Article pubs.acs.org/Langmuir
High-Frequency Rheological Characterization of Homogeneous Polymer Films with the Quartz Crystal Microbalance Garret C. DeNolf, Lauren F. Sturdy, and Kenneth R. Shull* Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208-3108, United States ABSTRACT: We utilize quartz crystal resonators operating at multiple resonant harmonics to measure the high-frequency rheological properties of materials with a broad range of viscoelastic properties. The technique is demonstrated with poly(tbutyl acrylate) films in the vicinity of the calorimetrically determined glass transition and with rubbery polyisoprene films. The technique is a noncontact technique that can be used to quantify the temperature or time-dependent viscoelastic response in homogeneous films with thicknesses in the micrometer range. This work complements the ability of the resonators to quantify the viscoelastic behavior of viscoelastic polymer solutions and simple Newtonian liquids. For each material we obtain the density−shear modulus product and the viscoelastic phase angle at frequencies of 5 and 15 MHz. A standardized analysis protocol is described that enables this information to be obtained reliably and accurately. The polyisoprene data are found to be in good agreement with measurements obtained by dynamic mechanical analysis using extrapolated temperature shift factors.
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INTRODUCTION Most mechanical property measurements of films and coatings require that some mechanical contact be made with the top surface of the coating. Microindentation or nanoindentation are perhaps the most well-known examples. These are powerful techniques but are difficult to apply in a rapid fashion to evaluate time-dependent changes in mechanical properties associated with curing or aging of a material. Additionally, contact approaches cannot easily be applied to very soft or liquidlike layers, where adhesive interactions with a probe tip become problematic. In these situations, noncontact methods for determining the mechanical properties of the layer are preferred. A common method for assessing the mechanical properties of coatings involves the propagation of acoustic waves in the coating and the reflection of these acoustic waves at the coating substrate and coating/air interfaces.1−5 The connection to the linear viscoelastic properties of the material is through the complex acoustic impedance, Z*. The acoustic impedance has units of stress/velocity and is given by the square root of the product of the density and the appropriate elastic modulus of the material. For the propagation of transverse shear waves relevant to our experiments, Z* = (ρG*)1/2, where G* is the complex shear modulus. 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* and hence the complex shear modulus, G*, as well.3−6 A problem with many acoustic wave measurements is that customized substrates are required, an issue that can be problematic when high-throughput methods that can be applied to large numbers of samples are desired. Of the possible substrate choices, quartz crystal resonators are the most practical alternative. Because quartz crystal resonators are commonly used as mass sensors,7,8 the crystals and the © 2014 American Chemical Society
equipment needed to measure the appropriate resonance properties of these crystals are readily available and are quite inexpensive. A quartz crystal used as a mass sensor is commonly referred to as a quartz crystal microbalance (QCM) . We use QCM as a general abbreviation for the approach when applied to either simple mass sensing or the quantification of the rheological properties of a coating that is placed onto the surface of a quartz crystal resonator. The method enables the in situ measurement of coating properties without bringing an external probe into contact with the film of interest. In this way, the difficulties of contact mechanics-based techniques, such as probe adhesion and area of contact uncertainty, are avoided. Qualitative versions of the method have been used to study cross-linking kinetics,9−11 swelling,12−14 and aging behavior.15,16 A quantitative approach to rheological measurement with the QCM was developed by Lucklum et al. in 1997.1,2 In this earlier work, the rheological response of the material required that the thickness and density of the film be determined by independent means. In the present paper we describe an analysis method based on the use of multiple harmonics that enables properties of the material to be obtained without the need for separate measurements of any of the film properties. Use of the technique requires that the film thickness be within a specified range, typically between 1 to 10 μm. For films that are too thin, the response of the crystal is sensitive only to the mass thickness of the coating and not to the rheological properties of the material. For thicker films the crystal resonance can be too strongly damped. With suitable experimental design and data interpretation, a remarkably diverse range of materials can be investigated using a robust and straightforward analysis procedure that gives very accurate results. We have previously Received: June 6, 2014 Revised: July 13, 2014 Published: July 14, 2014 9731
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described in the excellent review article by Johannsmann.19 Here we summarize the elements that are most relevant to our own analysis. The fundamental relationship is a simple linear relationship between the shift in the complex resonance frequency Δf *n and the corresponding shift in the complex load impedance, ΔZ*n :20
applied the technique to curing and aging studies of rigid paint coatings.17,18 Our focus in the present paper is on the extension of the technique to rubbery materials and on the experimental validation with more traditional rheometric methods. In the following section we describe the theoretical background for the technique. The remainder of the paper provides experimental examples for glassy and rubbery polymer films. Quartz Crystal Resonators. The QCM consists of a single-crystal quartz disk cut at a specific angle so that it oscillates in shear when an alternating voltage is applied to electrodes deposited on opposite sides of the crystal. As the crystal oscillates, a shear wave is excited that propagates into any material that is deposited onto one of the electrode surfaces of the crystal. The setup is shown schematically in Figure 1.
