Anal. Chem. 2005, 77, 335-343

Film Resonance on Acoustic Wave Devices: The Roles of Frequency and Contacting Fluid C. M. Lagier,† I. Efimov,‡ and A. R. Hillman*,‡

Facultad de Ciencias Bioquı´micas y Farmace´ uticas, Universidad Nacional de Rosario, Suipacha 531, AR-2000-Rosario, Argentina, and Department of Chemistry, University of Leicester, Leicester LE1 7RH, United Kingdom

The dynamics of composite films of polypyrrole and sodium poly(styrenesulfonate) were studied by means of the electrochemical quartz crystal microbalance. Admittance spectra recorded after successive cycles of electrodeposition showed dramatic changes, which were interpreted in terms of acoustic resonance of the film. Reports of this phenomenon are rare and unquantified, presenting a unique opportunity for the first test of a recently reported theoretical model. The model, valid at frequencies in the vicinity of film resonance, is represented in terms of an equivalent electrical circuit with parallel LCR elements in the motional arm of the resonator. Since it was developed for viscoelastic films exposed to a vacuum, this provides an opportunity to test the importance of the fluid necessarily present in in situ electrochemical experiments. Measurements at the fundamental frequency and at higher harmonics reveal the sensitivity of film resonance effects to frequency and provide insights into film dynamics through the variation of shear moduli with time scale (frequency). Acoustic wave devices offer novel means of exploring polymer dynamics in the context of thin films, relevant to utilization of these materials in a huge range of applications in sensors, actuators, energy storage and conversion, and electronic/optical devices. Commonly, insight into polymer dynamics is deduced somewhat indirectly, for example, through influence on transport of mobile species (ions and solvent), on electronic behavior, or through device “ageing” phenomena; this is far from ideal. Here we use high-frequency acoustic wave devices to probe polymer dynamics in a more direct manner through film viscoelasticity. In particular, we explore a rather dramatic phenomenon, film resonance, which occurs for a specific combination of film properties, sample size, and observational frequency. Despite the fact that the criterion for film resonance has previously been defined, observations of the phenomenon are rare and there has been no previous test of the only available theoretical model. Although this investigation is focused within the context of electrochemical control for the case of a specific polymer, we suggest that the acoustic wave methodology has generic value for understanding fundamental aspects of polymeric films whose dimensions are not that dis* Corresponding author. Tel: +44-116-252-2144, Fax: +44-116-252-3789. E-mail: [email protected]. † Universidad Nacional de Rosario. ‡ University of Leicester. 10.1021/ac0489061 CCC: $30.25 Published on Web 12/01/2004

© 2005 American Chemical Society

similar to total chain length and to which the characteristic length and time scales of conventional macroscopic mechanical devices are not suited. Depositing electroactive polymers on electrode surfaces has been demonstrated to efficiently influence interfacial properties, leading to potential technological exploitation in such diverse applications as transistors, rechargeable batteries, sensors, and actuators. These opportunities make polymer-modified electrodes one of the most active research areas in electrochemistry.1-4 Successful implementation of such electrodes is not widespread, however, largely due to apparent variations in materials properties, for example, with time, use, or ambient environment. The consequence of this has been a major input of effort into characterization of film properties, of which one of the more recent inclusions has been electromechanical properties. Despite the wide appreciation of their importance in controlling film dynamicss with notable consequences generically for sensor response time and sensitivity4 and, specifically in the case of acoustic wave devices, response interpretationsquantitative studies are rare. This paper addresses the need to understand the performance of an electrode-modifying film in terms of its rheological characteristics. In a general sense, the notion of “assembling” desirable electronic and mechanical properties by using two materials in a composite is widely appreciated: the simplest assumption is that these properties can be viewed as additive. Here we show that this approach, exemplified by a composite of one of the most promising and extensively studied conducting polymers, polypyrrole (PPy),5-7 with poly(styrenesulfonate) (PSS), can in fact yield quite dramatic and unexpected phenomena. One powerful tool to study polymer-modified electrodes is the electrochemical quartz crystal microbalance (EQCM). Through the converse piezoelectric effect displayed by quartz, application of an alternating voltage between the electrodes of the QCM induces mechanical vibrations in the crystal, the nature of which (1) Gale, R. J. Spectroelectrochemistry; Plenum Press: New York, 1988. (2) Molecular Design of Electrode Surface; Murray, R. W., Ed.; John Wiley and Sons: New York, 1992. (3) Calvo, E. J.; Echenique, R.; Pietrasanta, L.; Wolosyuk, A.; Danilowicz, C. Anal. Chem. 2001, 73, 1161-1168. (4) Hillman, A. R. In Electrochemical Technology of Polymers; Linford, R., Ed.; Elsevier: London, 1987. (5) Ohtsuka, T.; Wakabayashi, T.; Einaga H. J. Electroanal. Chem. 1994, 377, 107-114. (6) Yang, H.; Kwak, J. J. Phys. Chem. B 1997, 101, 774-781. (7) Smela, E.; Kariis, H.; Yang, Z.; Mecklenburg, M.; Liedberg, B. Langmuir 1998, 14, 2984-2995.

