Opportunities arising from the use of the rheometric quartz crystal microbalance (RheoQCM) as a fixed frequency rheometer operating at 15 MHz are discussed. The technique requires the use of films in a specified thickness range that depends on the mechanical properties of the material of interest. A regime map quantifying the appropriate thicknesses is developed, based on the properties of a highly crosslinked epoxy sample that is representative of a broad class of polymeric materials. Relative errors in the measured film properties are typically in the range of several percent or less and are minimized by using a power law model to relate the rheological properties at two different resonant harmonics of the quartz crystal. Application of the RheoQCM technique is illustrated by measuring the temperature- and molecular weight-dependent properties of polystyrene and poly(methyl methacrylate) in the vicinity of the glass transition.

## I. INTRODUCTION

A unique aspect of quartz crystal resonators is their ability to accurately probe the viscoelastic properties in the megahertz frequency regime. Part of the appeal of the technique is its experimental simplicity. The experimental geometry, illustrated schematically in Fig. 1, consists of a single crystal quartz disk with electrodes deposited on either side, and with one of these electrode surfaces in contact with the material of interest. The crystals are excited in a shear mode by an oscillating voltage, producing a response at the mechanical resonance condition for the crystal that is extraordinarily sensitive to the surface boundary condition at the resonance condition. If a film with an appropriate thickness is deposited on the crystal, this sensitivity enables the rheological properties and film mass per unit area to be determined at a fixed reference frequency that is typically close to 15 MHz. Because of the well-known sensitivity of the technique to the film mass, it is commonly referred to as a quartz crystal microbalance (QCM). The rheological adaptation of the method central to the proposed research is referred to as the RheoQCM. The strength of the technique is rooted in three aspects of the method.

### A. The frequency of the measurement

The high frequency of the technique complements traditional rheological methods that operate at much lower frequencies. This technique is particularly useful if one is interested in local segmental dynamics in systems where time-temperature equivalence cannot be used because of an underlying phase transition. Aqueous polymer solutions are in this category.

### B. The geometry of the measurement

The QCM is a surface loaded rheometer, with physical contact made on only one side of the sample. For this reason, it is ideally suited for investigations of environmental effects, which require that the material be exposed to a well-defined environment. Examples include exposure to oxygen, water vapor or solvent vapor, or different solution environments when immersed in liquid, typically an aqueous solution.

### C. The thickness of the polymer samples used in the measurement

The rheometric mode of the QCM requires the use of samples in a relatively narrow thickness range that depends on the properties of the sample of interest. This thickness is in a very convenient range for a variety of experiments: thick enough so that the “bulk” properties of the material are being probed, but thin enough so that equilibration with the local environment throughout the sample is assured.

## II. BACKGROUND

Use of the quartz crystal microbalance (QCM) to characterize the properties of soft materials is well established,^{1–8} with the book by Johannsmann^{8} being perhaps the most complete source of information on this topic. There are, however, some specific points that need to be addressed when using the QCM as a quantitative rheometer, and these points are illustrated in this paper. We begin with a general overview of the technique, with a description of the design of the experiment and the elements involved in extracting rheological information from the experimental data. A key element in the analysis involves the comparison of the measured quantities to predictions from a detailed model, and a summary of key elements of the model is presented as well. This section concludes with a discussion of the importance of the thickness of the film being investigated, since using films of the correct thickness is essential for obtaining accurate film properties from the technique.

### A. Experimental measurables and technique overview

Information is obtained from the QCM by analyzing the electrical response of the crystal circuit in the vicinity of its mechanical resonance. Frequency and time domain versions of the experiment are both commonly performed and are equivalent to one another. In the frequency domain experiment employed in the proposed research, the admittance spectrum of the crystal is measured in the vicinity of the resonant frequency. Two parameters are obtained from the crystal resonating at its $n$th harmonic: the resonant frequency, $fn$, defined as the frequency at which the conductance is maximized, and the half-bandwidth, $\Gamma n$. (Note that the subscript $n$ indicates properties corresponding to the $n$th harmonic throughout this proposal, with only the odd harmonics being active.) Both quantities are illustrated in Fig. 2. In a time domain or “ringdown” experiment, the dissipation factor, $D$, is related to the time constant of the decay of the resonance after the driving voltage has been turned off. These two measures of the energy dissipation are equivalent to one another and are related by the following expression:^{6,8}

