Nanoscale Contact-Radius Determination by Spectral Analysis of

Quantitative characterization of surface topography using spectral analysis. Tevis D B Jacobs , Till Junge , Lars Pastewka. Surface Topography: Metrol...
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Nanoscale Contact-Radius Determination by Spectral Analysis of Polymer Roughness Images Armin W. Knoll* IBM Research - Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland ABSTRACT: In spite of the long history of atomic force microscopy (AFM) imaging of soft materials such as polymers, little is known about the detailed effect of a finite tip size and applied force on the imaging performance on such materials. Here we exploit the defined scaling of roughness amplitudes on amorphous polymer films to determine the transfer function imposed by the imaging tip. The finite indentation of the nanometer-scale tip into the comparatively soft polymer surface leads to a finite contact area, which in turn effectively acts as a moving average filter for the surface roughness. In the power spectral density (PSD), this leads to an attenuation of the roughness amplitudes related to the Airy pattern known from light diffraction of a circular aperture. This transfer function is affected by the roughness-induced local modulation of the tip height and contact area, which is studied by performing simulations of the polymer roughness and the imaging process. We find that for typical polymer parameters and sharp tips the contact radius of the tip−sample contact can be recovered from the roughness spectrum. We experimentally verify and demonstrate the method by measuring the nanoscale contact radius as a function of applied load and travel distance on a highly cross-linked model polymer. The data are consistent with the Johnson−Kendall− Roberts (JKR) contact model and verifies its applicability at the nanometer scale. Using the model, quantitative values of the elastic sample parameters can be determined.

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INTRODUCTION The quantitative analysis of data collected by atomic force microscopes (AFM) is often hampered by the limited knowledge of the precise shape of the scanning tip at its apex. The impact of this challenge ranges from tip-shape convolution effects in topographical/roughness measurements1 to difficulties in obtaining quantitative measurements of mechanical properties,2 friction,3 adhesion,4,5 or tip wear.6 Several strategies currently exist to determine the tip shape ex situ or in situ. Ex situ measurements include scanning/ transmission electron microscopy (SEM/TEM) imaging of the tip, which is cumbersome and potentially results in unwanted carbon depositions at the tip surface. A recently developed in situ method7 is to measure the critical free amplitude in amplitude modulation AFM, which triggers the transition from attractive to repulsive imaging regimes.8 Other in situ measurements typically involve the use of tip-shape calibration samples9,10 and either deconvolution algorithms for known sample features or blind tip-shape reconstruction.11 The calibration samples are made from hard materials to prevent unwanted sample deformation during imaging. On soft materials, however, these methods cannot be applied because the van der Waals force between tip and sample is sufficient to significantly deform the surface. The deformation leads to the formation of an area of direct contact between tip and sample, the so-called contact area.12 The contact area is an intrinsic parameter that depends on the applied load and therefore cannot be measured ex situ. Without knowledge of the contact area, contact models need to be applied to interpret © 2013 American Chemical Society

the results of some of the techniques mentioned above. This pertains in particular to measurements of physical quantities directly related to the contact area, such as friction3 and thermal13 or electrical conductance.14 One direct, albeit elaborate, way to solve this challenge is to integrate an AFM tool into an electron microscope.15 It has been suggested before that imaging rough surfaces with known roughness scaling allows the recovery of tip-shape information.16,17 In that approach, however, deformations of tip and sample are ignored, and the pure convolution of tip shape and roughness is considered. Therefore, that work is applicable only for hard sample (and tip) materials. Here we exploit the well-defined roughness spectrum and the soft nature of amorphous polymer films to determine the contact area in situ as a function of applied conditions.

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EXPERIMENTAL SECTION

Polymer Films. For the experiments discussed here we used two variants of poly(aryl−ether−ketone) (PAEK) polymers with a molecular weight of Mn ≃ 4000 g/mol and relative amounts of cross-linker monomers of 18% (PAEK-18, see Figure 1A) and 21% (PAEK-21). These polymers were chosen because of their technical relevance in data storage applications and their low wear properties in the context of contact-mode imaging.18 Thin films of ≃100 nm Received: August 13, 2013 Revised: October 1, 2013 Published: October 23, 2013 13958

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thermal time constant of the sensor of ≈6 μs,21 and the sharpness of the tip. We adjusted the imaging speed to 60 μs per pixel to ensure that the imaging resolution is limited solely by the shape of the tip.

