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Brownian Fluctuation Spectroscopy Using Atomic Force Microscopes Huilian Ma, Jorge Jimenez, and Raj Rajagopalan* Department of Chemical Engineering, University of Florida, Gainesville, Florida 32611-6005 Received August 3, 1999. In Final Form: November 1, 1999 We examine the use of the thermal fluctuations of the cantilever of an atomic force microscope to study the microrheological behavior of fluids near a solid/liquid interface. A model-independent approach is used for the analysis of power spectral densities and to extract frequency-dependent dissipative and inducedmass contributions of the fluid to the force experienced by the cantilever. The approach provides a framework for the calibration of AFM cantilevers using thermal fluctuations in viscous fluids and for extracting the loss and storage moduli of viscoelastic fluids. The results show that in viscous fluids the excess zerofrequency viscous dissipation (i.e., relative to the magnitude in the bulk) caused by a nearby surface scales inversely with the distance between the cantilever and the surface, in contrast to the inverse cubic scaling assumed in the literature. Interestingly, the observed scaling is practically identical to what is expected for the increase in the hydrodynamic drag on a sphere descending normally toward a flat surface at low Reynolds numbers.
Introduction The analysis of the microscopic viscoelasticity or microrheology of complex fluids has received increased attention in recent years.1-9 Most of these studies have taken advantage of the thermal fluctuations of either the constituents of the fluids or a probe particle immersed in the fluids as a source of perturbations. The measured thermal fluctuations are then analyzed for extracting the dissipative or viscoelastic response of the fluid to fluctuations. In view of the use of the thermal noise in the analysis, we call this class of techniques Brownian Fluctuation Spectroscopy (BFS) here. Studies of biological samples,3-5 polymer solutions,4,7,8 and microemulsions7-9 have established the suitability of this technique for studying local small-scale structure of complex fluids and to complement the results obtained from macroscopic or conventional rheological measurements (i.e., macrorheology). In a typical experiment based on thermal noise, the local viscous or viscoelastic behavior is obtained from an analysis of the power spectrum of the thermal fluctuations of the Brownian probe. These fluctuations are monitored and recorded using an appropriate technique (e.g., laser interferometry,4 video tracking,5 diffusing wave spectroscopy,7 laser deflection particle tracking,6 conventional dynamic light scattering7,9); one then interprets the measurements using, implicitly or explicitly, a Langevin formalism in either Laplace or Fourier space. The use of * Corresponding author. E-mail:
[email protected]. (1) Nhan, D. Brownian Fluctuation Spectroscopy Using Atomic Force Microscopy. MS Thesis, University of Houston, Houston, TX, 1998. (2) Jaganathan, A. Direct Measurement of Polymer-Induced Forces. MS Thesis, University of Florida, Gainesville, FL, 1999. (3) Gittes, F.; Schnurr, B.; Olmsted, P. D.; MacKintosh, F. C.; Schmidt, C. F. Phys. Rev. Lett. 1997, 79, 3286. (4) Schnurr, B.; Gittes, F.; MacKintosh, F. C.; Schmidt, C. F. Macromolecules 1997, 30, 7781. (5) Amblard, F.; Maggs, A. C.; Yurke, B.; Pargellis, A. N.; Leibler, S. Phys. Rev. Lett. 1996, 77, 4470. (6) Mason, T. G.; Ganesan, K.; van Zanten, J. H.; Wirtz, D.; Kuo, S. C. Phys. Rev. Lett. 1997, 79, 3282. (7) Mason, T. G.; Weitz, D. A. Phys. Rev. Lett. 1995, 74, 1250. (8) Mason, T. G.; Gang, H.; Weitz, D. A. J. Opt. Soc. Am. A 1997, 14, 139. (9) Mason, T. G.; Gang, H.; Weitz, D. A. J. Mol. Struct. 1996, 383, 81.
