Influence of Viscoelasticity and Interfacial Slip on Acoustic Wave. Sensors

Transcription

1 Postprint Version G. McHale, R. Lüclum, M.I. Newton and J.A. Cowen, Inluence o viscoelasticity and interacial slip on acoustic wave sensors, J. Appl. Phys., (000); DOI:0.063/ The ollowing article appeared in Journal o Applied Physics and may be ound at Copyright 000 American Institute o Physics. File reconstructed rom Pre-print ile: May not be the inal version. Inluence o Viscoelasticity and Interacial Slip on Acoustic Wave Sensors G. M c Hale **, R. Luclum, M.I. Newton * and J.A. Cowen * * Department o Chemistry and Physics, The Nottingham Trent University, Cliton Lane, Nottingham NG 8NS, UK. Institute or Micro- and Sensor Systems, Faculty o Electrical Engineering and Inormation Technology, Otto-von-Guerice University, P.O.B. 40, D-3906 Magdeburg, Germany. * glen.mchale@ntu.ac.u; Tel: ; Fax:

2 Abstract Acoustic wave devices with shear horizontal displacements, such as quartz crystal microbalances (QCM) and shear horizontally polarised surace acoustic wave (SH-SAW) devices provide sensitive probes o changes at solid-solid and solid- liquid interaces. Increasingly the suraces o acoustic wave devices are being chemically or physically modiied to alter surace adhesion or coated with one or more layers to ampliy their response to any change o mass or material properties. In this wor, we describe a model that provides a uniied view o the modiication in the shear motion in acoustic wave systems by multiple inite thicness loadings o viscoelastic luids. This model encompasses QCM and other classes o acoustic wave devices based on a shear motion o the substrate surace and is also valid whether the coating ilm has a liquid or solid character. As a speciic example, the transition o a coating rom liquid to solid is modelled using a single relaxation time Maxwell model. The correspondence between parameters rom this physical model and parameters rom alternative acoustic impedance models is given explicitly. The characteristic changes in QCM requency and attenuation as a unction o thicness are illustrated or a single layer device as the coating is varied rom liquid-lie to that o an amorphous solid. Results or a double layer structure are given explicitly and the extension o the physical model to multiple layers is described. An advantage o this physical approach to modelling the response o acoustic wave devices to multilayer ilms is that it provides a basis or considering how interacial slip boundary conditions might be incorporated into the acoustic impedance used within circuit models o acoustic wave devices. Explicit results are derived or interacial slip occurring at the substrate-layer interace using a single real slip parameter, s, which has inverse dimensions o impedance. In terms o acoustic impedance, such interacial slip acts as a single-loop negative eedbac. It is suggested that these results can also be viewed as arising rom a double-layer model with an ininitesimally thin slip layer which gives rise to a modiied acoustic load o the second layer. Finally, the diiculties with deining appropriate slip boundary conditions between any two successive layers in a multilayer device are outlined rom a physical point o view. Keywords: Acoustic waves, quartz crystal microbalances, sensors, slip, contact angles, wetting.

3 I. Introduction Acoustic wave devices provide a simple and eective means or probing changes at the interace between a solid ilm or a liquid. An example o such a device is the quartz crystal microbalance (QCM) which uses a transverse-shear mode oscillation. Sauerbrey showed that the decrease in resonant requency o the device due to loading the surace by a thin ilm o a rigidly coupled material is proportional to the mass (change) o the ilm, m, and to the square o the requency,. In this case, no change in damping o the resonance occurs and so only an energy storage is occurring. When a QCM is operated in a liquid the oscillation o the device surace is coupled into the liquid and induces an oscillation in the liquid. This oscillation does not extend throughout the bul o the liquid, but is damped within a small distance, ( ) δ = η ωρ where η is the viscosity, ρ is the density and ω = π is the angular requency. As a consequence o this viscous entrainment o the liquid, the requency decrease is related to the square root o the density viscosity product and to a lower power o the requency,3, 3/. In addition, a damping o the QCM resonance, also related to the square root o the density viscosity product occurs indicating that energy loss is occurring. In recent years, the application o acoustic wave devices has been extended to include in-situ monitoring o ilm deposition, e.g. in electrochemistry 4,5, and chemical and biological sensing, e.g. sensing with polymer coated devices 6-9. A common eature o these applications is that the device suraces are coated with materials that are neither purely rigidly coupled mass nor simply Newtonian type liquids. It has thereore become essential that models o acoustic device response to layers o viscoelastic materials be developed. Moreover, these models need to allow or multiple and inite thicness coatings and operation o devices in liquid. A general approach is to consider the acoustic impedance, L, at the interace between the acoustic device and the coating. This surace acoustic impedance summarizes the overall acoustic load acting on the acoustic device and can be applied to single and multilayer arrangements 0. With some approximations, this model can be translated into equivalent 3