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RESULTS Theoretical Background. The roughness of amorphous polymer films is caused by thermally activated capillary waves which are excited at the film surface in the liquid or molten state22,23 and frozen in during the transition into the glassy state.24 In our experiment, we used thermal annealing and cross-linking of the polymer films at 400 °C with rapid cooling to room temperature. The capillary waves form at the elevated annealing temperature and are frozen in during the cooling process. When imaging the roughness, an average spectral power G(q) = ⟨|Aq|2⟩ is observed:19,21 G(q) =

kBT γq2

(1)

where γ is the surface tension at the temperature T at which the polymer reaches the glassy regime. Equation 1 is valid for wavenumbers lower than that of molecular dimensions25 qmax ≃ 10 nm−1 and greater than qmin = (A/2πγt4)1/2 ≤ 10−4 nm−1, where A ≃ 10−20 J is the Hamaker constant of the system and t denotes the film thickness,23 in our case t ≈ 100 nm. In addition, a low q cutoff has been predicted26 and observed27 for molten polymer brushes, however, outside the range of q values discussed here. Figure 1B depicts a topographical image of a highly crosslinked poly(aryl−ether−ketone) (PAEK-18) polymer film. The scan size of the image is 500 × 500 nm2, sampled with 1800 × 1800 pixels. The inset in Figure 1B shows a zoom into the 2D power spectral density (2D-PSD) plot of the data on a logarithmic color scale. The spectrum shows an elevated noise floor along the y-axis, which is attributed to low frequency noise and the line-by-line offset subtraction used for flattening. Figure 1C depicts the azimuthally averaged power spectral density (PSD) of the polymer roughness; for details see ref 19. To avoid this elevated noise floor in the spectrum along the y-axis, the averaging procedure is restricted to a sector of ±45° opening angle as depicted in Figure 1B. The amplitudes in the PSD spectrum drop much faster than the q−2 behavior expected for the polymer roughness for q > 0.1 nm−1. This drop originates from the finite size of the tip imaging the surface roughness as is discussed in the following. The PSD amplitudes correspond to the measured height of the tip, which is modulated by the polymer roughness as a function of position. Thus, the effect of the surface roughness on the tip height needs to be considered. Unlike on a flat surface, the physical problem given by the contact of a tip to a rough surface is not straightforward to solve. First, the contact area is not well-defined if the tip penetration depth is similar to or less than the surface roughness amplitudes. Second, the roughness introduces roughness asperities within the contact area. The asperities deform nonlinearly and modify the contact behavior in comparison to a flat surface. To tackle the problem, we will start out by investigating the effect when translating a contact from a flat to a rough surface in the case when both effects remain small. For a flat surface the elastic contact12 between a sphere and the surface is sketched in Figure 1D. For a given load P0, the tip deforms the medium elastically by an amount δ. In lateral direction the sphere is in conformal contact with the sample up to the contact radius a. In our case, Young’s modulus of the

Figure 1. Imaging of a polymer surface. (A) Chemical structure of the PAEK-18 polymer. (B) Topographical image of a PAEK-18 polymer surface and 2D-PSD plot of the data (inset). (C) PSD spectrum obtained by azimuthal averaging. A characteristic drop from the q−2 behavior is observed. A fit of an Airy pattern to the data is shown by the red curve. (D) Sketch of a spherical indenter of radius R in contact with a polymer surface that deforms the surface to a depth δ due to the applied load P0. The sphere is in contact with the polymer up to the contact radius, a, and the contact depth, d, giving rise to the contact area Ac. (E) Sketch of the situation when transforming the contact from a flat to a rough surface. In the case of small roughness amplitudes, the finite roughness leads to an average shift of the surface within Ac by h̅, giving rise to a change Δz = h̅ of the tip height (see text). thickness were spin-coated from anisole solution onto highly doped silicon wafers and cured at 400 °C for 1 h in a nitrogen atmosphere. AFM Imaging. AFM images were obtained using home-built equipment described in detail elsewhere.19 In short, we used cantilever-style force sensors with a fundamental mode resonant frequency of ≈50 kHz and typical spring constants of 0.1 ± 0.05 N/m. To obtain the spring constants of the two levers used for the experiments, we measured the thickness of the levers in an SEM and performed finite-element simulations of the lever geometry. The levers comprise integrated thermal sensors for topographic sensing. The AFM is operated in contact mode using contact forces below 10 nN as determined by the approach distance beyond the snap-in position and the spring constant. For reliable roughness imaging, we record the signal of the thermal sensor, which provides a direct measure of the height of the sensor above the surface.20 The sensor signal is recorded as a function of position without feedback loop for the tip−sample distance. The soft cantilever spring deflects, allowing the tip to follow the corrugation of the substrate with minimal force modulation. The peak-to-peak corrugation of the films investigated here is about 2 nm, and the tilt of the sample was compensated so that it contributed less than 5 nm modulation of the out-of-plane component. Therefore, the load force on the tip varies by less than 1 nN across an image. Because of the direct imaging method, the setup reliably captures high spatial frequencies. The resolution of the topographic image is limited by the fundamental mechanical response time of the lever of ≈20 μs, the 13959