The frequency domain technique is used in the proposed experiments, both because the equipment is more readily available and adaptable, and because it is better suited for samples with a high dissipation, including the samples used in our proposed experiments. Equation (1) is useful because it makes a connection to the information obtained from the time domain techniques popularized by the widespread use of the commercial Q-Sense instrument. The online database of published papers including data obtained with the Q-Sense instrument^{9} contains many hundreds of papers, and the technique has been extensively applied to polyelectrolyte complex films formed by layer-by-layer deposition.^{10–18} When Eq. (1) is used to convert $D$ to $\Gamma $, the analysis we describe here can be applied to these datasets as well.

For a given value of $n$, the values of $fn$ and $\Gamma n$ are determined by the following three experimental parameters, which are the generalized outputs of a rheometric QCM experiment:

$d\rho $: The product of the film thickness ($d$) and its density ($\rho $), more commonly reported as the mass per unit area of the film.

$|Gn\u2217|\rho $: The product of the magnitude of $Gn\u2217$, the complex shear modulus of the film at a frequency of $fn$, and the film density.

$\varphi n$: The phase angle of $Gn\u2217$ at a frequency of $fn$.

Note that the $G\u2032$ and $G\u2032\u2032$, the storage and loss moduli, respectively, are given as $G\u2032=|Gn\u2217|cos\u2061\varphi $ and $G\u2032\u2032=|Gn\u2217|sin\u2061\varphi $.

Because the crystal response at a given harmonic depends on three independent properties, but only two quantities are measured at a given resonant harmonic of the crystal, data need to be obtained at multiple harmonics, making an assumption about the frequency dependence of $|G\u2217|$ and $\varphi $. A commonly used assumption is embedded in the so-called Voigt model, where $G\u2032$ is assumed to be frequency independent and $G\u2032\u2032$ is assumed to increase linearly with the frequency. As pointed out by Reviakine *et al.*,^{19} this assumption is almost never valid and can, in fact, lead to substantial errors in the extracted values for the viscoelastic analysis. In this paper, we compare errors associated with the Voigt model with the following power law form:

This approximation is precise for a material exhibiting power law behavior and significantly reduces the error of the calculations in comparison to results from the Voigt model.

### B. Mathematical model

The physical basis of quartz crystal rheometry is rooted in the nature of acoustic wave propagation in the coating and the reflection of acoustic waves at the coating substrate and coating/air interfaces.^{2,3,20–22} The technique is one of a family of acoustic techniques that can be used to extract material properties.^{20–26} The connection to the linear viscoelastic properties of the material is through the complex acoustic impedance, $Z\u2217$. For the transverse shear waves relevant to our experiments, $Z\u2217=(\rho G\u2217)1/2$, where $G\u2217$ is the complex shear modulus and $\rho $ is the density. Measurements of the reflection of acoustic waves through a bulk sample, or measurements of the reflection of acoustic waves from interfaces of materials with known values of the acoustic impedance, can be used to determine $Z\u2217$, and hence the complex shear modulus, $G\u2217$, as well.^{20–22,25}

For convenience, one can define a complex resonant frequency, $fn\u2217$, given by $fn+i\Gamma n$. Applications of the QCM, including our rheometric adaptation, are based on the relationship between $fn\u2217$ and the load impedance, $ZLn\u2217$, associated with the material that is in contact with the crystal surface. Our starting point for the most complete treatment of the QCM response is determined by the solution to the following expression, derived in a different form by Lu and Lewis,^{27} and referred to here as the Lu–Lewis equation,

