Langmuir 2006, 22, 431-439
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QCM Studies of Gel Spreading: Kraton Gels on Polystyrene Surfaces F. Nelson Nunalee and Kenneth R. Shull* Department of Materials Science and Engineering, Northwestern UniVersity, 2220 Campus DriVe, EVanston, Illinois 60208-3108 ReceiVed July 21, 2005. In Final Form: October 12, 2005 Contact of a polymer gel made from a styrene/ethylene-butene/styrene triblock copolymer in mineral oil was investigated by bringing the gel into contact with the coated surface of a quartz crystal microbalance (QCM). The experimental apparatus enabled simultaneous measurement of the load, displacement, and contact area, in addition to the resonant frequency and dissipation of the oscillating quartz crystal. The QCM response was determined by the linear viscoelastic properties of the gel at the frequency of oscillation. A geometric correction factor involving the contact area provided a means for quantitatively determining these viscoelastic parameters as the gel spread over the QCM surface. When the gel was removed from the surface, a thin solvent layer was left behind. The thickness of this solvent layer was determined from the QCM response and was compared to predictions from a simple model involving the disjoining pressure of the film and the osmotic pressure of the gel. Qualitative agreement with the model required that tensile, adhesive forces at the perimeter of the gel/QCM contact area were taken into account when calculating the film thickness.
1. Introduction When a highly compliant elastic solid comes into contact with a rigid surface, the compliant material spreads over the surface. The equilibrium spreading is determined by a balance between adhesive forces, which drive the spreading, and elastic restoring forces, which oppose the spreading. If we characterize the adhesion energy by the work of adhesion, W, and the elastic response by the shear modulus, G0, we expect that the ratio of these two quantities, W/G0, will determine the amount of spreading that is observed at equilibrium. This ratio, which we define as the adhesion length, l adh, appears in many adhesion problems. Consider, for example, the elastic spreading of an incompressible, elastic sphere of radius R on a flat surface. If no external loads are placed on the sphere, and the weight of the sphere is neglected, the equilibrium radius, a0, of the circular contact patch between the sphere and flat surface is given by the following expression, derived originally by Johnson, Kendall, and Roberts:1
(
a0 πW ) R 32G0R
)
1/3
) 0.461
( ) l adh R
1/3
(1)
Equation 1 is valid in the case in which a0/R, and hence l adh/R, is small. This requirement is often met quite easily. If we use a value of 0.05 J/m2 for W, which is a typical value for organic materials adhering to one another in air, we have l adh ) 50 nm for a typical elastomer with G0 ) 106 Pa, and l adh ) 50 µm for a soft gel with G0 ) 1000 Pa. Because the contact radius scales as l adh1/3, measurable values of a0 are still obtained, despite these very small values of the adhesion length. For R ) 1 mm, for example, eq 1 gives a0 ) 17 µm for the elastomer and a0 ) 170 µm for the gel. While polymer gels are elastic materials with well-defined moduli, a majority of their volume is generally made up of a small molecule solvent. It is useful, therefore, to introduce the terminology that is used to describe the adhesion of purely liquid * To whom correspondence should be addressed. Phone: 847-467-1752. Fax: 847-491-7820. E-mail:
[email protected]. (1) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London Ser. A 1971, 324, 301.
systems. Since liquid systems have no elastic contribution, the adhesion length diverges to infinity, and the shape of a droplet is determined by the contact angle, θ, that the droplet makes with the surface. The work of adhesion is related to this contact angle by the familiar Young-Laplace equation:
W ) γ(1 + cos θ)
(2)
where γ is the surface energy of the liquid. For cases in which the adhesive interactions of an elastic gel are dominated by the small molecule component, the value of W appearing in eq 1 is closely approximated by the value that would be obtained from a contact angle measurement that utilizes the gel solvent. By replacing W in eqs 1 and 2 with the energy release rate, the treatment can be extended to situations in which the contact angle is not necessarily equal to the equilibrium contact angle.2 Adhesion hysteresis in the contact of an elastic gel is conceptually related to contact angle hysteresis of the corresponding liquid, although the molecular origins of energy dissipation in the two cases are generally different.3 The picture presented above is that the spreading process is driven by the tendency of the solvent to spread over the surface. The elastic polymer network is simultaneously deformed and limits the degree to which the solvent is able to spread. The implicit assumption in this picture is that the gel is osmotically incompressible, so that the polymer concentration remains spatially uniform, even in regions of high tensile stresses that exist near the perimeter of an adhesive contact.4 At a macroscopic level, this assumption is valid when the osmotic pressure of the gel is substantially larger than its shear modulus. At a microscopic level, however, this assumption must break down at sufficiently small length scales. For example, when a gel is pressed against a surface and then removed, one generally finds that a very thin layer of solvent will be transferred to the surface. In this paper, we use the quartz crystal microbalance (QCM) to study the spreading behavior of a model polymer gel on a rigid surface. The elastic component of the gel is a Kraton triblock (2) Maugis, D.; Barquins, M. J. Phys. D Appl. Phys. 1978, 11, 1989. (3) Shull, K. R.; Chen, W.-L. Interface Sci. 2000, 8, 95. (4) Shull, K. R. Mater. Sci. Eng. R 2002, R36, 1.