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are determined by the orientation of the crystal cut. Attaching a film onto one of the electrodes alters the mechanical oscillation frequency of the crystal and, due to the piezoelectric properties of quartz, the electrical characteristics of the QCM. By measuring the resonant frequency or the spectrum at frequencies close to crystal resonance, this technique allows the monitoring of polymer deposition, extraction of film viscoelastic parameters, and study of mobile species exchange between the modified electrode and the bathing electrolyte.8-11 If the attached film is thin and rigid (termed acoustically thin), the acoustic wave launched from the quartz is transmitted across the layer with low phase shift in the film, and the only change is a shift of the resonance peak frequency; for thin films, this frequency shift is proportional to the mass change associated with the relevant process, e.g., film deposition.12,13 However, if the film becomes thick or deviates from rigidity (in either instance, termed acoustically thick), both the frequency and the shape of the resonance peak change. Now, the changes are a function not only of the attached mass of the film but also its rheology.11,14,15 Physically, the quartz resonator oscillation that is imposed on the polymer immediately adjacent to it undergoes changes in acoustic phase and amplitude during transmission through the film. The outgoing wave launched from the quartz resonator/polymer interface also undergoes an interference effect with the acoustic wave reflected from the outer film boundary. The properties that govern the interference effect are film thickness and the material’s shear modulus, G. In the context of film deposition (the case studied here), increasing film thickness during the polymer deposition process initially results in destructive interference and diminution of the amplitude. However, when the phase shift across the film reaches π/2 rad, the total phase shift between the outgoing and reflected waves becomes 2π (π/2 during transmission in each direction and π upon reflection at the quartz/film interface), leading to coherence between the outgoing and reflected acoustic waves. This phenomenon, termed film resonance (as distinct from the resonance of the crystal), has been described for a small number of polymers; manipulation of the phase shift to achieve the resonance conditions has been accomplished via changing film thickness or temperature (and thereby shear modulus).16-18 The facility to exploit applied potential as a control parameter is a major advantage of the electrochemical context. However, (8) Oyama, N.; Ohsaka, T. Prog. Polym. Sci. 1995, 20, 761-818. (9) Bruckenstein, S.; Hillman, A. R. In Handbook of Surface Imaging and Visualization; Hubbard, A. T., Ed.; CRC Press: Boca Raton, FL, 1995; pp 101-113. (10) Ballantine, D. S.; White, R. M.; Martin, S. J.; Ricco, A. J.; Zeller, E. T.; Frye, G. C.; Wohltjen, H. Acoustic Wave Sensors; Academic Press: New York, 1997. (11) Hillman, A. R. The electrochemical quartz crystal microbalance. In Encyclopaedia of Electrochemistry; Bard, A. J., Stratmann, M., Eds.; John Wiley and Sons: New York, 2003; Vol. 3, pp 230-289. (12) Sauerbrey, G. Z. Phys. 1959, 155, 206-222. (13) Lu, C.; Lewis, O. J. Appl. Phys. 1972, 43, 4385-4390. (14) Kanazawa, K. K.; Gordon, J. G., II. Anal. Chem. 1985, 57, 1770-1771. (15) Bandey, H. L.; Hillman, A. R.; Brown M. J.; Martin, S. J. Faraday Discuss. 1997, 107, 105-121. (16) Domack, A.; Prucker, O.; Ruhe, J.; Johannsmann, D. Phys. Rev. E 1997, 56, 680-689. (17) Hillman, A. R.; Brown, M. J.; Martin, S. J. J. Am. Chem. Soc. 1998, 120, 12968-12969. (18) Saraswathi, R.; Hillman, A. R.; Martin, S. J. J. Electroanal. Chem. Interfacial Electrochem. 1999, 460, 267-272.

