynolds number (Re) in liquid ∼ 10 4 , indicating viscous forces are negligible relative to in-ertial forces. The modest decrease in  Q -value from 61 to40 upon complete immersion in ethanol is a unique propertyof macro-scale cantilever sensors that makes possible real-time in-liquid measurements as we demonstrate in the follow-ing sections. In contrast, a typical PZT-5A micro-cantilever(  L  =  10  µ m,  w  =  1  µ m,  t   =  0.1  µ m), which is of similaraspect ratio to the macro-cantilever fabricated in this study,would vibrate in fundamental bending mode at ∼ 470 kHz ac-cording to Eq. (5). For such conditions, the Reynolds number (Re) in liquid ( ρ  = 1000 kg/m 3 ,  η  = 10 –3 Pa s) is ∼ 0.1, in-dicating that viscous forces are significant compared to in-ertial forces. Thus, micro-cantilevers are highly damped inliquid, making in-liquid biosensing difficult relative to macro-cantilevers. C. In-liquid characterization by small density changesin a flowing liquid The effective mass of a cantilever in a fluid increases be-cause the fluid adjacent to the cantilever also moves with theoscillating surface. The resonant frequency in a fluid is givenby 15  f  fluid  =  1 + πρ  f  w 4 ρ c t   − 1 / 2  f  vac ,  (6)where  f  vac  and  f  fluid  are in vacuum and in a medium of den-sity  ρ  f  ;  t   is cantilever thickness and  ρ c  is cantilever density.For a submerged cantilever in DIW ( ρ  =  0.998 g/mL) themodel predicts the first three bending modes at 3.9, 24.4, and68.1 kHz, respectively. Experimentally, the modes in DIWwere at 4.3 ± 0.5, 26.2 ± 2.1, and 73.0 ± 3.5 kHz ( n = threedevices) indicating agreement between theory and experimentwithin 9%.  In situ  experiment in a flow format avoids measurementambiguities and results in reliable and repeatable responses.Thus, for measuring small density changes, the sensor wasinstalled in a flow cell and a flow rate of 0.6 ± 0.05 mL/minwas used. The resonant frequency of the first three modes wasmonitored continuously as the running fluid was changed be-tween DIW and PBS solutions (2.5 to 10 mM). Upon chang-ing the flowing fluid from DIW to higher density PBS solu-tions, the resonant frequency decreased for all three modes.The shifts were successively higher with mode number andwith higher density differences. Typical density change re-sponse and a compilation of measured frequency changes as afunction of liquid density are shown in Fig. 7. The model pre- dicts the first bending mode to decrease by 3, 5, and 9 Hz forchanges between DIW and 2.5 mM ( ρ = 1.001 g/mL), 5 mM( ρ  =  1.004 g/mL), and 10 mM ( ρ  =  1.008 g/mL) PBS, re-spectively. Experimentally, successive changes between DIWand 2.5 mM, 5.0 mM, and 10 mM PBS caused the first modeto decrease by 4  ±  1, 7  ±  2, and 13  ±  2 Hz ( n  =  threesensors), respectively. For the second mode, the model pre-dicts a decrease of 16, 32, and 54 Hz for the same changes,and experimental results were in agreement being 17  ±  5,32  ±  8, and 65  ±  9 Hz ( n  =  three sensors), respectively. 45.6545.7045.7545.8045.85020406080    R  e  s  o  n  a  n   t   f  r  e  q  u  e  n  c  y ,   f   [   k   H  z   ] Time, t [min] A DI Water( ρ =0.998 g/mL)5 mMPBS( ρ =1.004 g/mL)DI Water( ρ =0.998 g/mL) 02040608010012014011.0021.0041.0061.0081.01    R  e  s  o  n  a  n   t   f  r  e  q  u  e  n  c  y  s   h   i   f   t ,   f    D   I   –   f       ρ    [   H  z   ] Fluid density, ρ [g/mL] Third Mode B Second ModeFirst Mode FIG. 7. (Color online) (A) Real-time sensor response is shown over thecourse of a liquid density change experiment. The sensor was initially sta-bilized in 10 mM PBS creating a baseline for the experiment. After a base-line was reached, the flowing liquid was changed form 10 mM PBS to DIW( ρ = 0.010 g/mL) and allowed to reach new resonant frequency value. Thechange caused the frequency to increase by 65 Hz. After reaching steady statein DIW, the flowing liquid was then switched to a less concentrated PBS so-lution (5 mM,  ρ  = 0.006 g/mL) resulting in a resonant frequency decreaseof 33 Hz. (B) The decrease in resonant frequency values for changing fromDIW to liquids of higher density is linear with respect to density of the higherdensity liquid. The overall change in density investigated is 0.8%, which issmall enough that a linear response is seen. The model described in the text(Eq. (6)) predicts a square root relationship. Note that sensitivity increases with mode number. The third mode, in comparison, gave the largest response. Themodel predicted values were in good agreement with experi-ments, 45 versus 41 ± 6, 89 versus 76 ± 10, and 149 versus120 ± 11 Hz, respectively;  n = three sensors. D. In-liquid characterization by molecular bindingin a flowing liquid Although density-change experiments give a measureof sensitivity, detection sensitivity is best tested by bindingexperiments in liquid. Therefore, molecular chemisorptionof 1-dodecanethiol (DDT, 202 Daltons) in ethanol (EtOH)was conducted in flow format. Gold (Au) sputtered on thesensor (0.75 mm 2 per side, 100 nm thick) gave Au  < 111 > sites. X-ray diffraction confirmed  > 95 % Au < 111 > . 3 The
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