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Validity of the Rumpf and the Rabinovich adhesion force models for alumina substrates with nanoscale roughness
Powder Technology 246 (2013) 545–552
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Powder Technology journal homepage: www.elsevier.com/locate/powtec
Validity of the Rumpf and the Rabinovich adhesion force models for alumina substrates with nanoscale roughness O. Laitinen b,⁎, K. Bauer a, J. Niinimäki b, U.A. Peuker a a b
Technical University Bergakademie Freiberg, Institute of Mechanical Process Engineering and Minerals Processing, Germany University of Oulu, Fibre and Particle Engineering Laboratory, Finland
a r t i c l e
i n f o
Article history: Received 4 March 2013 Received in revised form 24 May 2013 Accepted 31 May 2013 Available online 18 June 2013 Keywords: Adhesion Alumina Atomic force microscope Colloidal probe Roughness
a b s t r a c t Adhesion forces between alumina substrates and spherical alumina particles were measured using an Atomic Force Microscope (AFM) colloidal probe technique. It was shown experimentally how nanoscale roughness of alumina substrates affects AFM adhesion force measurements. The adhesion force decrease was approximately five-fold between 1.5 and 12.0 nm root mean square (rms) surface roughness. Determining the roughness, which describes the actual geometry of the investigated surfaces, as accurately as possible, is crucial for predicting interaction forces. It was proven, that using a realistic value of the nanoscale rms is one of the most important parameters for accurately predicting adhesion forces between substrates. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Interaction forces between particles and surfaces affect a lot in technical materials. The major focus of this study is the determination of surface forces between alumina particles and alumina filter materials. Since the atomic force microscope was introduced by Binning et al. [1], it has been used to directly measure interaction forces between various surfaces. The principal part of the AFM is a small silicon cantilever with a specific spring constant. This cantilever is moved slowly towards a sample surface. The deflection of the cantilever, which is determined by the interaction between the cantilever tip and the surface of the sample, is detected by the reflection of a laser beam by the backside of the cantilever. Since the deflection is a function of the interaction force and the spring constant, the force between the tip and the sample surface can be calculated. Measurements in ambient air enable the determination of the van der Waals force as the adhesion force. The van der Waals force depends mainly on the material specific Hamaker Constant, the distance between the surfaces and the roughness of the interacting surfaces. Invention and development of colloidal probe techniques have had major effects on studies of adhesion and effects of nanoscale roughness [2]. To explain experimentally observed adhesion on rough surfaces,
⁎ Corresponding author at: University of Oulu, Fibre and Particle Engineering Laboratory, P.O. Box 4300, 90014 Oulun yliopisto, Finland. Tel.: +358 503504960. E-mail address: ossi.laitinen@oulu.fi (O. Laitinen). 0032-5910/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.powtec.2013.05.051
both analytical [3–5] and computational [6,7] approaches have been used. While computational methods produce results that are comparable to experimental results [8,9], these methods are complex and not easily applied to quickly estimate adhesion for specific systems. In addition, analytical equations depend on several parameters that are easily determined from surface topography, which ensures that these equations are useful for estimating adhesion forces [10]. Measurement of surface/particle adhesion with the colloid probe technique gives lower adhesion values than expected for simple sphere/plate geometries, and this difference is attributed mainly to surface roughness. This study focuses on measuring adhesive forces for surfaces with nanoscale roughness. Adhesion force measurements were performed between very pure spherical alumina particles and polished alumina filter materials. Adhesion forces measured by AFM are compared to the modified Rumpf model and the Rabinovich model, which takes surface asperities into account. The quantitative calculation of adhesion forces between a particle and a rough surface can be difficult for many reasons. The size, shape, homogeneity, mechanical properties, and distribution of asperities (deviations from an ideal planar surface) influence the actual contact area and, therefore, directly affect the adhesion force [11]. The main target of this study was to clarify whether it is possible to measure reliable adhesion force results using a roughness profile that is as accurate as possible for the substrate surface. The Rabinovich roughness model estimated the same magnitude of adhesion forces as the experimental AFM force measurements. The modified Rumpf model estimated adhesion forces that were approximately 10 times lower than expected for the surface roughness (1.5–12.0 nm).
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2. Material & methods 2.1. Materials Samples were produced by deposition of the origin alumina filter material slurry on planar substrate. Substrates were polished in a stepwise procedure with a commercial Struers laboratory polishing machine to obtain several roughness values. Samples were cleaned, dried and mounted in the AFM. Adhesion forces were measured at points that were chosen to provide reliable adhesion results. Topographies of measurement points were scanned using the AFM in contact mode with special Contact Cantilevers TYPE 905 M-NSC36. Based on these scans, the root mean square (rms) surface roughness and surface asperities of substrates were determined. The surface points used for adhesive force measurements had rms roughness between 1.5 and 12.0 nm, and examples are presented in Fig. 1. Alumina particles (Al2O3) with 99.9975% purity from IMPRATEX were used in this study (see Fig. 2). Particle sizes were in the range from 5 to 70 μm. Alumina has a high Young's modulus of E = 370 GPa and according Götzinger and Peukert [12] plastic deformation can be neglected for adhering particles without external forces. Since the used particles were aggregates of fine alumina particles the plastic deformation of the used particles were investigated by SEM pictures. As a conclusion no significant plastic deformations on the interaction surface could be observed (see Fig. 3). Based on several AFM topography analyses, mean roughness values for alumina particles were approximately 8 nm. Fig. 4 presents an example roughness profile based on performed AFM topography scanning. According to many references in the literature, Hamaker constants of alumina particles in ambient atmosphere are between 14.5 and 16.0 · 10−20 J [12–17]. 2.2. Methods Adhesive forces were measured experimentally with a Park Systems XE-100 atomic force microscope. To measure the adhesion force between a particle and a substrate, a colloidal probe cantilever was prepared. A particle was glued to the top of a Budget Sensors tipless All-in-One type B cantilever. In this study, alumina spheres with radii of 5, 10 and 15 μm were attached to the cantilevers. Fig. 3 shows an SEM photograph of a typical 15 μm alumina particle on a colloidal probe cantilever. The force constant of each cantilever was determined with the thermal method suggested by Park Systems. The thermal method calibrates the spring constant of a cantilever by fitting the power spectral density of the cantilever fluctuations with a known Lorentzian curve. Colloidal probes have been added to the cantilever before spring constants have been determined. This means in practice that the weight of the particle is taken into account the all adhesion measurements. The specifications of the cantilevers used in this study are presented in Table 1. Force–distance curves were determined by measuring the cantilever deflection while approaching, attaching to and detaching from a
