Langmuir 2003, 19, 7933-7940
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Nanorheological Properties of the Perfluoropolyether Meniscus Bridge in the Separation Range of 10-1000 nm Junho Choi* and Takahisa Kato Institute of Mechanical Systems Engineering, National Institute of Advanced Industrial Science and Technology (AIST), Namiki 1-2-1, Tsukuba, Ibaraki 305-8564, Japan Received March 3, 2003. In Final Form: May 15, 2003 In the present study, the nanorheological properties of a perfluoropolyether (PFPE) meniscus bridge, which is formed between two 20-µm-diameter glass spheres, have been investigated using a nanorheometer developed recently. The PFPE meniscus-bridge system is dominated by an elastic force at low frequencies and by a viscous force at high frequencies, and this behavior can be explained by the Voigt arrangement. The elastic force is predominant at a large separation. At high frequencies and small separations, the behavior of the PFPE meniscus bridge approaches that of the bulk polymer liquid in the “terminal zone”, whereas the PFPE film shows excess stiffness and damping at low frequencies and large separations originating from the meniscus bridge. The storage and loss moduli of the meniscus-bridge system are quantitatively modified from the bulk values in terms of the frequency and the separation. The retardation time can be used as a measure of the dynamic property of the liquid meniscus bridge.
1. Introduction Perfluoropolyether (PFPE) film has great potential as a nanoscale lubricant because of its good viscosity characteristics, low surface energy, low volatility, good thermal stability, and good affinity for the surface, which allows it to be used in many applications such as the magnetic hard disk industry,1,2 microelectromechanical system,3 and space. The bulk rheological properties of PFPE using a conventional rheometer have been reported by Kono et al.4 However, the PFPE films are generally placed in a narrow gap, such as 10 nm, and sheared at a high rate in the magnetic storage disk, where the physical and mechanical properties of the confined PFPE films are fundamentally altered from those of the bulk liquid.5,6 Particularly, the liquid meniscus bridges formed between a magnetic head and disk play an important role in the dynamic behavior of the films. Most of the nanorheological studies of polymer liquids have been carried out by a surface forces apparatus (SFA)7-10 and an atomic force microscope (AFM).11-13 It is known that the ratio of the surface separation to the contact area in the SFA is very small so that the meniscus force with respect to the surface * Author to whom correspondence should be addressed. (1) Homola, A. M. IEEE Trans. Magn. 1996, 32, 1812. (2) Waltman, R. J.; Pocker, D. J.; Tyndall, G. W. Tribol. Lett. 1998, 4, 267. (3) Eapen, K. C.; Patton, S. T.; Zabinski, J. S. Tribol. Lett. 2002, 12, 35. (4) Kono, R. N.; Izumisawa, S.; John, M. S.; Kim, C. A.; Choi, H. J. IEEE Trans. Magn. 2001, 37, 1827. (5) Granick, S. Science 1991, 253, 1374. (6) Hu, H. W.; Carson, G. A.; Granick, S. Phys. Rev. Lett. 1991, 66, 2758. (7) Israelachvili, J. N.; McGuiggan, P. M.; Homola, A. M. Science 1988, 240, 189. (8) Tonck, A.; Georges, J. M.; Loubet, J. L. J. Colloid Interface Sci. 1988, 126, 150. (9) Hu, H. W.; Granick, S. Science 1992, 258, 1339. (10) Luckham, P. F.; Manimaaran, S. Macromolecules 1997, 30, 5025. (11) Kajiyama, T.; Tanaka, K.; Ohki, I.; Ge, S. R.; Yoon, J. S.; Takahara, A. Macromolecules 1994, 27, 7932. (12) O’Shea, S. J.; Welland, M. E.; Pethica, J. B. Chem. Phys. Lett. 1994, 223, 336. (13) Overney, R. M.; Leta, D. P.; Pictroski, C. F.; Rafailovich, M. H.; Lin, Y.; Quinn, J.; Sokolov, J.; Eisenberg, A.; Overney, G. Phys. Rev. Lett. 1996, 76, 1272.
