A comparison of JKR-based methods to analyze quasi-static and dynamic indentation force curves

The Johnson-Kendall-Roberts (JKR) theory of elastic contact, extended to take viscoelastic effects into account, is used to evaluate work of adhesion and modulus of elastomeric films. In this paper, we present a comparison of five approaches to analyze quasi-static and dynamic JKR force curve data obtained using instrumented indentation. The load-displacement experiments were performed using a 200-microm radius borosilicate glass sphere against poly(dimethyl siloxane) (PDMS). By applying a small oscillation to the tip during indentation, dynamic stiffness vs load data were also obtained for frequencies between 25 and 160 Hz. Direct curve fitting as well as simplified 2- and 3-point analysis methods were used to compare modulus values obtained from load-displacement and stiffness-load data. Fit methods not requiring determination of the initial point of tip-sample contact ("zero" displacement) provided modulus values closest to those obtained by direct curve fitting. The dynamic stiffness-load data revealed a frequency dependent modulus; load-displacement measurements obtained simultaneously were consistent with the relaxed, or low-frequency, modulus of the PDMS sample. These experiments demonstrate that both the frequency dependent and relaxed modulus can be obtained from a single experiment.     

Oscillatory loading of a viscoelastic adhesive contact

The cycle of loading and unloading of a spherically-tipped probe against an adhesive, viscoelastic plane specimen is studied by numerical integration of the relations between crack speed and apparent surface energy previously found for a linear 3-element viscoelastic solid with a Maugis-Dugdale law of force across the crack. It is found that even when the rate of loading is so slow that the loading and unloading curves almost coincide, suggesting purely elastic behaviour, the pull-off force can be appreciably greater than the elastic (JKR) value. When the normal force is modulated with a small amplitude sinusoidal variation during unloading--in order to find the contact stiffness--the contact radius barely changes, and the stiffness is close to that for a rigid flat punch instead of having the expected JKR value.     

Evaluation of the adhesion properties of inorganic materials with high surface energies

With the aim of checking the validity of methods for characterizing the adhesion between inorganic materials with high surface energies, the properties of the adhesion between an inorganic material (indium tin oxide (ITO)) and model surfaces with various surface energies (Cl-, NH2-, CH(3)-, and CF3-functionalized surfaces) were evaluated using atomic force microscopy (AFM) and the Johnson-Kendall-Roberts (JKR) apparatus. For this purpose, the AFM tip and the JKR lens were modified with ITO using radio frequency (rf) magnetron sputtering. The work of adhesion between the ITO coating and each model surface was estimated using AFM and the JKR apparatus and compared with the result obtained from contact angle measurements. The adhesion forces determined from the force-displacement curves (AFM) were found to agree with the predictions of the Derjaguin-Muller-Toporov (DMT) theory. The JKR equation used in the interpretation of the JKR experiments was modified by taking into account the differences between the surface and bulk moduli of the ITO-coated poly(dimethylsiloxane) (PDMS) lens. The ratio of the surface modulus to the bulk modulus we used in this modified JKR equation was obtained by determining the slope of the attracting part of the force-displacement curve. The values of the work of adhesion calculated using the modified JKR equation were also found to agree with the values obtained from contact angle measurements. We conclude that the two methods using AFM and the JKR apparatus can be used in the evaluation of the work of adhesion between inorganic materials with high surface energies such as metal and metal oxide surfaces.     

RECENT PROGRESS OF NANO-MECHANICAL MAPPING

Nano-mechanical mapping by atomic force microscopy has been developed as an useful application to measure mechanical properties of soft materials at nanometer scale.To date,the Hertzian theory was used for analyzing force- distance curves as the simplest model among several contact mechanics between elastic bodies.However,the preexisting methods based on this theory do not consider the adhesive interaction in principle,which cannot be neglected in the ambient condition.A new analytical method was introdu...     

RECENT PROGRESS OF NANO-MECHANICAL MAPPING

 Nano-mechanical mapping by atomic force microscopy has been developed as an useful application to measure mechanical properties of soft materials at nanometer scale. To date, the Hertzian theory was used for analyzing force-distance curves as the simplest model among several contact mechanics between elastic bodies. However, the preexisting methods based on this theory do not consider the adhesive interaction in principle, which cannot be neglected in the ambient condition. A new analytical method was introduced to estimate the elasticity and the adhesive energy simultaneously by means of the JKR theory, describing adhesive contact between elastic materials. Poly(dimethylsiloxane) (PDMS) and isobutylene-co-isoprene rubber (IIR) were analyzed to verify the applicable limit of the JKR analysis. For elastic samples such as PDMS, the force-deformation plots obtained experimentally were consistent with JKR theoretical curves. Meanwhile, for viscoelastic samples, especially for IIR, the experimental plots revealed large deviations from JKR curves depending on scanning velocity and maximum loading force. Some nano-rheological arguments were employed based on the difference between these specimens.     

