Langmuir 2004, 20, 7083-7089
7083
Contact Mechanics Studies with the Quartz Crystal Microbalance: Origins of the Contrast Factor for Polymer Gels and Solutions F. Nelson Nunalee and Kenneth R. Shull* Department of Materials Science and Engineering, Northwestern University, 2220 Campus Drive, Evanston, Illinois 60208-3108 Received April 19, 2004. In Final Form: June 4, 2004 A contact mechanics methodology utilizing the quartz crystal microbalance (QCM) has been applied to study the spreading behavior of polymer solutions and gels. Changes in the resonant frequency and in the dissipation are monitored as these materials are brought into contact with the electrode surface of the QCM. The primary application is in studies of elastic polymer gels, where spreading over the surface of the QCM is limited by the elasticity of the gel. Simultaneous measurement of the applied loads and displacements, along with measurement of the QCM/gel contact area, the frequency shift, and the dissipation, enable us to calibrate the QCM as a contact sensor. While changes in the frequency and dissipation both depend linearly on the contact area, measurements of the dissipation provide a more reliable indicator. The relationship between the dissipation and the contact area is determined by the solvent viscosity and by the high-frequency intrinsic viscosity of the system of interest. This result is consistent with previous results on the high-frequency rheological behavior of polymer solutions.
Introduction Quartz crystal resonators are remarkably sensitive devices for analyzing surface effects in a wide variety of situations.1 These resonators are based on the fact that appropriately cut quartz crystals exhibit a converse piezoelectric effect, where an applied electric field results in a directional mechanical deformation.2 In the case of an AT-cut quartz crystal disk, an alternating voltage applied through electrodes on each side of the disk induces an acoustic shear wave. Crystal resonance occurs when a certain thickness-frequency condition is met, corresponding to internal constructive interference of the shear wave
f ) nvq/2d
(1)
where f is the resonant frequency of the standing wave, d is the crystal thickness, vq is the speed of sound in ATcut quartz (3328 m/s), and n is an odd integer that denotes the existence of higher harmonics.1 In 1959, Sauerbrey published the famous result that the deposition of a thin, solid film on the surface of such a crystal leads to a decrease in the resonant frequency that is proportional to the mass of the film.3 This discovery has led to countless experiments in which quartz crystals are used as mass-sensing devices, hence the name quartz crystal microbalance (QCM).4 Typical QCM devices operate at resonant frequencies of 5 MHz or greater and possess a mass sensitivity on the order of 10 ng Hz-1 cm-2. In 2000, Flanigan et al. introduced a novel way of using the QCM, showing that the change in contact area between a hemispherical gel cap and the crystal could be linearly * To whom correspondence may be addressed. Phone: (847) 4671752. Fax: (847) 491-7820. E-mail:
[email protected]. (1) Janshoff, A.; Galla, H.-J.; Steinem, C. Angew. Chem., Int. Ed. 2000, 39, 4004. (2) Nye, J. F. Physical Properties of Crystals: Their Representation by Tensors and Matrixes; Clarendon Press: Oxford, U.K., 1985. (3) Sauerbrey, G. Z. Phys. 1959, 155, 206. (4) O’Sullivan, C. K.; Guilbault, G. G. Biosens. Bioelectron. 1999, 14, 663.