separation stays almost constant; that is, the meniscus force is negligibly small compared to other forces.14,15 In case of the AFM, it is very difficult to determine the quantitative rheological values because the surface separation and exact area of contact are unknown.11,16 As a result, the obtained results are qualitative. In a recent study,15 Friedenberg and Mate measured the amplitude and phase responses of a poly(dimethylsiloxane) bridge constrained between a flat silicon wafer and a tip using an AFM, where the tip was modified by a 44-µm-diameter glass sphere. They found that the dynamic behavior of the liquid meniscus bridge is different from the bulk value, which can usually be measured using the SFA. Although their study was very successful, the quantitative analysis of the rheological properties of the liquid meniscus-bridge system is not sufficient. In the present study, we investigated the nanorheological behavior of the PFPE meniscus-bridge system and quantitatively determined the complex moduli when the surface separation between the two surfaces is in the range of 10-1000 nm. 2. Experimental Section 2.1. Liquid Sample. The PFPE lubricant used was Fomblin Zdol (molecular weight ) 2000 g/mol) produced by Ausimont, which has a bulk viscosity of 144 mPa‚s at room temperature and a surface energy of 21.4 mN/m. The chemical formula of Zdol is as follows: HOCH2-(CF2CF2O)p-(CF2O)q-CH2OH, where p/q ≈ 2/3. 2.2. Experimental Setup. The schematic diagram of the nanorheometer is shown in Figure 1. The optical-beam-deflection technique17 was used for detecting the interaction forces between the two glass spheres. The interaction force is manifested as a displacement of a cantilever, where the magnitude of the force is a product of the displacement and the spring constant of the cantilever. To measure the cantilever displacement, the direction of a laser beam reflected off the backside of the cantilever is monitored with a four-quadrant photodiode as a position sensor. (14) Montfort, J. P.; Tonck, A.; Loubet, J. L.; Georges, J. M. J. Polym. Sci., Polym. Phys. Ed. 1991, 29, 677. (15) Friedenberg, M. C.; Mate, C. M. Langmuir 1996, 12, 6138. (16) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: New York, 1992. (17) Meyer, G.; Amer, N. M. Appl. Phys. Lett. 1990, 57, 2089.
10.1021/la034365j CCC: $25.00 © 2003 American Chemical Society Published on Web 08/12/2003
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Figure 1. Schematic diagram of the experimental setup.
Figure 3. Schematic diagram of a normal oscillatory experiment. The sphere mounted on the piezo actuator oscillates with driving amplitude Y, and then the other sphere mounted on the cantilever oscillates with amplitude X and phase shift φ.
Figure 2. Glass spheres mounted on (top) the cantilever and (bottom) the piezo actuator. The spring constant of the cantilever (Olympus, Japan) used in this study was 0.75 N/m, which was corrected because a glass sphere on the cantilever decreases the effective length of the cantilever.18 The experimental setup is equipped with two twophase lock-in amplifiers so that the amplitude and phase of the probe can be monitored simultaneously in both the normal and the lateral directions. The 20-µm-diameter borosilicate glass spheres (Duke Scientific, Palo Alto, CA) used as solid surfaces are shown in Figure 2. The glass spheres were glued to the cantilever and the piezo actuator by an adhesive, epoxy resin. The cantilever-sphere assembly was rinsed in toluene, acetone, and ethanol in turn before the experiments. 2.3. Force Modulation. Figure 3 shows a schematic diagram of a dynamic experiment. PFPE meniscus bridges were formed between two glass spheres by dropping a droplet onto the sphere (18) Sader, J. E.; White, L. R. Rev. Sci. Instrum. 1995, 66, 3789.
