Direct, Robust Technique for the Measurement of ... - ACS Publications

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A Direct, Robust Technique for the Measurement of Friction between Microspheres Nicolas Fernandez,† Juliette Cayer-Barrioz,‡ Lucio Isa,¶ and Nicholas D. Spencer∗,† E-mail: [email protected]

Abstract Friction between microscopic objects controls many macroscopic phenomena. For instance, the friction between micro-asperities determines the tribology of rough surfaces in contact and in relative motion. Additionally, the friction between micro-particles

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is responsible for many aspects of the rheological response of granular media, ranging from micro-scale contacts at the single-particle level to macroscopic flow properties of sheared, dry granular systems and non-Brownian suspensions. We propose a new, precise and robust method, based on lateral force microscopy, to measure the coefficient of friction between microspheres quantitatively and without complex data processing.

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We have successfully applied this method to the contact between silica spheres in liquid with and without polymer coating. To whom correspondence should be addressed Laboratory for Surface Science and Technology, Department of Materials, ETH Zurich, Zurich, Switzerland ‡ Laboratoire de Tribologie et Dynamique des Systèmes - UMR 5513 CNRS, École Centrale de Lyon, Ecully, France ¶ Interfaces, Soft Matter and Assembly, Department of Materials, ETH Zurich, Zurich, Switzerland ∗ †

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Introduction Contacts between micrometer-sized objects are extremely common at any solid-solid interface and determine a significant part of the macroscopic mechanical response of materials in many natural phenomena and industrial processes. Micrometer-scale contacts play a key role in 5

the tribology (friction) of two contacting surfaces in relative motion, as found in a huge variety of processes, including in car engines or cutting processes, and in the flow of granular media, with an impact on diverse phenomena ranging from the formation of sand piles to the processing of cement and ceramic slurries. In tribology it has been demonstrated that upon contact between two solids, only a

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fraction of their micro-asperities actually interacts with each other. 1 These asperity contacts therefore determine most of the macroscopic properties of the sliding process. For example, their plastic deformation is directly involved in the wear mechanism 2 and local creep can lead to stick-slip instabilities. 3,4 Similarly, local dissipation represents the largest part of the observed macroscopic friction in boundary lubrication. 5 Based on this, several models have

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been developed linking the properties of these micro-contacts to macroscopic tribology. 6–9 Even in the very common and relatively simple case of hard, smooth polished surfaces, such as industrial steel or ceramics, in which the asperities can be modeled as elastic spheric sections, 10 testing and exploiting these models require careful measurement of the properties of the contact between two micropheres.

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Similarly, most of the properties of any granular media, however complex they may be, are the direct consequence of relatively simple microparticle-pair interactions. Recent significant increases in computing capacity have led to an extensive body of work that clarifies the link between particle properties (e.g. friction, stiffness, shape, charge...) and their assembly behavior. 11 In particular, the frictional dissipation between two contacting grains

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is one of the most studied parameters in numerical simulations 12–19 and is directly related to the yield stress, 20 the viscosity 21 and also the shear-thickening in dense non-Brownian suspensions 22–24 of granular media. 2


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Unfortunately, experimental validations of such results, in both in the areas of tribology of rough surfaces and granular-media rheology, are considerably lagging behind the calculations, mainly due to the lack of robust and reliable measurements of particle-pair friction for particles below the millimeter range. Despite its key relevance, a direct, robust, and simple 5

technique to measure the friction between two particles at the microscale has not yet been reported. For a long time, the microscopic scale could not be accessed in standard tribometeric measurements. Some important steps forward were taken by the introduction of the surface forces apparatus (SFA) (by Tabor 25,26 and by Georges 27,28 ), which allows contact forces

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to be measured in micrometer-scale contacts. Unfortunately, this is restricted to specific solid surfaces (mica for Tabor’s SFA) or non-aqueous lubricating liquids (for Georges’s SFA - molecular tribometer). Moreover sphere-on-sphere contacts are not readily implemented in an SFA and consequently these instruments are not really suited to the investigation of microparticle tribology. These limitations are overcome by the recent development of