related to the corresponding change in resonant frequency.5 A fracture mechanics analysis of adhesive contact, such as the one formulated by Johnson, Kendall, and Roberts (JKR), requires knowledge of the contact area between two objects.6,7 With a signal decay length typically corresponding to several tenths of a micrometer, the QCM preserves the sensitivity to the “bulk” mechanical properties of a material, while still probing length scales that are small enough so that it can legitimately be described as a sensor of surface contact. In this paper we extend our previous work with this contact geometry by measuring both the frequency shift and the dissipation of the QCM. More importantly, we quantify the appropriate contrast factor that relates these measurable quantities to the contact area between the gel and the QCM surface. QCM Background To provide a context for our experiments, we begin with a brief overview of the response of the quartz crystal to mechanical loading. An attractive feature of the QCM is that the impedance behavior around its resonant frequency can be modeled by a simple, four-component electrical circuit, known as the Butterworth-van Dyke (BVD) equivalent circuit.8 Figure 1 shows a BVD circuit, which is composed of a capacitor in parallel with a series combination of an inductor, capacitor, and resistor. The electric branch of the circuit includes the capacitance C0. This quantity represents the capacitance of the crystal disk plus any stray capacitive effects, and it dominates the impedance behavior away from resonance. The other three components make up the acoustic branch, so named because these components dominate around the resonant frequency of the acoustic shear wave in the crystal. La is akin to the crystal’s inertia (an inductor opposes changes (5) Flanigan, C. M.; Desai, M.; Shull, K. R. Langmuir 2000, 16, 9825. (6) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324, 301. (7) Shull, K. R. Mater. Sci. Eng., R: Rep. 2002, R36, 1. (8) Bandey, H. L.; Martin, S. J.; Cernosek, R. W.; Hillman, A. R. Anal. Chem. 1999, 71, 2205.
10.1021/la049015r CCC: $27.50 © 2004 American Chemical Society Published on Web 07/20/2004
7084
Langmuir, Vol. 20, No. 17, 2004
Nunalee and Shull
with loading, ∆Γ, is an extremely useful parameter because it accounts for the dissipation of the acoustic shear wave and is directly related to the increased resistance, R2
∆Γ ) R2/4πL1
Figure 1. Butterworth-van Dyke equivalent circuit approximating the acoustic resonance of the QCM. Table 1. Equivalent Circuit Parameters of the Unloaded QCM in Terms of Physical Constants of AT-Cut Quartza expression
C0
L1
22Ao dq
3
d q Fq 8Aoe262
mechanical analog
inertial mass
Ca 8Aoe262 2
π dqµq elasticity
R1 dqηqπ2 8Aoe262 viscosity
a Key: , dielectric constant; A , electrode area; d , crystal 22 o q thickness; Fq, density; e26, piezoelectric coupling constant; µq, elastic constant; ηq, viscosity.
in current as a mass opposes changes in velocity), Ca is related to elastically stored energy, and Ra represents dissipation of acoustic energy similarly to viscous dissipation of shear stresses in fluids.9 For an unloaded crystal, the BVD circuit parameters may be represented purely in terms of the physical properties of AT-cut quartz.1 Table 1 summarizes these analytical expressions, with L1 and R1 used to respectively denote La and Ra for the unloaded crystal case. Upon loading one or both of the crystal electrodes with some mass or viscous substance, the circuit must be modified. Fortunately, the modification involves only the addition of an inductor, L2, and a resistor, R2, in series with the acoustic branch, i.e., La ) L1 + L2 and Ra ) R1 + R2. The total motional impedance, Zm, of the equivalent circuit takes into account all components of the acoustic branch and is given by8
Zm ) (R1 + R2) + iω(L1 + L2) + (1/iωCa)
(2)
where ω is very close to the angular resonant frequency. The series resonant frequency is defined as the frequency where the imaginary component of the motional impedance vanishes, which is equivalent to requiring the phase angle to be zero
i[ωs(L1 + L2) - (1/ωsCa)] ) 0
(3)
and
fs )
1 2π(Ca(L1 + L2))1/2
(4)
Therefore, the change in resonant frequency of a quartz crystal upon loading is simply
∆f ) -fsL2/2L1
(5)
At this point, it is useful to define a second frequency value, Γ, which describes the half width at half maximum of the impedance minimum around fs. The change in Γ (9) Laschitsch, A.; Johannsmann, D. J. Appl. Phys. 1999, 85, 3759.
