Tribol Lett (2012) 45:185–194
`DOI 10.1007/s11249-011-9870-y
`
`M E T H O D S P A P E R
`
`Optical In Situ Micro Tribometer for Analysis of Real Contact
`Area for Contact Mechanics, Adhesion, and Sliding Experiments
`
`Brandon A. Krick • Jennifer R. Vail •
`Bo N. J. Persson • W. Gregory Sawyer
`
`Received: 1 July 2011 / Accepted: 26 September 2011 / Published online: 30 October 2011
`Ó Springer Science+Business Media, LLC 2011
`
`Abstract An instrument has been developed that allows
`in situ optical analysis and tribological measurements for
`contacts between solid bodies; an interferometric optical
`analysis can be used to measure and observe contact size,
`contact geometry, near contact topography, tribofilm for-
`mation, tribofilm motion, tribofilm thickness, wear debris
`formation, and wear debris morphology. The optical
`arrangement is in such a way that a 0th order interference
`fringe highlights the real contact area of contact, while near
`contact
`regions are height-mapped with higher order
`Newton’s rings interference fringes. Images synchronized
`with force and position measurements allow for
`the
`potential to test and validate models for contact mechanics,
`adhesion, and sliding. The contact and friction measure-
`ment between a rough rubber sphere and a polished glass
`counterface were studied over a range of loads from 1 to
`50 mN.
`Keywords Contact area In situ Optical Tribometer
`Tribology Contact mechanics Rubber Friction
`
`1 Introduction
`
`In situ tribometry is a powerful tool used by materials
`tribologists to study the interaction between surfaces dur-
`ing contact and sliding [1, 2]. There are various pathways
`
`B. A. Krick J. R. Vail B. N. J. Persson W. G. Sawyer (&)
`Department of Mechanical and Aerospace Engineering,
`University of Florida, Gainesville, FL 32611, USA
`e-mail: wgsawyer@ufl.edu
`
`B. N. J. Persson
`IFF, FZ-Ju¨lich, Ju¨lich 52425, Germany
`
`for in situ analysis of a surface. In situ spectroscopies such
`as Raman spectroscopy have been used to analyze the
`chemical nature of the interactions by examining the wear
`surface or transfer films during sliding or just after it exits
`the contact without changing the environment [3, 4]. In situ
`electron microscopy is increasingly popular; Varenberg
`used a scanning electron microscope to analyze the inter-
`action at the interface from the side [5]. Marks showed a
`liquid-like transfer of gold with in situ transmission elec-
`tron microscope experiments [6]. The state-of-the-art in in
`situ tribology was recently reviewed by Sawyer and Wahl
`[1, 2].
`One phenomenon that has been historically probed with
`in situ techniques is the real area of contact between solids.
`In situ tribology and contact mechanics experiments are not
`entirely unprecedented, especially when examining the real
`contact area between solids [7–13]. Contact area has been
`indirectly monitored in situ by contact resistance mea-
`surements [8, 9] and optical methods of examining the
`contact through a transparent counter sample [7, 10–16];
`Dyson and Hirst examined the real area of contact of
`metallic films with a phase contrast microscope through a
`glass disk [7]. Federle used in situ optical techniques to
`explore contact mechanics and adhesion in the feet of
`frogs, ants and other insects [14–16]. McCutchen examined
`the contact area between a polyvinyl chloride surface and
`an optically transparent counter surface using two optical
`methods: frustrated total
`internal reflection and optical
`interference of the Newton’s rings type [10]. The Newton’s
`rings interference can be used to measure contact because
`the destructive 0th order interference occurs at contact with
`higher order fringes radiating out
`in the near contact
`regions. The higher order fringes can also be used to map
`the near surface separations, and for closely spaced solids
`the distribution of interfacial separation, which is of crucial
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`Tribol Lett (2012) 45:185–194
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`importance for topics such as sealing, mixed lubrication,
`and the contact heat resistance. These techniques have been
`used for in situ tribology experiments to explore both film
`thicknesses and contact geometries [10, 12, 13].
`Here, we have designed and constructed a new instru-
`ment that uses microtribological probes and methods to
`perform indentation, friction, and wear experiments while
`making high fidelity optical interferometric measurements
`of the contact area. This instrument is capable of measuring
`normal and friction forces ranging from 10 lN to over 2 N.