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Figure 2. Simulation of a polymer surface imaging. (A) Comparison of an experimental image (left) and a simulated roughness image with (bottom right) and without (top right) tip imaging effects. (B) Linear elastic model used for the simulation. Top: on a flat surface, the tip forms a contact with contact radius ai and displaced volume V. Bottom: V is conserved on the rough surface. (C) Transfer function of the tip imaging process for contact radii on a flat surface of ai = 3 (bottom) and 9 nm (top) and tip radii of R = 10, 18, 32, 56, and 100 nm. The insets show the probability of contact for the tip as a function of the radial distance r. (D) Examples of contact points (black pixels) in comparison to contact area on a flat surface (red circles). (E) Contact radii a determined from a fit of the transfer functions (see text) to the Airy pattern for ai = 3, 5, 7, and 9 nm (dotted lines). The black line denotes a contact depth d equal to the RMS value of the surface roughness σ. Shaded areas denote the range of values observed in the experiments. (F) For d/σ < 1, the estimated contact radius a deviates significantly from ai.

transform of a convolution of two images in real space can be expressed as the pointwise multiplication of the Fourier transforms of the images. If F denotes the 2D Fourier transform operator, we can write

polymer is much smaller than that of silicon, and we can assume that only the polymer deforms. Within the contact area Ac, the polymer deforms according to the shape of the tip. For a spherical tip, the depth of direct contact d is given by d = R − (R2 − a2)1/2 (see Figure 1D). The topography image is determined by recording the height of the tip Δz relative to the average position of the surface z = 0 (see Figure 1E). The surface roughness h(x) is the source of Δz and also modifies the contact between tip and surface a, δ = f C(R, P, ...), described by a contact function f C on a flat surface. As starting point we assume that the high q (q > π/a) surface roughness components are averaged across the contact area and produce negligible net average pressure such that a, δ = f C approximately holds. Even then, the vertical displacement Δz between tip and sample is still influenced by the long wavelength roughness components. The interference of the surface waves leads to a localized accumulation or depletion of material given by the mean height h̅ within Ac (see Figure 1E). The tip establishes the contact to this locally elevated surface. Thus, Δz at the tip position xt is equal to the average height of the sample surface h̅ across the contact area Ac at position xt: Δz(x t) ≃ h ̅ =

F {h∗w} = F {h} ·F {ζ }

where ∗ and · denote the convolution and the pointwise multiplication, respectively. The PSD can be expressed as the square of the Fourier transform. Thus, its attenuation due to the convoluting contact area is the normalized square of the Fourier transform of ζ. For the simple case of a spherical tip (see Figure 1E), the contact area is a disk of radius a; i.e., ζ(r) = 1 for r ≤ a and ζ(r) = 0 for r > a. In this case, the squared modulus of the Fourier transform of ζ is the so-called Airy pattern well known from the light diffraction of a spherical aperture and has a radial component as a function of the wave vector q of D(q): ⎛ 2J (qa) ⎞2 ⎟ D(q) = ⎜ 1 ⎝ qa ⎠

∫A ,x h(x) dx = ∫∞ ζ(x − x t)h(x) dx c

(3)

(4)

where J1(x) is the Bessel function of the first kind of order one. The PSD spectrum of the resulting image Gi is given by the square of its Fourier transforms combining eqs 1, 3, and 4:

t

(2)

where ζ(x) = 1 if x lies within Ac and ζ(x) = 0 otherwise. Equation 2 describes a convolution of the surface roughness with a function defined by the extent of the contact area Ac. In general, the convolution theorem states that the Fourier

Gi(q) = 13960

2 4kBT J1(qa)

γq2 (qa)2

(5)