(6)
Knowledge of ∆f and ∆Γ is enough to characterize the changes in the QCM upon loading because the former is related to L2 and the latter to R2. It is now appropriate to discuss some of the specific kinds of loading conditions that a QCM may encounter. Here, only the final equations will be presented as the derivations have been detailed previously.8 Table 2 includes a summary of the effect of loading conditions on ∆f and ∆Γ. The simplest kind of load is a thin layer of a purely elastic substance, such as a metal deposited from vapor or a polymer cast from solution. This kind of loading was discussed by Sauerbrey and results in a linear relationship between ∆f and mass.3 Since the mass layer is thin and elastic, there is no dissipation of the shear wave and ∆Γ ) 0. If the contacting material is a Newtonian liquid, the shear wave amplitude decays quickly into the liquid to a point beyond which the QCM can no longer detect the presence of the material. The attenuation of the shear wave is given by a decay length, d, which is a function of the liquid’s density, Fl, and viscosity, ηl10
d ) (ηl/πfsFl)1/2
(7)
Knowing Fl and ηl for water, its decay length at 5 MHz is calculated to be 0.25 µm. For a Newtonian liquid, -∆f ) ∆Γ, and each of these quantities is proportional to (Flηl)1/2. This dependence has been verified by testing the frequency response of mixtures of water with different percentages of glycerol added.10,11 More generally, we can consider the loading effect of a semi-infinite viscoelastic material with a complex shear modulus, G* ) |G|eiδ. Here, “semi-infinite” refers to the assumption that the material thickness is much greater than the decay length of the shear wave. In this case, the decay length is12
d)
(|G|/Fv)1/2 2πfs sin(δ/2)
(8)
where Fv is the density of the viscoelastic material and δ is the phase angle describing the linear viscoelastic response. Note from Table 2 that if |G| approaches G′′ (δ ) 90° and G′′ ) ωηl), then ∆f and ∆Γ for the viscoelastic case reduce simply to the Newtonian liquid case, as expected. However, if |G| approaches G′ (δ ) 0°), the assumption of a semi-infinite medium fails because as δ approaches zero, the decay length in eq 8 diverges to infinity. In other words, the equations for the QCM loaded by a viscoelastic material must be modified when the decay length of the acoustic shear wave is larger than the material thickness. The modification to ∆f and ∆Γ is complex, but it will be noted that for very thin films, the Sauerbrey equation is recovered.8 The equations in Table 2 assume full coverage of one QCM electrode. If the electrode is only partially covered by a contacting material, a correction factor must be (10) Martin, S. J.; Spates, J. J.; Wessendorf, K. O.; Schneider, T. W.; Huber, R. J. Anal. Chem. 1997, 69, 2050. (11) Janshoff, A.; Wegener, J.; Sieber, M.; Galla, H. J. Eur. Biophys. J. 1996, 25, 93. (12) Reddy, S. M.; Jones, J. P.; John Lewis, T. Faraday Discuss. 1997, 107, 177.
Spreading Behavior of Polymer Solutions and Gels
Langmuir, Vol. 20, No. 17, 2004 7085
Table 2. Summary of QCM Loading Equationsa
a Key: m, thin layer mass; |G|, magnitude of complex shear modulus; δ, phase angle of complex shear modulus; “l” subscripts denote “liquid”; “v” subscripts denote “viscoelastic”.