the cantilever of an atomic force microscope (AFM) in this context was first suggested by Thundat and co-workers10 in an attempt to develop a method for measuring viscosities of relatively small samples. Johannsmann and coworkers11-13 subsequently realized that the advantage of the use of an AFM cantilever as the probe was in the analysis of the local microrheological behavior near a solid/liquid interface (e.g., a bare or polymer-coated surface). The use of AFM in this context is appealing since one can probe the compliance and relaxational behavior of interfaces. The thermal-noise-induced fluctuations of an AFM cantilever are also important in the standard uses of AFM,14 e.g., for developing feedback procedures in imaging applications of AFM, as well as in micromechanical experiments, as pointed out by Gittes and Schmidt.15 However, the analysis of the thermal fluctuations in this case is much more complicated than when Brownian particles are used as probes. In particular, a proper interpretation of the dissipative response of the fluids to an oscillating cantilever requires the inclusion of frequency effects since the dynamic Reynolds numbers appropriate for the oscillations of AFM cantilevers can be much larger than unity. The objective of the present paper is to attempt a closer examination of the fluctuations of AFM cantilevers in order to examine the nature of the drag force acting on the cantilever, that is, to identify the inertial and the frequency-dependent contributions to viscous dissipation. A proper accounting of these contributions is essential for obtaining more than a qualitative description of the response of the fluid to the thermal fluctuations and for proper calibration of the cantilevers for subsequent uses in the analyses of thermal noise and microrheological responses of the fluids. We shall also (10) Chen, G. Y.; Warmack, R. J.; Thundat, T.; Allison, D. P.; Huang, A. Rev. Sci. Instrum. 1994, 65, 2532. (11) Roters, A.; Gelbert, M.; Schimmel, M.; Ru¨he, J.; Johannsmann, D. Phys. Rev. E 1997, 56, 3256. (12) Roters, A.; Schimmel, M.; Ru¨he, J.; Johannsmann, D. Langmuir 1998, 14, 3999. (13) Gelbert, M.; Roters, A.; Schimmel, M.; Ru¨he, J.; Johannsmann, D. Surf. Interface Anal. 1999, 27, 572. (14) Sarid, D. Scanning Force Microscopy; Oxford University Press: Oxford, 1994. (15) Gittes, F.; Schmidt, C. F. Eur. Biophys. J. 1998, 27, 75.
10.1021/la991059q CCC: $19.00 © 2000 American Chemical Society Published on Web 01/08/2000
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suggest a simpler, model-independent method for the analysis of thermal power spectra so that the physical parameters can be extracted from the fluctuations of the cantilever in an efficient and robust manner. The paper is divided into four sections. The following section presents the Langevin formalism and the basic theoretical background needed for modeling the thermal fluctuations of the cantilever. Some of the details on the experimental technique and the analysis of the results obtained in water at a number of surface-to-cantilever separations then follow. These results are compared with theoretical predictions in order to shed light on the forces that affect the fluctuations of the cantilever. We also comment on the implications of this work to current efforts to study the local microrheological behavior near solid/ liquid interfaces. The final section concludes with a brief summary and some of the broader implications of the approach and results presented here.
cantilever in the frequency domain to a unit applied force has been described in various contexts in a number of references (see, for example, Landau and Lifshitz17 and Reichl18). Schnurr et al.4 have used such a formulation to extract the viscoelastic behavior of F-actin solutions from the thermal fluctuations of micron-sized particles dispersed in the solution. Here, we follow the description of Nhan1 for analyzing the fluctuations of an AFM cantilever. More recently, Gilbert et al.13 have advocated such an approach in their examination of viscoelastic behavior of polymer brushes using an AFM cantilever as an overdamped harmonic oscillator. A general expression for the response of the fluid to the motion of an immersed body, appropriate for both viscous and viscoelastic fluids, is given by
Relating Probe Fluctuations to Rheological Response of the Fluid
where ζ(t - t′) is the so-called memory function. The Langevin equation can then be rewritten as
As well-known, the thermal fluctuations of a probe of mass m immersed in a viscous fluid can be described as a stochastic process that satisfies a second-order differential equation, i.e., a Langevin equation,
d2z(t)
m
dt2
) Ft(t) ) Fd(t)+ Fext(t)+ Fr(t)
(1)
where z(t) is the position vector of the particle at time t and Ft(t) is the sum of all forces acting on the probe particle. The total force Ft(t) includes the drag force, Fd(t); any relevant external force, Fext(t); and a random force, Fr(t), due to the constant molecular bombardments exerted by the surrounding fluid. This equation applies rigorously to particles that are executing translational Brownian motion, and in what follows we shall restrict our attention to the one-dimensional version of the above equation. The description of the motion of a cantilever (which has one of its ends fixed) is, however, generally much more complicated since, in principle, the elastodynamic equation for the (clamped) cantilever must be solved along with the relevant Navier-Stokes equation.16 Nevertheless, a common approximation, which has been sufficiently successful, is to consider the cantilever as a body of effective “geometric” mass, mg, executing “free” oscillations in an “external” harmonic potential with spring constant, kc, i.e., Fext(t) ) -kcz(t). Even within this approximation, an important point in the analysis that requires close examination is the choice of appropriate expressions for the drag and inertial forces of the fluid to the oscillations. These forces can have significant frequency-dependent contributions for typical cantilevers used in atomic force microscopy. In what follows we describe two different, but related, approaches one can take in analyzing the observed deflections of the cantilever. The first is a general approach1,13 which basically makes no assumptions about the functional form of the drag and inertial forces. The second considers detailed expressions or models for the forces based on analytical solutions for oscillations of objects of simple shapes. The advantages and disadvantages of each method will be discussed in the subsequent discussion of the experimental results. General Formulation for the Response Function. The general formulation outlined in this section to obtain the real and imaginary components of the response of the (16) Sader, J. E. J. Appl. Phys. 1998, 84, 64.
