# Stratified Layer Systems

Chapter
Part of the Soft and Biological Matter book series (SOBIMA)

## Abstract

Samples, which are homogeneous in the resonator plane, can be modeled as acoustic multilayers. The deformation pattern is a plane wave. Thin films exposed to air behave as predicted by Sauerbrey. For somewhat thicker films, there is a viscoelastic correction scaling as the square of the film’s mass. For films exposed to a liquid, the viscoelastic correction is independent of thickness. If the layer is soft, the correction can be substantial, even for molecularly thin films. Under certain conditions, the film’s elastic compliance, Jf′, can be calculation from the ratio of ΔΓ and (–Δf). Thick films display a film resonance.

## 10.1 Viscoelastic Film in Air

The geometry of the film in air is shown in Fig. 10.1. The acoustic wave is reflected at the outer edge of the film and returns to the resonator surface. The stress at the resonator surface is
\begin{aligned} {\hat{\upsigma }}_{S} & = - \tilde{G}_{f} \left. {\frac{{{\text{d}}\hat{u}\left( z \right)}}{{{\text{d}}z}}} \right|_{{z = d_{q} }} = - \tilde{G}_{f} \left( { - {\text{i}}\tilde{k}_{f} } \right)\left( {\hat{u}_{f}^{ - } - \hat{u}_{f}^{ + } } \right) \\ & = {\text{i}}\tilde{G}_{f} \frac{\upomega}{{\tilde{c}_{f} }}\left( {\hat{u}_{f}^{ - } - \hat{u}_{f}^{ + } } \right) \\ & = {\text{i}}\upomega\tilde{Z}_{f} \left( {\hat{u}_{f}^{ - } - \hat{u}_{f}^{ + } } \right) \\ \end{aligned}
(10.1.1)
Note the minus sign before G̃f in line 1. It occurs because $${\hat{\upsigma }}_{S}$$ is exerted by the resonator onto the sample. For a film in air one has $$\tilde{Z}_{bulk} \approx 0$$ and the reflectivity at the film-air interface is $$\tilde{r}_{f,bulk} = (\tilde{Z}_{f} - \tilde{Z}_{bulk} )/(\tilde{Z}_{f} + \tilde{Z}_{bulk} ) = 1$$. $$\hat{u}_{f}^{ + }$$ then is related to $$\hat{u}_{f}^{ - }$$ by
\begin{aligned} \hat{u}_{f}^{ + } & = \exp \left( { - {\text{i}}\tilde{k}_{f} d_{f} } \right)\tilde{r}_{f,bulk} \exp \left( { - {\text{i}}\tilde{k}_{f} d_{f} } \right)\hat{u}_{f}^{ - } \\ & = \exp \left( { - 2{\text{i}}\tilde{k}_{f} d_{f} } \right)\hat{u}_{f}^{ - } \\ \end{aligned}
(10.1.2)
\begin{aligned} \tilde{Z}_{L} & = \frac{{{\hat{\upsigma }}_{S} }}{{{\hat{\text{v}}}_{S} }} = \frac{{{\text{i}}\upomega\;\tilde{Z}_{f} \left( {\hat{u}_{f}^{ - } - \hat{u}_{f}^{ + } } \right)}}{{{\text{i}}\upomega\left( {\hat{u}_{f}^{ - } + \hat{u}_{f}^{ + } } \right)}} = \tilde{Z}_{f} \frac{{1 - \exp \left( { - 2{\text{i}}\tilde{k}_{f} d_{f} } \right)}}{{1 + \exp \left( { - 2{\text{i}}\tilde{k}_{f} d_{f} } \right)}} \\ & = \tilde{Z}_{f} \frac{{\exp \left( {{\text{i}}\tilde{k}_{f} d_{f} } \right) - \exp \left( { - {\text{i}}\tilde{k}_{f} d_{f} } \right)}}{{\exp \left( {{\text{i}}\tilde{k}_{f} d_{f} } \right) + \exp \left( {{\text{i}}\tilde{k}_{f} d_{f} } \right)}} \\ & = {\text{i}}\tilde{Z}_{f} \tan \left( {\tilde{k}_{f} d_{f} } \right) \\ \end{aligned}
(10.1.3)
The relation $$\tan (x) = - {\text{i}}(\exp ({\text{i}}x) - \exp ( - {\text{i}}x))/(\exp ({\text{i}}x) + \exp ( - {\text{i}}x))$$ was used in line 3. The frequency shift is
$$\frac{{\Delta \tilde{f}}}{{f_{0} }} = \frac{ - 1}{{\uppi\,Z_{q} }}\tilde{Z}_{f} \tan \left( {\tilde{k}_{f} d_{f} } \right)$$
(10.1.4)
In the derivation of Eq. 10.1.4 we have applied the SLA. The corresponding equation derived without invoking the SLA (but still neglecting piezoelectric stiffening) is
$$\tilde{Z}_{q} \tan \left( {\tilde{k}_{q} d_{q} } \right) = - \tilde{Z}_{f} \tan \left( {\tilde{k}_{f} d_{f} } \right)$$
(10.1.5)
Equation 10.1.5 follows from Eqs. 10.1.3 and 4.6.8. Using $$\tilde{k}_{q,ref} \approx \tilde{k}_{q,OC} = n\uppi/d_{q}$$ as well as $$\tan (n\;\uppi + x) = \tan (x)$$ for integer n, this amounts to
$$\tan \left( {\left( {\Delta \tilde{k} + \tilde{k}_{ref} } \right)d_{q} } \right) = \tan \left( {\Delta \tilde{k}d_{q} } \right) = \tan \left( {\frac{{2\uppi d_{q} }}{{\tilde{c}_{q} }}\Delta \tilde{f}} \right) = - \frac{{\tilde{Z}_{f} }}{{\tilde{Z}_{q} }}\tan \left( {\frac{{2\uppi d_{f} }}{{\tilde{c}_{f} }}\tilde{f}} \right)$$
(10.1.6)
Equation 10.1.6 is usually ascribed to Lu and Lewis (Ref. [1], for the extension to layers with finite viscosity see Ref. [2]). In the context of film-thickness monitors, the use of Eq. 10.1.6 is called “Z-match method” [3]. The user provides the wave impedance of the coating to be deposited. Properly accounting for a nontrivial Z-ratio $$(Z_{f} /Z_{q} \ne 1)$$, the instrument calculates a value for the deposited mass, which is more accurate than the Sauerbrey mass. Tabulated values of Zf for some common coating materials can be found in Ref. [4].
It is instructive to rewrite Eq. 10.1.4 as
\begin{aligned} \frac{{\Delta \tilde{f}}}{{f_{0} }} & = \frac{ - 1}{{\uppi\;Z_{q} }}\tilde{Z}_{f} \tan \left( {\tilde{k}_{f} d_{f} } \right) \\ & = \frac{{ - \tilde{Z}_{f} }}{{\uppi\;Z_{q} }}\tan \left( {\frac{\upomega}{{\tilde{c}_{f} }}d_{f} } \right) = \frac{{ - \tilde{Z}_{f} }}{{\uppi\;Z_{q} }}\tan \left( {\frac{{\upomega\sqrt {\uprho_{f} } }}{{\sqrt {\tilde{G}_{f} } }}d_{f} } \right) = \frac{{ - \tilde{Z}_{f} }}{{\uppi\;Z_{q} }}\tan \left( {\frac{\upomega}{{\tilde{Z}_{f} }}\uprho_{f} d_{f} } \right) \\ & = \frac{{ - \tilde{Z}_{f} }}{{\uppi Z_{q} }}\tan \left( {\frac{\upomega }{{\tilde{Z}_{f} }}m_{f} } \right) \\ \end{aligned}
(10.1.7)
Equation 10.1.7 shows that the acoustic properties of a film are fully specified by two parameters, which are its acoustic wave impedance, $$\tilde{Z}_{f}$$, and its mass per unit area, mf. (Strictly speaking, there are three parameters, which are $$Z_{f}^{{\prime }} ,Z_{f}^{{\prime \prime }}$$, and mf.) Equation 10.1.4 misleadingly suggests that one might be able to separately determine the film thickness and the density of the film. Independent information is needed to separate thickness from mass.
Figure 10.2 shows $$\Delta f$$ and $$\Delta \Gamma$$ as predicted by Eq. 10.1.4 versus film thickness. The arrows indicate four different regimes, which are the Sauerbrey regime at very low thickness, the “thin-film regime”, in which the viscoelastic correction is small compared to the Sauerbrey-term, the film resonance(s), [5, 6], and the limit of infinite thickness. In the limit of $$d_{f} \to 0$$, one expands the tangent as $$\tan ({\text{x}}) \approx x$$ and recovers the Sauerbrey equation. In the limit of $$d_{f} \to \infty$$, the complex tangent approaches −i. (Remember that $$k^{\prime\prime} > 0$$ and $$\tilde{k} = k^{\prime} - {\text{i}}k^{\prime\prime}$$, Eq. 4.1.6). Equation 10.1.4 then approaches the Gordon-Kanazawa-Mason result (Eq. 9.1.5).
The maximum in $${\Delta \Gamma }(d_{f} )$$ occurring at $$k_{f} d_{f} =\uppi/2$$ is called “film resonance”. At the film resonance, Δfincreases with increasing thickness. Δf can even turn positive, if the film resonance is sharp enough. The fact that the film resonance is a coupled resonance in the sense of Sect. 4.6.3 becomes clear upon approximating $$\tan (\tilde{k}_{f} d_{f} )$$ by $$(\tilde{k}_{f} d_{f} -\uppi/2)^{ - 1}$$ in the vicinity of $$\tilde{k}_{f} d_{f} =\uppi/2$$:
\begin{aligned} \frac{{\Delta \tilde{f}}}{{f_{0} }} & = \frac{ - 1}{{\uppi\;Z_{q} }}\tilde{Z}_{f} \tan \left( {\tilde{k}_{f} d_{f} } \right) \\ & \approx \frac{ - 1}{{\uppi\;Z_{q} }}\tilde{Z}_{f} \frac{1}{{\tilde{k}_{f} d_{f} - \frac{\uppi}{2}}} \\ & = \frac{{ - \tilde{Z}_{f} }}{{\uppi Z_{q} }}\frac{2}{\uppi }\frac{1}{{\frac{{\uppi \tilde{k}_{f} d_{f} }}{2} - 1}} \\ & = \frac{{\tilde{Z}_{f} }}{{\uppi Z_{q} }}\frac{2}{\uppi }\frac{{\tilde{\upomega }_{FR} }}{{\tilde{\upomega }_{FR} - \upomega }} \\ \end{aligned}
(10.1.8)
The index FR denotes the Film Resonance. The film’s resonance frequency, ωFR, is given by the condition that $${\tilde{\upomega }}_{FR} /\tilde{c}_{f} d_{f} = \tilde{k}_{f} d_{f} =\uppi/2$$. Equation 10.1.8 has the same structure as Eq. 4.6.24.
Figure 10.3a sketches why the film itself forms a resonator at $$k_{f} d_{f} =\uppi/2$$. At $$k_{f} d_{f} =\uppi/2$$, the film thickness is one quarter of the wavelength of sound. The displacement pattern is a standing wave with an antinode at the surface and a node at the interface with the resonator. Because there is a node at the substrate-film interface, a small amplitude at the resonator surface translates to a large amplitude at the film-air interface, which is characteristic of a resonance. The film can be viewed as a viscoelastic plate rigidly clamped on one side. The film is not strictly clamped because the resonator has a finite elastic compliance. However, since most polymer films are much softer than quartz, one can approximate the film as being clamped. If the film is not much softer than quartz, Eq. 10.1.4 must be replaced by Eq. 10.1.6.