surface. The measurement signal is the location-sensitive laser beam intensity on the detector. The distance to the surface is detected by piezo actuators. Since the deflection of the cantilever is a function of the force and the spring constant, the force–distance curve can be calculated. There is a typical linear range in the force–distance curves for alumina materials, and the adhesion force can be calculated easily from Hooke's law (see Eq. 1) as suggested by Götzinger and Peukert [12] and reported by Yang et al. [17]. F adhesion ¼ k⋅z:
ð1Þ
The distance, z, to lift off a particle can be calculated from the difference between the “snap-in” and the “pull-off” events in an AFM force– distance curve. A simplified schematic description of the force–distance curve is presented in Fig. 5. Based on a repeatability test consisting of 150 parallel measurements with colloidal probe cantilevers, the variation (i.e., two times the standard deviation) of the experimentally measured adhesion force was approximately 5%. No clear upward or downward trend was observed during adhesion measurements, indicating no significant permanent deformation of the surfaces. Fig. 6 presents a histogram of the AFM repeatability test results where interactions between an alumina substrate and spherical alumina particles were measured in ambient air (see conditions in Table 2). Fig. 6 indicates also that adhesion force measurements do not exactly follow normal distributions, because so many things affect adhesion measurements. Fig. 6 shows that local variations of substrate topography at the measurement point can influence the adhesion force by 10%. In our experience, the statistical variation (the difference between the minimum and the maximum values) of a single measurement of the adhesion force is approximately 10%, which is comparable to the measurement error from Rabinovich et al. [4,5]. Each reported data point is the average of at least 10 measurements at specific locations on the substrate. After every adhesion measurement, surface roughness (i.e., rms1 and rms2 for large-scale and small-scale roughness, respectively) of the measurement location was determined to obtain the most accurate roughness information. 3. Theory In general, the adhesion force, Fad, is a combination of the electrostatic force, Fel, the van der Waals force, FvdW, the meniscus or capillary force, Fcap, and forces due to chemical bonds or acid-base interactions, Fchem. An equation for the adhesion force is: F ad ¼ Fel þ FvdW þ Fcap þ Fchem :
ð2Þ
Under gaseous conditions, contributions from electrostatic forces are significant on insulators and at very low humidity, where charge dissipation is ineffective. Under aqueous conditions, most surfaces become charged due to surface group dissociations, and electrostatic
Fig. 1. Scanned topographical images of some used measure places. Note, that only places (marked with red arrows), which have enough smooth surfaces (i.e. no pores and holes), were used in the measurements.
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Fig. 2. An SEM photograph of spherical alumina particles.
forces are important, but their magnitude also depends on electrolyte concentration. The van der Waals force always contributes to adhesion, and it is attractive in most cases. Under ambient conditions, a water neck forms between the AFM tip and the substrate due to capillary condensation and adsorption of thin water films at surfaces. This attractive interaction depends on relative humidity and hydrophilicity of the tip and the sample. Depending on functional groups present on the tip and the substrate, chemical bonds or other specific chemical interactions (e.g., receptor–ligand interactions) may form during contact. When these interactions occur, they dominate the adhesion force [2]. In many AFM studies of the adhesion force, conditions are chosen where van der Waals forces dominate the adhesive force. In this case, Fad should be given by Hamaker constants of the AFM probe and the sample and the contact geometry. Some quantitative comparisons of these experiments with theoretical predictions are complicated by the following factors: • Surface roughness has a pronounced influence on the adhesive force that is difficult to quantify. • It is too difficult to determine the precise contact geometry. • Adsorption of contaminants on solid surfaces leads to chemically inhomogeneous surfaces. 3.1. Effect of humidity
Fig. 3. SEM photograph of a spherical alumina particle with a 15 μm radius attached to a cantilever.
Many studies present that relative humidity levels during AFM measurements are very critical [18–22]. He et al. [20] introduced three adhesive force humidity regimes. The force–humidity spectrum
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Fig. 4. Example of roughness profile of an alumina particle based on AFM topography scanning.
with hydrophilic interfaces is divided into three regimes: (I) a van der Waals regime at low RH, (II) a mid-RH regime dominated by capillary forces, and (III) a mixed repulsive–attractive regime at high RH. The force discontinuity is caused by the minimum thickness required for a water film to form a capillary neck. He et al. [20] showed that when relative humidity is below 20% with calcium fluoride, or 40% with silicon, the water film is too thin to form a capillary neck on a hydrophilic tip. Pakarinen et al. [21] and Chen et al. [22] reported that the van der Waals force was much smaller than the capillary force under high relative humidity conditions (in practice, RH >30%). We focused on performing all measurements in very constant conditions. In this study humidity and temperature inside the AFM chamber were maintained as low and as constant as possible during measurements to obtain reliable adhesion results. The relative humidity and the temperature were 26.5% ± 1.5% and 27 °C ± 1 °C, respectively, during all AFM measurements.
3.2. Effect of roughness The pull-off force, which is applied to separate two elastically smooth surfaces in adhesive contact, can be analysed with general contact mechanics. Two basic theories; one from Johnson, Kendall, and Roberts (JKR-model, [23]) and another from Derjaguin, Müller, and Toporov (DMT-model, [24]), use surface energy approximations to estimate adhesion between two surfaces in contact. Nevertheless,
Table 1 Specifications of cantilevers used in this study. Parameter
Specifications
Cantilever length Cantilever mean width Cantilever mean thickness Spring constant, CP 5 μm Spring constant, CP 10 μm Spring constant, CP 15 μm
210 μm 30 μm 2.7 μm 3.52 N/m 2.52 N/m 2.71 N/m
a b
± ± ± ± ± ±
10 μma 5 μma 1 μma 0.1 N/mb 0.1 N/mb 0.1 N/mb
Specifications provided by the cantilever manufacturer, Budget Sensors. Force constants determined with the described thermal method in this study.