10.1021/la051980+ CCC: $33.50 © 2006 American Chemical Society Published on Web 11/24/2005
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Figure 1. Representation of the QCM resonance in conductance space for a clean crystal and a crystal loaded by a polymer gel with a small interfacial contact area. Graphical definitions of the resonant frequency, f3, and the dissipation, Γ3, are indicated.
copolymer with polystyrene end blocks and an ethylene-butene midblock. Gels are formed in mineral oil, which is an excellent model solvent because of its low volatility. The QCM is used as a contact probe because of its inherent surface sensitivity and because of its adaptability to traditional contact mechanics experiments. Since the QCM is a relatively new tool in studies of elastic contact and liquid spreading, we begin in the following section with a background description of loading effects on the QCM behavior. We then describe the specific experimental apparatus used in our experiments and present results obtained for the Kraton gels spreading on polystyrene surfaces.
2. Background 2.1. Effect of Loading on QCM Resonance. The QCM consists of a thin, quartz disk cut at a crystallographic angle that results in an induced transverse shear wave when an alternating voltage is applied through electrodes on the top and bottom of the crystal. The acoustic shear wave experiences sharp resonances at certain frequencies, corresponding to internal constructive interference within the crystal. These resonance conditions are determined by the following relationship:
fn )
()
n n µq Vq ) 2dq 2dq Fq
1/2
(3)
where fn represents the resonant eigenfrequencies, n is an odd integer, dq is the thickness of the crystal, Vq is the speed of sound in quartz (3328 m/s), µq is the shear modulus of quartz (2.95 × 1010 Pa), and Fq is the density of quartz (2.65 g/cm3).5 The lowest resonant frequency (n ) 1) is called the fundamental frequency and is denoted by f1. The utility of the QCM lies in the fact that its resonant frequencies change in predictable ways when loaded by an external material.6 The resonance behavior of the QCM is well-illustrated by the measured real admittance, or conductance, across the QCM at frequencies near its resonant frequency, as shown in Figure 1. One finds that at most frequencies, the conductance across the QCM is rather small because quartz is a dielectric material. (5) Janshoff, A.; Galla, H.-J.; Steinem, C. Angew. Chem. Int. Ed. 2000, 39, 4004. (6) Bandey, H. L.; Martin, S. J.; Cernosek, R. W.; Hillman, A. R. Anal. Chem. 1999, 71, 2205.
However, at the resonant frequency, there is a sharp increase in the measured conductance. The frequency of maximum conductance is defined as the resonant frequency, fn, and the half width at half-maximum of the conductance curve is the crystal’s dissipation, Γn. In Figure 1, examples of the resonance for both a clean crystal and a crystal loaded by a polymer gel with a small interfacial contact area are shown for the third harmonic (n ) 3; f3 ) 15 MHz). All data in this paper are collected at the third harmonic because of increased stability of the resonant frequency. This enhanced stability is generally ascribed to a more efficient confinement of the shear amplitude to the center of the quartz crystal, as quantified in more detail below.7 Note that the change in resonant frequency upon gel loading, ∆f3, is negative, while the change in dissipation, ∆Γ3, is positive. This general trend is common when loading the QCM with soft, solid materials or liquids.6,8,9 Certain loading conditions, such as very stiff indenters or very small contact areas, may result in positive changes in the resonant frequency, but these conditions are not met in the experiments described here.10,11 2.2. Radial Sensitivity of the QCM. In this section, we describe some of the equations used to characterize the QCM, with a particular focus on the radial sensitivity of quartz crystals with circular electrodes. It is assumed that the QCM has a circular electrode area, A0, over which it is sensitive to changes in loading conditions. It is often convenient to define ∆fn and ∆Γn as the real and imaginary components of a complex frequency shift, ∆f /n, as follows:12
∆f /n ) ∆fn + i∆Γn
(4)
For an established contact area, A, between a hemispherical gel and the QCM surface, the complex frequency shift is related to the contact area and the complex acoustic impedance of the gel, Z*:
A if1 Z* ∆f /n ) KA A0 π Zq
(5)
where Zq ) (Fqµq)1/2 is the acoustic impedance of the quartz and KA is a contact area-dependent sensitivity factor. Equation 5 is an adaptation of the result obtained for uniform spatial loading of the QCM, where KAA/A0 is equal to one.12 The gel impedance, Z*, is equal to (FvG*)1/2, where Fv is the density of the viscoelastic gel and G* is its complex shear modulus at the frequency of oscillation. By expressing G* in terms of its magnitude, |G*|, and phase angle, φ, i.e., G* ) |G*|eiφ, the following expression for ∆f /n is obtained:
A if1 (F |G*|)1/2eiφ/2 ∆f /n ) KA A0 πZq v
(6)
From the trigonometric relationship eiφ/2 ) cos(φ/2) + i sin(φ/2) and the definition of the complex frequency shift given in eq 4, we obtain the following expressions for ∆fn and ∆Γn for the semi-infinite, viscoelastic loading condition: (7) Plunkett, M. A.; Wang, Z. H.; Rutland, M. W.; Johannsmann, D. Langmuir 2003, 19, 6837. (8) Flanigan, C. M.; Desai, M.; Shull, K. R. Langmuir 2000, 16, 9825. (9) Nunalee, F. N.; Shull, K. R. Langmuir 2004, 20, 7083. (10) Laschitsch, A.; Johannsmann, D. J. Appl. Phys. 1999, 85, 3759. (11) Borovsky, B.; Krim, J.; Syed Asif, S. A.; Wahl, K. J. J. Appl. Phys. 2001, 90, 6391. (12) Johannsmann, D. Macromol. Chem. Phys. 1999, 200, 501.
