Viscoelastic Study Using an Atomic Force Microscope Modified to

Both spectroscopic (harmonic response as a function of drive frequency) and ... corresponding to the maximum phase response is found to coincide with ...
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Langmuir 2003, 19, 5254-5260

Viscoelastic Study Using an Atomic Force Microscope Modified to Operate as a Nanorheometer James W. G. Tyrrell and Phil Attard* School of Chemistry F11, University of Sydney, New South Wales 2006, Australia Received August 13, 2002. In Final Form: March 18, 2003 A commercial atomic force microscope (AFM) has been modified to operate as a nanorheometer using only a PC-based hardware component and a custom software interface. Measurements on an agar gel model substrate reveal a viscoelastic response that is well described by a recently proposed theory for viscoelastic deformation. Both spectroscopic (harmonic response as a function of drive frequency) and impulsive (relaxation following a step drive) measurements are made. The reciprocal of the frequency corresponding to the maximum phase response is found to coincide with the step relaxation time in agreement with theory. A relaxation time of 16.6 ms is obtained for the 4% agar gel, and 9.3 ms for the 3% agar gel. Both theory and experiment give a phase lead which, although counterintuitive, has a physical explanation. Material properties measured using the modified AFM show broad agreement with data generated using a conventional parallel plate rheometer and illustrate a useful basis for the comparison of measurement techniques.

Introduction (AFM)1

is routinely emThe atomic force microscope ployed to extract topographical information, with up to subnanometer resolution,2 by profiling the sample surface with a microfabricated cantilever. However, by monitoring the response of the cantilever to a known stress profile, that is, considering the cantilever assembly instead as a load transducer, one can study the rheological behavior of the sample under test.3,4 This opens the way for contributions to the field of nanorheology,5,6 coupling the study of material behavior with the high spatial resolution of AFM apparatus. The aims of this study are to modify the AFM to act as a nanorheometer, to obtain the viscoelastic response of a soft material (agar gel), and to interpret those results in terms of a soft-contact viscoelastic theory. Nanorheological measurements performed using AFM apparatus are, due to the dimensions of the contact region, highly compatible with small sample quantities. This feature, in addition to the aforementioned gains in spatial resolution, offers advantages when analyzing a broad range of samples spanning biological matter and highvalue pharmaceutical compounds, through to polymer composites, blends, novel organic semiconductor networks, and even micromechanical assemblies. Such an opportunity for exploiting existing AFM architecture to advance the study of nanorheological behavior has been recognized and implemented by various workers7-14 over the past decade. Their approaches fall * Corresponding author. Phone: +61 2 9351 5878. Fax: +61 2 9351 3329. E-mail: [email protected]. (1) Binnig, G.; Quate, C. F.; Gerber, C. Phys. Rev. Lett. 1986, 56, 930. (2) Alexander, S.; Hellemans, L.; Marti, O.; Schneir, J.; Elings, V.; Hansma, P. K.; Longmire, M.; Gurley, J. J. Appl. Phys. 1989, 65, 164. (3) Maivald, P.; Butt, H.-J.; Gould, S. A. C.; Prater, C. B.; Drake, B.; Gurley, J. A.; Elings, V. B.; Hansma, P. K. Nanotechnology 1991, 2, 103. (4) Butt, H.-J.; Jaschke, M.; Ducker, W. Bioelectrochem. Bioenerg. 1995, 38, 191. (5) Meyer, E.; Overney, R. M.; Dransfeld, K.; Gyalog, T. Nanoscience: Friction and Rheology on the Nanometer Scale, 1st ed.; World Scientific: River Edge, NJ, 1999; Chapter 6. (6) Mukhopadhyay, A.; Grannick, S. Curr. Opin. Colloid Interface Sci. 2001, 6, 423. (7) Radmacher, M.; Tillmann, R. W.; Gaub, H. E. Biophys. J. 1993, 64, 735.