Δf n* =
πZq
(2)
In eq 2 Zq is the acoustic impedance of the quartz (i.e., Zq = (μqρq)1/2), where μq is the shear modulus of quartz in the plane of the quartz disc used as the oscillator and ρq is the density of quartz. Quartz has a density of 2.66 g/cm3. The shear modulus for the AT cut quartz used in our experiments is 2.95 × 1010 Pa, giving Zq = 8.84 × 106 kg/m2s. The complex impedance is defined as the ratio of the interfacial shear stress, σn*, to the velocity of the shear wave, u̇n*: ΔZn* ≡
σn* uṅ*
(3)
In our case σ*n and u̇*n correspond to the shear stress and shear velocity at the film/QCM interface. In the following subsection we consider the well-known thin film limit, where the device is sensitive to mass and not to the rheological properties of the coating. This is followed by a discussion of the bulk limit, where the device is sensitive to the viscoelastic properties of a thick liquid layer that is put in contact with the crystal but is insensitive to the thickness of this layer. This limit is restricted to fluids with a relatively low complex viscosity so that the crystal resonance is still measurable. The most relevant section of the experiments reported in this work is the third subsection, where the layer is thick enough that the oscillator is sensitive to the viscoelastic properties of the layer but not so thick that a high-modulus coating completely damps the crystal resonance. Thin Film Limit. The simplest situation to consider is the thin film limit, where the load impedance, ΔZn*, is due to inertial effects. In this case the interfacial stress (equal to the added mass times the interfacial acceleration) is completely out of phase with the interfacial velocity. The angular frequency of the oscillation at the nth harmonic is 2πnf1, so in the inertial limit ΔZ*n = 2πinf1MA. Here MA is the mass per unit area, assumed to be uniformly distributed over one of the crystal faces. When combined with eq 2 we see that Δf n* = −Δfsn, with Δfsn given by the following expression:
Figure 1. Schematic representation of a quartz crystal microbalance where an alternating voltage is applied across two gold electrodes (a). Spectrum of electrical conductance measured from the QCM with the corresponding resonance frequency ( f n) and dissipation (Γn) labeled (b). The solid line corresponds to data from an unloaded crystal, and the dashed line corresponds to data from a loaded crystal.
The resonance properties of the crystal can be determined by either time domain or frequency domain experiments.19 In our case we work in the frequency domain, with the resonance frequency defined as the frequency where the electrical conductance of the circuit is maximized. The fundamental resonance of the quartz disks utilized in these studies is at 5 MHz. Additional resonances are obtained at the odd harmonics of this fundamental frequency. As described in more detail below, measurement at multiple harmonics is essential to the analysis that we have used. Lorentzian peak fits to the conductance curve yield two components of the complex resonance frequency f n*: the resonance frequency of the quartz (f n) where the conductance is maximized and the dissipation (Γn), which is the half-maximum half-width of the conductance peak. The frequency and dissipation are referenced to the values obtained from the oscillator circuit before the coating of interest was added, enabling us to define a complex resonance frequency shift, Δf *n , where Δf n is the real component and ΔΓn is the imaginary component: Δf n* = Δfn + iΔΓn
if1 ΔZn*
Δfsn =
2nf12 Zq
ΔMA =
2nf12 Zq
ρd (4)
We assume throughout this paper that the properties of the film are uniform in both the thickness and lateral directions so that ΔMA = ρd, where ρ is the density and d is the thickness of the layer deposited on the QCM surface. A version of eq 4 was derived by Sauerbrey,21 and this thin-film limit for which it applies is typically referred to as the Sauerbrey limit. As described in more detail below, Δfsn is a natural normalization for shifts in both frequency and dissipation for more complicated situations. Bulk Limit. For a very thick material ΔZn* reduces to the acoustic shear impedance of the material, which we designate here simply as Z*n :
(1)
Note that the subscript n that we use throughout our treatment refers to measurements made at the nth harmonic (i.e., n = 1 for the fundamental resonance frequency of 5 MHz, n = 3 at the 15 MHz harmonic). A generalized approach is valid when the frequency shift is small in comparison to the resonance frequency itself, which is true in virtually all uses of the QCM. The general background is
Zn* = (ρGn*)1/2 9732
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Figure 2. Visual representations of eq 14 (real part in (a) and imaginary part in (b)), eq 15 (c), and eq 16 (d).