included. This arises from the fact that there is a Gaussian distribution of shear wave amplitudes at the electrode surface that has a maximum at the center and decays toward the outer edge of the electrode.13,14 As a result of this nonuniformity, the QCM is most sensitive at the center of its electrodes. This effect can be quantified by introducing a sensitivity factor, KA, which is a function of the fractional coverage, A/A0. Here A is the actual coverage and A0 is the area of the circular quartz electrode. For partial electrode coverage, we assume that the QCM equations in Table 2 are multiplied by KA and by the fractional contact area. General expressions for ∆f and ∆Γ can be written as follows
A fsL2 ∆f(A) ) -KA A0 2L1
(9)
A R2 ∆Γ(A) ) KA A0 4πL1
(10)
and
In the full coverage case (A g A0) we have KA ) 1. For small coverages (A/A0 f 0), the QCM response is approximately twice as sensitive as in the full coverage case, i.e. KA ≈ 2.15 Experimental Section Polymer Gels and Solutions. A triblock copolymer consisting of poly(methyl methacrylate) (PMMA) endblocks (weight average molecular weight, Mw ) 20 000 g/mol) and a poly(tertbutyl acrylate) (PtBA) midblock (Mw ) 150 000 g/mol) was synthesized by anionic living polymerization. Details of the polymerization technique have been discussed in previous publications.16,17 Gels were formed by dissolving the triblock copolymer in 2-ethylhexanol at 80-90 °C and cooling to room temperature. At ∼60 °C, the solution underwent a sharp sol-gel transition due to the insolubility of the PMMA endblocks at these lower temperatures. The PMMA endblocks aggregate into spherical domains, forming a networked structure. This general behavior has been illustrated with a variety of alcohols;18 2-ethylhexanol was chosen in our case because it does not evaporate significantly over the course of a single experiment. (13) McKenna, L.; Newton, M. I.; McHale, G.; Lucklum, R.; Schroeder, J. J. Appl. Phys. 2001, 89, 676. (14) Josse, F.; Lee, Y.; Martin, S. J.; Cernosek, R. W. Anal. Chem. 1998, 70, 237. (15) Lin, Z.; Hill, R. M.; Davis, H. T.; Ward, M. D. Langmuir 1994, 10, 4060. (16) Mowery, C. L.; Crosby, A. J.; Ahn, D.; Shull, K. R. Langmuir 1997, 13, 6101. (17) Flanigan, C. M.; Crosby, A. J.; Shull, K. R. Macromolecules 1999, 32, 7251. (18) Drzal, P. L.; Shull, K. R. Macromolecules 2003, 36, 2000.
Figure 2. Apparatus for experiments in which a polymer gel cap is brought into contact with a polymer-coated quartz crystal. Load, displacement, contact images, and QCM data are all simultaneously recorded during the test. The inset shows an image taken during gel contact: the large white circle is the Au electrode; the dark circle is the outline of the gel cap; and the small white circle is the area of contact. For impedance analysis, the oscillator circuit is replaced by an impedance analyzer. Elastic moduli of the gels ranged from 102 to 104 Pa, depending on the concentration of polymer in the solvent. In these experiments, concentrations of 0.05, 0.10, 0.15, 0.20, and 0.25 g/mL were tested. Hemispherical caps of the gels were formed by heating to 90 °C and transferring ∼0.1 mL of the solution to the surface of a clean glass slide. In addition to the triblock copolymer, PtBA homopolymer (Mw ) 300 000 g/mol) was also synthesized by anionic polymerization. Solutions in 2-ethylhexanol were prepared at several concentrations. QCM Apparatus. Figure 2 is a schematic diagram of the apparatus employed for QCM adhesion testing. The motion of the apparatus was controlled by a piezoelectric stepping motor (IW-702-00, Burleigh Instruments, Fishers, NY) in series with a 50 g load transducer (FTD-G-50, Schaevitz Sensors, Hampton, VA) capable of resolving tensile and compressive loads of ∼0.1 mN. A fiber optic displacement sensor (RC100-GM2OV, Philtec, Annapolis, MD) was used to record the movement of the gels relative to the quartz crystal. When a gel came into contact with the QCM, circular contact areas were formed, as shown in the inset of Figure 2. A microscope and camera were used to capture images of these contact areas, which were later quantified using an image analysis program (ImageJ, National Institutes of Health). All motion control and data acquisition were automated using a LabVIEW programming interface (National Instruments, Austin, TX). A typical experiment involved bringing the gel into contact with the QCM at a motor velocity of 5 µm/s, increasing contact up to a maximum compressive load of 1-2 mN, and then pulling the gel away at a similar velocity until full detachment occurred. Quartz crystals (131223-5, International Crystal Manufacturing, Oklahoma City, OK) were 1.37 cm in diameter, 333 µm in
7086
Langmuir, Vol. 20, No. 17, 2004
Nunalee and Shull