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Langmuir, Vol. 22, No. 1, 2006 433
A f1 ∆fn ) -KA (F |G*|)1/2 sin (φ/2) A0πZq v
(7)
A f1 ∆Γn ) KA (F |G*|)1/2 cos(φ/2) A0 πZq v
(8)
Direct expressions for |G*| and φ are obtained by rearrangement of eqs 7 and 8:
|G*| )
( )
2 2 2 1 A0 2π Fqµq (∆fn) + (∆Γn) KA A Fv f2
φ ) 2 tan-1
48 000 upon gel loading in this example, and the maximum shear displacement is reduced to approximately 25 Å. The sensitivity of the QCM is related to the energy imparted into the gel from the vibrating quartz crystal, which scales as the square of the surface shear displacement, u2(r).15,20,21 Since the form of u(r) is known from eq 11, the sensitivity factor, KAA/A0, can be analytically defined as the ratio of the integrated squared surface shear displacement across the contacted area to that over the total quartz crystal:
A KA ) A0
(9)
1
( ) -∆fn ∆Γn
(10)
Equations 7 and 8 predict a linear relationship between ∆fn or ∆Γn and A/A0 when KA and the gel’s material properties are independent of contact area. This linearity has been demonstrated in previous work, with the experimental constraint that A/A0 is small.8,9 Fractional contact areas in those previous experiments were all less than 0.2 and were often much smaller. In reality, the sensitivity of the QCM is not constant over its electrodes, and thus the linear relationship between frequency shifts and A/A0 must break down at larger contact areas.13 Because of the physical condition that the shear displacement amplitude at the QCM surface must decay to zero beyond the outer perimeter of its electrodes, there is a radial distribution of shear amplitudes, u(r), across the surface that is empirically fit by a Gaussian function:5,14-16
( )
r2 u(r) ) umax exp -β 2 r0
fn umax ) cnV 2Γn
(12)
where cn is a harmonic-dependent proportionality constant. For 5 MHz AT-cut quartz oscillators operating at their fundamental frequency (n ) 1), the proportionality factor has been shown experimentally to be 1.4 ( 0.1 pm/V,18 a value that is consistent with the piezoelectric strain coefficient for quartz.14 Johannsmann has recently suggested that cn must be inversely proportional to n2,19 in which case we expect c3 ) 0.16 pm/V. In our experiments, the crystal is driven at a voltage amplitude of 315 mV. In the case of an unloaded QCM with n ) 3 (Figure 1), Q ) 250 000, and eq 12 gives umax ≈ 125 Å. The quality factor decreases to (13) Ullevig, D. M.; Evans, J. F.; Albrecht, M. G. Anal. Chem. 1982, 54, 2341. (14) Martin, B. A.; Hager, H. E. J. Appl. Phys. 1989, 65, 2630. (15) Lin, Z.; Hill, R. M.; Davis, H. T.; Ward, M. D. Langmuir 1994, 10, 4060. (16) Lin, Z. X.; Ward, M. D. Anal. Chem. 1996, 68, 1285. (17) Kanazawa, K. K. Faraday Discuss. 1997, 77. (18) Borovsky, B.; Mason, B. L.; Krim, J. J. Appl. Phys. 2000, 88, 4017. (19) Johannsmann, D., personal communication, 2005.
(13)
q
where a is the radius of the area of contact between the gel and the QCM surface and rq is the radius of the entire quartz crystal. Integration of eq 13 gives the following:
A KA ) A0
( )
a2 1 - exp -2β 2 r0
( )
rq2 1 - exp - 2β 2 r0
(14)
When the radius of the quartz crystal (rq) substantially exceeds the electrode radius (r0), as is the case in our experiments, the denominator in eq 14 approaches unity. By making the substitutions A ) πa2 and A0 ) πr02, eq 14 can be rewritten in terms of the fractional contact area, assuming the area is circular and concentric with the center of the electrodes:
(
)
A A KA ) 1 - exp -2β A0 A0
(11)
Here, umax is the maximum displacement at the center of the electrodes (r ) 0), r0 is the radius of the smaller electrode, and β is the Gaussian shape factor. Note that umax and β are both expected to depend to a certain degree on the harmonic index, n. The value of umax is proportional to the quality factor, Q, of the QCM multiplied by the amplitude of the alternating voltage applied across the crystal, V.14,17-19 In our formalism, Q ) fn/ 2Γn, so a general expression for umax is
∫0a 2πru2(r)dr ∫0r 2πru2(r)dr
(15)
By combining eqs 7 and 8 with eq 15, one obtains the following results for the changes in resonant frequency and dissipation upon loading the QCM with a viscoelastic material with contact area A/A0:
( (
( )) ( ))
A f1 (F |G*|)1/2 sin(φ/2) (16) ∆fn ) - 1 - exp -2β A0 πZq v A f1 ∆Γn ) 1 - exp -2β (F |G*|)1/2 cos(φ/2) (17) A0 πZq v For small contact areas, KA reduces to 2β and the assumption of a linear dependence of ∆f and ∆Γ on A/A0 is sensible. Conversely, when A/A0 is large, KAA/A0 approaches unity and the QCM equations describing the response to uniform crystal loading are recovered. Previous research indicates that β ≈ 1 at the fundamental resonant frequency, and a similar value is obtained from our experiments with n ) 3.8,9,15 3. Experimental Section 3.1. Kraton Gels. Kraton G polymer was purchased from an industrial supplier. It is an ABA triblock copolymer with poly(styrene) (PS) endblocks and a midblock composed of a random arrangement of poly(ethylene) (PE) and poly(1-butene) (PB). The midblock composition arises from the random 1,2 and 1,4 addition a of butadiene monomer, followed by hydrogenation of the resulting polymer. The molecular weight of the entire polymer was determined (20) Ward, M. D.; Delawski, E. J. Anal. Chem. 1991, 63, 886. (21) Josse, F.; Lee, Y.; Martin, S. J.; Cernosek, R. W. Anal. Chem. 1998, 70, 237.