`This design facilitates
`synchronized measurement of
`externally applied contact
`force (including adhesive),
`friction force, penetration depth, deformation, and in situ
`optical imaging of the contact with a spatial resolution
`limited by the diffraction limit. Preliminary optical in situ
`loading/unloading and sliding experiments were performed
`between a nitrile rubber half sphere and glass, with the aim
`to provide experimental measurements that can be com-
`pared to the available theories and models on contact area
`[8, 11, 17–29].
`
`2 Description of the In Situ Optical Micro Tribometer
`
`The in situ optical micro tribometer is capable of per-
`forming load/unload, friction, and wear experiments with
`in situ optical capabilities (Fig. 1). In an experiment, a
`sample is mounted directly to a load head; the load head
`measures both normal and frictional forces and is mounted
`to a piezoelectric stage that displaces the sample toward
`and away from a counter sample. Beneath the sample and
`counter sample is a microscope objective facing upward
`toward the sample; between the sample and objective is a
`transparent counter sample (in this case a flat optical
`window). The transparent counter sample is mounted to a
`piezoelectric stage that generates sliding between the
`samples. Figure 1 shows a schematic with important
`components.
`is
`A microscope objective (typically 59 or 109)
`mounted beneath the transparent counter sample. The
`objective has unique, low profile optical path to a 5 mega
`pixel Sony XCL-5005CR CCD camera. The sample is
`
`Fig. 1 a Schematic of optical in situ micro tribometer: the sample (1)
`is slid against a transparent counter sample (2). The sample is
`mounted directly to a calibrated cantilevered force transducer flexure
`(3). Capacitance probes (4 and 5) measure the displacement of a
`target (6) mounted on the cantilevered flexure; with the calibration of
`the flexure,
`these displacements provide the normal and friction
`forces. A microscope objective (7) mounted directly beneath the
`transparent counter sample held by a sample holder (8). b Schematic
`
`of optical pathway. A monochromatic coherent light source passes
`through the microscope objective and up through the transparent
`counter sample. The light is reflected off of the surfaces of the sample
`and counter sample back through the microscope objective and
`ultimately to a CCD camera. In the image, there is a 0th order
`destructive interface representing contact and higher order fringes
`surrounding contact
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`When the sample is lowered into contact with the trans-
`parent counter sample a contact area may be observed
`through interference. At the most basic level, there is a
`destructive interference fringe of 0th order everywhere the
`samples are in contact with each other; this allows obser-
`vation of contact area and geometry (Fig. 1b). In addition
`to the 0th order fringes, higher order fringes exist when the
`samples are separated; further information can be deter-
`mined from the fringes, such as thickness of a film between
`the solids and relative distance between the solids. The
`fringe pattern oscillates from dark to light as the separation
`gap between the samples is increased. The separation dis-
`tance required for destructive interference (Eq. 1) and
`constructive interference (Eq. 2) is a function of wave-
`length, k, and order.
`ddestructive ¼ m
`2
`dconstructive ¼ ð2m þ 1Þ
`
`ð1Þ
`
`ð2Þ
`
`k
`
`k
`
`4
`
`The optical fringe pattern phenomenon is caused by
`interference between light reflected from the interface
`
`Fig. 2 Post processing
`technique schematic: a image
`N of contact at an applied load.
`Contact, represented by the dark
`0th order destructive
`interference, is surrounded by
`constructive and destructive
`interference fringes and other
`features that are inherent
`impurities on the glass. b The
`image being analyzed, the
`previous image and the next
`image are averaged to
`preferentially weight contact
`and discount higher order
`fringes. c A background image
`acquired before loading began is
`subtracted from the averaged
`image. d A threshold is applied
`to the image revealing the
`contact area and geometry
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`Tribol Lett (2012) 45:185–194
`
`187
`
`illuminated through the objective with an interchangeable
`LED; for this experiment an LED of 595 nm with a mea-
`sured FWHM of 16 nm was selected.
`
`2.1 Imaging Methodology and Contact Area Analysis
`
`between the transparent counter sample and air and the p
`phase shifted light reflected off of the pin sample.
