Langmuir 2008, 24, 10817-10824
10817
Effect of Solution Concentration, Surface Bias and Protonation on the Dynamic Response of Amplitude-Modulated Atomic Force Microscopy in Water Yan Wu,† Chaitanya Gupta,‡ and Mark A. Shannon*,†,‡ Mechanical Science and Engineering, UniVersity of Illinois, Urbana, Illinois 61801, Chemical and Biomolecular Engineering, UniVersity of Illinois, Urbana, Illinois 61801 ReceiVed April 25, 2008. ReVised Manuscript ReceiVed July 3, 2008 The dynamic response of amplitude-modulated atomic force microscopy (AM-AFM) is studied at the solid/water interface with respect to changes in ionic concentration, applied surface potential, and surface protonation. Each affects the electric double layer in the solution, charge on the tip and the sample surface, and thus the forces affecting the dynamic response. A theoretical model is developed to relate the effective stiffness and hydrodynamic damping of the AFM cantilever that is due to the tip/surface interaction with the phase and amplitude signals measured in the AM-AFM experiments. The phase and amplitude of an oscillating cantilever are measured as a function of tip-sample distance in three experiments: mica surface in potassium nitrate solutions with different concentrations, biased gold surface in potassium nitrate solution, and carboxylic acid-terminated self-assembled monolayers (SAMs) on gold in potassium nitrate pH buffers. Results show that, over the range where the higher harmonic modes of the oscillation are negligible, the effective stiffness of the AFM cantilever increases to a maximum as the tip approaches the surface before declining again as a result of the repulsive electrical double layer interaction. For attractive electrical doublelayer interactions, the effective stiffness declines monotonically as the tip approaches the surface. Similarly, the hydrodynamic damping of the tip increases and then decreases as the tip approaches the solid/water interface, with the magnitude depending on the species present in the solution.
Introduction Amplitude-modulated atomic force microscopy (AM-AFM) has become a powerful tool for the nanometer scale characterization of a wide variety of materials. In AM-AFM, a microcantilever with a sharp tip at its free end is excited near its free resonance frequency. The oscillation amplitude of the cantilever is maintained at a constant value by a feedback controller to track the surface topography. The information that AM-AFM technology can deliver is not limited to surface topography. Recently, extensive efforts have been made to use the phase information in the AM-AFM signal for identifying material properties.1 It has been shown that the phase information can be related to energy dissipation of the tip-sample interaction in an ambient environment.2-4 The application of AM-AFM in liquids is important since it allows biological samples to be studied under conditions close to their native environments. In this paper, we study the feasibility of using phase and amplitude signal in AM-AFM to characterize the tip-sample interaction in aqueous electrolyte solutions. In particular, we are interested in the surface charge at the solid/water interface, since it is fundamental to a variety of fields, including tribology, colloidal systems, electrochemical cells, biomolecular interactions, and electrokinetic transport through micro- and nanofluidic devices. There have been many reports on studying the electrical double layer interactions at the solid/water interface using static force measurements with AFM,5-10 where the deflection signal of a * Corresponding author. E-mail:
[email protected]. † Mechanical Science and Engineering. ‡ Chemical and Biomolecular Engineering.
(1) Garcı´a, R.; Magerle, R.; Pe´rez, R. Nat. Mater. 2007, 6, 405. (2) Cleveland, J. P.; Anczykowski, B.; Schmid, A. E; Elings, V. B. Appl. Phys. Lett. 1998, 72, 2613. (3) Tamayo, J.; Garcı´a, R. Appl. Phys. Lett. 1998, 73, 2926. (4) Paulo, 940 > A. S.; Garcı´a, R. Phys. ReV. B 2002, 66, 041406. (5) Butt, H.-J.; Cappella, B.; Kappl, M. Surf. Sci. Rep. 2005, 59, 1.
nonoscillating cantilever is measured as a function of tip sample distance. Methods for mapping surface charge density with AFM in aqueous solutions with static force measurements have been proposed, such as force volume mode,5 pulsed force mode,11 and fluid electric force microscopy.12 Compared with the methods based on static force measurements, AM-AFM has the potential advantage to achieve surface charge density mapping simultaneously with topography measurement because the effective stiffness of the vibrating cantilever increases/decreases due to repulsive/attractive interaction forces between the tip and the sample, leading to a change in the phase signal. However, in liquids, the hydrodynamic damping on cantilever vibration is much greater than in air, and it is distance dependent, which will also affect the phase and the amplitude of the signal. Unless we know how to decouple the distance-dependent electrical doublelayer interaction and the distance-dependent hydrodynamic damping in the phase and the amplitude of the cantilever vibration signal, we can not unambiguously attribute the phase contrast in AM-AFM imaging to surface charge states on the sample. In order to study the feasibility of mapping surface charge density with AM-AFM, we measured the dynamic response of AM-AFM as we changed ionic strength, surface bias, and surface protonation in experiments. Each affects the surface charge on the tip and the sample surface, and thus the forces affecting the dynamic response. The amplitude and phase signal were measured as an oscillating AFM cantilever is brought near the surface of (6) Ducker, W. A.; Senden, W. A.; Pashley, R. M. Langmuir 1992, 8, 1831. (7) Hu, K.; Bard, A. J. Langmuir 1997, 13, 5114. (8) Kane, V.; Mulvaney, P Langmuir 1998, 14, 3303. (9) Hiller, A. C.; Kim, S.; Bard, A. J. J. Phys. Chem. 1996, 100, 18817. (10) Barten, D; Kleijin, J. M.; Duval, J.; Leeuwen, H. P. v.; Lyklema, J.; Cohen Stuart, M. A Langmuir 2003, 19, 1133. (11) Miyatani, T.; Horii, M.; Rosa, A.; Fujihira, M.; Marti, O Appl. Phys. Lett. 1997, 71, 2632. (12) Johnson, A. S.; Nehl, C. L.; Mason, M. G.; Hafner, J. S. Langmuir 2003, 19, 10007.