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AFM-based tribometry 29 . This approach consists of attaching a microparticle to an atomic force microscope cantilever, which can be scanned over a second particle at a controlled speed and applied normal load. The vertical deflection of the cantilever is related to the interaction forces between the two particles and it has already been used to characterize the adhesion between particle pairs 30–32 and the interaction potential between charged particles

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in electrolyte solutions. 33–37 Measuring the torsional deflection of the cantilever during sliding is a route to determining the frictional forces between the particles. This lateral force microscopy approach (LFM) with a “colloidal probe" is a now a well-established technique to measure friction between a probe sphere and a flat surface, also in the presence of surface modifications, 38–40 but its

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extension to particle-on-particle friction measurements is far from trivial. In fact, for particle-friction measurements, the complexity arises not only from the small size of the objects but also from their shapes; the exact geometry of the contact deeply

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influences friction measurements. 41–43 Considering model spherical particles both simplifies the geometry and is is also relevant for practical applications, given that a large range of microparticles are indeed spherical as a consequence of their synthesis method. To date the most successful friction measurement between pairs of microspheres has been demon5

strated by Ling and co-workers. 44 In their seminal work, these authors attached a 3-4µm silica sphere onto an AFM cantilever and measured its friction in air against other identical silica microparticles attached to a flat surface as a function of various preparation protocols. Their measurements are based on a pixel-by-pixel analysis of the LFM images. This requires complex image post-processing to produce a local map of the point-by-point friction

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between the two surfaces in contact. However, the high level of symmetry of a sphere-onsphere contact can be exploited to design a much simpler technique that readily allows the measurement of the average friction between the two spherical surfaces in relative motion, which is, ultimately, the friction coefficient that is relevant for experiments. In this paper, we explain how the sliding friction between two rigid homogeneous micro-

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spheres can be readily extracted from a 2D-AFM tribology experiment in a single, simple, and robust calculation step. We demonstrate that our approach can be used to measure the friction between two silica microspheres, with and without polymer coating in aqueous media, exemplifying the cases of low and high friction, respectively. Finally, we quantify the accuracy of the results in terms of the most common experimental errors.

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Theory We first derive a mathematical relation between the measured signal, the location and the forces in the contact in a system consisting of two spheres and the AFM cantilever. Subsequently, we will show how this relation can be simplified, exploiting the symmetry of the problem.

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As shown in Fig.1, the system is composed of two rigid homogeneous spheres in contact:

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the center of the moving sphere, O2 , is O2 = (x · Ra , y · Ra , z · Ra ) with (x, y, z) being its dimensionless position parameters. For geometrical reasons, the following calculation is restricted

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to x2 + y 2 + z 2 < 4. The distance between O2 and the torsional axis of the

cantilever is called αR2 . In the case studied here, there is no spacer (eg, tip or excess of 5

glue) between the colloid and the cantilever, so α = 1 + 2Rt 2 with t being the thickness of the cantilever. Sphere 2 is connected to the rest of the system by two mechanical joints. The contact with Sphere 1 at the point C1 is a frictional point contact, and thus it transmits a repulsive force that is normal to the surface, FN , and a friction force, FF (here, FN is not corrected

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for the adhesion). The direction of FF can be unambiguously determined by noting that FF is antiparallel to the velocity of Sphere 2 and therefore, perpendicular to ey and FN . In the orthonormal coordinate system, we have: 



FN

sin φ cos θ     = sin φ sin θ FN   cos φ 

∓cos φ

(1)



 √1−sin2 φ sin2 θ      FF =   FF 0    ±sin φ cos θ  √

(2)

1−sin2 φ sin2 θ

With θ and φ, the polar and azimuthal angles as defined in Fig.1, ± means "+" in the 15

trace direction and "−" in the retrace direction and ∓ means the opposite. The joint with the cantilever is a rigid joint and thus has no degrees of freedom. Because this joint is rigid, it can be positioned rather arbitrary without modifying the mathematical validity of the analysis. We define C2 as the intersection point of the symmetry axis of the cantilever and the vertical line (i.e. parallel to ez ) going through O2 . We decided to position 1

see the paragraph “Necessary requirements for the image size” for a stricter definition.

