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Quartz Crystal Microbalance Studies of the Contact between Soft, Viscoelastic Solids Miriam Kunze,† Kenneth R. Shull,*,‡ and Diethelm Johannsmann*,† Institute of Physical Chemistry, Clausthal UniVersity of Technology, Arnold-Sommerfeld-Str. 4, 38678 Clausthal-Zellerfeld, Germany, and Department of Material Science and Engineering, Northwestern UniVersity, 2220 Campus DriVe, EVanston 60208-3108, Illinois ReceiVed June 30, 2005. In Final Form: September 19, 2005 Mechanical contact between a viscoelastic lens and a viscoelastic film has been probed by means of a quartz crystal microbalance operated in the impedance analysis mode. The frequency shift induced by the formation of the contact decreases with increasing film thickness because of the finite penetration depth of the acoustic shear wave. The dependence of frequency and bandwidth on film thickness and contact area is described within a sheet-contact model, which can be employed to quantitatively analyze mechanical contact in a wide range of materials problems. The model was tested by bringing a quartz crystal coated with an elastomeric gel into contact with a hemispherical cap of a similar gel. Both gels consisted of the thermoreversible gel Kraton G swollen in mineral oil. The experiments support the model well.
Introduction Quartz crystal resonators can be used to probe the properties of soft matter in a number of different ways, with microgravimetry being the most widely used application.1,2 Generally speaking, a quartz crystal resonator responds to any kind of sample touching its surface. The complex frequency shift ∆f* ) ∆f + i∆Γ, where ∆f and ∆Γ are the respective changes in the resonant frequency and half-band half width, is proportional to the average stressspeed ratio at the crystal surface, a quantity that is defined as the “load impedance”.3 A relatively straightforward averaging scheme is applicable for a nonuniform contact of liquids or soft elastic solids across the electrode surface of the disk.4,5 In these cases, the frequency shift is proportional to the area of contact, which is rationalized by assuming that the integrated stress is proportional to the fractional area covered by the lens. A more detailed analysis (necessary for fractional coverages of the electrode surface larger than ∼ 0.1) shows that a weight function has to be introduced in the averaging process, which accounts for the variation of the oscillation amplitude over the crystal surface. If the acoustic impedance of the sample is known, this technique allows for the determination of the area of contact of two materials by acoustic means. Conversely, if the contact area is known independently, the complex frequency shift can be converted to a high-frequency shear modulus of the material at the crystal surface. This kind of near-surface rheology can thus give insight to the mechanical properties within the contact zone. Here, we report on an extension of this formalism to a geometry where the contact to the crystal is made across a film coated onto the * To whom correspondence should be addressed. (D.J.) E-mail:
[email protected]. Phone: + 49-5323-72-3768. Fax: + 495323-72-4835. (K.R.S.) E-mail:
[email protected]. Phone: +1-847-491-3537. Fax: +1-847-491-7820. † Clausthal University of Technology. ‡ Northwestern University. (1) Arnau, A., Ed. Piezoelectric Transducers and Applications; Springer: Heidelberg, Germany, 2004. (2) Lu, C., Czanderna, W., Eds. Applications of Piezoelectric Quartz Crystal Microbalances; Elsevier: Amsterdam, 1984. (3) Johannsmann, D.; Mathauer, K.; Wegner, G.; Knoll, W. Phys. ReV. B 1992, 46, 7808-7815. (4) Flanigan, C. M.; Desai, M.; Shull, K. R. Langmuir 2000, 16, 9825. (5) Nunalee, F. N.; Shull, K. R. Langmuir 2004, 20, 7083.
surface. We can thus investigate the adhesion of gels to arbitrary materials, provided that these can be coated onto the crystal surface as thin films.
