ARTICLE pubs.acs.org/Langmuir
In Situ Hydrodynamic Lateral Force Calibration of AFM Colloidal Probes Sangjin Ryu and Christian Franck* School of Engineering, Brown University, Providence, Rhode Island, United States
bS Supporting Information ABSTRACT: Lateral force microscopy (LFM) is an application of atomic force microscopy (AFM) to sense lateral forces applied to the AFM probe tip. Recent advances in tissue engineering and functional biomaterials have shown a need for the surface characterization of their material and biochemical properties under the application of lateral forces. LFM equipped with colloidal probes of well-defined tip geometries has been a natural fit to address these needs but has remained limited to provide primarily qualitative results. For quantitative measurements, LFM requires the successful determination of the lateral force or torque conversion factor of the probe. Usually, force calibration results obtained in air are used for force measurements in liquids, but refractive index differences between air and liquids induce changes in the conversion factor. Furthermore, in the case of biochemically functionalized tips, damage can occur during calibration because tipsurface contact is inevitable in most calibration methods. Therefore, a nondestructive in situ lateral force calibration is desirable for LFM applications in liquids. Here we present an in situ hydrodynamic lateral force calibration method for AFM colloidal probes. In this method, the laterally scanned substrate surface generated a creeping Couette flow, which deformed the probe under torsion. The spherical geometry of the tip enabled the calculation of tip drag forces, and the lateral torque conversion factor was calibrated from the lateral voltage change and estimated torque. Comparisons with lateral force calibrations performed in air show that the hydrodynamic lateral force calibration method enables quantitative lateral force measurements in liquid using colloidal probes.
’ INTRODUCTION As the feature size or characteristic length scale of various materials approaches the nano and atomic scales, atomic force microscopy (AFM) has become a benchmark characterization tool for investigating materials on that scale. In particular, lateral force microscopy (LFM), which is a specific mode of AFM, has been widely used to study frictional, wear and tear, and even molecular effects under the application of shear loads on the micro- and nanoscales.13 In LFM, the AFM probe is scanned over a sample surface in the direction perpendicular to the probe’s long axis. Because of frictional forces between the probe tip and the sample surface, the probe cantilever is twisted during the scan. This torsional deformation of the probe is amplified by the optical lever system of the AFM: the position-sensitive photodiode (PSPD) of the AFM detects changes in the laser beam path reflected off the back of the probe. Thus, LFM can convert torsional deformations of the AFM probe to changes in the PSPD output voltage. This lateral voltage output can then be converted to an actual force via lateral force calibration methods.48 Through this calibration, LFM can be used to measure lateral forces quantitatively. LFM studies utilize a variety of tip geometries ranging from sharp pyramidal to spherical colloidal tips. In particular, the colloidal probe9,10 has been widely used in AFM applications because it has several advantages: first, the spherical tip offers a r 2011 American Chemical Society
large contact area between the tip and sample. Therefore, the colloidal probe enables quantitative measurements of forces with a higher sensitivity.11 Second, the tip of the colloidal probe presents a well-defined geometry.12 Third, colloidal probes offer flexibility in material choices and properties:13 in the case of sharp tip probes, the tip and cantilever are made of the same material, which is usually Si or SiN3, providing only a limited ability to investigate interactions between the tip and various sample surfaces. In contrast, tip materials of colloidal probes can be chosen depending on necessity; these include glass, silica, polystyrene, and even deformable matter such as cells and liquid drops. Additionally, colloidal probes accommodate the easily controllable functionalization of the tip.12 For these reasons, colloidal probes have been used for indentation studies,14,15 tribology studies,1618 studies on interaction forces among particles, drops and surfaces,1922 ligandreceptor interaction studies,23,24 and hydrodynamic studies.2527 As previously mentioned, quantitative measurements of lateral forces require a lateral force calibration of the AFM probe. 4 Similar to normal force calibrations, lateral force calibrations usually consist of two steps: the measurement of the Received: March 18, 2011 Revised: September 2, 2011 Published: September 09, 2011 13390
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Figure 2. Scanning electron microscopy image of probe 4. See Table 1 for probe information.
Table 1. Colloidal Probes Used for the Hydrodynamic Lateral Force Calibration Figure 1. Schematic diagram of the hydrodynamic lateral force calibration method for the AFM colloidal probe.
probe
cantilever
nominal spring constant
tip radius
no.
type
a kN ^ (N/m)
R (μm)
torsional/lateral sensitivity of the optical lever system (in rad/V or μm/V) and the determination of the torsional/lateral spring constant of the probe (in μN 3 m/rad or nN/μm). In contrast, one-step lateral force calibration methods measure the lateral torque/force conversion factor (in μN 3 m/V or nN/V), which is the product of the spring constant and the optical lever sensitivity, by applying lateral forces of the known magnitude to the probe tip. In both calibration methods, the optical lever sensitivity and conversion factor depend on the AFM measurement conditions whereas the spring constant is an intrinsic property of the AFM probe. One of those conditions is the medium in which the lateral force measurements are performed. Because many LFM applications are directed toward biological and chemical research, lateral force measurements have been performed in aqueous or liquid environments.2830 However, the optical lever sensitivity determined in air cannot be directly used in liquid because the laser beam path of the optical lever system differs depending on the refractive index of the media.3133 To account for refractive index differences during lateral force calibrations, a simple model was proposed33
1
V-shaped
0.1
58.8
2
V-shaped
0.03
40.1
3 4
V-shaped V-shaped
0.03 0.01
41.1 44.8
5
rectangular
0.02
36.7
6
rectangular
0.02
35.4
7
rectangular
0.02
44.0
ð1Þ
)
)
Sa nl ¼ a l n S )
where S is the lateral optical lever sensitivity, n is the refractive index, and superscripts a and l denote the air and liquid media, respectively. However, eq 1 is based on a particular AFM configuration. Therefore, it is best to perform lateral force calibrations in situ by applying a lateral force of the known magnitude to the probe tip. Additionally, because the tipsurface contact may damage the tip or surface during calibration, contact-based calibration methods are not desirable, especially for functionalized AFM probe tips. A nondestructive force calibration method was proposed using electrostatic forces, but its application is limited to conductive tips and dielectric media.34 Here we noticed that the liquid environment itself can be exploited for the lateral force calibration because of the naturally occurring viscous loading forces applied to the tip during scanning. Using the well-defined colloidal tip geometry and a hydrodynamic model, we show how to accurately determine the hydrodynamic forces that act on the tip during scanning. Thus, we present an in situ hydrodynamic method for calibrating the lateral torque conversion factor for colloidal AFM probes (Figure 1). In this method, the AFM colloidal
a
Nominal spring constant values of tipless cantilevers as provided by the manufacturer.