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the only theoretical analysis reported to date19 is for films exposed to gas environments. The fundamental issue then is the role of the solvent in the electrochemical context. Intuitively, one might view the absence/presence of fluid as imposing very different outer interface boundary conditions, resulting in very different film responses. However, an additional requirement for resonance19 is that the reflectivity coefficient at the outer film/solution interface be close to unity; were this not so, there would be insufficient reflected intensity for interference effects to be significant. In this case, one might argue that the nature of the external medium has less impact upon film response. Here we set out to explore which of these arguments most closely represents the physical situation, by comparing novel data for composite PPy/PSS films with the previously untested predictions of Martin’s19 lumped-element model for a viscoelastically loaded quartz resonator near film resonance. THEORY Overview. The goal is an understanding of film mechanical properties. Experimentally, these are accessed via resonator electrical response. The two are unambiguously linked by the piezoelectric effect, such that the response can be expressed interchangeably in terms of either representation;10 for convenience, we will use the electrical equivalent circuit representation. As will become clear from the data (below), progressive loading of the resonator during polymer electrodeposition results in decreases in the frequencies associated with both the crystal and film resonances. However, the thickness dependences of the two resonances are very different: film resonance moves much more sharply with polymer loading.19 Thus, observations in the vicinity of crystal resonance initially correspond to a situation far from film resonance. As polymer is deposited, the film resonance sweeps rapidly down and crosses through the crystal resonance. We therefore need to consider two cases of crystal resonance: far from and near to film resonance. Electrical Equivalent Circuit Models. The electrical input impedance, Z, for a quartz crystal resonator can be represented by a lumped-element model15,19,20 involving a static arm and a motional arm. The static arm comprises a capacitance, C0*, which is the sum of a capacitance (C0) arising between the electrodes located at opposite sides on the surface of the quartz wafer and a parasitic capacitance (Cp) associated with the electrode shape and configuration and with the supporting fixture. The static arm dominates the resonator response at frequencies far from resonance. By contrast, the motional arm dominates the resonator response in the vicinity of resonance. The impedance, Z1, of the motional arm has been modeled in various ways, of which the simplest is a series combination of elements L1, R1, and C1, known as the Butterworth-Van Dyke (BVD) model (see Figure 1a).20 Through the piezoelectric effect, each of the electrical elements represents a mechanical effect: L1 is the electrical manifestation of inertial mass, R1 symbolizes energy dissipation by viscous effects and internal friction, and C1 represents mechanical elasticity, through energy stored in the resonator during elastic deformation. (19) Martin, S. J.; Bandey, H. L.; Cernosek R. W.; Hillman, A. R.; Brown, M. J. Anal. Chem. 2000, 72, 141-149. (20) Bandey, H. L.; Martin, S. J.; Cernosek, R. W.; Hillman, A. R. Anal. Chem. 1999, 71, 2205-2214.

2πf), K2 ) 7.74 × 10-3 is the electromechanical coupling coefficient of quartz, φq is the acoustic phase shift across the quartz, and ξ ) ZS/Zq. ZS and Zq are the surface mechanical impedance and the characteristic shear wave impedance of the quartz, respectively. Zq ) (Fqµq)1/2 ) 8.83 × 105 g cm-2 s-1, where the density of quartz, Fq ) 2.651 g cm-3 and the quartz shear stiffness, µq ) 2.947 × 1011 dyn cm-2. ZS depends on the nature and properties of the loading.20 The relationship between the electrical input impedance Z, described in eq 1, and the lumped model described previously is given by Z ) 1/Y ) (jωC0* + 1/Zm)-1, where Y is the admittance and Zm, the motional impedance, is given by

Z m ) Z1 + Z 2

(2)

where Z1, the impedance of unperturbed resonator, is given by

Z1 ) R1 + jωL1 + 1/jωC1

(3)

Near resonance, at ω ≈ (L1C1)-1/2 ≡ ωs the following equality holds:

Z1 ) R1 + 2jL1(ω - ωs)

Figure 1. (a) Butterworth-Van Dyke circuit model; (b) modified Butterworth-Van Dyke equivalent circuit model for the resonator plus a loading near crystal resonance and far from film resonance; (c) equivalent circuit model18 for a film-loaded resonator, near crystal and film resonance.

System Near Crystal Resonance and Far from Film Resonance. The BVD model can be modified to account for a system consisting of a load attached to the surface of one of the electrodes and immersed in a liquid medium. These perturbations of the resonator are modeled by adding a series impedance Z2 to the motional arm, which represents the addition of a mass onto the piezoelectric substrate. If the excitation frequency of the resonator is near the quartz crystal resonance but far from film resonance, the load can be represented by a series arrangement of an inductance, L2, and a resistance, R2 (see Figure 1b). (Although one can invoke a more sophisticated transmission line model, it reduces to the BVD model under conditions of low loading of a resonator near resonance, i.e., the case considered here.20) It has been demonstrated elsewhere21-23 that the complex electrical input impedance for a thickness-shear mode resonator can be described by

Z)

[

]

1 K2 2 tan(φq/2) - jξ 1jωC0* φq 1 - jξ cot(φq)

(1)

where j ) (-1)1/2, ω is the angular excitation frequency (ω )

(4)

where the radial resonance frequency, ωs ) 2πfs, and fs is the series resonance frequency. System Near Crystal and Film Resonances. The model we wish to test19 for a resonator with an attached viscoelastic load proposes that, near film resonance, Z2 can be represented by a parallel combination of an inductance L2, a capacitance C2, and a resistance R2, as depicted in Figure 1c. For the case of a viscoelastic film, Z2 is given by19

Z2 )

8 (N'π)2

ωf3AhfFf

[ ( )] 2

(5)

