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High Frequency Rheometry of Viscoelastic Coatings with the Quartz Crystal Microbalance Garret C. DeNolf,† Larry Haack,‡ Joe Holubka,‡ Ann Straccia,‡ Kay Blohowiak,§ Chris Broadbent,§ and Kenneth R. Shull*,† †
Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, United States Ford Motor Company, Dearborn, Michigan, United States § Boeing, Seattle, Washington, United States ‡
ABSTRACT: We describe a quantitative method for using the quartz crystal microbalance (QCM) to characterize the high frequency viscoelastic response of glassy polymer coatings with thicknesses in the 510 μm regime. By measuring the frequency and dissipation at the fundamental resonant frequency (5 MHz) and at the third harmonic (15 MHz), we obtain three independent quantities. For coatings with a predominantly elastic response, characterized by relatively low phase angles, these quantities are the mass per unit area of the coating, the density-shear modulus product, and the phase angle itself. The approach was demonstrated with a model polyurethane coating, where the evolution of these properties as a function of cure time was investigated. For fully cured films, data obtained from the QCM are in good agreement with results obtained from traditional dynamic mechanical analysis.
1. INTRODUCTION Quartz crystal resonators have been used for many applications, with the most common involving detection of very small mass changes.1 The high sensitivity to changes in mass on the surface has been utilized to characterize thin film deposition processes,2,3 DNA attachment,4,5 and molecular adsorption.68 These applications most commonly utilize the Sauerbrey equation, which relates shifts in the resonant frequency to the physical properties of the quartz to the density and thickness of a layer that is deposited on its surface:9 Δf ¼ Δfsn ¼ nΔfs1 ,
Δfs1
2f1 2 Fd Zq
ð1Þ
Here Δf is the shift in resonant frequency, n is the order of the harmonic (n = 1, 3, 5, ...), f1 is the fundamental frequency of the quartz crystal, Zq is the shear acoustic impedance of the quartz (8.84 106 kg m2 s1), F is coating density, and d is the coating thickness. Because most applications of quartz crystal resonators are based on eq 1, it is common to refer to these instruments as quartz crystal microbalances (QCMs). We use this convention in this paper, even though we are not using the resonators as straightforward mass sensors. Note that for the commonly used quartz crystals with a thickness of 330 μm and f1 = 5 MHz, a 1 mg/m2 increase in Fd (equating to a thickness increase of 1 nm for a material with F = 1 g/cm3) corresponds to a change in Δfs1 of 5.6 Hz. The Sauerbrey equation is accurate for layers that are thin and rigid, but this relation breaks down if the layer is viscoelastic,1012 too thick,13,14 or nonuniform.15 However, the deviation from the Sauerbrey equation for viscoelastic materials can be accurately r 2011 American Chemical Society
modeled,1621 enabling a wide variety of additional applications for these devices.2123 In this paper, we apply a generalized model of the QCM response for a uniform viscoelastic layer to characterize the cure behavior of a model paint coating. By measuring information at the first and third harmonics (n = 1, 3), we demonstrate a procedure for extracting quantitative viscoelastic parameters from the late stages of cure, where the material is highly elastic but has physical properties that evolve with time. The ability to measure these viscoelastic parameters, which are closely coupled to relevant mechanical properties such as the toughness of the coating, is important in a wide range of coating applications.
