# Effect of Noise on Determining Ultrathin-Film Parameters from QCM-D Data with the Viscoelastic Model

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## Abstract

**:**

## 1. Introduction

_{res}/2 The additional information contained in the sets of Δf/n and ΔΓ/n gives access to certain nongravimetric parameters [4,5], such as the mechanical properties of the film.

- (A)
- A thin film in air;
- (B)
- A stiff, thin film in a liquid;
- (C)
- A semi-infinite Newtonian liquid with slightly altered viscosity close to the resonator surface;
- (D)
- A soft film in a liquid.

- Data from the fundamental often show irregular and erratic behavior. The fundamental is therefore usually discarded.
- The QCM produces artifacts on the high overtones when applied to samples which are known to be Newtonian liquids [16]. The imaginary part of the viscosity as reported by the QCM is sometimes negative, which is unphysical.

## 2. Target Parameters of a Viscoelastic Analysis

^{′}+ iG

^{″}, is the ratio of shear stress to shear strain. In the following, the tilde denotes a complex parameter (mostly a viscoelastic response function). $\tilde{G}$ is popular in polymer research [17]. The shear compliance, $\tilde{J}$ = 1/$\tilde{G}$ = J′− iJ″, is the ratio of strain to stress. QCM-D experiments are most easily analyzed in terms of J’ and J″ because the trivial case (Sauerbrey-type behavior) corresponds to $\tilde{J}$ = 0 (rather than G′ = ∞ or G″ = ∞). Additionally, the recipes from Section 3.2 and Section 3.3 relate J′ and J″ to the characteristic features of the plots as opposed to G′ and G″. The viscosity, $\tilde{\eta}$ = $\tilde{G}$/(iω) = η′ − iη″, is useful when the layer under study has a viscosity of η′ ≈ η′

_{bulk}and small elasticity (η″ ≈ 0). The bulk in the following is assumed to be a Newtonian liquid (η′ = const, η″ = 0). In all three cases ($\tilde{G}$, $\tilde{J}$, or $\tilde{\eta}$), the real part and the imaginary part may be replaced by absolute values (|$\tilde{G}$|, |$\tilde{J}$|, or |$\tilde{\eta}$|) and the loss tangent (tan δ = G″/G′ = J″/J′ = η′/η″). The loss tangent is actually independent of whether the quantification of viscoelasticity occurs with $\tilde{G}$, $\tilde{J}$, or $\tilde{\eta}$. If the loss tangent has a peak at some frequency, the medium under study undergoes relaxations with rates similar to the frequency of the peak in tan δ.

_{ovt}frequencies, plus the layer thickness. The problem of inversion would then be underdetermined because the experiment only reports 2n

_{ovt}parameters; however, viscoelastic spectra are usually smooth. In the limited frequency range covered by the QCM, |$\tilde{G}$| and tan δ (f) can be approximated with fair accuracy by power laws (Figure 3). These are of the following form:

_{cen}is a frequency in the center of the QCM’s range (often f

_{cen}≈ 30 MHz). |G

_{cen}| and (tan δ)

_{cen}are the values at this frequency. If J′ and J″ are used rather than |$\tilde{G}$| and tan δ, one may write the following:

_{cen}, and two power law exponents. The stiffness at f

_{cen}may be quantified with the pair {|$\tilde{G}$|, tan δ}, quantified with the pair {J′, J″}, or with some other pair. Given that viscoelastic dispersion cannot be ignored for soft matter (β ≠ 0, β ≠ 0), any realistic model taking viscoelasticity into account is bound to have at least five free parameters. Fewer parameters amount to assumptions.

## 3. Background

#### 3.1. Underlying Equations

^{1/2}is the wave number; $\tilde{Z}$= (ρ$\tilde{G}$)

^{1/2}is the shear wave impedance; the subscripts “f” and “q” denote the film and the resonator, respectively; and f

_{0}is the frequency of the fundamental. Equation (3) applies to all thicknesses, at least in principle. It does not involve a Taylor expansion in film thickness, d

_{f}. (It does, however, make use of the small-load approximation [5].) Equation (3) was used to produce the fits shown in Section 3.5.