Here G*n is the complex shear modulus of the material at the frequency corresponding to the nth harmonic. We choose to express the complex modulus in terms of its magnitude, |Gn*|, and phase angle, ϕn:
Gn* = |Gn*|exp(iϕn)
use of quartz crystal resonators as sensors immersed in bulk fluids. For an AT cut quartz crystal operating at a fundamental resonance frequency of 5 MHz exposed on one side to water (η = 10−3 Pa s and ρ = 1 g/cm3), ΔΓ = −Δf = 713 Hz. As a general rule, we have found that high-quality data can be obtained when Γ is less than ∼20 kHz, corresponding to solution viscosities of less than about 0.5 Pa s (for ρ = 1 g/ cm3). In order to quantify the properties of higher-viscosity materials, including almost any useful polymeric coating beyond the very early stages of the curing process, thin films must be utilized, using an analysis based on the description provided in the following subsection. Intermediate Regime. In the intermediate-film-thickness regime the following general equation for the complex load impedance of the film must be used:23
(6)
Once these quantities are known, the storage and loss moduli, G′n and G″n , can be determined: Gn′ = |Gn*| cos(ϕn) Gn″ = |Gn*| sin(ϕn)
(7)
The following expressions are obtained for the frequency and dissipation shifts: Δfn = −Δfn ∞ ; Δfn ∞ =
ΔΓn = ΔΓn ∞ =
f1 πZq
f1 πZq
(ρ|Gn*|)1/2 sin(ϕn/2)
1/2
(ρ|Gn*|)
cos(ϕn/2)
ΔZn* = iZn* tan(kn*d) (8)
Here kn* is the complex wavenumber, which is inversely proportional to the acoustic shear impedance of the material:
(9)
kn* =
For a Newtonian liquid, ϕ = 90° and |G*n | = 2πηnf1, where η is the viscosity of the liquid. In this case the frequency and dissipation shift are given by the following equation: ΔΓ n90∞ = −Δf n90∞ =
2πnf1 ρ 2π = [1 − i tan(ϕn/2)] λn Zn*
(12)
The assumption here is that the top surface of the film is in contact with a layer of zero impedance, e.g., air or vacuum. The quantity λn is the wavelength of the shear wave, given as follows:
f13/2 ⎛ nηρ ⎞1/2 ⎜ ⎟ Zq ⎝ π ⎠
(11)
(10)
Δf 90 n∞
Note that for n = 1 the expression for reduces to the wellknown expression given by Kanazawa and Gordon.22 Equation 10 can be used to determine some general guidelines for the
λn = 9733
1/2 1 ⎛ |Gn*| ⎞ 1 ⎟ ⎜ fn ⎝ ρ ⎠ cos(ϕn/2)
(13)
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angle, ϕ, is unchanged for the different harmonics that are measured. This assumption is equivalent to assuming that the magnitude of the shear modulus has a power law frequency dependence over a reasonably broad frequency range25,26 so that the dependence of |G*| on the harmonic order, n, is given as follows:
These equations can be combined to give the following generalized expression for the complex frequency shift:24 Δf n* Δfsn
=
−tan{(2πd /λn)(1 − i tan(ϕn/2))} (2πd /λn)(1 − i tan(ϕn/2))
(14)
We can also normalize by the bulk values of Δf n and ΔΓn given by eqs 8 and 9, from which we obtain the following: ⎡ −2 tan{(2πd /λ)(1 − i tan(ϕ /2))} ⎤ n ⎥ = Re⎢ ⎢⎣ ⎥⎦ sin(ϕn)(1 − i tan(ϕn/2))
(15)
⎡ −tan{(2πd /λ )(1 − i tan(ϕ /2))} ⎤ ΔΓn n n ⎥ = Im⎢ 2 ⎢⎣ cos (ϕn/2)(1 − i tan(ϕn/2)) ⎥⎦ ΔΓ∞ n
(16)
Δfn Δf∞ n
|Gn*| ∝ nϕ /90
Here we continue our convention of expressing ϕ in degrees. We show in the Results and Discussion section that use of this constant phase angle approximation does not introduce significant error into the measurement. From eqs 13 and 18 the shear wavelength scales with the harmonic order as follows: λn ∝ nφ /180 − 1
(19)
An appropriate choice of film thickness enables Δfs1, d/λ1, and ϕ all to be obtained from the measured data. These quantities can be arranged algebraically to obtain dρ, |G*1 |ρ, and ϕ. The detailed numerical procedure utilizes MATLAB as described previously17 and requires that data be obtained at two different harmonics, n1 and n2. In our previous work we used n1 = 1 (the fundamental resonance at 5 MHz) and n2 = 3 (the third harmonic at 15 MHz). A more generalized procedure that utilizes different harmonics is as follows: (1) A quantity that we call the harmonic ratio, rh(n1, n2), is obtained experimentally by comparing the frequency shifts at n = n1 and n = n2:
In Figure 2 the resonance frequency and the dissipation are plotted with two different normalizations. The first normalization (Figure 2a,b) is based on the Sauerbrey shift from eq 14 which is relevant to thin polymer films. The second normalization (Figure 2c,d) is based on the behavior in the bulk limit from eqs 8 and 9. The plot of Δf n/Δfsn approaches a limit of −1 for very small values of d/λn, consistent with the Sauerbrey equation. For d/λn ≈ 0.25 there is a very large increase in the dissipation, and the frequency shift becomes very large in magnitude. This is the so-called film resonance condition, where at the QCM/film interface the shear wave reflected from the film surface is exactly out of phase with the oscillation of the crystal itself. The dissipation is much too large for a stable resonance to be obtained under these conditions. However, the increase in the magnitude of Δf n and ΔΓn as d/λn is increased toward 0.25 provides the sensitivity that is needed to extract quantitative mechanical property data, as described in more detail below. The normalizations with respect to ΔΓn∞ and Δf n∞ are included to illustrate the thickness regime for which these limiting expressions for the dissipation and frequency can be reliably used. The relevant length scale here is the decay length of the shear wave propagating into the material in contact with the electrode surface of the QCM. This decay length, δn, is given by the following expression: λ 1 = n cot(ϕn/2) δn = * 2π Im(kn )