into the following general categories: (I) analysis of conventional force-distance curves,10,13,14 (II) modulation of the tip either directly8 at the probe (force modulation) or (III) indirectly9 at the root of the cantilever, and finally (IV) modulation of existing (ramp) sample piezo drive signals.7,11,12 Although category I nanorheology can be conducted without any subsequent modification to the existing AFM apparatus, the pitfalls associated with this technique are numerous and center particularly around the nonlinear and velocity-dependent motion of the piezo15-18 and uncertainties in the constant compliance calibration.18 Category II, III, and IV measurements all require decreasingly complex modifications to the basic AFM setup. Category II measurements (direct modulation of the tip, e.g., through magnetic or electromechanical coupling) are desirable in that the magnitude of the applied force is generally well-known following the accurate precalibration of the force sensor, but this has to be balanced against the complexity of necessary modifications to key hardware components. Category III measurements (indirect modulation of the tip) are often achieved using an upgraded version of the standard commercial cantilever holder embedded with a miniature piezo-crystal. However, it is arguably modulation of the sample piezo (category IV) that offers the most cost-effective method of probing sample rheology without succumbing to the pitfalls of force (8) Jarvis, S. P.; Oral, A.; Weihs, T. P.; Pethica, J. B. Rev. Sci. Instrum. 1993, 64, 3515. (9) Overney, R. M.; Leta, D. P.; Pictroski, C. F.; Rafailovich, M. H.; Lui, Y.; Quinn, J.; Sokolov, J.; Eisenberg, A.; Overney, G. Phys. Rev. Lett. 1996, 76, 1272. (10) Nakajima, K.; Yamaguchi, H.; Lee, J.-C.; Kageshima, M.; Ikehara, T.; Nishi, T. Jpn. J. Appl. Phys. 1997, 36, 3850. (11) Braithwaite, G. J. C.; Luckham, P. F. J. Colloid Interface Sci. 1999, 218, 97. (12) Notley, S. M.; Craig, V. S. J.; Biggs, S. Microsc. Microanal. 2000, 6, 121. (13) Opdahl, A.; Somorjai, G. A. J. Polym. Sci., Part B: Polym. Phys. 2001, 39, 2263. (14) Gillies, G.; Prestidge, C. A.; Attard, P. Langmuir 2002, 18, 1674. (15) Libioulle, L.; Ronda, A.; Taborelli, M.; Gilles, J. M. J. Vac. Sci. Technol., B 1991, 9, 655. (16) Hues, S. M.; Draper, C. F.; Lee, K. P.; Colton, R. J. Rev. Sci. Instrum. 1994, 65, 1561. (17) Tyrrell, J. W. G.; Cleaver, J. A. S. Adv. Powder Technol. 2001, 12, 1. (18) Rutland, M.; Tyrrell, J. W. G.; Attard, P. To be submitted.

10.1021/la0207163 CCC: $25.00 © 2003 American Chemical Society Published on Web 05/21/2003

Viscoelastic Study Using a Modified AFM

curve analysis, and it is a variation on this approach that is presented here. Although data logging is generally PC based, external hardware is often required to generate piezo modulation signals (signal generators) and facilitate phase detection (lock-in amplifiers), which adds to the cost and complexity of the apparatus. This paper exploits recent advances in on-board processing power which now enable analogue input/output (I/O) boards, coupled with a modest host computer, to compete with external hardware. Such a configuration reduces apparatus costs and often improves functionality, for example, the capacity to output custom (piezo drive) waveforms, the real-time analysis of the response, and the possibility of using feedback on the analyzed response to modify the drive signal. It does have a possible disadvantage compared to using a signal generator and lock-in amplifier in that the frequency range for spectroscopic measurements may be reduced. Nevertheless, providing sample piezo drive and photodiode circuits are accessible, it is possible to modify an AFM for use as a nanorheometer using only (i) an I/O board, (ii) a host computer, and (iii) a compatible software environment. The overall utility of any apparatus is dependent on the ability to both process and interpret the collected data. One of us has recently developed a theory19,20 to describe the interaction and deformation of viscoelastic particles and substrates. The theory builds on earlier analysis21,22 of elastic particles and numerically solves the viscoelastic equations for the deformation and local pressure selfconsistently as a function of time. What is unique about the theory is that it takes into account the extended range of real surface forces, in contrast to contact elastic theories such as Johnson-Kendall-Roberts (JKR) or Hertz and their viscoelastic extensions.23-26 The approach eschews the empirical spring and dashpot models of rheometry in favor of a molecular model. Because of this difference and in view of the fact that the deformations are calculated at the nanometer level, the approach may be called a nanorheometric theory. The theory19,20 utilizes an exponential form for the viscoelastic response function with three parameters, the short-time elastic response E0, the long-time elastic response E∞, and the relaxation time τ. The questions are, does this model describe real viscoelastic materials and can the model viscoelastic parameters be extracted from experimental measurements? Both questions have been answered in the affirmative for the case of cross-linked poly(dimethylsiloxane) colloid particles, at least for a triangular force measurement.14 Below, the present paper reports quantitative nanorheometric measurements performed on agar gel using different measurement protocols (impulsive and harmonic) and a different geometry (planar substrate). The purpose of this study is threefold: (i) to prove the feasibility of modifying an AFM for use as a nanorheometer using only PC-based I/O hardware, (ii) to obtain specific results for a model experimental system, namely, agar gel, and (iii) to validate the qualitative theoretical behavior described recently by Attard.19 (19) Attard, P. Phys. Rev. E 2001, 63, 061604. (20) Attard, P. Langmuir 2001, 17, 4322. (21) Attard, P.; Parker, J. Phys. Rev. A 1992, 46, 7959. (22) Attard, P. J. Phys. Chem. B 2000, 104, 10635. (23) Falsafi, A.; Deprez, P.; Bates, F. S.; Tirrell, M. J. Rheol. 1997, 41, 1349. (24) Hui, C.-Y.; Baney, J. M.; Kramer, E. J. Langmuir 1998, 14, 6570. (25) Lin, Y. Y.; Hui, C.-Y.; Baney, J. M. J. Phys. Appl. Phys. 1999, 32, 2250. (26) Barthel, E.; Roux, S. Langmuir 2000, 16, 8134.