If the film has little internal dissipation, the film resonance amounts to a sharp peak in a plot ΔΓ versus df. Note, however, that the SLA does not apply under these conditions. Δf and ΔΓ plotted versus n only form a resonance curve of their own, if the damping factor of the coupled resonance is large enough (cf. Eq. 4.6.26). For samples with low internal dissipation, one has two separate modes around $$k_{f} d_{f} =\uppi/2$$ [7]. Two separate modes have been seen with Langmuir-Blodgett films, sequentially deposited on the resonator surface [8]. One also can find the resonance frequency to discontinuously jump upwards in swelling experiments. In this case the instrument at some point looses the lower branch because the respective resonance becomes to weak. It then searches a resonance and finds the upper branch. Only sufficiently viscous films follow Eq. 10.1.4 and Fig. 10.2.

Figure 10.3b shows the same data as Fig. 10.2, plotted in polar form (ΔΓ versus Δf). Clearly there is a spiral, characteristic for the damped coupled resonance. Contrasting to the case of a sphere on a plate (Sects. 4.6.3 and 11.4), the spiral can make more than one turn. The subsequent turns are the higher overtones of the film resonance. An experimental example, which follows Eq. 10.1.4 particularly well, is shown in Fig. 10.3c [9]. Salomaki and Kankare display their results in polar form and they convert Δf and ΔΓ to $$\tilde{Z}_{L}$$ using Eq. 4.6.1. The authors use the metric Rayl as the unit of the impedance (cf. Sect. 4.2). The metric Rayl is equal to 1 kg m−2 s−1. The agreement between Fig. 10.3c and the model is quite remarkable. Figure 10.3c is a rare example, where the higher overtones of the film resonance are seen. The spiral makes 1.5 turns. (The “higher film resonances” have $$k_{f} d_{f} = m\;\uppi/2$$ with m an odd integer).

Figure 10.4 shows a second experimental example of a film resonance [10]. In this case, a polymer film was swollen in solvent vapor. Upon exposing the film to the vapor phase, it becomes thicker and softer. The highest overtone shown (n = 5 at 25 MHz) reaches the film resonance first because it has the shortest wavelength. The third overtone (15 MH) follows later. For the fundamental, the film resonance is barely reached.
The film resonance experimentally confirms the multilayer formalism, but it is of limited practical relevance, otherwise. The thin-film limit is much more important. Many samples show small viscoelastic effects without showing the film resonance itself. In the “thin film regime”, one expands tan ($$\tilde{k}_{f} d_{f}$$) from Eq. 10.1.4 to third order in $$\tilde{k}_{f} d_{f} (\tan (\varepsilon ) \approx \varepsilon + \varepsilon^{3} /3$$), which results in
\begin{aligned} \frac{{\Delta \tilde{f}}}{{f_{0} }} & = \frac{ - 1}{{\uppi\;Z_{q} }}\tilde{Z}_{f} \tan \left( {\tilde{k}_{f} d_{f} } \right) \approx \frac{ - 1}{{\uppi\;Z_{q} }}\tilde{Z}_{f} \left( {\tilde{k}_{f} d_{f} + \frac{1}{3}\left( {\tilde{k}_{f} d_{f} } \right)^{3} } \right) \\ & = \frac{ - 1}{{\uppi\;Z_{q} }}\upomega\;m_{f} \left[ {1 + \frac{{\left( {n \uppi} \right)^{2} }}{3}\frac{{\tilde{J}_{f} }}{{\uprho_{f} }}Z_{q}^{2} \left( {\frac{{m_{f} }}{{m_{q} }}} \right)^{2} } \right] \\ & = \frac{ - 1}{{\uppi\;Z_{q} }}\upomega\;m_{f} \left[ {1 + \frac{{\left( {n \uppi} \right)^{2} }}{3}\frac{{Z_{q}^{2} }}{{\tilde{Z}_{f}^{2} }}\left( {\frac{{m_{f} }}{{m_{q} }}} \right)^{2} } \right] \\ \end{aligned}
(10.1.9)
The relations $$\tilde{k}_{f} = \upomega /\tilde{c}_{f} = \upomega (\uprho \tilde{J}_{f} )^{1/2}$$ and $$\upomega = 2\uppi nf_{0} = \uppi nZ_{q} /m_{q}$$ were used in line 2. The relation $$\tilde{Z}_{f} = (\uprho_{f} /\tilde{J}_{f} )^{1/2}$$ was used in line 3.
At this point we recall three results from Sects. 6.1.4 and 6.2:
1. (a)

Revisiting the approximations inherent to the SLA, one finds that the term $$(\tilde{J}_{f} /(\uprho_{f} )Z_{q}^{2}$$ needs to be replaced by $$(\tilde{J}_{f} /(\uprho_{f} )Z_{q}^{2} - 1$$:

\begin{aligned} \frac{{\Delta \tilde{f}}}{{f_{0} }} & \approx \frac{{ -\upomega\,m_{f} }}{{\uppi\,Z_{q} }}\left[ {1 + \frac{{\left( {n\uppi} \right)^{2} }}{3}\left( {\frac{{Z_{q}^{2} }}{{Z_{f}^{2} }} - 1} \right)\left( {\frac{{m_{f} }}{{m_{q} }}} \right)^{2} } \right] \\ & = \frac{{ -\upomega\,m_{f} }}{{\uppi\,Z_{q} }}\left[ {1 + \frac{{\left( {n\uppi } \right)^{2} }}{3}\left( {\frac{{\tilde{J}_{f} }}{{\uprho_{f} }}Z_{q}^{2} - 1} \right)\left( {\frac{{m_{f} }}{{m_{q} }}} \right)^{2} } \right] \\ \end{aligned}
(10.1.10)
The difference between Eqs. 10.1.9 and 6.2.8 matters when the film’s acoustic wave impedance is comparable to Zq.
1. (b)

The parameter mq in the denominator is half the plate’s mass within the parallel plate model, but it turns into the “modal mass” when accounting for deviations from that model (Sect. 6.1.4). The true modal mass as a function of overtone order can be determined with thin rigid films. An overtone dependence of Δf/n can only be attributed to viscoelastic effects after the modal mass has been corrected for.

2. (c)

A dependence of Δf/n on n2 can also result from electrode effects. See the text around Eq. 6.2.15 for details.