these models do not take into account roughness variations, which cause non-uniform pressure distributions across actual contact areas. One way to estimate adhesion force is by means of Hamaker's approximation. This model assumes that only van der Waals forces act between interacting surfaces. A Hamaker-based approach for calculating the van der Waals force between surfaces with nanoscale roughness was described by Rumpf [3]. The Rumpf model is based on contact between a single hemispherical asperity on the surface and interacting with a much larger spherical particle along a normal line connecting their centers. The geometry of the Rumpf model is shown in Fig. 7. The adhesion (pull-off) force Fad can be described by: F ad ¼
A rR R þ 2 rþR 2 ð1 þ r=H 0 Þ 6H0
ð3Þ
where Fad is the adhesion (pull-off) force and A is the Hamaker constant, R and r are the radii of the adhering particle and asperity, respectively, and H0 is the distance of closest approach between the two surfaces, which can be approximated by 0.3 nm [4,5]. Rabinovich et al. [4] identified two principal limitations of the Rumpf model. First, the center of the hemispherical asperity must be located on the planar surface. Second, the asperity radius cannot be easily measured experimentally. Rabinovich et al. [4] introduced a modified Rumpf model (see Eq. 4) where the adhesion force can be expressed as a function of the roughness. These rms values can be measured with an AFM. The modified Rumpf model assumes that rms roughness treats peaks and valleys equivalently. The work by Rabinovich et al. also revealed that the Rumpf model does not accurately describe surfaces in the nanoscale regime. F ad ¼
AR 1 1 þ : 6H20 1 þ ðR=1:48rms1 Þ ð1 þ 1:48rms2 =H 0 Þ2
ð4Þ
The total surface roughness in Eq. (4) can be calculated by Eq. 5: rms ¼ rms∑ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rms21 þ rms22
ð5Þ
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Fig. 5. This schematic diagram shows the detector positions corresponding to a force–distance curve. The curve includes trace and retrace parts.
where rms1 and rms2 are the average root-mean-square values of the long and short peak-to-peak distances, respectively. Eq. 4 is applicable when rms2 is much less than or much greater than rms1. Rabinovich extended the modified Rumpf model by introducing the peak-to-peak distance, λ, between asperities. During experimental work, Rabinovich
et al. noticed that many surfaces exhibit two scales of roughness. The first type of roughness, rms1, is associated with longer peak-to-peak distances, λ1. A second smaller roughness, rms2, is associated with smaller peak-to-peak distances, λ2. The geometry for the Rabinovich model is shown in Fig. 8.
Fig. 6. A histogram of the results of 150 parallel measurements of colloidal cantilever probes at the same point on an alumina substrate. The histogram included superimposed normal distribution.
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O. Laitinen et al. / Powder Technology 246 (2013) 545–552 Table 2 Conditions during AFM repeatability test. Condition
Value
Temperature, T Humidity, RH Particle radius, R Roughness of substrate, rms1 Roughness of substrate, rms2
27.6 28.0 10.0 12.0 2.5
°C μm nm nm Fig. 8. The geometry for surfaces that have two scales of roughness. Modified from Rabinovich et al. [4].
The adhesion force can now be calculated by the following equation: 2 F ad ¼
3 2 H0
AR 6 1 1 7 þ 2 þ 4 5 2Þ 6H 20 1 þ 58Rðrms ð1 þ 1:82ðrms1 þ rms2 ÞÞ2 2 1 þ 58Rðrms1 Þ 1 þ 1:82rms2 λ2
λ21
H0
ð6Þ
where Fad is the adhesion (pull-off) force and A is the Hamaker constant, H0 is the distance of closest approach between surfaces, rms1 and rms2 are roughness values of surfaces and peak-to-peak distances, and λ1 or λ2, are associated with both types of superimposed roughness. When the peak-to-peak distance, λ1, is comparable to the particle radius, R, the third term of Eq. (6) can be neglected. Surface roughness reduces adhesion forces due to reduced contact areas [3–5,10,12,25]. Adhesion forces are very sensitive to small variations in surface roughness. Adhesion forces decreased significantly with only small increases of 1–2 nm in surface roughness for a variety of samples [4,5]. In addition, adhesive forces for hard materials such as alumina are very sensitive to asperity surface geometry because deformations are negligible [12]. Therefore, we assumed that local deformations of particles and/or rough substrates are very small in this study. All of the models described above take into account two components of the interaction, contact between asperities on spheres and surfaces and noncontact interactions between spheres and substrates below asperities. 4. Results The measured roughness of the used alumina substrates is between 1.5 and 12.0 nm. The focus of this study was the evaluation of existing roughness models for technically rough materials. The models introduced above and used in this study take roughness and surface asperities into account. Nevertheless there has to be a notice that presented models are not taken into account the roughness of added colloidal particles. If the surface at the measurement point is too rough or has large pores, the adhesion force cannot be determined accurately and it is not reasonable to measure adhesion at that point. The above models require an actual representation of geometry and roughness of interacting surfaces for more accurate predictions of adhesion forces. Table 3 presents the values that were used in these calculations, which were based on AFM topography measurements, SEM analyses and literature sources.
4.1. Comparison of roughness models Fig. 9 shows plots of adhesive forces for different particle sizes calculated with the modified Rumpf model (see Eq. 4). Values used in these calculations are presented in Table 3. The modified Rumpf model did not work well for rough material (rms2 > 1.5 nm) because calculated adhesion forces were too low (between 6 and 240 nN), around 10 times lower than could be expected. Fig. 10 shows plots of adhesion forces for different particle sizes calculated with the Rabinovich model (see Eq. 6). Values used in these calculations are presented in Table 3. The Rabinovich model provides much better estimates of adhesion forces for rough surfaces than the modified Rumpf model. 4.2. Adhesion measurements Fig. 11 presents graphs of measurements of adhesion forces for different colloidal probe particle sizes (Radius: 5, 10 and 15 μm). All three colloidal probe sizes lie on the same curve and no clear differences between probes were observed. 4.3. Comparison of the Rabinovich model and experimental data Because no clear differences between different size colloidal probes were observed in this study, all data were combined and then compared to the Rabinovich model. According to references in the literature, Hamaker constants of alumina particles in ambient atmosphere are between 14.5 and 16.0 · 10− 20 J [12–17]. Hamaker constant (16.0 · 10− 20 J) provided the best estimation compared to measured adhesion forces in our research case. The mean curve was calculated for all of the colloidal probes. When the experimental adhesion forces and the forces calculated with the Rabinovich model (i.e., the mean curve for 5 μm, 10 μm and 15 μm particles) were compared, they correlated very closely, and the fitted curves had the same form (see Fig. 12). 5. Discussion For the theoretical prediction of adhesion forces, van der Waals attractions are assumed to be primarily responsible for particle adhesion [2,26]. Apart from adsorbed water, additional adhesion forces might arise from liquid bridges formed by condensed liquid capillaries and
Table 3 Values used in calculations (see also Fig. 8). Parameter
Value
Hamaker constant, A The smallest distance, H0 Particle radius, R Roughness of substrate, rms1 Roughness of substrate, rms2 Peak-to-peak distance, λ1 Peak-to-peak distance, λ2
16.0 · 10−20 J a 0.3 nm a 10.0 μm (5–15 μm) 12.0 nm c 1.5–12.0 nm c 10.2 μm c 0.7 μm c
a
Fig. 7. The geometry of the Rumpf model of particle adhesion to rough surfaces. Modified from Rabinovich et al. [4].