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Figure 1. Schematic diagram showing the signal flow (boxed arrows) between key elements of the experimental apparatus. A dashed arrow represents the optical interaction between the displacement sensor and the piezo element contained within the AFM microscope head.

Experimental Section Materials. Agar gel was selected for use as a model viscoelastic sample. Agar is a polysaccharide consisting of agarose and agaropectin, extracted from seaweed. It is a versatile material, highlighted by the myriad of industrial applications. In this study, agar gel was an ideal candidate to test hardware, software, and theory because changing the solids concentration modifies the material properties of the sample.27 Agar gel solutions (2%, 3%, and 4% agar by weight) were prepared 24 h in advance by dissolving powdered agar (food grade) in purified water (Elga UHQ) approaching boiling point. The resulting solution was poured into a Petri dish and allowed to cool to room temperature. Once at room temperature, the Petri dish was covered. Immediately prior to measurement, a gel sample approximately 10 × 10 × 1 mm in size was prepared and loaded into the AFM fluid cell assembly. The cell was then filled with UHQ water (advice concerning the filling procedure is given elsewhere28,29) to prevent sample dehydration during the measurement process. The experiments consisted of driving the piezo-crystal upon which the agar gel sample was mounted and measuring the response of the AFM cantilever, which was in contact with the sample via its pyramidal tip. Measurements were made using a noncontact silicon microfabricated AFM cantilever. Preliminary studies established the requirement for applied loads in the range of 8-10 µN in order to elicit a measurable viscoelastic response from the gel sample. This necessitated the use of noncontact mode silicon, as opposed to contact mode silicon nitride (Si3N4), cantilevers as their greater stiffness allowed the application of a larger applied load for a given deflection. The spring constant of the cantilever used in this study (k ) 47 N/m ( 20%) was determined by measuring its resonant frequency (f ) 353 kHz) in air and then cross-referencing this value against the manufacturer’s batch calibration curve. Measurements. AFM. Measurements were performed using a Nanoscope IIIa (Digital Instruments) AFM retrofitted with a Signal Access Module (Digital Instruments) to provide electrical contact with the sample piezo drive and photodiode circuits. The Signal Access Module (SAM) enabled the application of custom piezo drive trajectories and independent monitoring of the AFM optical head photodiode response. Importantly, this setup (a schematic representation is shown in Figure 1) offered a departure from the constraints of the instrument’s standard operational software and hence afforded a high degree of experimental freedom. Custom piezo drive signals (step and sinusoidal) were software generated and applied to the Signal Access Module via the digital-to-analog (D/A) component of the installed I/O board (PCI DAS 1602, supplied by ComputerBoards Inc.). The data acquisition card had 16 bit analogue to digital resolution and a 200 kHz sample rate. Prior to use, the I/O board was autocalibrated against an on-board precision voltage reference (InstaCal). (27) Nitta, T.; Haga, H.; Kawabata, K.; Abe, K.; Sambongi, T. Ultramicroscopy 2000, 82, 223. (28) Mahnke, J.; Stearnes, J.; Hayes, R. A.; Fornasiero, D.; Ralston, J. Phys. Chem. Chem. Phys. 1999, 1, 2793. (29) Tyrrell, J. W. G.; Attard, P. Langmuir 2002, 18, 160.