probe was immersed in a viscous liquid and placed close to the substrate surface. The lateral scan motion of the surface generated a creeping Couette flow, and the flow twisted the cantilever of the colloidal probe by exerting viscous drag forces and torque on the spherical tip. Sensed by the laser beam of the optical lever system, the torsional deformation of the cantilever was converted to a lateral voltage change. This voltage difference increased as the lateral scan speed increased and/or the tip-to-surface distance decreased. The lateral torque conversion factor of the colloidal probe was calibrated by relating the measured lateral voltage change to the total estimated applied torque on the probe. For comparison, the lateral torque conversion factor of the colloidal probe was also calibrated in air using the diamagnetic lateral force calibrator (D-LFC) introduced by Li et al.35 With the refractive index difference between the media taken into account, both calibration results showed reasonable agreement. Therefore, the in situ hydrodynamic lateral force calibration method enables quantitative measurements of lateral forces in liquid environments.
’ METHODS AFM Colloidal Probe Fabrication. Colloidal probes were fabricated by attaching silica beads (OPS Diagnostics, Lebanon, NJ) to the ends of tipless cantilevers (MLCT-O10, Veeco, Camarillo, CA) with epoxy (5 minute epoxy, ITW Devcon, Danvers, MA) using a micromanipulator. Each bead diameter was measured from images taken with a light microscope or scanning electron microscope (Figure 2 and Table 1). Following complete curing of the epoxy for more than 24 h at room temperature, all probes were washed in 70% ethanol in deionized water and UV cleaned for 515 min (PSD-UV, Novascan, Ames, IA). Then, each probe was loaded onto the AFM (MFP-3D, Asylum Research, Santa Babara, CA), and the whole system was warmed up for at least 2 h prior to measurements to reduce thermal drift. 13391
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Figure 3. Lateral force calibration with the diamagnetic lateral force calibrator (D-LFC). (A) Photograph of the D-LFC. (B) Schematic of the colloidal probe engaged on top of the D-LFC (O, tip center; A, contact point). See the text for the other symbols.
Figure 4. Flow chart of the hydrodynamic lateral force calibration method.
)
Lateral Force Calibration in Air. Prior to the hydrodynamic lateral force calibration, the lateral force calibration of the colloidal probes was performed in air using the D-LFC technique35 (Figure 3A). The D-LFC system consisted of four neodymium magnets (1/8 in. 1/8 in. 1/16 in., K&J Magnetics, Jamison, PA) and one pyrolytic graphite (PG) sheet (scitoys.com). When displaced laterally from its equilibrium position, the levitating PG sheet tended to return to its original position because it was trapped in the magnetic field formed by the magnets. This homing tendency was quantified by the D-LFC spring constant kDLFC (nN/μm) determined by the vibration frequency and mass of the PG sheet.35 To calibrate our colloidal probes, we used two PG sheets with respective spring constants of 9.38 and 22.82 nN/μm. To perform the calibration, the colloidal probes were engaged near the center of the PG sheet, and the magnet base of the calibrator was moved laterally (in the x direction as shown in Figure 1) for a scan size of 210 μm at a usual scan speed of 1 μm/s. More than 50 lateral voltage curves were recorded. Because the PG sheet exerted a lateral force on the probe tip (i.e., at point A in Figure 3B) that is proportional to its spring constant and the lateral scan size, it was possible to calibrate the lateral force conversion factor k (nN/V) of the probe from the averaged lateral voltage curve using ΔX ΔV
ð2Þ
)
)
k ¼ kDLFC
where tc is the thickness of the probe cantilever (μm), R is the radius of the colloidal tip (μm), and θ is the inclination angle of the cantilever (11). Because the cantilever was inclined by θ as shown in Figure 3B, the D-LFC exerted lateral forces not at the apex of the tip but at point A close to the apex.36 This resulted in a total torque arm length of L = 1/2tc + R(1 + cos θ) as shown in Figure 3B.13 Because the nominal thickness of the cantilevers was 0.50.6 μm (R . tc), tc in eq 3 was assumed to be negligible. Hydrodynamic Lateral Force Calibration. Following the lateral force calibration in air, we determined the lateral torque conversion factor of the same AFM colloidal probe in glycerol (Mallinckrodt, Phillipsburg, NJ). The calibration procedure is shown schematically in Figure 4. After the probe was submerged in a glycerol solution contained in a Petri dish (Falcon 50 9 mm style, Becton Dickinson Labware, Franklin Lakes, NJ), the laser beam path was adjusted to compensate for the offset of the output voltage. To minimize thermal drift, calibrations were started about an hour after immersion. The probe was placed near the bottom surface of the dish by lowering the AFM head. To initiate lateral force calibration, the dish was moved laterally (in the x direction) to cover a certain scan distance at a given scan speed (90 μm at 10 μm/s, 45 μm at 5 μm/s, 9 μm at 1 μm/s, and 0.9 μm at 0.1 μm/s). For each scan, 64 curves of the lateral voltage output were recorded, with each curve containing 1024 data points. After the lateral scanning measurements were completed, five deflection voltagedistance curves were obtained for the tip-to-surface distance determination. The probe was moved vertically down (in the z direction) at 0.51.0 μm/s with a preset deflection voltage trigger point. Starting from the z position where the lateral force calibration was performed, the probe was bent upward because of the presence of a viscous drag force. Because this drag force increased as the probe approached the bottom surface, the trigger point was reached before the tip was engaged. On the basis of the relationship between the magnitude of the drag force and the separation between the tip and the surface, the tip-to-surface distance could be determined from the deflection voltagedistance curves37 (Hydrodynamic Models section). After all necessary voltage