434 Langmuir, Vol. 22, No. 1, 2006 to be approximately 170 000 g/mol from gel permeation chromatography, with a polydispersity index of 1.1. The midblock was determined to be 86% PE and 14% PB by 13C NMR. The overall PS content of the copolymer was not quantified independently, but it is expected to be approximately 30%, which is typical of this series of commercial Kraton polymers. Gels were formed by mixing desired concentrations of Kraton G polymer in light mineral oil (330779-1L, Sigma-Aldrich, St. Louis, MO), heating to 175 °C for 1 h, and then cooling to room temperature. The Kraton gels were viscoelastic with very temperature-dependent viscosities; moderate polymer concentrations (0.10-0.25 g/cm3) were quite elastic at room temperature but flowed like liquids at temperatures above approximately 100 °C, making them easily moldable into any desired shape. Gels with concentrations of around 0.05 g/cm3 or less flowed even at room temperature, while those with concentrations of 0.30 g/cm3 or more were too viscous to be conveniently molded at elevated temperatures. Thus, the experiments described below utilized Kraton gels with concentrations of 0.10, 0.15, 0.20, and 0.25 g/cm3. The density of the gels was taken to be 0.84 g/cm3, which is the density of the majority component, mineral oil. The gels were made into hemispheres with several different radii of curvature, ranging from 1.5 to 6.0 mm, by heating the gels into the liquid state, pouring them into poly(dimethyl siloxane) (PDMS) (Sylgard 184, Dow Corning, Midland, MI) molds, allowing the gels to cool, and then removing them from the molds. 3.2. JKR-QCM Apparatus. Because a groundbreaking analysis of the adherence of soft spheres was developed by Johnson, Kendall, and Roberts,1 this experimental geometry is commonly referred to as the JKR geometry. We utilize this notation in our current paper and refer to the apparatus that combines mechanical measurements with the QCM as the JKR-QCM apparatus, even though it can be applied to a variety of different geometries. The implementation of the QCM into a micromechanics experiment has been described in detail previously,9 but the QCM itself has been updated to more accurately determine values of fn and Γn for a variety of harmonics. AT-cut quartz crystals (149257-1, Maxtek, Inc., Santa Fe Springs, CA) were designed to operate at a fundamental resonant frequency of 5.0 MHz and had a total diameter of 2.54 cm. The as-received crystals had circular gold electrodes above a titanium adhesion layer. The diameters of the top and bottom gold electrodes were 1.27 and 0.64 cm, respectively. Since the bottom electrode is the smaller of the two, it is taken as the relevant electrode area of the QCM, A0, which is 31.7 mm2. In each experiment, the top half of the crystal was coated with a thin polystyrene (PS) film (Mw ) 193 kg/mol), typically 100 nm thick, cast from anhydrous toluene (244511-1L, Sigma-Aldrich) using a commercial spin coater (1-EC101-CB15, Headway Research, Inc., Garland, TX). The crystals were held in a custom holder (CHC-100, Maxtek, Inc.), which provided good electrical contact between the gold electrodes and the coaxial wires leading to the network analyzer. Scans of real and imaginary admittance of the crystals were taken with a network analyzer designed specifically to operate at the high frequencies required by the QCM (250B-1, Saunders & Associates, Phoenix, AZ). The network analyzer was controlled by a software program created by Prof. Diethelm Johannsmann (QTZ, Resonant Probes, Goslar, Germany). The QTZ program determined the resonant frequency and bandwidth (dissipation) of the QCM at any given time by fitting the Lorentzian form of the resonance condition to the measured real and imaginary admittance of the crystal. In addition to the fundamental resonant frequency, the QTZ program is able to monitor up to 25 harmonics. For a JKR-QCM experiment, the quartz crystal and its holder were placed below a hemispherical polymer gel. The gel was attached to a glass slide that was rigidly bound to a piezoelectric stepping motor (IW-702-00, Burleigh Instruments, Fishers, NY) in series with a load sensor (FTD-G-50, Schaevitz Sensors, Hampton, VA). The vertical motion of the gel was followed using a fiber-optic displacement sensor (RC62-GLMORV, Philtec, Annapolis, MD). Images of the gel-QCM interface were recorded using a microscope (Zoom 6000, Navitar, Inc., Rochester, NY) and CCD camera (KPM2AN, Hitachi Kokusai Electric, Inc., Japan). The motion of the
Nunalee and Shull
Figure 2. Schematic representation of the JKR-QCM experiment, during which a hemispherical Kraton gel is brought into contact with a PS-coated quartz crystal with gold electrodes. Inset: image of the circular contact area between a gel and the QCM, which appears as the central bright region. motor and the acquisition of load, displacement, and image data were all controlled automatically using a customized LabVIEW program (LabVIEW 7.0, National Instruments, Austin, TX). During a typical JKR-QCM experiment, a gel was brought into contact with the PS-coated QCM surface at a velocity of 3 µm/s and compressed up to an arbitrary maximum load. At that point, the direction of the motor was reversed and the gel was pulled away from the QCM until detachment occurred. Note that during the experiment, load, displacement, contact area, ∆fn, and ∆Γn were all recorded automatically. Figure 2 shows a simplified schematic diagram of the JKR-QCM setup. The motor, load sensor, and displacement sensor are omitted from the diagram. The inset of Figure 2 is an image of a Kraton gel in contact with the QCM, with the light circle in the middle indicating the area of contact between the gel and the PS-coated QCM.
