Langmuir 2000, 16, 9825-9829
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Contact Mechanics Studies with the Quartz Crystal Microbalance Cynthia M. Flanigan, Manishi Desai, and Kenneth R. Shull* Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208 Received May 22, 2000. In Final Form: August 31, 2000 A quartz crystal microbalance (QCM) is used to measure the contact area between a gold-coated quartz crystal and a low-modulus elastic solid. When combined with an appropriate contact mechanics analysis, such as the well-known method of Johnson, Kendall, and Roberts, quantitative information about the adhesion of the elastic solid to the electrode surface of the quartz crystal can be obtained. We use a hemispherical, elastic gel to demonstrate that the QCM can be used to acquire contact area information directly, without visually monitoring the area of contact between the gel and the electrode surface. Provided that the area of contact is confined to the central portions of the gold electrode, a linear relationship is observed between the crystal’s resonant frequency and the gel/electrode contact area. The coefficient describing this linear relationship depends on the viscoelastic properties of the gel. A simple expression for this coefficient is derived.
Introduction The quartz crystal microbalance (QCM) has been used extensively to determine a variety of interactions between materials, ranging from the wetting behavior of liquids on solid surfaces to the spreading of cells on coated surfaces.1,2 Studies with the quartz crystal microbalance became popular after Sauerbrey’s demonstration in 1959 that a quartz crystal could be used as a mass sensor.3 These applications are based on a “converse” piezoelectric effect, where a voltage is applied across a quartz crystal that is held between two metal electrodes.4 In response to this alternating electric field, transverse shear waves propagate across the crystal thickness and are reflected at the crystal surfaces. Constructive interference of these shear waves leads to a very pronounced resonance. When an additional layer is deposited on the crystal, wave propagation through this layer increases the effective thickness of the crystal, thereby changing its resonant frequency.5 The utility of the QCM as a sensor arises from its very high sensitivity to small mass loadings. A simple oscillator circuit is typically used to measure the resonant frequency of the crystal. Interpretation of the frequency shift is straightforward when the added mass is rigidly coupled to the crystal, but the analysis is complicated considerably by nonidealities including viscoelastic losses and slippage at the crystal surface.4,5 Recently, impedance spectroscopy has been introduced as a complementary technique for analyzing results obtained with the quartz crystal microbalance.6-8 In these experiments much more information is obtained about the detailed frequency response of the crystal, either in the vicinity of a single resonance, or at higher harmonics of the fundamental resonance. (1) Wegener, J.; Janshoff, A.; Galla, H.-H. Eur. Biophys. J. 1998, 28, 26. (2) Rodahl, M.; Hook, F.; Fredriksson, C.; Keller, C. A.; Krozer, A.; Brzezinski, P.; Voinova, M.; Kasemo, B. Faraday Discuss. 1997, 107, 229. (3) Sauerbrey, G. Z. Phys. 1959, 155, 206. (4) Buttry, D. A.; Ward, M. D. Chem. Rev. 1992, 92, 1355. (5) Ward, M. D.; Buttry, D. A. Science 1990, 249, 1000. (6) Hillman, A. R. Solid State Ionics 1997, 94, 151. (7) Kanazawa, K. K. Faraday Discuss. 1997, 107, 77. (8) Calvo, E. J.; Etchenique, R.; Bartlett, P. N.; Singhal, K.; Santamaria, C. Faraday Discuss. 1997, 107, 141.
Figure 1. Schematic representation of quartz crystal microbalance experiment where a hemispherical gel cap is pressed against a quartz crystal. Changes in resonant frequency and corresponding contact area are measured during the test.
Here we demonstrate an approach for applying the quartz crystal microbalance as a sensor for quantifying adhesion between a low-modulus elastomer and a rigid substrate. The underlying principle is that the contact of two elastic solids is determined by a balance between adhesive forces and elastic restoring forces. The specific geometry used in our experiments is shown in Figure 1. An elastomeric hemisphere is brought into contact with the gold electrode surface of the quartz crystal. The effects of adhesive forces for this geometry were considered initially by Johnson, Kendall, and Roberts9 in 1971, and were placed in the context of a more generalized fracture mechanics approach by Maugis and Barquins.10,11 These investigators developed relationships between the applied load, P, the resulting displacement, δ, and the contact radius, a. These relationships involve the elastic modulus of the lens, E, and the adhesion energy, G, characterizing the interface between the lens and the substrate. These theories all assume that the contact radius is small in comparison to the other dimensions of the sample, an assumption that can be relaxed.12 To measure the adhesion energy, the contact radius must be measured, along with either the applied load or the displacement. For an (9) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London A. 1971, 324, 301. (10) Maugis, D.; Barquins, M. J. Phys. D: Appl. Phys. 1978, 11, 1989. (11) Maugis, D. Langmuir 1995, 11, 679. (12) Shull, K. R.; Ahn, D.; Chen, W.-L.; Flanigan, C. M.; Crosby, A. J. Macromol. Chem. Phys. 1998, 199, 489.