For small mf/mq (neglecting the second term in square brackets), Eq. 10.1.10 reduces to the Sauerbrey equation. For slightly thicker films, there is a viscoelastic correction proportional to n2, $$m_{f}^{2}$$, and $$\tilde{J}_{f}$$. Importantly, the viscoelastic contribution is large for soft films (with large $$\tilde{J}_{f}$$). The non-gravimetric QCM plays out its strength when the sample is softer than quartz. Viscoelastic effects disappear in the thin-film limit because the film’s own inertia is the only source of shear stress; the sample is not clamped from the other side. It is difficult to determine the softness of a film thinner than about 10 nm and it is quite impossible to determine the softness of a monomolecular adsorbate. This contrasts to the situation in liquids. A liquid exerts a stress onto the film from the outer edge and thereby makes viscoelastic effects visible even for adsorbed monolayers.

A plot adapted to the structure of Eq. 10.1.10 has n2 along the x-axis and the normalized shifts of frequency and half-bandwidth (Δf/n) and ΔΓ/n) along y. An example is shown in Fig. 10.5. If $$\tilde{J}(\upomega)$$ were constant, straight lines would result and the compliances (elastic and viscous) would be proportional to the slopes. However, $$\tilde{J}(\upomega)$$ usually does depend on frequency. For J′′ this is not a problem because $$J^{\prime\prime}(\upomega)$$ can be explicitly derived from the ΔΓ/n. However, there is a problem for the shear compliance because one needs to extrapolate Δf(n) to n = 0 in order to determine the mass, mf. An approximate law for $$J^{\prime}(\upomega)$$ must be assumed in order to do the extrapolation. Typical would be a power law as in Eq. 10.4.1.
For thin films in air, ΔΓ can be converted to $$J^{\prime\prime}$$ using [11]
$$\Delta \Gamma \approx \frac{8}{{3\uprho_{f} Z_{q} }}f_{0}^{4} m_{f}^{3} n^{3}\uppi^{2} J^{\prime\prime} = \frac{{8f_{0}^{4} n^{3}\uppi^{2} }}{{3Z_{q} }}\uprho_{f}^{2} d_{f}^{3} J^{\prime\prime}$$
(10.1.11)
mf can be estimated from the Sauerbrey equation ($$m_{f} \approx - m_{q}\Delta f/f_{r}$$), resulting in
$$\frac{{J_{f}^{\prime \prime } }}{{\uprho_{f} }} \approx \frac{3}{{\left( {n\uppi} \right)^{2} }}\frac{1}{{Z_{q}^{2} }}\left( {\frac{f}{{\Delta f}}} \right)^{2} \left( {\frac{{\Delta \Gamma }}{ - \Delta f}} \right)$$
(10.1.12)

## 10.2 Viscoelastic Film in a Liquid

Start the analysis from Fig. 10.1, but let the bulk medium be a liquid rather than air. Equation 10.1.1 still applies, but the reflectivity at the outer interface now is $$\tilde{r}_{f,bulk} = (\tilde{Z}_{f} - \tilde{Z}_{liq} )/(\tilde{Z}_{f} + \tilde{Z}_{liq} )$$. The amplitude of the reflected wave at the resonator surface, $$\hat{u}^{ + }$$, is given as
$$\hat{u}^{ + } = \exp \left( { - 2i\tilde{k}_{f} d_{f} } \right)\frac{{\tilde{Z}_{f} - \tilde{Z}_{liq} }}{{\tilde{Z}_{f} + \tilde{Z}_{liq} }}\hat{u}^{ - }$$
(10.2.1)
Inserting Eq. 10.2.1 into Eq. 10.1.1 leads to
\begin{aligned} \tilde{Z}_{L} & = \tilde{Z}_{f} \frac{{1 - \exp \left( { - 2{\text{i}}\tilde{k}_{f} d_{f} } \right)\frac{{\tilde{Z}_{f} - \tilde{Z}_{liq} }}{{\tilde{Z}_{f} + \tilde{Z}_{liq} }}}}{{1 + \exp \left( { - 2{\text{i}}\tilde{k}_{f} d_{f} } \right)\frac{{\tilde{Z}_{f} - \tilde{Z}_{liq} }}{{\tilde{Z}_{f} + \tilde{Z}_{liq} }}}} \\ & = \tilde{Z}_{f} \frac{{\tilde{Z}_{f} \left( {\exp \left( {{\text{i}}\tilde{k}_{f} d_{f} } \right) - \exp \left( { - {\text{i}}\tilde{k}_{f} d_{f} } \right)} \right) + \tilde{Z}_{liq} \left( {\exp \left( {{\text{i}}\tilde{k}_{f} d_{f} } \right) + \exp \left( { - {\text{i}}\tilde{k}_{f} d_{f} } \right)} \right)}}{{\tilde{Z}_{f} \left( {\exp \left( {{\text{i}}\tilde{k}_{f} d_{f} } \right) + \exp \left( { - {\text{i}}\tilde{k}_{f} d_{f} } \right)} \right) + \tilde{Z}_{liq} \left( {\exp \left( {{\text{i}}\tilde{k}_{f} d_{f} } \right) - \exp \left( { - {\text{i}}\tilde{k}_{f} d_{f} } \right)} \right)}} \\ & = \tilde{Z}_{f} \frac{{\tilde{Z}_{f} 2{\text{i}}\sin \left( {\tilde{k}_{f} d_{f} } \right) + \tilde{Z}_{liq} 2\cos \left( {\tilde{k}_{f} d_{f} } \right)}}{{\tilde{Z}_{f} 2\cos \left( {\tilde{k}_{f} d_{f} } \right) + \tilde{Z}_{liq} 2{\text{i}}\sin \left( {\tilde{k}_{f} d_{f} } \right)}} \\ \end{aligned}
(10.2.2)
Upon dividing numerator and denominator by $$2\cos (\tilde{k}_{f} d_{f} )$$ and inserting the result into the SLA, one finds [12, 13],
$$\frac{{\Delta \tilde{f}}}{{f_{0} }} = \frac{{ - \tilde{Z}_{f} }}{{\uppi Z_{q} }}\frac{{\tilde{Z}_{f} \tan \left( {\tilde{k}_{f} d_{f} } \right) - {\text{i}}\tilde{Z}_{liq} }}{{\tilde{Z}_{f} + {\text{i}}\tilde{Z}_{liq} \tan \left( {\tilde{k}_{f} d_{f} } \right)}}$$
(10.2.3)
As in Eq. 10.1.7, one can express $$\tilde{k}_{f} d_{f}$$ as $$(\upomega/\tilde{Z}_{f} )d_{f}$$, which shows that the thickness and the mass per unit area of a film cannot be derived independently. With regard to shear waves, the properties of a film are fully specified by the parameters $$\tilde{Z}_{f}$$ and mf.
Similar to the film in air, the Gordon-Kanazawa-Mason result is recovered in the limit of $$d_{f} \to \infty$$. At intermediate thickness, there again is a film resonance. The film resonance has been seen in liquids; Fig. 10.6 shows an example. The importance of the film resonance in liquid sensing is limited and the discussion therefore continues with the thin-film limit.
Expanding Eq. 10.2.3 to first order in df, one finds
$$\frac{{\Delta \tilde{f}}}{{f_{0} }} \approx \frac{\text{i}}{{\uppi\,Z_{q} }}\left[ {\tilde{Z}_{liq} + {\text{i}}\tilde{Z}_{f} \tilde{k}_{f} d_{f} \left( {1 - \frac{{\tilde{Z}_{liq}^{2} }}{{\tilde{Z}_{f}^{2} }}} \right)} \right]$$
(10.2.4)
The two terms can be attributed to the bulk viscosity and to the adsorbed mass. For a sufficiently stiff film $$(|\tilde{Z}_{f}^{2} | \gg |\tilde{Z}_{liq}^{2} |)^{2}$$ the second term in square brackets is the Sauerbrey term:
\begin{aligned} \frac{{\Delta \tilde{f}}}{{f_{0} }} & \approx \frac{\text{i}}{{\uppi\,Z_{q} }}\left( {\tilde{Z}_{liq} + {\text{i}}\tilde{Z}_{f} \tilde{k}_{f} d_{f} } \right) = \frac{\text{i}}{{\uppi\,Z_{q} }}\left( {\sqrt {{\text{i}}\upomega \uprho _{liq} \tilde{\upeta }_{liq} } + {\text{i}}\upomega\,m_{f} } \right) \\ & = \frac{{ - 1 + {\text{i}}}}{{\sqrt 2\uppi\,Z_{q} }}\sqrt {\upomega \uprho _{liq} \tilde{\upeta }_{liq} } - \frac{{2nf_{0} }}{{Z_{q} }}m_{f} \\ \end{aligned}
(10.2.5)
The frequency shift is a sum of a viscous contribution (following the Gordon-Kanazawa-Mason equation) and an inertial contribution (following Sauerbrey). For stiff films the QCM works as a gravimetric sensor even in the liquid phase. Additivity of the two contributions was found experimentally in Ref. [14].
The way in which viscoelasticity enters the equations of the thin-film limit deserves a closer look. The frequency shift is usually determined with respect to a reference state, where the quartz is already immersed in liquid. Referencing the measurement to the bare quartz in the liquid, one obtains [13, 14, 15, 16]
\begin{aligned} \frac{{\Delta \tilde{f}}}{{f_{0} }} & \approx \frac{\text{i}}{{\uppi\,Z_{q} }}(\tilde{Z}_{tot} - \tilde{Z}_{liq} ) = \frac{ - 1}{{\uppi\,Z_{q} }}\tilde{Z}_{f} \tilde{k}_{f} d_{f} \left[ {1 - \frac{{\tilde{Z}_{liq}^{2} }}{{\tilde{Z}_{f}^{2} }}} \right] \\ & = \frac{{ -\upomega\,m_{f} }}{{\uppi\,Z_{q} }}\left[ {1 - \frac{{\tilde{Z}_{liq}^{2} }}{{\tilde{Z}_{f}^{2} }}} \right] = \frac{{ -\upomega\,m_{f} }}{{\uppi\,Z_{q} }}\left[ {1 - \frac{{\tilde{J}_{f} \left(\upomega \right)}}{{\uprho_{f} }}{\text{i}}\upomega \uprho _{liq}\upeta_{liq} } \right] \\ & = \frac{{ -\upomega\,m_{f} }}{{\uppi\,Z_{q} }}\left[ {1 - 2\uppi\,in\frac{{\tilde{J}_{f} \left(\upomega \right)}}{{\uprho_{f} }}f_{0}\uprho_{liq}\upeta_{liq} } \right] \\ \end{aligned}
(10.2.6)
The prefactor has the same form as the right-hand side of the Sauerbrey equation; the term in square brackets is the viscoelastic correction. As opposed to experiments air, softness increases the resonance frequency. It decreases the apparent Sauerbrey mass (the mass obtained when naively analyzing experimental data with the Sauerbrey equation).
An experimental example is shown in Fig. 10.7. These data were obtained on a layer of adsorbed vesicles. The details are unessential. Of course vesicles are discrete objects, but the layer appears like a soft film to the QCM. Characteristically for a soft film in a liquid, the plot of Δf/n versus n slopes upwards. The larger the slope, the larger is the sample’s viscous (!) compliance. Note: When a film is immersed in a liquid, a finite elastic compliance mostly affects the bandwidth, while a finite viscous compliance mostly affects the frequency. It is the other way round in air.