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the rigid joint in C2 , because it is the point where the forces are measured experimentally. This rigid joint transmits not only a force FC = (Fx , Fy , Fz ) but also a moment M = (Mx , My , Mz ). The impact of the cantilever torsion on the position of the contact point is neglected, so we have: 



sin φ cos θ   −−−→   C1 C2 =  sin φ sin θ R2   cos φ + α

(3)

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In contrast to the sphere-on-plane case, the relations between these forces and moments depend on the relative location of the spheres and FN and Fz are almost never equal. In order to clarify this dependence, the mechanical equilibrium of Sphere 2 at C1 , i.e. the contact point with Sphere 1, can then be projected on ex and ez for the forces and on ey 10

for the moment: ±cosφ Fx = p FF −sinφ cosθ FN 1−sin2 φ sin2 θ ∓sinφ cosθ

(4)

FF −cosφ FN

(5)

My =− sinφ cosθ Fz +(α+cosφ) Fx R2

(6)

Fz = p

1−sin2 φ sin2 θ

In the limit of R2 being small compared to the length of the cantilever, 59 My and Fz can be deduced from the torsion and the flexion of the cantilever. By substitution of Fx and Fz in Eq.6 using Eq.4 and 5, a direct link between the forces in the contact, FN and FF , and the measurable quantities My and Fz can be written as: 

  (1−sin2 φ sin2 θ)+α cosφ √ ±  (α+1) 1−sin2 φ sin2 θ = ∓ √ sinφ 2cosθ 2

My  R2 (α+1) 

Fz

1−sin φ sin θ

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

−α sinφ cosθ (α+1) 

−cosφ




FF FN

 

(7)

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one obtains:

γ=

p p 4−y 2 ±(2α 4−x2 −y 2 +4−y 2 )µ(FN ) p p (α+1)( 4−x2 −y 2 4−y 2 ±2xµ(FN ))

−αx

(12)

In order to take advantage of the symmetry of the problem and the homogeneity of the surfaces, we define an average value of γ over a square area2 : 1 Γ= 2 8a

Z aZ

a

[γT race (x,y)−γRetrace (x,y)]dydx

(13)

−a −a

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with a, the half-length of the image subsection normalized by Ra , i.e. a =

Image Size . 2Ra

Since My is proportional to the lateral deflection and since Fz is kept constant during the experiment, Γ is experimentally obtained by averaging the trace and the retrace lateral deflection image and dividing by the load. 10

Summarizing, the friction coefficient µ is calculated from the measured averaged value of the apparent friction as obtained from Eq.13. This constitutes a major difference from the technique proposed by Ling et al. 44 in which the coefficient of friction is calculated at each single measurement point, at the cost of complex image processing.

Necessary requirements for the image size 15

Because of the specific geometry, the scanned area cannot be arbitrarily large. In this paragraph, the size limit will be expressed in terms of the image size, which is the relevant experimental parameter. One can notice that, due to the curvature of the particles, the point of contact moves less than the center of the scanning sphere, O2 , (see Fig.5) and thus the real scanned area on the Sphere i of radius Ri (i = 1 or 2), is not the image size, i.e. 2 The difference between trace and retrace signals is not strictly needed. Nevertheless, it is commonly used in order to remove any offset in the lateral deflection that can arise from initial setting and from the position of Sphere 2 on the cantilever. It also significantly reduces the error coming from a miscentered image.