10.1021/la801295c CCC: $40.75 2008 American Chemical Society Published on Web 09/03/2008
10818 Langmuir, Vol. 24, No. 19, 2008
Wu et al.
Figure 1. Schematic diagram showing the experimental setup in amplitude-modulated dynamic force microscopy. The tip oscillates with an average distance D from the sample surface with amplitude A.
interest. Insight into the dynamic response of the tip-sample interaction at the solid/water interface is the first step for understanding how a pixel is formed when imaging with AMAFM in aqueous solution. To use the phase contrast in an AMAFM image to get surface charge density mapping, one needs to know the changes in the effective interaction stiffness and the changes in hydrodynamic damping due to ion/surface/tip interactions. The purpose of this paper is to show how to obtain the effective interaction stiffness and hydrodynamic damping from the dynamic response measurements, and in particular the first order change in stiffness and damping due specifically to the interaction. A theoretical model is developed to relate the effective interaction stiffness and hydrodynamic damping sensed by the AFM cantilever with the phase and amplitude signals measured in the experiments. The magnitude and sign of the effective interaction stiffness are determined by the magnitude and polarity of the surface charge density. The experimental results are presented in terms of the change in interaction stiffness over the cantilever stiffness and the change in hydrodynamic damping over that far from the surface found from the model. The limitations of the model are also discussed.
The equation of motion of an oscillating cantilever in a liquid, shown in Figure 1, can be approximated in low Reynold’s number flow to the first order by a point mass versus beam system,13,14 such that
(1)
where z is the position of the tip at any time, and z0 is the position at zero cantilever deflection. The coefficient m is the effective mass of the cantilever, including the additional fluid mass being accelerated,14 and kc is the spring stiffness of the cantilever. The tip-sample interaction force, Fi, consists of the electrical double layer force and van der Waals force. The hydrodynamic damping, b, and the tip-sample interaction force, Fi, are all distance dependent.14 The cantilever is driven at frequency ωd/2π, with constant excitation amplitude, ad, at its base. Experimentally, the tip displacement is monitored through the cantilever deflection signal. For magnetically driven cantilevers (where the base does not oscillate), it is usually safe to assume that the tip displacement equals the cantilever deflection. For acoustically driven cantilevers (where the base oscillates at amplitude ad, as shown in Figure 1), the cantilever deflection is the result of both base displacement and tip displacement. In this case, the cantilever deflection will only be approximately the same as the tip displacement when (13) Basak, S.; Raman, A Appl. Phys. Lett. 2007, 91, 064107. (14) O’Shea, S. J.; Welland, M. E. Langmuir 1998, 14, 4186.
z)D+
∑ An cos(nωdt - φn)
(2)
ng1
where D is the mean value of the tip-sample distance D, An and φn are the amplitude and phase shift of the oscillation at the nth harmonics, respectively. Previous studies5,18 on static force measurements have shown that the electrical double-layer interaction force Fel for the constant surface charge condition can be approximated by
Fel ≈ gσSσTLD exp(-z/LD)
Theoretical Basis
mz¨ ) -bz˙ - kc(z - z0) + kcad cos(ωdt) + Fi
that displacement is much greater than the base displacement. Previous studies15,25 have shown in liquid environments that the behavior of the resonance curves of cantilevers depends greatly on the cantilever driving method, because of the difference between the cantilever defection and the tip displacement. In this work, we found that attributing cantilever deflection solely to tip displacement will lead to a misinterpretation of the data when the amplitude of tip displacement is comparable to or less than the base amplitude, as we shall discuss in the Results and Discussion section. Compared with the operation of AM-AFM in ambient environments, there are two significant differences of cantilever oscillation in liquids. First, as a result of the viscous interaction between the fluids and the cantilever, the quality factor, Q, of the oscillator is much lower (at least 2 orders of magnitude) than in gas or vacuum atmospheres, and higher variation modes are more significantly present, as previously observed by several researchers.13,16 Second, unlike applying dynamic force measurements in gas atmosphere where the change in phase of the cantilever oscillation can be attributed primarily to change of interaction effective stiffness, 17 the response of phase signal in liquid is a distance-dependent function of both the hydrodynamic damping and effective stiffness of the AFM cantilever, due to the double-layer interaction. Including the higher harmonic modes in the steady state solution of eq 1 and the oscillation of the cantilever tip can be approximated as
(3)
where σS and σT are the surface charge densities of the sample and the AFM tip, respectively, LD is the Debye length of the electrolyte solution, and g can be considered as a constant when z is of the same order of magnitude of the tip radius. If we assume a conical tip with a spherical end and a perfectly flat surface, the constant g can be calculated from the half-opening angle R of the tip cone and the radius of curvature R of the sphere. 18 In dynamic measurements, if the oscillation amplitude is small (A1 < D) and the Reynolds number, Re, for the oscillations is small (Re < 1), we can assume tip-sample interaction force, Fi, is a single value function of tip position z, and can be approximated by
Fi(z) ≈ Fi(D) +
∂Fi (z - D) ) 〈Fi 〉 - ki(z - D) ∂z
(4)
where 〈Fi〉 is the averaged interaction force over one cycle, and ki is the force gradient or the effective stiffness of the tip-sample interaction. If the electrical double-layer force dominates in the total tip-sample interaction force, according to eq 3, we find for ki that (15) Herruzo, E. T.; Garcia, R. Appl. Phys. Lett. 2007, 91, 143113. (16) Tamayo, J. Appl. Phys. Lett. 1999, 75, 3569. (17) Dianoux, R.; Martins, F.; Marchi, F.; Alandi, C.; Comin, F.; Chevrier, J. Phys. ReV. B 2003, 68, 045403. (18) Butt, H.-J. Nanotechnology 1992, 3, 60.