4 circuit models used in electrical engineering -3. The imaginary part o the impedance gives the requency shit and the real part gives the damping. An alternative approach, applied to the change in response o a QCM due to a Newtonian liquid, was developed by Rodahl and Kasemo 4. They considered the motion o the unloaded QCM surace to be a damped simple harmonic oscillator and then deduced the increase in damping due to shear stress on the surace arising rom immersion in a liquid. They have recently extended this theory to two polymer layers with the viscoelastic material modelled as a Voigt element 5. McHale et al 6 have also considered the extension o the simple harmonic oscillator model to include a single polymer layer, but using a Maxwell model or the viscoelasticity and allowing or a range o acoustic wave devices. An implication o this extension is that the results o the simple harmonic oscillator model are valid not only or QCM devices, but also all classes o acoustic wave devices based on shearing o the surace loadings. This alternative physical approach to modelling acoustic wave device response has the advantage o providing access to the physical boundary conditions. One aspect o QCM response that has generated controversy is the possible role o interacial slip. It appears clear rom the wor o Krim et al 7,8 that when certain atoms are adsorbed onto the surace o a QCM in ultra high vacuum conditions they can loc together as the layer coverage approaches a monolayer and slide on the surace. However, whether slip occurs when a QCM is operated in a liquid or with a polymer coating is ar less certain. Several authors have reported an apparent dependence o the acoustic impedance on the contact angle o the liquid, and hence surace wettability 9-. Martin et al,3 have argued that apparently anomalous results or the acoustic impedance can arise rom surace roughness. The argument is that air or liquid can be trapped within small pits on a device surace. Trapped liquid could act as a rigid mass loading in the Sauerbrey manner while trapped air prevents such a mechanism. This would distort the response that would be expected i only viscous entrainment, in the Kanazawa and Gordon manner, occurred. Contact angle dependence then arises in the acoustic response as liquid penetration into small surace aspherities and is 4

5 determined by surace wettability. Whilst it is clear air trapping could occur, it is not clear that all anomalies in measured acoustic impedances are due to this mechanism. An obvious alternative mechanism or anomalous acoustic response that could depend on interacial energy is slip. It is anticipated that should slip be a mechanism inluencing QCM response it could have its greatest impact in biological sensor applications where the hydrophobicity and hydrophilicity o the interace may change. A particular diiculty in assessing whether slip is occurring when QCM devices are operated in liquids is that ew models exist that can predict how device response would be altered. Such models need to be suiciently lexible to allow or both bare devices operated in liquid and or coated devices. Whilst circuit models o QCM response are useul, and can be extended to multiple viscoelastic layers, they do not oer a simple and transparent way o including interacial slip. Hayward and Thompson 4,5 have presented a model that included both interacial slip and multiple viscoelastic layers. However, the inclusion o slip involved complex slip parameters that were not explicitly related to the impedance o the layers. In this wor, we irst review the physical model 6 o acoustic wave device response with a single viscoelastic layer. Previously presented expressions are reormulated in the orm o the general complex shear modulus and the acoustic impedance, since it is this impedance which is oten most directly related to the values measured in experiments (Section IIa). The model is then extended to include multiple viscoelastic layers and so provide a simple, but general, ramewor in which to consider interacial slip (Section IIb). The no-slip boundary condition at the device-viscoelastic irst layer interace is then relaxed using a Stoes riction law, as previously suggested or a QCM in contact with a Newtonian luid 4. This boundary condition introduces a single real slip parameter, s, and results in a simple recipe or the inclusion o slip in the response o any acoustic wave device based on shearing motion induced in viscoelastic layers (Section IIc). The physical signiicance o the slip parameter is discussed and an acoustic load concept is introduced as an interpretation o interacial slip in a single layer 5