Contrasting to the dry state, the viscoelastic correction is independent of film thickness in a liquid environment. For films in air the viscoelastic correction scales as the square of the mass (Eq. 10.1.9) because the film surface is stress-free in air. The film only shears under its own weight. This is different in liquids because the liquids exerts a stress onto the film. The viscoelastic correction may be sizeable, even for molecularly thin films.

Equation 10.2.6 predicts –Δf to increase with overtone order, as shown in Fig. 10.7. In experiment, one can also find –Δf decreasing with n. The reasons are unclear; roughness may play a role.

Because Eq. 10.2.6 is linear in mass, it also holds in an integral sense [17];
$$\frac{{\Delta \tilde{f}}}{{f_{0} }} \approx - \frac{{ -\upomega}}{{\uppi\,Z_{q} }}\int\limits_{0}^{\infty } {\left( {\frac{{\tilde{Z}^{2} (z) - \tilde{Z}_{liq}^{2} }}{{\tilde{Z}^{2} (z)}}} \right)\uprho(z){\text{d}}z} \approx - \frac{{\uprho_{liq}\upomega}}{{\uppi\,Z_{q} }}\int\limits_{0}^{\infty } {\left( {\frac{{\tilde{G}(z) - \tilde{G}_{liq} }}{{\tilde{G}(z)}}} \right){\text{d}}z}$$
(10.2.7)
In step 2, ρ(z) was assumed to be about constant and equal to ρliq. Equation 10.2.7 will be central importance in Sects. 10.6 and 16.2.
Two notes concerning the literature:
• The Voigt model from Ref. [18] is equivalent to the acoustic multilayer formalism. For the proof see the appendix of Ref. [13]. Although not readily apparent, the “Voigt model” also makes use of the SLA. At the end of the derivation, Voinova et al. invoke an approximation from a previous paper of the group (Ref. [19]), which essentially is the SLA.

• The “missing-mass effect” from Ref. [15] is equivalent to the viscoelastic correction in Eq. 10.2.6 (Ref. [11]).

## 10.3 Equivalent Mass and Equivalent Thickness

The interpretation of QCM measurements in liquids is somewhat of a challenge. Practitioners often just apply the Sauerbrey equation (Eq. 8.1.2) to their data and call the resulting areal mass density the “Sauerbrey mass”. The corresponding thickness is the “Sauerbrey thickness”. The Sauerbrey thickness can certainly serve to compare different experiments, but it must not be naively identified with the geometric thickness. Here is a list of worthwhile considerations:
1. (a)

The QCM always measures a film’s areal mass density, never its geometric thickness. The conversion from areal mass density to thickness requires the physical density as independent input.

2. (b)

One can account for the viscoelastic correction by fitting the experimental sets of {Δf(n)}and {ΔΓ(n)}, with in Eq. 10.2.6. The parameter mf then is one of the fit parameters. The fits are not trivial though, there sometimes is a question with regard to uniqueness.

3. (c)

If full-fledged fitting shall be avoided, one can extrapolate the function Δf(n) to n = 0. At n = 0, the viscoelastic correction disappears.

4. (d)

If the correction factor is significantly different from unity, it also affects the bandwidth and makes −Δf/n depend on overtone order. If, conversely, such effects are absent (if $$\Delta \Gamma \ll ( -\Delta f) \, {\rm{and}} \, \Delta f/n \approx const)$$, one may assume $$1 - \left| {\tilde{Z}_{liq}^{2} /\tilde{Z}_{f}^{2} } \right| \approx 1$$. The Sauerbrey equation then is trustworthy.

5. (e)

If the viscoelastic correction is insignificant as discussed in (c), this does not imply that the film would not be swollen in the ambient liquid. It only implies that the (swollen) film is much more rigid than the ambient liquid. The amount of swelling can only be inferred from the comparison of wet and dry thickness. QCM data taken on the wet sample alone do not allow to infer the degree of swelling (cf. Sect. 16.2).

6. (f)

Soft samples often have a fuzzy interface with the ambient liquid. Fuzzy interfaces usually are soft interfaces. They will induce significant viscoelastic effects (ΔΓ > 0, Δf/n a function of n). In the absence of such effects, one may conclude that the border of the film with the liquid is sharp.

7. (g)

The degree of swelling of a film in aqueous solution can be inferred from H2O/D2O exchange experiments, see Ref. [20].

## 10.4 Determination of Viscoelastic Constants

In principle, both $$J_{f}^{{\prime }}$$ and $$J_{f}^{{\prime \prime }}$$ can be extracted from the n-dependence of Δf and ΔΓ. A few difficulties must be kept in mind, though. Firstly and importantly, $$J_{f}^{{\prime }}$$ and $$J_{f}^{{\prime \prime }}$$ themselves depend on n. Secondly, $$J_{f}^{{\prime }}$$ and $$J_{f}^{{\prime \prime }}$$ cannot be derived explicitly because the mass is not a priori known. Finally, films in a liquid environment can show an overtone dependence of Δf/n because of roughness (Sects. 9.5 and 12.2). (And of course there is always the possibility of compressional waves having an influence.)

Rheological spectra usually are smooth. They are displayed on logarithmic scales (Sect. 3.7). Since the experimentally accessible range of frequencies is only about a decade, it is fair to approximate the frequency dependence of $$J_{f}^{{\prime }}$$ and $$J_{f}^{{\prime \prime }}$$ by a power law with power law exponents $${\upbeta^{\prime}}$$ and $${\upbeta^{\prime\prime}}$$
\begin{aligned} J_{f}^{{\prime }} \left( f \right) & \approx J_{f}^{{\prime }} \left( {f_{ref} } \right)\left( {\frac{f}{{f_{ref} }}} \right)^{{{\upbeta^{\prime}}}} \\ J_{f}^{{\prime \prime }} \left( f \right) & \approx J_{f}^{{\prime \prime }} \left( {f_{ref} } \right)\left( {\frac{f}{{f_{ref} }}} \right)^{{{\upbeta^{\prime\prime}}}} \\ \end{aligned}
(10.4.1)
Note: The index “ref” here denotes a frequency somewhere in the middle of the frequency range of the QCM (as opposed to a reference state of the crystal without a sample). Power laws analogous to Eq. 10.4.1 can be formulated for other choices of viscoelastic parameters ($$G_{f}^{{\prime }}$$ and $$G_{f}^{{\prime \prime }}$$ or $$G_{f}^{{\prime }}$$ and $$\upeta_{f}^{{\prime }}$$). Unfortunately, power laws in $$J_{f}^{{\prime }}$$ and $$J_{f}^{{\prime \prime }}$$ do not turn into power laws in $$G_{f}^{{\prime }}$$ and $$G_{f}^{{\prime \prime }}$$ when transformed with Eq. 3.7.4. A set of power laws in $$J_{f}^{{\prime }}$$ and $$J_{f}^{{\prime \prime }}$$ therefore is not strictly equivalent to a similar set in $$G_{f}^{{\prime }}$$ and $$G_{f}^{{\prime \prime }}$$.