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ki ≈ gσSσT exp(-D⁄LD)
(5)
Note that ki is positive for repulsive interaction and negative for attractive interaction. The magnitude and sign of the effective stiffness are determined by the magnitude and polarity of the surface charge densities on the sample and the tip according to eq 5, and it is a function of average tip-sample distance D. As detailed in the Supporting Information, we can relate the amplitude and phase signals in dynamic response curves of an oscillating AFM cantilever to changes in its effective stiffness due to the electrical double-layer interaction between the AFM tip and surface, and the hydrodynamic damping of the tip to the ionic solution. For the case where Re , 1, the flow around the tip and the beam of the cantilever is creeping with reversible streamlines, allowing the flow to be approximated by potential functions. For oscillating tips, Re ) (ωd/2π)A1(Lc/ν), where A1 is the fundamental mode amplitude, ν is the kinematic viscosity of the liquid, and Lc is the characteristic length, which can be approximated either by the boundary layer length across either the tip height or around the beam width. For the experiments in this work, Re ranges from 10-6 (at the tip radius) to 10-3 (at the widest beam width), both meeting the criteria. Thus, the forces arising from interactions between the AFM tip, liquid, and surfaces can be superimposed, and the virial theorem19 can be used to estimate the time averaged forces. At steady state, the virial theorem for the point mass model is (1/2)m〈z˙2〉 ) -(1/2)〈Fz〉, where F is the total conservative forces, and the power balance at steady state, 〈kcad cos(ωdt)z˙〉 + 〈-bz˙2〉 + 〈Fiz˙〉 ) 0, giving
( )
( )
ad cos φ1 ki ωd 2 1 1 n2An2 ) + 1 2 An2 2 A1 kc ω A1 ng1 r A1 ng1 (6a)
∑
∑
and
bωd ad sin φ1 ) n2An2 A1 k A 2 ng1
∑
(6b)
c 1
where ωr is the cantilever’s resonance frequency, and it equals (kc/m)1/2. Equation 6 shows that cos φ1 is related to the conservative interaction (the electrostatic double-layer force, ki), and sin φ1 is related to the dissipative interaction (hydrodynamic damping, b). This interpretation of the cos φ1 and sin φ1 signal is in agreement with previous work on applying dynamic force measurements in air.2-4 When imaging using the AM-AFM in these experiments, the amplitude A1 is maintained at a constant value using a close loop controller. Thus the contrast in the sine and cosine of the phase signal can be used to obtain changes in the effective stiffness due to the double-layer interaction and the damping at the same time as the topography is acquired. If the amplitudes at higher harmonics are negligible, i.e., A1 is much greater than An for n > 1, eq 6 can be simplified to
1-
( )
ωd 2 ki ad + ) cos φ1 ωr kc A1
(7a)
ωdb ad ) sin φ1 kc A1
(7b)
and
Equation 7 shows that the two major experimental observable quantities, namely, the amplitude A1 and the phase φ1 in AM(19) Goldstein, H. Classical Mechanics, 2nd ed.; Addsion-Wesley: Reading, MA, 1981; Chapter 3.