`At a given magnification,
`the real contact area and
`geometry is given by the 0th order interference which
`manifests as a dark area on the digital image for these
`material sets. Thresholding techniques of the digital image
`can be applied to calculate this contact area. Unfortunately,
`additional features such as fringes surrounding the contact
`and impurities in the glass produce false contact spots in
`the analysis. To accommodate this error we apply a post
`processing technique illustrated in Fig. 2. The image that
`we are analyzing the contact area, image N, is an intensity
`profile with pixels correlating to contact spots, higher order
`fringes, background impurities, and a background level
`intensity (Fig. 2a). Contact is represented by the dark 0th
`order destructive interference. Surrounding the contact are
`constructive and destructive interference fringes and other
`features that are inherent impurities on the glass.
`By simply applying a threshold to the image, one cannot
`accurately separate contact area from the background
`intensity because the higher order destructive fringes often
`produce false contact areas. To reduce this error, the image
`being analyzed is averaged with the image taken directly
`before and directly after the image of interest. Higher
`order, non-contact fringes will change as a result of a
`displacement of the rubber ball toward or away from the
`optical flat. If there is a change in the separation distance in
`the near contact zones, then the higher order fringes occupy
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`different pixels from frame to frame. Consequentially,
`higher order dark fringes will not overlap throughout the
`three images as shown in Fig. 2b. This effectively removes
`higher order fringes, but does not remove impurities in the
`surface of the glass that result in false contact pixels.
`Background impurities are removed by subtracting a
`background image acquired before loading from the aver-
`aged image (Fig. 2c). Finally a threshold can be applied to
`calculate the contact area (Fig. 2d). This contact area can
`be calculated by multiplying the number of pixels in con-
`tact by the square of the calibration constant 0.35 lm/pixel.
`The accuracy of the reported contact area is limited by the
`diffraction limit of light and other optical effects in very
`thin separations.
`
`2.2 Force Measurements and Positioning Metrology
`
`The load head is responsible for holding the sample,
`measuring normal and tangential (friction) forces, and
`applying the normal
`load by bringing the sample into
`contact with the counter sample with a piezoelectric stage.
`It consists of a cantilever that is instrumented with two
`capacitance probes; one capacitance normal to the contact
`and one in the sliding direction.
`The capacitive probes are aligned with a conductive
`target that is fixed to the end of the loading cantilever
`assembly. The probes are calibrated to monitor the change
`in distance between the capacitance probe and the target
`that is fixed to the cantilever. The cantilever assemblies
`consist of two double-leaf cantilevers that are mounted
`parallel to one another to constrain the flexures to recti-
`linear displacements. The use of a one double-leaf flexure
`would result in a change in slope at the capacitive target
`and would produce adverse effects [30]. The resolution of
`normal and tangential
`forces is only limited by the
`capacitance probe resolution and the stiffness of
`the
`interchangeable cantilevers.
`Through cantilever selection, normal loads of 2 N or
`more can be applied for high load cantilevers and normal
`loads of less than 10 lN can be applied. This particular
`combination of cantilevers and of capacitive probes can
`measure forces with uncertainties better than 50 lN and
`resolution which exceed that by a factor of 10.
`Piezoelectric stages are used for both loading displace-
`ments and sliding displacements. The piezoelectric stage
`responsible for bringing the sample into contact and
`modulating the normal load, the ‘‘loading piezo,’’ has a
`range of 100 lm, resolution of 0.4 nm and repeatability of
`±1 nm. The piezoelectric stage that produces the sliding
`motion, the ‘‘lateral piezo,’’ has a range of 1,500 lm,
`resolution of 3 nm and repeatability of ±14 nm. All of the
`stages are operated under closed loop positioning control.
`
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`Experimental control and acquisition is achieved
`through LabView. All force and position measurement
`signals are conditioned externally and interface with Lab-
`View with 16 bit analog-to-digital acquisition and control.
`Force versus displacement measurements are typically
`taken at 1,000 samples per second.
`Images are also
`acquired with LabView and are time-synched with all force
`and position data at as quickly as 15 frames per seconds.
`
`3 Materials
`
`Commercially available Buna-N nitrile rubber spheres of
`4.8 mm diameter were used in this study. The Buna-N
`rubber has a supplier specified durometer of 70A corre-
`sponding to a modulus of approximately 5.5 MPa. The
`spheres were cut in half with a razor blade and attached to
`the end of the cantilever. Prior to experiments, the half
`sphere was characterized with a Veeco Dektak 8 Advanced
`stylus profilometer, a Veeco Wyko NT9100 scanning white
`light interferometer (Fig. 3a–c), and an ASYLUM MFP-
`3D atomic force microscope; the measured RMS roughness
`of the rubber spheres was 5.2 lm. The surface roughness
`power spectrum was determined from this characterization
`(Fig. 3d)
`[29] and the resulting fractal dimension is
`Df * 2.