mounted on the piezo actuator using a microsyringe and then allowing the wetted sphere to approach the sphere mounted on the cantilever. The oscillatory motion in the direction of the meniscus-bridge axis is applied to the glass sphere mounted on the piezo actuator, and, thus, the other glass sphere mounted on the cantilever oscillates as a result of the coupling of the two spheres by the meniscus and the viscous forces. The induced oscillation of the cantilever-sphere assembly is detected via a two-phase lock-in amplifier with respect to the modulation frequency and the surface separation. The amplitude and the phase shift of the cantilever-sphere assembly relative to the motion of the sphere mounted on the piezo actuator represent the characteristics of the complex modulus of the PFPE film. The amplitude response is interpreted as elasticity, and the phaseshift response is interpreted as viscosity. By solving the equation for the motion of the sphere mounted on the cantilever, the storage modulus G′ and the loss modulus G′′ of the PFPE film are obtained as shown in eqs 1 and 2 (which are derived from the equation for motion in Appendix I). G′ is defined as the stress in phase with the strain in a sinusoidal deformation divided by the strain, and it is a measure of the energy stored and recovered per cycle. G′′ is defined as the stress 90° out of phase with the strain divided by the strain, and it is a measure of the energy dissipated or lost as heat per cycle of the sinusoidal deformation.19
2D h k 3πR2 2D h G′′ ) ωb 3πR2 G′ )
(1) (2)
where D h is the mean separation between the two spheres and
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Figure 4. Meniscus force FM and the meniscus force constant kM as a function of the surface separation. R is the radius of the sphere. k and ωb are the elastic and viscous force constants of the system, respectively, and are described as follows:
k)
(kc - mω2)(A cos φ - 1) (A cos φ - 1)2 + (A sin φ)2
ωb )
+ kM
(kc - mω2)(A sin φ) (A cos φ - 1)2 + (A sin φ)2
(3)
(4)
where kc is the spring constant of the cantilever, m is the mass of the cantilever-sphere assembly, ω is the radian frequency, A ()Y/X) is the inverse of the amplitude ratio, and φ is the phase shift between the two spheres. The elastic force constant consists of two components. One is the contribution of the viscous force (the first term on the right side of eq 3), and the other is the contribution of the meniscus force. The meniscus force constant, kM, is obtained from the slope of the meniscus force (FM) profile as a function of the mean separation D h , as is shown in eq 5.
kM(D h ) ) dFM(D h )/dD h
(5)
In this study, the modulation frequency was varied from 2 to 1000 Hz, and the modulation amplitude of the piezo actuator was 1.5 nm. The experiment was conducted at an ambient temperature of 27 ( 2 °C and a relative humidity of 28 ( 5%.
3. Results and Discussion 3.1. Meniscus Force Constant. Figure 4 shows the measured meniscus force and the meniscus force constant as a function of the surface separation. As the separation decreases, the meniscus force becomes large, and the maximum meniscus force is detected at separation, D h ) 0. The minus sign means that the meniscus force is attractive. The theoretical value of the maximum meniscus force of the liquid confined between two spheres of radii R is calculated by the equation FM,theoretical ) -2πRγ, where γ is the surface energy.16 In the case of Zdol used in this study, the theoretical value of the maximum meniscus force is -1.34 µN. This value agrees well with the experimental value of -1.38 µN. The meniscus force constant, kM(D h ), is obtained by differentiating the equation of the meniscus force, FM(D h ) ) -3 × 10-5D h 2 + 0.4361D h1399, which is obtained from the polynomial regression analysis of the measured meniscus force curve in Figure 4. The value decreases from 0.44 to 0.37 N/m in the surface separation range of 10-1000 nm. 3.2. Amplitude Ratio and Phase Shift. Figure 5a,b shows the representative raw data for the amplitude ratio and the phase shift as a function of the surface separation (19) Ferry, J. D. Viscoelastic Properties of Polymers; John Wiley & Sons: New York, 1980.
Figure 5. Representative raw data for (a) the amplitude ratio and (b) the phase shift as a function of the surface separation.
at various modulation frequencies. The plots were obtained from measuring the response of the sphere mounted on the cantilever when the oscillating sphere mounted on the piezo actuator at a constant frequency was pulled off from the sphere mounted on the cantilever. Because the adhesive used to bind the glass spheres to the cantilever and the PZT actuator and the apparatus itself possess viscoelastic properties, the measured liquid properties were corrected by measuring the dynamic response of the sphere mounted on the cantilever at dry adhesive contact between the two spheres (the correction method is explained in Appendix II).20,21 As the separation increases from the solid contact, the amplitude ratio decreases from unity and the phase shift increases from 0. In Figure 6a,b, the amplitude ratio and the phase shift between the spheres are shown as a function of the modulation frequency at various surface separations. At low frequencies, the amplitude ratio decreased to a finite value. This low-frequency response is attributed to the effect of the meniscus force, as was studied by Friedenberg and Mate,15 and is different from the studies using the SFA, where the meniscus effect is negligible compared to other forces. The data have the trend that, in the lowfrequency regime, the smaller the separation, the higher the amplitude ratio. This in-phase response is attributed (20) Reiter, G.; Demirel, A. L.; Peanasky, J.; Cai, L. L.; Granick, S. J. Chem. Phys. 1994, 101, 2606. (21) Dhinojwala, A.; Granick, S. Macromolecules 1997, 30, 1079.