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The AFM controller was used to automatically engage the cantilever with the substrate following the input of a cantilever deflection set-point. This process gave the user control over the initial load applied to the sample. Following successful engagement, the feedback element of the AFM control loop was subsequently disabled. The sample stage piezo was then excited by either a step or sinusoidal drive signal generated by the D/A component of the I/O board and sent via the SAM. The AFM photodiode signal, representing the response of the cantilever and the sample under load to the movement of the sample stage, was logged on an auxiliary PC (Intel Pentium I, 75 MHz processor, running LabView R6.0) and stored as a data file pending future analysis. The measurement procedure was repeated for a range of deflection set-points, that is, applied loads, and frequencies (when a sinusoidal drive signal was selected). Calibration of the photodiode (i.e., conversion of measured voltage to cantilever deflection) was performed by monitoring the response of the cantilever to the movement of a rigid substrate (glass slide). The piezo was excited using a step drive. This procedure allowed the cantilever deflection signal to be expressed in nanometers. Calibration of the actual piezo element18 (i.e., conversion of applied voltage to drive distance) was confirmed using a fiber-optic displacement sensor (model D20, Philtec Inc. fiber-optic displacement sensor, supplied by Applied Measurement Pty. Ltd.). This sensor exploits changes in the intensity of reflected light with separation and has a resolution of several nanometers over a working range of 15 µm. Rheometer. Sample properties were also measured using a conventional rheometer (Rheometrics SR5000) to enable a comparison in performance between the modified AFM apparatus and standard industry protocols. A 40 mm parallel plate geometry was employed, and samples (3% and 4% agar by weight) were prepared as described previously. However, in this case the samples were not sectioned prior to use. Petri dishes were instead filled to a level of 1.7 mm to complement the AFM sample thickness. Dehydration of the sample was suppressed due to the confining parallel plate geometry of the rheometer. Measurements were made at 21 °C commensurate with the AFM laboratory temperature. To yield a measurable response, it was necessary to apply a larger applied stress (100 Pa) during the 4% agar sample frequency sweep test than for the 3% agar sample (1 Pa). This no doubt reflects the structural contribution of the higher concentration of agar within the sample. Data Processing and Analysis. In the case of the spectroscopic measurements, where the sample stage was driven in a sinusoidal trajectory with time, data at discrete frequencies were analyzed to construct a frequency response profile (amplitude and phase) for the sample under test. The phase shift between the original piezo drive signal and the response of the sample, represented by the AFM photodiode signal received in proportion to the cantilever deflection, was established through crosscorrelation of the corresponding data arrays. Each data array comprised nine complete cycles. The first cycle in each data array was excluded to screen out transient behavior associated with the start-up process, and the following eight cycles were crosscorrelated to improve noise rejection.30 The cross-correlation of the respective signals was coded into a software routine (the corresponding output is shown in Figure 2). To determine the baseline response due to the electromechanical characteristics of the instrument, measurements were also performed against a rigid substrate. This baseline phase shift was found to be negligible on the time scale of the gel’s viscoelastic response. The data analysis protocol limits the instrument to a piezo driving frequency range of 0.1-40 Hz for a 7200 × 2 datapoint array. Naturally, this range could be extended through a combination of software optimization and hardware upgrades (at the upper limit of the current frequency range the A/D circuit is operating at only 28.8% of its maximum sampling rate, suggesting that the auxiliary PC is a prime candidate for upgrade). The lower limit is governed by the mechanical stability of the piezo and its inherent tendency to drift over long time periods. (30) Macosko, C. Rheology: Principles, Measurements and Applications, 1st ed.; Wiley-VCH: New York, 1994; Chapter 8.

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Figure 2. Screen output from the data analysis interface showing the response of a 4% agar gel sample to the sinusoidal (15 Hz) excitation of the AFM piezo along its z-axis. The top two plots show acquisition data corresponding to the drive signal (solid line) and the response of the sample (individual points). In the middle plot, the sample response has been offset by the phase shift revealed by the location of the central peak in the cross-correlation curve (bottom plot).