6 device. Finally, boundary conditions or slip occurring at the interace between any two layers in a multiple layer device are considered. II. Theory IIa. Single layer and no slip Experimental measurements can involve oscillators measuring the requency shit and electrical ampliication to compensate damping, conigurations measuring the damping o an oscillation ollowing excitation at the resonant requency or impedance analysers measuring the complex electrical impedance. It is thereore useul to note that in sensor applications changes in (angular) requency, ω, acoustic energy dissipation, D, and electrical impedance,, are caused by surace mechanical impedance (or acoustic load) changes. The surace mechanical impedance o the ilm, L, can be deined by, L F = () q& s In a linear approximation these relations or a QCM are given by, ω = ρ t q q F Im q& s = ρ t q q Im [ ] L () and D = ωρ t q q F Re q& s = ωρ t q q Re [ ] L (3) where ρ q and t q are the density and thicness o the substrate. Eqs. () and (3) can be derived within the damped simple harmonic oscillator model. In terms o a QCM the surace impedance is related to the quartz substrate parameters by, α q = L (4) ωc 0 4K q where q and acoustic impedance o the quartz substrate, α q is the wave phase shit in the quartz substrate and K is a value o electromechanical coupling 0,6,7 F is the orce exerted by 6

7 the ilm on the quartz substrate per unit area and q& s is the speed o displacement o the quartz substrate 4. In this paper, we will use the real part o the impedance and dissipation or energy loss as interchangeable terminology. Similarly, the imaginary part o the impedance will also be reerred to as requency shit or energy storage. To evaluate the orce exerted by a Newtonian liquid on the substrate it is necessary to obtain a proile o luid low by solving the Navier-Stoes equations or a viscous and incompressible luid. This approach can be extended to other classes o acoustic wave devices by replacing the quartz density by the substrate density, ρ s, and the quartz thicness t q by ξλ where λ is the acoustic wavelength in the (unloaded) substrate and ξ is a parameter representing the depth o substrate oscillating 6. For a QCM ξ=½ and or a shear horizontal surace acoustic wave (SH- SAW) ξ. Fig. shows the relative direction o oscillations or a QCM, a SH-SAW and the in-plane component o a Rayleigh acoustic wave (SAW). For a SH-SAW or SAW propagating along a path covered by a luid Eq. 3 can be replaced by a loss per unit length, L o L = 0ω (log0 e) F Re ρsξv q& s (5) where v is the (unloaded) speed o the acoustic wave. In our previous wors we included viscoelasticity in the model by a complex shear modulus, G, or by modiying the Navier-Stoes equations and the equation or the shear stress in the luid. The equation or luid low is then, iωv G = v (6) iωρ where ρ is the density o the luid, v is the luid velocity, and a time dependence e iωt has been assumed. Eq. (6) is the wave equation or bul shear waves propagating in a viscoelastic medium and a solution can be ound using the velocity proiles, 7

8 iωt QCM v = ( v ( z) e,0,0) (7) i( ω x) SH-SAW (0, ( ) t v = v z e,0) i( ωt x) i( ωt x) SAW v = ( v ( z) e,0, u ( z) e ) (8) (9) and the two boundary conditions, v ( z = 0 ) = q& (0) s and G v iω z u + = 0 () x Eq. (0) is the no-slip condition at the ilm-substrate interace and Eq. () is the continuity o shear stress at the ree surace o the ilm, which is o thicness t. The solution or the luid speed v (z) is identical or all three types o device provided v sh <<v, where v sh is the shear speed o the ilm, and is given by, v z= t ωη ( z t ) cosh G δ ( z) = q& s () ωη t cosh G δ The corresponding impedance is then given by, ρ = ρ G tanh iωt (3) G which is exactly the same result as derived rom the transmission line model or the acoustic load o a single (viscoelastic) ilm. 0,3 The Maxwell model or viscoelasticity views the total rate o strain as a spring and a dashpot connected in series. The complex shear modulus is then, G iωη = (4) + iωτ 8

9 where τ=η /µ is the relaxation time, and µ is the high requency shear modulus. In the notation o Behling 6,7, ωτ=/κ where κ is the loss actor o the polymer and is the ratio o the viscous to elastic contributions to the shear modulus. The Maxwell model contains the limits o Newtonian liquids and rigid solid overlayers and so the theory can represent a large number o device conigurations that are relevant to sensing applications. In this ormulation we can deine a complex eective penetration depth, δ, o, δ δ = (5) + iωτ which gives more physically obvious orms or the luid speed, v ( z) = q& s i ( z t ) cosh δ it cosh δ (6) and the ilm impedance, δ = δ iωρ η it tanh (7) δ IIb. Multiple layers and no slip Model deinition In the case o an acoustic wave device with n multiple layers o thicness t, we simpliy solve the general problem by assuming that v sh <<v, where v sh is the shear wave speed in any one layer. The oscillation in each layer induced by the substrate motion will satisy the equivalent o Eq. (6), G d v ( z) iω v ( z) = (8) iωρ dz where ρ is the density o the layer and G is the complex shear modulus o the layer. The general solution or each layer is then o the orm, α z α [ z ] v ( z) = q& A e + B e s (9) 9