curves were obtained at a given z position, measurements of the deflection/lateral voltage curves were repeated at a new z position. This calibration procedure was repeated for four different z positions. Saved PSPD output voltage files were converted to txt format and then processed in MATLAB (MathWorks, Natick, MA). Viscosity Measurement. Because hydrodynamic lateral force calibrations were performed at room temperature, the viscosity of glycerol was measured at 25 C with a viscometer (AR2000, TA Instruments, New Castle, DE). A 60-mm-diameter 1 acrylic cone plate was used for the measurements. Because the measured viscosity showed little dependence on the shear rate (0.1100 s1), it was determined that the glycerol used in this study could be treated as a Newtonian fluid with a viscosity of 0.909 ( 0.003 Pa 3 s. To account for any temperature dependence of glycerol, empirical formulas were evaluated to determine the viscosity μ and density F as38 TðT 1233Þ μ ¼ 12:1 exp ðPa 3 sÞ ð4Þ 70T þ 9900 and F ¼ 1277 0:654T ðkg=m3 Þ
)
)
Here, X is the lateral scan displacement (μm), ΔX is the lateral scan size, and ΔV is the magnitude of the lateral voltage change (V). Because the lateral voltage signal showed a linear dependence on the lateral displacement of the D-LFC base, ΔX/ΔV was approximated by the
)
)
slope of the fitting line between the lateral displacement and the measured lateral voltage (Figure 9B). The above lateral force conversion factor was then converted to the lateral torque conversion factor kT (μN 3 μm/V) by k 1 k kT ¼ Rð1 þ cos θÞ ð3Þ tc þ Rð1 þ cos θÞ ≈ 1000 2 1000
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ð7bÞ
)
ð8aÞ
T u ¼ 8πμR 2 Uw t u
ð8bÞ
)
F u ¼ 6πμRUw f u
where f and t are the force and torque friction coefficients, respectively. These friction coefficients take into account the effects of the wall and/or shear rate on the viscous drag. Although these coefficients can be solved for exact forms,40,41 we employed the following interpolation formulae42 that have been shown to be more computationally efficient i 5 ho ho c1i log ð9aÞ fu ¼ þ c2i R R i¼0
∑
t ¼ u
5
∑
i¼0
i ho ho c3i log þ c4i R R
)
)
F s ¼ 6πμR
Uw ðR þ ho Þf s H Uw s t H
)
T s ¼ 4πμR 3
ð10bÞ
∑
ts ¼
26
∑
i¼0
d2i
R R þ ho
i ð11bÞ
where d1 and d2 are the interpolation coefficients (Table S2). Finally, the total torque applied to the cantilever could be written as Ttot = F R + T , assuming the contribution of the cantilever thickness to be negligible (R . tc). Tip-to-Surface Distance Determination Model. We determined the tip-to-surface distance ho from the recorded deflection voltagedistance curves. The drag force acting on the colloidal probe consisted of the drag force acting on the cantilever and the drag force acting on the colloidal tip. While the cantilever drag 13393
)
where ho is the tip-to-surface distance, Uw is the speed of the moving surface, and H is the distance between the two surfaces (∼1.9 mm). As the AFM head was lowered or raised to change ho, H was also changed. However, H was assumed to be constant because the change in H was smaller than 50 μm. Because the maximum Reynolds number of the current study was about 9.7 108 (R < 60 μm, ho < 50 μm, and Uw < 10 μm/s), the Couette flow around the probe was in the Stokes flow regime. In the flow, the spherical tip experienced not only a viscous drag force but also a drag torque due to the asymmetric flow profile and proximity to the surface. To estimate the drag force and torque on the tip, we decomposed the flow around the tip into two components: a uniform flow and an inverse Couette flow with the top surface moving (Figure 5BC). This decomposition was necessary because no analytical solution was available for the flow shown in Figure 5A. In the case in which the AFM tip is moving instead of the bottom scanner, decomposition
ð10aÞ
Similarly, instead of determining the exact solutions of the friction coefficients, we used their interpolation formulae given as42 i 26 R fs ¼ d1i ð11aÞ R þ ho i¼0
)
ð6Þ
ð9bÞ
where c1, c2, c3, and c4 are the interpolation coefficients (Table S1). For the fixed sphere in the inverse Couette flow (Figure 5C), we used the following drag force and drag torque formulae for F s and T s : 43 )
Torque Estimation Model. To estimate the lateral torque that the viscous fluid flow exerted on the AFM colloidal probe, we developed a hydrodynamic drag model. Previous studies observed that the lateral voltage change of tipless cantilevers was negligible compared to that of colloidal probes under similar flow conditions27 (Figure S1). Therefore, we assumed that the viscous loading on the probe cantilever was negligible compared to the viscous drag acting on the tip and thus that the deformations of the probes were mostly due to the tip drag. We also assumed that the viscous loading on the colloidal tip was applied to the center of the tip. Because the fluid motion around the tip near the substrate surface was asymmetric, the hydrodynamic force center of the tip might not be identical to its geometric center. It was suggested that this offset between centers should be considered if the ratio of the tip-to-surface distance to the tip radius is smaller than 0.05.27 In our measurements, this ratio was greater than 0.15, so the center offset was not considered. Under these assumptions, the colloidal probe could be assumed to be a solid sphere in the Couette flow generated by the moving substrate (Figure 5A). In this case, the shear particle Reynolds number Re was defined as39
)
)
where superscripts u and s represent uniform flow and shear flow quantities, respectively. It should also be noted that the wall effect from the top surface was assumed to be negligible because it was too far away from the probe tip when compared to the bottom surface (H . ho). For the fixed sphere in the uniform flow (Figure 5B), the viscous drag force F u and torque T u were determined using
’ HYDRODYNAMIC MODELS
2FRðR þ ho ÞUw Re ¼ μH
)
)
where T is the temperature in C. Our viscosity measurements agreed well with the value predicted by eq 4 (0.906 Pa 3 s), thus we used eq 5 to obtain the density of glycerol (1.261 g/cm3).