Figure 3. Calculated cantilever deflection (symbols, k ) 100 N/m) in response to step or harmonic piezo drive (solid line) of a viscoelastic substrate (E0 ) 1010 N m-2, E∞ ) 109 N m-2, τ ) 1 ms). The interaction consists of an electrical double-layer force (P0 ) 107 N m-2, κ-1 ) 1 nm) and a steric force (Ps ) 108 N m-2, λ ) 0.2 nm), the radius is R ) 10 µm, and the initial load is L0 ) 1.1 µN, which corresponds to a nominal separation of h0 ) -5 nm. The amplitude is 0.1 nm for the step and 0.2 nmpp for the harmonic drive, and the angular frequency is ω ) 1250 rad/s.

Results and Discussion Theory. Figure 3 shows the predicted response of an AFM cantilever when either a step or oscillatory drive is applied to a model viscoelastic substrate. It can be seen that following a sudden increase in the applied load, the cantilever deflection initially increases and then over time relaxes. The relaxation is exponential with a time constant of 0.8 ms, which is 20% less than the viscoelastic relaxation time, τ ) 1 ms. The viscoelastic relaxation time is a parameter in the theoretical model whose value is specified in the calculations; the results for other values can be obtained by using it as the unit of time. The deflection of the cantilever changes by 0.07 nm following the 0.1 nm step in amplitude, which indicates that the viscoelastic material has deformed or flattened by 0.03 nm. The amplitude of the harmonic response is likewise less than that of the drive (0.16 nmpp compared to 0.2 nmpp). What is striking about the results in Figure 3 is that the harmonic response leads the applied drive signal (i.e., the phase difference is +0.12 rad). This paradoxical result,

Viscoelastic Study Using a Modified AFM

Figure 4. Phase lead (open symbols, divided by 10) and relative amplitude (closed symbols) in response to an harmonic piezo drive (0.2 nmpp amplitude). The triangles correspond to L0 ) 1.1 µN and k ) 10 N/m, the circles denote L0 ) 0.2 µN and k ) 100 N/m, and the squares signify L0 ) 1.1 µN and k ) 100 N/m; all other parameters are as in the preceding figure.

which appears to violate causality, can also be seen in the experimental measurements shown in Figure 2. The behavior may be understood as a competition between the rate at which the applied force increases and the rate at which the viscoelastic substrate relaxes. When the former dominates (e.g., when the applied load is changing fastest), then the cantilever moves in the same direction as the drive. However when the change in force slows (e.g., near the extremities), then the relaxation dominates and the cantilever moves in the opposite direction to the driven load. The result is that the response appears to anticipate the change in direction of the applied load. This prescience is of course an illusion; the viscoelastic substrate is not anticipating the reversal that will occur after the peak, but rather it is responding to the previously rapid increase in force and its currently slow rate of change, as shown in the figure. Figure 4 shows the model spectroscopic response of the viscoelastic material. Broadly speaking, there is a single well-defined peak in the phase difference and a corresponding step change in amplitude, similar to what has been found previously.19,20 The frequency at which the phase peak occurs and at which the step in amplitude occurs is independent of the applied load and of the instrumental compliance (the cantilever spring constant). Previously, it was shown19 that for weak loads, when the surfaces are out of contact and interact with a pure doublelayer interaction, the peak position depends on the load. Here the results show that for moderate and large loads, when the surfaces are definitely in intimate contact, the position of the phase peak is independent of applied load. However, the magnitude of the amplitude of the response and the magnitude of the phase lead do vary with the applied load and with the instrumental compliance. In particular, more deformation (i.e., greater departure of the relative response amplitude from unity) occurs with increasing spring constant. Also, the substrate appears more compliant (decreased response amplitude) at lower applied loads. The magnitude of the phase lead is independent of the applied load but decreases with decreasing spring constant. The position of the phase peak and the amplitude step in Figure 4 is ωmax ) 1250 rad/s, or 1/ωmax ) 0.8τ, which is identical to the macroscopic relaxation following an impulse, Figure 3. Increasing the applied load does not change the relaxation time following a step (not shown) or the position of the phase peak. Hence a strong prediction of the theory is that this material parameter can be equally

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Figure 5. Rheological phase eq 1 for the computed harmonic response (symbols are as in the preceding figure).