2. QCM THEORY 2.1. General Relationships. The interpretation of data obtained from the quartz crystal microbalance has been summarized in an excellent review article by Johannsmann.15 Here we provide a brief review of the technique and refer readers to this previous reference for a more detailed summary. The quartz crystal microbalance uses a quartz crystal disk with an oscillating voltage applied across its thickness. The piezoelectric nature of the quartz causes the disk to oscillate transversely and propagate a shear wave into the sample above. The setup is shown schematically in Figure 1a. Using a network analyzer, the complex admittance of the system is determined. We are interested in the conductance, which is the real component of the admittance, Received: February 18, 2011 Revised: May 16, 2011 Published: July 18, 2011 9873
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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 resonant frequency (fn) and dissipation (Γn) labeled (b).25 The solid line corresponds to data from an unloaded crystal, and the dashed line corresponds to data from a loaded crystal, in this case by bringing it into contact with a gel over a portion of its surface.25
in the vicinity of the crystal resonance. When the frequency of excitation from the electrodes matches the acoustic resonant frequency of the quartz, the conductance increases, giving Lorentzian conductance peaks in the frequency domain.24 These resonant peaks are also observed at odd multiples of the fundamental resonant frequency. Fitting of the conductance curve yields two values. The first of these is the frequency corresponding to the maximum conductance, which is our criterion for defining the resonant frequency, fn. The second of these is the half-maximum half-width of the conductance peak, Γn. This quantity is directly related to energy dissipation and is referred to simply as the “dissipation” throughout this paper. The “n” subscript in our notation is a reminder that data obtained are obtained at the odd harmonics of the fundamental resonant frequency, corresponding to n = 1, 3, etc. The resonant frequency and dissipation are the components of the complex resonant frequency, fn*. Typically, measurements are made based on a shift in these values from a bare crystal: Δfn
¼ Δfn þ iΔΓn
ð2Þ
A typical conductance peak is shown in Figure 1b, with the corresponding resonant frequency and dissipation labeled. Note that equivalent properties can be obtained from commonly employed time domain “ring down” experiments, where the decay time of the excited oscillations is recorded.26 When the quartz crystal is loaded on the surface, there is a shift in the complex resonant frequency, and both the resonant frequency and dissipation change as a result of this change in impedance. The basis of the technique is a coupling between Δf*n and the load impedance, Zn*, associated with the film. This load impedance is defined as the ratio of the shear stress over the shear velocity at the coating/QCM interface. If this load impedance is small in comparison to the impedance of the quartz crystal, a linear relationship exists between Δf*n and Z*: n
Δfn iZ ¼ n f1 πZq
ð3Þ
Equation 3 requires that the load impedance is much less than the impedance of the quartz (Z*n , Zq), or equivalently, Δf*/f n 1 ,1. This “small load approximation” is valid in virtually all applications of the QCM,15 and is certainly the case in the experiments described in this paper. If we assume that the film in contact with the QCM electrode surface has uniform viscoelastic properties, the load impedance is given by the following expression:15,21 2 !1=2 3 1=2 F 5 ð4Þ Zn ¼ i FGn tan42πfn d Gn where d is the layer thickness, F is the layer density, and Gn* is its complex shear modulus evaluated at a frequency fn. In this work, we express Gn* in terms of its magnitude |Gn*| and phase angle ϕn:
Gn ¼ jGn j expðiϕn Þ
ð5Þ
The wavelength, λn, of the shear wave in the coating is also related to the viscoelastic properties of the material and can be written in terms of ϕn and |Gn*|:16,20,27 ! 1=2 1 jGn j 1 ð6Þ λn ¼ nf1 F cos ϕn =2 Equations 1, 3, 4, and 6 can be combined to give the following expression for complex frequency shift:16
tanfð2πd=λn Þð1 itanðϕn =2Þg Δfn ¼ ð2πd=λn Þð1 itanðϕn =2Þ Δfsn
ð7Þ
From eq 7, we see that the shear wavelength is the natural normalization for the thickness of the layer that is deposited on the QCM surface, and that the Sauerbrey shift is the natural normalization for the shift in the resonant frequency and dissipation. In Figure 2, we show contour plots to illustrate how Δfn and ΔΓn depend on the phase angle and thickness of the 9874
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ratio, provided that d/λn can be obtained from a complementary measurement. Indeed, for highly elastic films with low phase angles, this information can be obtained by appropriately analyzing the response at higher harmonics, as described in the following subsection. For sufficiently thin films, the following expression describes the dissipation: 2 1 ΔΓn d d2 F ¼ 0:229 ¼ 0:229n2 f1 2 ð8Þ ϕn Δfn λn jGn j Figure 2. Contour plots of the normalized frequency shift and dissipation for films deposited on a QCM, as obtained from eq 7.