#### 3.2. Determination of Softness Is Possible for Thin Films in Air

_{bulk}= 0), Equation (3) simplifies to the following:

_{f}results in the following:

_{f}(the mass per unit area) is equal to ρ

_{f}d

_{f}. m

_{q}= Z

_{q}/(2f

_{0}) is the mass per unit area of the resonator plate. For the reasons discussed in Ref. [18], a slightly better approximation is as follows:

_{f}, J

_{f}′, J

_{f}″, β′, β″} from an experiment. The error bar on βʹ may be substantial (see the third bullet point below), but the error occurs on only this one parameter. Because the errors are not cross-correlated, the values of the other parameters remain robust. The scheme is sketched in Figure 4. The rules are based on Equation (5).

- The film thickness, d
_{f}(more precisely, the mass per unit area, m_{f}), is obtained from a plot of Δf/n versus n^{2}(Figure 4A). A line (possibly with a slight curvature) is fitted to the data. The intercept of this line with the y-axis is proportional to m_{f}. More specifically, one has m_{f}= −Δf_{intercept}/(nf_{0})m_{q}. - The elastic compliance, J
_{f}′, is obtained from the slope of this line. - The power law exponent of the elastic compliance, β′, is obtained from the curvature of this line. This curvature is often determined with considerable uncertainty. The error bars on β’ are correspondingly large.
- The viscous compliance, J
_{f}″, is obtained from the ratio between the bandwidth shift and the frequency shift following the relationship below:

_{f}). It rather scales as m

_{f}

^{2}. This is not a problem, because m

_{f}is determined with fair accuracy from the intercept with the y-axis in a plot of Δf/n vs. n

^{2}; it may be a problem when m

_{f}is small. Nongravimetric effects are difficult to detect for layers which are only a few nanometers thick. These layers shear under their own inertia, and the inertial forces are weak for thin films. On the positive side, the approximation underlying Equation (8) (namely that the real part of the right-hand side in Equation (6) is about unity) can always be reached by making the film thin enough.

- The power law exponent of the viscous compliance, β″, is obtained from the slope in a log–log plot of ΔΓ/(−Δf)/n
^{2}versus n (Figure 4B).

#### 3.3. Determination of Elastic Softness Is Possible for Thin, Stiff Films in a Liquid

_{f}(with $\tilde{Z}$

_{bulk}≠ 0) results in the following:

_{f}, and replaces ${\tilde{Z}}_{bulk}^{2}$ by iωρη

_{bulk}:

_{f}≈ ρ

_{bulk}≈ ρ). Because Equation (10) is linear in thickness, it also holds in an integral sense:

_{f}inside the film) is sometimes questionable. The QCM will always report apparent parameters because it does not recognize the profile.

_{f}in Equation (10) is zero, it reproduces the Sauerbrey result. If the layer is noticeably soft, the viscoelastic correction (the term in square brackets) lowers the value of −Δf/n. This decrease in apparent thickness (in thickness naively derived with the Sauerbrey equation) is sometimes called the “missing-mass effect” [20].

_{f}″<< J

_{bulk}″ or, equivalently, as η

_{f}′>>η

_{bulk}. Again, Equation (12) only applies if this condition is met.

- The film thickness, d
_{f}, is obtained from a plot of Δf/n versus n. A line (possibly with a slight curvature) is fitted to the data. The intercept of this line with the y-axis is proportional to the thickness. - The viscous compliance, J
_{f}″, is obtained from the slope of this line. Note: For the dry film, the elastic compliance is derived from the slope. It is the viscous compliance here. - The power law exponent of the viscous compliance, β″, is obtained from the curvature of this line. Error bars are often large because the curvature is determined with poor accuracy.
- The elastic compliance, J
_{f}′, is obtained from the acoustic ratio, following Equation (12). - The power law exponent of the elastic compliance, β′, is obtained from the slope in a log–log plot of ΔΓ/(−Δf)/n versus n (Figure 5B).