Fd(t) )
meq
d2z(t) 2
dt
+
∫0tζ(t - t′)
∫0tζ(t - t′)
dz(t′) dt′ dt′
dz(t′) dt′ + kcz(t) ) Fr(t) dt′
(2)
(3)
where the subscript “eq” on the mass m draws attention to the fact that the mass obtained by fitting the data to the above model is an “equivalent” mass that will typically include induced-mass contributions (see the next section). As well-known, the analysis of this equation is best carried out in the frequency domain (i.e., Fourier or Laplace space). The response function in the frequency domain relating the fluctuations of the cantilever, z˜ (ω), to the external perturbations or random force, F ˜ r(ω), is defined as the susceptibility, χ(ω), which, in view of eq 3, is given by
χ′(ω) χ′′(ω) 1 ) -i ) -meqω2 - Γ ˜ (ω) + kc χ(ω) |χ(ω)|2 |χ(ω)|2
(4)
˜ (ω), equal to iωζ˜ (ω), is related to the where i ) x-1 and Γ relaxation modulus of the fluid in the frequency domain. The modulus Γ ˜ (ω) as defined above depends on the geometry of the probe. Its relation to the viscoelastic modulus G ˜ (ω) of the fluid (a material property of the fluid) is in general complicated. Some additional discussion on this topic may be found in Mason et al.9 A generalized Stokes-Einstein approximation has been found useful in some contexts for relating the viscoelastic modulus G ˜ (ω) to Γ ˜ (ω) (and hence to ζ˜ (ω)). In the case of purely viscous fluids, the real part of Γ ˜ (ω), representing the storage of energy, becomes zero and the imaginary part becomes proportional to the dissipation of energy, with the proportionality coefficient being dependent on the geometry of the probe. The imaginary part of the susceptibility, χ′′(ω), is related to the spectral density of the fluctuations, |z˜ (ω)|2 (i.e., the Fourier transform of the autocorrelation function of z(t)), through the fluctuation-dissipation theorem
|z˜ (ω)|2 )
2kBT χ′′(ω) ω
(5)
where kB is the Boltzmann constant and T is the temperature of the system. The real part of the suscep(17) Landau, L. D.; Lifshitz, E. M. Statistical Physics, 3rd ed.; Pergamon Press: New York, 1980; Part 1. (18) Reichl, L. E. A Modern Course in Statistical Physics, 2nd ed.; Wiley: New York, 1998.
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tibility, χ′(ω), and the imaginary part, χ′′(ω), are not independent quantities but are related through the socalled Kramers-Kronig relations (see Landau and Lifshitz17 and Reichl18):
∫0∞χ′′(ξ)ξ2 -ξ ω2 dξ
(6)
∫0∞χ′(ξ)ω2 ω- ξ2 dξ
(7)
2 χ′(ω) ) P π and
2 χ′′(ω) ) P π
where the notation P in front of the integrals specifies that the Cauchy principal values of the integrals are to be taken. This result, a direct consequence of the linearity and causality of the system, allows one to calculate χ′(ω) from χ′′(ω) once the latter is obtained from the fluctuations through eq 5. Decomposing the complex modulus Γ ˜ (ω) in eq 4 into its real and imaginary parts
Γ ˜ (ω) ) Γ′(ω) + iΓ′′(ω) ′′(ω),
one then has for Γ
(8)
the loss modulus,
Γ′′(ω) )
χ′′(ω) |χ(ω)|2
(9)
It is important to emphasize that the procedure described above allows one to calculate the loss modulus, Γ′′(ω), from the spectral density of the probe fluctuations without any prior assumption on the functional form of Γ′′(ω). Dynamical Models for the Drag and Inertial Forces. In the case of viscous fluids, a more direct dynamical formulation for the dissipative term represented by eq 2 is possible. The details of the expression for the drag force depends on the geometric model used for representing the cantilever and on the Reynolds number. Even for the case of a spherical particle the drag force has a complicated form with two limiting expressions depending on the magnitude of the amplitude of the fluctuations relative to the radius of the sphere and on the Reynolds number. The same applies for other cases as well. Below, we consider two cases, one corresponding to a sphere and the other to a cylindrical rod (of infinite length), both oscillating freely (i.e., without any anchoring) in a fluid of infinite extent (i.e., without any confining boundaries). 1. Oscillating Spheres. In the low-frequency limit, one can define the Reynolds number simply as Re ) [R(dz(t)/ dt)F]/η. For spherical objects at low Reynolds numbers (i.e., Re , 1), the drag force is given by the Stokes law,19 i.e.,
Fd(t) ) -6πηR
dz(t) dt
(10)
where R is the radius of a sphere, and F and η are the density and viscosity of the fluid, respectively. In the case of high-frequency oscillations, it is more appropriate to define a kinetic Reynolds number, Reω, as Reω ) R2ωF/η, where ω is the circular frequency in rad/s. For Reω . 1 and δ , R, where δ is the penetration depth of the shear waves, the drag force has an inertial part as (19) Landau, L. D.; Lifshitz, E. M. Fluid Mechanics; Pergamon Press: Oxford, UK, 1959.