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Equation 5 has been used to fit the data shown in Figure 1C. It fits the data with high accuracy for a filter attenuation of more than 1 order of magnitude. However, the amplitudes at higher attenuation cannot be reproduced. In particular, the characteristic fringes of the quasi-periodic Bessel function are not apparent in the measured data. In a more realistic scenario, the modulation of the tip height due to the surface roughness will locally also modify the contact area AC. In particular for low contact forces, d becomes comparable to the root-mean-square (RMS) value of the surface roughness σ ≈ 0.4 nm. In this case, a height modulation of the tip on the order of the surface roughness will affect AC locally and will lead to a filter function that deviates from the pure Airy behavior. However, as the height modulation depends on the cumulative action of the surface waves, it cannot easily be separated and analyzed in a plane-wave expansion. Instead, we use a simulation of the scanning process of a spherical tip on a simulated polymer surface. Simulation. We generate simulated roughness images assuming thermally excited capillary waves as the only source of roughness. The ingredients are roughness amplitudes which on average depend on the wave vector q according to eq 1. Furthermore, because the capillary waves are thermally excited, the distribution of roughness amplitudes follows the Boltzmann distribution, i.e., A(q,ϕ) ∝ exp(−EA/kBT), where EA = 1 /2⟨Aq2⟩γq2 is the average energy of a surface mode.19 With this information and using random phases for the waves, a twodimensional Fourier spectrum of the roughness amplitudes is synthesized. The simulated topography is obtained from an inverse Fourier transformation. Figure 2A depicts such a synthesized topography in the top right corner. It is compared with an experimental image on the left and a simulated scanned image on the bottom right of the same panel. To simulate the tip imaging process, a linear elastic model closely related to the so-called elastic foundation model12 was employed. In this model, the sample is viewed as being composed of vertical springs supporting the surface (see Figure 2B). A local deformation of one of the springs does not influence the neighboring spring. It has been shown12 that the constants of the model can be renormalized to yield the right elastic constants for a Hertzian contact and that the error for the contact radius a is less than 7%. Here we use the model to simulate the vertical motion of the scanning tip. On a planar surface an applied load P0 deforms the sample, giving rise to a finite contact radius ai and contact depth d. The applied load on the tip is compensated by the accumulated action of the deflected springs, which is proportional to the displaced volume V. For a given contact radius ai on a flat substrate the displaced volume is V = πd2(3R − d)/3. For rough surfaces, we assume that V is the same as on a flat surface and stays constant at each tip position. A simulated roughness image is created by virtually moving the tip from one pixel position to the next, calculating the distance between tip and sample surface and determining the shift Δz in tip height that is needed to keep V constant. The relative tip height Δz as a function of the pixel position yields the simulated scanned topography. The model implies that we extend the first simple approach to a description in which the contact area does not need to be well-defined. Thus, we can study the effect of a spatially varying contact area and even contact areas which are split in multiple point-like contacts to surface asperities. Since the elastic model is linear, the volume constraint we use implies that we still take the average value of the surface roughness within this spatially

varying contact area to determine the height of the tip. Thus, the simulation still neglects the nonlinear deformation behavior of the roughness asperities. For obtaining meaningful spectra from the simulation, we simulate surfaces consisting of 2000 × 2000 pixels with a pixel size of 0.5 nm. Δz has to be determined 4 million times. The computation takes about 1 h on a standard desktop computer for one image. A more appropriate solution of the contact problem could be obtained using finite element methods; however, this approach would be computationally much more expensive. We take the linear model as a useful working compromise. Simulations were performed for tip radii R of 10, 18, 32, 56, and 100 nm and input contact radii ai of 3, 5, 7, and 9 nm. After simulation, the PSD of the simulated scanned image was obtained and divided by the PSD of the simulated surface. This yields the transfer function of the filter created by the scanning tip. The simulation was performed for two values of the RMS surface roughness of 0.44 pm and 0.44 nm. The latter value was observed in the experiments, while the first corresponds to the idealized case of small roughness amplitudes discussed above. For this case, pure Airy patterns were obtained for all input parameters in the simulation (data not shown), which corroborates our interpretation for low surface roughness values described above. For the higher value of the surface roughness two sets of the obtained filter spectra are shown in Figure 2C for a contact radius of ai = 9 nm (top) and ai = 3 nm (bottom). Corresponding exemplary contacts from the same tip position in the simulations are shown in Figure 2D. The spectra deviate little from the Airy pattern for the contact radius ai = 9 nm and low q values. For high q, however, the Airy fringes are replaced by a power-law decay. This effect becomes stronger with increasing tip radius, corresponding to a decreasing tip penetration d. For a tip radius of R = 100 nm, the shape of the filter has changed to a monotonously decreasing function. At the same time the probability p(r) of the tip being in contact at a radial position r drops below 1 at r = 0 (see insets in Figure 2C). Also, the type of contact (see Figure 2D) changes from a well-defined circular contact for R = 10 nm to a noncircular contact with additional contact points outside the contact area on a flat surface for R = 100 nm. For the smaller contact radius, ai = 3 nm, d is reduced and is typically comparable to or less than the RMS value of the roughness σ ≈ 0.44 nm (see black curve in Figure 2E). Only for R = 10 nm does a small dip remain at the position of the first zero of the Airy pattern. All other curves show a power law decay until the noise level of the simulation is reached. The onset of the initial drop of the curves shifts to lower q values with increasing R. At the same time the contact changes from a contact confined approximately to the contact area on a flat surface for R = 10 nm into multiple point-like contacts spread over a much larger area for R = 100 nm (see lower panels in Figure 2D). Irrespective of the change in the curves at higher q values, the shape of the initial drop up to an attenuation by 1 order of magnitude is conserved, and this range of the transfer function can be fitted with high accuracy to the Airy pattern given by eq 4. Figure 2E shows the result of the fit for the single free parameter, the apparent contact radius a, in comparison with the input contact radius ai (dotted lines) determined on a flat surface. For small tip radii and high values of ai, the agreement of a and ai is better than 10%. For small ai and large R, the contact radii a obtained deviate significantly from the input 13961