Here, $Zq\u2217$ is the shear acoustic impedance of the quartz, $dq$ is the thickness of the quartz crystal, $kq\u2217$ is the complex wavenumber of the shear wave in the quartz, and $ZLn,front\u2217$ and $ZLn,back\u2217$ are the respective load impedances at the front and back surfaces of the crystal. Equation (3) is similar to Eq. (4.5.9) in Ref. 8 and is discussed at some length by Petri and Johannsmann in the supplementary information of Ref. 28. In our adaptation, we neglect the effect of the back electrode on the response, setting $ZLn,back\u2217$ to zero. We also have simplified Eq. (3) by neglecting the term that counts for piezoelectric stiffening, having concluded (as do Petri and Johannsmann^{28}) that inclusion of this term does not measurably affect the analysis. The load impedance is obtained from the impedance and complex wave number within each layer, using a matrix formulation that can be extended to treatment to any number of layers.^{4,8,29} The wave vector in each layer is evaluated at the complex resonant frequency, including the frequency shift. The wave vector for the *n*th harmonic within the quartz crystal, $kqn\u2217,$ is given by the following expression:

Our method of the Lu–Lewis equation [Eq. (3)] is a direct adaptation of the Python code provided in the supplementary information of Ref. 28.

Numerical solution of the Lu–Lewis equation is readily accomplished via numerical methods, but the complicated nature of the procedure is not completely appealing. As discussed extensively by Johannsmann,^{4,6,8} the situation becomes notably simpler when $\Delta fn\u2217\u226af1,$ in which case the small load approximation (SLA) can be employed, and the Lu–Lewis equation simplifies to the following:

Here, $\Delta ZLn\u2217$ is the difference in load impedance between the reference state (without the film of interest) and the sample with the film. An important aim of this paper is to assess the accuracy of the validity of the small load approximation and to assess the degree to which its use introduces errors into measured properties of the film, relative to errors that must necessarily be made regarding the frequency dependence of the viscoelastic properties. For completeness, some essential results from the small load approximation are included in Secs. II C and II D. The results correspond to the “single layer” case, where the electrode is included in the reference state, and where there is no overlayer. Corresponding expressions for the case where an overlayer exists (most commonly a bulk layer of water) have been provided elsewhere.^{4,8,30,31}

#### 1. Thin film limit

For the simplest cases where the film is very thin, the load impedance is purely inertial. The dissipative contribution of the impedance can be ignored (so that $\Delta \Gamma n$ is very small), and the resonant frequency decreases by the Sauerbrey shift, $\Delta fsn$, given by the following expression:^{32}

Here, $\rho $ is the film density and $d$ is its thickness, with the product of these two quantities being the mass per unit area of the film.

#### 2. Bulk limit

For a very thick material, $\Delta ZLn\u2217$ reduces to the acoustic shear impedance of the material, $(\rho G\u2217)1/2$. In this case, $\Delta fn$ and $\Delta \Gamma n$ can be used to obtain the viscoelastic properties of the material at the measured resonant frequencies, utilizing the following expressions:

Equations (7) and (8) are the basis for the use of the QCM to obtain the viscosity of relatively low viscosity liquids (viscosity below $\u22480.1$ Pa s), the most well-established application of the QCM for mechanical property determination.^{33–36}

#### 3. Generalized case

The parameter that determines where the thin film and bulk limits apply is the decay length of the shear wave, $\delta n$, which is given by the following expression:

The thin film, Sauerbrey limit applies when the film thickness, $d$, is much less than $\delta n$ and the bulk limit applies in the opposite limit, where $d\u226b\delta n$. In almost all situations of relevance to the proposed work, film thicknesses are in the intermediate regime where quantitative deviations from either one of these limits are observed. The complex frequency shift in the intermediate regime differs from the Sauerbrey prediction in a manner that depends on two parameters: $\varphi n$, the viscoelastic phase angle of the film, and $d/\lambda n$, the film thickness divided by wavelength of a shear wavelength. Within the small load approximation, the full expression for the complex resonant frequency in this case is given by the following expression:^{6,37}

By measuring the frequency response at multiple harmonics and making a physically based assumption regarding the frequency dependence of $\lambda n$, one can determine $\varphi n$ and $\lambda n$. The magnitude of the complex shear modulus, $|Gn\u2217|$ is then obtained from the following expression for the shear wavelength:

### C. Regime maps for polymer melts

Use of the QCM to accurately quantify the rheological properties of a polymer film requires that the film be thick enough so that measurable deviations from the Sauerbrey equation [Eq. (6)] are obtained, but thin enough so that the bandwidth of the crystal resonance is not too large to measure. These criteria can be combined to form a regime map illustrating the appropriate thickness range for different polymer properties. Because the QCM response depends on the magnitude and phase of the viscoelastic phase angle, we use the relationship between $\varphi $ and $|G\u2217|$ for a representative model system to eliminate $\varphi $ as an independent variable, using $|G\u2217|$ as the variable to quantify the viscoelastic behavior of the material. A particularly useful dataset for this purpose is from a crosslinked epoxy system given in parameterized form by Simon *et al.*^{38} shown in Fig. 3. The rheological master curves shown in parts a and b of this figure are characteristic of crosslinked polymers in general. Figure 3(c), referred to as a van Gurp–Palmen plot,^{39,40} is simply a cross-plot relating $\varphi $ to $|G\u2217|$. A modified form is often utilized in presenting results from the QCM, where $|G\u2217|$ is replaced by $|G\u2217|\rho $.

A RheoQCM regime map, illustrating the sample thickness range for which quantitative rheological information can be obtained from the QCM, is shown in Fig. 4. This regime map is based on the behavior of $\Delta f3$ and $\Delta f5,$ the frequency shifts at the third and fifth harmonics, and $\Delta \Gamma 5,$ the bandwidth of the fifth harmonic. We generally use $n=3$ and $n=5$ in our calculations of the film properties because the frequency shifts for $n=1$ are less reliable, being more strongly affected by the boundary conditions at the edge of the crystal, which are not completely reproducible. The shear displacement fields are more strongly localized at the center of the crystal for higher harmonics, an effect referred to as “energy trapping” that is discussed extensively by Johannsmann.^{41} In Fig. 4(a), we plot the thickness dependence of $\Delta f3$, $\Delta f5$, and $\Delta \Gamma 5$, assuming $|G3\u2217|=108$ Pa and $\rho =1g/cm3$ and using Fig. 3 to obtain the value of $\varphi $ corresponding to $|G3\u2217|=108$ Pa. Values of these three quantities are used to identify the Sauerbrey, viscoelastic, and overdamped regimes. The boundaries between these different regimes are plotted as a function $|G3\u2217|\rho $ in Fig. 4(b), again assuming $\rho =1g/cm3$ and using the data shown in Fig. 3(c) to extract the appropriate value of $\varphi $ for each value of $|G3\u2217|$. In general, the following four different regimes are obtained in a QCM regime map, with the first three of them appearing in Fig. 4:

#### 1. Sauerbrey regime

For films that are sufficiently thin, the QCM response is determined by the inertial response due to the film mass. In this region, $\Delta fn\u221dn$ and $\Delta \Gamma $ is small. The mass is obtained from the Sauerbrey equation [Eq. (6)], but sufficient information to quantify the rheological properties of the film is not available.

#### 2. Viscoelastic regime

This regime is the desired regime for the RheoQCM to be utilized as a high-frequency rheometer. Sufficient differentiation between values of $\Delta fn/n$ for different harmonics exists so that the viscoelastic analysis can be performed. In addition, $\Delta \Gamma n$ for the measurements is sufficiently large to be accurately determined, but small enough so that the resonant properties of the crystal circuit can still be obtained from an impedance scan.

#### 3. Overdamped regime

In this regime, $\Delta \Gamma n$ is large enough so that a well-defined crystal resonance can no longer be obtained. No information can be obtained from a QCM experiment in this regime.