well measured by a single step measurement as by spectroscopy. This prediction is confirmed by the experimental measurements reported later in this section. To make some comparison of these nanorheometric results with those of traditional rheometry, one identifies the stress as the deflection of the cantilever and the strain as the deformation, which is the difference between the deflection and the drive distance. The complex viscoelastic modulus, which consists of a real storage modulus G′ and an imaginary loss modulus G′′, is the ratio of the stress to the strain. What is often reported is the tangent of the phase angle (tan δ), the so-called rheological lag, which is the ratio G′/G′′. This can be expressed as11,31

tan δ )

sin θ G′′ ) G′ cos θ - γ

(1)

in terms of the nanorheometric parameters, γ (the ratio of cantilever to drive amplitudes) and θ (the phase difference between drive and received signals; note that here a phase lead is defined as θ > 0). Figure 5 shows the phase data of Figure 4 in terms of this rheological lag or phase shift. Interestingly enough, the phase shift is independent of the system compliance, but the magnitude is strongly dependent upon the applied load, which is in contrast to the representation of Figure 4. The frequency of the maximum phase shift coincides in all three cases and agrees with that of the maximum phase lead shown in Figure 4. This appears to be the extent to which one can make quantitative comparison between traditional rheometry and the present nanorheometry. Experiment. The phase response of the 4% agar gel to sinusoidal forced oscillations is shown in Figure 6. The frequency of the oscillations ranged from 0.1 to 40 Hz, with each test being conducted at a single frequency (sometimes referred to as a “discrete frequency test”). A third-order polynomial fit to the data was used to identify the maximum phase response, and more importantly its frequency location. For the 4% agar gel, the maximum phase response corresponds to a drive frequency of 60.3 rad/s (9.6 Hz). Results for the 3% agar gel (shown in Figure 7) are qualitatively similar, with the maximum phase lead corresponding to a frequency of 107.2 rad/s (17.0 Hz). In agreement with the theoretical data shown in Figure 4, the location of the maximum phase response shows little variation with applied load. The amplitude response of the 4% agar sample (Figure 6, inset) is seen to increase by about a factor of 2.5 as the (31) Burnham, N. A.; Gremaud, G.; Kulik, A. J.; Gallo, P.-J.; Oulevey, F. J. Vac. Sci. Technol., B 1996, 14, 1308.

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Figure 6. The magnitude of the phase response of a 4% agar gel sample to sinusoidal displacement (66 nmpp) of the piezo along its z-axis at discrete frequencies (0.2-40 Hz, base 10 logarithmic scale). Crosses are for a load of 9.87 µN, and circles are for a load of 14.80 µN. A third-order polynomial fit to the data is shown as a solid line. The amplitude response of the sample is provided in the inset.

Figure 7. The phase and amplitude response of a 3% agar gel sample. All other parameters are as described in the preceding figure.

phase shift approaches its maximum value. The theory predicts a step change in amplitude at the frequency of maximum phase shift. Despite the evident scatter in the data, the inset of Figure 6 confirms this. The increase in the amplitude response seen at higher frequencies indicates a stiffening of the sample. One can also just notice a slight increase in response amplitude at increased mean load. The increase in the amplitude response of the 3% agar sample (Figure 7, inset), although less than for the 4% agar sample, is still evident as the phase shift approaches its maximum value. Using the analysis discussed above, which describes the relationship between the viscoelastic relaxation time and the frequency of the maximum phase response, rheological information can be extracted from the spectroscopic response of the material. In summary, the macroscopic viscoelastic relaxation time in response to a step impulse, τstep (seconds), is given by

τstep )

1 ωmax

(2)

where ωmax is the frequency (radians/second) corresponding to the maximum phase response. As was alluded to above and will be demonstrated next, this viscoelastic relaxation time can then be used to describe independently the response of the same material to a step change in applied load (achieved by suddenly displacing the sample piezo).

Figure 8. The piezo response (gray) to a step change in drive signal (black). The measurement was made by placing the AFM cantilever in contact with a rigid substrate (glass slide) coupled to the piezo.

Figure 9. The response of a 4% agar gel sample to a step change in the displacement (66 nm) of the sample piezo. The applied load (from top to bottom) was 4.93, 9.87, and 19.74 µN. The response of the agar gel sample is shown in gray, where the solid line passing through the gray data points corresponds to an exponential fit, as predicted by the viscoelastic theory (ref 19), using the measured spectroscopic relaxation time, τ ) 16.6 ms. The response curves have been offset by 0, 40, and 80 nm for clarity.