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values ai. This is the case when the penetration depth of the tip on a flat surface is less than the magnitude of the surface roughness. In fact, we obtain a master curve if we plot a/ai vs d/ σ (see Figure 2F). The black curve in Figure 2E marks the parameter pair of a and R when d = σ = 0.44 nm. The simulations suggest that the height modulation of the tip due to the surface roughness leads to a suppression of the higher-order fringes of the Airy pattern transfer function, which are gradually replaced by a power-law decay. The contact radius can be recovered from a roughness image if the penetration depth of the tip is greater than the RMS value of the surface roughness.

law spectrum, and there is no sign of fringes similar to those obtained in the simulations. We will return to this point in the Discussion section below. For now, we adapt our fitting function from eq 5 to a truncated Airy pattern, which is extended by a power-law decay in a continuous and differentiable manner: Gi(q ≤ qc) =

2 4kBT J1(qa)

γq2 (qa)2

(6)

Gi(q > qc) = Gi(qc)(q/qc)s

■

(7)

where s is the exponent of the power-law function and qc is the lowest q value at which the derivative of eq 6 equals the doublelogarithmic slope s. The data sets were analyzed by fitting a total roughness power amplitude Gtot = Gi + Gnoise, where the amplitudes Gi are given by eqs 6 and 7 and Gnoise is a fixed white noise level of Gnoise = 2 × 10−2 nm4, caused by the electronic noise in the setup.19 From a fit to the average of all curves, the surface energy gamma was determined to be 26 mN m−1 at 400 °C, the curing temperature of the PAEK polymers. The fit parameters used for the individual curves were the contact radius a and the exponent s. The quality of the fit is excellent for all curves obtained (see Figure 3A). The crossover at qc from Airy to power law decay is marked by the dashed line in Figure 3A. A constant slope of s = −5.2 ± 0.4 was observed for all curves. The resulting values for a are plotted against the load P in Figure 3B. The contact radius values corresponding to the PSD spectra shown in Figure 3A are marked in Figure 3B using open symbols. The standard error of a as obtained from the fit was smaller than the marker size in Figure 3B. The contact radius a increases monotonically with increasing contact load, as expected. The curve levels off at a load of about 4 nN, corresponding to a contact radius of 10.3 ± 0.2 nm. This value is in good agreement with the overall radius of the tip as determined by SEM imaging. The inset in Figure 3B shows the SEM micrograph, and we can estimate an tip-apex radius of 11 ± 1 nm (indicated by the circle in the inset). Thus, we may infer that the constant value of the contact radius observed at high forces is due to the geometry of the tip. At lower loads, the data can be fitted using a JKR contact model. The JKR model predicts the following scaling of the contact radius:28

EXPERIMENTS To test the concept, two sets of experiments were performed. In the first set, the surface roughness of a PAEK-21 sample was measured as a function of applied load. The scan size was 1.6 × 1.6 μm using 1800 × 1800 pixels for imaging. In total, 19 images were acquired using a cantilever with a spring constant of k ≈ 0.1 N/m and applied forces ranging from −4.4 to 9.7 nN. At applied loads of less than −4.4 nN, the adhesion was insufficient to maintain contact between tip and sample during imaging. Figure 3A depicts the PSD curves for four selected images recorded at −3.4, −2.4, 0, and 4.8 nN. The effect of the higher load is clearly apparent in the spectral curves: for higher loads the curves shift to lower q values, as expected. All curves show a smooth crossover to a power-law decay in the power

a3K = P + 3πwR + R

6πwRP + (3πwR )2 −1

(8)

K is an effective elastic modulus defined by K = /4(1 − ν )/ E, where ν is the Poisson ratio, E is Young’s modulus, and w is the energy of adhesion. The red curve in Figure 3B corresponds to the best fit to that model using a fixed tip radius R = 11 nm. The value for w = 79 ± 2 mN/m is as expected for a contact of a silicon tip and a polymer.23 Assuming a Poisson ratio of 0.35, typical for amorphous polymers, the value for K = 0.25 ± 0.02 GPa suggests a Young’s modulus of E ≃ 0.17 GPa. A second set of experiments monitored the change in the contact parameters obtained from the roughness measurements of a PAEK-18 film as a function of the travel distance due to tip wear. To speed up the tip-wear rate, we split the experiment into series of five scans at zero applied load to determine the contact radius at zero load (see Figure 1B as example), followed by an exponentially increasing number of scans at a higher load of 8 nN. In total, 379 images were obtained and analyzed: 312