Figure 3. Change in resonant frequency upon loading the QCM with centered droplets of water with varying sizes. The lines represent the slopes of ∆f vs fractional area for both small (dotted line) and total (solid line) electrode coverages. thickness, and polished to an optically smooth finish. The electrodes were 0.681 cm in diameter and were composed of a 10 nm Cr adhesion layer under 100 nm Au. Prior to testing, thin layers of PMMA (∼100 nm) were spin-coated onto one side of the quartz crystals in order to create the desired polymer surface. Copper wires were then soldered to the edges of the Au electrodes and connected to either the QCM oscillator circuit (35366, International Crystal Manufacturing) or an impedance analyzer (1260, Solartron, Hampshire, U.K.). For experiments involving PtBA solutions, a short glass tube was glued to the QCM to create a reservoir. No motion or image acquisition was required in these liquid experiments because coverage was over the entire electrode surface. The oscillator circuit was designed to drive a quartz crystal at its resonant frequency (∼5.0 MHz), and offered two pieces of information: the frequency at which the impedance phase angle is equal to zero, and a voltage that is related to Γ.19 Resonant frequencies were measured with a frequency counter (53181A, Hewlett-Packard, Palo Alto, CA), and voltages were recorded with a source meter (2400, Keithley Instruments, Cleveland, OH). Calibration curves relating the oscillator output voltage to Γ were created by subjecting the QCM to various loading conditions. For each load, the voltage of the oscillator circuit was recorded in conjunction with a frequency sweep using the impedance analyzer. BVD equivalent circuit fits to the impedance spectra yielded values of Ra, and eq 6 was used to convert R2 to ∆Γ.
Results and Discussion QCM Results. Figure 3 shows the response of the QCM in terms of ∆f as centered water droplets are added to an electrode surface. Note that the magnitude of the slope of the ∆f vs fractional area curve begins to decrease as more of the electrode is covered. Quantitatively, we can use eq 9 in conjunction with the data from Figure 3 to determine the accuracy of our assumptions of the boundary conditions for KA. The form of eq 9 specific to Newtonian liquid loading is
∆f(A) ) -fs3/2KA
( )
A Flηl A0 πFqµq
1/2
(11)
The solid line in Figure 3 represents the relationship between ∆f and A/A0 for full electrode coverage (KA ) 1). Using this slope and assuming Fl ) 1000 kg/m3, the (19) Wessendorf, K. O. IEEE 1993, 1-711.
Figure 4. Impedance magnitude and phase angle for the QCM in air (circles), H2O (diamonds), and 2-ethylhexanol (squares). Solid lines represent fits to the data using the equivalent circuit model shown in Figure 1. The resonant frequency for each loading condition is highlighted by the box.
viscosity of H2O is calculated to be 0.84 cP, in good agreement with the actual value of 0.89 cP at 25 °C. Likewise, the dashed line in Figure 3 illustrates the frequency-contact area proportionality for very small, centralized contact areas. In this case, an assumption of KA ) 2 yields a viscosity of 0.95 cP. The measured viscosity values in both cases deviate from the actual value by less than 10%. Therefore, in our experiments involving contact between hemispherical gels and the QCM, the contact areas are small enough so that KA ) 2 is a good assumption. Conversely, KA ) 1 will be used for experiments involving full liquid coverage of the electrode. Figure 4 shows plots of impedance magnitude and phase angle around fs for three different conditions: the QCM in air, with one electrode submerged in water, and with one electrode submerged in 2-ethylhexanol (η ) 10 cP at 25 °C). These spectra were taken using an impedance analyzer, which recorded the complex impedance behavior of the QCM over the given frequency range. The local minima in impedance correspond closely to the series resonant frequencies, while the local maxima occur near the parallel resonant frequencies. Solid lines reflect BVD circuit fits to the data. Note that the magnitudes of both ∆f and ∆Γ are larger for 2-ethylhexanol than for water, as predicted by the equations for a Newtonian liquid in Table 2. This result can be seen graphically by noting the decrease in the resonant frequency (highlighted by box) and the broadening of the resonance with the addition of
Spreading Behavior of Polymer Solutions and Gels
Langmuir, Vol. 20, No. 17, 2004 7087
Figure 6. ∆Γ vs contact area for PMMA-PtBA-PMMA gels in 2-ethylhexanol with five different polymer concentrations in contact with the QCM. The data represent the loading portion of a full contact experiment. Solid lines are linear fits to the data.