ωf G′′ +j 1- 2 ω G′ ω 2

where G′ ) Re(G) and G′′ ) Im(G), respectively, are the real (storage) and the imaginary (loss) components of the shear modulus, G, hf is the film thickness and Ff is the film density. The constant A includes only parameters of the piezoelectric quartz resonator:

A ) Nπ/4K2ωsC0*Zq

(6)

where N is the resonator harmonic number (an odd integer). The angular frequency, ωf, at which film resonance occurs is expressed in terms of film properties and the film harmonic number, N′ (an (21) Granstaff, V. E.; Martin, S. J. J. Appl. Phys. 1994, 75, 1319-1329. (22) Martin, S. J.; Granstaff, V. E.; Frye, G. C. Anal. Chem. 1991, 63, 22722281. (23) Lucklum, R.; Behling, C.; Cernozek, R. W.; Martin, S. J. J. Phys. D: Appl. Phys. 1997, 30, 346-356.

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odd integer; note the physical distinction between resonator and film harmonics, N and N′, respectively), by

ωf )

N′π|G| 2hfxFfG′

(7)

The passive circuit elements in Figure 1c can also be expressed in terms of film properties:19

R2 ) 2A|G|2/(ω hf G′′)

(8a)

C2 ) hfG′/(2A|G|2)

(8b)

L2 )8AhfFf /(N′π)2

(8c)

Strictly, this model considers the film as the only loading, i.e., does not consider the effect of a bathing fluid. Although this model differs from the physical situation of an electrochemical experiment, we explore its ability to represent film resonance since this is synonymous with the reflection of most of the acoustic energy back into the film, i.e., relatively low coupling to any fluid that is present. If this apparently radical strategy can be shown to be an appropriate approximation, the simplicity of the model promises a number of attractions for investigating film dynamics. In particular, this includes a novel means to explore the frequency dependence of film properties (G′ and G′′) through the frequency dependence of the circuit elements described by eq 8. EXPERIMENTAL SECTION. Equipment. The instrumentation has been described in detail elsewhere.15,17 Au-coated 10-MHz AT-cut polished quartz crystals were used, with piezoelectric and electroactive areas of 0.21 and 0.23 cm2, respectively. The three-electrode cell consisted of one of the gold contacts to the quartz acting as the working electrode, a Ag/AgCl reference electrode, and a platinum gauze counter electrode. Crystal impedance spectra were recorded at the first, third, and fifth harmonics (nominally 10, 30, and 50 MHz), using a Hewlett-Packard HP8751A network analyzer connected via a 50-Ω coaxial cable to a HP8512A transmission/reflection unit. Materials and Procedures. The deposition medium was an aqueous solution of 0.1 M pyrrole (Aldrich; distilled under vacuum and stored under nitrogen) and 0.3 mM poly(sodium 4-styrenesulfonate) (Aldrich; MW ∼70 000; as received). Composite PPy/ PSS films were deposited potentiodynamically (between -0.75 and +0.75 V), with acquisition of admittance data after each cycle. Data are presented in detail for a scan rate, v ) 20 mV s-1. Summary data for a film prepared using v ) 10 mV s-1 are given in Supporting Information; changing conditions alters quantitative film properties, but the same phenomenasthe primary focus of this studyswere observed. Spectra at the fundamental (nominally 10 MHz) and at higher harmonics were obtained; since the deposition was interrupted between potentiodynamic cycles for acquisition of these data sets, we will refer to these fundamental and higher harmonic data as being acquired “simultaneously”. For viscoelastic films, the QCM frequency response is a complex function of thickness. A general approach to treatment of resonator responses for acoustically thick films has been 338 Analytical Chemistry, Vol. 77, No. 1, January 1, 2005

Figure 2. Real part of crystal admittance spectra at the resonator fundamental frequency (nominally 10 MHz) for a PPy/PSS composite film during deposition (potential sweep rate 20 mV s-1). Spectra were recorded after each deposition cycle, but for presentation purposes, only selected spectra are shown; figures indicate number of deposition cycles. Inset shows response after 25 cycles, when the two peaks are of similar amplitude.

described elsewhere;24 the strategy involves consideration of acoustically thin films to deduce film density and to establish a means of determining thickness from an external measurand, here electrochemical charge, Q. In the present case, the situation is relatively simple, since all the film components have densities very close to unity, so we can assign a value of Ff ) 1 g cm-3, regardless of the level of solvation. Furthermore, independent studies25 have established the coulometric relationship for film deposition: with PSS as the counteranion, the charge consumed on reduction of the film corresponds to one electron per five monomer units. RESULTS AND DISCUSSION Overview of Raw Data. Figure 2 shows selected crystal admittance (Y ) 1/Z) spectra acquired during deposition of a composite polymeric PPy/PSS film; analogous data with a different deposition protocol are shown in Supporting Information, Figure S1. Initially, we focus on these spectra acquired at the fundamental frequency of the resonator (nominally 10 MHz); corresponding responses at the third and fifth harmonics (nominally 30 and 50 MHz, respectively) are discussed later. As deposition proceeds (moving forward in Figure 2), the primary admittance peak diminishes in amplitude and there is the rise of a second peak at higher frequency. Interpretation involves consideration of the individual contributions to the overall response of the resonator and film motional impedances, Z1 and Z2.19 When the unperturbed angular resonance frequency, ωs, is lower than the film resonance angular frequency, ωf, the relatively narrow bandwidth of the resonator means that the effect of the film on the spectrum is not noticeable. Thus, at the outset of the deposition, the spectrum (24) Hillman, R. A.; Jackson, A.; Martin, S. J. Anal. Chem. 2001, 73, 540-549. (25) Baker, C. K.; Qiu, Y.; Reynolds, J. R. J. Phys. Chem. 1991, 95, 4446-4452.