Figure 3. Roughness analysis as a function of applied load. (A) Selected PSD spectra from images recorded with applied forces of −3.4, −2.4, 0, and 4.8 nN. Red curves mark fits according to Airy/ power-law model. The dashed black line mark the crossover at qc from the Airy pattern to the power law. (B) Contact radii determined from the PSD analysis for applied loads P ranging from −4.4 to 9.7 nN. Open symbols mark the contact radii for which PSD spectra are shown in panel A. A fit to the initial rise of the curve according to the JKR contact model is marked by the solid line. At loads of about 5 nN, the curve levels off at a contact radius of ≈10.3 nm, corresponding well to the tip radius of 11 nm as determined by SEM inspection (see inset). 13962

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The observed behavior of ah and al with the travel distance is interpreted using the JKR model. Initially, the tip radius is too small to be supported by the soft polymer: The tip sinks into the polymer by more than the radius of the tip. In this case the JKR model is not valid because it predicts a contact radius larger than the tip radius. In other words, the experimentally measured contact radius is determined by the tip radius. Initially, this is true for the high and the low loads, and therefore the same contact radius is measured for ah and al. During the experiment, the tip radius increases, and ah and al begin to separate because the contact radius at low load al is no longer limited by the tip radius. At a travel distance of ≈30 mm, also ah is no longer limited by R. For longer travel distances the predicted values for ah are in good agreement with the measured data. As in the first part of the experiment, the adhesion related to each image was measured by obtaining 4 force−distance curves and measuring the distance between the snap-in and snap-out position. The adhesion was determined by the average distance multiplied by the spring constant k = 0.083 N/m of the lever. The values are plotted in Figure 4B as a function of the contact radius. For comparison, using the parameters of the JKR fit, the adhesion can be obtained from the contact radius at zero load as a03 = 6πwR2/K, yielding a adhesion force of Fadh = 3/2πRw. The red line plotted in Figure 4D shows this relation and the data obtained. Similar to the result obtained before, the adhesion is always somewhat higher than predicted by the JKR model. However, the slope of the curve representing the power law with an exponent of 1.5 is well reproduced.

at high and 67 at zero load. A selection of the spectra obtained is shown in Figure 4A, namely, 4 spectra at high load after 2, 15,

Figure 4. Roughness analysis as a function of travel distance. (A) PSD spectra for 4 exemplary images obtained at zero load (light blue dots) and 8 nN load (black dots). Fits of a Airy/power-law model using a noise level of 1.5 × 10−3 nm4 to the data are shown as blue and red lines, respectively. (B) Contact radii determined from the PSD analysis at zero load (light blue dots) and 8 nN load (black dots). Open symbols mark data points for which spectra are shown in panel A. (C) SEM micrographs of the tip before and after the experiment, shown at the same scale. The tip-apex radius increased approximately from 5.5 to 16 nm as marked by the circles. (D) Adhesion as a function of contact radius. The adhesion values measured are ≈50% higher than predicted by the JKR model.

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DISCUSSION The simulations show that the roughness induced tip-height modulation leads to a modulation of the contact area and therefore to a local variation of the filter function. As a consequence, the higher-order fringes of the Airy pattern are smoothed and gradually replaced by a monotonously decreasing power-law function. For q-values lower than the first-order fringe at q ≤ π/a, the effect of the tip height modulation on the roughness spectrum depends on the penetration depth of the tip. If the penetration depth is less than the RMS value of the surface roughness, the onset of the attenuation in the spectra drops to lower q values than expected from the Airy model. As a consequence, a simple fit to a (truncated) Airy pattern results in an overestimate of the contact radius. Accordingly, this regime relies on the contact between a few surface roughness asperities and the tip, which are distributed on a much larger area than the contact area on a flat surface (see Figure 2D). Here the elastic deformation is small, and the regime is more closely related to the tip deconvolution problem of contacts between hard materials which has been treated by Lechenault et al.17 For penetrations of the tip larger than approximately the RMS value of the surface roughness, the transfer function is much less affected for wave numbers up to q ≈ π/a. This q range can be fitted with a truncated Airy pattern, and the fit reproduces the contact radius with an accuracy of better than 5%. Assuming spherical tip ends in the experiments, we find that all the experiments were done in this regime (see shaded areas in Figure 2E). For the parameters of the experiments, the simulations predict a well-defined contact area and a corresponding pronounced dip in the PSD at the position of the first Airy