Theory The model which we apply to the film-lens contact is an extension of the sheet-contact model that combines the ideas set form in refs 4 and 6. The sheet-contact model states that the frequency response of the crystal is given by the following expression: / ∆f* iZ load Ac ) K ff πZq AA0
(1)
where ∆f* is the complex frequency shift, ff is the frequency of the fundamental resonance, Zq ) 8.8 × 106 kg m-2s-1 is the acoustic impedance of AT-cut quartz, Ac is the area of contact, and A0 is the active electrode area, defined as the area of the smaller of the two circular electrodes on the quartz disk. In our case, this smaller electrode is the back electrode, defined here as the noncontacting side of the crystal. The quantity Z /load is the load impedance, which in general is not a material constant but also contains geometrical factors as described in more detail below. The quantity KA in eq 1 is a sensitivity factor that accounts for the variation of the oscillation amplitude across the crystal surface. It is defined in the following way for a system with radial symmetry:
A0 KA(rc) ) Ac
∫0r u2(r)2πr dr ∫0r u2(r)2πr dr c
q
(2)
where rc is the contact radius (Ac ) πrc2), rq is the radius of the disk, and u(r) is the oscillation amplitude at the distance r from the axis of symmetry. The back electrode confines the area of oscillation to the center of the crystal via energy trapping, giving a displacement distribution that can generally be approximated (6) Johannsmann, D. Macromol. Chem. Phys. 1999, 200, 501-516.
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G/l ) G′l + iG′′l ) |G/l |exp(iφl)
by the following Gaussian expression:7,8
( )
r2 u(r) ) u0 exp -β 2 r0
(3)
where r0 is the radius of the active electrode area (A0 ) πr02). For oscillation at the fundamental frequency, β has a value that is close to one. Somewhat larger values of β are obtained for the higher harmonics, where enhanced energy trapping leads to greater confinement of the shear displacement. The efficiency of energy trapping increases with overtone order because the ratio of the thickness of the back electrode to the wavelength of shear sound increases. The argument can be substantiated with calculations based on the theory by Stevens and Tiersten.9,10 Because the displacement amplitude decays rapidly for r > r0, rq can generally be replaced by infinity in eq 2, so that the quantity KAAc/A0 is equal to 1 - exp(-2βAc/A0). For sufficiently small values of Ac/A0, one has KA ≡ KA,0 ) 2β. One might suspect that touching the crystal from the front changes the strength of energy trapping and thereby modifies the amplitude distribution. The experiments show that this is indeed the case, but that, on the other hand, this effect is so small that it may be ignored in practice. In our experiments, we are interested in layered systems where the layers may have different acoustic properties. In our notation the acoustic impedance of material l is defined as Z /l . This quantity is a materials property, and is related to the density, Fl and complex shear modulus, G/l , of the material by:
Z /l ) (FlG/l )1/2
(4)
Note that complex quantities are starred in order to distinguish them from quantities that are purely real. Several additional parameters describe the propagation of shear waves in a homogeneous material, and are useful additions to our discussion. The first of these is the complex wave vector, k/l , characterizing the propagating shear wave
k /l )
2πfFl Z /l
) 2πf(Fl/G/l )1/2
(5)
where f ) nff is the actual vibrational frequency of the crystal. Here n is the order of the vibrational overtone, and is equal to an odd integer. Note that the imaginary part of k is negative: k ) k′ - ik. The wavelength, λl, and decay length, δl, of the shear wave in the film are related to the respective real and imaginary components of k/l
( )
/ 1 |Gl | 2π ) λl ) Re(k/l ) f Fl
δl )
λl 1 ) cot(φl/2) / |Im(kl )| 2π
k/l )
2π [1 - i tan(φl/2)] λl
(6)
(7)
where |G/l | and φl are the magnitude and phase angle of the complex shear modulus, that is (7) Lin, Z.; Hill, R. M.; Davis, H. T.; Ward, M. D. Langmuir 1994, 10, 4060. (8) Josse, F.; Lee, Y.; Martin, S. J.; Cernosek, R. W. Anal. Chem. 1998, 70, 237. (9) Seydel, E. Private communication. (10) Stevens, S. D.; Tiersten, H. F. J. Acoust. Soc. Am. 1986, 79, 1811.