Oscillating drop/bubble tensiometry: effect of viscous forces on the measurement of interfacial tension

The oscillating drop/bubble technique is increasingly popular for measuring the interfacial dilatational properties of surfactant/polymer-laden fluid/fluid interfaces. A caveat of this technique, however, is that viscous forces are important at higher oscillation frequencies or fluid viscosities; these can affect determination of the interfacial tension. Here, we experimentally quantify the effect of viscous forces on the interfacial-tension measurement by oscillating 100 and 200 cSt poly(dimethylsiloxane) (PDMS) droplets in water at small amplitudes and frequencies ranging between 0.01 and 1 Hz. Due to viscous forces, the measured interfacial tension oscillates sinusoidally with the same frequency as the oscillation of the drop volume. The tension oscillation precedes that of the drop volume, and the amplitude varies linearly with Capillary number, Ca=DeltamuomegaDeltaV/gammaa(2), where Deltamu=mu(D)-mu is the difference between the bulk Newtonian viscosities of the drop and surrounding continuous fluid, omega is the oscillation frequency of the drop, DeltaV is the amplitude of volume oscillation, gamma is the equilibrium interfacial tension between the PDMS drop and water, and a is the radius of the capillary. A simplified model of a freely suspended spherical oscillating-drop well explains these observations. Viscous forces distort the drop shape at Ca>0.002, although this criterion is apparatus dependent.     

Characterization of adhesion at solid surfaces: Development of an adhesion-testing device

A custom-built adhesion-testing device (ATD) is described in this paper, which was developed to study energetics of various solid (polymeric) interfaces. A review is also given of the main techniques of adhesion and adherence measurements, including non-destructive and destructive methods, with major emphasis on the evolution and applications of contact mechanics techniques. Using the Johnson-Kendall-Roberts (JKR) theory of contact mechanics in the elastic deformation regime, the interfacial energy of solid surfaces can be obtained by measuring the contact radius, loading force, and vertical displacement between an (elastic) sphere (lens) and a flat surface (one of which, or both, coated with the sample of interest). The parameters needed for JKR analyses were determined by our custom-built device. Based on the JKR theory, the values of work of adhesion, combined elastic modulus and interfacial energy were determined from the loading and unloading curves on poly(dimethylsiloxane)-poly(dimethylsiloxane) (PDMS) systems. Cumulative adhesion hysteresis and elastic modulus were also calculated. The results obtained agree well with literature data measured by different methods. These measurements on compliant PDMS-PDMS model systems can also serve as validation and verification of the adhesion-testing devices described in this study.     

Viscoelastic characterization of polymers using instrumented indentation. I. Quasi‐static testing*

The use of instrumented indentation to characterize the mechanical response of polymeric materials was studied. A model based on contact between a rigid probe and a linear viscoelastic material was used to calculate values for the creep compliance and stress relaxation modulus for two glassy polymeric materials, epoxy and poly(methyl methacrylate), and two poly(dimethyl siloxane) (PDMS) elastomers. Results from bulk rheometry studies were used for comparison with the indentation stress relaxation results. For the two glassy polymers, the use of sharp pyramidal tips produced responses that were considerably more compliant (less stiff) than the rheometry values. Additional study of the deformation remaining in epoxy after indentation creep testing as a function of the creep hold time revealed that a large portion of the creep displacement measured was due to postyield flow. Indentation creep measurements of the epoxy with a rounded conical tip also produced nonlinear responses, but the creep compliance values appeared to approach linear viscoelastic values with decreasing creep force. Responses measured for the unfilled PDMS were mainly linear elastic, with the filled PDMS exhibiting some time‐dependent and slight nonlinear responses in both rheometry and indentation measurements. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 1794–1811, 2005     

QCM studies of gel spreading: Kraton gels on polystyrene surfaces

Contact of a polymer gel made from a styrene/ethylene-butene/styrene triblock copolymer in mineral oil was investigated by bringing the gel into contact with the coated surface of a quartz crystal microbalance (QCM). The experimental apparatus enabled simultaneous measurement of the load, displacement, and contact area, in addition to the resonant frequency and dissipation of the oscillating quartz crystal. The QCM response was determined by the linear viscoelastic properties of the gel at the frequency of oscillation. A geometric correction factor involving the contact area provided a means for quantitatively determining these viscoelastic parameters as the gel spread over the QCM surface. When the gel was removed from the surface, a thin solvent layer was left behind. The thickness of this solvent layer was determined from the QCM response and was compared to predictions from a simple model involving the disjoining pressure of the film and the osmotic pressure of the gel. Qualitative agreement with the model required that tensile, adhesive forces at the perimeter of the gel/QCM contact area were taken into account when calculating the film thickness.     