well as a dissipative part,19
Fd(t) ) -Cv
dz(t) d2z(t) - Cmmd dt dt2
(11)
where Cv is the damping coefficient (or, also known as the drag coefficient) and Cmmd is the added or induced mass, expressed in multiples of displaced mass, md (equal to the mass of the fluid displaced by the volume of the oscillating object). The induced mass mI )Cmmd adds significantly to the effective mass of the particle. For the specific case of a sphere executing high-frequency translatory oscillations in a viscous fluid, a more detailed expression can be obtained from the expressions for Cv, Cm, and δ presented by Landau and Lifshitz19
dz(t) dt 2 2 3 2ηF d z(t) πR F + 3πR2 (12) 3 ω dt2
Fd(t) ) -[R0(η) + (3πR2x2ηFω)]
(
x )
where R0(η) represents the zero-frequency (Stokes) coefficient. Note that the induced mass mI has a frequencyindependent part, which we shall denote by mI,0 (equal to (2/3)πR3F), and a frequency-dependent part (3πR2 x2ηF/ω), which we shall write as mI,ω/xω. 2. Oscillating Cylinders. For an infinitely long cylinder of radius R executing free translatory oscillations in a direction normal to its axis (in an unbounded fluid), the Navier-Stokes equations can be solved analytically, as shown recently by Kirstein et al.20 One can then write the drag force for a length L of the cylinder as
Fd(t) ) -[R0(η; Reω) + (2πRLx2ηFω)]
(
dz(t) dt 2 2ηF d z(t) (13) ω dt2
x )
πR2LF + 2πRL
As well-known, in contrast to the case of spherical objects, the zero-frequency drag coefficient for cylinders, i.e., R0(η; Re), is a function of the velocity and has a logarithmic singularity in the Reynolds number. The geometry of typical cantilevers used in experiments is generally much more complicated than the simple geometries we have considered above, and in principle, modeling the deflections of a cantilever is more difficult. A simpler approach is to model the cantilever as a collection of spheres of appropriate radius arranged in a way to mimic the overall shape of the cantilever, as suggested by Hosaka et al.21 Regardless of the approach used, it is reasonable to assume, in view of the similarity of the expressions presented above, that the following form of the equation for Fd(t) captures the essential features of the force on a cantilever:
dz(t) dt mI,ω(F,η) d2z(t) mI,0 + (14) dt2 xω
Fd(t) ) -[R0(η) + Rω(F,η)xω]
(
)
where mI,0 represents the frequency-independent part of the induced mass and mI,ω(F,η) is the coefficient of the (20) Kirstein, S.; Mertesdorf, M.; Schonhoff, M. J. Appl. Phys. 1998, 84, 1782. (21) Hosaka, H.; Itao, K.; Kuroda, S. Sens. Actuators A 1995, 49, 87.