AFM, can be directly related to the effective stiffness of doublelayer interaction and hydrodynamic damping at solid/water interfaces if the contribution of the higher harmonics to cantilever oscillation is negligible. In general, the following two conditions are necessary if we can safely neglect the higher harmonics without losing much insight to the physics of the interaction. First, the mechanical stiffness of the cantilever needs to be much greater than the effective stiffness of the tip-sample interaction. Second, the oscillation amplitude needs to be small compared to the characteristic range of the interaction, which is the electric double layer that is approximately equal to the Debye length of the electrolyte solution. We later experimentally check the limitation of this simplification by measuring the power spectral density (PSD) of the oscillating cantilever as it approaches the surface. We further normalize the measured effective stiffness and hydrodynamic damping in eq 7 using the phase, φ∞, and amplitude A∞, when the AFM probe is far away (6 µm away in the experiments) from the solid/water interface, leading to
(
ki cos φ1 cos φ∞ ) ad kc A1 A∞
)
(8a)
and
A∞ sin φ1 b ) b∞ A1 sin φ∞
(8b)
where ki/kc is the normalized effective interaction stiffness of the tip near the surface, and b/b0 is the normalized effective damping of the solution to the tip. Since the amplitude of acoustic excitation ad to the cantilever is not monitored by the laser in AFM, we approximate ad to be A∞/Q∞, where Q∞ is the Q factor of the cantilever resonance curve measured at a distance far away from the surface.
Experimental Methods Three types of experiments were conducted to study the dynamic AFM response in liquids. (1) The effect of ionic concentration and shielding of the surface by the solution is tested over a nearly atomically smooth cleaved mica surface, which is reported to have a negative surface charge density in aqueous solutions,20 in potassium nitrate solutions with three different concentrations (0.5, 1, and 5 mM). (2) The effect of surface potential is tested over a gold surface on silicon, where the gold surface is electrically biased to different voltages with respect to a 0.1 mM potassium nitrate solution. (3) The effect of surface protonation is tested over a carboxylic acidterminated self-assembled monolayer (SAM) on gold on glass, where the surface charge density in aqueous solutions varies with the pH of a potassium nitrate buffer solution.7,8 Preparation of Electrolyte Solutions. Potassium nitrate solutions were freshly prepared before each experiment from reagent-grade chemicals (Sigma-Aldrich) without further purification in 18 MΩ deionized (DI) water (Direct_Q, Millipore Corp.). The concentrations of the electrolyte solution ranged from 0.1 to 5 mM, with corresponding Debye lengths from 30.4 to 4.3 nm, respectively. The solution pH was monitored with a commercial pH and conductivity meter (Accumet, Fisher Scientific). All the potassium nitrate solutions for experiments 1 and 2 have pH values around 6 as made, and did not go through further pH adjustment. The pH values of potassium nitrate solutions for experiment 3 were adjusted from 5 to 9 with potassium hydroxide right before each measurement. Preparation of Gold Surfaces for Use in Electrochemical Cell. Chromium (5 nm) and gold (200 nm) were deposited on both sides of a commercial n-type silicon wafer by DC magnetron sputtering after removing the native oxide in dilute hydrofluoric acid solutions. (20) Scales, P. J.; Grieser, F.; Healy, T. W. Langmuir 1990, 6, 582.
10820 Langmuir, Vol. 24, No. 19, 2008 The sample was then annealed at 375 °C in a tube furnace with nitrogen atmosphere to form ohmic contact at the metal/silicon interface. The gold surface was biased from the backside of the silicon wafer via a potentiostat (FAS2 Fentostat, Gamry Instruments, Warminster, PA), with a silver/silver chloride microelectrode (MI402, Microelectrodes Inc., Londonderry, NH) as the solution reference electrode, and a gold wire (Alfa Aesar, Ward Hill, MA) as the counter electrode in the solution. Preparation of Monolayer Surfaces. 11-Mercaptoundecanoic acid was purchased from Sigma Aldrich for use in these experiments for the SAM. These thiol molecules were dissolved in absolute ethanol (Pharmaco-Aaper, Shelbyville, KY) to form a dilute (1 mM) incubation solution. Glass disks that fit the fluid cell were coated with sputtered films of chromium (5 nm) and gold (100 nm). These disks were washed in a heated SC-1 bath (100 mL DI water/25 mL H2O2/2 mL NH4OH) and rinsed thoroughly with DI water and ethanol. These gold surfaces were left to incubate in a parafilm sealed beaker containing a solution of thiol molecules for 48 h within a class 100 cleanroom. The thiol-coated surfaces were rinsed thoroughly with ethanol and DI water and blow-dried with dry nitrogen before use. AFM Measurements. The experiments were performed using an Asylum Research MFP 3D AFM, with the probe in a fluid cell (experiments 1 and 3) or an electrochemical cell (experiment 2). Figure 1 shows the schematic diagram of the experimental setup. The amplitude and phase delay (with respect to the cantilever driving signal) of the cantilever deflection was detected by a lock-in amplifier at the same time the mean deflection was measured. There are two kinds of AFM probes used in the experiments. For the experiments with mica and gold surfaces, triangular shaped silicon nitride probes (NP Series, Veeco) with a nominal spring constant of 0.1 N/m were used. For the Au/glass surface with carboxylic acid-terminated SAMs, rectangular gold-coated silicon nitride probes (Biolever, Olymus) with a nominal spring constant of 0.06 N/m were used. Both kinds of the probes become negatively charged in electrolyte solutions used.8,21 The spring constants of the cantilever were calibrated and reported for every experiment using a thermal noise method, where the power spectrum of the cantilever thermal fluctuation in air was fitted using a simple harmonic oscillator model.22,23 When driving the cantilever acoustically in liquids, there can be multiple peaks in the resonance curve due to the fluid-structure interactions and the convolution of the cantilever dynamics with the dynamics of the fluid cell hardware and the surrounding fluid.24,25 The thermal spectrum of cantilever in liquids was measured as a guide to peak the drive frequency at the resonance frequency of cantilever for acoustic excitation. 