)
T ¼ Tu ð TsÞ
)
ð7aÞ
)
F ¼ Fu Fs
)
Figure 5. Flow decomposition to estimate drag force and torque on the spherical tip. (A) Shear flow occurred through the lateral motion of the bottom surface. (B) Uniform flow. (C) Shear flow occurred through the motion of the top surface. In all cases, the sphere is immobilized. See the text for symbols. These images are not to scale.
)
)
is unnecessary. The flow decomposition was possible because of the linearity of the Stokes equation governing the flow. The drag force F and the drag torque T on the tip could be divided into two components as
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Figure 6. Determination of the tip-to-surface distance ho. (A) Schematic diagram of the colloidal probe approaching the substrate surface. The undeflected probe is shown with gray dashed lines. See the text for symbols. (B) Flow chart to determine ho.
was distributed over the total cantilever area, the tip drag was applied to the cantilever as a point load.44 As the probe approached the substrate surface, the contribution of the tip drag was more significant than the contribution from the cantilever because the cantilever was always ∼h + 2R away from the surface whereas the tip was h + R away (Figure 6A). The tip drag became even more significant as the tip-to-surface distance was decreased below the tip radius.37 Under such conditions, the colloidal probe approaching the substrate surface could be represented by a solid sphere moving toward the surface with a tip velocity of Ut. Because the Reynolds number (2FRUt/μ) was smaller than 1.7 107 (R < 60 μm and Ut < 1 μm/s), the solid sphere was moving in the Stokes flow regime. Thus, the normal drag force on the sphere F^ could be calculated as45,46 F^ ¼ 6πμRUt λ
deflection voltage can be related to eq 12a by k^ Sl^ ðV^ V^o Þ ¼ 6πμRUt λ
V^ V^o ¼
Ut ¼ Z_ d_ ¼ Up Sl^ V_ ^
ð12bÞ α ¼ cosh1
R þ h R
6πμR Ut λ ¼ CUt λ k^ Sl^
ð14Þ
where C is the fitting constant. Because the cantilever is being deflected, Ut 6¼ Up, where Up is the probe velocity. Instead, Ut can be calculated from the deflection voltage and the probe’s travel distance as
∑
ð13Þ
where k^ is the normal spring constant of the probe, S^ is the normal optical lever sensitivity, V^ is the deflection voltage output, and V^o is the initial value of V^. Therefore, V^ is proportional to Utλ by
ð12aÞ
∞ 4 iði þ 1Þ sinh α 3 ð2i 1Þð2i þ 3Þ i¼1 " # 2 sinhð2i þ 1Þα þ ð2i þ 1Þsinh 2α 1 4 sinh2 ði þ 1=2Þα ð2i þ 1Þ2 sinh2 α
λ¼
Figure 7. Representative tip deflection curves for the tip-to-surface distance determination (probe 5). (A) Deflection voltagedistance curves with increasing ho. ho increases from the right-hand side to the left-hand side. (B) Fitting result of the curve with the largest ho in part A (solid line, λ of eq 12b; dot, (V^ V^o)/UtC from eq 14). hmin = 0.93 μm and ho = 23.48 μm. (Inset) Fitting result of the curve with the smallest ho. hmin = 0.95 μm and ho = 8.26 μm.
ð12cÞ
where λ is the wall effect correction factor and h is the distance between the sphere and the surface (its initial value is ho). Equation 12 shows that the sphere experiences more viscous resistance as it approaches the surface and that this increase in resistance can be accounted for by the wall effect correction factor. The drag force acting on the spherical tip deflects the cantilever (d = Sl^(V^ V^o) in Figure 6A) such that the recorded
ð15Þ
where Z is the z position of the probe and the dot notation denotes the time derivative. The determination of Ut requires Sl^ to be known. Sl^ can be assumed from the normal optical lever sensitivity measured in air Sa^,33 but Sa^ can introduce significant error because of the severe stickslip behavior observed between the colloidal tip and substrate surface13 (Figure S2). Instead, we determined ho by finding the best linear fit between V^ and Utλ following the procedure described in Figure 6B; we assumed a set of minimum tip-to-surface distances hmin and optical sensitivities Sl^. For a given pair of hmin and Sl^, Ut was calculated from eq 15 and the known z position Z, and h was 13394
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where Z o is the initial value of Z. Raw data from the deflection voltage curves were downsampled to reduce the overall calculation time, and only 30% of the data (the higher z side, Figure 7B) was used for fitting purposes to satisfy the surface proximity condition.