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(2aRa )2 , but (aRi )2 . Γ is a function of α, µ and a. To maintain the contact between the spheres, we must √ keep a < 2. Moreover, a can be chosen to be arbitrarily small in the limit of the AFM scanning capabilities. The choice of a is affected by several requirements. 5

From an experimental point of view, the scanned area should be large enough to average out the local heterogeneity of the surface. Nevertheless, the range of z and z-speed that can be reached by the cantilever is usually limited. Therefore, depending on the particle size, only the topmost part of the spheres can be used. In addition, far from the top of the particle, the contact sliding speed starts

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differing from the half of scanning speed and also starts depending on the position due to the sphere curvature. From a theoretical point of view, the validity of the formulae that will be provided in the following paragraphs is restricted to a < 1, to keep the normal force FN closely related to the vertical force Fz , and to have aµ ≪ 1, to limit the nonlinear coupling between the

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vertical force Fz and frictional force FF . According to the range of particle radii that can be used with the present technique (1 to 102 µm), the most relevant scan size is in the range of a ∈ [0, 0.4], which corresponds to scanning areas from 400x400nm2 to 40x40µm2 , depending on the particle size. We note that a can easily be reduced after the experiment by cropping the image.

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Determination of the friction coefficient In this part, the coefficient of friction is assumed to be constant over the range of normal forces that is applied during a single scan (see Supporting Information∗ for derivation of the normal force range). The case of variable coefficient of friction, which can occur in the case of high adhesion or during a transition between lubrication regimes, is also addressed in the

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Supporting Information∗ . Due to the complexity of the mathematical relation in Eq.13, the link between Γ and µ 10


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cannot be analytically expressed. Consequently, in the rest of the paper, γ will be approximated by its (x, y)-Taylor series in (0; 0). After integration, Γ can then be developed as a Taylor’s series of aµ that is arbitrary small (i.e. aµ ≪ 1). The errors arising from this approximation are detailed in the dedicated paragraph below. This leads to two formulae, 5

depending on the value of the friction coefficient as described below: high and low friction. High friction Without further hypotheses concerning µ and keeping the second order in aµ, one can determine the friction coefficient by solving the following relation numerically with experimentally measured values:

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a2 (3α + 1) 2 1+ a µ + µ3 = Γ 24(α + 1) 12

(14)

Low friction If, in addition, the friction coefficient is low (i.e. µ ≪ 1), Equation 14 becomes: µ=

(3α + 1)a2 Γ 1− 24(α + 1)

(15)

Due to the restrictions on the image half-size a and because α & 1 if the colloid is directly fixed on the cantilever (i.e. no spacer), these equations also show that even the first-order relative correction is very small (∼ 10−2 ). Consequently, this correction is often smaller than 15

the common uncertainties arising from the cantilever calibration and can often be neglected and therefore in many experimental situations µ ≃ Γ. Nevertheless, this formula should not be confused with the sphere-on-plane formula since the present one is only valid for the average of the measured quantities over a centered image.

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dishes are sonicated in Milli-Q water to detach the particles, forming the second and upper layers. These particles will be used as scanning spheres. The sample is then cleaned in Piranha solution (same protocol as previously) in order to remove any organic contaminant and restore silanol groups at the silica surface. Due to the absence of any organic compound 5

in this sample, the risk of contamination of the contact is lowered compared to previous techniques. 44 Moreover, between particles, the glass surface provides a clean, flat, rigid surface that can also be used for the cantilever calibration. Scanning spheres The particles that detach themselves during the Petri dish sonication are collected from

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the surrounding fluid by centrifugation. The sediment is cleaned in Pirahna solution (same protocol as before with centrifugation between rinsing steps) and dried for 2 days at low pressure (50 mbar, static, in air). One of these spheres is glued onto a rectangular tipless cantilever (NSC 36 copper-gold coated, Spring constant around 1N/m, from MikroMasch, Tallinn, Estonia) using a small amount of epoxy glue Araldite Standard (Huntsman, The