#### 4. Bulk regime

In the bulk regime, where the medium in contact with the crystal is thicker than the decay length of the shear wave [Eq. (9)], no mass information is obtained. However, in this region, the viscoelastic properties are reliably determined using the combination of $\Delta f$ and $\Delta \Gamma $, through the use of Eqs. (7) and (8).^{7,31,35,36} The bulk regime does not exist for the epoxy systems, although it exists for softer materials investigated with the QCM. An example regime map for a model system based on polyelectrolyte complexes in equilibrium with an overlayer of water has been given previously.^{42}

The boundaries between the different regimes in Fig. 4 are somewhat arbitrary and are determined by the specific criteria used in their definition. In Fig. 4, the boundary between the Sauerbrey and viscoelastic regimes is defined by the criterion that $\Delta f3/3$ and $\Delta f5/5$ differ by 100 Hz, and the boundary between the viscoelastic and overdamped regime occurs where $\Gamma 5=20$ kHz. Simpler versions of these criteria can also be used to provide guidance in experimental design. An example is the simple rule of thumb described in Ref. 31, where the borderline between the Sauerbrey and viscoelastic regime is defined by the criterion that $d/\lambda 3=0.05.$ Also, not all portions of the viscoelastic regime are equally desirable. Uncertainties in the extracted material properties depend on the specific thickness and are generally minimized for thicknesses in the upper range of the viscoelastic regime, close to the border with the overdamped region. In our comparison in the section of different QCM models, we assume an optimized thickness corresponding to the dashed line in Fig. 4(b), where $\Delta \Gamma 5=10$ kHz.

### D. Uncertainty analysis and model comparison

Two sources of uncertainty enter into the determination of rheological properties from a QCM experiment. The first of these is experimental, originating from nonuniformities in the polymer film that is being investigated. The film is assumed to be uniform in properties throughout its thickness and is also assumed to be laterally homogeneous over the active area of the quartz crystal, with typical lateral dimensions of $\u22481$ cm. While some creativity is sometimes required in order to make sure these conditions are met for some of the systems of interest, including relatively low viscosity polymer precursors at the early stages of cure, a wide variety of films meeting these criteria can be obtained by spin-casting. In these cases, the primary source of uncertainty originates from the uncertainty in the measured values of $fn$ and $\Gamma n$. The resonant peaks are often complicated by the presence of anharmonic sidebands at frequencies slightly higher than the resonance peak of interest. The spectrum shown in Fig. 2 is an example, showing a small sideband for the fifth harmonic. Appropriately accounting for these sidebands is particularly important when the bandwidth of the peak is large, where substantial peak overlap may occur.

Uncertainties in the measured values of the $\Delta fn$ and $\Delta \Gamma n$ propagate to an uncertainty in the film properties that are extracted by fitting to the predictions from the predicted QCM response. A linearized uncertainty analysis is used here, illustrated with a 3:5,3 calculation, where values of $d\rho $, $\varphi $, and $|G3\u2217|\rho $ are extracted from measured values of $\Delta f3$, $\Delta f5$, and $\Delta \Gamma 3$. A linearized analysis of the property uncertainty involves the following steps:

Determination of $\Delta f3err$, $\Delta f5err$, and $\Delta \Gamma 3err$, and the uncertainties in $\Delta f3$, $\Delta f5$, and $\Delta 3$ based on the admittance fits illustrated in Fig. 2. As a rule of thumb, these uncertainties are $\u22483$% of the value of $\Gamma $ for the harmonic of interest. This is the assumption made in our analysis.

Evaluation of the sensitivity of each of the extracted parameters ($d\rho $, $\varphi $, and $|G3\u2217|\rho $) to each of the measured quantities ($\Delta f3$, $\Delta f5$, and $\Delta \Gamma 3$). These are expressed as a matrix of partial derivatives obtained from the Python routine that fits the measured quantities to the predictions from the mathematical model. For example, the sensitivity of the extracted value of $|G3|\rho $ to the measured value of $\Delta f3$ is $\u2202(|G3|\rho )/\u2202\Delta f3$.

Extraction of the total error for a given property by addition in quadrature the contributions from $\Delta f3err$, $\Delta f5err$, and $\Delta \Gamma 3err$. As an example, the following is obtained for the error in $|G3|\rho $:

Application of the error analysis is illustrated in Fig. 5, which includes data for a rubbery polymer system (polyisoprene with a high 3–4 content) investigated above its glass transition temperature.^{43} Note that in order to obtain the material response across a broad temperature range, a series of samples with different thickness generally needs to be utilized.