(8)
The sheet-contact model requires that the acoustic waves radiated into the lens are plane waves, which in turn requires that the contact diameter be larger than the wavelength of sound in the crystal. Otherwise, acoustic scattering from the contact will be significant. The wavelength of the shear wave in the quartz crystal λq is equal to 2h/n, where h is the crystal thickness (h ≈ 330 µm for crystals with ff ) 5 MHz). In our experiments, the electrode area is 32.2 mm2 (r0 ) 3.2 mm), and we acquire data at the third harmonic (n ) 3). The plane wave requirement that 2rc exceed λq is therefore valid for Ac > 0.04 mm2. Typical contact dimensions in our experiments are larger than this value, and are therefore in the range where the plane-wave-approximation is very reasonable. In the experiments reported here, the contact is established between an elastomeric lens and a film coated onto the crystal (rather than the bare crystal). By coating the crystal with a film, studies of adhesion become possible for substrates other than the electrode material. For this geometry, the same argument applies. As long as the plane-wave picture holds, the frequency shift can be calculated as in the standard viscoelastic model for layered systems, augmented by a factor KA,0 Ac/A0 in order to account for the partial coverage of the crystal by the lens. In most cases of practical interest, the acoustic properties of the film and the lens will be different, and are represented by setting the subscript l in eqs 4-8 to either 1 or 2. The two different layers can have different thicknesses, d1 and d2, with d1 representing the thickness of the layer that is deposited directly onto the QCM electrode surface. We begin with the following general expression for Z /1,2, which is the value of Z /load for this general two-layer system:119
Z /1,2
Z /1 tan(k/1d1) + Z /2 tan(k/2d2)
)i 1 - (Z /2/Z /1) tan(k/1d1) tan(k/2d2)
(9)
Equation 9 can be applied to a variety of geometries that can be modeled as bilayer systems. Three special cases are particularly relevant to our case of a semi-infinite material (d2 ) ∞) in contact with a layer of thickness d1 that is already placed on the QCM surface. Before contact of the semi-infinite material, the load impedance is completely determined by the first layer, and we can calculate the load impedance by setting d2 ) 0
Z /1,0 ) iZ /1 tan(k/1d1)
1/2
1 cos(φl/2)
These expressions can also be used to write k/l in terms of λ and φ
(10)
where the 1,0 subscript indicates that the first layer has a finite thickness and the second layer has a thickness of zero. Expressions involving infinitely thick layers can be obtained from the asymptotic behavior of the tangent of a complex argument. For |k/2d2| . 1 we have tan(k/2d2) ) -i. We obtain the following / expressions for Z 1,∞ (semi-infinite layer 2):
Z /2 + iZ /1 tan(k/1d1) / ) Z /1 / Z 1,∞ Z 1 + iZ /2 tan(k/1d1)
(11)
If the second layer remains very thick, but the first layer is removed (d1 ) 0), we recover the appropriate load impedance value for
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a single, semi-infinite layer / Z 0,∞ ) Z /2
(12)
We define the contact impedance, ∆Z /c as the change in impedance associated with the addition of the second, semiinfinite layer
[
/ - Z /1,0 ) Z /2 ∆Z /c ) Z 1,∞
]
Z /1 + Z /1 tan2(k/1d1) Z /1 + iZ /2 tan(k/1d1)
(13)
In our case, this change in impedance occurs only within the contact region, so the change in the complex frequency associated with the contact is given by eq 1, with Z /load ) ∆Zc
[
]
∆f /c iKA Ac / Z /1 + Z /1 tan2(k/1d1) ) Z ff πZq A0 2 Z / + iZ / tan(k/d ) 1 2 1 1
(14)
Equation 14 is the central theoretical result for the “extended sheet-contact model” that describes the contact of thick, viscoelastic material (layer 2) with a thin viscoelastic film (layer 1). The thick layer represents any material with a thickness substantially larger than the decay length of the shear wave in that material. If both materials have identical properties, so that Z /1 ) Z /2, eq 14 simplifies to the following: / (1 - i tan(k*df)) ∆f /c ) ∆ fc + i∆Γc ) ∆f 0,∞
(15)