This equation is accurate to within 15% for d/λn values less than 0.08. Rearrangement of eq 8 gives ϕn d 2 F 4:36 ΔΓn ¼ jGn j fn 2 Δfn
Figure 3. Normalized dissipation ratio as a function of normalized thickness on log (a) and linear (b) scales. The phase angles range from 1° to 75°.
uniform coating deposited on the surface. These plots illustrate the complete QCM response and are quantitatively accurate whenever the magnitude of Δf* is small in comparison to fn. 2.2. Effect of Viscoelastic Dissipation on the Response. Equation 7 is quite general and is the basis for the detailed analysis of experimental results that are presented in this paper. A substantial portion of the QCM literature has been devoted to the development of approximations of eq 7 (or equivalent representations of it) that do not require the evaluation of the tangent of a complex argument. Our approach is different and is motivated by the fact that modern software tools (MATLAB in our case) are able to directly evaluate eq 7 so that additional approximations are not necessary. The frequency shift and dissipation are obtained simply as the real and imaginary components of Δfn*. Nevertheless, analytic approximations can be a useful tool to more clearly illustrate the connection to the physical properties of the coating. Here we develop an approximation for the dissipation that is useful when the deviations from the Sauerbrey expression are not too large. Specifically, we consider the way in which the dissipation ratio ΔΓn/Δfn depends on d/λn. When the curves obtained for different phase angles are divided by their respective phase angle, they collapse onto a single master curve, provided that d/λn is sufficiently small. This result is shown in Figure 3, where we plot (1/ϕn)ΔΓn/Δfn as a function of d/λn. Curves corresponding to ϕn = 1° and ϕn = 20° are virtually indistinguishable from one another, apart from some small differences near the film resonance condition at d/λn =0.25. In addition, the curves remain very similar for very large phase angles as well, with a phase angle of 75° (well into the liquid regime) giving a very similar curve for d/λn < 0.1. An important consequence of this result is that the phase angle describing the viscoelastic response can be obtained with a high degree of accuracy from a measurement of the dissipation
ð9Þ
This equation relates four physical properties of the layer (ϕn, |G*|, n F, and d) to measured quantities obtained directly from the QCM. If three of these quantities are known, the fourth can be calculated. The requirement that d/λn < 0.08 can be expressed in either of the following forms: !1=2 F < 0:08 ð10Þ nf1 d jGn j ΔΓn ϕn Δfn
!1=2 < 0:038
ð11Þ
Use of eq 8 is well suited for thin, glassy coatings because they are often in the regime where this approximation is valid. 2.3. Use of Multiple Harmonics. Additional information can be obtained from the experiments by simultaneously measuring the resonant frequency and dissipation for multiple harmonics. This approach allows different regions of the response maps shown in Figure 2 to be probed simultaneously because of the frequency dependence of λn exhibited in eq 6. We show here that the ratio of frequency shifts at the first and third harmonics can be used to quantify the correction to the film thickness obtained from the Sauerbrey equation and to quantify the actual value of d/λ1. Because |G*| n and ϕn both depend on frequency, it is formally necessary to make an assumption about the frequency dependence of these quantities in order to obtain λ3/λ1 from eq 6. A particularly useful expression is obtained by assuming that |G*| n has a power law frequency response, in which case ϕn is independent of frequency. In our subsequent analysis, justified by our dynamic mechanical results described in section 4.1, we assume that ϕ is independent of the harmonic. We refer to the phase angle simply as ϕ, without the subscript. The power law describing the frequency dependence is related to ϕ in the following way:28,29
jGn j µ nΛ ,
Λ¼
ϕ 90
ð12Þ
where we continue our convention of expressing ϕ in degrees. From eqs 6 and 12, the ratio of shear wavelengths for the nth and the first harmonic is given by the following expression: λn ¼ nΛ=2 1 λ1
ð13Þ
This ratio can be used in conjunction with eq 7 to obtain the ratio of frequency shifts at n = 1 and n = 3. The result is in Figure 4a as 9875
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Figure 4. Normalized film thickness as a function of the frequency shift ratio (a). Frequency shift normalized by the Sauerbrey shift as a function of the frequency shift ratio (b). Δf1 was multiplied by 3 so a perfectly elastic layer has a ratio of 1.
the normalized thickness d/λ1 is plotted as a function of the ratio of frequency shifts for ϕ = 0°, ϕ = 5°, and ϕ = 10°. Note that for these low phase angles relevant to the polymer films in the glassy regime, the correction factors are not strongly dependent on the exact phase angle. From this plot, the normalized thickness d/λ1 was determined for each sample. Figure 4b shows the resonant frequency shift normalized by the Sauerbrey shift as a function of the ratio of frequency shifts at n = 1 and n = 3. The first harmonic was multiplied by 3 in the normalization because the Sauerbrey frequency shift is proportional to n.