67, and 311 wear cycles and 4 spectra at zero load after 0, 1, 7, and 212 wear cycles. The imaged area was 500 × 500 nm2 in size and sampled with 1800 × 1800 pixels. Because of the smaller scan size, the apparent electronic noise level is lower19 than for the data shown in Figure 3. The noise level for the fit was set to 1.05 × 10−2 nm4. The quality of the fit was excellent for all spectra, and the standard error of a obtained from the fit is similar to or smaller than the size of the symbols in the plot. Figure 4B plots the contact radius obtained from the analysis as a function of the travel distance. Initially, the contact radii at high (ah) and low load (al) are at the same level of ah = al ≈ 5 nm. Up to a travel distance of 30 mm, ah and al diverge. For higher travel distances, the gap between the radii reduces again on the logarithmic scale. The data for al approximately follows a power law with an exponent of 0.11 (see red line in Figure 4B). Again, we apply the JKR framework to analyze the data. Because the chemical natures of both polymers are similar, we assume the same work of adhesion of w = 79 mN/m. The tip radii obtained from SEM micrographs shown in Figure 4C before (inset) and after the experiment are ≈5.5 and 16 nm, respectively. Good agreement of the JKR model with the data can be reached by fixing the elastic modulus to K = 0.39 GPa, which is the only free parameter left. This value is slightly higher than that of the PAEK-21 polymer. We attribute this to batch to batch variations in the synthesis of the polymers. 13963

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area. Thus, as long as the contact area is well-defined, the error of the determined contact radii remains small. The experimental results obtained in this study corroborate this interpretation. The JKR model can be used to fit the load dependence of the contact radius with high accuracy. Also, the constant value of the contact radius at higher loads is consistent with the diameter of the tip as determined by SEM. The JKR fit was done assuming that the radius of the tip as determined by the SEM inspection is 11 nm. The model then yields values for the effective elastic modulus K and the work of adhesion w. The values for w are in the range of what can be expected for contact of silicon (oxide) tips and polymers.23,30 At first glance the value for K seems to be low for a glassy amorphous polymer, for which typical values of Young’s modulus are 3−6 GPa. However, one needs to consider that we probe the elastic constant in a very thin surface layer. Its thickness is given by the depth penetration of the stress field into the elastic medium, which is comparable to the contact radius of the tip of about 10 nm.12 It is known that for polymers the elastic modulus at the surface may be reduced by several orders of magnitude.31,32 In particular, polymers with a highly flexible backbone exhibit this effect.33 The polymer used here was specifically tuned to have a low glass transition temperature of about 150 °C at extremely high cross-link densities.18 To achieve this, highly flexible double-resourcinol units have been integrated into the backbone of the polymer. Therefore, it is not surprising that with such a backbone architecture also the elastic properties at the free surface are strongly modified. Similarly, in the tip-wear experiments we achieve excellent agreement of the tip parameters measured in the SEM before and after the experiment and the increasing values of the contact radii obtained at high and low load. The JKR model can be used to explain the observed behavior of the contact radii measured. Again, a low value for the elastic modulus is obtained. We can use the parameters obtained to test the validity of the contact model. One measure is Maugis’ parameter λ = 2σ0/(π wK2/R)1/3, where σ0 = 1.03w/Z0 is the theoretical stress and Z0 is the equilibrium distance between atoms.28 For the range of tip radii and elastic constants investigated here, we get λ = 5−9, which confirms that the JKR model is appropriate. The JKR model can also be used to predict the adhesion of the system. The values obtained from force−distance curves exhibit the expected scaling behavior but are shifted to higher values, possibly because of viscoelastic effects on the polymer surface. From the above considerations, we conjecture that the analysis of the imaged surface roughness provides a good estimate of the contact radius on soft surfaces. Using the relation obtained in the simulations for the tip penetration necessary, we can estimate an upper bound for the elastic properties of the polymer for which the method is valid. For a typical contact-mode scanning experiment, we assume a low contact force of about P = 5 nN (including adhesion), a tip of R = 5 nm radius, and a surface roughness of σ = 0.4 nm. For the contact depth to be equal to the roughness, we arrive at a contact radius of a = 2 nm. Such a contact radius is predicted by the JKR and DMT contact models for Young’s moduli of up to Emax ≈ 6−7 GPa, according to the JKR model, and Emax ≈ 3−4 GPa, according to the DMT model.28 Similarly, for Greenwood and Tripp’s model the crossover to asperities controlled mechanics occurs at an elastic modulus of Emax ≈ 6 GPa. Thus, the method is valid on most amorphous polymers.

fringe. Such a dip, however, could not be observed in the experiments. In the following we argue that the linear nature of the simulation leads to an overexposure of the Airy fringes, in particular at high attenuations. To see this, we reconsider eq 2 for the more general case of roughness amplitudes influencing the contact but keeping the contact area defined, i.e. d > σ; we can separate the surface roughness within the contact area Ac into an average height h̅ across the contact area and a fluctuation around this value Δh: h = h ̅ + Δh

(9)