`A borosilicate float glass optical window was used as the
`transparent counter sample. The windows were 25 mm in
`diameter and 3 mm thick. The manufacturer specified
`modulus of the glass is four orders of magnitude higher
`than the rubber sample at approximately 64 GPa. The glass
`samples have RMS roughness of 2.06 nm measured with
`the stylus profilometer; that is more than three orders of
`magnitude less than the roughness of the rubber. These
`large differences in roughness and modulus make the glass
`appear infinitely stiff and perfectly smooth when compared
`to the rubber sample.
`
`4 Description of Loading and Sliding Experiments
`
`For these experiments, a 2.4 mm radius nitrile rubber half
`sphere was pressed against and slid against borosilicate
`glass windows. The nitrile spherical cap was brought into
`contact with the glass window to a prescribed force; the
`piezo was commanded to move at a constant rate of
`2.75 lm/s during loading and unloading. Four different
`experiments were run with target loads of 5, 10, 25, and
`50 mN. Images were acquired at half second intervals and
`were synchronized with the experiments;
`these digital
`images were then processed to compute the measured real
`contact area.
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`Fig. 3 Surface profilometry of
`nitrile spherical cap 2.4 mm
`radius. a Surface profile
`acquired from a Veeko Wyko
`NT9100 Scanning White Light
`Interferometer. b and c line
`scans taken across sample
`indicated by C1 and C2.
`d Surface roughness power
`spectrum of nitrile ball. The
`fractal exponent of the surface is
`1.8; this determined by the slope
`of the log C versus log
`q relationship
`
`reciprocating sliding experiments were per-
`Linear
`formed on the nitrile half sphere. For each test, the sample
`was brought
`into contact at
`the desired normal force
`loading: 25 and 50 mN; a static image was acquired before
`the onset of sliding. Sliding experiments were performed at
`sliding velocities of 20 and 50 lm/s over a stroke of
`800 lm; images of the contact were acquired before and
`during sliding. Images were acquired at six frames per
`second.
`
`5 Results and Discussion
`
`Figure 4 shows the contact area as a function of externally
`applied normal load for nitrile rubber half spheres, where
`each data point represents a processed image file. There is a
`nearly linear increase in contact area with increasing force
`over the range of the experiments, as predicted by contact
`mechanics theories (see, e.g., Ref. [28]), as long as the area
`of contact, A, is small compared to the nominal contact
`area AO. This linearity means that the (average) pressure is
`the area of real contact is nearly constant.
`In all cases, there is a strong hysteresis in the contact
`area plotted against externally applied force in the loading
`versus unloading of the rubber against the glass. This has
`been explained by considering the loading scenario as a
`crack closing between the rubber and the glass, and the
`unloading and breaking of contact as a crack opening: At a
`distance r away from the tip of a propagating crack the
`rubber experiences time-dependent deformation charac-
`terized by a frequency t/r, where t is the crack tip velocity.
`
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`
`Fig. 4 Measured real contact area versus externally applied force of
`the nitrile spherical cap pressed against the borosilicate glass window.
`a Processed contact area images for loadings of 1, 5, 10, 25, 40, and
`50 mN. b Contact area plotted against externally applied force for
`four loading and unloading profiles
`
`Rubber-like materials are viscoelastic, and a large energy
`dissipation in the rubber may occur at a distance r from the
`crack tip where the perturbing frequency t/r is close to the
`frequency where tan d ¼ ImE=ReE is maximal. For a fast
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`moving opening crack this may give rise to an effective
`energy G = c0(1 ? f(v)), which may be
`interfacial
`enhanced by a factor of 100 or 1,000, while for a closing tip
`the interfacial energy may be reduced by a similar number
`(see Ref. [31, 32]). This is the origin of the strong hys-
`teresis seen in our experiments.