Figure 8 shows the piezo drive response to a step change in drive voltage measured by placing the cantilever in hard mechanical contact with a rigid substrate (glass slide). The displacement of the piezo drive follows closely the characteristics of the input drive signal. It is essential to identify the piezo drive and photodiode response in order to confirm that the measured response of the gel sample is genuine. Notice that there is almost no phase lag evident on this scale between the drive signal and the photodiode response to the piezo movement. One can discern in the data a brief transient creep after the step increase and decrease in voltage, and also a small hysteresis on resetting the voltage to zero (i.e., the photodiode response does not return precisely to zero). The gel response to a sudden increase in applied load (66 nm drive step) is shown in Figure 9 (for a 4% agar sample) and in Figure 10 (for a 3% agar sample). The 2% agar samples were also measured, but the response was comparatively rapid and appeared to be dominated by the response of the rigid substrate. Three different applied loads (i.e., the rest position of the piezo drive before the step in and after the step out) are shown in each figure. In the figures, both the short-time peak and the long-time relaxed response increase with an increase in applied load, indicating a stiffer sample. In contrast, the relaxation times (τstep ) 16.6 ms for the 4% agar sample and τstep ) 9.3 ms for the 3% agar sample) remain approximately

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Figure 10. The response of a 3% agar gel sample to a step change in the displacement of the sample piezo. The applied load (from top to bottom) was 4.93, 9.87, and 19.74 µN. The response of the agar sample is shown in gray, with the solid line being an exponential fit using the measured spectroscopic relaxation time, τ ) 9.3 ms. The response curves have been offset by 0, 40, and 80 nm for clarity.

constant for the respective agar concentrations and are independent of the applied load. This is consistent with the theoretical predictions. As mentioned previously, τstep is not equal to the microscopic relaxation time. In the case of the 4% agar sample (Figure 9), the deflection of the cantilever decreases from 20 nm (at the highest applied load, 19.74 µN) to 10 nm (at the lowest applied load, 4.93 µN) following the 66 nm step in amplitude, which indicates that the viscoelastic material has deformed or flattened by 46-56 nm. That is, the gel substrate appears more compliant at lower applied loads (as found above). The amplitude of the harmonic response (Figure 6) is likewise less than that of the drive (5-20 nmpp response, increasing with increasing frequency, compared to 66 nmpp drive). Similar trends are displayed by the 3% agar data (Figures 7 and 10). The viscoelastic theory predicts an exponentially decaying relaxation (Figure 3).

δ(t) ) δfinal + (δinitial - δfinal)e-t/τstep

Figure 11. The response of a 4% agar gel sample to a stress sweep at a fixed oscillation (1 Hz). These data were measured using a commercial rheometer. The triangles correspond to the storage modulus, G′, the squares denote the loss modulus, G′′, and the circles signify the rheological phase lag, tan δ.

Figure 12. The response of a 3% agar gel sample to a stress sweep at a fixed oscillation (1 Hz). These data were measured using a commercial rheometer. Symbols are as in the preceding figure.

(3)

To fit the theoretical response (δ) to the data, only two fitting parameters are required: (i) an initial displacement (δinitial) and (ii) a final relaxed displacement (δfinal). Importantly, the viscoelastic decay time, τstep, is derived independently via the frequency response profile (shown in Figures 6 and 7) using the relationship provided in eq 2. The agreement between theory and experiment exhibited in Figures 9 and 10 confirms the theoretical prediction that τstep ) 1/ωmax. Deviations from the theoretical response exhibited by the 3% agar sample under high load (Figure 10, bottom curve) likely reflect the consequential loss of structure within the sample (see also the conventional rheological measurements discussed below). In fact, under high load the behavior of the 3% agar sample is close to that of the 4% agar sample. The relaxation time of the 4% agar sample (16.6 ms) is greater than that of the 3% agar sample (9.3 ms). This behavior is expected given the greater resistance to material flow offered by the more concentrated internal structure of the 4% agar sample. Figures 11 and 12 show the respective responses of 4% and 3% agar gel samples to a stress sweep performed at a fixed oscillation (1 Hz) using a conventional rheometer. The structure of the 4% gel begins to break down at an applied stress of 150 Pa, indicated by a fall in the in-phase component (G′ modulus) representing the elasticity of the sample. Equivalent behavior occurs at an applied stress

Figure 13. The response of a 4% agar gel sample to a frequency sweep at a fixed stress (100 Pa). The filled circles correspond to the rheological phase lag, (tan δ)/10, derived from data measured using the modified AFM apparatus. All other data were measured using a commercial rheometer (symbols are as in the preceding figure; note that open circles correspond to the rheological phase lag, tan δ).