where df ) d1 is the film thickness and ∆f0,∞ is the complex frequency shift for a semi-infinite layer in contact with the bare crystal surface at the same value of Ac. Equation 15 can also be derived directly from eqs 12 and 10 by subtraction. As outlined in the Appendix, the following expressions for the normalized frequency shift and change in bandwidth can be obtained directly from eq 15:
[
∆fc ) Re (1 - i cot(φ/2)) × ∆f0,∞
(
1 - i tan
[
∆Γc ) Im (i - tan(φ/2)) × ∆Γ0,∞
(
1 - i tan
(
))]
(16)
(
))]
(17)
2πdf (1 - i tan(φ/2)) λ
2πdf (1 - i tan(φ/2)) λ
Note that values for ∆fc and ∆Γc are both normalized by the response that is obtained in the absence of layer 1. These expressions are plotted in Figure 1 for three different values of the phase angle, φ. The curves in Figure 1 illustrate the decay in the sensitivity of the QCM as the thickness of the layer 1 is increased. By normalizing the film thickness by the wavelength, λ, of the acoustic shear wave, we are able to obtain a series of universal plots that depend only on the value of the phase angle, φ. Because the magnitude of the effect is very sensitive to the value of the phase angle, we have broken Figure 1 into two parts. In part a, we show the response for films with relatively large values of the phase angle (60° or larger), whereas the response for more elastic films (phase angles of 45° or less) is illustrated in part b. Note that these phase angles correspond to the frequency of oscillation (15 MHz in our case) and at these high frequencies, the larger values of the phase angle are typical of the gels used
in our experiments. The sensitivity to contact with the lens initially decreases with increasing film thickness because of the finite penetration depth of the shear wave. Equations 16 and 17 provide a quantitative description of this effect, and also show that, for a certain range of film thicknesses, the signs of ∆fc and ∆Γc are opposite to what is obtained for contact with a bare crystal. This effect is attributable to the existence of multiple reflections between the upper and the lower surface of a thin layer that can lead to a “film resonance” for films in the appropriate thickness range.12 At thicknesses somewhat below the thickness corresponding to film resonance, the magnitude of the frequency shift induced by the film is significantly larger than what is predicted by the Sauerbrey equation.13 When the gel lens contacts the film, it reduces the reflection of the wave at the film surface and thereby largely removes the film resonance. This effect can outweigh the effect of added mass and lead to an increase of frequency after contact with the gel lens. Experimental tests of these predictions are described below. Experimental Section Figure 2 is a sketch of the experimental setup. AT cut quartz crystals with a fundamental frequency of 5 MHz and diameter of 1 in. (# 149257-1, Maxtek Inc., Santa Fe Springs, CA) were employed. The crystals were coated with gold electrodes. We checked for a frequency shift induced by bending under a localized vertical force applied to the center of the crystal14 and found it to be insignificant. The crystals were mounted in the holder CHC-100C from Maxtek Inc. (Santa Fe Springs, CA). Frequency and bandwidth were determined by impedance analysis using the 250B network analyzer from Saunders & Associates (Saunders & Ass., Inc., Scottsdale, AZ) in conjunction with the software package QTZ (Resonant Probes, Goslar, Germany). The accuracy in the determination of frequency and bandwidth is below 1 Hz. Both the lens and the film consisted of the thermoreversible polymer gel Kraton G 1650 E (Kraton Polymers GmbH, Eschborn, Germany). Kraton G is a triblock copolymer with middle blocks of polyethylene/polybutylene and end blocks of polystyrene. The polystyrene end blocks form micelles that act as physical cross links. For the preparation of the lens, the polymer was dissolved in mineral oil (Mineral Oil, light, white, product # 568821, Sigma Aldrich USA) at a temperature of 120 °C and a concentration of 25 wt %. Above the sol-gel transition temperature, the solution flows. Hemispherical caps with radii of 1.5 mm were formed by pouring the mixture into a PDMS mold. Gelation