Article pubs.acs.org/Langmuir
Study of Adhesion and Friction Properties on a Nanoparticle Gradient Surface: Transition from JKR to DMT Contact Mechanics Shivaprakash N. Ramakrishna, Prathima C. Nalam, Lucy Y. Clasohm, and Nicholas D. Spencer* Laboratory for Surface Science and Technology, Department of Materials, ETH Zurich, Wolfgang-Pauli-Strasse 10, 8093 Zurich, Switzerland ABSTRACT: We have previously investigated the dependence of adhesion on nanometer-scale surface roughness by employing a roughness gradient. In this study, we correlate the obtained adhesion forces on nanometer-scale rough surfaces to their frictional properties. A roughness gradient with varying silica particle (diameter ≈ 12 nm) density was prepared, and adhesion and frictional forces were measured across the gradient surface in perfluorodecalin by means of atomic force microscopy with a polyethylene colloidal probe. Similarly to the pull-off measurements, the frictional forces initially showed a reduction with decreasing particle density and later an abrupt increase as the colloidal sphere began to touch the flat substrate beneath, at very low particle densities. The friction−load relation is found to depend on the real contact area (Areal) between the colloid probe and the underlying particles. At high particle density, the colloidal sphere undergoes large deformations over several nanoparticles, and the contact adhesion (JKR type) dominates the frictional response. However, at low particle density (before the colloidal probe is in contact with the underlying surface), the colloidal sphere is suspended by a few particles only, resulting in local deformations of the colloid sphere, with the frictional response to the applied load being dominated by long-range, noncontact (DMT-type) interactions with the substrate beneath.
1. INTRODUCTION Surface roughness and interfacial adhesion are two major factors that affect friction. The effect of adhesion in terms of surface energy and the effect of roughness in terms of the real contact area1−4 on the frictional properties of engineering systems are especially enhanced at reduced scales such as for micro/nanoelectro-mechanical systems (MEMS/NEMS). Because of the increase in the surface-to-volume ratio of these devices, an estimation of adhesion and frictional forces and a thorough understanding of their dependence on the applied load and real contact area are necessary to optimize the mechanical performance.5−12 This understanding, however, is still very incomplete. To eliminate stiction effects, nanostructured surfaces are employed in MEMS and robotic devices to tune both the interfacial friction and adhesion. Studies conducted on textured surfaces4,13,14 have shown that the friction at the interface is influenced by the geometry and distribution of the features and the material properties at the interface. Friction and adhesion properties have been observed to vary differently when the scale of measurement changes from the macroscale to the nanoscale,15−17 and the influence of adhesion on friction properties18,19 at different length scales has been discussed over the last few decades. On a continuum scale, the classical Amontons’ law20 predicts the frictional force between two macroscopic sliding objects to be directly proportional to the applied load and independent of the apparent contact area. However, on rough surfaces Bowden and Tabor21 showed that the force required to overcome the static friction between two © 2012 American Chemical Society
sliding surfaces is strongly dependent on the real area of contact, which occurs at micrometer/submicrometer scale asperities. Later, the Greenwood−Williamson model,22 extended by Fuller and Tabor,23 takes the random distribution of asperity heights (Gaussian) into account to estimate the total contact area, contact load and their dependence on adhesion. Adhesive contact between elastic surfaces is usually described by single-asperity theories such as Johnson−Kendall−Roberts (JKR)24 or Derjaguin−Mü ller−Toporov (DMT).25 JKR continuum theory applies to materials with a high surface energy that undergo large elastic deformations when in contact. Strong, short-range adhesion forces dominate the surface interactions, and the effect of adhesion is included within the contact zone. At the other extreme, the DMT model is applied to materials with low surface energy, which are stiff and resistant to deformation upon contact. In this case, the adhesion is caused by the presence of weak, long-range attractive forces felt outside the contact zone. The behavior of materials between the JKR and DMT extremes can be described by parameters such as Tabor’s dimensionless parameter26 or the Maugis transition parameter.27 Nanoscale, adhesion-induced friction has been discussed extensively in the literature.28,29 Surface chemistry and surface roughness30,31 are known to influence the interfacial properties. Under low-adhesion conditions, the measured friction shows a Received: August 16, 2012 Revised: December 9, 2012 Published: December 10, 2012 175