’ RESULTS The proposed hydrodynamic lateral force calibration method successfully determined the lateral torque conversion factor from the lateral voltage change due to viscous tip drag forces. The lateral voltage change was determined from the difference between the trace and retrace lateral voltage curves, and it reflected the torsional deformation of the probe. The total probe torque from the viscous flow was estimated using the presented hydrodynamic torque estimation model. Because the torque estimation model requires a knowledge of the tip-to-surface distance, the successful application of the current method depends on the determination of the tip-to-surface distance. Determination of Tip-to-Surface Distances. The hydrodynamic drag force model to determine the tip-to-surface distance ho was applied to the deflection voltage curves, and curves of probe 5 are shown as a representative example in Figure 7A. Four deflection voltage curves obtained at different probe positions show that, for a given trigger point, the probe traversed longer distances if the probe initially started farther away from the surface. During the approach, the probe was deflected upward (V^ > 0) because of the viscous drag force, and the deflection increased in magnitude as the probe moved closer to the substrate surface. In contrast, the deflection voltage became negative during retraction because the drag force was acting in the opposite direction, deflecting the probe downward. hmin of each of the curves was expected to be similar regardless of the initial probe position because near the surface the probe experienced the same drag force magnitude as determined by eq 12. Figure 7B shows the fitting results for the lowest and highest probe positions, and in the fitting zone, the wall effect correction factor λ and the normalized deflection voltage (V^ V^o)/UtC are in good agreement. The tip-to-surface distances hmin = 0.93 μm and ho = 23.48 μm were determined from the curve of the highest probe position, whereas hmin = 0.95 μm and ho = 8.26 μm were determined from the other curve. As expected, values of hmin are similar between the two curves. Additionally, estimated tip-to-surface distances show negligible deviations among obtained deflection voltage curves: hmin = 0.92 ( 0.01 μm and ho = 23.43 ( 0.04 μm for the highest probe position, and hmin = 0.86 ( 0.05 μm and ho = 8.11 ( 0.10 μm for the lowest probe position. Therefore, the hydrodynamic model for the deflection of the
)
ð17Þ
)
ho ¼ hmin þ Zmax Zo Sl^ ðV^max V^o Þ
)
Once λ was calculated from h and eq 12b, a linear least-squares fitting between V^ and Utλ was performed, and the norm of the fitting residual was computed for the pair. This yielded a set of residual norms for different combinations of h min and S l^ from which the minimum residual and corresponding h min could be determined. Once h min for the best fit was found, ho was determined by
)
ð16Þ
)
¼ hmin þ Zmax Z Sl^ ðV^max V^ Þ
)
h ¼ hmin þ Zmax Z ðdmax dÞ
AFM colloidal probe enabled the determination of the tip-tosurface distance. Additionally, it was possible to determine the normal spring constant of the probes along with the tip-to-surface distance determination. As eq 14 shows, the denominator of the fitting constant C is the normal force conversion factor of the probe. Because Sl^ and C were determined during the fitting process described in Figure 6B, the normal spring constant of the probe 14 = 6πμR/CSl^. As shown in Table 2, the could be calibrated with keq ^ hydrodynamically determined normal spring constant values of the colloidal probes were greater than the nominal values of the tipless cantilevers provided by the manufacturer. There are several reasons that the attachment of a colloidal tip would increase the normal spring constant of a cantilever. For example, the attachment of the colloidal tip can decrease the effective length of the cantilever because the colloid is not always placed at the free end of the cantilever. Second, by adding the colloidal tip, the overall inertia of the AFM probe is increased, resulting in smaller deformations as compared to those of a tipless cantilever. Analysis of Lateral Voltage Signals. Once ho was determined, the magnitude of the total torque applied to the probe could be calculated on the basis of the probe geometry and flow conditions. In addition to the total torque, the lateral force calibration required the measurement of the lateral voltage change due to the total torque. Figure 8 shows the lateral voltage profiles of probe 1 measured at different scan speeds and tip-tosurface distances. The viscous loading from the Coutte flow was great enough to deform the probe, as manifested by measured differences between the trace and retrace lateral voltage curves. As expected from the torque estimation model, the probe experienced more viscous drag if it was placed closer to the substrate surface and if the surface moved at a higher speed. Therefore, the voltage difference increased with increasing scan speed and decreasing tip-to-surface distance. As marked in Figure 8, the voltage difference between the trace and retrace curves was equal to twice the lateral voltage change due to the Coutte flow (|V ,trace V ,retrace| = 2ΔV ). This is because the probe experienced a torque of the same magnitude but different directions during trace and retrace. To calculate the average value of the lateral voltage change from the obtained lateral voltage curves, we tried two methods. In the first method, the voltage difference was calculated at a certain X point in the curve, and the differences were averaged along the curve. Individual voltage differences did not show much deviation (2ΔV = 0.498 ( 0.003 V from the top left graph of Figure 8). In the second method, linear fittings of the trace and retrace curves were performed (from the same graph, V = 8.495 104X 2.643 for the trace curve and V = 9.113 104X 2.145 for the retrace curve), and the voltage difference was obtained from the difference in the constants of the fitting lines (2ΔV = 2.643 2.145 = 0.498 V). The slope of the fitting lines showed that the trace and retrace curves were approximately parallel to each other. In both methods, more than 60 lateral voltage curves were ensemble averaged, and the lateral voltage differences obtained from the two methods agreed well with each other. Although it was possible to determine the lateral voltage difference because the trace and retrace curves were approximately parallel as expected, the trace and retrace lateral voltage curves were slanted as shown in Figure 8. It was expected that the lateral voltage curves would be horizontal and that the )
calculated by
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Figure 8. Representative averaged lateral voltage curves measured at different scan speeds and tip-to-surface distances (probe 1).