10 where, α ω ρ = (0) G To obtain speciic solutions it is necessary to impose boundary conditions at the interace between the substrate and the irst layer, all intermediate interaces and at the ree surace o the top most layer. Assuming no-slip the appropriate boundary conditions are, v ( z = 0) = q& s () G v ( t ) = v ( t ) p= dv G dz = p + p= dv dz + + z= t p z= t p p= p= p () (3) G n dv n dz n x= t p p= = 0 (4) Eqs. () and (3) represent a set o equations with = to n- and the summations simply deine the z location o the relevant interace. The acoustic impedance presented to the substrate by the combined overlayers is given by the shear stress acting at the substrate-irst layer interace, = F G = q& iω& q s s dv dz z= 0 (5) In the no-slip case, is related to the irst coeicient in the solution, Eq. (9), through the actor A -. Eqs. (9)-(5) deine the problem or the acoustic impedance or an arbitrary number o inite thicness viscoelastic layers. Solving these equations enables the calculation o energy storage and energy loss due to shearing motion induced in the viscoelastic overlayers by a range o acoustic wave devices. The transmission line model relates the overall acoustic impedance o a multilayer arrangement to the surace acoustic impedance at the device-irst ilm interace with a chain matrix technique 0,3, starting rom the ront 0

11 acoustic port. The acoustic impedance or the -th layer with the acoustic load, +, at the outer surace is 8 : + j tan ( ω( ρ c ) t ) ( ω( ρ ) t ) = + c c (6) c + j + tan c where c = ρg is the characteristic acoustic impedance o the th layer. The impedance calculated or = is the surace acoustic load impedance, L. Solution or two layers It is useul to obtain the analytical solution or a system with two arbitrary thicness overlayers so that the consistency o the method can be veriied. Ater some algebra we ind, L + = + ρg (7) where ρ = ρ G tanh iωt =, (8) G is the acoustic impedance o each individual ilm. Within the Maxwell model, the relationship to a luid-lie layer can be made more obvious by using the analogous relation to Eq. (7), which explicitly introduces a viscous penetration depth through α = i δ. Eqs. (7) and (8) agree with results rom transmission line models 7. In general, the total acoustic impedance or a multilayer system is not simply the sum o the individual impedances o each layer (Eq. (8)). Luclum et al 9,30 have discussed how the existence o the denominator in Eq. (7) can be used to obtain ampliication in sensors by maing the irst layer a rubbery polymer. IIc. Slip boundary condition Device-layer one interace

12 One diiculty with the transmission line approach to modelling the acoustic impedance is that the method or inclusion o slip at interaces is not obvious. The transmision line model assumes continuity o mechanical tension and particle velocity. With the physical model interacial-slip can be accounted or by altering the boundary conditions, eqs. ()-(4), in the solution or the motion o the layers. In the model or Newtonian luids on QCM s, Rodahl and Kasemo 3, argued that slip at the device interace could be related to the shear stress acting on the substrate due to the liquid. They invoed a Stoes law and suggested that Eq. () be replaced by, (& ( )) χmml qs v z = 0 = F (9) where χ is the coeicient o riction between the ilm and the surace and m ML is the mass per unit area o a monolayer o the ilm. Within the context o viscoelastic layers o inite thicness we replace the actor o χm ML occurring in Eq. (9) by a single real parameter /s. Applying this boundary condition to a multilayer model we ind the simple result that the total acoustic impedance calculated without slip can be mapped to the total acoustic impedance allowing or slip by a simple replacement rule or the irst layer, (30) + s The essential idea in introducing slip is that a discontinuity in the speed occurs so that the speeds obtained by approaching the device-layer one interace rom either side no longer match. Hayward and Thompson 4,5 have used a slip boundary condition in a multilayer system, but their solution involves simply scaling one speed by a complex slip actor. This type o approach has several limitations. Each interace involves a complex slip actor and hence two parameters. More importantly, the relationship o the slip actor to material and drive parameters, such as the shear modulus, layer thicness and density, and oscillation requency are not well-deined within the approach. In the Stoe s law approach, the eect o Eq. (9) on the speeds at the device-layer one interace is to introduce a mis-match o, v q& s ( z = 0) = (3) + s