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Figure 1. AFM topographical images (2.5 μm × 2.5 μm) taken at a few positions along the length of the silica wafer covered with silica nanoparticles. Images are stitched together to represent the gradient in the particle density. The sample is prepared by the immersion method where the rough end corresponds to the high particle density and the smooth end corresponds to the low particle density. The number below the image corresponds to the particle density (average of the three measurements). The height of the silica nanoparticles is obtained as 6 ± 2 nm. dried in a stream of nitrogen gas and was then immediately transferred to a furnace. The temperature inside the furnace was ramped to 1080 °C and held at this temperature for 2 h before it was cooled to room temperature at a constant rate. This heating step not only burns away the PEI from the surface but because the sample was held a few degrees below the glass-transition temperature of silica the nanoparticles sinter to the SiO2 on the wafer, becoming firmly attached and thus resistant to removal during subsequent friction measurements. Topographical images were taken along the length of the sample by means of AFM (Dimension 300, Veeco, Digital Instruments (cantilever from Olympus, Japan, AC160, KN ≈ 36 N/m, f ≈ 300 kHz), and the particle heights (after sintering) and particle densities were measured along the gradient. Figure 1 shows the AFM images (2.5 μm × 2.5 μm) of silica nanoparticles along the gradient. 2.2. Adhesion and Frictional Force Measurements. Adhesion and friction measurements were carried out by means of atomic force microscopy (MFP3D, Asylum Research, Santa Barbara, CA). Measurements were performed under perfluorodecalin (Aldrich, purity 95%). This nonpolar, low-refractive-index (1.313) liquid with a dielectric constant of 1.8 effectively amplifies the van der Waals forces between the probe and the sample.31,34 A polyethylene microsphere with a diameter of 18 μm (Kobo Products, Inc. South Plainfield, N.J., USA) was glued to a tipless cantilever (CSC12, Micromasch, Tallinn, Estonia) with Araldite glue using a home-built micromanipulator. The normal spring constant calibration of the cantilever was carried out by the thermal noise method35 prior to the attachment of the colloidal sphere. The torsional spring constant was measured using Sader’s method36 with Sader’s online calibration applet.37 The lateral sensitivity (SL) of the cantilever (probe) was obtained using the test-probe method, as described by Cannara et al.38 A silicon wafer was cut along its ⟨100⟩ crystal plane and glued to a glass slide such that the smooth edge of the wafer (along the crystal plane) was used as a vertical wall to twist the cantilever laterally. The lateral sensitivity for a cantilever (test) with a colloid probe of diameter 35 μm (a diameter larger than the width of the cantilever) was obtained using the lateral deflection versus piezo distance curve. The lateral sensitivity for the probe cantilever was deduced from the obtained lateral sensitivity value of the test cantilever, as described in detail by Cannara et al. The normal and torsional spring constants of the cantilever were determined to be KN = 0.05 N/m and KT = 1.45 × 10−9 N·m, respectively. The torsional spring constant was converted to the lateral spring constant by means of the relation KL = KT/h2, where h is the torsional arm length (equal to the sum of the diameter of the sphere and half the thickness of the cantilever beam). The measured friction signal was converted to force with a conversion factor (α) defined as KL/SL.38 Before acquisition of the data, the PE colloidal probe was rubbed against a smooth silicon surface for about 20 min at high load (40 nN) until the micro/nanoasperities on the colloidal probe were worn out and consistent friction loops were obtained. The very low surface roughness on the colloidal sphere after this running-in process was
linear dependence on the applied load, whereas when tip− surface adhesion becomes significant the friction shows a nonlinear dependence on the applied load. According to Gao29 et al., the adhesive strength at the interface is a function of the number of molecular interactions, which can be estimated from the real contact area between the probe and the surface. They showed that adhesive (JKR) sliding between two smooth surfaces undergoes a transition to Amontons’ behavior when the surfaces are damaged (and therefore develop multiple asperities). Mo32 et al. showed a linear-to-sublinear transition, with an increase in adhesion for a multiasperity nanoscale contact. The contact area is defined to be proportional to the number of interacting atoms. Recently, the Leggett31 group studied the influence of solvent on the interaction energies between contacts covered with a monolayer. They observed that the friction at the interface arises from the shearing of dispersion interactions. They noticed the linear dependency of friction with applied load for a highly solvated film. As for nonsolvated films, the friction varies with the applied load in accordance with JKR or DMT contact, depending on the strength of adhesion. In this article, the effect of nanoparticle density (i.e., number of contacts) on adhesion and friction properties is studied by means of a polyethylene colloid probe and a silica nanoparticle density gradient substrate under perfluorodecalin. A direct correlation between adhesion and frictional forces as a function of applied load across different particle densities is observed. A simple multiasperity model (including adhesion) is developed to estimate the real contact area at the interface for different particle densities, and this was compared to the experimental data. A transition from JKR- to DMT-type adhesion between the particles and colloid probe was observed while moving from the high-particle-density end to the low-particle-density end of the gradient.