While the largest contributions to the property uncertainty generally originate from uncertainties in the measured values of $\Delta fn$ and $\Delta \Gamma n,$ some situations exist where errors may be introduced by the small load approximation itself, in which case the analysis based on the solution to the Lu–Lewis equation [Eq. (3)] needs to be used in place of the simpler analysis based on the small load approximation [Eq. (5)]. To understand the significance of the small load approximation and also to understand the relative merits for using the power law and Voigt models for approximating the frequency dependence of the rheological response of the material, we consider a thin film with the viscoelastic properties shown in Fig. 3. These properties are replotted as the solid lines in the top panels of parts (a) and (b) of Fig. 6. The solid line in the top panel of Fig. 6(c) shows the optimum thickness for which $\Gamma 5=10$ kHz. The Lu-Lewis equation was solved using these property values to obtain corresponding values of $\Delta f3$, $\Delta f5$ and $\Delta \Gamma 3$. Because the Lu–Lewis equation gives the most accurate representation of the QCM response, these calculated values for the frequency and dissipation shifts were taken as the values that would be obtained in an actual experiment. These values were then used as inputs to three different approximation schemes for calculating the properties. These approximations are designated as LL, SLA, and Voigt in Fig. 6 and correspond to the following:

LL: Lew–Lewis equation with the power law rheological model.

SLA: small load approximation with the power law rheological model.

Voigt: small load approximation with the Voigt rheological model.

Relative errors associated with each of these approximations are plotted at the bottom of Fig. 6. In addition, properties extracted from the small load approximation with the power law rheological model are plotted with the actual properties in the top panels of this figure. Two primary conclusions can be drawn from these comparisons. First, the similarity between the relative errors for the LL and SLA cases in Fig. 6 indicates that the main source of error in the calculation originates from the approximation used to relate viscoelastic properties at the third and fifth harmonics and are not intrinsic to the use of the small load approximation. Also, we see that errors associated with the use of the power law rheological model are much lower than errors introduced by use of the Voigt rheological model. Given both the accuracy and the simplicity of the power law model, our view is that it should always be used in preference to the Voigt model when using the QCM to determine rheological properties of materials from data obtained at two different harmonics. Even with this approximation, however, an error of up to 25% in both $|G3\u2217|$ and $\varphi $ is obtained in the high temperature regime, where the factor of 5/3 difference in frequency between the third and fifth harmonics corresponds to a measurable increase in the phase angle. For the model data investigated here, the largest errors occur at the low-frequency side of the peak in the phase angle, where $|G3\u2217|\u2248107$ Pa, and where the difference in phase angle for $n=3$ and $n=5$ can be as large as 8$\xb0$. For stiffer materials, corresponding to the high-frequency side of the phase angle peak where the materials is either glassy or nearly glassy, the maximum error in the magnitude and phase of the complex shear modulus is around 8%.

## III. CASE STUDY: POLYMER BEHAVIOR IN THE GLASS TRANSITION REGIME

In order to illustrate the use of the QCM as a fixed frequency rheometer, we describe its use here for the characterization of glassy polymer films plasticized with the solvent from which they were spuncast. These experiments illustrate the benefits of the thin film nature of the experiment, since equilibration through the film thickness is relatively fast. They also illustrate the benefit of the surface loading geometry, which allows the film to be equilibrated with its environment. We discuss solvent effects in Sec. III A, and proceed to a discussion of the behavior of polymers in the glass transition regime for the case where solvent has been completely removed from the polymer films.