(18)
rh(n1 , n2) ≡
n2 Δfn1 n1 Δfn
2
(20)
The harmonic ratio is equal to unity in the Sauerbrey regime. Deviations from this value provide a measure of d/λn, as illustrated in the contour plot shown in Figure 2a. (2) A second quantity, referred to here as the dissipation ratio, rd(n3), is defined as the ratio of the shift in dissipation to the shift in resonance frequency for n = n3: rd(n3) ≡ ΔΓn3/Δfn
3
(21)
(3) Equations 14 and 19 are used to express rh(n1, n2) and rd(n3) in terms of ϕ and d/λn1. These equations are solved numerically, adjusting ϕ and d/λn1 so that the predicted and experimentally determined values for rh and rd are in agreement with one another. (4) Once d/λn1 is known, dρ can be determined from eqs 4 and 14 and the measured value of Δf n1. If appropriate film thicknesses are chosen, then accurate results can be obtained for films spanning the full regime of polymer materials behavior, including low-viscosity liquids, highly dissipative elastomeric films, and glassy polymer films. In our discussion of the results we use the notation n1:n2,n3 to refer to the values of the harmonics used to calculate the film properties. For example, a 1:3,1 calculation refers to the case where the harmonic ratio is obtained from the first and third harmonics, and the dissipation ratio is determined from the first harmonic. Error Analysis. In general we measure four quantities (Δf n1, Δf n2, Δf n3, and ΔΓn3), and we use these quantities to extract three physical properties of the coating (dρ, ϕ, and ρ|G*n1|ρ). Some experimental uncertainty exists in each of the measured
(17)
From this equation it is evident that for a Newtonian liquid with ϕn = 90° the decay length is λn/2π. The response of viscous systems with d ≫ δn (d/λn ≫ 0.16) can therefore by approximated well by eqs 8 and 9. The effects of the reflections from the film surface may still be important, however, so the correction factors represented by eqs 15 and 16 should be used if the film thickness is not substantially larger than λn. In the work that follows we are generally interested in films with thicknesses thinner than the film resonance condition (d/λn < 0.25). In this regime the Sauerbrey normalization is much more convenient because Δfsn does not depend on the rheological properties of the film, which is changing as either the temperature or the cure state of the film changes during an experiment. Method of Solution. Because the wavelength depends on the frequency as described by eq 13, collecting data at two harmonics enables us to simultaneously probe two different values of d/λn on the response maps shown in Figure 2. We make the simplifying approximation that the viscoelastic phase 9734
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Temperature Ramp Profile and Calibration. A temperature calibration step was necessary because shifting temperatures affect the resonance frequency of the uncoated quartz crystals. The quartz crystals used for these experiments were AT cut, commonly used for measurements with the QCM because of the small temperature dependence of the fundamental resonance frequency near room temperature. However, the temperatures for these experiments were in the range where the shift in resonance frequency for the needed higher harmonics was significant enough to justify the calibration. The effect of temperature on the dissipation was smaller than the experimental uncertainties at any fixed temperature (values listed in Table 1). After determining the bare crystal resonance frequency at room temperature, an unloaded crystal was mounted in the holder and placed in an oven for a temperature ramp. The measurements were taken in 10° increments for 30 min at each temperature to allow the device to equilibrate for each measurement. QCM measurements were taken continuously during the temperature ramp, and the average values in the final minute at each temperature were recorded. The calibration was performed three times for two different quartz crystals to confirm repeatability. This temperature calibration was used to determine the values of Δf n for each experiment. The results of the temperature calibration are shown in Figure 3.