b c
Based on a literature source [12]. Based on the authors' SEM analyses. Based on the authors' AFM topography measurements.
b
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Fig. 9. Adhesion forces for different particle sizes calculated with the modified Rumpf model.
Fig. 11. Adhesion forces measured by AFM. Each reported data point is the average of at least 10 measurements at specific locations on the substrate.
electrostatic adhesion. Liquid bridge formation between surfaces in contact, which may be equal or larger in magnitude than van der Waals forces, was not clearly observed during the present investigation, possibly because the relative humidity was low enough (RH b30%) and the surfaces (rms >1.5 nm) were not atomically smooth. The influence of electrostatic adhesion was not observed as well, because the humidity level was low and constant. In addition, there was enough time (over 10 days) after alumina particle production for discharge to occur, according to Götzinger and Peukert [12]. The technical surface topography of the particles was not hemispherical. The geometry in the actual contact zone is not sufficiently regular, which can be a real challenge. Nevertheless, in this study, the Rabinovich model gave moderately good estimates of adhesion forces for experimental surfaces with roughness in the range 1.5–12.0 nm.
The most probable reason is that particle and substrate roughness were great enough to reduce surface contact areas below the level where they affect adhesion.
5.1. Comparison of roughness models The modified Rumpf model does not work well when the roughness of particles or substrates is over 2 nm, as in this study. This finding is consistent with the results reported by Götzinger and Peukert [12] and Rabinovich et al. [4,5]. The Rabinovich model provides better estimates of adhesion forces for rough surfaces because the Rabinovich model takes into account roughness on two scales. 5.2. Adhesion measurements Significant differences were not observed in measurements of adhesion forces with different size colloidal probes (probe radius: 5–15 μm) in these tests. In addition there was no clear trend according to particle size, and any differences were within experimental scatter.
Fig. 10. Adhesion forces for different particles size calculated with the Rabinovich model.
5.3. Comparison of Rabinovich model to measured data Measurements during this study were performed under constant conditions, and differences between measured and estimated forces were constant for every measurement. Adhesion forces are very sensitive to small variations in surface roughness. Experimental adhesion forces were quite regular 50–60% higher than forces estimated with the Rabinovich model, which indicated the presence of systematic errors. The most likely reason for the differences between estimated and measured data was that the Rabinovich model takes into account only the roughness of the sample surface, and does not include colloidal probe roughness. The roughness values for alumina particles used in this research were determined to be approximately 8 nm. Because there is roughness on both contact surfaces (substrate and particle), this will lead to an increased actual contact area, and therefore to higher measured adhesion forces. Additionally the AFM adhesion measurement is more sensitive to small variation in roughness values from 1 nm to 4 nm and small local variation in roughness profile leads to higher scatter in measured adhesion values. Furthermore, the experimental surfaces were not spherical, and the forces may be larger because the experimental surface areas or numbers of contact points were larger than in the models. The AFM measurements of surface roughness could also deviate from the actual values. During measurements, other factors can also influence adhesion forces. For example, electrostatic or capillary forces and deformation
Fig. 12. Experimental adhesion forces vs. forces calculated with the Rabinovich model. Each reported data point is the average of at least 10 measurements at specific locations on the substrate.
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sometimes have sometimes significant effects. Even if water does not exist in a continuous phase and form a meniscus under these measurement conditions and capillary forces do not affect adhesion, water monolayers on the surfaces can influence adhesion. In practice, capillary forces might also affect observed adhesion forces. After all, the differences between measured and estimated forces can be explained primarily by nanoscale roughness variations (both substrate and particle), and irregular profile variations of contact surfaces. 6. Conclusions In this study, the adhesion forces between alumina surfaces of defined roughness (1.5–12.0 nm) were measured. It was experimentally proven that the adhesive force decreased approximately five-fold between 1.5 and 12.0 nm rms roughness levels. Nevertheless, the magnitude of the adhesion force decrease did not strongly depend on the colloidal probe radius sizes studied here (Radius: 5–15 μm) under constant AFM measurement conditions. The Rabinovich roughness model estimated adhesion forces with the same magnitudes as AFM force measurements. The modified Rumpf model estimated adhesion force that were approximately 10 times lower than could be expected for the researched surface roughness (1.5–12.0 nm). The roughness level, which describes the true geometry of the surfaces as accurately as possible, is critical for predicting interaction forces. In this study, an analysis of substrate topography by AFM, a mathematical description was determined for asperity height and breadth, which are expressed as rms roughness and peak-to-peak distance, respectively. It was necessary to use a realistic value of the nanoscale rms to accurately predict adhesion forces between substrates. It was also noticed, that the Rabinovich model even provided acceptable fits to very irregular, technical rough surfaces like in this study. Acknowledgement The authors would like to thank the German Research Foundation (DFG) for the financial support within the framework of the Collaborative Research Centre “Multi-Functional Filters for Metal Melt Filtration — A Contribute towards Zero Defect Materials” (SFB 920). References [1] G. Binnig, C.F. Quate, C. Gerber, Atomic force microscope, Physical Review Letters 56 (9) (1986) 930–934. [2] H.-J. Butt, B. Cappella, M. Kappl, Force measurements with the atomic force microscope: technique, interpretation and applications, Surface Science Reports 59 (2005) 1–152.