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Figure 6. (a) Amplitude ratio and (b) phase shift as a function of the modulation frequency at various surface separations.
to the meniscus force, which increases with decreasing separation. The phase shift approaches 180° at low frequencies and large separations. This is surprising, because in the case of the bulk liquid, the phase shift approaches 90° when the frequency goes to 0 and the separation goes to infinity. This result implies that there is another damping mode originating from the meniscus bridge in addition to the hydrodynamic contribution, and it will be discussed later in detail. At high frequencies, particularly at a small separation, the amplitude ratio approaches unity, and the phase shift goes to 0 as a result of the coupling of the two spheres by the viscous and meniscus forces. 3.3. Complex Modulus. Figure 7a,b shows the elastic force constant, k, and the viscous force constant, ωb, which are calculated by eqs 3 and 4, respectively, plotted against the modulation frequency on a log-log scale, and Figure 8 shows the relative contribution of the viscous force against the elastic force with respect to the separation and frequency. At low frequencies, the elastic forces show higher values than the viscous forces, that is, ωb - k < 0 (closed circles in Figure 8), whereas the contribution of the elastic force decreases with frequency, resulting in ωb - k > 0 (open circles in Figure 8). Dhinojwala and Granick21 measured the viscous forces due to drainage of a good solvent between the solid surfaces coated with a polymer brush when the separation is smaller than the brush thickness. They found that the elastic forces due to the resistance of the brush layers to interpenetrate dominate when the frequency is low, but the viscous forces grow in relative magnitude as the frequency increases.20 This behavior can be explained by the Voigt arrangement of a spring and a dashpot in parallel, where the elastic force caused by the polymer brush and the viscous force undergo the same strain when a force is applied. This is similar to the dynamic behavior of the meniscus system in the present study. In the case of the polymer-brush
Choi and Kato
Figure 7. (a) Elastic force constant, k, and (b) viscous force constant, ωb, plotted against the modulation frequency on a log-log scale.
Figure 8. Relative contribution of the viscous force against the elastic force as functions of the separation and frequency: open circles, ωb - k < 0; closed circles, ωb - k > 0.
system, the elastic contribution is initiated when the surface separation is comparable to the polymer-brush length, whereas the meniscus-bridge system contains the elastic force due to the meniscus bridge as long as the liquid meniscus bridge is present. The interesting thing is how the dynamic behavior of the meniscus-bridge system changes with the surface separation. At small separations, the viscous force dominates at high frequencies and the elastic force dominates at low frequencies. At large separations, the elastic force dominates the system over the entire frequency, that is, ωb - k < 0, and this trend is enhanced as the frequency decreases. This indicates that the relative contribution of the elastic force compared
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Figure 9. (a) Storage modulus, G′, and (b) loss modulus, G′′, as a function of the modulation frequency.
Figure 10. Contributions of (a) the viscous force G′v and (b) the meniscus force G′M in the storage modulus G′.
to the viscous force is enhanced at large separations and the system behaves as a viscoelastic solid. Figure 9a,b shows the storage modulus, G′, and the loss modulus, G′′, which are calculated from eqs 1 and 2, as a function of the modulation frequency. At small separations, G′ shows a gradual increase with the frequency, and the slope of the G′ plots approaches 2. The G′′ plots have slopes of unity at high frequencies. It is known that the bulk polymer liquid in the “terminal zone” shows G′ ∝ ω2 and G′′ ∝ ω,19 which indicates that the dynamic behavior of the PFPE meniscus-bridge system approaches that of the bulk liquid at a high frequency and small separation. On the other hand, the slopes of the G′ and G′′ plots become small compared to the those of the bulk lines as the frequency decreases, which indicates that the meniscus system shows excess stiffness and excess damping compared to the bulk. The excess stiffness and damping comes from the presence of the liquid meniscus bridge, and the excess stiffness is directly related to the meniscus force, as is shown in Figure 10, which shows the contributions of the viscous force G′v (eq A24) and meniscus force G′M (eq A25) in the storage modulus G′. G′M is calculated using the meniscus force constant kM in Figure 4. G′v approaches 0 as the frequency decreases, whereas G′M stays constant with the frequency. The excess damping originates from the damping characteristics of the meniscus bridge, which is related to wetting dynamics. When an oscillatory motion is applied to the sphere mounted on the piezo actuator, the meniscus bridge formed between the spheres is deformed by the liquid flow, and then the contact line, which is associated with the three-phaseboundaries (liquid, air, and solid phases) overlap, moves on the sphere surface.22 The moving contact line induces the line friction as a result of the surface heterogeneities,