10.1021/la000701+ CCC: $19.00 © 2000 American Chemical Society Published on Web 11/15/2000
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incompressible material with a known elastic modulus, the adhesion energy can be obtained from the relationship between the contact radius and displacement from the following expression:
{
2
( )}
G 2(δ′ - δ) a a ) 1 + 2.67 + 5.33 E 3πa h h
()
3
(1)
Here h is the thickness of the lens (see Figure 1) and δ′ is the nonadhesive displacement, given by the following expression:
δ′ )
a2 (0.4 + 0.6 exp(-a/h)) R
(2)
where R is the radius of curvature of the lens. Zero displacement is defined as the point of initial contact between the elastic lens and the substrate, which is assumed here to be completely rigid. Derivation and use of these expressions have been discussed in previous publications.12,13 Note that when a/h is very small, eqs 1 and 2 reduce to the more well-known expressions utilized by Maugis and Barquins.14 The contact radius is typically obtained by visual inspection of an image of the contact area. While automation of this process is possible, it involves image analysis, which is not always straightforward. Since the QCM detects changes in the resonant frequency of the crystal as it comes into contact with another material, changes in this frequency can be used to determine the contact radius. In addition the relative displacement, δ, of the two bodies can be easily measured with a displacement sensor. Simultaneous acquisition of signals from the QCM and from the displacement sensor can therefore provide all of the information necessary to obtain G/E from eqs 1 and 2. In this paper we explore the possibility of utilizing this approach to measure the adhesive properties of model thermoreversible gels. We use the QCM in conjunction with our axisymmetric adhesion test apparatus to obtain calibrations between the frequency change of the crystal and corresponding contact area of a gel cap pressed against the crystal surface. We also develop an expression connecting the relationship between the resonant frequency and contact area to the viscoelastic properties of the gel. Experimental Method Materials. Triblock copolymers with poly(methyl methacrylate) endblocks and a poly(n-butyl acrylate) midblock were dissolved in 2-ethyl hexanol, a selective solvent for the midblock. At temperatures above about 60 °C, these solutions are freely flowing liquids. At low temperatures, however, the PMMA endblocks aggregate to form ideally elastic solids. The more detailed characteristics of these polyacrylate gels have been described previously.13,15 Briefly, the copolymer used in this study is a triblock copolymer with a polydispersity index of 1.12. The PMMA endblocks have a molecular weight of 15 000 g/mol and the PNBA midblock has a molecular weight of 137 000 g/mol. Gel caps were formed on circular, etched regions on fluorinated glass cover slips. Glass cover slips were dipped in 9 wt % solutions of 1H, 1H, 2H, 2H-perfluorodecyltricholorosilane (PCR, Inc.) in hexane and dried at room temperature for 1 day. To etch a circular region through the fluorination to expose the glass surface, a pipet was used to place a drop of a 10% potassium hydroxide/ ethanol solution onto the fluorinated glass slide. The residual (13) Mowery, C. L.; Crosby, A. J.; Ahn, D.; Shull, K. R. Langmuir 1997, 13, 6101. (14) Maugis, D.; Barquins, M. J. Phys. D: Appl. Phys 1983, 16, 1843. (15) Flanigan, C. M.; Crosby, A. J.; Shull, K. R. Macromolecules 1999, 32, 7251.