2. MATERIALS AND METHODS 2.1. Preparation and Characterization of Nanoparticle Roughness Gradient. The silica-nanoparticle density gradients were prepared by the immersion method.33 Oxidized silicon wafers were rendered positively charged by the adsorption of poly(ethylene imine) (PEI). The PEI-coated silicon wafer was mounted on a linearmotion drive and gradually immersed in a silica nanoparticle (diameter ≈ 12.1 nm, CV < 15%, Microspheres-Nanospheres, Cold Spring, New York, USA) suspension (0.002 wt %) in deionized water. By slowly dipping the wafer into the suspension (i.e., varying the immersion time in the nanoparticle suspension), a gradual variation of silica nanoparticle density along the sample was obtained. The sample was 176
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μm2), high adhesion with a large standard deviation is obtained. Adhesive forces decrease while moving toward the smooth end of the gradient as a result of the decrease in the real contact area between the colloid probe and the underlying particles, as previously discussed.6 However, the lowest adhesion is not obtained at the smooth end of the gradient but at an intermediate particle density (at around 230−245/μm2), at which the PE sphere is supported by the minimum number of nanoparticles. At very low particle densities, the colloid sphere comes into contact with the underlying silicon wafer substrate, resulting in increased pull-off forces with high standard deviations (below 150/μm2 particle density). At the very smooth end (no particles on the silicon wafer surface), the PE colloidal probe experiences very high (van der Waals) adhesion forces in the presence of perfluorodecalin. These are so strong that they cannot be measured using a soft (KN = 0.05 N/m) cantilever. Friction loops were obtained in the same region as those in which the adhesion forces were recorded. Figure 4 shows representative friction loops obtained at various particle densities. The friction loop at high particle density, such as at 450/μm2, shows perturbations (irregularities in the trace and retrace scans) that are in contrast to the regular stick−slip motion often found on rough surfaces. The nanoparticles present underneath the colloid probe cause these perturbations in the sample, and thus their frequency is high in the friction loop at a high particle density of 450/μm2 (Figure 4a) in comparison to that obtained at a particle density of 245/μm2 (Figure 4b). At a particle density of 245/μm2, the colloid is resting on the lowest possible number of nanoparticles (no contact with the underlying substrate), resulting in a reduced magnitude of the friction loop (trace−retrace). The decrease in the total contact area between the colloid and the particles at 245/μm2, in comparison to that at 450/μm2, results in lower frictional forces at the interface. At particle densities lower than 245/μm2 (e.g., 150/μm2, Figure 4c) at the load employed, the friction loop shows sharp spikes whenever the PE sphere comes in contact with the silica substrate beneath. The frictional forces obtained in this area thus have high standard deviations. Figure 4d shows the friction loop recorded at a particle density of 110/ μm2 where the PE sphere is mostly in contact with the flat silica surface and a friction loop with a high shear force is obtained. A 3D plot of frictional force as a function of load at different particle densities along the particle gradient is shown in Figure 5. The 3D friction plot shows similar behavior to that of the measured adhesion on the gradient substrate as a function of load.6 At high particle densities, high frictional forces are measured. With increasing load, the number of particles that come into contact with the colloidal probe increases, leading to an increase in the measured frictional force. A decrease in the frictional forces is observed as the particle density decreases, and a minimum frictional force is obtained at the same point on the gradient substrate where the minimum adhesion force was measured (particle density of around 245/μm2). Below this particle density, the friction starts to increase abruptly as the colloidal probe starts to come into direct contact with the substrate beneath. With increases in the load, the deformation of the colloidal probe against the flat silica substrate increases, resulting in an increase in the contact area and hence the frictional forces. 3.1. Dependence of Contact Area/Frictional Force on Applied Load for Multiple-Asperity Contact. When a sphere comes in contact with a flat surface, the resulting contact