3. MODEL POLYURETHANE COATING The analysis described above has been applied to investigate the cure process of a model polymeric coating. Polymeric coatings are used in a variety of applications to protect the surface from corrosion and abrasion while also providing color and gloss. Polyurethanes are one type of coating commonly used in these applications because of their weatherability as well as chemical and mechanical resistance.30 They also have great versatility because of the large selection of monomeric materials for the isocyanate chemistry, as well as additives and adhesion promoters.30,31 Traditional rheometric techniques have been utilized to measure the cure behavior of a wide variety of systems.3236 Rheological monitoring of the cure behavior of thin coatings is more challenging. One successful example of early stage cure monitoring of a polymeric coating involved the contact of an oscillating T-bar with the coating.37 An important advantage of the QCM approach is that it is a noncontact method that can easily be applied in a variety of situations. By measuring the resonant frequency as well as dissipation, information about the rheological and physical properties of the coating can be obtained. The coating system used in our experiments was a two component polyurethane topcoat system from PPG. The acrylic base was Deltron DC3000 high velocity clearcoat, and the isocyanate cross-linker was Deltron DCH3070 low temperature hardener. The DC3000 consists primarily of an acrylic polymer, and the DC3070 consists primarily of di-isocyanate. The DC3000 system contains carrier solvents including xylene, acetone, and toluene at a combined weight fraction of 0.5 ( 0.2 that evaporates after the mixed system is deposited. The optimal volume fraction of the cross-linking solution suggested by the manufacturer is 0.2, and a schematic of the chemistry is shown in Figure 5. After mixing the two components, eight drops were spin coated onto a quartz crystal at 3000 rpm for 40 s, and the coated crystal was mounted in the QCM immediately after spinning.
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Figure 5. Schematic of the basic chemistry of a model polyurethane coating used in the experiments.
The resonant frequency and dissipation were obtained by continuously monitoring the conductance peak near the 5 MHz and 15 MHz resonant frequency of the quartz crystal. Three volume fractions of the cross-linking solution (R = 0.05, 0.2, and 0.35) were used to determine the stoichiometric effect on the curing of the coating at room temperature. Temperature effects were evaluated by curing a R = 0.2 coating at room temperature and at 50 °C. Timetemperature superposition on the coating was measured using dynamic mechanical analysis (DMA). The samples were prepared by creating a 1 mm deep trough from glass slides and then pouring the premixed coating solution into the trough. The excess solution was skimmed off the top using a glass slide and allowed to cure for 4 days. Afterward, the coating was gently peeled out of the trough to limit deformation of the coating. Samples were 400 μm thick after curing and were cut to size for the DMA using a razor blade.
4. EXPERIMENTAL RESULTS 4.1. Dynamic Mechanical Analysis. The magnitude of the complex shear modulus |Gn*| is dependent on the frequency at which it is measured, which for our QCM experiments is either 5 MHz (n = 1) or 15 MHz (n = 3). To determine the modulus for each coating at this frequency, timetemperature superposition was performed on samples of cured films 400 μm thick using DMA from 60 to 60 °C in 10° increments. Frequency sweeps were obtained at frequencies ranging from 0.1 to 80 Hz, and the shift factor was set to aT = 1 at 20 °C. The shifted curves of the storage modulus in tension, E0 , and phase angle, ϕ, are shown in Figure 6 for the sample with the optimum cross-linker volume fraction, R, of 0.2. The temperature dependence of the shift factors used to generate these master curves is shown in Figure 7. The activation energy obtained from this plot is in good agreement with values obtained from other thermosetting polymers.38 The storage modulus G0 was determined for frequency of 5 MHz and using a Poisson’s ratio of 0.3, which is typical for glassy polymers.39 The equation used to convert from the elastic modulus to shear modulus is
G¼
E 2ð1 þ νÞ
ð14Þ
These shear modulus values were compared to the values obtained from the QCM experiments at long annealing times. We have made the assumption that the rheology of the films used for the DMA is comparable to the thinner films measured on the QCM. Using this analysis with the data plotted in Figure 6, the phase angle was determined to be ϕ = 2.6° and G ∼ 0.85 GPa at the frequency of 5 MHz. 9876
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Figure 9. Phase angle (a), dF (b), and |G1*|F (c) for a coating with R = 0.2 as calculated using the procedure from section 2.3. Figure 6. Master curves of the storage modulus (a) and phase angle (b) for a polyurethane coating with a cross-linker volume fraction of 0.2, at a reference temperature of 20 °C. The dashed line corresponds to a phase angle of 2.6°.