`The area and shape of the contact changes during slid-
`ing; the shear stress at the contact distorts the contact area
`and shears it in the direction of sliding. Figure 5 shows the
`contact geometry for the nitrile rubber for static and sliding
`conditions for loads of 25 and 50 mN and sliding velocities
`of 20 and 50 lm/s. At
`these loads and velocities, an
`increase in contact area is observed during sliding. This can
`be explained by the strong increase in the effective inter-
`facial energy at the opening crack (at the exit of the contact
`region), see below. In this context, it is interesting to note
`that for silicone rubber, which can be considered as purely
`elastic with respect to the type of experiments discussed
`here, the opposite effect is observed, namely the static (or
`low sliding velocity) contact region is larger (and given by
`
`the JKR theory) than at higher velocities where the contact
`region is smaller and given accurately by the Hertz contact
`theory (e.g., negligible influence of adhesion) [33]. See
`also the discussion below.
`The area of real contact depends, A(f) in general on the
`resolution f (or magnification) of the instrument used to
`study the system. This is illustrated in Fig. 6, which shows
`an elastic block (dotted area) in adhesive contact with a
`rigid rough substrate (dashed area).The substrate has
`roughness on many different length scales and the block
`makes partial contact with the substrate on all
`length
`scales. When a contact area is studied at low magnification,
`it appears as if complete contact occurs, but when the
`magnification is increased it is observed that in reality only
`partial contact has taken place. The true (or atomic) contact
`area A(f1) is obtained at
`the highest magnification f1,
`corresponding to atomic resolution. The dependency of the
`area of contact, A(f), on the magnification f is of funda-
`mental importance in many applications.
`We have used the Persson contact mechanics theory [28,
`34] to calculate the variation of the contact area with the
`magnification. In this theory, the surface roughness enters
`only via the surface roughness power spectrum C(q). In
`Fig. 7, we show the logarithm (with 10 as the basis) of the
`surface power spectra as a function of the logarithm of the
`wave-vector, as obtained from AFM and line scan topog-
`raphy data (from Fig. 3) with a linear fit to the data cor-
`responding to a root-mean-square roughness hrms = 6 lm
`and the fractal dimension Df = 2 (or Hurst exponent
`H = 1).The fit curve corresponds to a surface with the
`
`Fig. 5 Processed contact area images during static loading (left) and
`sliding (right) for loads of 25 and 50 mN and sliding speeds of 20 and
`50 lm/s. There is a noticeable increase in contact area during steady
`state sliding
`
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`
`Fig. 6 An elastic block (dotted area) in adhesive contact with a rigid
`rough substrate (dashed area). The substrate has roughness on many
`different length scales and the block makes partial contact with the
`substrate on all length scales. When a contact area is studied, at low
`magnification it appears as if complete contact occurs, but when the
`magnification is increased it is observed that in reality only partial
`contact has taken place
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`Fig. 7 The logarithm (with 10 as basis) of the surface power spectra
`as a function of the logarithm of the wave-vector. The plot contains
`AFM and line scan topography data, and a fit
`to the data
`corresponding to a root-mean-square roughness hrms = 6 lm and
`the fractal dimension Df = 2 (or Hurst exponent H = 1). The fit
`corresponds to a surface with a root-mean-squared slope of 0.31 and
`surface area Atot = 1.14A0
`
`root-mean-squared slope of 0.31 and the surface area
`Atot = 1.14A0 (where A0 is the nominal (flat) surface area).
`In Fig. 8, we show the calculated relative contact area A/
`A0 (where A0 is the nominal contact area) as a function of the
`logarithm (with 10 as basis) of the magnification for models
`with adhesion [34]and the without adhesion [28]. In the
`calculation, we have assumed the power spectra given by the
`fit line in Fig. 7 (with q0 \ q \ q1, q0 = 2 9 104 m-1,
`q1 = 109 m-1), and the rubber elastic modulus E = 5 MPa
`and squeezing pressure p = 0.6 MPa. The curve denoted
`‘‘adhesion’’ is calculated using the theory of Ref. [34]
`assuming the work of adhesion c0 = 0.08 J/m2.
`The resolution of the optical instrument we have used to
`study the contact between the rubber ball and the flat glass
`surface is of order k & 1 lm, which corresponds to the
`magnification f & (p/k)/q0 & 100. According to our cal-
`culations (see Fig. 8) at
`this magnification, adhesion
`already manifests itself and increasing the magnification
`even more does not decrease the contact area. That is, at
`length scales shorter than &1/(10q0) (see Fig. 8) the
`adhesion pulls the surfaces into complete contact so that
`increasing the magnification above 10 does not result in a
`decrease in the contact area as would be the case without
`adhesion (in Fig. 8).