quantities, which we represent with a superscript e. For example, the uncertainty in Δf n1 is denoted as Δf en1. Our procedure for extracting the error caused by each of these uncertainties is to calculate sensitivity functions relating each of the extracted quantities to each of the measured input quantities. Suppose, for example, that we want to understand how sensitive an extracted quantity Q (where Q can be dρ, |Gn*1|ρ, or ϕ) is to the measured value of Δf n1. The appropriate sensitivity function in this case is calculated by obtaining solutions for values of Δf n1 that are slightly different from the measured value and using these values to estimate the partial derivative, ∂Q/∂Δf n1. A similar procedure can then be used to calculate the partial derivatives with respect to Δf n2, Δf n3, and ΔΓn3. The total error in Q is estimated by summing the uncertainties in quadrature: ⎞2 ⎛ ⎞2 ⎛ ⎞2 ⎛ Q Q Q ∂ ∂ ∂ e e e Δf n ⎟⎟ Δf n ⎟⎟ + ⎜⎜ Δf n ⎟⎟ + ⎜⎜ (Q ) = ⎜⎜ 1 2 3 ⎝ ∂Δfn1 ⎠ ⎝ ∂Δfn2 ⎠ ⎝ ∂Δfn3 ⎠ e 2
⎛ ∂Q ⎞2 e + ⎜⎜ ΔΓ n3⎟⎟ ⎝ ∂ΔΓn3 ⎠
(22)
Note that if n3 is equal to n1 or n2 then the term involving ∂Q/ ∂Δf n3 is redundant and is not used in the calculation of Qe. The data given in the following sections includes error bars with a length of 2Qe. Values for Δf en and ΔΓen were obtained from the measured reproducibility for these quantities, obtained when the unloaded quartz crystals are removed from the holder and then reinserted. These values are listed below in Table 1. When Table 1. Experimental Uncertainties in the Frequency and Dissipation Shifts n
Δf en (Hz)
ΔΓen (Hz)
1 3 5
45 120 270
11 22 14
Figure 3. Temperature calibration of bare quartz crystal from room temperature to 80 °C for the resonance frequency. Frequency shifts are relative to the values measured at 25 °C.
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RESULTS AND DISCUSSION In order to illustrate the ability of the QCM to measure the mechanical response over a broad parameter space, we include data from 25 to 80 °C for polyisoprene and poly(t-butyl acrylate) (PtBA), amorphous polymers with very different glass-transition temperatures. Results for both of these polymers are described below. Poly(t-butyl acrylate). Poly(t-butyl acrylate) (PtBA) is glassy at room temperature with a calorimetrically determined glass-transition temperature of ∼50 °C.27 Measured values of Δf n and ΔΓn for n = 1 and 3 are shown in Figure 4, and the film properties extracted from the analysis outlined above are shown in Figure 5. In the Sauerbrey limit, the normalized frequency shifts (Δf n/n) are independent of n and do not depend on temperature. Deviations from the Sauerbrey limit are evident from the temperature dependence of Δf n/n and by the fact that Δf n/n is larger for n = 3 (the 15 MHz harmonic) than it is for n = 1 (the fundamental, 5 MHz harmonic). Rheological information is extracted from the details of these deviations, which become larger as the film thickness increases. For this reason, the accuracy of the method increases as the film thickness increases. If the films become too thick, however, the dissipation becomes large enough (>20 kHz) that a resonance
no error bars are shown, the length of the calculated error bars is smaller than the symbols used to represent the data. Note that our calculated errors include only uncertainties in the measured values of ΔΓn and Δf n and do not include additional errors introduced by the use of inhomogeneous polymer films.