which is related to the energy dissipation in the region. As a result, the system shows excess damping compared to the bulk. Crassous et al. studied the damping characteristics of a simple liquid meniscus bridge by SFA.23,24 They also found that the meniscus system shows excess damping and is related to the friction due to the moving contact line. They reported that the line friction depends essentially on the meniscus displacement and much less on its velocity (at a modulation frequency below 80 Hz and a modulation amplitude below 1.3 nm) and then concluded that the contribution to the line friction is quasistatic. In the present study, where the maximum velocity is as large as 1 order of magnitude compared to that in the work of Crassous et al.,24 we found that the excess damping due to the meniscus bridge becomes negligibly small in a high-velocity (frequency) regime; that is, the line friction is related to the meniscus velocity. A more detailed analysis of the velocity dependency of the line friction would be required for a better understanding of the damping characteristics of the liquid meniscus bridges. As is shown in Figure 9, the storage and loss moduli are modified quantitatively from the bulk line in the direction of the arrows in terms of the frequency and surface separation. From this behavior, it would be possible to obtain a master curve using a series of curves obtained at constant separation and estimate the dynamic properties of the meniscus-bridge system in terms of the separation and frequency, resembling the WilliamsLandel-Ferry theory. In the case of the magnetic hard disk industry, a low flying height below 10 nm and a high disk rotating speed are essential to achieve much higher recording densities, but it is very difficult to obtain the
(22) Dussan, V. E. B. Annu. Rev. Fluid Mech. 1979, 11, 371.
(23) Crassous, J.; Charlaix, E.; Gayvallet, H.; Loubet, J. L. Langmuir 1993, 9, 1995. (24) Crassous, J.; Charlaix, E.; Loubet, J. L. Phys. Rev. Lett. 1997, 78, 2425.
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Figure 11. Viscous damping coefficient, b, as a function of the modulation frequency.
dynamic properties of a liquid meniscus bridge at such high-velocity and small-separation ranges. By formulating the meniscus behavior describing the separation-frequency relationship, it may be possible to estimate the behavior in such extreme ranges. 3.4. Viscosity Measurement. Figure 11 shows the viscous damping coefficient, b, as a function of the modulation frequency. The viscous damping coefficient at a small separation of 10 nm is larger than that at a large separation of 1000 nm. It is well-known that the motion of an oscillating sphere in a fluid becomes highly damped as the sphere approaches the other solid surface. The bulk viscous damping coefficient between the two spheres as a function of the mean separation D h is obtained from eq 6, expected from the lubrication theory:25
b ) 3πR2η/2D h
(6)
where η is the bulk viscosity of the liquid and R is the radii of the spheres. The dotted and solid lines in Figure 11 respectively show the bulk viscous damping coefficients at separations of 10 and 1000 nm calculated from eq 6. It is found that the viscous damping coefficient of the PFPE meniscus-bridge system at a separation of 10 nm is almost constant relative to the frequency and agrees well with the bulk value but increases slightly as the frequency decreases. In the case of the separation of 1000 nm, the damping coefficient coincides with the bulk value at a very high frequency and increases steeply as the frequency decreases, where the damping due to the meniscus bridge is remarkable, as is discussed in Figure 8. Figure 12 shows the dynamic viscosity as a function of the surface separation. The dynamic viscosity, η′, is related to the loss modulus by the equation η′ ) G′′/ω and describes the viscous properties of the material. For a purely viscous liquid, the dynamic viscosity is equal to the bulk viscosity η.26 The dynamic viscosity has an almost constant value at high frequencies and small separations, as is shown in Figure 12. We calculated a viscosity of 133 mPa‚s at a small separation of 10 nm and in the frequency range of 150-1000 Hz, which shows good agreement with the bulk viscosity of 144 mPa‚s. This indicates that the systems can be described as a purely viscous liquid. However, the dynamic viscosity increases steeply as the frequency decreases, and the viscosity is enhanced with the separation, as was expected from the results of the complex modulus shown in Figure 9. (25) Moore, D. F. The Friction and Lubrication of Elastomers; Pergamon Press: New York, 1972.