occurred upon cooling. The modulus of the gel can be adjusted via the polymer content. The low-frequency shear modulus of this material was determined as G′ ) 6.8 × 104 Pa with conventional rheology (Bohlin Instruments, Gemini Advanced Rheometer, Pforzheim, Germany). Flat films with a thickness below 1 µm cannot be prepared in this way because spin coaters cannot be operated at elevated temperatures. For the spin-casting process, a ternary mixture of toluene, Kraton G 1650, and mineral oil was used. The relative weight fraction of the polymer and the mineral oil was maintained constant at a value of 1:3. The toluene content was varied between 88 wt % and 98 wt %, thereby varying the viscosity of the solution and the thickness of the film after spin-coating. Using a speed of rotation of 2800 rpm, film thicknesses of 5, 98, 294, 414, and 595 nm were obtained from solutions with a toluene content of 2, 5, 9.5, 11, and 12 wt %, respectively. After the coating process, the sample was heated to 120 °C for 5 min in order to remove the toluene. The film thickness was determined by quartz crystal microgravimetry and was crosschecked with an optical film-analyzer (Film Analyzer F-20, Filmetric Inc., San Diego, CA). The optical and the gravimetric thicknesses (11) Domack, A.; Prucker, O.; Ru¨he, J.; Johannsmann, D. Phys. ReV. E 1997, 56, 680. (12) Granstaff, V. E.; Martin, S. J. J. Appl. Phys. 1994, 75, 1319. (13) Sauerbrey, G. Z. Phys. 1959, 155, 206. (14) Heusler, K. E.; Grzegozewski, A.; Ja¨ckel, L.; Pietrucha, J. Ber. BunsenGes. Phys. Chem. 1988, 92, 1218.
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Figure 1. (a) Normalized shift of frequency and bandwidth for a semi-infinite viscoelastic material in contact with a thin layer with the same viscoelastic properties for viscoelastic materials with a predominantly viscous character (phase angles for the linear viscoelastic response at the frequency of oscillation of 60° or more). (b) Continuation of part a, but for viscoelastic materials with a stronger elastic character (phase angles of 45° or less).
Figure 2. Sketch of the experimental setup. agreed within 10%. The gravimetrically determined thicknesses were used in the further analysis. It should be noted that the structure of the thinnest film (5 nm) presumably is affected by surface roughness (which is of the order of 2-3 nm). As our data show, even such a thin, and laterally heterogeneous, film screens the interaction between the substrate and the lens to some extent. The gel lens was displaced vertically with a stepping motor (IW702-00, Burleigh Instruments, Fishers, NY). The load and the displacement were measured with a load transducer (FTD-G-1, Schaevitz Sensors, Hampton, VA) and a fiber optic displacement sensor (RC100-GM2OV, Philtec, Annapolis, MD), respectively. The device was controlled using a LabView program (National Instruments, Austin, TX). Images of the areas of contact were acquired by a microscope imaging the sample from above. The area of contact
Figure 3. Frequency shift (top) and bandwidth (bottom) at the 3rd overtone (15 MHz) versus the area of contact Ac for films with thicknesses of 0 (0), 5 (O), 98 (4), 294 (3), 414 (]), 595 (left facing triangle), and 1400 (right facing triangle) nm. The solid are linear fits according to eq 1. was extracted from these images by utilizing ImageJ image analysis software (NIH, Maryland).
Results and Discussion Figure 3 shows a typical set of raw data for the contact experiments, where frequency shift and bandwidth are plotted as a function of the contact area that the gel lens makes with gel
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Langmuir, Vol. 22, No. 1, 2006 173
has remained inside the film. This conjecture is corroborated by the measurement of the high-frequency value of viscous compliance J′′ of the film, where the latter is based on the shifts of bandwidth induced by coating the film onto the bare substrate.6 The details show that the viscoelastic parameters of the film can unfortunately not be uniquely derived from our experimental data, and we therefore leave it with the statement that the film is softer than the gel lens.