Figure 5. Load, ∆f, and ∆Γ vs displacement for a 0.10 g/mL PMMA-PtBA-PMMA gel in 2-ethylhexanol contacting a PMMA-coated quartz crystal in air.
the liquids. Spectra such as these give a complete analysis of the QCM resonance under a specific loading condition, but their usefulness is limited by the fact that significant time is required to complete a full impedance scan. We wish to collect load, displacement, contact area, and QCM data in situ during gel contact experiments, so the use of the oscillator circuit has been employed to allow for this. The results of a representative contact experiment between a 0.10 g/mL triblock copolymer gel and a PMMAcoated quartz crystal in air are shown in Figure 5. Note that load, ∆f, and ∆Γ are all simultaneously recorded as a function of the displacement. Contact areas were also measured but are not shown in this figure. Just as the two surfaces meet, there is a jump into contact that is apparent by the slightly negative load at zero displacement. At this point, ∆f drops and ∆Γ rises suddenly due to the abrupt increase in contact area. After the maximum load is reached, the motor switches directions and the contact area begins to decrease. All three plots indicate a measure of hysteresis, which arises from the details of the adhesive interaction between the two surfaces. Note that after final detachment, ∆f does not immediately go back to zero. This effect is likely due to condensed 2-ethylhexanol on the QCM surface, which then evaporates after a few seconds. Since this deposited liquid is so thin, it obeys the Sauerbrey condition;8 this explains why ∆Γ does not undergo the same phenomenon upon detachment (see Table 2). Experiments with different gel concentrations on PMMA-coated quartz crystals yielded similar trends. Another way to plot the above results is to show ∆Γ as a function of contact area, which is done in Figure 6 for gels with five different polymer concentrations. For clarity, only the loading portion of the experiments is shown. The linear relationship between ∆Γ and contact area is clearly evident from Figure 6. A similar linearity between ∆f and the contact area is also observed, as discussed previously
by Flanigan et al.5 It should be noted that this linear relationship is valid only when the contact area between the gel and substrate is much less than the total electrode area, as previously highlighted in the discussion of Figure 3. While ∆f and ∆Γ both depend linearly on contact area, the slope from one experiment to another is much more consistent for the dissipation, ∆Γ, than for the frequency shift, ∆f. The origins of this effect are still under investigation, but the result is consistent with previous statements that frequency shifts are much more sensitive than the dissipation to uncontrolled parameters such as the detailed stress state of the crystal itself, particularly when operating at the fundamental resonant frequency.20,21 The potentially important role of residual stresses is evident in the work of Borovsky et al., where frequency shifts were recorded while the crystal was brought into contact with a rigid indenter.22 In this case, the resonant frequency increased with the applied load at a rate of 10-30 Hz/mN. These frequency shifts are in the opposite direction to the shifts due to the mechanical loading by a liquid or soft gel that are described in Table 2. While the effect is relatively small in comparison to the frequency shifts plotted in Figure 5, it points to the potential importance of stress effects in the frequency data. In our experiments, the loads applied from the gel are low, but the crystal itself is not always supported in a completely reproducible manner. Because the dissipation represented by ∆Γ is quite reliably reproduced and is in quantitative agreement with the expected response for a Newtonian liquid, we utilize primarily the results obtained from the dissipation in our qualitative analysis. High-Frequency Behavior of Dilute Polymer Solutions. When reliable values of ∆f and ∆Γ are both obtained, the ratio between these quantities can be used to extract the phase angle of the complex shear modulus, as suggested by the equations listed in Table 2. Because of the issues related to the determination of ∆f that exist in our case, we are unable to measure the phase angle (20) Plunkett, M. A.; Wang, Z. H.; Rutland, M. W.; Johannsmann, D. Langmuir 2003, 19, 6837. (21) Johannsmann, D. Personal communication. (22) Borovsky, B.; Krim, J.; Syed Asif, S. A.; Wahl, K. J. J. Appl. Phys. 2001, 90, 6391.