)
)
)
)
lateral force calibration performed in air with the D-LFC technique. As expected, the lateral voltage output showed a linear dependence on the lateral force of the D-LFC with good agreement between the trace and retrace curves (Figure 9B). The torque conversion factor calibrated in air was 7.50 μN 3 μm/ V, which was greater than that in liquid. The ratio of torque conversion factors is equal to that of optical lever sensitivities (kaT/klT = kTSa/kTSl = Sa/Sl ). Therefore, the torque conversion factor ratio is equal to the refractive index ratio according to eq 1. The laser wavelength of the AFM was 860 nm, and the refractive indices of air and glycerol at this wavelength were 1.00 and 1.47, respectively.48,49 The theoretical prediction of the torque conversion factor ratio was 1.47, and the conversion factor ratio obtained from Figure 9 was 1.64. The measured ratio is reasonably close to the predicted ratio considering that most lateral force calibration methods have 1020% uncertainties in their results.4 Table 2 shows the lateral force calibration results of seven different colloidal probes. Overall, the standard deviations of the calibrated torque conversion factors were less than 10% of all average values. Ranging from 1.39 to 1.91, the kT ratios of the probes were in reasonable agreement with the theoretical prediction of eq 1 although their nominal normal spring constants differed by one order of magnitude. )
deflection voltage difference between trace and retrace would be negligible if the colloidal probe experienced pure torsional deformation. An examination of the deflection voltage signal showed changes according to viscous loading conditions (Figure S3). We suspect that coupling or crosstalk between normal and lateral voltage signals produced the observed nonhorizontal curves. In general, there are three main sources of crosstalk: mechanical, optical (or geometric), and electronic.4,47 Mechanical crosstalk is mainly due to the positional offset of the probe shear center from the geometric center of the probe. The AFM colloidal probes are prone to mechanical crosstalk because it is difficult to align the colloid exactly at the center of the cantilever. However, mechanical crosstalk may not solely account for the observed behavior because even tipless probes showed voltage curves similar to that of the colloidal probe (Figure S1). Therefore, we suspect that optical crosstalk and electric crosstalk are also reasons for the nonhorizontal voltage curves. Optical crosstalk is caused by a rotational misalignment of the PSPD or a mounting error of the AFM probe, and electronic crosstalk is due to interference between signals in the electronic components of AFM. Because it is difficult to eliminate optical and electronic crosstalk,47 it is desirable to perform a lateral force calibration in situ. Hydrodynamic Lateral Force Calibration. Figure 9A shows the lateral force calibration results obtained by applying the hydrodynamic model to the data shown in Figure 8. The plot shows that the lateral voltage difference increased linearly with respect to the total torque applied to the probe. The slope of the fitting line in Figure 8 is the lateral torque conversion factor (klT = dTtot/dΔV ). Calibration results at four different ho values showed a relatively small standard deviation (klT = 4.58 ( 0.22 μN 3 μm/V). This calibration result was compared to the
’ DISCUSSION We showed that our hydrodynamic lateral force calibration method can calibrate the torque conversion factor of the AFM probes of large colloidal tips. Hydrodynamically calibrated torque conversion factors were compared to those calibrated in air using the D-LFC technique based on eq 1. Because the torque con13396
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Figure 9. Representative lateral force calibration result (probe 1). (A) Hydrodynamic calibration in glycerol (red line: fitting line between the lateral voltage change and the total torque applied to the probe). klT = 4.58 ( 0.22 μN 3 μm/V. (B) Calibration with the D-LFC in air. kaT = 7.50 μN 3 μm/V.
)
)
)
)
ð18Þ
)
Sa^ nl cos θ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ na Sl^ 1 ðnl =na Þ2 sin2 θ
and their measured normal sensitivity ratios showed ∼10% deviations from the prediction based on eq 18. Pettersson et al.32 corroborated eq 1 by measuring the lateral optical lever sensitivity of a tipless rectangular cantilever in air and water from the pure rotation of the probe: Sa = 2.84 104 rad/V and Sl = 2.14 104 rad/V, so Sa/Sl = 1.33 and nl/na = 1.33. However, it is noticeable that they derived a different form of the lateral optical sensitivity ratio and that the calculated ratio was 1.28 (Appendix in Pettersson et al.32). This difference seems to be due to the optical path difference between the two optical lever systems. Recently, Wagner et al.50 measured the lateral force conversion factor of sharp tip rectangular probes in air, water, and ethanol from the torsional thermal noise spectrum: ka/kl = 1.6 ( 0.3 (1.151.85) and nl/na = 1.33 or 1.36 (ethanol). They also measured the normal optical lever sensitivity in air and liquids (Sa^ = 298.5 nm/V and Sl^ = 327.9 nm/V for water and Sl^ = 263.9 nm/V for ethylene glycol) and the measured ratios of the normal optical lever sensitivity were ∼30% smaller than those predicted by eq 18. Because Wagner et al. used the same kind of AFM that was used in this study, their lateral force conversion factor ratio can serve as a reference to establish an error estimate for our measurements: their measurements showed ∼35% differences from the predictions whereas ours showed ∼30% differences. Hence, our proposed hydrodynamic calibration method enables quantitative measurements of the lateral forces in liquids. The calibrated torque conversion factors shown in Table 2 had standard deviations of 210% of their mean values. This is because calibrated torque conversion factors were slightly different at different ho positions as Figure 9A shows. One of the reasons for the deviation of klT is the uncertainty of the determined tip-to-surface distance. It is evident from the hydrodynamic models that the uncertainty of klT depends on the uncertainty of ho. There are two contributions for the uncertainty in ho. First, the probe did not seem to return to the same z position after each traverse during the determination of ho. Although the z position of the probe was feedback-controlled, this deviation in ho was inevitable because of the nonlinearity and creep of the z-piezo actuator. In this study, the standard deviation of ho was usually smaller than 0.2 μm, and this uncertainty could cause an ∼1% deviation in klT (uncertainty analysis in Supporting Information). Second, ho determination based on the drag formula fitting has an inherent uncertainty because the cantilever drag was ignored. Because measured deflection voltage profiles will follow eq 12 better when the probe is in closer proximity to the substrate, uncertainty in the determination of ho depends on how much of the total data is used for fitting. We tested the )
version factor ratio between air and liquid is equal to the ratio of lateral optical lever sensitivities, eq 1 can provide a prediction of the torque conversion factor ratio. In our case, the value of eq 1 was 1.47, and the measured torque conversion factor ratio was approximately 1.62 ( 0.19 as Table 2 shows. To the best of our knowledge, this study is the first of its kind to measure the torque conversion factor ratio between air and liquid by directly applying lateral forces to the AFM probe tip. Although eq 1 was employed to compare the torque conversion factor ratios, Tocha et al.33 inferred the validity of eq 1 from the validity of the normal optical lever sensitivity relation
Table 2. Summary of Lateral Force Calibration Results kT (μN 3 μm/V)
k^ (N/m) probe no.