Conclusion Figure 4. Normalized values of ∆fc and ∆Γc from the experimental data (symbols) and predicted values for φ ) 70.2°. A wavelength of 6.6 µm was obtained for the shear wave by forcing agreement of the measured values of ∆fc with the theoretical prediction.
films of increasing thicknesses. Both frequency shift and bandwidth are proportional to the contact area, which is consistent with previous results indicating that the sensitivity factor, KA is roughly constant for small values of Ac/A0.4 Calculation of the magnitude of the complex shear modulus requires that the sensitivity factor be known, but the phase angle can be obtained directly from the ratio of the real and imaginary components of the complex frequency shift for the lens that is directly in contact with the electrode surface of the QCM
φ ) 2 arctan
(
)
- ∆f0,∞ ∆Γ0,∞
(18)
The value of the phase angle for the gel as calculated from eq 18 is 70.2°. If we assume that the properties of the gel film and gel lens are identical, we can use this value of φ in eqs 16 and 17 to obtain a plot of the reduced frequency and dissipation as a function of the reduced film thickness. The reduction variables for ∆fc and ∆Γc are the experimentally measured values of ∆f0,∞ and ∆Γ0,∞. The reduction variable for the film thickness is the wavelength, λ, which we use as an adjustable parameter to force agreement between the measured and predicted values. The comparison, illustrated in Figure 4, yields a value of 6.6 µm for λ. From eq 6, we obtain |G*| ) 5.4 × 106 Pa, using the value of λ determined from contact from the thin film, and the value of φ determined by direct contact of the lens with the QCM surface. As a final check on the validity of our assumptions, we can use these values of φ and |G*| to calculate the value of the sensitivity factor KA,0. Using eqs 1 and 4 with Z /load ) Z0,∞ ) Z /2, we obtain KA ≈ 4, a value that is somewhat larger than the value of 2 that is typically obtained. Comparison of the results for different overtone orders (data not shown) indicate that the variability of the parameter β with overtone order (cf. eq 3) is not a likely reason for the high value of KA. A possible explanation for the discrepancy is that the lens is somewhat stiffer than the layer, perhaps as a result of the different processing conditions for these two materials. A much better match between the data and the theory (using KA ) 2) can be made by assuming that the film is somewhat softer than the gel because some of the solvent
We have developed an extended sheet-contact model that describes the change in resonant frequency and dissipation obtained when a viscoelastic material is brought into contact with a QCM surface that is already coated with a thin layer of a second, viscoelastic material. The decay in sensitivity to the lens contact provides a very sensitive measure of the viscoelastic properties of the thin layer. In addition, the extended sheet contact model extends the range of contact problems that can be investigated with the QCM. Application of the model was demonstrated by comparing to experimental results obtained from a model gel system. The extended sheet contact model works well and, in particular, captures the sign reversal in the frequency shift at intermediate film thickness, which is related to the film resonance. Acknowledgment. We thank F. Nelson Nunalee, David Brass, and Binyang Du for helpful discussions. Work at Northwestern University was supported by grants from the National Science Foundation (DMR 0214146) and by the National Institutes of Health (R01 DE14193). M. K. acknowledges support by the DAAD.
Appendix: Derivation of Equations 16 and 17 / With ∆f 0,∞ ) ∆f0,∞ + i∆Γ0,∞, we can write eq 15 in the following form:
∆fc + i∆Γc ) (∆f0,∞ + i∆Γ0,∞)(1 - i tan(k*df)) (19) Expressions for ∆f0,∞ and ∆Γ0,∞ are obtained by the following rearrangements of eq 18:
∆f0,∞ ) -∆Γ0,∞ tan(φ/2), ∆Γ0,∞ ) -∆f0,∞ cot(φ/2) (20) Substitution of these two equations into eq 19 gives the following:
∆Γc ∆fc +i ) (1 - i cot(φ/2))(1 - i tan(k*df)) (21) ∆f0,∞ ∆f0,∞ i∆Γc ∆fc + ) (i - tan(φ/2))(1 - i tan(k*df)) (22) ∆Γ0,∞ ∆Γ0,∞ Equation 16 is obtained by equating the real parts of eq 21, and eq 17 is obtained by equating the imaginary parts of eq 22. In both cases, we use eq 8 to express k* in terms of λ and φ. LA051757C