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Powder Technology journal homepage: www.elsevier.com/locate/powtec
Validity of the Rumpf and the Rabinovich adhesion force models for alumina substrates with nanoscale roughness O. Laitinen b,⁎, K. Bauer a, J. Niinimäki b, U.A. Peuker a a b
Technical University Bergakademie Freiberg, Institute of Mechanical Process Engineering and Minerals Processing, Germany University of Oulu, Fibre and Particle Engineering Laboratory, Finland
a r t i c l e
i n f o
Article history: Received 4 March 2013 Received in revised form 24 May 2013 Accepted 31 May 2013 Available online 18 June 2013 Keywords: Adhesion Alumina Atomic force microscope Colloidal probe Roughness
a b s t r a c t Adhesion forces between alumina substrates and spherical alumina particles were measured using an Atomic Force Microscope (AFM) colloidal probe technique. It was shown experimentally how nanoscale roughness of alumina substrates affects AFM adhesion force measurements. The adhesion force decrease was approximately five-fold between 1.5 and 12.0 nm root mean square (rms) surface roughness. Determining the roughness, which describes the actual geometry of the investigated surfaces, as accurately as possible, is crucial for predicting interaction forces. It was proven, that using a realistic value of the nanoscale rms is one of the most important parameters for accurately predicting adhesion forces between substrates. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Interaction forces between particles and surfaces affect a lot in technical materials. The major focus of this study is the determination of surface forces between alumina particles and alumina filter materials. Since the atomic force microscope was introduced by Binning et al. [1], it has been used to directly measure interaction forces between various surfaces. The principal part of the AFM is a small silicon cantilever with a specific spring constant. This cantilever is moved slowly towards a sample surface. The deflection of the cantilever, which is determined by the interaction between the cantilever tip and the surface of the sample, is detected by the reflection of a laser beam by the backside of the cantilever. Since the deflection is a function of the interaction force and the spring constant, the force between the tip and the sample surface can be calculated. Measurements in ambient air enable the determination of the van der Waals force as the adhesion force. The van der Waals force depends mainly on the material specific Hamaker Constant, the distance between the surfaces and the roughness of the interacting surfaces. Invention and development of colloidal probe techniques have had major effects on studies of adhesion and effects of nanoscale roughness [2]. To explain experimentally observed adhesion on rough surfaces,
⁎ Corresponding author at: University of Oulu, Fibre and Particle Engineering Laboratory, P.O. Box 4300, 90014 Oulun yliopisto, Finland. Tel.: +358 503504960. E-mail address: ossi.laitinen@oulu.fi (O. Laitinen). 0032-5910/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.powtec.2013.05.051
both analytical [3–5] and computational [6,7] approaches have been used. While computational methods produce results that are comparable to experimental results [8,9], these methods are complex and not easily applied to quickly estimate adhesion for specific systems. In addition, analytical equations depend on several parameters that are easily determined from surface topography, which ensures that these equations are useful for estimating adhesion forces [10]. Measurement of surface/particle adhesion with the colloid probe technique gives lower adhesion values than expected for simple sphere/plate geometries, and this difference is attributed mainly to surface roughness. This study focuses on measuring adhesive forces for surfaces with nanoscale roughness. Adhesion force measurements were performed between very pure spherical alumina particles and polished alumina filter materials. Adhesion forces measured by AFM are compared to the modified Rumpf model and the Rabinovich model, which takes surface asperities into account. The quantitative calculation of adhesion forces between a particle and a rough surface can be difficult for many reasons. The size, shape, homogeneity, mechanical properties, and distribution of asperities (deviations from an ideal planar surface) influence the actual contact area and, therefore, directly affect the adhesion force [11]. The main target of this study was to clarify whether it is possible to measure reliable adhesion force results using a roughness profile that is as accurate as possible for the substrate surface. The Rabinovich roughness model estimated the same magnitude of adhesion forces as the experimental AFM force measurements. The modified Rumpf model estimated adhesion forces that were approximately 10 times lower than expected for the surface roughness (1.5–12.0 nm).
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2. Material & methods 2.1. Materials Samples were produced by deposition of the origin alumina filter material slurry on planar substrate. Substrates were polished in a stepwise procedure with a commercial Struers laboratory polishing machine to obtain several roughness values. Samples were cleaned, dried and mounted in the AFM. Adhesion forces were measured at points that were chosen to provide reliable adhesion results. Topographies of measurement points were scanned using the AFM in contact mode with special Contact Cantilevers TYPE 905 M-NSC36. Based on these scans, the root mean square (rms) surface roughness and surface asperities of substrates were determined. The surface points used for adhesive force measurements had rms roughness between 1.5 and 12.0 nm, and examples are presented in Fig. 1. Alumina particles (Al2O3) with 99.9975% purity from IMPRATEX were used in this study (see Fig. 2). Particle sizes were in the range from 5 to 70 μm. Alumina has a high Young's modulus of E = 370 GPa and according Götzinger and Peukert [12] plastic deformation can be neglected for adhering particles without external forces. Since the used particles were aggregates of fine alumina particles the plastic deformation of the used particles were investigated by SEM pictures. As a conclusion no significant plastic deformations on the interaction surface could be observed (see Fig. 3). Based on several AFM topography analyses, mean roughness values for alumina particles were approximately 8 nm. Fig. 4 presents an example roughness profile based on performed AFM topography scanning. According to many references in the literature, Hamaker constants of alumina particles in ambient atmosphere are between 14.5 and 16.0 · 10−20 J [12–17]. 2.2. Methods Adhesive forces were measured experimentally with a Park Systems XE-100 atomic force microscope. To measure the adhesion force between a particle and a substrate, a colloidal probe cantilever was prepared. A particle was glued to the top of a Budget Sensors tipless All-in-One type B cantilever. In this study, alumina spheres with radii of 5, 10 and 15 μm were attached to the cantilevers. Fig. 3 shows an SEM photograph of a typical 15 μm alumina particle on a colloidal probe cantilever. The force constant of each cantilever was determined with the thermal method suggested by Park Systems. The thermal method calibrates the spring constant of a cantilever by fitting the power spectral density of the cantilever fluctuations with a known Lorentzian curve. Colloidal probes have been added to the cantilever before spring constants have been determined. This means in practice that the weight of the particle is taken into account the all adhesion measurements. The specifications of the cantilevers used in this study are presented in Table 1. Force–distance curves were determined by measuring the cantilever deflection while approaching, attaching to and detaching from a