a
cantilever type
tip radius R (μm)
kN ^
14 keq ^
kT ratiob
klTa
kaT
kaT/klT
1
V-shaped
58.8
0.1
0.29
4.58 ( 0.22
7.50
1.64
2
V-shaped
40.1
0.03
0.10
15.54 ( 0.55
21.53
1.39
3
V-shaped
41.1
0.03
0.10
4.57 ( 0.11
6.51
1.43
4
V-shaped
44.8
0.01
0.08
3.18 ( 0.20
5.76
1.81
5 6
rectangular rectangular
36.7 35.4
0.02 0.02
0.10 0.05
2.64 ( 0.18 6.27 ( 0.56
4.28 9.44
1.62 1.51
7
rectangular
44.0
0.02
0.03
2.69 ( 0.07
5.14
1.91
Mean ( standard deviation. b The kT ratio from eq 1 is 1.47. 13397
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)
)
)
)
dependence of the torque conversion factor on the fitting portion size using the data shown in Figure 7 (Table S3). As the fitting portion increased from 30 to 80%, ho increased by no more than 0.5 μm and klT changed by ∼1%. Another reason for the deviations in klT is that any contributions of the cantilever were ignored in estimating the viscous torque applied to the colloidal probe. Because the torsional deformations of tipless cantilevers in the Couette flow were negligible (Figure S1), the colloidal probe was regarded as a solid sphere in the flow to determine the total torque on the probe. However, it is obvious that the cantilever played a role as a boundary condition for the flow around and motion of the colloidal tip. This contribution of the cantilever is expected to depend on the tip-to-surface distance and to be responsible for the deviation in the calibrated klT values. We also suspect that any temperature changes of glycerol could cause deviations. Although the temperature of glycerol was not enforced to be constant, the maximum temperature fluctuation recorded was within (1 C, yielding an ∼9% change in the viscosity of glycerol according to eq 4. The torque conversion factor is expected to change proportionally to the change in viscosity. The proposed hydrodynamic lateral force method has three distinct advantages for practical LFM applications. First, because the whole calibration procedure is performed in situ in the fluid, there is no need for separate calibrations in air and a compensation for the refractive index difference between media. Second, the method is a one-step calibration. Therefore, there is no need for additional procedures to calibrate the lateral optical lever sensitivity or lateral/torsional spring constant. Finally, our hydrodynamic calibration method does not require any contact of the probe tip with the underlying surfaces. This can be especially advantageous when using functionalized colloidal probes to avoid tip damage or contamination during calibration. To take full advantage of the hydrodynamic lateral force calibration method, soft cantilevers and large colloidal tips in highly viscous liquids are preferred.27 We investigated colloidal probes with smaller tip sizes (R ≈ 10 μm) and found that the measured lateral voltage change posed challenges to the hydrodynamic lateral force calibration method depending on the experimental conditions. However, for many experimental designs, the scan speed and tip-to-surface distance can be adjusted to obtain adequate viscous loading forces. As eqs 8 and 10 show, the total torque is proportional to the scan speed and friction coefficients. As the tip-to-surface distance decreases, the viscous forces from the uniform flow component become more dominant, and the friction coefficients f u and t u increase exponentially (Figure S5). Therefore, increasing the scan speed and/or decreasing the tipto-surface distance will increase the total torque and hence the lateral output voltage change. For scenarios in which the hydrodynamic lateral force calibration method is not applicable, lateral force/torque conversion factors may be calibrated in air first and may then be converted with eq 1 for lateral force measurements in liquids. However, it should be noted that eq 1 was derived for a particular AFM optical path, hence eq 1 may cause significant errors in lateral force measurements. Instead, Sa/Sl , ka/kl , or kaT/klT can be measured for a particular AFM setup using any possible calibration methods, and their values can be used to convert lateral force/ torque conversion factors calibrated in air.