The mean height h̅ (see also eq 2) corresponds to a shift of the mean surface zero position within the contact area with respect to the overall mean value of the surface roughness. Thus, h̅ directly acts on the tip height Δz independent of the contact model. Furthermore, h̅ is composed mainly of roughness waves with low q < π/a wave vectors (according to the Airy pattern), and therefore the initial drop of the Airy pattern is conserved as long as Ac is well-defined. Δh is mainly composed of waves with q > π/a and describes the asperities formed in the contact area. Because we use the displaced volume V as a conserved quantity, the linear model averages Δh to zero across the contact area. Thus, the effect of Δh on Δz is clearly underestimated, and the Airy fringes are overexposed in the simulation. Any nonlinear deformation effect of the asperities would lead to deviations in Δz and contributes amplitude modulations to the PSD. The deviations are expected to remain small because the nonlinear deformation behavior of the asperities is averaged accross the contact area. Thus, predominantly the highly attenuated amplitudes at wave vectors q > π/a are significantly affected. This contribution leads to a reduction of the sharp features of the Airy pattern but does not significantly modify the behavior of the high amplitudes in the PSD for q < π/a. The elevated noise floor along the y-axis has similar amplitudes of ≈10−2 nm4 than the amplitude of the first Airy fringe (see Figure 1C). This noncoherent noise floor screens the signals at high attenuations and masks the Airy fringes. From these considerations we conclude that the effect of the Airy fringes is weaker than predicted from the linear model and hidden by noise in the experiments. This reasoning is corroborated when considering previous work describing the contact to rough surfaces. It has been shown in the pioneering work of Greenwood and Tripp29 that the nonlinear deformation characteristics of the asperities can be neglected (i.e., bulk deformation dominates), when the dimensionless load T exceeds T = 2P/[σE(2Rσ)1/2] > 2. For the values obtained in the experiments, i.e. K = 0.39 GPa, σ = 0.44 nm, R < 16 nm, and using the adhesion force as minimal load, P > 4 nN, we get T ≥ 20, which is well in the bulkdominated mode. Also for Greenwood and Tripp’s model, we can establish the relation to parameters on a flat surface: The model by Greenwood and Tripp assumes Hertzian contact mechanics. If we replace the load P by the Hertzian relation P = (4/3)(a3E/R) and also a = (R/δ)1/2, we obtain T = (4√2/ 3)(δ/σ)3/2. Thus, the limit for observing bulk deformation is given by δ/σ > 1.5, similar to the value obtained from the simulations. Note that also Greenwood’s result depends only on the ration of δ and σ and thus is independent of the asperity geometry. The good agreement between the simulations and the results by Greenwood and Tripp suggests that the source of the observed shift in the PSD at q < π/a for weak tip penetrations and is given by the increased apparent contact 13964

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CONCLUSION Imaging the surface roughness of (soft) glassy polymer films with nanometer-scale tips in contact mode can be used to determine the contact radius of the tips at the imaging conditions. The attenuation of the roughness amplitudes by the imaging tip with a finite tip radius is described by an Airypattern behavior at low spatial frequencies and a power-law behavior at higher spatial frequencies. The surface roughness causes a local variation of the contact area, which leads to this transition. A reliable measure of the contact area on flat surfaces is obtained if the penetration depth of the tip into the material is larger than the RMS value of the surface roughness. This condition is satisfied for typical imaging parameters and typical material properties of glassy amorphous polymers. The validity of the result is supported by two experiments in which the contact radius is obtained as a function of load and travel distance. Both experiments predict contact radii that are consistent with the tip radii measured ex situ and the JKR contact model. In combination with the JKR model, the method establishes a direct link to the material parameters which can be assessed quantitatively. Dynamic mode AFM is typically used to image soft surfaces to minimize material wear. In this case, the contact radius obtained from the PSD is more difficult to interpret and would constitute a weighted average of the contact over a tiposcillation cycle. In general, the method enables one to determine the contact radius on the nanometer scale as a function of conditions applied. This has direct application in polymer-based scanning probe microscopy and lithography techniques34,35 for monitoring the state of the tip in the instrument. Furthermore, the method provides a calibration scheme enabling a variety of quantitative AFM measurements. Testing mechanical material properties and mechanical contact models has been demonstrated here and could be expanded by simultaneously acquiring the penetration depth of the tip into the sample. Other measurements on polymers include heat or current transport into (conductive) polymeric samples, friction force microscopy, quantitative tip wear, etc.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The author thanks C. Rawlings, U. Duerig, and B. Gotsmann for extended discussions and fruitful ideas. I gratefully acknowledge the invaluable support of the probe storage team at the IBM Research - Zurich. I thank Russell Pratt and James Hedrick from IBM Research - Almaden for the synthesis and characterization of the polymer materials, Ute Drechsler for the preparation of samples and cantilevers, and Michel Despont and Walter Riess for their support.

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