There are limits on $$\upbeta^{{\prime }}$$ and $$\upbeta^{{\prime \prime }}$$. If expressing viscoelasticity in terms of compliance ($$J_{f}^{{\prime }}$$ and $$J_{f}^{{\prime \prime }}$$ as in Eq. 10.4.1), one has $$- 2 <\upbeta^{{\prime }} < 0$$ and $$- 1 <\upbeta^{{\prime \prime }} < 1$$. If moduli are used ($$G_{f}^{{\prime }}$$ and $$G_{f}^{{\prime \prime }}$$), one has $$0 <\upbeta^{{\prime }} < 2$$ and $$- 1 <\upbeta^{{\prime \prime }} < 1$$. Users of the QCM-D often express viscoelasticity in terms of $$\upmu = G_{f}^{{\prime }}$$ and $$\upeta = G_{f}^{{\prime \prime }} /\upomega$$. The power law exponent for η is between −2 and 0.

## 10.5 The ΔΓ/Δf-Ratio and a Film’s Elastic Compliance

The term in square brackets in Eq. 10.2.6 is the viscoelastic correction to the Sauerbrey equation. The film thickness only enters as a prefactor. In the limit of thin films, the viscoelastic correction is independent of film thickness and the thickness can be eliminated from the equations by taking the ratio ΔΓ to (–Δf) (the “ΔΓ/Δf-ratio”). The ΔΓ/Δf-ratio is an intrinsic property of the adsorbed material. It is related to the material’s density and viscoelastic constants as
$$\frac{{\Delta \Gamma }}{{ -\Delta f}} = \frac{{ - \text{Im} \left( {1 - \tilde{Z}_{liq}^{2} /\tilde{Z}_{f}^{2} } \right)}}{{\text{Re} \left( {1 - \tilde{Z}_{liq}^{2} /\tilde{Z}_{f}^{2} } \right)}}$$
(10.5.1)
It turns out that the ΔΓ/Δf-ratio mostly is a measure of a film’s softness. To see this more quantitatively, we make further approximations. Assume the liquid to be Newtonian. The shear-wave impedance of this liquid is $$\tilde{Z}_{liq} = ({\text{i}}\upomega \uprho _{liq}\upeta_{liq} )^{1/2}$$, which leads to
$$\frac{{\Delta \Gamma }}{{ -\Delta f}} = \frac{{ - \text{Im} \left( {1 - {{\text{i}\upomega \uprho _{liq}\upeta_{liq} \tilde{J}_{f} } \mathord{\left/ {\vphantom {{\text{i}\upomega \uprho _{liq}\upeta_{liq} \tilde{J}_{f} } {\uprho_{f} }}} \right. \kern-0pt} {\uprho_{f} }}} \right)}}{{\text{Re} \left( {1 - {{\text{i}\upomega \uprho _{liq}\upeta_{liq} \tilde{J}_{f} } \mathord{\left/ {\vphantom {{\text{i}\upomega \uprho _{liq}\upeta_{liq} \tilde{J}_{f} } {\uprho_{f} }}} \right. \kern-0pt} {\uprho_{f} }}} \right)}} = \frac{{\upomega \uprho _{liq}\upeta_{liq} J_{f}^{{\prime }} }}{{\uprho_{f} - \upomega \uprho _{liq}\upeta_{liq} J_{f}^{{\prime \prime }} }}$$
(10.5.2)
The densities in soft-matter experiments often are similar. Approximating ρf by ρliq yields
$$\frac{{\Delta \Gamma }}{{ -\Delta f}} \approx \frac{{\upomega \upeta _{liq} J_{f}^{{\prime }} }}{{1 - \upomega \upeta _{liq} J_{f}^{{\prime \prime }} }}$$
(10.5.3)
Most films of interest are substantially more rigid than the ambient liquid. Even films, which are soft at low frequencies, often appear as rigid in the MHz range. The denominator in Eq. 10.5.3 can be rewritten as $$1 - J_{f}^{{\prime \prime }} /J_{liq}^{{\prime \prime }}$$ where $$J_{liq}^{{\prime \prime }} = (\upomega \upeta _{liq} )^{ - 1}$$ is the viscous compliance of the liquid. If the viscous compliance of the film is much smaller than the viscous compliance of the liquid (given as $$(\upomega \upeta _{liq} )^{ - 1}$$), the denominator is close to unity and one has [17]
$$\frac{{\Delta \Gamma }}{{ -\Delta f}} \approx\upomega \upeta _{liq} J^{\prime}_{f} = 2\uppi\,nf_{0}\upeta_{liq} J^{\prime}_{f}$$
(10.5.4)
This is one of the cases, where the shear compliance as a parameter quantifying viscoelastic response is more convenient than the shear modulus. One can easily determine $$J^{\prime}_{f}$$ from experiment, but not $$J^{\prime\prime}_{f}$$. Because the conversion to shear modulus requires knowledge of both $$J^{\prime}_{f}$$ and $$J^{\prime\prime}_{f}$$ (Eq. 3.7.4), neither $$G^{\prime}_{f}$$ nor $$G^{\prime\prime}_{f}$$ can be derived (but $$J^{\prime}_{f}$$ can).
If the density of the film and the liquid are unequal, one can still neglect the second term in the denominator in Eq. 10.5.3, which leads to
$$\frac{{\Delta \Gamma }}{{ -\Delta f}} \approx \frac{{\uprho_{liq} }}{{\uprho_{f} }}\upeta_{liq}\upomega\,J^{\prime}_{f} = \frac{{\uprho_{liq} }}{{\uprho_{f} }}2\uppi\,nf_{0}\upeta_{liq} J^{\prime}_{f} = \text{Re} \left( {\frac{{\left| {Z_{liq}^{2} } \right|}}{{\tilde{Z}_{f}^{2} }}} \right)$$
(10.5.5)
Again: Eq. 10.5.1 only holds for thin films. The detailed calculations show that Eqs. 10.5.4 and 10.5.5 should only be applied if the thickness is a few nanometers at most.

The Gizeli group has carried this concept even further [21, 22]. They take the ratio ΔD/(–ΔF) (D the dissipation factor and ΔF = Δf/n the normalized frequency shift) and call this parameter the “acoustic ratio”. While the acoustic ratio clearly is equivalent to what is called ΔΓ/Δf above, the meaning given to it by the Gizeli group is different. These workers attached DNA strands to the QCM surface. They find that acoustic ratio to be constant up to a certain limit in coverage. From the fact that ΔD/(–ΔF) is independent of coverage, they conclude that the acoustic ratio is a property of the molecules under investigation (as opposed to a property of the layer). Evidently, this argument only holds in the range where ΔD/(−ΔF) is indeed independent of ΔF. The acoustic ratio was found to depend on the length of the DNA strands adsorbed. It increased with increasing length. In the discussion, the group draws an analogy to the intrinsic viscosity of a dilute solution, [η], which is defined as [η] = (η−η0)/(η0c), to be evaluated in the limit of low c. η0 is the viscosity of the pure solvent, c is the concentration of the solute. In the limit of low concentration, intermolecular interactions do not affect the viscosity and [η] is a property of the solute/solvent pair.

Two more remarks:
• From a mathematical point of view, it is not necessarily clear that the ΔΓ/Δf-ratio should be proportional to the intrinsic viscosity of the adsorbate molecules. The Gizeli group has established experimentally that there is a correlation between the ΔΓ/Δf-ratio and the intrinsic viscosity of the same molecules in the bulk. In principle, the ΔΓ/Δf-ratio might relate to other molecular parameters, as well. That would not invalidate the argument as such. Whatever the parameter is, it would still be a property of the solute/solvent pair.

• A ΔΓ/Δf-ratio independent of −Δf is not always observed. Figure 12.8 shows counter examples. The ΔΓ/Δf-ratio only is a property of the molecules involved, as long as it is independent of coverage.