General Equations Describing Elastic Indentation Depth and Normal Contact Stiffness versus Load

Continuum mechanics models describing the contact between two adhesive elastic spheres, such as the JKR and DMT models, provide a relationship between the elastic indentation depth and the normal load, but the general intermediate case between these two limiting cases requires a more complex analysis. The Maugis-Dugdale theory gives analytical solutions, but they are difficult to use when comparing to experimental data such as those obtained by scanning force microscopy. In this paper we propose a generalized equation between elastic indentation depth and load that approximates Maugis' solution very closely. If the normal contact stiffness can be described as the force gradient, that is the case of the force modulation microcopy, then a generalized equation between normal contact stiffness and load can be deduced. Both general equations can be easily fit to experimental data, and then interfacial energy and elastic modulus of the contact can be determined if the radius of the indenting sphere is known. Copyright 2000 Academic Press.     

Forces between a stiff and a soft surface

The contact between a sphere and a planar half space, one being rigid and the other elastic (or between two elastic spheres), can be described by the JKR theory of Johnson, Kendall and Roberts (Proc. R. Soc. Lond. A 1971, 324, 301). One assumption of JKR theory is that the characteristic length scale L ≈ w/E is much smaller than the radius R of the sphere; where w is the work of adhesion and E is the Young's modulus of the soft, elastic body. Relative deformations for a mechanical contact increase with increasing L and decreasing particle size R. Experiments show that up to at least L/R = 0.2, JKR theory predicts the correct dependencies between the contact radius, the indentation and the load. However, when R ≫ L is no longer satisfied, the change in total free surface area due to deformation needs to be considered. Then, elastocapillary effects start playing a significant role. In addition to discussing theory and experiments of pure solid contacts, the effect of elastic deformation on capillary and hydrodynamic forces is discussed. Finally, we consider the interaction of hollow capsules as one example of a deformable body that is still formed from a stiff material.     

Study on the viscoelastic properties of the epoxy surface by means of nanodynamic mechanical analysis

The viscoelastic properties of the epoxy surface have been investigated by nanodynamic mechanical analysis (nano‐DMA). Both a Berkovich tip and a conospherical tip were used under the condition of different forces (i.e., different penetration depths) in the frequency range of 10–200 Hz. Loss tangent and storage modulus are characteristics that describe the viscoelastic properties. The effect of force frequency, penetration depth, and tip shape on the viscoelastic properties is studied and discussed according to the features of microstructures and mobility of molecular chains. The experimental results show important variations when the penetration depth is shallow (<30 nm). As the depth becomes deeper, the results tend to be stable and become almost constant over 120 nm. The two kinds of indenter tip can cause a slight difference of the storage modulus. A “master curve” of the storage modulus as a function of force frequency is established. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 281–288, 2008     

Adhesion of amorphous polymers as a function of temperature probed with AFM force-distance curves

Force-displacement curves have been obtained with a commercial atomic force microscope at different temperatures and probe rates on a thick film of poly(n-butyl methacrylate) and on two films of polystyrene with different molecular weight. In a previous publication [B. Cappella, S.K. Kaliappan, H. Sturm, Macromolecules 38 (2005)1874] the analysis of force-displacement curves has been focused on the stiffness and on the Young's modulus of the samples. In the present publication we consider the temperature dependence of the work of adhesion. We have obtained master curves of the work of adhesion at fixed maximum loads and, by comparing the results of the two analysis, we show that the work of adhesion follows the Williams-Landel-Ferry equation with the same coefficients previously found for the Young's modulus. Furthermore, we show that the temperature dependence of the work of adhesion of the polymers is a consequence of the temperature dependence of the tip-sample contact area and in the end of the temperature dependence of the stiffness and of the elastic-plastic properties of the samples.     

Adhesive Contact Deformation of a Single Microelastomeric Sphere

This paper reports on an experimental study of the adhesive contact of a single microscopic (about 300 μm) elastomer sphere compressed between two smooth parallel glass platens at small imposed deformations. An experimental arrangement that allows the simultaneous measurement of the compressive displacements and the reaction forces is described. A number of interesting phenomena, including the pull-off separation and the “jump” contact phenomena of the microsphere and the moving platen supported by a cantilever, are shown in the experimental force-displacement curve of a loading and unloading cycle. The pull-off forces are demonstrated to not depend upon the applied dimensionless approach (compressive displacement/initial particle diameter), while they increase with the increasing rate at which the interfaces are separated. The predictions of an established contact mechanical adhesive theory, Johnson–Kendall–Roberts (JKR) theory, in which the influence of the surface energy on the contact has been taken into account, are in good agreement with these experimental results. An application of the JKR analysis to the pull-off force provides a reasonable estimate of the interfacial free energy of the contact.     