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EXPERIMENTAL METHODS
Sample Preparation. Poly(t-butyl acrylate) (PtBA, Mw = 52 000 g/mol) was used as a model glassy polymer film, and cis-1,4polyisoprene was used as a model rubbery material. For both polymers, the QCM was used to extract film properties at temperatures from room temperature to 80 °C. Both polymers were dissolved in toluene and spun cast directly onto the electrode surface of the quartz crystals. For the experiments with dynamic mechanical analysis, sulfur crosslinked polyisoprene was created using the following procedure. One gram of polyisoprene and 0.01 g of antioxidant was added to 24 mL of toluene, and the solution was stirred for 2 to 3 days until the rubber was fully dissolved. Next, 0.01 g of sulfur, 0.0075 g of accelerator, and 8 mL of carbon disulfide were added to the solution and stirred for 5 min. The film was formed by casting from solution, creating a film ∼0.8 mm thick, and the cross-linking in the film was performed by placing the sample in the oven for 20 min at 100 °C, followed by 40 min at 150 °C. 9735
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calculated value of ΔΓ3 is also in good agreement with the measured value, providing confirmation on the consistency of our treatment. Similar agreement is observed for the measured value of ΔΓ1 and the value of ΔΓ1 determined from the 1:3,3 analysis. The error bars for the 1:3,3 analysis are smaller than those for the 1:3,1 analysis because the dissipation is higher for the third harmonic and is less sensitive to any uncertainties in ΔΓ. In general we find that the most accurate values of the film properties are obtained from the 1:3,3 analysis. Polyisoprene. The glass-transition temperature of polyisoprene used in these studies was approximately −60 °C as determined by differential scanning calorimetry. We investigated two samples: a spun-cast thin film investigated with the QCM and a sulfur-cross-linked version of the same polymer that was investigated with traditional dynamic mechanical analysis (DMA). Dynamic Mechanical Analysis. The dynamic mechanical data obtained from the cross-linked polyisoprene sample are shown in Figure 6. In part (a) we show the frequency dependence of |G*|, obtained from the tensile experiment by dividing the dynamic Young’s modulus by a factor of 3 (assumed Poisson’s ratio of 0.5). Values for the viscoelastic phase angle, ϕ (ϕ = arctan (G″/G′)), are shown in part (b) These data were collected in a tensile geometry at frequencies of 0.1−10 Hz at 2° temperature increments from −66 to −28 °C. A master plot was obtained by multiplying by the temperature shift factors, aT, that are plotted in Figure 6c. These shift factors were fit to the following form of the Vogel equation, where the reference temperature (where aT = 1) is −50 °C
Figure 4. Measured values of the shifts in resonance frequency (Δf n) and dissipation (ΔΓn) for a thin PtBA film at temperatures from 25 to 80 °C.
peak can no longer be measured. For glassy polymers, this confines the layer thicknesses to values of less than about 10 μm. Since the mass per area remains unchanged as the temperature is changed, the variance of these values of about 0.2% is a measure of the accuracy of the mass measurement. This accuracy is not surprising, given the widespread use of the quartz crystal microbalance to measure very small mass changes in thin film samples. However, obtaining this degree of accuracy for films in this thickness range used in our experiments requires that the corrections to the Sauerbrey equation be accounted for in the way described in this paper. For the PtBA film, these corrections range from 1.5% for the first harmonic at 25 °C, where d/λ1 = 0.02, to 13% for the third harmonic at 80 °C, where d/λ3 = 0.09. In our approach we measure four separate quantities (Δf1, Δf 3, ΔΓ1, and ΔΓ3), but only three of these are used for any given determination of the film properties. In each case, the availability of the fourth parameter provides a consistency check of the analysis. For example, in the 1:3,1 analysis ΔΓ3 is not used to determine the film properties. Once these properties have been determined, however, ΔΓ3 can be calculated from eqs 4 and 14. In Figure 4 the lines represent the actual measured values of Δf and ΔΓ and the symbols represent the values determined from the analysis. In the 1:3,1 analysis, the values of Δf1, Δf 3, and ΔΓ1 determined from the analysis agree exactly with the measured values because our method of solution forces these values to be equal to one another. The
log aT = − 11.31 +
603 T + 103.3
(23)
with T in °C. Equation 23 was used to obtain values for the shift factors at the temperatures used for the QCM experiments, as indicated by the circles in Figure 6c. QCM Data. Measured values of ΔΓ and Δf are shown in Figure 7, and the extracted values of dρ, |G*|ρ, and ϕ are shown in Figure 8. As with the PtBA data, we observe good agreement between the measured value of the extra parameter, with the 1:3,1 solution accurately predicting the value of Γ3 and the 1:3,3 solution accurately predicting the value of Γ1. The shear moduli are in the range of 107 Pa, about 2 orders of magnitude below the shear modulus for the glassy (or near-glassy) PtBA
Figure 5. Values of |G*|ρ, dρ, and ϕ for the PtBA film, extracted from the measured values of Δf n and ΔΓn plotted in Figure 4. Values were obtained from the ratio of the frequency shifts at the first harmonic, in conjunction with the measured dissipation at n = 1 (the 1:3,1 data set) or with the measured dissipation at n = 3 (the 1:3,3 data set). Values of |G*| in each case correspond to the first harmonic at 5 MHz. 9736
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Figure 6. Dynamic mechanical data for the cross-linked polyisoprene sample. Values of |G*| and ϕ are plotted in parts (a) and (b) as a function of the temperature-shifted master frequency, aT f. The experimentally determined temperature shift factors, aT, are plotted in part (c), along with the Vogel fit (eq 23). The solid lines in (a) and (b) represent fits to the data that are used in Figures 9 and 11.