## 10.6 Multilayers and Continuous Viscoelastic Profiles

For samples consisting of multiple layers, the explicit equations predicting $$\Delta \tilde{f}$$ become long and complicated. For two layers in a liquid, the equation still fits into one line, which is [6]:
$$\frac{{\Delta \tilde{f}}}{{f_{0} }} = \frac{{ - \tilde{Z}_{1} }}{{\uppi Z_{q} }}\frac{{\tilde{Z}_{2} \left( {\tilde{Z}_{1} \tan \left( {\tilde{k}_{1} d_{1} } \right) + \tilde{Z}_{2} \tan \left( {\tilde{k}_{2} d_{2} } \right)} \right) + {\text{i}}\tilde{Z}_{liq} \left( {\tilde{Z}_{1} \tan \left( {\tilde{k}_{2} d_{2} } \right)\tan \left( {\tilde{k}_{1} d_{1} } \right) - \tilde{Z}_{2} } \right)}}{{\tilde{Z}_{2} \left( {\tilde{Z}_{1} - \tilde{Z}_{2} \tan \left( {\tilde{k}_{2} d_{2} } \right)\,\tan \left( {\tilde{k}_{1} d_{1} } \right)} \right) + {\text{i}}\tilde{Z}_{liq} \left( {\tilde{Z}_{1} \tan \left( {\tilde{k}_{2} d_{2} } \right) + \tilde{Z}_{2} \tan \left( {\tilde{k}_{1} d_{1} } \right)} \right)}}$$
(10.6.1)
The indices 1, 2, and liq denote the two films and the liquid, respectively. The author has made a software package for the analysis of multilayers available on the web [23]. This software calculates Δf for up to 4 layers. Of course the uniqueness of the fit must be checked carefully.
Continuous viscoelastic profiles can be approximated by a sequence of many thin layers, evaluated with the matrix method (Sect. 1.5.3). There is a second method. It is not necessarily difficult, but it requires the use of one of the advanced mathematical software packages like Mathematica or MATLAB. These can solve ordinary differential equations numerically. The respective command is a one-liner. If the functions $$\uprho(z),G^{\prime}(z)$$, and $$G^{\prime\prime}(z)$$ are given, one can calculate the displacement $$\hat{u}(z)$$ with Mathematica and infer the stress-velocity ratio from $$\hat{u}(z = 0)$$ and $${\text{d}}\hat{u}/{\text{d}}z(z = 0)$$ One has
$$Z_{L} = \frac{{\hat{\upsigma }_{S} }}{{\hat{\rm{v}}}_{S} } = \frac{{ - \tilde{G}\left. {\frac{\text{d}}{{{\text{d}}z}}\hat{u}\left( z \right)} \right|_{z = 0} }}{{\text{i}}\upomega\left. {\hat{u}\left( z \right)} \right|_{z = 0} }$$
(10.6.2)
$$\hat{u}(z)$$ is the solution to the wave equation, which in this case is
$$-\uprho\left( z \right)\upomega^{2} \hat{u}\left( z \right) = \frac{{{\rm{d}}\hat{\upsigma }}}{{\rm{d}}z} = \frac{{\rm{d}}}{{\rm{d}}z}\left( {\tilde{G}\left( z \right)\frac{{{\rm{d}}\hat{u}\left( z \right)}}{{\rm{d}}z}} \right)$$
(10.6.3)
Note that the shear modulus must be inside the second derivative in case $$\tilde{G}$$ itself is a function of z.

Mathematica code implementing this method is provided in Sect. 10.8. (This section also discusses the initial conditions). The input to such a model consists of two steps. First, an assumption is needed on the dependence of the polymer volume fraction on distance, φ(z), where φ is the polymer volume fraction. More difficult is the question of how the shear modulus should depend on φ. Remember: The model needs the shear modulus at MHz frequencies. The low frequency shear modulus might be determined with conventional rheology on bulk solutions of the respective material, but such low-frequency moduli are of little help. Input from polymer physics is needed. What is written down in Sect. 10.8 is just a guess.

If the overall thickness of the sample is much below the penetration depth, there are relations between the complex frequency shifts and certain integrals. This limit was called the thin-film limit in Sect. 10.2. Another term used in optics is the “long-wavelength limit” [24]. Applying Eq. 10.2.7 to viscoelastic profiles and assuming ρ(z) ≈ ρliq, one finds
\begin{aligned} \frac{{\Delta f}}{{f_{0} }} & \approx - \frac{{\uprho_{liq}\upomega}}{{\uppi\,Z_{q} }}\int\limits_{0}^{\infty } {\left( {1 -\upomega \upeta _{liq} J''(z)} \right){\rm{d}}z} \\ \frac{{\Delta \Gamma }}{{f_{0} }} & \approx \frac{{\uprho_{liq}\upomega}}{{\uppi\,Z_{q} }}\int\limits_{0}^{\infty } {\upomega \upeta _{liq} J'\left( z \right){\rm{d}}z} \\ \end{aligned}
(10.6.4)
The Sauerbrey thickness is given as
$$d_{app} = - \frac{1}{{\uprho_{liq} }}\frac{{Z_{q} }}{{2nf_{0}^{2} }}\Delta f \approx \int\limits_{0}^{\infty } {\left( {1 -\upomega \upeta _{liq} J^{\prime\prime}\left( z \right)} \right){\rm{d}}z}$$
(10.6.5)
The index app denotes the apparent thickness. Based on the ΔΓ/Δf-ratio (Sect. 10.5), one can define an apparent elastic compliance as $$J^{\prime}_{app} =\Delta \Gamma /( -\Delta f)/(\upomega \upeta _{liq} )$$. The apparent elastic compliance is related to the viscoelastic profile as
$$J^{\prime}_{app} = \frac{1}{{\upomega \upeta _{liq} }}\left( {\frac{{\Delta \Gamma }}{{ -\Delta f}}} \right) \approx \frac{{\int\limits_{0}^{\infty } {J^{\prime}\left( z \right){\rm{d}}z} }}{{\int\limits_{0}^{\infty } {1 - \upomega \upeta _{liq} J^{\prime\prime}\left( z \right){\rm{d}}z} }}$$
(10.6.6)

## 10.7 Slip

Slippage at an interface between a solid and a simple liquid has for a long time been regarded as a misunderstanding. As the textbooks explain, the solid-liquid interface obeys a no-slip condition because the attraction between the small molecules forming the liquid phase and a wall is at least as strong as the mutual attraction between the small molecules [25, 26]. At first glance, there appears to be little reason, why a material should flow more easily close to a wall than in the bulk. On the contrary: Adsorption of molecules to the surface should enhance the local viscosity. There are many examples of solidified layers of liquid molecules next to a solid surface. The opposite phenomenon, a layer of highly mobile molecules next to a wall, was long considered an exception, if not a phantom.

Today there is solid experimental evidence for slip even in simple liquids [27, 28, 29, 30, 31]. Different mechanisms have been put forward. For instance, nanobubbles can facilitate a flow. The hydrophobic interaction can also reduce the density of water near a hydrophobic surface, thereby reducing the local viscosity [29].

In passing we mention two other meanings of slip, different from what is discussed below. Firstly, slip can mean “solid-like sliding” in the sense of friction and tribology [32]. Some aspects of tribology with relevance to the QCM are discussed in Sect. 12.1. Secondly, slip can denote interfacial shear-thinning in complex fluids [33, 34]. A well-known example is tooth paste. When tooth paste is squeezed out of the tube, it experiences plug flow, meaning that bulk of the material remains undeformed and the shear gradient is concentrated close to the wall. In tooth paste, large stress causes structural transformations, which lower the viscosity. On start-up, the pressure-driven flow has a parabolic profile. Soon afterwards, the structural transformation sets in close to the wall, where the shear gradient is largest. There is positive feed-back, eventually leading to strong shear-thinning at the wall and movement without internal shear everywhere else. MHz shear waves are not suited to study shear-thinning in complex fluids. Firstly, the structural transformations are too slow to be induced by MHz excitation. Also, the fluids giving rise to plug flow all are viscous to the extent that they overdamp the resonator’s vibration. Further, the stress required to induce shear-thinning is too large to be exerted by a QCM.

We now return to simple liquids. Slip in simple liquids is of practical relevance, whenever the flow occurs on small spatial scales. An obvious example is flow in porous media [35], where the pore size may easily be comparable to the slip length. (For the definition of the slip length see Fig. 10.8). A second example is sedimentation [36], where the slip length must be compared to the particle diameter. Nanobubbles and slip also are of importance in electrochemistry, key words being gas diffusion electrodes [37] and electrophoretic deposition [38].

In simple liquids, the layer of anomalous viscosity (be it decreased or increased) is few nanometers thick, at most. Only molecules in the vicinity of the surface feel the effect, which the surface has on the material’s density and dynamics. Unless the viscosity in the denominator of Eq. 10.7.1 is close to zero, the slip length is a few nanometers, as well, and the measurement of the slip length by conventional macroscopic techniques is difficult. As has been pointed out by numerous authors, MHz shear waves can be exploited to this end [39, 40, 41, 42, 43]. Their penetration depth is around 100 nm. If the slip length is—for instance—1 % of the penetration depth, slip is readily detected.

The continuum model of slip in simple liquids amounts to a layer of decreased viscosity near a surface. The slip length, bsl, is defined by extrapolating the linear portion of the flow profile, v(z), to the plane of zero shear (Fig. 10.8) [44]. Again, the near-surface viscosity may be decreased or increased. It can be increased by a layer of adsorbed molecules. The distance between the surface and the plane of zero shear then (by definition) is the hydrodynamic thickness of the respective layer [45]. The slip length can be viewed as an apparent negative hydrodynamic thickness.