`The contact pressure we use in the calculations is similar
`to what prevail in the central region of the contact pictures
`in Fig. 4: the load FN = 50 mN is mainly distributed over
`a nominal contact area of 300 9 300 lm2 giving an
`average pressure of order p = 0.6 MPa in the central part
`of the contact region. In this region, the relative contact
`area A/A0 is of order 0.5, which is similar to what we
`observe in our calculation, see Fig. 8.
`
`Fig. 8 The calculated relative contact area A/A0 (where A0 is the
`nominal contact area) as a function of the logarithm (with 10 as basis)
`of the magnification for models both with and without adhesion. In
`the calculation, we have assumed the power spectra from Fig. 7
`(curve with q0 \ q \ q1, q0 = 2 9 104 m-1, q1 = 109 m-1), and the
`elastic modulus E = 5 MPa
`rubber
`and squeezing pressure
`p = 0.6 MPa. The curve denoted ‘‘adhesion’’ is calculated using the
`theory of Ref. [34] assuming the work of adhesion c0 = 0.08 J/m2
`
`When surface roughness occurs, in order for two solids
`to make adhesive contact, the surfaces must bend at the
`interface. This will result
`in (asperity induced) elastic
`energy stored at
`the interface which is, at
`least
`in
`part, given back during pull-off and helps to break the
`interfacial bond. This effect is described by the effective
`interfacial binding energy (see Ref. [34]) ceff(f)A0 = c0A(f1)
`- Ue1(f) where Ue1(f) is the elastic energy stored within the
`interface including only the roughnesses with wave-vector
`q [ q0f. In Fig. 9, we show the calculated effective inter-
`facial energy ceff (in units of the interfacial energy c0 for flat
`surfaces) as a function of the logarithm (with 10 as basis) of
`the magnification. The curve with adhesion is calculated
`assuming the work of adhesion c0 = 0.08 J/m2. Note that ceff
`(1) vanishes. This implies that the area of real contact will be
`proportional to the squeezing force even when adhesion is
`included [39]. This is illustrated in Fig. 10 which shows the
`calculated relative contact area A/A0 (where A0 is the nom-
`inal contact area), at the highest magnification, as a function
`of the applied pressure. Note that the area of real contact
`varies (nearly) linearly with the pressure or load even when
`adhesion is included, which is in good agreement with the
`experimental data shown in Fig. 4 during loading. During
`unloading this is no longer the case, because of the strong
`increase in the effective interfacial binding energy
`c0 ? G = c0 (1 ? f(v)) at
`the opening crack during
`unloading, see below.
`The area and shape of the contact changes during slid-
`ing; the shear stress at the contact distorts the contact area
`and shears it in the direction of sliding. Figure 5 shows the
`contact area and geometry for the nitrile rubber for static
`
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`rough surfaces. Thus, during sliding elastic energy is
`‘‘stored’’ at the interface, which helps to break the adhesive
`bonds, making the contact essentially non-adhesive at high
`enough velocity (this assumes that the rubber friction force
`increases with increasing sliding velocity). One such
`mechanism was studied by Savkoor et al. [38], and another
`in Ref. [39] where it was also shown that the quantitative
`theory to describe the observed effects may still be lacking.