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frequency-dependent part. Notice that we have ignored the fact that the cantilever is anchored at one end. Equation 14 may now be used in combination with the Langevin equation, i.e., eq 1, to obtain
(
mv +
)
mI,ω(F,η) d2z(t)
xω
2
dt
dz(t) + dt kcz(t) ) Fr(t) (15)
+ [R0(η) + Rω(F,η)xω]
where mv, the virtual mass of the cantilever, is the sum of the effective geometric mass, mg, and the frequencyindependent induced mass, mI,0, i.e., mv ) mg + mI,0. When written in the above form, the Langevin equation is relatively independent of any assumptions concerning the geometry of the cantilever, as long as the frequencydependent drag coefficient and the induced mass are of the forms used, as in the case of spheres and cylinders or an array of spheres. The susceptibility, χ(ω), can then be written as
[ (
R0(η) mI,ω(F,η) 3/2 z˜ (ω) ωω ) mv -ω2 - i χ(ω) ) mv mv F ˜ r(ω) Rω(F,η) 3/2 kc -1 i ω + (16) mv mv
)]
Substituting eq 16 into eq 5, one obtains
|z˜ (ω)|2 )
C1(C2 + C3ω1/2) (C4 - ω2 - C5ω3/2)2 + (C2ω + C3ω3/2)2
(17)
where
C1 )
2kBT mv
(18)
C2 )
R0(η) mv
(19)
Rω(F,η) mv
(20)
C4 )
kc ) ω02 mv
(21)
C5 )
mI,ω(F,η) mv
(22)
C3 )
and
The parameters Ci described above are then obtained from a nonlinear fit of the spectral density of the fluctuations. The physical significance of each of the coefficients Ci is readily apparent. The coefficient C4 is usually written as ω02 (as noted above) and is the square of the resonance frequency of the cantilever. The ratio xC4/C2 is commonly written as Q, the “quality factor” in AFM terminology, representing the width of the experimental power spectral density. Note that eq 17 holds good even at moderate or large values of Reω, as it includes the effects of frequency-dependent inertial and dissipative contributions (through C3 and C5). In the following section of the paper we examine the drag coefficients, R0(η) and Rω(F,η); the spring constant, kc; the virtual mass of the cantilever, mv; and the frequency-dependent induced-mass coefficient, mI,ω(F,η),
Figure 1. The power spectral density for the fluctuations of a tipless cantilever in water at a cantilever/surface distance of 110 µm.
obtained from the power spectrum of thermal fluctuations in two different ways: In the first, we avoid any explicit modeling of the dissipative and induced-mass contributions and instead rely on the known viscous nature of the fluid to uncover the appropriate dissipative and inertial terms in the Langevin equation. In the second, we use the standard procedure of fitting the power spectrum to eq 17. The mutual consistency of the two approaches and the advantages and disadvantages of each will then be discussed. Experimental Details All the experiments reported here were done with a Multimode Nanoscope III (Digital Instrument, CA) atomic force microscope and tipless V-shaped silicon nitride cantilevers, bought from Digital Instruments. The thermal fluctuations of the cantilever were measured at room temperature in distilled water at different separations of the cantilever with respect to a glass surface. The distance from the surface was adjusted with the z-piezo of the scanner, which has a step size of 0.1 µm. After setting the position of the cantilever, measurements were taken with both the scanner and the feedback circuit disconnected from the AFM unit. The thermal noise data were then captured under contact mode and transformed to the Fourier domain to obtain the noise spectrum. Experimental Results and Discussion Here an examination of experimental data collected using a tipless cantilever in water is presented using the formalism outlined above. As noted in the introductory part of the paper, in addition to investigating the inertial and dissipative contributions in the response of the fluid to thermal fluctuations, our primary focus will be to establish a framework for using power spectral densities of AFM cantilevers for probing microrheology. Model-Independent Analysis of the Data. We begin with an examination of the frequency dependence of energy dissipation. Figure 1 shows the power spectrum obtained for the cantilever in water at a separation of 110 µm from a glass surface. This spectrum was obtained from the discrete Fourier transform of a time series containing the deflections of the cantilever. Equation 5 allows one to transform the power spectral density to χ′′(ω), and the
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Figure 2. The real and imaginary parts of the susceptibility corresponding to the spectral density shown in Figure 1. The imaginary part, χ′′(ω), is obtained using the fluctuationdissipation theorem (eq 5). The real part, χ′(ω), is obtained from χ′′(ω) using the Kramers-Kronig relation (eq 6).
Figure 3. The loss modulus, Γ′′(ω), as a function of angular frequency, ω, for the data shown in Figures 1 and 2.
result is plotted in Figure 2, along with the χ′(ω) obtained from the Kramers-Kronig relation, (eq 6). The dynamical contributions to χ′′(ω) are more readily extracted from the loss modulus Γ′′(ω). Figure 3 shows the result obtained for Γ′′(ω) from eqs 5 and 9 for the χ′′(ω) shown in Figure 2. As the result thus obtained contains no assumptions as to the functional form of the coefficient of the dissipative contributions to oscillations of the cantilever, it can be used for testing the most appropriate expression for dissipation. For viscous fluids, one expects
Γ′′(ω) ∝η lim ωf0 ω
(23)
Further, in the absence of any frequency dependence of the dissipation coefficient, one expects Γ′′(ω)/ω to remain constant when plotted against ω. It is evident from Figure 3 that Γ′′(ω) is not linear and that one does need a frequency-dependent dissipation coefficientsan observation that has been recognized by others and is not surprising by itself. In Figure 4 we have plotted Γ′′(ω)/ω versus xω, showing that a drag coefficient of the form
Cv ) R0 + Rωxω
(24)
fits the data quite well, thus formally justifying the form
Figure 4. The loss modulus in Figure 3 plotted as Γ′′(ω)/ω against ω0.5. The figure illustrates the validity of the LandauLifshitz model for viscous dissipation shown in eq 14.