of only 10 Pa in the case of the 3% agar sample. The fact that the structure of the 3% agar sample breaks down at a much lower applied stress than does that of the 4% agar sample may explain the failure of the theoretical fit, using ωmax from Figure 7, to match the relaxation of the substrate at higher applied loads (Figure 10, bottom curve). Finally, Figures 13 and 14 attempt to reconcile data obtained from a conventional rheometer, storage (G′) and loss (G′′) moduli, with signals acquired from the modified

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Figure 14. The response of a 3% agar gel sample to a frequency sweep at a fixed stress (1 Pa). The filled circles correspond to the rheological phase lag, (tan δ)/5, derived from data measured using the modified AFM apparatus. All other data were measured using a commercial rheometer (symbols are as in the preceding figure; note that open circles correspond to the rheological phase lag, tan δ).

AFM apparatus. The rheological lag provides a useful basis for the comparison of different measurement techniques given that any instrumental compliance, accounting for differences in measurement architecture (which must be known for the absolute determination of G′ and G′′ moduli), cancels. This being said, the absolute magnitudes of the respective tan δ curves shown in Figures 13 and 14 differ considerably. However, such a discrepancy in the response may reflect differences in both the magnitude of the applied stress and the nature of its application, that is, tip (AFM) versus parallel plate (conventional rheometer) load distribution. Given this observed discrepancy in absolute magnitude, it is useful to recall the theoretical behavior displayed in Figure 5 that shows the rheological lag, as described by eq 1, to indeed exhibit a load dependence. Interestingly, there is a clear match in the frequency range over which the tan δ function shows a response, with both curves describing a broad maximum. In Figure 13, the 4% agar sample has a peak at 10 Hz (9.6 Hz in Figure 6) in the modified AFM and at 10-20 Hz in the rheometer. In Figure 14, the response of the 3% agar sample is noisy but nonetheless shows a peak at 10-20 Hz (17.0 Hz in Figure 7) in the modified AFM and at about 10 Hz in the rheometer. In agreement with the theoretical analysis, the frequency of the maximum phase response appears to be the extent to which one can make quantitative comparison between traditional rheometry and the present nanorheometry, given inherent uncertainties in the absolute contact geometry. Summary and Conclusions The key outcomes of this work can be summarized as follows:

Tyrrell and Attard

A commercial AFM has been modified using minimal external hardware for use as a nanorheometer, allowing the probing of material properties on the nanoscale. The coincidence of piezo drive and cantilever response data following the addition of an offset identified from the crosscorrelation analysis shows the software-based process to be effective in detecting changes in phase. The automated procedure is efficient and removes any bias or uncertainty inherent to a manual fitting process. The measured response of the model gel system has been described well by a recently proposed viscoelastic theory.19 In addition, the experimental data have shown that a single step (impulsive) measurement can yield viscoelastic information equivalent to that extracted from a sweep of discrete frequency (harmonic) tests. The time saved in adopting such a single-step protocol would of course be considerable for applications involving (i) multiple measurements (e.g., the generation of a viscoelastic map32,33 based upon the two-dimensional scan of a sample surface) and (ii) the characterization of long relaxation times. The validity of measurements performed using the nanorheometer has been corroborated. Frequency response profiles generated using a conventional shear flow rheometer show broad agreement with equivalent data derived from the modified AFM apparatus. Such a procedure appears to be a useful basis for the comparison of different measurement techniques, although one should be aware of the difficulties posed by the different geometries (shear flow versus squeeze flow) and applied loads. To conclude, these results show the potential of AFM apparatus for the study of a wide range of viscoelastic systems, responding to deformation on the order of nanometers. The proposed theory of viscoelasticity offers a robust, general, and computationally simple approach to the modeling of such systems, the validity of which has been demonstrated by the modified AFM apparatus. Compatibility with standard industry techniques is an important issue that has been embraced in this study by the promising agreement between AFM results and those measured using a conventional rheometer apparatus. Acknowledgment. We thank Igor Ametov for facilitating the measurement of bulk material properties, Shannon Notley and Jeff Honeymon for correspondence about the design of the AFM interface, and Mark Rutland for useful discussions during the measurements. LA0207163 (32) Radmacher, M.; Fritz, M.; Kacher, C. M.; Cleveland, J. P.; Hansma, P. K. Biophys. J. 1996, 70, 556. (33) Bowen, W. R.; Lovitt, R. W.; Wright, C. J. Biotechnol. Lett. 2000, 22, 893.