### A. Effects of solvent

In the case study presented here, we used atactic polystyrene (PS) with a number average molecular weight of 131 kg/mol and amorphous poly(methyl methacrylate) (PMMA) with a number average molecular weight of 100 kg/mole. In both cases, films were spuncast from toluene solutions directly onto the gold electrode surface of the 0.5 in. diameter quartz crystals used for the QCM experiments. The solution concentrations were adjusted to give films of the appropriate thickness. The crystals were mounted into holders designed to measure the impedance and admittance spectra of the crystals, and measurements began within several minutes of the conclusion of the spincoating process. The samples were heated to a maximum temperature of 150 $\xb0$C at a rate of 1 $\xb0$C/min and were cooled back to room temperature at this same rate. This heating/cooling cycle was then repeated a second time, giving the full temperature profile shown in Fig. 7. Information regarding the effect of solvent on the measured properties is obtained from the first heating cycle (shown as a red dashed line and labeled as phase “1” in Fig. 7). In this phase of the experiment, the properties depend not only on the temperature but also on the time, as the solvent is slowly removed from the sample by evaporation. In the subsequent phases of the experiment, plotted as the “x” symbols in Fig. 7 and labeled as phase “2,” “3,” or “4,”the solvent has been completely removed and the film properties depend only on the temperature.

The effect of residual solvent on both the polystyrene and PMMA films is shown in Fig. 8. At the beginning of the initial heating cycle, there is a substantial amount of toluene left in the films. The amount of residual solvent is quantified by the mass per unit area for the different films, which we specify in terms of $d\rho $, the product of the film thickness, and the density. As shown by an inspection of Figs. 8(a) and 8(d), the solvent mass fraction in the polystyrene film at the beginning of the experiment is $\u22480.06$. The initial solvent mass fraction in the PMMA film is higher, equal to about 0.11. The solvent also persists to higher temperatures in PMMA than it does in polystyrene.

For the polystyrene films, the solvent content is small enough so that the phase angle is unaffected, whereas a measurable increase in the phase angle is observed for the PMMA films. The correlation between the solvent content and the elastic and the modulus-density product is illustrated in Fig. 9. Here, we compare the value of $|G3\u2217|\rho $ at a given solvent weight fraction for the initial heating phase, where solvent is present at a given weight fraction, $ws$, to $(|G3\u2217|\rho )dry$, the value of $|G3\u2217|\rho $ at the same temperature, but where no solvent is present in the film. In other words, we are plotting the relative difference between the dashed, red line and the blue symbols in Figs. 8(a) and 8(c). The solid and dashed lines in Fig. 9 represent the following expression for the relative change in $|G3\u2217|\rho $:

with $K=1.7$ giving a good approximation to the PS data, and $K=8$ giving a good approximating to the PMMA data for low values of $ws$.

For this particular set of conditions, the effect of toluene on the modulus of PMMA is larger than the effect of toluene on the modulus of polystyrene. Some care needs to be taken when interpreting these results because of two complicating factors. The first of these concerns is the dynamic nature of the experiment, with the solvent leaving through the film by evaporation from the top surface. Concentration gradients in the film will exist if the rate of the solvent diffusion within the film is sufficiently slow in comparison to the evaporation rate. Also, because the temperature is changing during the experiment, the temperatures corresponding to a given value of $ws$ are different for the PS and PMMA films. Both of these complicating factors can easily be addressed by performing a more complete set of control experiments where the solvent content in the film is controlled more carefully than in the simple experiments reported here. Our point here is simply to illustrate the sort material property data that can easily be obtained in a RheoQCM experiment.

An additional comparison between the polystyrene and PMMA materials is shown in Fig. 10, where we plot all of the data from Fig. 8 as a modified van Gurp–Palmen plot, with $\varphi $ plotted as a function of $|G3\u2217|\rho $. van Gurp–Palmen plots, where $|G3\u2217|$ is generally not multiplied by $\rho $, were developed initially as a way testing the validity of time-time-temperature superposition for polymeric systems.^{39,40} In our case, they provide a convenient way of comparing data obtained at the much lower frequencies that are more commonly employed in rheometric measurements. In Fig. 10 we see that dry films of PMMA and PS have similar van Gurp–Palmen plots, but with values of $|G3\u2217|\rho $ being about twice as large for PMMA as they are for polystyrene. In Sec. III B, we discuss the behavior of the dry polystyrene films in more detail.

### B. Molecular weight dependence of the glass transition

Temperature-dependent data for dry polystyrene samples of different molecular weights are shown in Fig. 12. The temperature dependence of the viscoelastic properties for each of the polymers is similar, after accounting for the molecular weight dependence of the glass transition temperature. This similarity in behavior is most clearly seen in the modified van Gurp–Palmen plots for the three polystyrene samples shown in Fig. 11. An additional relaxation is observed in the high-modulus portion of the sample with the lowest molecular weight, but otherwise the curves overlap, indicating a similarity in the overall relaxation behavior.