25 The amplitude of cantilever oscillation was tuned to be around 5 nm when the tip is about 6 µm away from the sample surface, and the Q factor of the resonance curve was recorded. For each of the dynamic force curves, the averaged cantilever deflection signal, the amplitude signal, and the phase signal were measured as a function of the piezo displacement. We only studied the force curves when the cantilever approached the sample surface. The dynamic response when the cantilever retracts from the surface is complicated by the adhesion force and is not the focus of this paper. After the contact point (zero tip sample distance point) was found in the deflection versus piezo displacement curve, the tip-sample distance was calculated using the method described by H.-J. Butt et al.5 The amplitude versus tip-sample distance and the phase versus tip-sample distance were then replotted in terms of effective stiffness versus distance and damping versus distance using eqs 8a and 8b. We also measured the time sequence of the deflection signal by sampling at a rate of 5 MHz with an analog to digital converter (ADC) located at another identical controller of the Asylum Research MFP 3D system. The cantilever oscillation amplitude at the driving frequency was maintained at a fixed value by the controller when the time sequence was measured. A series of the time sequence (21) Raiteri, R.; Margesin, B.; Grattarola, M. Sens. Actuators B 1998, 46, 126. (22) Hutter, J. L.; Bechhoefer, J. ReV. Sci. Instrum. 1993, 64, 1868. (23) Proksch, R.; Scha¨ffer, T. E.; Cleveland, J. P.; Callahan, R. C.; Viani, M. B. Nanotechnology 2004, 15, 1344. (24) Revenko, I.; Proksch, R. J. Appl. Phys. 2000, 87, 526. (25) Xu, X.; Raman, A. J. Appl. Phys. 2007, 102, 034303.
Wu et al. was measured as the oscillating cantilever approached the sample surface. The PSD of the deflection signal was then calculated to check the range over which the higher harmonics are negligible.
Results and Discussion Mica Surface in Potassium Nitrate Solutions. To study the dependence of the ionic strength of electrolyte solutions on tip-surface interactions and hydrodynamic damping by the ionic solutions on the tip, mica surfaces are immersed in potassium nitrate solutions with concentrations from 0.5 to 5 mM, which corresponds to Debye shielding lengths of charges on surfaces from 13.6 to 4.3 nm. Cleaved mica is chosen for its near-atomic smoothness to negate the effects of local surface roughness on the effective stiffness and damping due to the surface and solution on the tip. Rough surfaces could potentially channel fluids in local grooves and/or confine them within a region of the tip, thereby altering the effect that ions might have on the results. Mica in aqueous solutions is negatively charged,20 and the silicon nitride tip is negatively charged.21 Hence, repulsive electrostatic forces are expected to act on the AFM tip, which will become stronger as shielding decreases in the solution. The average dynamic force versus distance curves were obtained by multiplying the mean deflection of the cantilever with the spring constant of the cantilever. As seen in the average force versus distance curves in Figure 2a, higher repulsion occurs as the tip approaches the surface in the lower concentration electrolytic solutions, which have larger Debye lengths. From the amplitude and phase data shown in Figure 2b,c, the effective stiffness versus distance and the damping versus distance curves can be calculated using eq 8 and are shown in Figure 3. In our theoretical analysis, we pointed out that validity of eq 8 is subject to several assumptions, including that the oscillation amplitude A1 is less than the mean tip-sample distance D, and the contribution of the higher harmonics to cantilever oscillation is negligible. Since the oscillation amplitude A1 is around 2.5 nm when the tip is within 30 nm from the surface (cf. Figure 2b) and the shortest Debye length in the study is 4.3 nm, the higher harmonics constraint breaks down before the tip-sample distance constraint, being more than the oscillation amplitude, is reached. For the comparison between solutions with difference Debye lengths, we plot the dynamic response results in the same range of tip-sample distance (from 0 to 30 nm) in Figure 3. The effect of higher harmonics on the dynamic response and the validity of eq 8 will be discussed specially in the part on PSD of the cantilever vibration. As shown in Figure 3a, the effective stiffness of the double-layer interaction is within 5% of the spring constant of the cantilever (0.109 N/m as calibrated), which satisfies the constraint that the effective stiffness of the tip-sample interaction force must be much less than the spring constant of the cantilever. Figure 3a shows that the electrical double layer interactions scale with the Debye length of the electrolyte solutions. For example, at a normalized effective stiffness of 0.02, the tip-sample distance is 10, 7, and 1 nm in solutions with Debye lengths of 13.6, 9.6, and 4.3 nm, respectively. The interaction range dependence of the double-layer force on the Debye length is consistent with previous studies using static force measurements.5,6 Comparing Figure 3a with Figure 2a, the effective stiffness versus distance curve represents the slope of the force versus distance curve, except that the effective stiffness reaches a maximum and starts to decrease as the distance decreases to zero. As shown in Figure 3a, the distance D at which the maximum occurs scales with the Debye length of the electrolyte. The loss of effective stiffness after the peak is not reflected in the average dynamic force versus distance curves since the slope of the average force versus distance curves continue to increase as the distance get closer. This decrease
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Langmuir, Vol. 24, No. 19, 2008 10821
Figure 3. Plots of normalized effective stiffness (a) and hydrodynamic damping (b) vs tip-sample distance over a mica surface immersed in 0.5, 1, and 5 mM of potassium nitrate solutions (pH ) 6). The corresponding Debye lengths, LD, of the solutions are also listed.