13 Using Eq. (3) as the deining slip boundary condition or a single layer device will recover the result o Eq. (30). The advantage o this boundary condition is that material and drive parameters are introduced in a well-deined manner by the acoustic impedance calculated rom the no-slip problem. In eect, Eq. (3) provides a deinition o the complex slip parameters used by Hayward and Thompson 4,5 through the single real parameter, s. When slip is wea, s, is small and the mis-match in speeds vanishes, thus recovering the no-slip boundary condition. When slip is strong, s is large and the speed o the layer vanishes so that the layer is decoupled rom the underlying motion o the substrate. It is worth cautioning that this last statement is a simpliied interpretation as the layer impedance is a complex number involving a tanh() unction with a complex argument. In essence, Eq. (3) can be viewed as introducing a irst order approximation using a single-loop negative eedbac system with a orward element o (the open-loop term) and a eedbac term o s as shown in Fig.. III. Results and Discussion IIIa. Viscoelastic overlayer with no-slip boundary condition Eqs. (3) and (7), or a device coated with a single layer o viscoelastic material, provide two complementary views o the acoustic impedance. For small relaxation times where the relaxation time is the ratio o the viscosity to high requency shear modulus, the eective penetration depth reduces to the penetration depth and the Kanazawa and Gordon result or Newtonian liquids is recovered rom Eq. (7). For large relaxation times and small argument (phase angle) in the tanh() unction o Eq. (3) we recover the Sauerbrey result or mass loading by a thin rigid mass ilm. In the case that the argument o the tanh() unction is not small the acoustic impedance provides a method o measuring the shear modulus o the ilm 3. In examining thic ilms care is needed as the argument o the tanh() unction is complex and this can lead to damped shear wave resonances. Experimentally, locating these resonances is one method by which the shear modulus o a ilm can be determined 6. One consequence o the physical basis presented in this wor is that the uniied interpretation provided or QCMs, 3

14 which typically operate in the -0 MHz range, and SH-SAW devices, which can operate in the 00 MHz to GHz range, allows a wider requency range to be investigated. In Figs. 3 and 4 we use a Maxwell model with a single relaxation time to show the generic changes in acoustic impedance that can be expected as the layer thicness and viscoelasticity are altered. The normalised impedance = ωρ η depends on only two norm parameters ωτ and t /δ,. For Newtonian liquids, which have ωτ small, both real and imaginary parts o the impedance increase with ilm thicness until a saturation occurs ater a ew penetration depths. For these thic Newtonian ilms the impedance is only sensitive to the properties o the liquid at the device interace; in this regime a device could also be used to sense the area o interace covered by the liquid. As ωτ increases the inluence o viscoelasticity increases, the eective penetration depth δ lengthens and oscillations occur in the impedance. For very large relaxation times the liquid appears as an amorphous solid and the oscillations tae on the character o sharp shear wave resonances, indicating the ability o the shear waves to travel to the ree ilm surace, to relect and interere constructively. In this amorphous solid limit and or very thin ilms (h/δ<<) the real part o the acoustic impedance (energy dissipation) vanishes and only energy storage (imaginary part o the impedance) occurs. This corresponds to the Sauerbrey equation or rigidly coupled mass in which requency decreases (i.e. positive Im[ ]) occur without loss o quality o the resonance. In assessing whether changes in the response o an acoustic wave device, such as a QCM, are due to changes in the viscoelasticity o the ilm it is necessary to compare the trends in the energy loss with those o the energy storage 9. For ilms much thinner than the viscous penetration depth increasing viscoelasticity results in decreases in the resonant requency and a broadening o the resonance (see Fig. 4 and Fig. 5). For ilms much thicer than the penetration depth the requency will increase bac towards the unloaded resonant requency as the viscoelasticity increases, but the attenuation initially increases and then decreases, so a broadening is ollowed by a sharpening o the resonance. In the region between these two 4