confirmed by obtaining SEM images (Zeiss, LEO 1530) from the apex of the sphere, as shown in Figure 2. Adhesion forces were measured
Figure 2. SEM image of the surface of a polyethylene microsphere after running in for 20 min on a bare silica wafer to remove micro/ nanoasperities from the surface. The inset shows the smooth part of the colloidal probe after the asperities were removed from the contact region. across the particle gradient with the colloidal probe by obtaining force−distance curves with an approach rate of 1 Hz. About 200 force curves were obtained over an area of 20 × 20 μm2, and histograms were plotted. Adhesion forces were measured over two different areas for each particle density. A Gaussian function was used to fit the histogram, and the mean and standard deviation of the measured pulloff forces were obtained. Friction measurements were carried out in the same region where the adhesion measurements were performed. Approximately 10 friction loops were recorded at each load (both during loading/unloading) with a scan speed of 1 Hz and a scan length of 20 μm. The friction loops were analyzed using IGOR software (version 6.22A), and the average and standard deviation of the measured friction loops were estimated.
3. RESULTS AND DISCUSSION Figure 3 shows the measured adhesion forces (pull-off forces) using the PE colloid probe along the 12 nm particle gradient. At the rough end of the particle gradient (i.e., at 450 particles/
Figure 3. Adhesion force vs particle density plot along the gradient substrate. Pull-off forces were measured under perfluorodecalin using a cantilever with a normal spring constant of KN = 0.05 N/m with an attached PE sphere of 18 μm diameter. Around 200 force curves were acquired over a 20 × 20 μm2 area, and the means and standard deviations of the measured adhesion forces were estimated. 177
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Figure 4. Representative friction loops obtained along the 12 nm particle gradient substrate at particle densities of (a) 450/μm2, (b) 245/μm2, (c) 150/μm2, and (d) 110/μm2. The friction loops are taken at a normal load of 29 nN. 2 1 − ν2 2 ⎞ 4 ⎛ 1 − ν1 ⎟ K= ⎜ + 3 ⎝ E1 E2 ⎠
−1
(3)
with E1, E2 and v1, v2 being the Young’s moduli and Poisson’s ratios of the sphere and the flat surface, respectively. However, in the presence of adhesion, the effect of surface energy on the contact area has to be considered. The dependence of the contact radius on the applied load for JKR- and DMT-type contacts is given as a(JKR) =
⎛R ⎜ L + 3πRγ + ⎝K
(
6πRγL + (3πRγ )2
⎞1/3 ⎟ ⎠
)
(4)
a(DMT) = Figure 5. Dependence of the frictional force on the normal load along the 12 nm particle gradient. The friction loops were acquired in the same areas where the adhesion forces had been measured. The scan length and scan rate of the friction loop are 20 μm and 1 Hz, respectively. The normal and torsional spring constants of the cantilever are KN = 0.05 N/m and KT = 1.45 × 10−9 N m.
⎛ LR ⎞1/3 ⎜ ⎟ ⎝K ⎠
where R is the effective radius at the contact R1R 2 R= R1 + R 2
(5)
where γ is the work of adhesion and is defined as γ = 2(γ1γ2)1/2 with γ1 and γ2 being the surface energies of the sphere and flat surface, respectively. In the Hertz and DMT model, the pressure distribution across a contact, when an elastic sphere is pressed against a rigid, flat surface, is given as39
area can be estimated using single-asperity contact mechanics models such as those of Hertz,39 Johnson, Kendall, and Roberts (JKR)24 or Derjaguin, Müller, and Toporov (DMT).25 According to Hertz, in the absence of adhesion, the contact radius between an elastic sphere and a flat surface can be written as a=
⎛R ⎞1/3 ⎜ (L + 2πRγ )⎟ ⎝K ⎠
⎛ 1 − r 2 ⎞1/2 P(r ) = P0⎜ 2 ⎟ ⎝ a ⎠
(6)
In the JKR model, the pressure distribution across the contact is given as ⎛ 1 − r 2 ⎞1/2 ⎛ 1 − r 2 ⎞−1/2 P(r ) = P0⎜ 2 ⎟ + P0′⎜ 2 ⎟ ⎝ a ⎠ ⎝ a ⎠
(1)
(7)
where a is the contact radius formed by sphere on a flat surface, r is the instantaneous radius along a, P0 is the maximum Hertzian pressure acting at the center of the contact (given by eq 8), and P0′ is the negative pressure acting at the contact because of adhesion (eq 9).