Figure 2. Plots of the average force (a), the amplitude (b) and the phase (c) as the tip approaches a mica surface immersed in 0.5, 1, and 5 mM of potassium nitrate solutions (pH ) 6). The corresponding Debye lengths, LD, of the solutions are also listed.
in apparent effective stiffness for the dynamic response is due to the effect of higher harmonics, as we shall discuss later with the PSD of the cantilever vibration. What the average force versus distance curve (cf. Figure 2a) cannot show is the dependence of hydrodynamic damping with tip-sample distance on the ionic strength of the solution, which
is shown in Figure 3b. Figure 3b shows that the hydrodynamic damping in the 0.5 mM potassium nitrate solution (Debye length 14.6 nm) remains at its bulk value b0 until the probe is within 15 nm from the surface. The damping increases rapidly as the probe approaches the surface, until it reaches a maximum. For the other two concentrations, the damping versus distance curves show similar behavior, and the onset distance of the increasing damping scales with the Debye length. The overall rapid increase in hydrodynamic damping can be attributed to a squeeze film effect, where fluid is constrained to flow through a narrow space. While the distance-dependent damping at the solid/water interface is well-known,14,26,27 the dependence of hydrodynamic damping on the ionic strength of the solution is relatively less reported, and we observe strong differences. For all the relatively low ionic concentrations tested, we would not expect there to be any appreciable differences in squeeze film damping, as this effect primarily pushes between the tip and surface of the trapped liquid, where the overwhelming dominate molecule is H2O. Figure 3b shows that at tip sample distance of 5 nm the normalized hydrodynamic damping is 2.7, 1.6, and 1.1 for 0.5, 1, and 5 mM (26) Nnebe, I.; Schneider, J. W. Langmuir 2004, 20, 3195. (27) O’Shea, S. J.; Lantz, M. A.; Tokumoto, H. Langmuir 1999, 15, 922.
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potassium nitride solutions, respectively. The decrease in the normalized damping with increasing electrolyte concentration at a fixed tip sample distance might be explained by the reduction of the diffuse layer size with increasing ionic strength. At the diffuse layers near the surfaces of the sample and the tip, there is a net imbalance of charge from the ions with respect to the bulk solution. As the tip approaches the surface from the bulk solution, the interaction between the AFM tip and the sample will occur at a larger separation between the two surfaces for a thicker diffuse layer, which is a characteristic of more dilute electrolytes. Thus, the damping effect should scale with the diffuse layer thickness. So et al. 28 also reported decreasing viscosity in silica particle suspensions with increasing ionic strength for a fixed particle volume fraction, which is in agreement with our observation. Again we observe a similar maximum in Figure 3b as in Figure 3a when the tip gets closer to the surface. However, the maxima in Figure 3b and Figure 3a did not happen at the same point. For example, the peak of the effective stiffness for the 0.5 mM concentration is at a distance of 7 nm, whereas the peak of the damping is at 4 nm. Clearly, the mechanisms for the maxima observed in the effective stiffness and the damping curves are not the same. Comparing Figure 3b with Figure 2b, we found that the maximum in the damping curve (cf. Figure 3b) occurs at the same point where there is an increase in the amplitude curve (cf. Figure 2b). According to eq 6b, the apparent damping would drop if the measured amplitude A1 at the driving frequency increases. We shall come back to this point later in our discussion of the PSD of the cantilever. Biased Gold Surface in 0.1 mM Potassium Nitrate Solutions. To study the effect of surface potential on the dynamic response, we need to be able to adjust the electric potential at the surfaces. Both of the silicon nitride probe19 (kc ) 0.107 N/m as calibrated) and the unbiased gold surface9,10 are negatively charged in potassium nitrate solutions at pH of 6, as a result of unprotonated surface sites. Thus, the double-layer force between them is repulsive. The polarity and magnitude of charge density on gold surface can be controlled by applying an external electrical field across the gold/solution interface. Figure 4a shows the response of effective stiffness to three different bias voltages (with respect to the Ag/AgCl reference electrode in the solution). As the applied bias changes from -0.2 V to +0.8 V, the total surface charge (summation of electronic charge and protonic charge) changes from negative to positive. As the polarity goes from negative to positive, corresponding to a change from repulsive to attractive double-layer interaction, the calculated effective stiffness using eq 8a changes from positive to negative at a distance of 5 nm, as seen in Figure 4a. Figure 4a shows that the dynamic response of the cantilever is sensitive to the magnitude and polarity change in the surface charge at distances from 5 to 10 nm. As the distance decreases to zero, all the effective stiffness curves become negative, which we could not conclusively attribute to an attractive interaction or is purely an artifact due to the higher harmonic modes of the cantilever, as we shall discuss later. There is also a significant dependence of damping curves on the applied bias, as shown in Figure 4b. The dependence is mostly due to the magnitude of the surface potential rather than its polarity. The proposed hypothesis to explain the dependence of the damping on the magnitude of the surface potential is that the damping comes from the displacement of the counterions as the probe approaches the surface. With greater surface potential (no matter in what polarity), there are more counterions to be displaced and thus more damping. (28) So, J-H.; Yang, S-M.; Kim, C.; Hyun, J. C. Colloids Surf. A 2001, 190, 89.