`There exist a second mechanism, which tends to
`increase the contact area, and which may be particular
`important in our applications. To explain this, note first that
`the theory of adhesive bonding can be formulated as a
`theory of
`interfacial cracks. During sliding,
`the line
`boundary between contact and non-contact will consist of
`closing cracks on the front side, and an opening crack on
`the exit side of the sliding ball. For viscoelastic materials
`such as rubber there may be a strong enhancement in the
`effective interfacial energy at opening cracks [32, 40]
`which effectively may increase the adhesive interaction
`and the contact area. This effect is particular important for
`a rubber with a high glass transition temperature, which
`behave highly dissipative already for relative low per-
`turbing frequencies (or low sliding velocities). On the
`contrary, rubber with low glass transition temperature may
`behave as a nearly perfect elastic material with respect to
`the perturbing frequencies involved in sliding at
`low
`velocities. We illustrate this in Fig. 11 which shows the
`calculated (using the theory presented in Ref. [32]) effec-
`tive interfacial crack propagation energy as a function of
`the crack tip speed for PDMS and bromobutyl rubber
`(filled). Unfortunately, we do not have the viscoelastic
`modulus for nitrile rubber, which enter in the calculation of
`ceff, but the glass transition temperature of nitrile rubber
`
`Fig. 9 The calculated effective interfacial energy ceff (in units of the
`interfacial energy c0 for flat surfaces) as function of the logarithm
`(with 10 as basis) of the magnification. In the calculation, we have
`assumed the power spectra from Fig. 7, an elastic modulus of
`E = 5 MPa, and the work of adhesion c0 = 0.08 J/m2
`
`and sliding conditions for loads of 25 and 50 mN and
`sliding velocities of 20 and 50 lm/s. At these loads and
`velocities, an increase in contact area is observed. When a
`rubber ball with smooth surface is sliding on a hard smooth
`substrate, or a hard smooth ball on a flat rubber surface, the
`area of real contact usually decreases with increasing
`velocity, roughly from the JKR (adhesive) theory limit for
`zero velocity to the Hertz (non-adhesive) limit for high
`enough velocity [33, 35–37]. This is usually attributed to
`the build-up of elastic deformation energy, due to the
`frictional shear stress at the sliding interface, which may
`reduce the adhesional interaction in a very similar way as
`the asperity-induced elastic energy reduce the adhesion on
`
`Fig. 10 The calculated relative contact area A/A0 (where A0 is the
`nominal contact area), at the highest magnification, as a function of
`the applied pressure for the cases of both with and without adhesion.
`In the calculation, we have assumed the power spectra from Fig. 2,
`and the rubber elastic modulus E = 5 MPa. The curve with adhesion
`is calculated assuming the work of adhesion c0 = 0.08 J/m2
`
`Fig. 11 The calculated effective interfacial crack propagation energy
`as a function of the crack tip speed for PDMS and bromobutyl rubber
`(filled). PDMS has a much lower glass transition temperature and
`bromobutyl rubber and much higher crack tip velocities are necessary
`for G to reach its high-velocity plateau value
`
`123
`
`Regeneron Exhibit 1164.008
`Regeneron v. Novartis
`IPR2021-00816
`
`

`

`Tribol Lett (2012) 45:185–194
`
`193
`
`(Tg & -26 °C) is much higher than that of bromobutyl
`rubber (Tg & -73 °C), which will result in even larger ceff
`in the studied velocity range. PDMS has a much lower
`glass transition temperature (Tg & -120 °C) than bro-
`mobutyl rubber, and much higher crack tip velocities are
`necessary for ceff
`to reach its high-velocity plateau
`value[31, 32].
`The fact that in most cases the contact area decreases
`rather than increases indicate that in most cases the first
`effect discussed above,
`involving the elastic energy
`‘‘stored’’ at the interface, may in most cases be the most
`important one for smooth surfaces. However, the opening
`crack propagation mechanism is proportional to the length
`L of the boundary line between contact and non-contact.
`For smooth surfaces, this length scales linearly with the
`diameter D of the contact area. Since the area scale as
`A & D2, we have L & A/D. Now assume that (at least)
`one of the solids has surface roughness. In this case, the
`contact area may consist of a large number, say N, of small
`contact spots (see Fig. 4). If d is the typical diameter of a
`contact spot and A is the total contact area, then N & A/d2
`and the total length of the region between contact and non-
`(crack tip line) becomes L & Nd & A/d [[
`contact
`A/D. Thus, surface roughness may strongly enhance the
`contribution from the opening crack to the change in the
`contact area with increasing sliding velocity. We believe
`this is the explanation for why we observe an increase in
`the contact area for our sliding system. We also note that
`nitrile butadiene rubber (which has much higher glass
`transition temperature than bromobutyl rubber), is much
`more dissipative at low frequency than silicon rubber used
`in most of the earlier studies. This too will tend to enhance
`the importance of the opening crack mechanism in our
`case, as compared to most of the earlier studies.
`Finally, let us note that there is a fundamental difference
`between having the roughness on the rubber side or on the
`hard counter surface. If a rubber block (e.g., a ball) is sliding
`on a hard rough substrate the surface asperities will exert
`pulsating deformations on the rubber surface which will lead
`to energy dissipation via the internal friction of the rubb