Acoustically thick film

In a number of cases, polymer films that are most certainly not acoustically thin have been studied the viscoelastic nature of some polymeric materials makes them acoustically thick even at thicknesses well below 1% of a wavelength. Because other relationships that contribute to the overall response are often nonlinear (e.g., the solubility of organic vapor in the polymer, particularly at high concentrations), the additional nonlinearity introduced by an acoustically thick polymer film does not invariably cause difficulty, provided calibration is carried out over the entire concentration range of interest. There are some situations, however, where an acoustically thick film can cause confusion, such as when (counterintuitive) positive frequency shifts occur as a result of an increase in the concentration of species sorbed by the film due to viscoelastic effects [13] in such cases, the fiequency change vs concentration plot can be multivalued, i.e., two or more very different concentrations of an analyte give an identical frequency shift [14]. 

In contrast, when loaded with a sufficiently thick film, a TSM resonator functions as a rheological probe of the properties of surface-attached species. An acoustically thick film is one whose combination of thickness and shear stiffness is such that the acoustic wave significantly deforms it. Consequently, displacement of the outer regions of the film exhibits a phase delay with respect to displacement of the underlying resonator. Multilayer adsorbates may behave in this manner if they are either intrinsically soft or are plasticized by the permeation of a solvent or other small molecules. This situation is more complicated and it is only recently that the theoretical aspects have been treated in quantitative fashion for fluid-immersed films. Accordingly, this case is described subsequently to the simpler gravimetric case and in greater detail. 

Fig. 6 Summary of data interpretation strategy (described in main text) for extracting viscoelastic parameters from film-loaded TSM resonator frequency response. Input parameters (at left of diagram) are resonator impedance and any selected parameter representative of film thickness (here, charge, interpreted using Faraday s law). Upper part of the scheme relates to acoustically thin films (yielding hf and pf). Lower part of diagram relates to acoustically thick films (yielding, with the help of hf and pf, G and C"). (Reproduced from Ref. [24] with permission from the American Chemical Society.)...

Theoretical treatments for the acoustically thick film case have only recently been developed (see Sect. 2.7.2.1.2) thus this topic has not been reviewed previously. The field is at an interesting stage of development the existence of viscoelastic phenomena in this context are generally accepted, the basic theoretical concepts have been delineated [18-20, 22, 23, 25],... 

Lee, C. C., Tsai, C. S., and Cheng, X. (1985). Complete characterization of thin- and thick-film materials using wideband reflection acoustic microscopy. IEEE Trans. SU-32, 248-58. [207]... 

Figure 3.28 Deformation generated by a SAW (a) in an acoustically thin (R << 1) film, in which in-plane displacement gradients (due to sinusoidal wave variation) dominate, and (b) in an acoustically thick (R 1) film, where cross-film gradients (due to inertial fllm lag) also arise. (Reprinted with permission. See Ref. [50]. O 1994 American CJiemical Society.)...

Next, we focus on a portion of the film that is small in lateral extent compared with the SAW wavelength. As the wave passes a fixed point, the lower surface of the film oscillates in response to the sinusoidal SAW surface displacement. If the film is acoustically thick (/7 1), the upper portions of the film tend to lag behind the driven substrate/film interface, inducing strains across the thickness of the film. This inertial deformation of the film results in nonuniform displacement across the film. 

In the somewhat rare instance that the coating (even when loaded with analyte), remains highly elastic, mass loading may be the only operative interaction mechanism. In this case, as the total coating-plus-anaiyte thickness reaches and exceeds several percent of one acoustic wavelength, the mass sensitivity deviates significantly from that derived from perturbation analyses for acoustically thin films, and is difficult to predict. 

Fig. 9 The acoustic contrast easily saturates. The figure shows a sketch of the contrast function (integrands in Eqs. 78 and 79) as a function of the polymer volume fraction of an adsorbed polymer film. It is assumed that both the shear modulus Gf and the dielectric constant Sf are roughly proportional to the polymer concentration. However, Gf increases much more strongly than gf. If, for instance, a swollen polymer film contains 50% water, this will not appreciably decrease the apparent acoustic thickness because the modulus of the film is still much larger than the modulus of water and (Gf - Gijq)/Gf remains about unity. This is different in optics because the contrast is roughly proportional to the concentration. As a consequence, the apparent optical film thickness is proportional to the product of concentration and thickness, which is the adsorbed amoimt. In acoustics, the apparent thickness is close to the geometric thickness. Trapped water appears as part of the film in acoustics...

However, with such high mass loads, the mechanical stability of the system quartz-crystal deposited thick films decreases. Thus the fact that materials with different elastic properties will obey different mass-frequency relations requiring correction by the acoustic impedance ratio Z = Zq / Zf, eqn. (5), is of less importance practically. 

A flow chart demonstrating the protocol is shown in Fig. 6. The procedure has been demonstrated for poly(3-methylthiophene) films, by analysis of frequency response as a function of time during film electropolymerization short (long) time responses represent the acoustically thin (thick) film scenario [24]. Film mass (whether or not directly accessible from A/ data) defines the product hypy, so (as shown in Fig. 6 [24]) a plot of hy versus py is a hyperbola. As film mass (polymer coverage) increases, a series of hyperbolae are generated. The acoustically thin film data (A/ and Q) define the unique solution (of the infinity of solutions on the hyperbola) for p as indicated in Fig. 6 [24] this value is projected across all the hyperbolae. 

As described in Sect. 2.7.2.1, there is no sharp distinction between rigid and nonrigid behavior, but rather a continuous variation between films that are acoustically thin h < S) and acoustically thick h > S). Since the decay length, S, is a function of shear modulus and thereby of film solvent content, medium effects are critical. The discussion of poly thiophene systems above was deliberately restricted to examples of acoustically thin films, but we now move to a consideration of acoustically thick polythiophene-based films. 

The EQCM also gives the thickness of films, but in terms of an acoustic thickness. Some reports showed that it is very useful to combine EQCM and SPR, as the acoustic thickness can be different from the optical thickness [78-80]. Thus, by knowing both the acoustic and the optical thickness, one can elucidate the mechanisms that govern changing in the structure of adsorbed species in time. This fact is quite important in biological samples, but also for conducting polymers. 

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