comprises a single peak, predominantly derived from Z1. Film deposition decreases ωf such that an overlapping region appears between resonator and film resonances. Therefore, the admittance spectrum changes its shape to a broader, damped peak. As the film thickness increases, this broad peak splits into two damped peaks as a result of the combination of impedances Z1 and Z2, both of which contribute significantly to the composite resonatorfilm motional impedance; a snapshot corresponding to this region is shown in the enlarged inset in Figure 2. Further deposition of polymer diminishes ωf even more, such that the trend in peak frequency shift is reversed and the peak admittance increases. Eventually, a single resonance peak emerges at a higher frequency than ωs. This sequence of events is the predicted qualitative signature of film resonance.19 A condition for peak splitting to be observable is that the impedance bandwidth associated with the film be narrower than that associated with that of the resonator. Physically, this requires that the film loss tangent (R ) G′′/G′) be small, a test that will subsequently be applied to the fitted data. Analysis of Resonance at First Harmonic. The equivalent circuit model proposed by Martin et al.19 to account for film resonance was designed for a loaded crystal resonator in contact with air, i.e., a single-layer structure over the resonator. Although the system here is a loaded resonator immersed in a liquid (the electrolyte solution), calculations by Martin et al. suggest that the mechanical impedance associated with the liquid may be significantly smaller than that associated with the film. Prompted by this, our first approach is to analyze the experimental data (represented by experiments such as that of Figure 2) in terms of this model,19 neglecting the impedance of the contacting liquid. The extent to which this vastly simplifying approximation (in terms of both a model and its practical application) can be made without incurring significant error is a key issue in this study. Application of the model19 to the data will be demonstrated in detail for a representative response, that of the inset in Figure 2; subsequently presented shear moduli data for different conditions were extracted by the same procedure. Each of the resonances (admittance peaks) corresponds to the condition Im(Z1 + Z2) ) 0. When the peaks are of equal height, the model indicates that ωf ) ωs. Equations 2-4 then lead to the following expression for the imaginary component:

L1 )

ωf2AhfFf 8 (N′π)2 (ωfR)2 + 4(ω1,2 - ωf)2

(9)

where ω1,2 ) 2πf1,2 are the radial frequencies corresponding to either peak in Figure 2. Considering the real component, when ω ) ωf ) ωs:

R2 )

8 ωfAhfFf (N′π)2 R

(10)

Using eqs 9 and 10, respectively, one can estimate the film loss tangent, R, from the peak separation, (f1 - f2), and the minimum of Re(Y) between the peaks in Figure 2. Experimental values of parameters for the nonloaded quartz resonator (see eq 3) are R1 ) 18 Ω and L1 ) 8.75 mH. At the resonator fundamental frequency (N ) 1; nominally, fs ) 10 MHz),

with C0 ) 4.11 pF, this generates a value of the constant A in eq 6 of 4.44 × 10-1 Ω g-1 s cm2. Since R1 , R2 (the resonant admittance in the presence of the film drops by well over 1 order of magnitude), Re(Y) ≈ 1/R2 in the minimum between the peaks in Figure 2: this yields R2 ) 18.2 kΩ (more than adequately justifying the approximation). From eqs 9 and 10, the film loss tangent can be estimated by

R)

( x( ) (

1 R2 ( 2 L1ωf

R2 L1ωf

2

- 16

))

ω1,2 - ωf ωf

2

(11)