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Woodlands, Texas) by means of a micromanipulator (DC-3K from Märzhäuser, Wetzlar, Germany) under an optical microscope. Lubricating fluid and polymer The lubricating fluid is a salty alkaline solution ([K2 SO4 ] = 80mmol/L and saturation of Ca(OH)2 ). The adsorbed polymer brush is composed of a comb copolymer: a random

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copolymer of polyethylene glycol (2kDa) methacrylate (22.7%) and ethylene glycol methacrylate phosphate (77.3%). Due to its phosphate anchors, this kind of polymer is known to adsorb from solution onto silica surfaces by calcium bridging at high pH. Moreover it is also representative of the ordinary admixtures that are used in the cement and ceramics industries. The polymer is diluted to 0.1%mass in the same salty alkaline solution.

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Methods AFM tribology calibration Friction measurements were carried out with an atomic force microscope (MFP-3D, Asylum Research, Santa Barbara, CA). The normal and lateral signals, measured using AFM, were 5

converted to forces using appropriate calibration constants. The normal spring constant calibration of the cantilever was carried out using the thermal-noise method 55 before the attachment of the colloidal probe and corrected from the colloid y-position compared to the cantilever apex. The torsional spring constant was estimated using Sader’s method 56 with the help of Sader’s online calibration applet 3 . The cantilevers were choosen to prevent

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in-plane bending, i.e. ǫ-ratio as defined by Sader et al. 58 of the order of 10−2 . The normal sensitivity was directly measured by contact with the flat glass surface. The lateral sensitivity (SL) of one cantilever was estimated using the test-probe method, as described by Cannara et al. 57 for rectangular cantilevers. The lateral sensitivity of all the other cantilevers was estimated from the measured lateral sensitivity of the test cantilever.

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Tribotest The surfaces were immersed in the lubricating solution (with or without polymer) 30 min before starting the tribotests. The whole measuring system was maintained a N2 atmosphere in order to prevent reaction of Ca2+ with dissolved CO2 . In this way, the pH was kept around 13 (measured by pH indicator) and no calcite crystal have been seen in the bulk solution. The cantilever particle was positioned on top of the center of the target scanned particle.

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An image of this particle obtained by the second one was recorded in contact mode with a scanning speed of 6µm/s (i.e. a contact point speed of around 3µm), perpendicular to the cantilever axis. The resolution of the image was set to 64x64 pixels. The size of the image was 6x6 µm2 , so the scanned region on each sphere was 3x3 µm2 (see Fig.5). Since the 25

radius of the contact between the two spheres ranged between 10 and 30nm according to 3

www.ampc.ms.unimelb.edu.au/afm/calibration.html#torsional

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Hertz theory, 1 in this case, the friction tracks (i.e. the scanning lines) did not significantly overlap but the full area was scanned. Since this discretization does not change the theory above, one can choose the separation or the overlap between the friction tracks according to each system and goals (as shown in Fig.5 ) 5

Another original experimental characteristic of this sphere-on-sphere contact is that the contact point moves during the test on both surfaces. A local contamination of the cantilever sphere can then be easily detected as an area of anomalous friction or topography in the image. Moreover, since a given point of one sphere is always in contact with the same point of the other sphere, it is unlikely that any contamination spreads over the whole friction

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area. By comparison, in the traditional sphere-on-plane test, the contact only takes place on the top of the ball. In this case, contamination of the top of ball cannot be detected (except by referring to external data) and this contamination will be spread over the full scanned area. For a single particle, the scanning process was repeated several times with different normal

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loads successively from low loads to high loads and back to low loads. No history dependance of the friction has been detected. A 2x2µm image of the top of the particle was manually cropped and used for the friction coefficient calculation.

