JKR apparatus

In these studies, a variety of experimental tools have been employed by different researchers. These include the JKR apparatus and the SFA. Each one of these tools offers certain advantages over the others. These experimental tools are briefly described in the following section. 

This theory appears to work adequately for most SFA experiments, and is the basis for the so-called JKR apparatus which measures the deformation of a rubbery polydimethylsiloxane (PDMS) hemisphere against a flat surface (see Figure 20.4, taken from ref. (78)) The difficulty with the JKR theory is the prediction of infinite stress at the contact boundary. The true situation lies somewhere between the DMT and JKR limits and rather complicated numerical models employing a Lennart-Jones potential to describe the surface forces resolve this issue (79, 80). As a general rule, the JKR theory is most appropriate for large R values and deformable materials, whereas the DMT approach is better for stiff materials and small deformations. 

Section 4.1 briefly describes some of the commonly employed experimental tools and procedures. Chaudhury et al., Israelachvili et al. and Tirrell et al. employed contact mechanics based approach to estimate surface energies of different self-assembled monolayers and polymers. In these studies, the results of these measurements were compared to the results of contact angle measurements. These measurements are reviewed in Section 4.2. The JKR type measurements are discussed in Section 4.2.1, and the measurements done using the surface forces apparatus (SFA) are reviewed in Section 4.2.2. 

In the JKR experiments, a macroscopic spherical cap of a soft, elastic material is in contact with a planar surface. In these experiments, the contact radius is measured as a function of the applied load (a versus P) using an optical microscope, and the interfacial adhesion (W) is determined using Eqs. 11 and 16. In their original work, Johnson et al. [6] measured a versus P between a rubber-rubber interface, and the interface between crosslinked silicone rubber sphere and poly(methyl methacrylate) flat. The apparatus used for these measurements was fairly simple. The contact radius was measured using a simple optical microscope. This type of measurement is particularly suitable for soft elastic materials. 

Fig. 7. Schematic of the apparatus used for JKR type adhesion measurements. For a constant applied displacement S, the contact radius a and the load P are measured.

The JKR theory predicts correct contact radii for relative soft surfaces with effective radii larger than 100 /an. This was shown in direct force measurements by the surface forces apparatus [217, 218] or specifically designed systems. For smaller spheres it was verified using the colloidal probe technique [219],... 

The JKR theory, similar to the Hertz theory, is a continuum theory in which two elastic semi-infinite bodies are in a non-conforming contact. Recently, the contact of layered solids has been addressed within the framework of the JKR theory. In a fundamental study, Sridhar et al. [32] analyzed the adhesion of elastic layers used in the SFA and compared it with the JKR analysis for a homogeneous isotropic half-space. As mentioned previously and depicted in Fig. 5, in SFA thin films of mica or polymeric materials ( i, /ji) are put on an adhesive layer Ej, I12) coated onto quartz cylinders ( 3, /i3). Sridhar et al. followed two separate approaches. In the first approach, based on finite element analysis, it is assumed that the thickness of the layers and their individual elastic constants are known in advance, a case which is rare. The adhesion characteristics, including the pull-off force are shown to depend not only on the adhesion energy, but also on the ratios of elastic moduli and the layers thickness. In the second approach, a procedure is proposed for calibrating the apparatus in situ to find the effective modulus e as a function of contact radius a. In this approach, it is necessary to measure the load, contact area... 

Fig. 11. Use of the JKR equation (Eq. 18) relating applied force, F, to radius of contact, a, to analyse results from the surface forces apparatus. Surface energy, y, of surfactant layers (DMPE), cf. (CTAB) (after Chen et al. [24]).

The JKR equation is usually employed to interpret the results of the Surface forces apparatus and of Atomic force microscopy, which may be employed to study adhesion between two surfaces. 

It has recently become common to use the JKR theory (Johnson, Kendall Roberts, 1971) to extract the surface and inteifacial energies of polymeric materials from adhesion tests with micro-probe instruments such as the Surface Force Apparatus and the Atomic Force Microscope. However the JKR theory strictly applies only to perfectly elastic solids. The paper will review progress in extending the JKR theory to the contact mechanics and adhesion of linear viscoelastic spheres. The observed effects of adhesion hysteresis and rate-dependent adhesion are predicted by the extended eory. 

In recent years it has been demonstrated that also adhesion (or adhesion hysteresis) plays an important role in friction. Israelachvili and coworkers could show that friction and adhesion hystereses are, in general, directly correlated if certain assumptions are fulfilled. These authors have proposed models based on data obtained by surface forces apparatus (SFA) experiments, e. g. the cobblestone model of interfacial friction (4). In addition, several groups described the application of continuum contact mechanics (e.g. Johnson-Kendall-Roberts (JKR) theory (5)) to describe friction data measured between flat surfaces and nanometer sized contacts (d). 

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