Rheological properties for un-cross-linked polyisoprene measured by the QCM are compared to properties obtained by dynamic mechanical analysis (DMA) of a cross-linked version of the same polyisoprene sample in Figure 9. All of the data are plotted against the temperature-shifted frequencies, using the temperature shift factors plotted in Figure 6c. The QCM data shown in Figure 9 are from the 1:3,3 calculation and correspond to a frequency of 15 MHz. A density of 0.93 g/cm3 was assumed for polyisoprene in order to convert the QCM values of |G*|ρ to values of |G*|. In comparing data from these two types of experiments, it is important to keep in mind that the cross-linked and un-cross-linked versions of the polymers are not exactly the same, so the observed quantitative differences are not surprising. In this case, time−temperature superposition works well because the polymer does not undergo any structural changes within the temperature range of the experiments. An important aspect of the QCM and related resonator-based methods is that they provide a direct measure of the high-frequency response that could not be obtained by traditional methods in cases where time− temperature superposition fails.28 A convenient way to represent data obtained from the QCM is to plot |G*|ρ and ϕ against one another, as illustrated in Figure 10. This plot includes both the PtBA and polyisoprene data for the QCM, in addition to the polyisoprene data obtained from the dynamic mechanical experiments. For consistency, all of the data plotted here are from the 1:3,3 calculation and correspond to a measurement frequency of 15
Figure 7. Measured values of the shifts in resonance frequency (Δf n) and dissipation (ΔΓn) for a thin polyisoprene film at temperatures from 25 to 80 °C.
films described above. Because the shear wavelength scales as the square root of the modulus (eq 13), values of the normalized thickness, d/λ1, are larger for the polyisoprene films, even though these films are substantially thinner than the PtBA films. For example, at 80 °C d/λ1 = 0.034 for the polyisoprene film with dρ = 4.53 g/m2. At the same temperature, d/λ1 = 0.10 for the much thinner polyisoprene film, which has dρ = 1.12 g/ m2. As a result, more substantial deviations from the Sauerbrey frequency shifts are observed for the polyisoprene film. These deviations are most noticeable for the values of Δf 3 at the higher temperatures. At 80 °C the magnitude of Δf 3 is reduced from the Sauerbrey value by 36%.
Figure 8. Values of dρ, |G*|ρ, and ϕ for the polyisoprene film, extracted from the measured values of Δf n and ΔΓn plotted in Figure 7. 9737
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Figure 9. Comparison of the magnitude of the complex shear modulus, |G*| (a), and phase angle, ϕ (b), at 15 MHz as a function of temperature for polyisoprene. Data included are from the QCM measurement of spun-case polyisoprene films (symbols) and from the DMA measurements of sulfur-cross-linked polyisoprene (solid lines).