Results Bare particles 20

We measured a linear dependance of the friction force on the load with a coefficient of friction of µBare = 0.9 (see Fig.6). The dependance on the load is consistent with the existence of a multi-asperity contacts that is expected since the theoretical indentation depth (around 0.2 nm according to Hertzian theory) is small compared to the surface roughness (Rq = 12.4nm, measured over 1µm2 and corrected from the sphere main curvature) and in the absence

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of strong adhesion. The adhesion tests performed both before and after the friction tests revealed an adhesion of the order of some nanonewtons. The high value of the coefficient 17


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of friction is probably due to the chemical activity at the surface of the silica particle, due to the high pH of the buffer and the extreme cleanliness of our system. This results are in contrast with the pH-induced lubrication previously reported by Taran et al., 49 which was attributed to the formation of a lubricating polysilicilic acid hydrogel. Nevertheless, this 5

hydrogel formation is dependent of the detailed surface-chemical properties and our system differs from Taran’s due to the production process of the silica surfaces, the surface-cleaning procedure 48 and the presence of absorbed calcium ions on our spheres. 53 The presence of adhesion in our system is consistent with the absence of polysilicilic hydrogel. Moreover, this differences in the production process also influences the topography of the surfaces, which

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are known to influence the adhesion and friction in the contact. 50–52 For the sake of comparison, some spheres have been exposed to ambient air for a long time before the tribotest, which led to contamination of their surface. The friction data with contaminated spheres in pH-neutral solution have also been plotted in Fig.6. In this case, the coefficient of friction (µDirty = 0.26) corresponds to the typical values for contaminated

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surfaces 54 and the values reported for contaminated silica microsphere pairs by Ling et al. 44 Polymer brush-coated particles The copolymer that has been used is known to adsorb on the surface via calcium bridging of its phosphate anchors, forming a hydrophilic polymer brush and reducing the coefficient of friction. 45,46 In our case, the adhesion tests revealed that both spheres repelled each other in

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close proximity, confirming the presence of the polymer at the surface. The friction coefficient is highly reduced (µP olymer ≃ 0.05) compared to the bare case.

Determination of the additional errors In this section, the link between the experimental errors and the uncertainty of the final results due to the sphere-on-sphere geometry, is detailed in the limit of small errors. These

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errors have to be added to the classical AFM-tribology errors, especially the tip-calibration error, which can be very large compared to those mentioned in the following.

Propagation of experimental errors Except for the error originating from the radius of the scanning sphere, these errors are 5

specific to the technique. They only refer to the conversion from vertical and horizontal forces to the coefficient of friction and are cumulative with, for example, the errors from cantilever calibration, which converts the deflections into the vertical and horizontal forces. Scanned-sphere radius Due to the presence of a large particle on the cantilever instead of a sharp tip, the radius of

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the scanned particle is not always precisely known. Since R1 only plays a role in the length normalization, this error in R1 , ǫR1 , is equivalent to an error in a in the Eq.15 or 14. Then, the relative error in the coefficient of friction due to ǫR1 is (highest non-zero order in a), in the low-friction case: 1 ǫR 3α + 1 2 ǫR1 µ = a µ 24(α + 1) Ra

(16)

and in the high-friction case: 1 ǫR µ = µ

µ2 3α + 1 + 24(α + 1) 12

a2

ǫ R1 Ra

(17)

In our experimental case, the standard deviation of the particle radius is 1.15µm and so

15 R ǫµ 1

µ

. 10−3 .

Scanning-sphere radius An error in R2 , ǫR2 , is much more problematic, since Sphere 2 also plays the role of a lever. Nevertheless, Sphere 2 has been intensively manipulated under microscopy in order to be 20


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glued on the cantilever and can, during this process, be precisely measured. The relative error on the coefficient of friction due to ǫR2 , independently of the friction coefficient, is (highest non-zero order in a): 2 ǫR ǫR µ = 2 µ R2

In our experimental case, the precision of our optical microscope (≈ 500nm), 5

(18) R

ǫµ 2 µ

≈ 10−1 .