4. Results and Discussion 4.1. Basic Features of the JKR-QCM Test. Figure 3 shows the results for a representative JKR-QCM experiment, where a Kraton gel of polymer concentration 0.15 g/cm3 in mineral oil is brought into contact with a PS-coated QCM. The point of initial contact occurs at a displacement of 0 µm. The compressive load is first increased to a maximum value of approximately 5 mN. The direction of the motor is then reversed and the gel is pulled away from the crystal until detachment occurs. Note that the fractional contact area, A/A0, increases during the compressive portion of the experiment and decreases when the motor direction is reversed. The hysteresis in the area vs displacement graph is present due to the excess energy necessary to overcome the adhesive forces between the gel and the substrate.4 Figure 3b shows the QCM response over the course of the same contact experiment. The change in dissipation of the QCM, ∆Γ3, has the same shape as the change in contact area during the tack test. Likewise, the negative change in resonant frequency, -∆f3, also shares this shape. These relationships illustrate one of the basic functional uses of the QCM in contact mechanics studies: the response of the QCM is proportional to contact area when A/A0 is small.9 The linearity between the resonant frequency and the contact area is more clearly demonstrated in Figure 4, which shows both ∆f and ∆Γ vs A/A0 for a Kraton gel with a polymer concentration of 0.20 g/cm3 in contact with a PS-coated QCM. For clarity, only the compressive loading portions of the curves are shown. Linear fits to the data are provided. In general, the resonant frequency decreases linearly and the dissipation increases linearly with increasing contact area. 4.2. Consideration of Larger Contact Areas. The linearity of both ∆f3 and ∆Γ3 with A/A0 in these and previous results includes the important assumption that the contact area is small.8,9 As discussed in Section 2.2, this linearity is a limiting condition
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Langmuir, Vol. 22, No. 1, 2006 435
Figure 5. Change in dissipation vs fractional contact area for several concentrations of Kraton gels in contact with a PS-coated QCM. Fits to the curves using eq 17 with the adjustable parameter |G*|1/2 cos(φ/2) are shown. A predicted line indicating the response to pure mineral oil is also provided.
shown in Figure 5. In these fits, we assume β ) 1 and Fv ) 0.84 g/cm3, and we use the factor |G*|1/2 cos(φ/2) as an adjustable parameter. Upon determining this factor, |G*| can be calculated since φ is known from ∆f3 and ∆Γ3 by eq 10. As a point of reference, a predicted line for pure mineral oil is shown in Figure 5. For a Newtonian liquid, the phase angle is 90° and |G*| ) 2πfnηl, where ηl is the liquid’s viscosity, so eqs 16 and 17 can be simplified to the following: Figure 3. (a) Fractional contact area and (b) changes in resonant frequency and dissipation as a function of advancing and receding displacement over the course of a JKR-QCM contact experiment with a 0.15 g/cm3 Kraton gel.
Figure 4. Changes in resonant frequency and dissipation vs fractional contact area for a 0.20 g/cm3 Kraton gel pressed against a PS-coated QCM. Linear fits to the data are provided.