7088
Langmuir, Vol. 20, No. 17, 2004
directly. Data exist, however, which suggest that the phase angle should be close to 90° for our gels and solutions at these frequencies. These data come from traditional rheological experiments involving extreme time-temperature superposition techniques that allowed frequencies of around 1 MHz to be sampled for polymer solutions such as polystyrene in a very high viscosity solvent.23 At these high frequencies the phase angle did indeed approach 90°, with a loss modulus that increased linearly with the angular frequency. The complex viscosity at this high frequency is dominated by its real component and has a value defined as η′∞ that is generally distinct from the solvent viscosity, ηs. Furthermore, the reduced highfrequency viscosity, log(η′∞/ηs), of the polystyrene solutions increased linearly with concentration.24 The slope of a plot of log(η′∞/ηs) vs the solution concentration is defined as the high-frequency intrinsic viscosity, [η]∞. This viscosity is independent of molecular weight but is sensitive to slight compositional changes in the polymer backbone. This behavior is in contrast to the low-frequency intrinsic viscosity, which is much more dependent on molecular weight than the backbone composition. Historically, it was observed that [η]∞ was always positive, i.e., increasing polymer concentration increased η′∞. A great deal of theoretical effort was made to describe the origins of this additional dissipation in polymer solutions at high frequencies.25-28 However, the discovery that some systems displayed a negative intrinsic viscosity nullified most of these theories.29 More recently, Lodge has proposed that the high-frequency behavior is governed primarily by the relaxation times of the solvent molecules and that these relaxation times are modified by the polymer itself.30 A negative intrinsic viscosity can be justified in cases where the time scale for segmental reorientation in a polymer is actually faster than a solvent’s reorientation times. The important result from our point of view is that dilute polymer solutions can behave like Newtonian liquids at high frequencies, with the solvent viscosity modified in some way by the presence of polymer. Returning to our results, we can use the equation for ∆Γ given in Table 2 for a viscoelastic material to find values of |G| in our gels (with KA ) 2). Additionally, we have assumed a phase angle of 90°, so the equation for a Newtonian liquid can be used, where ηl ) η′∞. Figure 7 shows reduced viscosity of a PMMA-PtBA-PMMA gel in 2-ethylhexanol as a function of polymer concentration. Multiple trials were run at each concentration. Data are also shown for solutions of PtBA in 2-ethylhexanol at the same polymer concentrations. These solution viscosities were found by flooding the QCM surface, determining ∆Γ, and again using the Newtonian liquid equations (with KA ) 1). In both the solution and gel cases, Fl was assumed to be that of pure 2-ethylhexanol (830 kg/m3). From Figure 7, it is immediately apparent that the linear relationship between reduced viscosity and concentration observed previously is obeyed in these QCM experiments as well. It is also noteworthy that the gels have a higher intrinsic viscosity than the PtBA solutions. This result is (23) Massa, D. J.; Schrag, J. L.; Ferry, J. D. Macromolecules 1971, 4, 210. (24) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; 1980. (25) Peterlin, A. J. Polym. Sci., Part B: Polym. Lett. 1972, 10, 101. (26) Fixman, M.; Kovac, J. J. Chem. Phys. 1974, 61, 4939. (27) Doi, M.; Nakajima, H.; Wada, Y. Colloid Polymer Sci. 1975, 253, 905. (28) Allegra, G. J. Chem. Phys. 1974, 61, 4910. (29) Morris, R. L.; Amelar, S.; Lodge, T. P. J. Chem. Phys. 1988, 89, 6523. (30) Lodge, T. P. J. Phys. Chem. 1993, 97, 1480.