15 limits, it is possible to ind a decreasing resonant requency (i.e. Im[ ] positive and becoming larger) and a sharpening o the resonance (i.e. Re[ ] positive and becoming smaller) as ωτ increases rom to 4. It is important to emphasise that changes in viscoelasticity o a ilm can, in principle, lead to all our combinations o increasing/decreasing resonant requency and increasing/decreasing attenuation. It is also worth noting that extreme viscoelasticity may even result in a resonant requency higher than the unloaded device as shown by the change in sign in Fig. 4. IIIb. Slip at the substrate interace For an acoustic wave device with a single layer we would expect slip to simply decouple the motion o the substrate rom the layer. In order to assess whether this might occur an order o magnitude estimate or the slip parameter, s, is useul. This parameter occurs in Eq. (3) in a combination (+s ) and we thereore anticipate that s will have dimensions o inverse impedance. Moreover the slip parameter must be related to both the layer and the device substrate and so a possible candidate is, s = (3) χ c where c = (ρ G ) is the characteristic impedance o the layer and χ, is a co-eicient o riction. For a glassy layer with G 0 9 and ρ =000 g m -3, c 0 6 Rayls and taing a typical coeicient o riction o the order between 0.0 and suggests that a slip parameter would have an order o magnitude o 0-5 to 0-7. For a liquid, the characteristic impedance would be several orders o magnitude lower and the slip parameter would be correspondingly smaller. Normalising the impedance containing the slip actor (Eq. (30)) gives, slip norm = + s norm norm (33) and this has a dependence on a parameter s = s ωρ η, which involves the requency in addition to the two parameters ωτ and t /δ. Using the density and viscosity o water and a 5 MHz oscillation requency gives s 5.6 s whereas using the density and viscosity o a 5

16 0 000 cst polydimethylsiloxane (PDMS) oil gives s.8x0 6 s. In Figs. 7-0, we have chosen s = 0, 0.5 and to illustrate the structure o changes to the acoustic impedance that interacial slip, within the model developed here, could produce; these values o s are not intended to be taen as suggested values or experiments. For the Newtonian liquid in Figs. 7 and 8 our previous order o magnitude estimate o s is ar lower than 0.5. In contrast, or PDMS the estimate is consistent with an s o the order o magnitude used in Fig. 9 and 0. Figs. 7 and 8 show the eect o slip with s = 0, 0.5 and on a QCM device coated with a single layer o a Newtonian liquid (i.e. ωτ=0). For this layer, when the thicness is much greater than the viscous penetration depth, the eect o slip is to reduce the acoustic impedance in the manner anticipated by a decoupling o the liquid rom the substrate. However, these curves show that the eect o slip is dierent when the ilm thicness is less than about one viscous penetration depth. In this case, the energy storage systematically decreases as slip increases, but in contrast the energy dissipation now increases rather than decreases. This model o slip may provide a basis or consideration o the eect o hydrophilic and hydrophobic suraces and o surace roughness through their inluence on the co-eicient o riction. Figs. 9 and 0 show the eect o slip with s = 0, 0.5 and on a device coated with a single layer having an amorphous solid character (i.e. ωτ=50). Close to the shear wave ilm resonance, the eect o slip is to dampen and broaden the acoustic impedance. For thin ilms the eect o increasing slip is to increase the real part o the acoustic impedance and reduce the imaginary part. In this respect, the qualitative behaviour o thin ilms is the same whether the ilm is o a liquid or solid character although the slope o the imaginary part o the acoustic impedance with layer thicness changes to concave rather than convex. Whilst Figs. 9 and 0 only show the thin ilm limit, the thic ilm limit or an amorphous solid is similar to the Newtonian liquid thic ilm limit. The predicted behaviour o the thin amorphous solid 6

17 ilm is consistent with the experimental observations 7 on sliding riction during the adsorption o a monolayer o Kr atoms on a gold electrode QCM at 80 K. Initially as atoms are adsorbed the requency shit increases, but as the coverage becomes greater the requency shit then decreases; this decreasing requency shit is interpreted as a decoupling o the adsorbed atoms rom the surace. IIIc. Acoustic load concept The treatment o interacial slip in this paper has been based on the physical boundary condition at the substrate-irst layer boundary and has used an approach based on mechanics. It is possible to interpret the results or slip in a single layer system (Eq. (30)) as arising rom a generalised acoustic load. To do so we imagine a substrate coated with two layers, an ininitesimally thin intermediate layer with acoustic impedance i immediately adjacent to the substrate and a second inite layer with acoustic impedance. Assuming a no-slip boundary condition, Eq. (7) gives a total acoustic load or this double layer system o, i + L = (34) i + ci This generalised acoustic load, L, will give the same result as Eq. (30) obtained or a single layer o acoustic impedance, directly in contact with the substrate, but with a slip boundary condition, provided we choose, i χ i = ci ci χ i ci (35) where ci is a characteristic impedance and χ i is dimensionless riction coeicient. Thus, in this interpretation, slip generates a certain impedance or an intermediate layer. The concept o an intermediate, or boundary, acoustic impedance then allows or an additional magnitude and phase change o the propagating wave, which is presented as an additional magnitude and phase change o the overall acoustic load 3. 7