This error is not characteristic of our method but common to all colloidal lateral force microscopy. This error can be further reduced by measuring the size of the scanning sphere more accurately by means of electron microscopy. Miscentered glueing of scanning sphere An error in the position of Sphere 2 relative to the longitudinal axis of symmetry of the can-

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tilever (miscentered glueing) generates a coupling between the normal force and the flexion of the cantilever. 47 Nevertheless, since this artifact is constant during the measurement and since we subtract the trace image from the retrace image, this does not have a significant impact on the measured friction in this technique. Image miscentering

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An error in the position of Sphere 1 relative to Sphere 2 (miscentered image) is equivalent to integrating γ over a non-zero centered interval. For an error of ǫx Ra in the trace direction in the other direction in low friction case: ǫxµ 2α + 1 2 = ǫx µ 8(α + 1)

(19)

and for high friction case: ǫxµ 2α + 1 = µ 8(α + 1)

2(α + 1) 2 1+ µ ǫx 2 2α + 1 21


(20)

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For an error of ǫy Ra in the other direction, independently of the friction coefficient: ǫyµ α = ǫy 2 µ 8(α + 1)

(21)

In our experiments, keeping the top of the particle in the center quarter (ǫx , ǫy < a2 ) of the image was a straightforward task. Then, in this case,

y ǫx µ +ǫµ µ

≈ 5.10−3 .

Limitation of the servo-loop control of FN 5

The limitations of the piezo-positioning in terms of z-velocity and the structure or parameters of the servo-loop that keeps the normal deflection constant could lead to a variation of the normal deflection with the position in case of fast scanning. We found in particular that it could depend on the x-position in a linear way with a constant slope ζ (see Fig.7 for example):

FN (x, y) = (1 ± ζx)FN (0, 0) 10

(22)

This leads to an underestimation of the coefficient of friction:

ǫζµ = −

α a2 ζ 6(α + 1)

(23)

Neglected terms in the theory Due to the restriction to the lowest orders of the aµ-Taylor series, the coefficient of friction, using the Eq.14 or 15, is underestimated by less than 1%. The highest error is achieved for large scanned areas and low friction.

15

Variation in the coefficient of friction The detailed derivation and the exact equation of the coefficient of friction in the case of a coefficient of friction that significantly varies in the range of normal force applied during a 22


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single image (using Eq.5:

∆FN FN

≃

a(4µ+a) ) 4

is given in the Supporting Information∗ . Never-

theless, one can interpret the variation of the coefficient of friction as a source of error. For dFF µ ˜ = dF µ − µ|, the error in the measured , the differential coefficient of friction and δµ = |˜ N A

coefficient of friction is given by

δ

a2 ǫµµ = µδµ µ 4 5

(24)

This formula is specifically useful for tribotests with loads that are small compared to the adhesion between the surfaces. In such situations, the apparent coefficient of friction, µ, varies significantly over a small range of loads and diverges at zero.

Conclusions Based on the mechanical analysis of a sphere-sphere contact, we propose a novel method to 10

quantify the sliding friction between homogenous rigid microspheres in dry or liquid environments. We have applied this technique to bare and polymer-coated particles in solution. The technique is simple and robust with respect to the most common experimental errors. Our method therefore offers the opportunity to significantly extend the characterization of frictional contacts at the single-particle-pair level and thus to further our understanding of

15

the many phenomena in which such microscale contacts play a crucial role.

Acknowledgement The authors thank Hélène Lombois-Burger, Martin Mosquet, David Rinaldi and Clément Cremmel for scientific support as well as Abdelaziz Labyad for technical support. This work was financially supported by Lafarge LCR. Lucio Isa is financially supported by the SNSF 20

grants PZ00P2_142532/1 and PP00P2_144646/1 ∗

Supporting Information Available: Detailed mathematical derivations. This material is

available free of charge via the Internet at http://pubs.acs.org .

23


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