Equation 11 has two sets of solutions for N′ and R: N′1 ) 2.44, R1 ) 0.025, and N′2 ) 4.5, R2 ) 0.007 35. Selection of the physically relevant set is made by fitting the experimental responses to eqs 2-5. The relatively rich information content of the two-peaked admittance spectra (at this first harmonic) allows one to employ an unconstrained fitting procedure, which is a stern test of the model in the sense that it must generate physically reasonable parameters. We only constrained R1 ) 18 Ω; as the value for the nonloaded crystal, this is uncontentious. We then varied L1, ωs () 2πfs), C2, ω f () 2πff), and R2 in the components of eq 2, together with a shunting resistance and a capacitance, which uniformly shift the baselines for Re(Y) and Im(Y), respectively. C1 and L2 were calculated as C1 ) 1/(ωs2L1) and L2 ) 1/(ωf2C2). Panels a and b of Figure 3, respectively, compare the experimental and fitted real, Re(Y), and imaginary, Im(Y), components of the admittance. This is illustrated for the representative case of the inset in Figure 2; all other spectra were analyzed similarly. Between the two peaks representing film resonance, there is a small peak unrelated to film resonance; we attribute this to imperfections in the crystal cut, resulting in a minor contribution from some other mode. Aside from this unrelated phenomenon, the fits are good; the resulting electrical parameters are listed in Table 1. Application of the data in Table 1 to eqs 8-11 yields the shear modulus components (G ) G′ + jG′′) and of N′ and R; we used Ff ) 1 g cm-3 throughout (see above) and hf values determined coulometrically (see above; 1.2 µm in the example shown). We find the film harmonic number, N′ ) 5.09; since this must be an (odd) integer, we deduce N′ ) 5. We find the film loss tangent, R ) 0.0055. These values of N′ and R are in fair agreement with the pair N′2 ) 4.5, R2 ) 0.007 35 obtained by the initial analytical estimation and are consistent with the requirement19 that the loss tangent, R, is small. G′ can then be calculated as follows:

G′ ) 4hf2Ffωf2/N′2(1 + R2)π2

(12)

We obtain G′ ) 8.94 × 105 dyn cm-2. We do not pursue the G′′ data in detail, since our experience in practice24 is that whichever is the smaller of the shear modulus components is determined accurately only if it is within 1 order of magnitude of the larger component. What is clear is that the observation of film resonance is a consequence of the low loss nature of the material. This is parametrized through the acoustic decay length, which we estimate to have the relatively large value of 55 µm. The results of applying this analysis to the film of Figure 2 at other points Analytical Chemistry, Vol. 77, No. 1, January 1, 2005

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Figure 4. Storage modulus, G′, as a function of PPy/PSS composite film thickness for the experiment of Figure 2 at the fundamental frequency (nominally 10 MHz).

Figure 3. (a) Real part, Re(Y), and (b) imaginary part, Im(Y), of resonator admittance as a function of excitation frequency in the vicinity of the resonator fundamental frequency for a PPy/PSS film showing film resonance. Solid lines: experimental data (from the experiment of Figure 2; the spectrum marked *, corresponding to a film with hf ) 1.2 µm). Dotted lines: model fit at fixed R1 ) 0.018 Ω. Table 1. Values of Electrical Equivalent Circuit Parameters Obtained by Fitting Observed Response (Nominal 1.2-µm Film; See Figure 2) to Model of Martin et al.19 parameter (units)

value

R1 (Ω) L1 (mH) fs (MHz) R2 (kΩ) L2 (µH) C2 (pF) ff (MHz)

18 (fixed) 8.989 9.987312 19.01 1.664 152.7 9.989873

during the deposition process are shown in Figure 4 in the form of plot of G′ as a function of film thickness. Note that the range of film thickness values considered is restricted to the regime where N′ is constant; see below. Analysis of Resonance at Higher Harmonics. Responses at higher harmonics were obtained effectively “simultaneously” to those at the fundamental (see above). Although the anticipated anomalous frequency shifts associated with resonance were observed, instrumental factors and the bandwidths involved mean that we do not see both peaks within the observational frequency window. Essentially, adequate frequency resolution precludes obtaining data over the required frequency span. From the perspective of interpretation using eqs 9-11, which uses the radial 340 Analytical Chemistry, Vol. 77, No. 1, January 1, 2005

frequencies corresponding to both peaks, this decreased informational content (as compared to the fundamental resonance response) means that we are not able to float all the parameters and obtain unique solutions. Thus, fitting was carried out keeping parameters for the quartz constant and only varying those of the growing film. For the third harmonic (N ) 3), R1 ) 47.71 Ω, and L1)10.716 mH, while for the fifth harmonic (N ) 5), R1 ) 378.8 Ω and L1 ) 11.062 mH. The effective resonance frequency of the fundamental harmonic for the nonloaded quartz was found to be ∼9.985 MHz; it is then realistic to choose fs ) 29.955 and 49.925 MHz for N ) 3 and N ) 5, respectively. Representative data, acquired “simultaneously” to the inset in Figure 2, for these higher harmonics are shown in Figures 5 and 6. In each case, we compare the experimental and fitted curves for the real (Re(Y); panels a) and imaginary (Im(Y); panels b) parts of the admittance. Although the fitting is more constrained than in the case of the N ) 1 data, the agreement between the observed data and the model fits is gratifying. Shear modulus components were also determined at these higher frequencies; their values and significance are discussed in the next section. Discussion and Verification of the Model. In this section, we explore three issues: the nature of the film resonance effect, the significance of the shear modulus values, and the implications of the presence and (non)participation of the liquid. Although they are interactive to some extent, for presentational purposes we discuss them separately as far as is possible. According to the compound resonator model we have used, the resonant frequency of the nonloaded quartz crystal, fs, must be independent of film thickness, hf, whereas the film resonance frequency, ff, should be inversely proportional to hf. We therefore calculated values of fs and ff for each stage (hf value) of the deposition process. The same fitting procedure as described above was used, keeping R1 fixed at 18 Ω. We restricted our attention to film thickness values in the vicinity of film resonance, since the model19 we are testing is only valid in this regime; far from