of the more complete contact area dependence of the QCM response. Figure 5 shows curves of ∆Γ3 vs A/A0 for four different concentrations of Kraton gels brought into contact with a PScoated QCM. Again, only the compressive loading portions of the curves are given. The curves in Figure 5 are no longer linear because the contact areas are much larger than those sampled in Figure 4. Fits to the dissipation data using eq 17 are also
(
(
)) ( )
A f1 fnFlηl -∆fn ) ∆Γn ) 1 - exp - 2β A0 Zq π
1/2
(18)
where F has the subscript "l" to indicate that the load is now purely that of a liquid.22 The calculated solid line shown in Figure 5 was drawn using eq 18, assuming β ) 1, Fl ) 0.84 g/cm3, and ηl ) 26 mPa-s. This value of viscosity was obtained experimentally by covering the upper electrode of the QCM with mineral oil (KAA/A0 ) 1), measuring ∆f and ∆Γ, and solving for ηl using eq 18. The phase angle of the complex shear modulus calculated from eq 10 was 89°, thus supporting the assertion that mineral oil acts as a Newtonian liquid even at these high frequencies. Figure 5 is useful because it illustrates the dependence of the QCM on contact area as elasticity is increasingly built into an initially viscous system. 4.3. Stress Effects. Up to this point, it has been assumed that the QCM response with respect to contact area is independent of whether the gel is in compression or tension. While this is a fair approximation in most cases, one finds that there is a slight difference between the loading and the unloading portions of the curves, a feature that is illustrated in Figure 6 for a 0.15 g/cm3 Kraton gel. Note that ∆Γ3 is lower in tension than it is in compression for the same contact areas. Therefore, while the response of the QCM is controlled primarily by contact area for a given gel-substrate system, the contact history is also important. We do not expect that the hysteresis observed in Figure 6 is due to the hysteresis in the load itself, because these small loads are not expected to affect the response of the crystal.10,23 To verify this, a 0.25 g/cm3 Kraton gel was made in a cylindrical (22) Kanazawa, K. K.; Gordon, J. G. Anal. Chem. 1985, 57, 1770. (23) Heusler, K. E.; Grzegorzewski, A.; Jackel, L.; Pietrucha, J. Ber. Bunsen Phys. Chem. 1988, 92, 1218.
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Figure 7. Normal tensile stress profile for adhesive contact between a hemispherical, elastic body and a flat, rigid substrate according to eq 19 with a/R ) 0.1 and (1 - ν) G /G0a ) 0.005.
Figure 8. Illustration of solvent transfer during compression and subsequent unloading of a Kraton gel on a PS-coated QCM.
Figure 6. (a) Change in frequency and (b) change in dissipation for a 0.15 g/cm3 Kraton gel in contact with a PS-coated QCM over the course of a typical JKR-QCM test. Data points are connected by lines to guide the eye.
shape so that the contact area between the gel and the QCM would remain constant. The gel was pressed into the QCM with a compressive force of 50 mN and then held in tension up to 100 mN. Both forces are much larger than those imposed on the crystal during any of the other experiments discussed in this paper, yet no changes in the QCM response were noted. On the other hand, it was shown in Figure 5 that the QCM is sensitive to changes in the mechanical properties of the contacting material, so it is possible that the hysteresis seen in Figure 6 is a result of the local changes in composition of the gel at its surface due to the nonuniform stress-state of the gel. The nature of the nonuniform stress state of the gel can be illustrated by considering the radial distribution of σzz, the normal tensile stress in the contact plane. For a soft, hemispherical material in adhesive contact with a flat, rigid substrate, σzz is given by the following:4,24
{
( )}
(1 - ν)σzz r 2 1/2 4a )1+ G0 πR a (1 - ν)G 2 πG0a
(
){ 1/2
1-
(ar) }
2 -1/2
(19)
Here, G is the energy release rate resulting from the adhesive interactions between the gel and the substrate, and G0 and ν are (24) Johnson, K. L. Contact Mechanics; Cambridge University Press: Cambridge, UK, 1985.
the shear modulus and Poisson’s ratio for the hemispherical gel. Note that these are the low-frequency elastic constants that determine the load/displacement relationship for the gel and not the high-frequency values corresponding to the frequency of oscillation of the crystal itself. We refer to the low-frequency modulus as G0 to distinguish it from the high-frequency complex modulus, G*, which determines the response of the QCM and corresponds to the appropriate resonant frequency, fn. The first term on the right-hand side of eq 19 is the compressive stress required to form a contact radius of a in the absence of adhesive interactions, and the second term is the tensile stress due to adhesive forces. The relative importance of this adhesive term is determined by the dimensionless ratio [(1 - ν) G R]/[G0a2], which is most prevalent for small contact areas. Figure 7 is a representative plot of the overall normal tensile stress distribution given by eq 19 with units of (1 - ν)σzz/G0 vs r/a. In generating this plot, we have assumed typical values for our experiments of a/R ) 0.1 and (1 - ν) G /G0a ) 0.005. 4.4. Solvent Transfer. Because of the osmotic compressibility of the gels, the solvent will tend to flow from regions of low to high hydrostatic tension, which has a spatial variation in the contact region that is similar to the normal tensile stress shown in Figure 7. As a result, the solvent is forced to the edge of the contact zone, where the local hydrostatic tension is highest.25 In addition, when the gel is removed from the substrate, a certain amount of the solvent (in this case mineral oil) will be transferred to the PS surface. Figure 8 illustrates this situation schematically. As a result of the solvent being expelled from the gel and remaining on the PS substrate, ∆fn does not return to its original value after detachment, which can be seen clearly in Figure 6a. This residual oil film is thin enough to act as an “ideal mass layer”, for which there is no change in dissipation and the change in resonant frequency is given by the Sauerbrey equation.6 It is (25) Webber, R. E.; Shull, K. R.; Roos, A.; Creton, C. Phys. ReV. E 2003, 68.