Nunalee and Shull
Figure 7. Reduced viscosity vs polymer concentration for PMMA-PtBA-PMMA gels (circles) and PtBA solutions (squares) in 2-ethylhexanol. Linear fits are shown, with their slopes being equivalent to the high-frequency intrinsic viscosity of the material.
interesting because the PMMA domains occupy only a small fractional volume of the overall gel, yet they have a significant effect on the measured response. Nonetheless, it is clear that for a given polymer gel or solution in contact with the QCM the magnitude of frequency shifts observed upon loading can be described by [η]∞. Experiments are currently planned to further investigate the nature of highfrequency viscosity measurements as probed by the QCM, including the use of alternative methods that can give a more reliable measure of ∆f. Gels in Liquid Environments. Ultimately, we wish to use the QCM to probe gel contact when submersed in liquids. The need for accurate adhesion measurements in liquid environments is increasing, particularly as efforts are made to synthesize and test biologically useful adhesives in aqueous surroundings.31 QCM experiments require that some contrast mechanism exist so that the gel can be distinguished from the surrounding environment. Ample contrast exists when the surrounding environment is air, as indicated by the data in Figure 6. In liquid environments, however, this contrast is reduced considerably. In the specific case where a solvent-rich gel is placed in the corresponding solvent (a hydrogel in water, for example), the contrast factor is determined by the highfrequency intrinsic viscosity, which can be quite low. The situation is further complicated by features of the QCM that are inherent when operating in liquids. For instance, the QCM generates compressional waves normal to the electrode surface when submersed in a liquid.13 These waves are longitudinal and thus do not decay. Upon reflection at a liquid-air or liquid-solid interface, the compressional waves can travel back to the QCM surface and modulate the crystal resonance. This effect is generally not pronounced in quiescent experiments that include measurements of polymer adsorption from solution.20,32 In our situation, however, the relative motion of two surfaces in our mechanical experiments results in the presence of moving interfaces. The net result is the introduction of an effective noise level in the frequency or dissipation measurement that is often comparable to (31) Lee, B. P.; Huang, K.; Nunalee, F. N.; Shull, K. R.; Messersmith, P. B. J. Biomater. Sci., Polym. Ed. 2004, 15, 449. (32) Ho¨o¨k, F.; Kasemo, B.; Nylander, T.; Fant, C.; Sott, K.; Elwing, H. Anal. Chem. 2001, 73, 5796.
Spreading Behavior of Polymer Solutions and Gels
changes due to the increasing contact area itself. These effects are a topic of ongoing investigation. The importance of our present work is that we have established the role of the high-frequency intrinsic viscosity in the determination of the contrast factor that enables the QCM to be used in these types of contact mechanics experiments. Summary An experimental apparatus that combines a traditional contact mechanics approach with the use of the quartz crystal microbalance has been developed. While changes in the frequency and dissipation are both linearly related to the contact area, the dissipation imparted by the viscous response of the solutions and gels is found to be more reliable. The relationship between the contact area and the measured dissipation is determined by a combination of the solvent viscosity and the high-frequency intrinsic
Langmuir, Vol. 20, No. 17, 2004 7089
viscosity for the system of interest. For situations where the gel is immersed in the corresponding solvent, the contrast factor describing the sensitivity of the QCM to contact with the gel is directly related to this highfrequency intrinsic viscosity. These results compare favorably to the existing literature and provide a basis for further uses of the quartz crystal microbalance in determining adhesive interactions between soft materials. Acknowledgment. This work was funded by an NSF Graduate Research Fellowship and by grants from NIH (R01 DE14193) and NSF (DMR 0214146). We also acknowledge Dr. Diethelm Johannsmann for a variety of helpful comments regarding practical issues related to the use of the QCM. LA049015R