Figure 5. (a) Re(Y) and (b) Im(Y) as a function of excitation frequency at the third harmonic (nominally 30 MHz). Data were acquired during the experiment of Figure 2 and correspond to the point in the experiment represented by the data of Figure 3 (hf ) 1.2 µm). Solid lines, experimental data. Dashed lines, fits to model.

this regime Martin’s model does not apply, and traditional models do not cover film resonance at all. Figure 7 shows the best-fit values of the frequencies fs and ff for the fundamental harmonic (N ) 1) response as functions of hf. Although there is clearly a little scatter on the data, the behavior predicted above is followed: fs increases by 5 kHz while ff decreases by 50 kHz in the same interval of hf. One can see clearly how ff passes through fs, giving rise to the interaction of the two resonances. Another important feature of the model is that N′ is a “quantum number” of the system, which should remain constant in the vicinity of film resonance. For the data of Figure 7 (hf ) 1.15 ( 0.15 µm), we find N′ ) 5.0 ( 0.5, satisfying this requirement. When one moves further from film resonance, i.e., outside this thickness range, deviations in the apparent value of N′ occur; this simply reflects that the calculational method is no longer applicable (approximations associated with the implementation of eqs 2-4 are not valid). Equivalent calculations were made for the third harmonic (N ) 3) results (Supporting Information; Figure S2), except that here (for reasons described above) fs was fixed and ff alone was fitted. As expected, ff decreases with film growth, crossing fs at hf ) 1 µm. The film harmonic number, N′, over the interval of hf from 0.96 to 1.2 µm was N′ ) 5.0 ( 0.5; given the poorer signal quality here, this is an acceptable result. Using the same arguments as for the fundamental, the range of film thickness (∼1 µm) for which N′ is constant is 5λ/2, so λ ) 0.4 µm. An interesting questions is why we see the N′ ) 5 film resonance, not the lower third harmonic (N′ ) 3) or the fundamental (N′ ) 1). There is no direct

Figure 6. (a) Re(Y) and (b) Im(Y) as a function of excitation frequency at the fifth harmonic (nominally 50 MHz). Data were acquired during the experiment of Figure 2 and correspond to the point in the experiment represented by the data of Figure 3 (hf ) 1.2 µm). Solid lines, experimental data. Dashed lines, fits to model.

Figure 7. Fitted resonant frequency data at the fundamental harmonic for the experiment of Figure 2 at different stages during the deposition process, corresponding to the film thickness range 0.95 e hf/µm e 1.32. ([) fs, nonloaded quartz; (0) ff, film. Corresponding N′ values, 5.0 ((0.5).

evidence here, but we make one speculation, based on the fact that these lower resonances would (for the same shear modulus values) occur at proportionately smaller film thickness. If the film, at an earlier stage in its deposition, were less uniform in thickness, more diffuse at its outer interface, or both, the film/solution interface would give a less coherent acoustic reflection. Another Analytical Chemistry, Vol. 77, No. 1, January 1, 2005

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Figure 8. Storage modulus normalized with respect to effect of frequency by dividing by N2 (see eq 13, involving G′/ω2) as functions of film thickness for the experiment of Figure 2 and analogues at higher harmonics. N ) 1 ([); 3 (0), and 5 (2).

possibility is associated with the fact that the film impedance Z2 (as defined in eq 5) varies with the resonance order in the film as 1/N′2, whereas Z1 is independent of N′. Therefore, the resonance peaks, defined by Im(Z1 + Z2) ) 0, are far apart at lower N′ and approach each other as N′ grows. At N′ ) 5, they become sufficiently close as to be observable simultaneously within the observational frequency window of the instrument under the experimental conditions. The mathematical arguments developed above clearly indicate that a description involving a compound resonatorsincluding two independent oscillators to represent the quartz crystal and the film loading but excluding the liquid impedancesallows one to apply Martin’s model to in situ (liquid contacting) situations. We now need to consider the physical implications of this. It is interesting that both the fundamental (N ) 1) and third harmonic (N ) 3) responses yield a film resonance number N′ ) 5 at the same value of film thickness. Regardless of the absolute value of hf, this means that acoustic wavelength is the same at both frequencies; i.e., the velocity of sound is linearly related to the frequency. Since the wavelength is related to the square root of the shear modulus, this implies that G′ must be increasing with ω2, here by a factor of 9 as we increase N from 1 to 3. In the case of media with a single relaxation time, τ, a classical model26,27 for shear modulus has