For steady flows, the slip length is given by [29]
$$b_{sl} = \left( {\frac{{\upeta_{liq} }}{{\upeta_{SL} }} - 1} \right)d_{SL}$$
(10.7.1)
dSL is the thickness of the layer with reduced viscosity (the Slipping Layer, solid line in Fig. 10.8) and ηSL is the viscosity inside this layer. Since 10.7.1 is linear in dSL, it also holds for continuous profiles, η(z), (dashed line in Fig. 10.8) in an integral sense:
$$b_{sl} = \int\limits_{0}^{\infty } {\left( {\frac{{\upeta_{liq} }}{{\upeta\left( z \right)}} - 1} \right)} {\text{d}}z.$$
(10.7.2)
The frequency shift induced by a layer with reduced viscosity can be predicted from an adapted version of Eq. 10.2.7, which is [46]:
\begin{aligned} \frac{{\Delta \tilde{f}}}{{f_{0} }} & = - \frac{2{\upomega}}{{{\pi}Z_{q} }}\int\limits_{0}^{\infty } {\left( {1 - \frac{{Z_{liq}^{2} }}{{\tilde{Z}^{2} (z)}}} \right)\uprho(z){\text{d}}z} \\ & = - \frac{2f}{{Z_{q} }}\int\limits_{0}^{\infty } {\left( {1 - \frac{{{\text{i}}\upomega \uprho _{liq}\upeta_{liq} }}{{{\text{i}}\upomega \uprho (z)\upeta(z)}}} \right)\uprho(z){\text{d}}z} \\ & = \frac{2f}{{Z_{q} }}\uprho_{liq} \int\limits_{0}^{\infty } {\left( {\frac{{\upeta_{liq} }}{{\upeta(z)}} - \frac{{\uprho(z)}}{{\uprho_{liq} }}} \right){\text{d}}z} \\ \end{aligned}
(10.7.3)
If the density of the slipping layer is the same as the density of the bulk liquid, (ρ(z) ≈ ρliq), one has
$$\frac{{\Delta \tilde{f}}}{{f_{0} }} \approx \frac{{2f\uprho_{liq} }}{{Z_{q} }}\int\limits_{0}^{\infty } {\left( {\frac{{\upeta_{liq} }}{{\upeta(z)}} - 1} \right){\text{d}}z} \approx \frac{{2f\uprho_{liq} }}{{Z_{q} }}b_{sl}$$
(10.7.4)
With constant density, slip looks like a negative Sauerbrey mass, where the slip length is equal to the negative Sauerbrey thickness.
If the densities of the slipping layer and the bulk liquid are unequal, the situation is more complicated. There is an “acoustic slip length”, bsl,ac, given as
$$b_{sl,ac} = \int\limits_{0}^{\infty } {\left( {\frac{{\upeta_{liq} }}{{\upeta\left( z \right)}} - \frac{{\uprho\left( z \right)}}{{\uprho_{liq} }}} \right) \, } {\text{d}}z$$
(10.7.5)
When slip is caused by a nanobubbles, the slipping layer’s average density is less than the density of the bulk. The acoustic slip length then is different from the conventional slip length.

Equations 10.7.2 and 10.7.4 ignore roughness and heterogeneities [47]. Bubbles and nanobubbles do constitute such heterogeneities. Nanobubbles do not necessarily induce slip. Since nanobubbles constitute a laterally structured sample, we defer their discussion to Sect. 12.4.

## 10.8 Appendix: Modeling Viscoelastic Profiles Using the Wave Equation

In Sect. 10.6 it was stated that the complex frequency shift induced by a sample with continuous $$\tilde{G}(z)$$ and ρ(z) can be found by solving the wave equation numerically. The calculation amounts to a few lines of code in Mathematica and we provide an example below.

The following assumptions were made to set up the model:
• The polymer volume fraction was chosen as φ = φ0 1/2(1 − tanh(z − zi)/w)). φ0 is the polymer volume fraction at z = −∞, zi is the point of inflection, and w is the width of the interface.

• The density, ρ, was assumed to be constant and equal to 1 g/cm3.

• The shear modulus was assumed to obey a Maxwell model, meaning that $$\tilde{G}$$ is given as $$\tilde{G} = {\text{i}}\upomega \upeta _{liq} + G_{\infty } {\text{i}}\upomega \uptau /(1 + {\text{i}}\upomega \uptau )$$. G is the elastic shear modulus at infinite frequency and τ is a relaxation time. τ was made to depend on polymer volume fraction as τ = 0.1 ns × 103 φ/φ0. At z = ∞, the shear modulus equals the (complex) shear modulus of the bulk liquid.

There is subtlety with regard to the initial condition of the problem, which is the amplitude at infinity (where “infinity” is some value zmax far outside the region of interest). Mathematica integrates the wave equation from zmax down to the resonator surface. Since the wave equation contains the second derivative, initial conditions are needed for both $$\hat{u}$$ and $${\text{d}}\hat{u}/{\text{d}}z$$ at z = zmax. It turns out that all initial conditions lead to the same frequency shift. Consider $$\hat{u}(z_{\hbox{max} } )$$ first. Because what is analyzed eventually is the ratio of stress to velocity, $$\hat{\upsigma }_{S} /\hat{\rm{v}}_{S}$$, one is free to pick the amplitude $$\hat{u}(z_{\hbox{max} } )$$ arbitrarily. Multiplying $$\hat{u}(z)$$ by some complex number does not change the ratio of $$\hat{\upsigma }_{S} /\hat{v}_{S}$$. Somewhat surprisingly, the value of the first derivative of $$\hat{u}(z)$$ at zmax has no influence on $$\hat{\upsigma }_{S} /\hat{v}_{S}$$, either. Some value must be chosen in order to get Mathematica running, but all choices lead to the same $$\Delta \tilde{f}$$. While one might think that $${\text{d}}\hat{u}(z)/{\text{d}}z$$ must be chosen as $$- {\text{i}}k\hat{u}(z)$$, any other choice is fine as well. This is so because any other choice will represent a superposition of two waves decaying towards positive and negative z. The second solution (decaying towards negative z) is unphysical because it grows to infinity far away from the resonator surface. However, the presence of this unphysical contribution is no problem if zmax is large enough. If zmax is far away from the surface, the unphysical part of the solution decays to zero at the resonator surface and has no influence on $$\Delta \tilde{f}$$.

The code below should not be misunderstood as a realistic model. It is a tool box, allowing users to develop their own model [49]1.

## Footnotes

1. 1.

A Mathematica file with the content given below is available for download at http://www.pc.tu-clausthal.de/en/forschung/ak-johannsmann/qcm-modellierung/

## Notes

### Glossary

Variable

app

As an index: apparent

bsl

Slip length

bsl,ac

Acoustic slip length (Eq. 10.7.5)

Speed of (shear) sound ($$\tilde{c} = (\tilde{G}/\uprho )^{1/2}$$)

d

Thickness of a layer

dq

Thickness of the resonator ($$d_{q} = m_{q} /\uprho_{q} = Z_{q} /(2\uprho_{q} f_{0} )$$)

f

Frequency

f

As an index: film

f0

Resonance frequency at the fundamental (f0 = Zq/(2mq) = Zq/(2ρqdq))

FR

As an index: Film Resonance

$$\tilde{G}$$

Shear modulus

G

Limiting storage modulus at high frequency

$$\tilde{J}$$

Shear compliance ($$\tilde{J} = 1/\tilde{G}$$)

$$\tilde{k}$$

Wavenumber ($$\tilde{k} = \upomega /\tilde{c}$$)

liq

As an index: liquid

m

Mass per unit area

mq

Mass per unit area of the resonator ($$m_{q} = \uprho_{q} d_{q} = Z_{q} /(2f_{0} )$$)

n

Overtone order

$$\tilde{r}$$

Amplitude reflection coefficient (reflectivity, for short)

ref

As an index: reference state of a crystal in the absence of a load or reference frequency for viscoelastic constants (Eq. 10.4.1)

S

As an index: Surface

SL

As an index: Slipping Layer

t

Time

$$\hat{u}$$

(Tangential) displacement

$${\hat{\rm{v}}}$$

Velocity

w

Width of a fuzzy interface (Sect. 10.8)

zi

Point of inflection of a segment density profile (Sect. 10.8)

$$\tilde{Z}_{liq}$$

Shear-wave impedance of a liquid ($$\tilde{Z}_{liq} = (\text{i}\upomega \uprho_{liq} \upeta_{liq} )^{1/2}$$)

$$\tilde{Z}_{L}$$

zmax

Limit of integration range (Sect. 10.8)

Zq

Acoustic wave impedance of AT-cut quartz (Zq = 8.8 × 106 kg m−2 s−1)

$$\upbeta^{\prime},\upbeta^{\prime\prime}$$

Power law exponents (Eq. 10.4.1)

$$\Gamma$$

Imaginary part of a resonance frequency

$$\updelta$$

Penetration depth of a shear wave (Newtonian liquids: $$\updelta = (2\upeta_{liq} /(\uprho_{liq} \upomega ))^{1/2}$$)

Δ

As a prefix: A shift induced by the presence of the sample

φ

Polymer volume fraction (Sect. 10.8)

$$\tilde{\upeta },\upeta$$

Viscosity $$\tilde{\upeta } = \tilde{G}/({\text{i}}\upomega )$$

ρ

Density

$$\hat{\upsigma }$$

(Tangential) stress

$$\hat{\upsigma }_{s}$$

Tangential stress at the surface, also: “traction”