(2)
and R1 and R2 are the radii of the sphere and the flat surface, L is the applied load, and K is the reduced modulus 178
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2aE* πR
⎛ 2γE* ⎞1/2 ⎟ P0′ = −⎜ ⎝ πa ⎠
Article
be noninteracting (i.e., they are considered to be independent of each other). However, when a spherical colloid probe comes in contact with the underlying nanoparticle substrate, the pressure experienced by the nanoparticles as a result of the colloid probe varies radially across the contact. Because of the presence of adhesion at the interface, both JKR and DMT pressure distributions have been considered for both cases (450/μm2 and 245/μm2). It is assumed that the contacting materials undergo only elastic deformation at the applied loads. Thus, the total contact area is a function of both the particle density and the applied load. The contact geometry of the PE sphere when compressed against the underlying nanoparticles is shown in Figure 7. The estimation of the real contact area between the PE colloid and the nanoparticles, considering adhesion at the interface, is described below.
(8)
(9)
where γ is the work of adhesion. 3 E* = K 4 The models discussed above were derived for contacts under normal compression. However, these models can also be applied to estimate frictional forces, especially for singleasperity contacts, for example, while measuring shear forces using the atomic force microscope or surface forces apparatus28,40 where frictional force can generally be assumed to be directly proportional to the contact area. (10) F = τA where F is the frictional force, A is the real contact area, and τ is the interfacial shear strength. Figure 6 shows the frictional force measured as a function of load at two particle densities: at the highest particle density
Figure 7. Geometry of the PE sphere in contact with the nanoparticles beneath. P0(PE−np) is the pressure on the nanoparticle in the center, and a is the apparent contact radius obtained by the PE sphere on a flat Si surface. Dividing the obtained apparent contact radius by the diameter of the nanoparticle yields the number of particles under the radial contact. Figure 6. Measured friction vs load plot for the 12 nm particle gradient sample at the highest particle density (450/μm2) and at the lowest particle density (245/μm2) for conditions where the colloid sphere does not come into direct contact with the substrate beneath. The solid and dotted lines are the JKR and DMT multiasperity fits to the measured frictional forces for the two particle densities. JKR (blue solid lines) and DMT (black dotted lines) models show the best fits to the experimental data for 450/μm2 and 245/μm2 particle densities, respectively.