G′ ) ReG(ω) )

G 0 + ω 2τ 2G ∞ 1 + ω2τ2

(13)

where G0 and G∞, respectively, are the low- and high-frequency limits of G′. From the form of eq 13, we see that G′ ∼ ω2 is the low-frequency limit (ωτ , 1) for the common case that G0 , ω2τ2G∞. Qualitatively, the rather low values of G′ (see Figure 4) are consistent with this view. Quantitatively, this is explored directly in Figure 8, which shows values of G′/ω2 (covering N ) (26) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; Wiley: New York, 1980; Chapter 17. (27) Aklonis, J. J.; MacKnight, W. J. Introduction to Polymer Viscoelasticity; John Wiley and Sons: New York, 1983; Chapter 7.

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1-5) as a function of film thickness: given that the values of G′ vary by over 1 order of magnitude, the normalization is good. The purpose of this study has been to describe and parametrize film resonance; in this we have been successful, through the vehicle of the model of Martin et al. The extension of thissa topic for future studysis an explanation of why the liquid exerts relatively little influence on the response. In other words, why is the quartz/film/gas model able to describe a quartz/film/liquid situation? Two possibilities that will be addressed in this future study are whether there is interfacial slip28 or whether the reflection of the acoustic wave (and associated energy) at the film/ liquid interface simply means that, under conditions leading to film resonance, the interaction with the external medium is relatively small. Above, we presented detailed analysis of a representative and striking case of film resonance, with split (but closely spaced) sharp admittance peaks. We have also observed film resonance in the PPy/PSS system for films deposited under different experimental conditions. One example of a small variation, shown in Figure S1, is for potentiodynamic deposition using a different scan rate (10 mV s-1). Film resonance is clearly visible, though less distinct as a consequence of somewhat greater viscous loss within the polymer, i.e., a larger G′′. Although exploring the effects of varying experimental conditions is beyond the scope of this report, analyses of these and similar data are achievable using the model employed here. The unsurprising outcome is that absolute values of film properties (notably shear moduli) depend on the conditions employed, but the primary generic observation is that of film resonance at the appropriate film thickness. CONCLUSIONS Composite films of PPy/PSS, formed by electropolymerization of pyrrole in the presence of PSS anion in the deposition solution, are viscoelastic. When deposited on acoustic wave resonators, the films show the phenomenon of film resonance at an appropriate value of film thickness (determined by the film properties, that are in turn a function of the detailed conditions of the experiment). The unequivocal signal of film resonance is the splitting of the admittance peak, as the film resonance crosses through the quartz resonance with increasing film thickness. Qualitatively, observation of film resonance implies that certain conditions are met: the film has sufficiently low loss characteristics that the acoustic wave can travel across the film without great attenuation, the film is sufficiently even that the resonance condition is simultaneously observed at all points on the interface, and the reflectivity coefficient for acoustic energy at the film/solution interface is not too different from unity. The quantitative details of film resonance were compared with the predictions of a recently proposed equivalent circuit model.19 The model was originally designed for the situation that the film is exposed to a gaseous environment, which is clearly different to the electrochemical context. However, it is able to describe the observed in situ admittance responses in the region of film resonance. We conclude that this unexpected simplification is a consequence of the acoustic energy being primarily reflected back (28) Daikhin, L.; Gileadi, E.; Tsionsky, V.; Urbakh, M.; Zilberman, G. Electrochim. Acta, 2000, 45, 3615-3621.

into the film at resonance; i.e., coupling to the external medium is small, so its nature is not so important. On the basis of the equivalent circuit model, the data were fitted to extract film properties. Under the experimental conditions, the fifth film harmonic was observed. The shear modulus is dominated by the real component; i.e., the loss tangent is small (<0.01). We find G′ ∼ 106 dyn cm-2, and measurements at higher resonator harmonics show that G′ increases with the square of angular frequency, ω. In terms of film resonance characteristics, this means that the acoustic wavelength in the film remains almost unchanged with frequency. In terms of film dynamics, this is consistent with the behavior of a classical mechanical model for a material with a single relaxation time, τ, under conditions where ωτ < 1, i.e., the low-frequency regime where G′ is far below its high-frequency limit, G∞.

ACKNOWLEDGMENT We thank the EPSRC (GR/N00968) and The Royal Society (Developing World Visit Program to the U.K.) for funding and Dr. S. J. Martin for helpful conversations. C.M.L. thanks “Antorchas Foundation” (A-13740/1-47, 14116-48), UNR and CONICET, Argentinean Research Council (PEI 6252) for funding to visit the University of Leicester. SUPPORTING INFORMATION AVAILABLE Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org. Received for review July 27, 2004. Accepted October 15, 2004. AC0489061

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