18 IIId. Slip at intermediate interaces in multilayer devices How to include slip at intermediate interaces, other than the device-layer one interace, in a multiple overlayer structure is not obvious as Stoe s law (Eq. (9)) does not easily generalise. At such an intermediate interace in a multilayer structure there are two boundary conditions, Eqs. () and (3), rather than the single boundary condition occurring at the device-layer one interace, Eq. (). We can derive a solution where the speeds are discontinuous, but nonetheless eeping the continuity o the shear stress. The alternative ormulation o Stoe s law involving a mismatch o the interace speeds (Eq. (3)) can be generalised to, v ( t ) = v ( t ) p p= p p= + s L (36) where L is the total load impedance o the th and higher layers calculated using a no-slip boundary condition. One approximation to L would be to sum the individual impedances o the (n-) top most layers. Whilst Eq. (36) is an obvious extension o Eq. (3) it is not the only choice that could be made. Alternatives include using the impedance o the th layer calculated using a no-slip boundary condition in place o L in the denominator in Eq. (36). It is liely that Eq. (36) will be a reasonable condition in a double layer system with slip at either o the two interaces. Solving the model or two layers using the boundary conditions given by eqs. (), (3), (4) and (36) gives the result, L + = + ρg + s + s ( + ) + ρg (37) The irst actor in the right hand side o Eq. (37) is the result or two-layers obtained using the no-slip boundary condition. The second actor is the correction allowing or slip. Examining Eq. (37), we ind that it arises rom the no-slip result or two layers using a simple replacement rule analogous to Eq. (30), i.e. 8

19 (38) + s The consistency o the result in Eq. (37) with the result or a single-layer on a device can be checed by imagining the thicness o the irst layer tends to zero. The result o Eq. (30) is then recovered. A second method o veriying the consistency is to imagine strong slip at the layer -layer interace. The slip parameter s then dominates the second actor in Eq. (37) and, ater cancellations, Eq. (37) reduces to the acoustic impedance o the irst layer. Physically, strong slip at this interace between the layers decouples the second layer and the device sees the same impedance as it would i it was covered by only a single layer. Using the single loop negative eedbac view, illustrated in Fig., the result or the two layer calculation suggests that slip at any layer, treated using the approximation represented by the slip boundary condition Eq. (36), will result in a replacement rule analogous to Eq. (38) with the index simply equal to the layer index. IV. Conclusion A physical model or viscoelastic contributions to the sensor response o a range o acoustic wave devices has been developed in terms o the surace acoustic impedance. Results or a device with a single viscoelastic layer have been shown to contain the limits o rigid mass loading and loading by Newtonian liquids. The eects o the viscoelasticity on the eective viscous penetration depth and the development o shear wave resonances with increasing viscoelasticity have been illustrated. All our combinations o requency increase/decrease and attenuation increase/decrease with changing viscoelasticity are shown to be possible. The extension o the model to multiple viscoelastic layers has been described and results explicitly derived or a two-layer system. These results are consistent with transmission line models o QCMs with two layers. The no-slip boundary condition at the interace between layers has been reconsidered. An alternative slip boundary condition based on a speed discontinuity related to the layer impedance calculated using a no-slip boundary condition has been introduced or a single layer system. For a system with a single thic layer, both liquid and 9

20 solid, slip reduces energy storage and energy loss whilst or thin liquid-lie layers, slip can increase the energy loss. A generalized acoustic load concept has been introduced as an interpretation o the results or slip in a single layer QCM device. The inclusion o slip at any interace in a multiple layer system is more complicated and a possible modiication o the boundary conditions has been suggested. Acnowledgements The authors grateully acnowledge inancial support rom the UK Engineering and Physical Sciences Research Council (EPSRC) and rom The British Council and DAAD. 0