Measurements of interfacial viscoelasticity with a quartz crystal microbalance: influence of acoustic scattering from a small crystal-sample contact

We discuss the influence of a limited contact size on measurements of high-frequency interfacial viscoelasticity performed with a combination of a quartz crystal microbalance (QCM) and the Johnson-Kendall-Roberts (JKR) apparatus. In this instrument, a sphere-plate contact is established between an elastomeric lens and a quartz resonator. The analysis is carried out in the frame of the sheet-contact model, which states that both the shift of resonance frequency and the bandwidth are proportional to the contact area as long as the contact area is much smaller than the crystal itself. In particular, the ratio of the shift in bandwidth and the shift in frequency (termed the D-f ratio) is predicted to be constant and independent of geometry. However, the experiment does show a slight increase in the D-f ratio with the contact radius when the contact radius is comparable to the wavelength of sound inside the crystal. This effect can be explained by acoustic scattering.     

"Contact" of nanoscale stiff films

We investigated the contact behaviors of a nanoscopic stiff thin film bonded to a compliant substrate and derived an analytical solution for determining the elastic modulus of thin films. Microscopic contact deformations of the gold and polydopamine thin films (<200 nm) coated on polydimethylsiloxane elastomers were measured by indenting a soft tip and analyzed in the framework of the classical plate theory and Johnson-Kendall-Roberts (JKR) contact mechanics. The analysis of this thin film contact mechanics focused on the bending and stretching resistance of thin films and is fundamentally different from conventional indentation measurements where the focus is on the fracture and compression of the films. The analytical solution of the elastic modulus of nanoscopic thin films was validated experimentally using 50 and 100 nm gold thin films coated on polydimethylsiloxane elastomers. The technical application of this analysis was further demonstrated by measuring the elastic modulus of thin films of polydopamine, a recently discovered biomimetic universal coating material. Furthermore, the method presented here is able to quantify the contact behaviors of nanoscopic thin films, effectively providing fundamental design parameters, the elastic modulus, and the work of adhesion, crucial for transferring them effectively into practical applications.     

CALCULATION OF THE YOUNG''''S MODULUS OF AN ADSORBED POLYMER LAYER

Polymer layers adsorbed to a surface or in a confined environment often change their mechanical properties. There is even the possibility of solidification of the confined layer. To judge the stiffness of such a layer, we used the Hertz model to calculate the Young's modulus of the polymer layer in the confinement of AFM experiments with silicon nitride tip with a radius of curvature of R≈50 nm and a glass sphere attached to the cantilever R = 5μm. Since there is no visible indentation of the layer in the AFM experiments, the layer is either penetrated very easily, or the indentation is too small to be seen in a force curve. The latter would be the case for a polymer layer with a Young's modulus above 4×108 Pa in case of an experiment with a silicon nitride tip and 4×105 Pa in case of a glass sphere.     

Measurement of interfacial adhesion between glassy polymers using the JKR method

The surface and interfacial energies of polymers are measured using the JKR-type experiments. A novel method has been developed to prepare samples of glassy polymers for adhesion measurements. A thin layer of a polymer is coated on the surface of an O₂-plasma modified cross-linked poly(dimethylsiloxane) [PDMS] spherical cap resulting in the formation of a composite. Using the JKR theory, the surface energies of polystyrene [PS] and poly(methyl methacrylate) [PMMA] are determined from the measurements of the contact radius as a function of applied load. The results of the JKR-type experiments are compared to adhesion measurements done using the surface forces apparatus (SFA). Adhesion hysteresis was observed for PS-PS contact as well as PMMA-PMMA contact. However, no hysteresis was observed for PDMS-PDMS, PDMS-PS, and PDMS-PMMA contacts. The exact origin of the hysteresis is not clear at present. The current evidence suggests that hysteresis is due to rearrangement of the interface during contact.     

Application of the JKR Method to the Measurement of Adhesion to Langmuir-Blodgett Cellulose Surfaces

The JKR method has been applied for studying adhesion between poly(dimethylsiloxane) (PDMS) caps and Langmuir–Blodgett cellulose surfaces including the substrate, hydrophobized mica, and two flat mineral surfaces, bare mica and glass. The self-adhesion of PDMS caps and oxidized PDMS caps are included as a reference to compare with literature data. The results of the measurements have been compared with previous studies using the surface force apparatus and similar systems. A satisfactory agreement is obtained for simple systems showing no, or very limited, hysteresis between loading and unloading curves. In several cases, however, a large hysteresis is found between loading and unloading curves, with a larger adhesion measured from the pull-off force than from the JKR-curve determined on loading. This is, for instance, the case for PDMS against cellulose. The situation is analogous to that found in wetting studies showing a large hysteresis between advancing and receding contact angles.