Figure 10. Error in the dissipation at n = 3 for a sample with R = 0.2 over the duration of the test.
Figure 7. Arrhenius plot of the shift factors used to create the master curves in Figure 6. The dashed line corresponds to an activation energy of 250 kJ/mol.
(4) |G*|F 1 is obtained from eq 6 and the known values of ϕ and dF. In this procedure, the gamma ratio ΔΓ3/ΔΓ1 could have been used instead of the frequency ratio from step 2, but we found the frequency data more reliable than the dissipation values. Instead, we used the fourth measured quantity, ΔΓ3, as a consistency check on the data, since the theoretical values for ΔΓ3 can be calculated from the information gathered in the procedure above. As a check of the calculated values that were obtained for the samples, the relative error between the measured value of the dissipation at n = 3 and the theoretical value was determined. The equation for the relative error is Γ3error ¼
Figure 8. Resonant frequency shift (a), dissipation ratio (b), and harmonic shift ratio (c) for a coating with R = 0.2.
4.2. QCM Results. The resonant frequency shift (Δf1), dissipation ratio (ΔΓ1/Δf1), and harmonic shift ratio (Δf3/3Δf1) for a coating with R = 0.2 are shown in Figure 8. In addition, we have as a function of curing time for each obtained ϕ, dF, and |G*|F n sample. These values are plotted in Figure 9 for the room temperature cured sample with R = 0.2. These three values are obtained from three measured quantities: Δf1, Δf3 and ΔΓ1. The more detailed procedure is as follows: (1) Equations for ΔΓ1/Δf1 and Δf3/3Δf1 are obtained from eqs 7 and 13. (2) These equations are solved with ϕ and d/λ1 as unknowns, so that the predicted values for ΔΓ1/Δf1 and Δf3/3Δf1 agree with the experimental values. MATLAB is used to solve these two coupled, nonlinear equations. (3) Once d/λ1 is known, dF can be determined from eqs 1 and 7 and the measured value of Δf1.
ΔΓ3measured ΔΓ3theoretical ΔΓ3measured
ð15Þ
A value closest to zero indicates good agreement between the measured and theoretical values of ΔΓ3. A plot showing these values is in Figure 10. From Figure 9, it is evident that the properties of the coatings were still evolving at times beyond 15 h. One of these properties is the mass per unit area of the sample, represented by the quantity dF. The sensitivity of the QCM to this quantity is well-known and is the basis for the use of the QCM as a mass measuring device, given the simple interpretation of the data through the Sauerbrey equation, that is, eq 1. Our analysis gives the correction to the Sauerbrey expression that becomes non-negligible for polymer films in the thickness range that we are using. For the example shown in Figure 8, Δf3/3Δf1 decreases from 1.25 to 1.11 during curing to 15 h. From Figure 4b, this harmonic shift ratio corresponds to a correction to the Sauerbrey equation of between 2 and 3%. While this correction is small, it is equivalent to the relative change in the coating mass throughout the 15 h cure (Figure 9b). An accurate understanding of the solvent loss that is responsible for the observed decrease in dF therefore requires that these corrections be taken into account. 9877
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Table 1. Properties Calculated from the Polyurethane Coatings after 15 h from Samples with Varying Volume Fractions of Crosslinker Are Shown Belowa
a
R
sample
T (°C)
dF (g/m2)
|G1*|F (GPa-g/cm3)
ϕ (deg)
0.05
1
RT
5.91
1.52
5.05
0.05
2
RT
5.97
1.40
6.28
0.2
1
RT
7.66
1.46
4.31
0.2
2
RT
8.01
1.20
3.20
0.35 0.35
1 2
RT RT
9.32 10.51
1.47 1.36
5.00 4.10
0.2
1
50
7.48
1.13
4.27
Samples were cured at room temperature (RT) and at 50°C.
Figure 11. Values for ϕ calculated for four samples (R = 0.05, 0.2, 0.35, and 0.2 at 50 °C) versus time.