The molecular weight dependence of the relaxation dynamics can be accounted for by measuring the temperature at which the phase angle reaches a fixed reference value. The molecular weight dependence of this quantity for a reference phase angle of 5$\xb0$ (referred to here as $T5)$ is shown in Fig. 12(c). Note that $Tg$ and $T5$ have the same temperature dependency, which can be expressed as follows and is represented by the solid line in Fig. 12(c):

Here, $Mn$ is the molecular weight in g/mol and $C=105Kmol/g$. The high-frequency nature of the RheoQCM technique is responsible for the fact that $T5\u221e$ is $\u224835$ K higher than $Tg\u221e$. This difference is consistent with the known temperature dependence for the polymer relaxation times, commonly expressed in the Vogel–Fulcher–Tamman form,

Using typical values for polystyrene of $B\u22483000$ and $T\u221e\u2248310\xb0$C,^{44} we find that the relaxation times decrease by a factor of $2.4\xd7107$ when the temperature is increased from 100 $\xb0$C to 135 $\xb0$C. Equivalently, the response at 135 $\xb0$C at a frequency of 15 MHz gives results comparable to a measurement performed at $\u2248100\xb0C$ at a frequency of 0.5 Hz. Collectively, these results indicate that the high-frequency measurements of the rheological properties are consistent with the expected behavior. The utility of the QCM technique originates from the fact that these dynamics can be accurately quantified for samples that are equilibrated with the sample environment, as illustrated by the solvent swelling experiments from Sec. III A.

## IV. DATA MANAGEMENT

An important issue with QCM experiments as commonly practiced is that the workflow outlined in Fig. 2 occurs as the measurement is being made, with no ability for subsequent reanalysis of the raw data (the admittance and susceptance curves in the case of our experiments). To address this limitation, we have developed Python-based interface for the RheoQCM, with a screenshot shown in Fig. 13. All data, including the raw admittance spectra for the sample and reference scans, time, temperature values obtained during different stages of the analysis ($\Delta fn$, $\Delta \Gamma n$, $\rho d$, $|G3\u2217|\rho $, $\varphi $, etc.) are saved in a single Hierarchical Data Format (HDF5)^{45} file. The software is open source and available on Github,^{46} and currently interfaces with vector network analyzers sold by Makarov instruments.^{47} This type of hardware platform is becoming increasingly available,^{48} with functional QCM instrumentation able to perform rheological measurements costing as little as several hundred dollars. Opportunities presented by this availability include the development of educational modules illustrating the measurement of viscosity, vitrification of glass, adsorption of water by a hydrophilic material, etc. The inexpensive combined hardware/software platform enables a variety of environmental monitoring applications, including those of interest to the museum science community, with whom we have an established relationship.^{49,50}

## V. SUMMARY

The aim of this paper has been to illustrate the use of the RheoQCM to quantify the viscoelastic properties of polymeric films at a fixed reference frequency, which for the experiments reported here is 15 MHz. Data from two resonant harmonics need to be obtained in order to extract the film properties, using a closure relationship relating the viscoelastic properties at these two harmonics. The largest source of error in the measurements originates from the approximate nature of this closure relationship. Nevertheless, an error of a 8% or less is obtained for the magnitude and phase of the complex shear modulus in the glassy or near-glassy regimes. A slightly larger error is obtained in the high temperature regime, where measurably different viscoelastic phase angles are observed for frequencies differing by a factor of 5/3. The technique was applied to polystyrene and poly(methyl methacrylate) thin films, illustrating the ability of the method to provide quantitative information from samples equilibrated with the ambient environment.

## ACKNOWLEDGMENTS

This work was supported by the National Science Foundation (NSF) (No. DMR-1710491) and by Financial Assistance Award No. 70NANB19H005 from the U.S. Department of Commerce, National Institute of Standards and Technology as part of the Center for Hierarchical Materials Design (CHiMaD).

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