τ

Relaxation time

ω

Angular frequency

### References

1. 1.
Lu, C.S., Lewis, O.: Investigation of film-thickness determination by oscillating quartz resonators with large mass load. J. Appl. Phys. 43(11), 4385 (1972)Google Scholar
2. 2.
Crane, R.A., Fischer, G.: Analysis of a quartz crystal microbalance with coatings of finite viscosity. J. Phys. D-Appl. Phys. 12(12), 2019–2026 (1979)
3. 3.
Benes, E.: Improved quartz crystal microbalance technique. J. Appl. Phys. 56(3), 608–626 (1984)
4. 4.
5. 5.
Johannsmann, D., Mathauer, K., Wegner, G., Knoll, W.: Viscoelastic properties of thin-films probed with a quartz-crystal resonator. Phys. Rev. B 46(12), 7808–7815 (1992)
6. 6.
Granstaff, V.E., Martin, S.J.: Characterization of a thickness-shear mode quartz resonator with multiple nonpiezoelectric layers. J. Appl. Phys. 75(3), 1319–1329 (1994)
7. 7.
Martin, S.J., Bandey, H.L., Cernosek, R.W., Hillman, A.R., Brown, M.J.: Equivalent-circuit model for the thickness-shear mode resonator with a viscoelastic film near film resonance. Anal. Chem. 72(1), 141–149 (2000)
8. 8.
9. 9.
Salomaki, M., Kankare, J.: Modeling the growth processes of polyelectrolyte multilayers using a quartz crystal resonator. J. Phys. Chem. B 111(29), 8509–8519 (2007)
10. 10.
Domack, A., Johannsmann, D.: Plastification during sorption of polymeric thin films: a quartz resonator study. J. Appl. Phys. 80(5), 2599–2604 (1996)
11. 11.
Johannsmann, D.: Viscoelastic analysis of organic thin films on quartz resonators. Macromol. Chem. Phys. 200(3), 501–516 (1999)
12. 12.
Domack, A., Prucker, O., Ruhe, J., Johannsmann, D.: Swelling of a polymer brush probed with a quartz crystal resonator. Phys. Rev. E 56(1), 680–689 (1997)
13. 13.
Johannsmann, D.: Viscoelastic, mechanical, and dielectric measurements on complex samples with the quartz crystal microbalance. Phys. Chem. Chem. Phys. 10(31), 4516–4534 (2008)
14. 14.
Martin, S.J., Granstaff, V.E., Frye, G.C.: Characterization of a quartz crystal microbalance with simultaneous mass and liquid loading. Anal. Chem. 63(20), 2272–2281 (1991)
15. 15.
Voinova, M.V., Jonson, M., Kasemo, B.: ‘Missing mass’ effect in biosensor’s QCM applications. Biosens. Bioelectron. 17(10), 835–841 (2002)
16. 16.
Kankare, J.: Sauerbrey equation of quartz crystal microbalance in liquid medium. Langmuir 18(18), 7092–7094 (2002)
17. 17.
Du, B.Y., Johannsmann, D.: Operation of the quartz crystal microbalance in liquids: Derivation of the elastic compliance of a film from the ratio of bandwidth shift and frequency shift. Langmuir 20(7), 2809–2812 (2004)
18. 18.
Voinova, M.V., Rodahl, M., Jonson, M., Kasemo, B.: Viscoelastic acoustic response of layered polymer films at fluid-solid interfaces: continuum mechanics approach. Phys. Scr. 59(5), 391–396 (1999)
19. 19.
Rodahl, M., Kasemo, B.: On the measurement of thin liquid overlayers with the quartz-crystal microbalance. Sens. Actuators A-Phys. 54(1–3), 448–456 (1996)
20. 20.
Craig, V.S.J., Plunkett, M.: Determination of coupled solvent mass in quartz crystal microbalance measurements using deuterated solvents. J. Colloid Interface Sci. 262(1), 126–129 (2003)
21. 21.
Tsortos, A., Papadakis, G., Gizeli, E.: Shear acoustic wave biosensor for detecting DNA intrinsic viscosity and conformation: a study with QCM-D. Biosens. Bioelectron. 24(4), 836–841 (2008)
22. 22.
Papadakis, G., Tsortos, A., Bender, F., Ferapontova, E.E., Gizeli, E.: Direct detection of DNA conformation in hybridization processes. Anal. Chem. 84(4), 1854–1861 (2012)
23. 23.
24. 24.
Lekner, J.: Theory of Reflection of Electromagnetic and Particle Waves. Springer, Berlin (1987)Google Scholar
25. 25.
Bernoulli, D.: Hydrodynamica (1738). http://en.wikipedia.org/wiki/Hydrodynamica. Accessed 15 June 2014
26. 26.
Larson, R.G.: The Structure and Rheology of Complex Fluids. Oxford University Press, New York (1998)Google Scholar
27. 27.
Vinogradova, O.I.: Slippage of water over hydrophobic surfaces. Int. J. Miner. Process. 56(1–4), 31–60 (1999)
28. 28.
Thompson, P.A., Troian, S.M.: A general boundary condition for liquid flow at solid surfaces. Nature 389(6649), 360–362 (1997)
29. 29.
Huang, D.M., Sendner, C., Horinek, D., Netz, R.R., Bocquet, L.: Water slippage versus contact angle: a quasiuniversal relationship. Phys. Rev. Lett. 101(22), 226101 (2008)Google Scholar
30. 30.
Barrat, J.L., Bocquet, L.: Large slip effect at a nonwetting fluid-solid interface. Phys. Rev. Lett. 82(23), 4671–4674 (1999)
31. 31.
Neto, C., Evans, D.R., Bonaccurso, E., Butt, H.J., Craig, V.S.J.: Boundary slip in Newtonian liquids: a review of experimental studies. Rep. Prog. Phys. 68(12), 2859–2897 (2005)Google Scholar
32. 32.
Bowden, F.P., Tabor, D.: Friction lubrication and wear—a survey of work during last decade. Br. J. Appl. Phys. 17(12), 1521–1524 (1966)
33. 33.
Tucker, C.L., Moldenaers, P.: Microstructural evolution in polymer blends. Annu. Rev. Fluid Mech. 34, 177–210 (2002)
34. 34.
Barnes, H.A.: A review of the slip (wall depletion) of polymer-solutions, emulsions and particle suspensions in viscometers—its cause, character, and cure. J. Nonnewton. Fluid Mech. 56(3), 221–251 (1995)
35. 35.
Lefevre, B., Saugey, A., Barrat, J.L., Bocquet, L., Charlaix, E., Gobin, P.F., Vigier, G.: Intrusion and extrusion of water in highly hydrophobic mesoporous materials: effect of the pore texture. Colloids Surf. A-Physicochem. Eng. Aspects 241(1–3), 265–272 (2004)
36. 36.
Boehnke, U.C., Remmler, T., Motschmann, H., Wurlitzer, S., Hauwede, J., Fischer, T.M.: Partial air wetting on solvophobic surfaces in polar liquids. J. Colloid Interface Sci. 211(2), 243–251 (1999)
37. 37.
Al-Fetlawi, H., Shah, A.A., Walsh, F.C.: Modelling the effects of oxygen evolution in the all-vanadium redox flow battery. Electrochim. Acta 55(9), 3192–3205 (2009)
38. 38.
Zhitomirsky, I.: Cathodic electrodeposition of ceramic and organoceramic materials. Fundamental aspects. Adv. Colloid Interface Sci. 97(1–3), 279–317 (2002)
39. 39.
Ferrante, F., Kipling, A.L., Thompson, M.: Molecular slip at the solid-liquid interface of an acoustic-wave sensor. J. Appl. Phys. 76(6), 3448–3462 (1994)
40. 40.
McHale, G., Lucklum, R., Newton, M.I., Cowen, J.A.: Influence of viscoelasticity and interfacial slip on acoustic wave sensors. J. Appl. Phys. 88(12), 7304–7312 (2000)
41. 41.
Ellis, J.S., Hayward, G.L.: Interfacial slip on a transverse-shear mode acoustic wave device. J. Appl. Phys. 94(12), 7856–7867 (2003)
42. 42.
Daikhin, L., Gileadi, E., Tsionsky, V., Urbakh, M., Zilberman, G.: Slippage at adsorbate-electrolyte interface. Response of electrochemical quartz crystal microbalance to adsorption. Electrochim. Acta 45(22–23), 3615–3621 (2000)
43. 43.
Zhuang, H., Lu, P., Lim, S.P., Lee, H.P.: Effects of interface slip and viscoelasticity on the dynamic response of droplet quartz crystal microbalances. Anal. Chem. 80(19), 7347–7353 (2008)
44. 44.
Tretheway, D.C., Meinhart, C.D.: Apparent fluid slip at hydrophobic microchannel walls. Phys. Fluids 14(3), L9–L12 (2002)
45. 45.
Klein, J., Kumacheva, E., Perahia, D., Mahalu, D., Warburg, S.: Interfacial sliding of polymer-bearing surfaces. Faraday Discuss. 98, 173–188 (1994)
46. 46.
Urbakh, M., Tsionsky, V.; Gileadi, E.; Daikhin, L.: Probing the solid/liquid interface with the quartz crystal microbalance. In: Steinem, C., Janshoff, A. (eds.) Piezoeletric Sensors. Springer, Heidelberg (2006)Google Scholar
47. 47.
Du, B.Y., Goubaidoulline, E., Johannsmann, D.: Effects of laterally heterogeneous slip on the resonance properties of quartz crystals immersed in liquids. Langmuir 20, 10617–10624 (2004)
48. 48.
Decher, G.: Fuzzy nanoassemblies: toward layered polymeric multicomposites. Science 277(5330), 1232–1237 (1997)
49. 49.
Wolff, O., Seydel, E., Johannsmann, D.: Viscoelastic properties of thin films studied with quartz crystal resonators. Faraday Discuss. 107, 91–104 (1997)