The utility of our method arises from the fact that we are able to accurately measure changes in ϕ and |G1*|F in addition to changes in the coating mass. To demonstrate the reproducibility of the results obtained for different coatings, we show data from two different room temperature cured samples for each of three different cross-linker volume fractions, R, in Table 1. Each of the reported values corresponds to a cure time of 15 h. Data for a sample with an optimal cross-linker volume fraction of 0.2 that was cured at 50 °C is also included in Table 1. A strength of our technique is that it is very sensitive to the low phase angles that are typical of glassy polymers. As a result, physical and chemical aging processes, such as shifts in the β relaxation40 that affect the fracture behavior of glassy polymers,41 can be readily investigated. The phase angles for each cross-linker ratio (sample 1 from Table 1 in each case) are plotted as a function of curing time in Figure 11. Some general features of the data are apparent. First, the optimal value of 0.2 for the crosslinker volume fraction that is specified by the manufacturer generally gives films with the lowest phase angles, consistent with the formation of the most highly cross-linked network. Also, the kinetics of the curing process are different at 50 °C than they are at room temperature, in ways that we would not have expected. For example, at 50 °C, more time is required in order to obtain a coating with properties that can be measured by the QCM than is required at room temperature. As a general rule of thumb, we find that resonances can be measured accurately once the value of Γ is less than about 10 kHz, although in some cases we are able to measure resonances for peaks with Γ as large as 20 kHz. The optimum thickness range in order to apply our analysis to glassy polymer films is ∼510 μm, assuming a normal
Figure 12. Values for ϕd2F/|G*| 1 calculated for three samples (R = 0.05, 0.2, and 0.35) versus time. The red line is the analytical approximation from eq 9.
polymeric density close to 1 g/cm3. For these film thicknesses, the dissipation for the first and third harmonics is in the measurable range for phase angles less than about 10°. This criterion determines the lowest cure times appearing in the data sets plotted in Figure 11. Phase angles for the longest cure times for the system with R = 0.2 are consistent with the phase angle obtained by dynamic mechanical analysis (Figure 6b). As a final comment, we return to the approximate expression for ϕd2F/|G*| 1 given by eq 9. In order to assess the validity of this expression, and the ability to obtain information solely from the first harmonic, we first use our detailed analysis to obtain ϕd2F/ |G1*|, constructing this quantity from the determined values of ϕ, dF, and |G*|F 1 from Figure 9. These values are plotted in Figure 12 for the room temperature cured samples (sample 1 in Table 1) with R = 0.05, 0.2, and 0.35. The lines in Figure 12 correspond to the values for ϕd2F/|G1*| obtained from eq 9 and the measured values of ΔΓ1/Δf1, and are in good agreement with the more rigorous treatment that we have proposed.
5. CONCLUSIONS We have shown that the QCM experiment, and the interpretation that we have developed, is highly sensitive to the relatively small changes in viscoelastic properties that are associated with the late stages of a curing process for a coating. For glassy coatings, the most appropriate thickness range is typically between 5 and 10 μm, where it is generally possible to obtain information from the first and third harmonics. Because information is obtained at two different frequencies, an assumption needs to be made regarding the frequency dependence of the viscoelastic response. For low phase angles, the interpretation is not strongly dependent on the assumption that is made. Our assumption is that the material obeys a power law response, with an exponent that is consistent with the phase angle itself. By measuring the shift in both the frequency and dissipation at the first (5 MHz) and third (15 MHz) harmonics, we are able to extract the following three quantities from the experiment. (1) ϕ, the phase angle describing the viscoelastic response, (2) dF, the product of the film thickness and its density, and the product of the magnitude of the dynamic shear (3) |G*|F, 1 modulus of the film at 5 MHz and its density. To validate these ideas, we compare QCM results obtained with a model polyurethane coating to results obtained on the same system by conventional dynamic mechanical analysis. Quantitative agreement was obtained for results obtained by the two methods. 9878
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’ ACKNOWLEDGMENT This work was supported by the FordBoeingNorthwestern Alliance and by the National Science Foundation (DMR0907384).
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(40) Lee-Sullivan, P.; Dykeman, D.; Shao, Q. Polym. Eng. Sci. 2003, 43, 369. (41) Cheng, W. M.; Miller, G. A.; Manson, J. A.; Hertzberg, R. W.; Sperling, L. H. J. Mater. Sci. 1990, 25, 1917.
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dx.doi.org/10.1021/la200646h |Langmuir 2011, 27, 9873–9879