base was left on the slide for 24 h before the slide was rinsed with acetone to clean the surface. Using this method, the size and adherence of the gel caps to the cover slip could be readily controlled. Following the procedure outlined previously, gel lenses with polymer volume fractions, Φp, of 0.05, 0.10, 0.15, and 0.25 were formed on the cover slips with radii of curvature in the range of 0.8-1.6 mm.13 Experiments were also conducted with a lightly cross-linked homopolymer of poly(n-butyl acrylate), used previously for adhesion studies.16 Quartz Crystal Microbalance Tests. AT-cut, circular quartz crystals with diameters of 1.37 cm and a resonant frequency of 5.0 MHz were obtained from International Crystal Manufacturing Co. Inc. These crystal resonators were sandwiched between circular gold electrodes with diameters of 0.95 cm, corresponding to an electrode area, Ao, of 0.71 cm2. A laboratory crystal oscillator by International Crystal Manufacturing Co. Inc. was used to measure the resonant frequency of the crystal during the adhesion test, with data acquired at 0.03 s intervals. A modification of the testing fixture described in previous adhesion tests16,17 allowed direct contact between the crystal and the gel cap, as contact area images were videotaped. The gel cap was aligned with the center of the electrode, since the sensitivity of the crystal varies in a Gaussian manner across the electrode surface.4 During a typical test, frequency measurements were taken as the gel cap was moved toward the quartz crystal at a crosshead velocity of approximately 5 µm/s. The gel was pushed into the crystal substrate until the circular contact area increased to ∼0.4 mm2. At this point, the gel lens was immediately retracted until it separated from the substrate. Prior to each run, the crystal was cleaned with acetone to ensure that the 2-ethyl hexanol solvent was not present on the quartz crystal surface. Resonant frequency changes of the quartz crystal were correlated to the contact area during the loading and unloading portions of the test. These videotaped contact areas were measured using a software package from Image Pro Plus.
Results and Discussion Contact Area Measurements with the QCM. Our main goal is to demonstrate the feasibility of using the quartz crystal microbalance as a sensor for quantifying adhesion in axisymmetric adhesion tests. To accomplish this task, we utilize a polymer gel model system to show that changes in resonant frequency of the quartz crystal may be calibrated to measure the contact area between a soft cap and the rigid crystal surface. These gels are ideally suited for this purpose because of their ideally elastic nature and low elastic moduli of approximately 104 Pa. An inherent concern with these systems is the effect of evaporation of solvent molecules onto the crystal before contact between the two bodies is achieved. Since the quartz crystals are particularly sensitive to minor mass changes, the small amount of condensation of 2-ethyl hexanol onto the crystal surface is reflected in a noticeable change in resonant frequency of the quartz crystal. These effects become even more significant for the lower polymer concentration gels, with Φp ) 0.05 and 0.10, since the solvent activity is relatively high for these materials. We begin by discussing the response of the crystal to this complicating factor and then show several representative relations between frequency change and contact area for gels with different polymer concentrations. Figure 2 illustrates the series of steps during the adhesion test and the corresponding effects on the change in the crystal’s resonant frequency. Here we plot representative curves of the frequency change versus time during the adhesion tests for a quartz crystal in contact with a gel lens with a polymer volume fraction of 0.15, and in contact with a PNBA lens. Figure 2a shows the results for a gel lens. As the gel lens approaches the crystal (16) Ahn, D.; Shull, K. R. Macromolecules 1996, 29, 4381. (17) Flanigan, C. M.; Shull, K. R. Langmuir 1999, 15, 4966.
QCM Contact Mechanics Studies
Figure 2. Change in frequency response of quartz crystal in contact with (a) swollen gel cap with Φp ) 0.15 and (b) a crosslinked poly(n-butyl acrylate) hemisphere.
surface (separation of 100 µm or less), there is a noticeable decrease in the resonant frequency due to the condensation of solvent molecules onto the opposing surface. Contact between the gel lens and substrate is marked by a pronounced decrease of this frequency, which continues as the gel is pressed to greater contact areas with the crystal. This behavior is due to the fact that the swollen polymers immediately form large contact areas with the substrate, due to the thermodynamic work of adhesion. The magnitude of this area, and hence the magnitude of the frequency change, is limited by the penalty associated with the elastic deformation of the lens. During the unloading portion of the test, the resonant frequency increases as the contact area between the lens and the electrode surface decreases. After detachment of the gel, corresponding to the reduction of the contact area to zero, the crystal gradually recovers its unperturbed resonant frequency as solvent evaporates from its surface. The original background frequency is achieved after about one minute of solvent evaporation. In contrast to the solvent effects shown in Figure 2a, Figure 2b demonstrates the response of a solvent-free poly(n-butyl acrylate) lens with Young’s modulus of 1.7 × 106 dyn/cm2. As seen in this plot, the resonant frequency simply decreases during the loading portion of the test and increases during the unloading portion of the test. In addition, the background frequency is immediately recovered upon separation of the two surfaces. We can investigate the application of the QCM as a sensor more clearly by reviewing several calibration tests between frequency change and contact area. Figure 3
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Figure 3. Frequency change of quartz crystal microbalance corresponding to change in contact area as a gel cap with Φp ) 0.15 is (a) loaded in contact with crystal (increasing contact area) and (b) unloaded from the crystal (decreasing contact area.) The four symbol types correspond to four different tests. The solid lines are fits to the data, and have slopes of 230 Hz/ mm2 (a) and 165 Hz/mm2 (b).