_{bulk}.

#### 3.4. Increased Near-Surface Viscosity (Case C in Figure 1)

_{f}as η

_{bulk}+ Δ$\tilde{J}$ and assume that |$\Delta \tilde{J}$

_{f}| << η

_{bulk}:

- Sauerbrey-type behavior occurs if Δ$\tilde{\eta}$ is mostly real (that is, if the double layer has increased Newtonian viscosity with negligible viscoelasticity). Double layer effects cannot be distinguished from adsorption. In ref. [28] it was argued that the distinction is possible based on the kinetics of the response to a voltage step. The distinction is not possible based on single sets of Δf/n and ΔΓ/n.
- The term in square brackets will often be smaller than unity. The thickness of the layer with increased viscosity can no longer be inferred from −Δf/n, meaning that the gravimetric information is lost. If η’ depends on z (which can be expected), the product d
_{f}(η_{f}′ − η_{bulk}) turns into an integral ∫(η_{f}′ − η_{bulk}) dz.

_{f}′ and J

_{f}″. For liquids, the absolute value of the viscosity, |$\tilde{J}$|, and the loss tangent, tan δ = η′

_{f}/η″

_{f}, are the more suitable parameters. (Arguably, an even more appropriate choice would be |${\tilde{J}}_{f}$|and the inverse loss tangent, because the inverse loss tangent is zero for a Newtonian liquid.)

_{f}| is similar to η

_{bulk}, the acoustic ratio is about half of the inverse loss tangent.

#### 3.5. The Soft Adsorbate (Case D in Figure 1)

_{f}= 1 g/cm

^{3}); however, it is far from certain (and even unlikely) that 2 nm is actually the geometric thickness at this time. There are two problems:

- Applying the Sauerbrey equation implicitly assumes a rigid layer. If no assumptions on the viscoelastic properties of the layer can be made, a wide range of viscoelastic constants—if combined suitably—can match an experimentally determined acoustic ratio. Figure 8 shows the predictions of Equations (15) and (18) as contour plots (A and B). Any set of parameters on a contour line will lead to the same value of ΔΓ/(−Δf). Only if additional assumptions are made (red boxes in Figure 8A) can the acoustic ratio be interpreted, e.g., in terms of Equation (12) (in the limit of small J″) or Equation (19) (in the limit of tan δ ≈ ∞ and η
_{f}′ ≈ η_{bulk}′ blue and red boxes in Figure 8B). - This uncertainty in the determination of viscoelastic parameters often spills over to the thickness. The point here is that the QCM cannot distinguish between a compact stiff layer and a dilute soft layer. The problem is illustrated in Figure 9.

_{f}= 9.8 nm, J

_{f}′ = 0.29 MPa

^{−1}, β′ = −1.61, J

_{f}″ = 1.68 MPa

^{−1}, and β″ = −0.91.

^{7}s

^{−1}) ≈ 20 ns. If fitted with |$\tilde{G}$|and tan $\mathsf{\delta}$, the fit parameters leading to Figure 7C are d

_{f}= 8.1 nm, |$\tilde{G}$

_{f}| = 1.0 MPa

^{−1}, β′ = −1.8, tan δ = 1.8 MPa

^{−1}, and β″ = −0.7. β″ is positive, indicative of fast relaxations.