First, the apparent contact radius (a) is estimated when the colloidal probe comes in contact with the flat surface (using both JKR and DMT, eqs 4 or 5) at each experimental applied load. The pressure distribution along the radius (a) between the PE colloidal probe and the nanoparticle placed at a distance (r) from the center is given as (by modifying eqs 6−9) ⎛ 1 − r 2 ⎞1/2 P(PE − np)(r ) = P0(PE − np)⎜ 2 ⎟ ⎝ a ⎠
(450/μm2) and at an optimum particle density (245/μm2) before the colloid sphere starts to come into contact with the silica substrate underneath. In both cases, a sublinear plot of friction as a function of load is obtained, which cannot be described by Amonton's law. At high particle density (450/ μm2), a significant frictional force was observed at zero normal load, in comparison to the behavior at a low particle density (245/μm2). However, a finite frictional force at zero normal loads is observed at both particle densities, which reflects the effect of adhesion on frictional forces. To extend these existing single-asperity contact-mechanics models to a case when a sphere comes in contact with multiple asperities, the real contact area can be estimated as the summation of contact areas formed by the PE colloid probe and all of the underlying silica nanoparticles present within the contact zone, where individual asperity contacts are assumed to
⎛ 1 − r 2 ⎞−1/2 ′ − np)⎜ 2 ⎟ + P0(PE ⎝ a ⎠
(11)
JKR contact ⎛ 1 − r 2 ⎞1/2 P(PE − np)(r ) = P0(PE − np)⎜ 2 ⎟ ⎝ a ⎠
(12)
DMT contact, where P0(PE − np) =
2a0(PE − np)E* πR (PE − np)
(13)
and 179
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⎛ 2γE* ⎞1/2 ⎟⎟ ′ − np) = −⎜⎜ P0(PE ⎝ πa0(PE − np) ⎠
(14)
P0(PE−np) and P0(PE−np) ′ are the Hertzian and the adhesive pressures on the particle that is at the center of the contact. Note that the effective radius is now defined as RPE−np obtained for the PE colloid and the nanoparticle and a0(PE−np) is the contact radius formed for each applied load when the PE colloid is pressed against a single nanoparticle placed at the center of the contact zone (calculated from eq 4 for JKR and eq 5 for DMT). By rewriting eq 13, we can obtain the contact radius between the colloidal probe and the nanoparticles underneath at a given radial distance as aPE − np(r ) =
πR (PE − np) 2E*
P(PE − np)(r )
Figure 8. Force−distance curves obtained at a high particle density (the rough end of the gradient) (450/μm2) and at a lower particle density (245/μm2). A high adhesion force is obtained at a particle density of 450/μm2, where the deformation of the colloidal probe is higher and the pull-off force is dominated by the short-range (JKRtype) contact interactions rather than the long-range noncontact forces. At a lower particle density (245/μm2), the contact adhesion with the nanoparticles is reduced and the interactions (long-range) from the underlying surface dominate the adhesion force.
(15)
Thus, the total real contact area between the colloidal probe and the underlying nanoparticles is given as A real =
2 π 3R (PE − np)ϕ
4(E*)2
a / Δr 2 ∑ P(PE − np)(r )2πri Δri i=0
(16)
Tabor’s transition parameter is used here to compare the interactions between the colloid probe and the silica surface with particle densities of 450/μm2 or 245/μm2 qualitatively. The Tabor transition parameter (μ) is a measure of the ratio of the elastic deformation of the contacts to the range of the surface force.
where ϕ is the particle density/m (calculated from AFM images) and Δr is the diameter of the nanoparticle (12 nm). The total real contact areas (Areal) were obtained for the particle densities (450/μm2 and 245/μm2) at each applied load using both JKR- and DMT-type contacts. The estimated real areas of contact are then converted to frictional forces using eq 10 and are compared to the experimentally measured data. The fit and the measured friction versus applied load data are shown in Figure 6. However, for a particle density of 450/μm2, JKR multiasperity contact mechanics (blue solid lines) and for a particle density of 245/μm2, DMT multiasperity contact mechanics (black dotted lines) show the best fits. The parameters used in the calculations are given in Table 1. 2
⎛ 16Rγ 2 ⎞1/3 μ = ⎜ 2 3⎟ ⎝ 9K z 0 ⎠
where K is the combined reduced modulus (eq 3), γ is the work of adhesion, R is the effective contact radius, and z0 is the equilibrium separation distance. For μ > 5, the JKR model is applicable, and when μ < 0.1, the DMT model is applied. If the value lies between 0.1 and 5, then it is said to be in a transition regime. For the case of a colloid sphere coming in contact with 450/ μm2 (at the rough end), the colloidal probe deforms over a large number of particles, resembling a contact situation in which a sphere is pressed against a flat surface having nanoroughness. When the PE colloid is in contact with a large number of particles, the interaction forces between the colloid probe and the particles are mainly dominated by shortrange contact forces. Thus, because of the larger elastic deformation involved (μ could potentially be >5) and the short-range (contact-type) forces, a JKR-type fit describes the situation reasonably well (Figure 9a). At low particle density (245/μm2), the colloidal sphere contacts fewer nanoparticles and therefore undergoes a local deformation at the contact. This is similar to a situation in which the PE sphere is supported by a single nanoparticle (μ value could potentially be