Figure 12. Dynamic viscosity, η′, as a function of the modulation frequency. The dynamic viscosity is related to the loss modulus by the equation η′ ) G′′/ω.
Figure 13. Retardation time, τR, plotted against the surface separation on a log-log scale. The retardation time is obtained from the inverse of the crossover frequency.
3.5. Retardation Time. Figure 13 shows the retardation time τR plotted against the surface separation on log-log scales. The retardation time τR is defined as a measure of the time required for the extension of the spring (elastic element) to its equilibrium length while retarded by the dashpot (viscous element) and is obtained from the inverse of the crossover frequency. The crossover frequency is the frequency at which the relative contributions of the elastic and viscous forces to the system are equal and is deduced from the intersections of the G′ and G′′ profiles. τR is about 10-1 s at a separation of 10 nm and decreases linearly on a log-log plot as the separation increases. At large surface separations, G′ is larger than G′′ because of the contribution of G′M; that is, the elastic force (spring) effect is larger than the viscous force (dashpot) effect, resulting in a small retardation time. On the other hand, G′ becomes small compared to G′′ at small separations because G′M is negligibly small, resulting in a large retardation time. We note that, though the retardation time is not an intrinsic property of the polymer liquid, it can be used as a measure of the dynamic properties of the liquid meniscus bridge. 4. Summary We have investigated the nanorheological properties of the PFPE meniscus bridge formed between two 20-µmdiameter glass spheres in the surface separation range of 10-1000 nm. The results are as follows: (1) The PFPE meniscus-bridge system is dominated by the elastic force at low frequencies (ωb - k < 0) and by the viscous force at high frequencies (ωb - k > 0); that is, the behavior can be explained by the Voigt arrangement. The elastic force is predominant at a large separation.
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(2) The behavior of the PFPE meniscus-bridge system at high frequencies and small separations approaches that of the bulk polymer liquid in the “terminal zone”, that is, G′ ∝ ω2, G′′ ∝ ω. The measured viscosity agrees well with the bulk value. (3) At low frequencies and large separations, the PFPE meniscus-bridge system shows excess stiffness and excess damping. The excess stiffness is directly related to the meniscus force and the excess damping is directly related to the line friction force due to the moving contact line on the sphere surface. (4) In the PFPE meniscus-bridge system, the storage and loss moduli are quantitatively modified in terms of the modulation frequency and surface separation from the bulk value. By formulating this relationship, it would be possible to estimate the rheological properties of the liquid nanomeniscus in an experimentally inaccessible range. (5) The retardation time can be used as a measure of the dynamic properties of the liquid meniscus bridge.
The Maxwell model predicts for a viscoelastic liquid
G(t) ) G0e-t/τ0
(A8)
where G0 is the relaxation modulus and τ0 is the relaxation time. By inserting eq A8 into eq A7 and using the Laplace transformation, we obtain eq A9.
∫∞te-t/τ e-iωt dt ) G01/τ0 1+ iω
G0
(A9)
0
From the relation of η0 ) τ0G0,
η0 1 ) G0 1/τ + iω 1/τ + iωτ0
(A10)
The Maxwell model allows us to express analytically η* with the classical relation
Appendix I The equation for the motion of the sphere mounted on the cantilever in the arrangement of Figure 3 is
η0 ) η* 1 + iωτ0
mx¨ + kcx + FH(t) + FM(D) ) 0
From the relation of eq A11, eq A6 is expressed as follows:
(A1)
where m is the mass of the cantilever-sphere assembly and kc is the spring constant of the cantilever. FH(t) is the hydrodynamic force of the viscoelastic liquid confined between two spheres as a function of time t, FM(D) is the meniscus force as a function of the surface separation D, and FH is described as follows:26
3πR2 FH(t) ) 2D(t)
∫
dD dt′ G(t - t′) -∞ dt′ t
iωD0eiωt
y)Y h + Yeiωt
(A3)
x)X h + Xei(ωt+φ)
(A4)
The surface separation, D, between two spheres is
∫∞0G(ξ)eiωξ dξ ) iωD0eiωtη*
(A12)
) iωη*(D - D h) That is,
FH ) -
(A2)
where G(t) is the relaxation modulus accounting for the past mechanical history of the liquid sample and R is the radius of the sphere. If the sphere mounted on the piezo actuator vibrates as Yeiωt about the mean position Y h , inducing the sphere mounted on the cantilever to vibrate as Xei(ωt+φ) about X h, then we may write at any instant t
(A11)
3πR2 iωη*(D - D h) 2D(t)
(A13)
The meniscus force is
h ) + kM(D - D h) FM(D) ) FM(D
(A14)
Inserting eqs A13 and A14 into eq A1, we obtain eq A15.