^{2}by using the remaining fit parameters, and plots the values of χ

^{2}and the fitted other parameters versus thickness. As Figure 11A shows, χ

^{2}sharply increases when the film thickness is lowered from its optimum value to below 1.8 nm. A film thickness of less than 1.8 nm is incompatible with the experiment; however, the increase in χ

^{2}is rather moderate when d

_{f}is moved upward into a range between 2 nm and 10 nm. The algorithm finds a good match with the experiment by compensating for a large thickness with a large J’ (grey bars in Figure 11A). No such problem occurs in Figure 11B. The χ

^{2}landscape now has a distinct minimum at d

_{f}≈ 10 nm. This film thickness is now a robust outcome of the fitting process. Again, the result is robust because the noise is low enough to let the curvatures in Figure 2C be quantified with confidence.

^{2}landscape, while the thickness was still well-defined. These questions need attention to detail in every single case.

## 4. The Software Package PyQTM

^{2}landscapes for evaluating the robustness of the fit parameters against experimental noise and, also, checks for the influence of a fuzzy interface between the film and the bulk. Figure 7B,C, as well as Figure 11, were created using PyQTM.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**A set of configurations in which the interpretation proceeds in different ways. For cases (

**A**,

**B**) the values of Δf/n and ΔΓ/n can be interpreted, respectively, at least approximately. For systems with increased near-surface viscosity (

**C**), Δf/n and ΔΓ/n have an interpretation, but the width of the range, inside which the viscosity is increased, is no longer accessible. Interpretation is difficult for soft adsorbates (

**D**).

**Figure 2.**For very thin films (

**A**) the fractional noise is too large to let a curvature in plots of Δf/n and ΔΓ/n versus n be determined reliably. Interpretation must rely on the offsets and the slopes for Δf/n and ΔΓ/n (totaling four parameters). If the model contains five parameters the problem is underdetermined. For thicker films (

**B**) the curvatures can be determined reliably; thus, five model parameters can also be derived reliably.

**Figure 3.**The shear moduli of viscoelastic materials depend on frequency. The plot shows a typical rheological spectrum of a solution of a long-chain linear polymer. The frequency scale extends over many decades, while the QCM only covers about one decade. In this limited frequency range, G′(f) and G″(f) can be approximated by power laws (dashed blue lines) in panel A. For the QCM, the representation with {J′(f), J″(f)} in panel B is more practical than the set {G′(f), G″(f)}. The graph discusses viscoelasticity in terms of either {Gʹ, Gʹʹ} or {J′, J″}. The same arguments apply to—for instance—the pair {|$\tilde{G}$|, tan δ} (see Equation (1)).

**Figure 4.**For a thin film in air, the set of parameters {d

_{f}, J

_{f}′, J

_{f}″, β′, β″} can be derived from plots of Δf/n versus n

^{2}(

**a**) and log(ΔΓ/(−Δf)/n

^{2}) versus log(n) (

**b**), as shown above. Arrows indicate which system parameters are derived from certain features of the plot.

**Figure 5.**A thin, stiff film in a liquid also allows viscoelastic parameters from the sets of Δf/n and ΔΓ/n to be extracted. The plot of Δf/n versus n yielded the thickness and the viscous compliance (

**a**). The plot of log(ΔΓ/(−Δf)/n) versus log(n) yields the elastic compliance and the power-law exponent β′ (

**b**). Arrows indicate which system parameters are derived from certain features of the plot.

**Figure 6.**An example, where an EQCM responds to changes in a liquid’s near-surface viscosity. A solution of electrochemically inert ions was subjected to voltage steps, as shown at the top. The current recharges the diffuse double layer. The shifts in frequency are much larger than the shifts in bandwidth, and Δf/n is similar on the different overtones. Typically, such a Sauerbrey-type behavior would be interpreted as being caused by adsorption, but increased viscosity in the diffuse double layer is equally possible. The kinetics suggests that altered viscosity makes a larger contribution to the overall frequency shift than adsorption. For details, see ref. [28]. The nonzero shift in ΔΓ shows that there is a small elastic component in the double layer’s response. Adapted from ref. [28].