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osmotic pressure of the gel and the disjoining pressure of the transferred oil film.29 We begin with an expression for the osmotic pressure, Πos, of a semidilute polymer solution, which can be described in terms of the polymer-polymer screening length, ξ:30
Πos )
kB T
(22)
ξ3
where kB is Boltzmann’s constant and T is the absolute temperature. In the semidilute regime, the scaling prediction for the screening length is:31
ξ ≈ bφp-3/4
Figure 9. Calculated thicknesses of residual oil films transferred to the QCM after Kraton gel contact using eq 21.
assumed that the area of the transferred oil layer is equal to the maximum contact area of the gel during compression, i.e., the total mass of the transferred oil is equal to m, and it is uniformly spread over an area equal to Amax. While spreading may occur in such an oil layer, the time scale for appreciable motion in films of nanometer thicknesses and viscosities similar to that of mineral oil is on the order of hours to months.26-28 Since the QCM measurements are taken only seconds after the maximum contact area has been established, spreading effects can safely be ignored. Thus, the following form of the Sauerbrey equation can be used to describe the change in frequency after detachment of the gel from the substrate:
(
(
))
Amax 2f1fn m ) A0 Zq Amax
∆fn ) - 1 - exp -2β
(
(
))
Here m/Amax ) Ff t, where Ff is the density of the oil film (0.84 g/cm3) and t is its thickness. Rearrangement of eq 20 gives the following expression for the thickness of the oil film:
(
(
))
Amax A0
t ) - 1 - exp -2β
-1
Zq
2nf12Ff
where φp is the polymer volume fraction and b is a systemdependent constant, typically on the order of 0.5 nm for small molecule solvents. Therefore, the osmotic pressure as a function of polymer volume fraction is
Πos )
∆fn
(21)
Figure 9 shows thickness values obtained from eq 21 for several gel samples at four different polymer concentrations. The value of the Gaussian shape factor, β, is again chosen to be unity. On the basis of these experiments, it can be seen that gels with lower polymer concentrations transfer more mineral oil to the substrate. This trend is sensible because gels with lower polymer concentrations have a higher thermodynamic activity of mineral oil. The relationship between polymer concentration and residual oil film thickness can be justified theoretically by examining the equilibrium state of the mineral oil at the ternary gel-PS-air interface. Here we present an approximate analysis, which ignores the elasticity of the gel and is based on an equality between the (26) Heslot, F.; Fraysse, N.; Cazabat, A. M. Nature 1989, 338, 640. (27) Heslot, F.; Cazabat, A. M.; Levinson, P.; Fraysse, N. Phys. ReV. Lett. 1990, 65, 599. (28) Daillant, J.; Benattar, J. J.; Leger, L. Phys. ReV. A 1990, 41, 1963.
kBT 9/4 φp b3
(24)
The disjoining pressure, Πdis, is the internal pressure necessary to support an oil film of thickness t between the PS substrate and the overlying air and is given by the following expression:32
Πdis )
-AHam
(25)
6πt3
Here, AHam is the Hamaker constant for the ternary PS-oilair system. By equating the osmotic pressure of the gel with the disjoining pressure of the oil film, the following relationship is obtained between φp and t:
φp )
Amax 2nf12 F t. (20) A0 Zq f
- 1 - exp -2β
(23)
() ( ) t b
-4/3
-AHam 6πkBT
4/9
(26)
From published values for AHam for similar systems of oligomeric hydrocarbon films on fused quartz in air, we expect AHam ≈ -10-20 J,32 so that eq 26 simplifies to φp ) 0.40(t/b)-4/3. With b ≈ 1 nm, values for φp obtained from the data in Figure 9 range from 0.01 at a polymer concentration of 0.1 g/cm3 to 0.1 at a polymer concentration of 0.25 g/cm3. These values of φp are much less than the polymer concentrations of the corresponding bulk gels. This result is consistent with an enhancement of the solvent concentration in the region of tensile stress at the periphery of the contact zone, where the film/gel equilibration process quantified by eq 26 is taking place. 4.5. Kinetics of Solvent Equilibration. The dynamics of solvent redistribution in polymer gels are governed by the following collective diffusion coefficient, Dc:33
Dc )
kBT 6πηsξh
(27)
where ξh is the hydrodynamic screening length and ηs is the solvent viscosity (≈ 25 mPa-s for the mineral oil used in our (29) Joanny, J. F.; Johner, A.; Vilgis, T. A. Eur. Phys. J. E 2001, 6, 201. (30) Rubinstein, M.; Colby, R. H. Polymer Physics; Oxford University Press: Oxford, UK, 2003. (31) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (32) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: San Diego, CA, 1992. (33) Adam, M.; Delsanti, M. Macromolecules 1977, 10, 1229.