mx¨ + kcx -
3πR2 h) + iωη*(D - D h ) + FM(D 2D(t) kM(D - D h ) ) 0 (A15)
By solving eq A15 in terms of the displacements of the two glass spheres y and x, we can obtain the relations as follows:
D ) y - x - 2R ) (Y h -X h - 2R) + (Y - Xeiφ)eiωt ) D h + D0eiωt (A5)
3πR2ω iη*(Y - Xeiφ)eiωt ) -mω2Xei(ωt+φ) + 2D(t) kc[Xei(ωt+φ)] + kM(Y - Xeiφ)eiωt (A16)
where D h is the mean separation and φ is the phase shift between the spheres. Substituting t - t′ ) ξ and inserting eq A5 into eq A2, we obtain eq A6.
If we set A ) Y/X, we can obtain eq A17 from eq A16.
∫
∫0 G(ξ)e
dD dt′ ) iωD0eiωt G(t - t′) -∞ dt′ t
∞
-iωξ
dξ
(A6)
In eq A6, let us consider the following term:
∫∞tG(t)e-iωt dt
(A7)
(26) Montfort, J. P.; Hadziioannou, G. J. Chem. Phys. 1988, 88, 7187.
[
] ]
(kc - mω2)(A cos φ - 1) 3πR2ω iη* ) + kM + 2D(t) (A cos φ - 1)2 + (A sin φ)2 (kc - mω2)(A sin φ) (A17) i (A cos φ - 1)2(A sin φ)2
[
The complex viscosity is expressed as follows:
η* ) η′ - iη′′
(A18)
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By comparing eqs A17 and A18, we can obtain the relations shown as follows:
η′ )
η′ )
[
] ]
(kc - mω2)(A sin φ) 2D h 3πR2ω (A cos φ - 1)2 + (A sin φ)2
[
(kc - mω2)(A cos φ - 1) 2D h + kM 3πR2ω (A cos φ - 1)2 + (A sin φ)2 η′ ) G′′/ω,
η′′ ) G′/ω
(A19)
(A20)
(A21)
From the relations of eq A21, the storage modulus G′(ω) and the loss modulus G′′(ω) of the system are obtained as follows:
G′ )
[
] ]
(kc - mω2)(A cos φ - 1) 2D h + kM 3πR2 (A cos φ - 1)2 + (A sin φ)2
(A22)
(kc - mω2)(A sin φ ) 2D h 3πR2 (A cos φ - 1)2 + (A sin φ)2
(A23)
G′′ )
[
We define the contributions of the viscous force and the meniscus force for the storage modulus, G′v and G′M, as follows:
G′v )
[
]
(kc - mω2)(A cos φ - 1) 2D h 3πR2 (A cos φ - 1)2 + (A sin φ)2 G′M )
2D h kM 3πR2
(A24) (A25)
The elastic force constant, k, and viscous force constant, G′′, of the system are obtained by omitting the hydrody-
Figure 14. Device compliance as a function of the modulation frequency, which was measured at dry adhesive contact (i.e., before the formation of the liquid meniscus bridge).
namic geometrical factor, 2D h /3πR2, from G′ and G′′, as follows:
k)
(kc - mω2)(A cos φ - 1) (A cos φ - 1)2 + (A sin φ)2
ωb )
+ kM
(kc - mω2)(A sin φ) (A cos φ - 1)2 + (A sin φ)2
(A26)
(A27)
Appendix II Adhesive is used to bind the glass spheres to the cantilever and the PZT actuator. Because the adhesive and the apparatus itself possess the viscoelastic properties, the measured liquid properties should be corrected. To obtain the correction factor, the viscoelastic responses of the sphere mounted on the cantilever are measured at dry adhesive contact between the two spheres. The results are shown in Figure 14. The experimental results are corrected as follows: (1) the phase shifts in Figure 14 are subtracted from the measured data of phase shifts and (2) the measured amplitudes are divided by the amplitudes in Figure A1 at the corresponding frequencies. LA034365J