**Figure 7.**Adsorption of a polymer brush as an example, where the material parameters cannot be determined without making assumptions (early in the adsorption process, “thin film”), but where such an analysis becomes possible once the thickness exceeds 10 nm (“thick film”); the fractional noise decreases correspondingly (

**A**). The raw data in (

**A**) have been pre-averaged. Every data point is an average of four adjacent points of the raw data. The bottom panels in (

**B**,

**C**) show the acoustic ratios. The fits are not based on the acoustic ratios, but rather on Δf/n and ΔΓ/n themselves. The fit was produced with the PyQTM program by using Equation (3).

**Figure 8.**Contour plots of the acoustic ratio versus the viscoelastic parameters. Combinations of values on the contour lines all lead to the same acoustic ratio. The bars indicate the regimes, in which the assumptions underlying Equation (12) (

**A**) and Equation (19) (

**B**) apply.

**Figure 9.**The QCM cannot determine whether an adsorbing polymer layer is initially thin and compact (

**A**) or extended and dilute (

**B**). The thickness in the later stages can be determined. The sketch is not meant to say that the configurations on the left-hand side (top and bottom) would lead to the exact same values of Δf/n and ΔΓ/n; if that was the case QCM-D could determine the coverage, which it cannot.

**Figure 10.**Viscoelastic spectra (red lines) and their approximation with power laws in a limited frequency range (dashed lines). The sign of the power law exponent pertaining to tan δ indicates whether the main relaxation is faster or slower than the frequency of the QCM. Β″ in the legend is the power law exponent of tan δ.

**Figure 11.**An analysis of the χ

^{2}landscape for the datasets shown in Figure 7B,C. The software prescribes values for the thickness in a certain range (0 to 10 nm in (

**A**) and fits the remaining parameters, which are J

_{f}′, β′, J

_{f}″, and β″. Given that the thickness is no longer a free parameter, χ

^{2}is larger than what is obtained with d

_{f}as a free parameter. As the top panel in (

**A**) shows, χ

^{2}is very large when d

_{f}is smaller than 2 nm, but is only marginally larger than the minimum value when d

_{f}is larger than 2 nm. The fit compensates for a large thickness with a large J′ and finds a good match with the experimental data (green bars on the left-hand side). The algorithm cannot distinguish between films that are either thin and stiff or thick and soft. The valley in the χ

^{2}landscape is not at all sharp, and a statement on the thickness is difficult. These problems are much alleviated in (

**B**) because the fractional errors have decreased in Figure 7C compared to Figure 7B. The valley in the χ

^{2}landscape is now sharp (top in (

**B**), light blue bar). On a qualitative level, the data now allow for a robust statement on the curvatures (Figure 7C). The interpretation can rest on a total of six robust experimental parameters (offsets, slopes, and curvatures in plots of Δf/n and ΔΓ/n versus $n$.) Five model parameters can be inferred from six experimental parameters. The fit problem is now overdetermined (as it should be).

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**MDPI and ACS Style**

Johannsmann, D.; Langhoff, A.; Leppin, C.; Reviakine, I.; Maan, A.M.C.
Effect of Noise on Determining Ultrathin-Film Parameters from QCM-D Data with the Viscoelastic Model. *Sensors* **2023**, *23*, 1348.
https://doi.org/10.3390/s23031348

**AMA Style**

Johannsmann D, Langhoff A, Leppin C, Reviakine I, Maan AMC.
Effect of Noise on Determining Ultrathin-Film Parameters from QCM-D Data with the Viscoelastic Model. *Sensors*. 2023; 23(3):1348.
https://doi.org/10.3390/s23031348

**Chicago/Turabian Style**

Johannsmann, Diethelm, Arne Langhoff, Christian Leppin, Ilya Reviakine, and Anna M. C. Maan.
2023. "Effect of Noise on Determining Ultrathin-Film Parameters from QCM-D Data with the Viscoelastic Model" *Sensors* 23, no. 3: 1348.
https://doi.org/10.3390/s23031348