’ CONCLUSIONS In this article, we present an in situ noncontact hydrodynamic lateral force calibration technique for AFM colloidal probes. The method exploited viscous drag forces of a creeping Couette flow generated by the lateral scanning motion of the AFM. The lateral torque conversion factor was determined by considering the viscous drag forces applied to the probe tip and the resulting lateral voltage change. In comparison with lateral force calibrations performed in air using the D-LFC technique, the hydrodynamic lateral force calibration method showed reasonably good agreement with the refractive index difference between the media considered. Therefore, the presented hydrodynamic lateral force calibration method can be employed for lateral force measurements in liquid using colloidal probes. ’ ASSOCIATED CONTENT
bS
Supporting Information. Values of the interpolation coefficient of eqs 9 and 11. Lateral and deflection voltage curves of AFM probes. Stickslip in a representative deflection voltagedistance curve. Ensemble-averaged deflection voltage profiles. Uncertainty analysis. Dependence of the torque conversion factor on the fitting portion size. Dependence of viscous drag components on the tipto-surface distance. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT We thank Prof. Eric Darling for assistance in using the AFM and Dr. Jinwoo Yi for helping in fabricating the colloidal probes and using the D-LFC technique. We also thank the reviewers for their constructive comments to improve the overall quality of this article. ’ REFERENCES (1) Bhushan, B.; Israelachvili, J. N.; Landman, U. Nature 1995, 374, 607–616. (2) Carpick, R. W. Chem. Rev. 1997, 97, 1163–1194. (3) Gnecco, E.; Bennewitz, R.; Gyalog, T.; Meyer, E. J. Phys.: Condens. Matter 2001, 13, R619–R642. (4) Munz, M. J. Phys. D: Appl. Phys. 2010, 43, 063001. (5) Liu, E.; Blanpain, B.; Celis, J. P. Wear 1996, 192, 141–150. (6) Gibson, C. T.; Watson, G. S.; Myhra, S. Wear 1997, 213, 72–79. (7) Perry, S. S. MRS Bull. 2004, 29, 478–483. (8) Palacio, M. L. B.; Bhushan, B. CRC Crit. Rev. Sol. State Sci. 2010, 35, 73–104. (9) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Nature 1991, 353, 239–241. (10) Butt, H.-J. Biophys. J. 1991, 60, 1438–1444. (11) Butt, H.-J.; Cappella, B.; Kappl, M. Surf. Sci. Rep. 2005, 59, 1–152. (12) Lorenz, B.; Keller, R.; Sunnick, E.; Geil, B.; Janshoff, A. Biophys. Chem. 2010, 150, 54–63. (13) Chung, K.-H.; Shaw, G. A.; Pratt, J. R. Rev. Sci. Instrum. 2009, 80, 065107. (14) Bremmell, K. E.; Evans, A.; Prestidge, C. A. Colloids Surf., B 2006, 50, 43–48. 13398
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ARTICLE
(15) Leporatti, S.; Gerth, A.; K€ohler, G.; Kohlstrunk, B.; Hauschildt, S.; Donath, E. FEBS Lett. 2006, 580, 450–454. (16) Gong, J.; Iwasaki, Y.; Osada, Y.; Kurihara, K.; Hamai, Y. J. Phys. Chem. B 1999, 103, 6001–6006. (17) Feiler, A.; Larson, I.; Jenkins, P.; Attard, P. Langmuir 2000, 16, 10269–10277. (18) Han, L.; Dean, D.; Ortiz, C.; Grodzinsky, A. J. Biophys. J. 2007, 92, 1384–1398. (19) Butt, H.-J.; Jaschke, M.; Ducker, W. Bioelectrochem. Bioenerg. 1995, 38, 191–201. (20) Kappl, M.; Butt, H.-J. Part. Part. Syst. Charact. 2002, 19, 129–143. (21) Leite, F. L.; Herrmann, P. S. P. J. Adhesion Sci. Technol. 2005, 19, 365–405. (22) Liang, Y.; Hilal, N.; Langston, P.; Starov, V. Adv. Colloid Interface Sci. 2007, 134135, 151–166. (23) Kokkoli, E.; Ochsenhirt, S. E.; Tirrell, M. Langmuir 2004, 20, 2397–2404. (24) Xu, L.-C.; Logan, B. E. Langmuir 2006, 22, 4720–4727. (25) Benmouna, F.; Johannsmann, D. J. Phys.: Condens. Matter 2003, 15, 3003–3012. (26) Bonaccurso, E.; Kappl, M.; Butt, H.-J. Curr. Opin. Colloid Interface Sci. 2008, 13, 107–119. (27) Vakarelski, I. U.; Dagastine, R. R.; Chan, D. Y. C.; Stevens, G. W.; Higashitani, K.; Grieser, F. Langmuir 2010, 26, 8002–8007. (28) Radmacher, M.; Tillmann, R. W.; Fritz, M.; Gaub, H. E. Science 1992, 257, 1900–1905. (29) Alessandrini, A.; Facci, P. Meas. Sci. Technol. 2005, 16, R65–R92. (30) Chang, D. P.; Abu-Lail, N. I.; Coles, J. M.; Guilak, F.; Jay, G. D.; Zauscher, S. Soft Matter 2009, 5, 3438–3445. (31) Haugstad, G.; Gladfelter, W. L. Ultramicroscopy 1994, 54, 31–40. (32) Pettersson, T.; Nordgren, N.; Rutland, M. W.; Feiler, A. Rev. Sci. Instrum. 2007, 78, 093702. (33) Tocha, E.; Song, J.; Sch€onherr, H.; Vancso, G. J. Langmuir 2007, 23, 7078–7082. (34) Chung, K.-H.; Pratt, J. R.; Reitsma, M. G. Langmuir 2010, 26, 1386–1394. (35) Li, Q.; Kim, K.-S.; Rydberg, A. Rev. Sci. Instrum. 2006, 77, 065105. (36) Heim, L.-O.; Kappl, M.; Butt, H.-J. Langmuir 2004, 20, 2760– 2764. (37) Craig, V. S. J.; Neto, C. Langmuir 2001, 17, 6018–6022. (38) Cheng, N.-S. Ind. Eng. Chem. Res. 2008, 47, 3285–3288. (39) Brooks, S. B.; Tozeren, A. Comp. Fluids. 1996, 25, 741–757. (40) O’Neill, M. E. Mathematika 1964, 11, 67–74. (41) O’Neill, M. E. Mathematika 1967, 14, 170–172. (42) Chaoui, M.; Feuillebois, F. Q. J. Mech. Appl. Math. 2003, 56, 381–410. (43) Goldman, A. J.; Cox, R. G.; Brenner, H. Chem. Eng. Sci. 1967, 22, 653–660. (44) Senden, T.; Ducker, W. Langmuir 1994, 10, 1003–1004. (45) Brenner, H. Chem. Eng. Sci. 1961, 16, 242–251. (46) Maude, A. D. Br. J. Appl. Phys. 1961, 12, 293–295. (47) Onal, C. D.; S€umer, B.; Sitti, M. Rev. Sci. Instrum. 2008, 79, 103706. (48) Ciddor, P. E. Appl. Opt. 1996, 35, 1566–1573. (49) Rheims, J.; K€oser, J.; Wriedt, T. Meas. Sci. Technol. 1997, 8, 601–605. (50) Wagner, K.; Cheng, P.; Vezenov, D. Langmuir 2011, 27, 4635–4644.
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