438 Langmuir, Vol. 22, No. 1, 2006
Nunalee and Shull
experiments). In our case, we can estimate Dc by assuming that ξh is close to the static screening length given by eq 23. With ξh ≈ 5 nm, we expect Dc ≈ 10-8 cm2/s for our gels. This estimated value for the solvent diffusion coefficient compares favorably with the value that can be estimated from the measured swelling kinetics of similar gels.34 Over the time scale of our experiment (minutes), solvent is able to redistribute over distances of several micrometers. These diffusion distances are small compared to the lateral dimensions of the gel but can be large in comparison to the decay length, ∆, of the shear wave in the gel, which is given by the following expression:
∆)
( )
1 |G*| 2πnf1 sin(φ/2) Fv
1/2
(28)
Values of |G*| and φ from the following section give decay lengths ranging from 1.5 to 3 µm. The characteristic time for solvent equilibration within the surface layer probed by the QCM, ∆2/Dc, is in the range of a few seconds. The characteristic time for solvent to be redistributed throughout the entire contact zone, a2/Dc, is orders of magnitude larger. In our experiments, we can therefore assume that solvent composition at the QCM surface is equilibrated and that the lateral solvent redistribution is limited to a width of several microns. The small region of enhanced solvent composition at the periphery of the contact zone is still important, however, in that the local polymer concentration in this region determines the amount of solvent that is transferred to the substrate as described above. 4.6. High-Frequency Viscoelastic Properties. The linear viscoelastic properties of the gels are fully described by φ and |G*|, which can be obtained from the measured values of the frequency shift and dissipation according to eqs 9 and 10. In Figure 10, φ and |G*| are plotted as a function of the polymer concentration in the gels. As expected, pure mineral oil has a phase angle that is very close to 90°, and the addition of polymer to the solvent gradually decreases the phase angle as the gels take on a more elastic character. Concurrently, the magnitude of the shear modulus increases as polymer concentration increases, with magnitudes in the gels ranging from 5 to 13 MPa. To verify that the methodology can also be applied to more elastic systems, we have loaded the QCM with a cross-linked poly(dimethyl siloxane) (PDMS) hemispherical lens, and a phase angle of 3° was obtained. The magnitudes of these moduli can be compared to the values obtained from the load/displacement relationship for the lens as it pushed into the QCM surface and then retracted. The JKR theory predicts the following relationship for a soft, hemispherical material with radius of curvature R in adhesive contact with a flat, rigid substrate:4
P)
(
4aG0 a2 δ1-ν 3R
)
(29)
where P is the compressive load and δ is the corresponding displacement. By controlling R and measuring P, δ, and a during a contact experiment, along with the assumption that the gel is nearly incompressible (ν ≈ 0.5), G0 can be calculated. Figure 11 shows values of G0, obtained from the measured loads and displacements, and the high-frequency storage modulus, G′, obtained from the QCM response. Values of G′ were obtained from the data in Figure 10 from the relationship G′ ) |G*| cos φ. As expected, both the JKR and QCM analyses reveal higher moduli as polymer concentration increases. In addition, at each (34) Quintana, J. R.; Diaz, E.; Katime, I. Polymer 1998, 39, 3029.
Figure 10. (a) Phase angle and (b) magnitude of the complex shear modulus for Kraton gels at a frequency of 15 MHz (n ) 3).
Figure 11. Comparison of storage moduli calculated from the JKR (G0) and the QCM (G′) for Kraton gels.
concentration the shear modulus calculated from the QCM is more than 2 orders of magnitude larger than that from the JKR analysis. This disparity in moduli is not surprising, since the QCM is oscillating at frequencies of 15 MHz, while the JKR measures mechanical responses at relatively small strain rates. Thus, the QCM can be used as a rheological tool to examine shear properties of materials in the megahertz frequency range. By testing multiple harmonic frequencies of the quartz, one can potentially measure frequencies from around 1 MHz to more
QCM Studies of Gel Spreading
than 100 MHz. The analysis requires that the overall dissipation, ∆Γn, be low enough so that a well-defined resonance is still obtained. Highly dissipative systems that would result in excessive values of Γn for full coverage of the quartz crystal can be studied by using relatively low values of A/A0. Hence, the QCM is able to determine the high frequency viscoelastic behavior of polymer solutions and gels at frequencies that have not been previously accessible.
5. Conclusions In this paper, we have studied polymer gels composed of a Kraton polymer in mineral oil with the JKR-QCM apparatus. The JKR portion of the experiment is a standard construction for determining the bulk mechanical and surface adhesive properties of a material. The addition of the QCM to the JKR apparatus provides sensitivity to the surface mechanical properties of the loading material. The particular findings of this investigation can be summarized as follows: 1. A formula for the nonlinear sensitivity factor has been utilized that describes the response of the QCM to a loading material with a variable contact area concentric with the crystal’s electrodes. Past results, which assumed a linear sensitivity factor, were found to be valid in the limiting case of small fractional contact areas. The validity of the sensitivity factor was verified
Langmuir, Vol. 22, No. 1, 2006 439
using Kraton gel hemispheres with varying polymer concentrations in contact with the QCM. 2. Films of mineral oil were transferred from the gels to a PS-coated QCM after a loading-unloading cycle. The thicknesses of these films were calculated from a generalized Sauerbrey relationship and were compared to a pressure balance involving the disjoining pressure of the film and the osmotic pressure in the gel. Qualitative agreement with the model was obtained if the region at the perimeter of the gel/QCM contact area was assumed to be enriched in solvent because of adhesion-related tensile stresses. 3. The high-frequency shear moduli and phase angles of the Kraton gels at different polymer concentrations were obtained from the complex frequency shift of the QCM resonance. The moduli were much larger than the low-frequency values determined from the load/displacement relationship for the polymer gel. Acknowledgment. We gratefully acknowledge Prof. Diethelm Johannsmann and Miriam Kunze for useful discussions regarding the QCM, as well as David A. Brass for his help in the characterization of Kraton G polymer. This work was funded by an NSF Graduate Research Fellowship and grants from NIH (R01 DE14193) and NSF (DMR 0214146). LA051980+