surface. The measurement signal is the location-sensitive laser beam intensity on the detector. The distance to the surface is detected by piezo actuators. Since the deflection of the cantilever is a function of the force and the spring constant, the force–distance curve can be calculated. There is a typical linear range in the force–distance curves for alumina materials, and the adhesion force can be calculated easily from Hooke's law (see Eq. 1) as suggested by Götzinger and Peukert [12] and reported by Yang et al. [17]. F adhesion ¼ k⋅z:
ð1Þ
The distance, z, to lift off a particle can be calculated from the difference between the “snap-in” and the “pull-off” events in an AFM force– distance curve. A simplified schematic description of the force–distance curve is presented in Fig. 5. Based on a repeatability test consisting of 150 parallel measurements with colloidal probe cantilevers, the variation (i.e., two times the standard deviation) of the experimentally measured adhesion force was approximately 5%. No clear upward or downward trend was observed during adhesion measurements, indicating no significant permanent deformation of the surfaces. Fig. 6 presents a histogram of the AFM repeatability test results where interactions between an alumina substrate and spherical alumina particles were measured in ambient air (see conditions in Table 2). Fig. 6 indicates also that adhesion force measurements do not exactly follow normal distributions, because so many things affect adhesion measurements. Fig. 6 shows that local variations of substrate topography at the measurement point can influence the adhesion force by 10%. In our experience, the statistical variation (the difference between the minimum and the maximum values) of a single measurement of the adhesion force is approximately 10%, which is comparable to the measurement error from Rabinovich et al. [4,5]. Each reported data point is the average of at least 10 measurements at specific locations on the substrate. After every adhesion measurement, surface roughness (i.e., rms1 and rms2 for large-scale and small-scale roughness, respectively) of the measurement location was determined to obtain the most accurate roughness information. 3. Theory In general, the adhesion force, Fad, is a combination of the electrostatic force, Fel, the van der Waals force, FvdW, the meniscus or capillary force, Fcap, and forces due to chemical bonds or acid-base interactions, Fchem. An equation for the adhesion force is: F ad ¼ Fel þ FvdW þ Fcap þ Fchem :
ð2Þ
Under gaseous conditions, contributions from electrostatic forces are significant on insulators and at very low humidity, where charge dissipation is ineffective. Under aqueous conditions, most surfaces become charged due to surface group dissociations, and electrostatic
Fig. 1. Scanned topographical images of some used measure places. Note, that only places (marked with red arrows), which have enough smooth surfaces (i.e. no pores and holes), were used in the measurements.
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Fig. 2. An SEM photograph of spherical alumina particles.
forces are important, but their magnitude also depends on electrolyte concentration. The van der Waals force always contributes to adhesion, and it is attractive in most cases. Under ambient conditions, a water neck forms between the AFM tip and the substrate due to capillary condensation and adsorption of thin water films at surfaces. This attractive interaction depends on relative humidity and hydrophilicity of the tip and the sample. Depending on functional groups present on the tip and the substrate, chemical bonds or other specific chemical interactions (e.g., receptor–ligand interactions) may form during contact. When these interactions occur, they dominate the adhesion force [2]. In many AFM studies of the adhesion force, conditions are chosen where van der Waals forces dominate the adhesive force. In this case, Fad should be given by Hamaker constants of the AFM probe and the sample and the contact geometry. Some quantitative comparisons of these experiments with theoretical predictions are complicated by the following factors: • Surface roughness has a pronounced influence on the adhesive force that is difficult to quantify. • It is too difficult to determine the precise contact geometry. • Adsorption of contaminants on solid surfaces leads to chemically inhomogeneous surfaces. 3.1. Effect of humidity
Fig. 3. SEM photograph of a spherical alumina particle with a 15 μm radius attached to a cantilever.
Many studies present that relative humidity levels during AFM measurements are very critical [18–22]. He et al. [20] introduced three adhesive force humidity regimes. The force–humidity spectrum
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Fig. 4. Example of roughness profile of an alumina particle based on AFM topography scanning.
with hydrophilic interfaces is divided into three regimes: (I) a van der Waals regime at low RH, (II) a mid-RH regime dominated by capillary forces, and (III) a mixed repulsive–attractive regime at high RH. The force discontinuity is caused by the minimum thickness required for a water film to form a capillary neck. He et al. [20] showed that when relative humidity is below 20% with calcium fluoride, or 40% with silicon, the water film is too thin to form a capillary neck on a hydrophilic tip. Pakarinen et al. [21] and Chen et al. [22] reported that the van der Waals force was much smaller than the capillary force under high relative humidity conditions (in practice, RH >30%). We focused on performing all measurements in very constant conditions. In this study humidity and temperature inside the AFM chamber were maintained as low and as constant as possible during measurements to obtain reliable adhesion results. The relative humidity and the temperature were 26.5% ± 1.5% and 27 °C ± 1 °C, respectively, during all AFM measurements.
3.2. Effect of roughness The pull-off force, which is applied to separate two elastically smooth surfaces in adhesive contact, can be analysed with general contact mechanics. Two basic theories; one from Johnson, Kendall, and Roberts (JKR-model, [23]) and another from Derjaguin, Müller, and Toporov (DMT-model, [24]), use surface energy approximations to estimate adhesion between two surfaces in contact. Nevertheless,
Table 1 Specifications of cantilevers used in this study. Parameter
Specifications
Cantilever length Cantilever mean width Cantilever mean thickness Spring constant, CP 5 μm Spring constant, CP 10 μm Spring constant, CP 15 μm
210 μm 30 μm 2.7 μm 3.52 N/m 2.52 N/m 2.71 N/m
a b
± ± ± ± ± ±
10 μma 5 μma 1 μma 0.1 N/mb 0.1 N/mb 0.1 N/mb
Specifications provided by the cantilever manufacturer, Budget Sensors. Force constants determined with the described thermal method in this study.