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Figure 4. Plots of normalized effective stiffness (a) and hydrodynamic damping (b) vs tip-sample distance over a biased gold surface in 0.1 mM potassium nitrate solution (pH ) 6).
Carboxylic Acid-Terminated Self-Assembled Monolayers in pH Buffers. To study the effect of surface charge on the dynamic response, we were able to adjust the charge on the surfaces in aqueous solutions. It has been shown previously with static force measurements7,8 that increasing pH leads more negative charge at the carboxylic acid-terminated SAMs due to chemical deprotonation of -COOH end group to -COO-. The reported pKa value7,8 for the dissociation reaction ranged from 6.3 to 7.7. However, the silicon nitride cantilever with a spring constant of ∼0.1 N/m and resonance frequency of 6.3 kHz did not give a sensitive enough signal to detect a surface charge change over the pH range from 5 to 8 in 0.5 mM potassium nitrate buffers. We switched to a cantilever with a lower spring constant (0.062 N/m as calibrated), but with a higher resonance frequency (9.7 kHz from thermal spectrum in liquid). The surface of the tip was coated with gold, and the surface charge of gold is negative in the range of pH 5 to 8 as previously reported by Barten et al. 9 The dynamic force measurement results with this kind of cantilever are shown in Figure 5. The increased sensitivity with the gold-coated cantilever was not likely due to the change of the surface coating (both silicon nitride and gold charge negatively in the pH range from 5 to 8). Instead, the increased sensitivity is due to its lower spring constant (higher ki/kc) and higher resonance frequency, which shifts the signal further away
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Figure 6. A series of PSD measurements as the oscillating probe is brought near the sample surface. The PSD has a single peak from 6 µm (A1/A∞ ) 1) to 200 nm (A1/A∞ ) 0.6) mean distance away from the surface. Higher harmonics start to show up as the probe approaches the surface, but the amplitudes of the higher harmonics are negligible until the oscillation of the cantilever is greatly damped (A1/A∞ < 0.3).
Figure 5. Plots of normalized effective stiffness (a) and hydrodynamic damping (b) vs tip-sample distance over the surface of carboxylic acidterminated thiols on gold for different pH in a 0.5 mM potassium nitrate solution.
from low frequency noises in the thermal spectrum leading to a greater signal-to-noise ratio. In Figure 5a the normalized effective stiffness ki increases with increasing pH of the solution before the maximum point is reached at the distance of 2 nm. The effective stiffness calculated using eq 8b correctly captures the change in surface charge magnitude before the oscillation of the cantilever was damped to the point that higher harmonic modes are no longer negligible. Again we observe significant dependence of hydrodynamic damping on the solution pH value in Figure 5b. Increasing solution pH leads to more negative charge on the surface and thus more counterions in the diffuse layer. The increase in hydrodynamic damping sensed by the tip is related to the increase of counterions to be displaced by the tip. PSD of the Cantilever Vibration. In this section, we discuss the effect of the higher harmonics of the cantilever oscillation on the dynamic response using a silicon nitride cantilever and a mica surface in a 0.5 mM potassium nitride solution as a representative example of the frequency response. Although the data presented are specific to this case, we have tested other cases, and the trend corresponds with all the cases in this paper. Figure 6 shows the PSD of the deflection signal as a silicon nitride probe (kc ) 0.109 N/m as calibrated) was brought close
to a mica surface in 0.5 mM potassium nitride solution. The estimated tip sample distance D of each spectrum is also listed. The initial vibration amplitude A∞ was about 5 nm at a mean distance of 6 µm away from the surface. The PSD at this point shows one distinct peak at a driving frequency of 6.3 kHz and no obvious peaks associated with higher harmonics. The amplitude A1 at the driving frequency was normalized to its initial value A∞ for each spectrum. When the probe was brought close to the surface (within 200 nm from the surface) and A1 was damped to 60% of its initial value, the PSD still maintained a single peak at driving frequency. As the probe was brought even closer to the surface after this point, higher harmonics started to show up, and the fundamental mode A1 decreased in magnitude with the next highest peak always at the second harmonic frequency. The higher harmonics are an indication of nonlinear tip-sample interactions on the cantilever dynamics. The calculated ratio of power amplitude at the first harmonic (at the driving frequency) over the one at the second harmonic is 84 and 20, when the tip-sample distances D are 10.7 and 6.6 nm, respectively. A1 is damped to 50 and 30% of its initial value at these two points, respectively. Based on this observation, we conclude that the contribution of higher harmonics to the cantilever dynamics is negligible (less than 5%) until the vibration is greatly damped (less than 30% of initial value). Therefore, the normalized effective stiffness and damping calculated from eqs 8a and 8b, respectively, are likely valid approximations of the actual values, at least for D beyond 7 nm for the case of mica surface in a 0.5 mM potassium nitride solution. For solutions with other concentrations in this study, we found that the onset distance for non-negligible higher harmonics scales with the Debye length of the electrolyte (cf. Figure 3a). The effect of the higher harmonics on the calculated effective stiffness using eq 8a can be predicted by eq 6a. As the amplitude of the higher harmonics (An, n > 1) continues to increase as the tip gets closer to the surface, the increase in the second term at the right-hand side of eq 6a will increase faster than the first term, leading to a decrease in the calculated effective stiffness using eq 8a. According to eq 5, the magnitude of ki increases as a result of stronger double-layer interaction as the tip gets closer to the surface. For repulsive interactions, ki is positive, and thus the calculated effective stiffness using eq 8a will increase to a maxima and then decrease. For attractive interactions, ki is negative, and thus the calculated effective stiffness using eq 8a will decrease monotonically as the tip approaches the surface.