Figure 5. The plot of χ′(ω)/|χ(ω)|2 against ω2 illustrating the need for frequency-dependent induced mass for describing spectral densities in the region of the resonance frequency.
of dissipative response adopted in eq 15 in the previous section. It is also evident from Figure 4 that the dependence of Cv on ω extends to rather low frequencies, well below the peak frequency.22 The fact that Γ′(ω) ) 0 for a viscous fluid allows one to use the real part of eq 4, i.e.,
χ′(ω) |χ(ω)|2
) -meqω2 + kc
(25)
to examine unambiguously if frequency-dependent induced mass is needed in the Langevin equation, since the left-hand side of the above equation is obtained directly from the experimental data without any models. It is evident from Figure 5, which shows χ′(ω)/|χ(ω)|2 as a function of ω2 for the data obtained at 110 µm, that the data do deviate from linearity for frequencies below the peak frequency, thereby indicating that using just the virtual mass (i.e., mg + mI,0) is insufficient to model the fluctuations accurately in the neighborhood of the peak in the power spectrum. This aspect has been overlooked in previous attempts to model the fluctuations of the (22) Figure 4 does not include data below about 3000 rad/s and above about 40 000 rad/s because of low signal-to-noise ratios in our current experimental setup. (23) Brenner, H. Chem. Eng. Sci. 1961, 16, 242.
Brownian Fluctuation Spectroscopy
Figure 6. Power spectra for a collection of cantilever/surface separations. All power spectra except for the one for H ) 110 µm have been shifted up to improve clarity. The (unmarked) open symbols correspond to the results for H ) 10, 30, 50, 70, 90, and 110 µm, respectively, from top down.
cantilever (which have been made typically to obtain a measure of the quality factor Q and the spring constant kc but not for extracting the rheological properties of the fluid from the fluctuations). In summary, the above discussion outlines a systematic way to calibrate the dynamic coefficients of AFM cantilevers and to calibrate their behavior using viscous fluids. The approach outlined above also allows one to extract the loss and storage components of the response function from the Brownian fluctuations, without recourse to any models for the power spectrum, as long as accurate data over a sufficiently large range of ω are available for employing the Kramers-Kronig relations. Hydrodynamic Influence of the Surface. We now turn our attention to the effects of the distance H of the cantilever from the surface on the extracted parameters. The hydrodynamic problem of resistance experienced by a vibrating body near an interface has not been analyzed theoretically so far in the literature, and therefore, there is no formal theoretical guidance for the analysis of the experimental data. Nevertheless, as illustrated in the above paragraphs, the experimental data can be examined without any recourse to theory in order to develop an understanding of the influence of the presence of a surface in the vicinity of the vibrating object. Figure 6 shows a series of noise spectra obtained in water at different positions of the cantilever from the glass surface. One first notices a shift in the peak frequency to lower values and an increase in the width of the distribution as the distance between the cantilever and the surface decreases (the separation distance H decreases from 110 to 5 µm). It is now instructive to examine Γ′′(ω)/ω versus xω as a function of H. Figure 7 shows that the frequencydependent coefficient Rω decreases as the cantilever approaches the surface and, within the statistical error of the experimental data, vanishes for H)5 µm. The values of R0 and Rω obtained from linear regressions of the data in Figure 7 are presented in Table 1. We shall return to the results obtained for R0 as a function of H shortly. A similar examination of χ′(ω)/|χ(ω)|2 as a function of ω2 for the various values of H, shown in Figure 8, indicates that at close separations the induced-mass effect, revealed by the departure of the data from linearity specified by eq 25, diminishes, again suggesting a decrease in the influence of the frequency-dependent contributions. Figure 8 also indicates that at close distances there is a small increase in the virtual mass of the cantilever.
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Figure 7. The ratio Γ′′(ω)/ω plotted against ω0.5 to extract the coefficients R0 and Rω as functions of the cantilever/surface separation H. The coefficients are listed in Table 1. For clarity, data for only five of the distances are shown. Table 1. Drag Coefficients Obtained from Plots of Γ′′(ω)/ω versus ω0.5 as Functions of Cantilever/Surface Separation H H, µm 5 10 20 30 50 90 100 110
R0, N s/m
Rω, N s1.5/m
10-6
∼0 2.0 × 10-9 5.2 × 10-9 6.8 × 10-9 9.2 × 10-9 9.8 × 10-9 9.9 × 10-9 1.0 × 10-8
8.7 × 4.7 × 10-6 2.7 × 10-6 1.9 × 10-6 1.3 × 10-6 1.0 × 10-6 9.5 × 10-7 9.5 × 10-7
Figure 8. The ratio χ′(ω)/|χ(ω)|2 versus ω2 as a function of the cantilever/surface separation H. For clarity, only three data sets are shown. The results for H g 20 µm are statistically indistinguishable from the result for H ) 110 µm.