shows this correlation for gel caps with polymer volume fractions of 0.15, with varying radii of curvature. Here we review four tests in which we have separated the loading and unloading data into two plots, Figures 3a and 3b, respectively. As seen in both figures, there is a linear relationship between the contact area and frequency, with an average slope of 4.2 mm2/Hz for loading and 6.0 mm2/ Hz for unloading. The presence of residual solvent on the quartz crystal influences the calibration between the contact area and frequency, as seen in the variation of slopes for the loading and unloading curves. Slopes obtained from the loading curves are the most useful because plots of the frequency change vs contact area extrapolate through the origin. This result indicates that solvent adsorption outside the contact area is not a problem for the loading portion of the test. As discussed previously, residual solvent is left on the electrode surface after pulloff, thereby increasing the effective area of contact. For this reason our more detailed analysis of the slopes of the frequency/area curves is restricted to the data obtained during loading. Gels with different polymer volume fractions each give straight lines when the resonant frequency is plotted against the contact area, but the slopes of these curves (∆f/A) are a function of Φp. These slopes are plotted as a function of Φp in Figure 4. For a given contact area between the gel cap and quartz crystal, a material with a higher
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where Fg is the density of the gel. For a Newtonian fluid, δ ) π/2 and |G| ) G′′ ) 2πfoηg, where ηg is the viscosity of the fluid. The decay length in a Newtonian fluid reduces to the following:
d)
( )
1/2
ηg πfoFg
(5)
The frequency change obtained by immersing the crystal in a viscous fluid is given by the following expression:
x
Fgηg πFqµq
∆f ) -fo3/2 Figure 4. Values of the frequency change/contact area slopes obtained from loading curves, as a function of the polymer volume fraction in the gel.
polymer concentration causes a greater decrease in the crystal’s resonant frequency than a material with a lower polymer volume fraction. The cross-linked PNBA homopolymer, which we consider as the limiting case with Φp ) 1, gave the highest frequency sensitivity. This material had ∆f/A ) 5 × 104 Hz/cm2. The data point corresponding to pure solvent, the opposite limiting case with Φp ) 0, was not measured directly because the solvent rapidly spread over the electrode and was not confined to the central region. The value of ∆f/A for this point was calculated by the method outlined in the following section. Relationship Between Frequency Response and Mechanical Properties of the Gel. As mentioned earlier, analyses using the quartz crystal microbalance have typically depended on some form of Sauerbrey’s equation for measuring the thickness of a film that is rigidly coupled to the quartz crystal.3,5 The Sauerbrey equation relates the frequency change ∆f, to the added mass, m:
∆f )
-2fo2m AoxFqµq
(3)
Here fo is the resonant frequency of the quartz crystal (5 MHz in our case), Ao is the active area of the crystal defined by the electrode surface, Fq is the density of quartz (2.648 g/cm3) and µq is its shear modulus (2.947 × 1011 dyn/cm2 for AT-cut quartz). If the quartz crystal is placed in a linear viscoelastic medium such as the gels used in our experiments, the shear wave decays into the gel with a decay length d. The magnitude of the shear displacement is also decreased by the presence of the viscoelastic medium. For an experimental geometry similar to that used in our experiments, Kanazawa calculates a shear displacement magnitude at the electrode surface that decreases from 1300 to 45 Å when the crystal is removed from air and immersed in water.7 These values are indicative of the expected shear amplitudes in our experiments as well. The decay length of the shear wave can be related to the complex dynamic shear modulus, G* ) G′ + iG′′ ) |G|eiδ, evaluated at the resonant frequency fo:18
d)
(|G|/Fg)1/2 2πfo sin(δ/2)
(4)
(18) Reddy, S. M.; Jones, J. P.; Lewis, T. J. Faraday Discuss 1997, 107, 177.