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Similarly, the effect of higher harmonics on the calculated hydrodynamic damping using eq 8b can be predicted using eq 6b. As the amplitude ratio of the higher harmonics (An, n > 1) over the fundamental mode A1 increases with decreasing tip-sample distance, eq 8b overpredicts the hydrodynamic damping sensed by the cantilever by (1 + (2A2/A1)2 + (3A3/A1)2 +...)-1. However, we can not attribute cantilever deflection solely to tip displacement when tip displacement is greatly damped since the base of the cantilever is shaking at the amplitude ad. In fact, the amplitude of the deflection signal does not drop to zero even when the tip is in contact with the sample surface. Instead, the amplitude A1 first decreases as the tip gets closer to the surface, and it increases again once the oscillation of the tip end becomes less than ad, as we observed in measured amplitude versus distance curves (c.f. Figure 2b). Thus the amplitude ratio of the higher harmonics (An, n > 1) over the fundamental mode A1 first increases but then decreases again, leading to a drop in the calculated hydrodynamic damping, as we observed in the experiments. If this were due purely to attractive interactions, the static curve in Figure 2a would decrease as the tip approaches the surface, which it does not.
Conclusions We demonstrated that the dynamic response of AM-AFM can be used to characterize the nature of the tip-sample interaction in aqueous solutions experimentally by varying the ionic strength of the solution, by altering the surface potential physically via external bias, and by altering the surface charge chemically through adjusting the pH value. The results show that dynamic measurements are not only sensitive to conservative interactions with the electrostatic double layer forces, but are also sensitive to dissipative interactions from hydrodynamic damping in aqueous solutions. The sensitivity is affected by the spring constant and the resonance frequency of the cantilever. Theoretically, the effective stiffness of double-layer interaction is related to the cosine of the phase signal and the hydrodynamic damping to the sine of the phase signal. The range over which the measured amplitude and phase signal can be used to calculate the effective stiffness and the hydrodynamic damping is limited by the condition that the contributions of the higher harmonics in the cantilever oscillation need to be negligible. We found experimentally through the power spectra density measurements that the contributions of higher harmonics to the cantilever dynamics
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are negligible until the vibration is greatly damped (less than 30% of initial value). In order to achieve surface charge density mapping in aqueous solutions, we need to know how the effective interaction stiffness of the oscillating cantilever in AM-AFM changes in response to the change of electrical double interaction and how the hydrodynamic damping changes as a result of the interaction of the tip with the surface and ions in the solution. Results show that the effective interaction stiffness is affected strongly by the ionic strength of the solution and by the potential/ charge of the surface. Results also show that the hydrodynamic damping close to the surface has a strong dependence on the ionic strength of the solution and a significant dependence on the potential and charge of the surface. In the imaging mode of AM-AFM, the amplitude signal is kept constant, and the ionic strength of the solution is usually fixed. We can decouple the effective stiffness of double-layer interaction and the hydrodynamic damping to the cosine of the phase signal and the sine of the phase signal as long as the onset distance for non-negligible higher harmonics remain relatively unchanged during scanning, which is true for electrolyte solution with fixed ionic strength. With this paper, the first step toward using the phase information in AM-AFM response to map the surface charge density in aqueous solutions has been demonstrated. However, whether it is possible to map surface charge density using AM-AFM is still subject to issues including the sensitivity of the cantilever and the spatial resolution of the image. Acknowledgment. This work is supported by the WaterCAMPWS, a Science and Technology Center of Advanced Materials for the Purification of Water with Systems program under the National Science Foundation (NSF), agreement number CTS-0120978. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the NSF. The AFM work was performed at the Frederick Seitz Materials Research Laboratory Central Facilities, University of Illinois, partially supported by the U.S. DOE under grants DE-FG0207ER46453 and DE-FG02-07ER46471. The authors would like to acknowledge Jonathan Wan for help in the sample preparation, and Dr. Scott MacLaren for assistance with the AFM experiments. Supporting Information Available: Additional equations for the cantilever in Figure 1. This material is available free of charge via the Internet at http://pubs.acs.org. LA801295C