Fitting of the Power Spectra Using the Langevin Equation. One can also obtain the parameters by directly fitting the experimental power spectra using eq 17, as done routinely in investigations aimed at AFM calibration and development of feedback techniques. The parameters obtained by directly fitting the power spectra are presented in Table 2. Although the results obtained with both methods are quite close to, and consistent with, each other, the model-independent approach outlined earlier allows one to analyze the power spectra and obtain the coefficients in a more efficient and robust manner than possible by fitting the power spectra directly using a relatively complicated and sensitive form such as the one in eq 17.
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Table 2. Parameters Obtained by Fitting the Power Spectrum Directlya H, µm 10 20 30 40 50 60 70 80 90 100 110
mv, kg 10-10
1× 1 × 10-10 1 × 10-10 8 × 10-11 7 × 10-11 8 × 10-11 1 × 10-10 9 × 10-11 8 × 10-11 8 × 10-11 8 × 10-11
σm, kg 10-11
5× 7 × 10-11 2 × 10-11 2 × 10-11 2 × 10-11 1 × 10-11 1 × 10-11 1 × 10-11 1 × 10-11 1 × 10-11 1 × 10-11
R0, N s/m
σR, N s/m
mI,ω, kg/s0.5
σmI, kg/s0.5
10-6
10-6
10-8
10-8
5.0 × 3.0 × 10-6 2.0 × 10-6 1.5 × 10-6 1.3 × 10-6 9.9 × 10-7 6.0 × 10-7 7.0 × 10-7 7.0 × 10-7 6.0 × 10-7 6.0 × 10-7
1.2 × 1.3 × 10-6 3.2 × 10-7 2.4 × 10-7 2.0 × 10-7 1.5 × 10-7 1.3 × 10-7 1.3 × 10-7 1.4 × 10-7 1.1 × 10-7 1.1 × 10-7
1.1 × 9.8 × 10-9 9.6 × 10-9 1.3 × 10-8 1.5 × 10-8 1.2 × 10-8 4.8 × 10-9 1.0 × 10-8 1.1 × 10-8 1.1 × 10-8 1.1 × 10-8
1.2 × 1.5 × 10-8 5.0 × 10-9 4.0 × 10-9 3.8 × 10-9 3.0 × 10-9 3.0 × 10-9 2.9 × 10-9 2.9 × 10-9 2.5 × 10-9 2.5 × 10-9
kc, N/m 10-1
1.1 × 1.1 × 10-1 1.1 × 10-1 1.1 × 10-1 1.1 × 10-1 1.1 × 10-1 1.0 × 10-1 1.1 × 10-1 1.1 × 10-1 1.0 × 10-1 1.1 × 10-1
σk, N/m 10-3
9.0 × 1.3 × 10-2 4.9 × 10-3 4.0 × 10-3 3.7 × 10-3 3.0 × 10-3 3.0 × 10-3 2.9 × 10-3 2.9 × 10-3 2.5 × 10-3 2.5 × 10-3
Rω, N s1.5/m
σRw, N s1.5/m
10-10
7.6 × 10-9 8.3 × 10-9 2.0 × 10-9 1.4 × 10-9 1.2 × 10-9 8.9 × 10-10 8.0 × 10-10 7.7 × 10-10 8.3 × 10-10 6.5 × 10-10 6.5 × 10-10
-5.0 × 4.5 × 10-9 7.7 × 10-9 8.8 × 10-9 9.9 × 10-9 1.1 × 10-8 1.3 × 10-8 1.2 × 10-8 1.2 × 10-8 1.2 × 10-8 1.2 × 10-8
a The σ values listed are the estimated standard deviations of the parameters, i.e., the values of the parameters are bounded roughly by (σ.
Influence of the Surface on Zero-Frequency Dissipation. Another issue that deserves examination is the behavior of the zero-frequency hydrodynamic dissipation as a function of the distance H. The drag coefficient R0 listed in Table 1 is plotted in Figure 9 against (1/H). Figure 9 shows that, interestingly, the increase in R0 as the probe approaches the glass surface is akin to Brenner’s result23 for a sphere descending normally toward a rigid surface. The enhancement in the zero-frequency drag force for a sphere of radius R moving vertically toward a rigid surface at which the no-slip condition for the fluid velocity applies can be described, within 5% of the exact solution, by the following approximate expression
R0(H) R ≈1+ R0,bulk H Figure 9. The zero-frequency dissipation coefficient, R0, as a function of the inverse of the cantilever/surface distance H. The results demonstrate that (R0 - R0,bulk) ∼ H-1.