(6)
This same expression can be obtained by assuming that a region of fluid with a thickness of d/2 is effectively coupled to the quartz crystal, i.e., by using eq 5 for d and substituting dFg/2 for m/Ao in eq 3.19 Here we approximate the effects of viscoelasticity by assuming that a similar substitution is valid when the phase angle is less than 90°. Using eq 4 for d and making the substitution of dFg/2 for m/Ao in eq 3 give the following general expression:
∆f )
x
-fo
Fg|G| Fqµq
2π sin(δ/2)
(7)
This expression assumes that the gel is in contact with the crystal across the entire electrode surface. In reality, the resonant frequency depends on the actual contact area, taking into account the nonuniform sensitivity of the crystal. If A is the actual contact area with the gel and Ao is the total electrode area, we can define a sensitivity factor, K(A/Ao) which is a function of the fractional coverage of the active area of the quartz crystal:
∆f )
-K(A/Ao)fo
x
Fg|G| A Fqµq Ao
2π sin(δ/2)
(8)
Consistency with eq 3 requires that K(1) ) 1. Lin et al. have obtained K(0) ≈ 2 for AT-cut quartz crystals used in a geometry very similar to ours.19 Thus, if the contact area is much smaller than the active area defined by the electrodes, the slopes of plots of frequency vs contact area are given by the following expression: 1/2 ∆f -Kg|G| ) A sin(δ/2)
(9)
with
Kg )
-fo πAo
x
Fg Fqµq
(10)
For the parameters corresponding to our experiments, Kg is equal to 2.30 g-1/2 s-3/2. Figure 5 shows values of |G|/ sin2(δ/2) obtained by applying eq 9 to the measured values of ∆f/A. Because the gels consist primarily of 2-ethylhexanol, we have used the density of this solvent (0.83 g/cm3) to estimate the value of Fg. The data point for pure solvent (Φp ) 0) was obtained from the known viscosity, ηg, of 0.10 Poise, as reported in the literature.20 The pure solvent is assumed to be a Newtonian fluid, with δ ) 90° and |G| ) 2πfoηg. (19) Lin, Z.; Hill, R. M.; Davis, H. T.; Ward, M. D. Langmuir 1994, 10, 4060. (20) Solvents Guide; Marsden, C., Mann, S., Eds.; Interscience: New York, 1963.
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proach is to develop an empirical calibration curve relating the contact area to the frequency shift. In cases where the rheological properties of the viscoelastic material are known, it may be possible to use these properties to directly calculate this calibration curve. Summary
Figure 5. Rheological data calculated from the measured values of ∆f/A, according to eqs 9 and 10. The pure solvent (Φp ) 0), was assumed to be a Newtonian fluid as described in the text.
The analysis presented in this section provides a simple method for obtaining rheological information from materials that are in contact with the electrode surface. Use of this approach to obtain the concentration dependence of |G|/sin2(δ/2) plotted in Figure 5 is an illustration of the type of rheological information that can be extracted from an analysis of the data obtained from the quartz crystal microbalance. The more important aspect of our results is that the quartz crystal microbalance is sensitive to the contact area between a viscoelastic solid and the substrate, and that the relationship between resonant frequency and contact area depends on the viscoelastic properties in a way that can be at least qualitatively taken into account. For most contact mechanics experiments, the best ap-
The quartz crystal microbalance provides a viable method for directly quantifying adhesion between soft, elastic materials and the gold electrode surface of a quartz crystal. The relationship between the contact area and the resonant frequency is quite linear, provided that the dimensions of the contact area are much smaller than the electrode dimensions of the quartz oscillator. Materials with different dynamic shear moduli at the oscillation frequency will give different calibration curves, however. The model presented here gives a good qualitative prediction of the relationship between the contact area and the resonant frequency. In cases where the required viscoelastic parameters are not known with sufficient precision calibration curves must be obtained experimentally. Once these calibrations are known, a quartz crystal microbalance can be used in conjunction with the appropriate contact mechanics analysis to obtain a wealth of information about adhesive processes. Acknowledgment. This work was supported by the Northwestern Universtity MRSEC, and by the NSF-DMR polymers program, grant DMR-9975468. Acknowledgment is also made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. LA000701+