value of |G*n | at the higher harmonics is assumed to follow the power law form of eq 18 that is consistent with our assumption of a constant phase angle over a reasonably broad frequency range. This assumption introduces very little error for small phase angles but may become more important for larger values of ϕ. In order to test the degree to which errors associated with the constant phase approximation might be affecting the analysis, we performed a consistency check based on the data obtained from the DMA experiments. Our approach was to use the measured rheological data to calculate values for Δf n and ΔΓn from eq 14, assuming dρ = 1.12 g/cm3. Our standard analysis using the constant phase angle approximation was then used to recalculate values of |G*|ρ, ϕ, and dρ from these frequency shifts. The agreement between the actual and recalculated values of these quantities is shown in Figure 11 and provides a check of the validity of the constant phase approximation. Calculated values of |G*n |ρ do not appear to be affected by this approximation. In addition, the phase angle itself is also obtained accurately, provided that it is assumed to correspond to the frequency at which the dissipation ratio was calculated, i.e., 5 MHz for a 1:3,1 calculation and 15 MHz for a 1:3,3 calculation. This is because the phase angle is most sensitively determined from the dissipation ratio, whereas the shear wavelength itself (and hence the harmonic ratio) is only weakly dependent on ϕ (eq 13). As a result, our simple procedure for obtaining the viscoelastic properties of the phase angle is quite reliable. The overall mass density of the film is also obtained accurately, with small deviations observed under conditions where the phase angle is relatively large and is
Figure 10. Magnitude of the complex shear modulus versus phase angle for PtBA and polyisoprene. For comparison, we show the properties of a cross-linked polyisoprene film measured with traditional dynamic mechanical analysis.
MHz. For a given material in Figure 10, different points in the figure were accessed by changing the temperature. Similar changes accompany the time-dependent curing or chemical degradation of a material, processes that the QCM technique is particularly well suited to investigate.17,18 Master plots of the sort shown in Figure 10 provide a useful way to track curing or aging processes. Significance of the Constant Phase Approximation. The values of |G1*| for the PtBA data plotted in Figure 5 correspond to the first harmonic at a frequency of 5 MHz. The
Figure 11. Comparison of the DMA data (solid lines) and the values obtained from the constant phase angle approximation. 9738
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Figure 12. Frequency and dissipation response maps corresponding to QCM measurements of PtBA (+) and polyisoprene (○) at 15 MHz.
decreasing with decreasing frequency (aT f > 1 s−1 in Figure 11). Indeed, the increase in the values of dρ obtained at the highest temperatures for the 1:3,3 calculation in Figure 8 is quantitatively consistent with the error in this quantity that is introduced by the constant phase approximation. Choosing the Film Thickness. The most appropriate thickness for a given material depends on the property range to be investigated and can be illustrated by considering the graphical representation of the QCM master equation (eq 14). In Figure 12 we plot the values obtained for ϕ3 and d/λ3 for the poly(t-butyl acrylate) and polyisoprene QCM measurements on the response maps for Δf 3 and ΔΓ3. We use the third harmonic because it is the behavior of the crystal resonance at this harmonic that determines the range of applicability of the technique. For glassy films with shear moduli near 109 Pa and low phase angles, films with thicknesses of between 5 and 10 μm are ideal (assuming a density close to 1 g/cm3). For materials that are either rubbery or well into the transition region toward rubber behavior (as with the polyisoprene in our example), thicknesses in the 1 μm range are generally appropriate. In all cases, experiments need to be run in portions of the response map where d/λ3 is large enough to give a measurable deviation from the Sauerbrey limit (d/λ3 of 0.05 or larger), while avoiding the large dissipation associated with the film resonance condition for d/λ3 = 0.25. Note that for high values of the phase angle, the film resonance is not as pronounced so that measurements can be made for values of d/ λ3 equal to or even larger than 0.25.
indicates that the QCM is able to access rheological properties of rubbery polymers in the transition regime between glassy and rubber behavior, where the phase angle is close to its maximum value of ∼60°. The method gives a remarkably accurate measurement that does not require any additional properties of the film to be measured independently.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS
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REFERENCES
This material is based upon work supported by the National Science Foundation through the Division of Materials Research (DMR-1241667) and through the Graduate Research Fellowship Program (DGE-1324585). The polyisoprene samples were provided by Goodyear, and the DMA data were provided by Dr. Charles Wood.
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CONCLUSIONS We have used the quartz crystal microbalance to measure the rheological properties of thin, viscoelastic films in the megahertz frequency regime. The method is based on the simultaneous measurement of the resonance frequency and bandwidth (or dissipation) at two resonant harmonics of the crystal, typically at f1 and 3f1, where f1 is the fundamental resonant harmonic of the quartz crystal used as a substrate for the film of interest. The technique has been demonstrated with poly(t-butyl acrylate films) for temperatures spanning the lowfrequency glass-transition temperature and for rubbery polyisoprene films. Our thin-film method complements the simpler QCM-based approaches that enable the properties of bulk low-viscosity fluids to be measured. A comparison of uncross-linked polyisoprene by the QCM and cross-linked polyisoprene by traditional dynamic mechanical analysis 9739
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