The results obtained from the latter approach are very sensitive to experimental errors, and as a consequence and because of the nonlinear form of eq 17, unphysical, negative values for the parameters can result (note, for example, the magnitude of Rω for H ) 5 µm in Table 2). Moreover, the model-independent approach clearly does not require any assumptions concerning the specific functional forms for the frequency-dependent terms. Therefore, we believe that the model-independent approach we suggest here is preferable over the usual method employed in the literature for the examination of the thermal noise in AFM measurements. The results reported in Table 2 also show that the cantilever behaves like an underdamped (i.e., ω0 . R0/mv) harmonic oscillator far from the surface. However, as the distance between the cantilever and the surface decreases, the damping increases (see Figure 9), and at distances below about 10 µm, the oscillations become overdamped. One notes, from Figure 8, a corresponding decrease in the range of frequencies over which the induced-mass is important. Moreover, it is evident from Figure 7 that the frequency-dependent term in dissipation, represented by the coefficient Rω, also diminishes in magnitude. In principle, one may then use a simpler form of the Langevin equation at very short distances. This conclusion is consistent with the observations of Gelbert et al.13 based on their experiments with standard AFM cantilevers in the vicinity of bare or polymer/coated surfaces. Finally, we note that the values shown in Table 2 for the spring constant, kc, at different cantilever/surface separations are relatively constant. The agreement between this value and the one reported by the manufacturer (i.e., kc ∼ 0.1 N/m) is excellent.
(26)
where H is the surface-to-surface distance between the sphere and the wall. Our experimental data follow the functional form shown in eq 26 remarkably well. It is important to note here that these results demonstrate that the drag force model usually employed by many investigators for analyzing AFM data (see, for example, O’Shea and Welland24), namely,
R0 ) R0,bulk +
C H3
(27)
where C is a proportionality constant, assumes a much stronger H-dependence than required. Concluding Remarks In summary, we have examined the use of the cantilever of an atomic force microscope as a probe for studying the mechanical and viscoelastic response of a fluid or an interface from the thermal noise spectrumsa technique we call Brownian fluctuation spectroscopy. The use of an AFM cantilever as the probe allows one to examine the microrheological behavior close to solid/liquid and liquid/ liquid interfaces and to measure the compliance of interfacial layers. The approach advocated in this paper can also serve as a systematic procedure for the calibration of AFM cantilevers and can provide important information for improving the AFM feedback when operating in fluids, as was recently suggested by O’Shea et al.25 The challenge in accomplishing the above resides in properly modeling the thermally induced deflections of the cantilever. The present work approaches this problem by focusing on the fluctuations of a tipless AFM cantilever in a viscous fluid (water) at different positions from a (24) O’Shea, S. J.; Welland, M. E. Langmuir 1998, 14, 4186. (25) O’Shea, S. J.; Lantz, M. A.; Tokumoto, H. Langmuir 1999, 15, 922.
Brownian Fluctuation Spectroscopy
glass surface. The known constitutive behavior of the fluid is then taken advantage of to examine the sufficiency of the basic approach and the functional forms of the dissipative and inertial contributions and their dependence on the cantilever/surface distance. We outline a model-independent approach for the analysis of power spectral densities and use the approach to extract frequency-dependent dissipative and induced-mass contributions of the fluid to the force experienced by the cantilever. The approach provides a framework for the calibration of AFM cantilevers using thermal fluctuations in viscous fluids and for extracting the loss and storage moduli of viscoelastic fluids using AFM. The results also show that the excess zero-frequency viscous dissipation (i.e., relative to the magnitude in the bulk) caused by a nearby surface scales as H-1, where H is the distance between the cantilever and the surface, in contrast to the inverse cubic scaling assumed in the literature. Interestingly, the observed scaling is essentially identical to what is expected for the increased hydrodynamic drag on a sphere descending normally toward a flat surface at low
Langmuir, Vol. 16, No. 5, 2000 2261
Reynolds numbers. This result suggests that it may be possible to model cantilevers of complex shapes as a collection of spheres of suitably chosen diameters. There still remain a number of issues open for examination, e.g., the scaling behavior of coefficients R0 and Rω with the dimensions and shapes of the cantilever and with density of the fluid (in the case of Rω and mI,ω) and the relation between Γ ˜ (ω) and the viscoelastic modulus G ˜ (ω) of complex fluids. We shall address these in subsequent publications. Acknowledgment. We thank the National Science Foundation (NSF-9896097) and the ERC (NSF-9402989) for partial support of the work reported here. We also thank Professor Mark Orazem of the University of Florida and Professor Aristide C. Dogariu of the University of Central Florida for many useful discussions concerning data analysis. LA991059Q