these models do not take into account roughness variations, which cause non-uniform pressure distributions across actual contact areas. One way to estimate adhesion force is by means of Hamaker's approximation. This model assumes that only van der Waals forces act between interacting surfaces. A Hamaker-based approach for calculating the van der Waals force between surfaces with nanoscale roughness was described by Rumpf [3]. The Rumpf model is based on contact between a single hemispherical asperity on the surface and interacting with a much larger spherical particle along a normal line connecting their centers. The geometry of the Rumpf model is shown in Fig. 7. The adhesion (pull-off) force Fad can be described by: F ad ¼
A rR R þ 2 rþR 2 ð1 þ r=H 0 Þ 6H0
ð3Þ
where Fad is the adhesion (pull-off) force and A is the Hamaker constant, R and r are the radii of the adhering particle and asperity, respectively, and H0 is the distance of closest approach between the two surfaces, which can be approximated by 0.3 nm [4,5]. Rabinovich et al. [4] identified two principal limitations of the Rumpf model. First, the center of the hemispherical asperity must be located on the planar surface. Second, the asperity radius cannot be easily measured experimentally. Rabinovich et al. [4] introduced a modified Rumpf model (see Eq. 4) where the adhesion force can be expressed as a function of the roughness. These rms values can be measured with an AFM. The modified Rumpf model assumes that rms roughness treats peaks and valleys equivalently. The work by Rabinovich et al. also revealed that the Rumpf model does not accurately describe surfaces in the nanoscale regime. F ad ¼
AR 1 1 þ : 6H20 1 þ ðR=1:48rms1 Þ ð1 þ 1:48rms2 =H 0 Þ2
ð4Þ
The total surface roughness in Eq. (4) can be calculated by Eq. 5: rms ¼ rms∑ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rms21 þ rms22
ð5Þ
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Fig. 5. This schematic diagram shows the detector positions corresponding to a force–distance curve. The curve includes trace and retrace parts.
where rms1 and rms2 are the average root-mean-square values of the long and short peak-to-peak distances, respectively. Eq. 4 is applicable when rms2 is much less than or much greater than rms1. Rabinovich extended the modified Rumpf model by introducing the peak-to-peak distance, λ, between asperities. During experimental work, Rabinovich
et al. noticed that many surfaces exhibit two scales of roughness. The first type of roughness, rms1, is associated with longer peak-to-peak distances, λ1. A second smaller roughness, rms2, is associated with smaller peak-to-peak distances, λ2. The geometry for the Rabinovich model is shown in Fig. 8.
Fig. 6. A histogram of the results of 150 parallel measurements of colloidal cantilever probes at the same point on an alumina substrate. The histogram included superimposed normal distribution.
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O. Laitinen et al. / Powder Technology 246 (2013) 545–552 Table 2 Conditions during AFM repeatability test. Condition
Value
Temperature, T Humidity, RH Particle radius, R Roughness of substrate, rms1 Roughness of substrate, rms2
27.6 28.0 10.0 12.0 2.5
°C μm nm nm Fig. 8. The geometry for surfaces that have two scales of roughness. Modified from Rabinovich et al. [4].
The adhesion force can now be calculated by the following equation: 2 F ad ¼
3 2 H0
AR 6 1 1 7 þ 2 þ 4 5 2Þ 6H 20 1 þ 58Rðrms ð1 þ 1:82ðrms1 þ rms2 ÞÞ2 2 1 þ 58Rðrms1 Þ 1 þ 1:82rms2 λ2
λ21
H0
ð6Þ
where Fad is the adhesion (pull-off) force and A is the Hamaker constant, H0 is the distance of closest approach between surfaces, rms1 and rms2 are roughness values of surfaces and peak-to-peak distances, and λ1 or λ2, are associated with both types of superimposed roughness. When the peak-to-peak distance, λ1, is comparable to the particle radius, R, the third term of Eq. (6) can be neglected. Surface roughness reduces adhesion forces due to reduced contact areas [3–5,10,12,25]. Adhesion forces are very sensitive to small variations in surface roughness. Adhesion forces decreased significantly with only small increases of 1–2 nm in surface roughness for a variety of samples [4,5]. In addition, adhesive forces for hard materials such as alumina are very sensitive to asperity surface geometry because deformations are negligible [12]. Therefore, we assumed that local deformations of particles and/or rough substrates are very small in this study. All of the models described above take into account two components of the interaction, contact between asperities on spheres and surfaces and noncontact interactions between spheres and substrates below asperities. 4. Results The measured roughness of the used alumina substrates is between 1.5 and 12.0 nm. The focus of this study was the evaluation of existing roughness models for technically rough materials. The models introduced above and used in this study take roughness and surface asperities into account. Nevertheless there has to be a notice that presented models are not taken into account the roughness of added colloidal particles. If the surface at the measurement point is too rough or has large pores, the adhesion force cannot be determined accurately and it is not reasonable to measure adhesion at that point. The above models require an actual representation of geometry and roughness of interacting surfaces for more accurate predictions of adhesion forces. Table 3 presents the values that were used in these calculations, which were based on AFM topography measurements, SEM analyses and literature sources.
4.1. Comparison of roughness models Fig. 9 shows plots of adhesive forces for different particle sizes calculated with the modified Rumpf model (see Eq. 4). Values used in these calculations are presented in Table 3. The modified Rumpf model did not work well for rough material (rms2 > 1.5 nm) because calculated adhesion forces were too low (between 6 and 240 nN), around 10 times lower than could be expected. Fig. 10 shows plots of adhesion forces for different particle sizes calculated with the Rabinovich model (see Eq. 6). Values used in these calculations are presented in Table 3. The Rabinovich model provides much better estimates of adhesion forces for rough surfaces than the modified Rumpf model. 4.2. Adhesion measurements Fig. 11 presents graphs of measurements of adhesion forces for different colloidal probe particle sizes (Radius: 5, 10 and 15 μm). All three colloidal probe sizes lie on the same curve and no clear differences between probes were observed. 4.3. Comparison of the Rabinovich model and experimental data Because no clear differences between different size colloidal probes were observed in this study, all data were combined and then compared to the Rabinovich model. According to references in the literature, Hamaker constants of alumina particles in ambient atmosphere are between 14.5 and 16.0 · 10− 20 J [12–17]. Hamaker constant (16.0 · 10− 20 J) provided the best estimation compared to measured adhesion forces in our research case. The mean curve was calculated for all of the colloidal probes. When the experimental adhesion forces and the forces calculated with the Rabinovich model (i.e., the mean curve for 5 μm, 10 μm and 15 μm particles) were compared, they correlated very closely, and the fitted curves had the same form (see Fig. 12). 5. Discussion For the theoretical prediction of adhesion forces, van der Waals attractions are assumed to be primarily responsible for particle adhesion [2,26]. Apart from adsorbed water, additional adhesion forces might arise from liquid bridges formed by condensed liquid capillaries and
Table 3 Values used in calculations (see also Fig. 8). Parameter
Value
Hamaker constant, A The smallest distance, H0 Particle radius, R Roughness of substrate, rms1 Roughness of substrate, rms2 Peak-to-peak distance, λ1 Peak-to-peak distance, λ2
16.0 · 10−20 J a 0.3 nm a 10.0 μm (5–15 μm) 12.0 nm c 1.5–12.0 nm c 10.2 μm c 0.7 μm c
a
Fig. 7. The geometry of the Rumpf model of particle adhesion to rough surfaces. Modified from Rabinovich et al. [4].