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22 8 Dayo, W. Alnasrallah, J. Krim, Phys. Rev. Letts. 80, 690 (998). 9 M. Yang, M. Thompson and W.C. Duncan-Hewitt, Langmuir 9, 8 (993). 0 N. Schmitt, L. Tessier, F. Patat, J.M. Chovelon and C. Martelet, Sensors and Actuators A46, 357 (995). L.M. Furtado, H. Su and M. Thompson, Anal. Chem. 7, 67 (999). S.J. Martin, G.C. Frye and A.J. Ricco, Anal. Chem. 65, 90 (993). 3 S.J. Martin, Faraday Discuss. 07, 463 (997). 4 B. A. Cavic, F. L. Chou, L. M. Furtado, S. Ghaouri, G. L. Hayward, D.P. Mac, M.E. McGovern, H. Su, M. Thompson, Faraday Discuss.07, 59 (997). 5 G.L. Hayward and M. Thompson, J. Appl. Phys. 83, 94 (998). 6 C. Behling, R. Luclum and P. Hauptmann, Meas. Sci. Technol. A6 (998) C. Behling, The non-gravimetric response o thicness shear mode resonators or sensor applications, PhD Thesis University o Magdeburg (999). 8 R. Luclum, C. Behling, and P. Hauptmann, Faraday Discuss. 09, 3 (997). 9 R. Luclum, C. Behling, and P. Hauptmann, Anal. Chem. 7, 488 (999). 30 R. Luclum, C. Behling, and P. Hauptmann, IEEE Ultrason., Ferroel., Freq. Contr. (000), in print. 3 C. Behling, R. Luclum, and P. Hauptmann, IEEE Ultrason., Ferroel., Freq. Contr. 46, 43 (999). 3 R. Luclum and G. McHale, 000 IEEE/EIA Int. Freq. Contr. Symp., Kansas City (MO).

23 Figure Captions Figure Three possible modes o oscillation o acoustic wave devices. The transverse waves in the QCM and SH-SAW devices and the longitudinal component o the Rayleigh SAW induce shearing o any viscoelastic overlayer. Figure The eect o the slip boundary condition can be interpreted as a single-loop negative eedbac system where the orward transer is the impedance calculated with a no-slip boundary condition,, and the eedbac bloc is s. 3

24 Figure 3 The change in the real part o the acoustic impedance o a single viscoelastic overlayer as a unction o thicness and relaxation time using a Maxwell model. The transition in the nature o the ilm rom liquid-lie to amorphous solid is accompanied by the onset o shear wave resonances. Figure 4 The change in the imaginary part o the acoustic impedance o a single viscoelastic overlayer as a unction o thicness and relaxation time using a Maxwell model. The change in sign o the imaginary part o the ilm impedance indicates that requencies larger than the unloaded resonant requency can occur when devices are operated as resonators. 4

25 Figure 5 Changes in dissipation or ωτ=0, and 4 (solid line, long dashes and short dashes, respectively). Increasing the relaxation time, and hence the viscoelasticity, increases the dissipation or thin ilms with thicness much less than the viscous penetration depth. However, or thicer ilms both increases and decreases can be seen. Figure 6 Changes in energy storage or ωτ=0, and 4 (solid line, long dashes and short dashes, respectively). Depending on ilm thicness, it is possible to have requency decreases or requency increases as the relaxation time increases. In both cases, the corresponding changes in loss (Fig. 5) can be either increases or decreases depending on ilm thicness. 5

26 Figure 7 The real part o the ilm impedance or a single layer Newtonian liquid (ωτ=0) with slip parameters o s = 0, 0.5 and (solid line, long dashes and short dashes, respectively). For thic layers, slip decouples the luid and the dissipation reduces. For thin layers slip can lead to increased dissipation. Figure 8 The imaginary part o the ilm impedance or a single layer Newtonian liquid (ωτ=0) with slip parameters o s = 0, 0.5 and (solid line, long dashes and short dashes, respectively). For thic layers, slip decouples the luid and the requency shit rom the unloaded value reduces. 6

27 Figure 9 The real part o the ilm impedance or a single layer o an amorphous solid (ωτ=50) with slip parameters o s = 0, 0.5 and (solid line, long dashes and short dashes, respectively). The eect o slip is to reduce the amplitude and broaden the shear wave ilm resonance. For thin layers slip increases the dissipation. Figure 0 The imaginary part o the ilm impedance or a single layer o an amorphous solid (ωτ=50) with slip parameters o s = 0, 0.5 and (solid line, long dashes and short dashes, respectively). The broadening o the ilm resonance is evident. For thin layers increasing slip reduces the energy storage and hence the requency shit. 7