b c
Based on a literature source [12]. Based on the authors' SEM analyses. Based on the authors' AFM topography measurements.
b
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Fig. 9. Adhesion forces for different particle sizes calculated with the modified Rumpf model.
Fig. 11. Adhesion forces measured by AFM. Each reported data point is the average of at least 10 measurements at specific locations on the substrate.
electrostatic adhesion. Liquid bridge formation between surfaces in contact, which may be equal or larger in magnitude than van der Waals forces, was not clearly observed during the present investigation, possibly because the relative humidity was low enough (RH b30%) and the surfaces (rms >1.5 nm) were not atomically smooth. The influence of electrostatic adhesion was not observed as well, because the humidity level was low and constant. In addition, there was enough time (over 10 days) after alumina particle production for discharge to occur, according to Götzinger and Peukert [12]. The technical surface topography of the particles was not hemispherical. The geometry in the actual contact zone is not sufficiently regular, which can be a real challenge. Nevertheless, in this study, the Rabinovich model gave moderately good estimates of adhesion forces for experimental surfaces with roughness in the range 1.5–12.0 nm.
The most probable reason is that particle and substrate roughness were great enough to reduce surface contact areas below the level where they affect adhesion.
5.1. Comparison of roughness models The modified Rumpf model does not work well when the roughness of particles or substrates is over 2 nm, as in this study. This finding is consistent with the results reported by Götzinger and Peukert [12] and Rabinovich et al. [4,5]. The Rabinovich model provides better estimates of adhesion forces for rough surfaces because the Rabinovich model takes into account roughness on two scales. 5.2. Adhesion measurements Significant differences were not observed in measurements of adhesion forces with different size colloidal probes (probe radius: 5–15 μm) in these tests. In addition there was no clear trend according to particle size, and any differences were within experimental scatter.
Fig. 10. Adhesion forces for different particles size calculated with the Rabinovich model.
5.3. Comparison of Rabinovich model to measured data Measurements during this study were performed under constant conditions, and differences between measured and estimated forces were constant for every measurement. Adhesion forces are very sensitive to small variations in surface roughness. Experimental adhesion forces were quite regular 50–60% higher than forces estimated with the Rabinovich model, which indicated the presence of systematic errors. The most likely reason for the differences between estimated and measured data was that the Rabinovich model takes into account only the roughness of the sample surface, and does not include colloidal probe roughness. The roughness values for alumina particles used in this research were determined to be approximately 8 nm. Because there is roughness on both contact surfaces (substrate and particle), this will lead to an increased actual contact area, and therefore to higher measured adhesion forces. Additionally the AFM adhesion measurement is more sensitive to small variation in roughness values from 1 nm to 4 nm and small local variation in roughness profile leads to higher scatter in measured adhesion values. Furthermore, the experimental surfaces were not spherical, and the forces may be larger because the experimental surface areas or numbers of contact points were larger than in the models. The AFM measurements of surface roughness could also deviate from the actual values. During measurements, other factors can also influence adhesion forces. For example, electrostatic or capillary forces and deformation
Fig. 12. Experimental adhesion forces vs. forces calculated with the Rabinovich model. Each reported data point is the average of at least 10 measurements at specific locations on the substrate.
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sometimes have sometimes significant effects. Even if water does not exist in a continuous phase and form a meniscus under these measurement conditions and capillary forces do not affect adhesion, water monolayers on the surfaces can influence adhesion. In practice, capillary forces might also affect observed adhesion forces. After all, the differences between measured and estimated forces can be explained primarily by nanoscale roughness variations (both substrate and particle), and irregular profile variations of contact surfaces. 6. Conclusions In this study, the adhesion forces between alumina surfaces of defined roughness (1.5–12.0 nm) were measured. It was experimentally proven that the adhesive force decreased approximately five-fold between 1.5 and 12.0 nm rms roughness levels. Nevertheless, the magnitude of the adhesion force decrease did not strongly depend on the colloidal probe radius sizes studied here (Radius: 5–15 μm) under constant AFM measurement conditions. The Rabinovich roughness model estimated adhesion forces with the same magnitudes as AFM force measurements. The modified Rumpf model estimated adhesion force that were approximately 10 times lower than could be expected for the researched surface roughness (1.5–12.0 nm). The roughness level, which describes the true geometry of the surfaces as accurately as possible, is critical for predicting interaction forces. In this study, an analysis of substrate topography by AFM, a mathematical description was determined for asperity height and breadth, which are expressed as rms roughness and peak-to-peak distance, respectively. It was necessary to use a realistic value of the nanoscale rms to accurately predict adhesion forces between substrates. It was also noticed, that the Rabinovich model even provided acceptable fits to very irregular, technical rough surfaces like in this study. Acknowledgement The authors would like to thank the German Research Foundation (DFG) for the financial support within the framework of the Collaborative Research Centre “Multi-Functional Filters for Metal Melt Filtration — A Contribute towards Zero Defect Materials” (SFB 920). References [1] G. Binnig, C.F. Quate, C. Gerber, Atomic force microscope, Physical Review Letters 56 (9) (1986) 930–934. [2] H.-J. Butt, B. Cappella, M. Kappl, Force measurements with the atomic force microscope: technique, interpretation and applications, Surface Science Reports 59 (2005) 1–152.
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