JKR

Carpick et al [M] used AFM, with a Pt-coated tip on a mica substrate in ultraliigh vacuum, to show that if the defonnation of the substrate and the tip-substrate adhesion are taken into account (the so-called JKR model [175] of elastic adliesive contact), then the frictional force is indeed proportional to the contact area between tip and sample. Flowever, under these smgle-asperity conditions, Amontons law does not hold, since the statistical effect of more asperities coming into play no longer occurs, and the contact area is not simply proportional to the applied load. 

Mangipudi V S ef a/1996 Measurement of interfacial adhesion between glassy polymers using the JKR method Macromoi. Symp. 102 131-43... 

The separation of two surfaces in contact is resisted by adhesive forces. As the nonnal force is decreased, the contact regions pass from conditions of compressive to tensile stress. As revealed by JKR theory, surface tension alone is sufficient to ensure that there is a finite contact area between the two at zero nonnal force. One contribution to adhesion is the work that must be done to increase surface area during separation. If the surfaces have undergone plastic defonnation, the contact area will be even greater at zero nonnal force than predicted by JKR theory. In reality, continued plastic defonnation can occur during separation and also contributes to adhesive work. 

Dutrowski [5] in 1969, and Johnson and coworkers [6] in 1971, independently, observed that relatively small particles, when in contact with each other or with a flat surface, deform, and these deformations are larger than those predicted by the Hertz theory. Johnson and coworkers [6] recognized that the excess deformation was due to the interfacial attractive forces, and modified the original Hertz theory to account for these interfacial forces. This led to the development of a new theory of contact mechanics, widely referred to as the JKR theory. Over the past two decades or so, the contact mechanics principles and the JKR theory have been employed extensively to study the adhesion and friction behavior of a variety of materials. 

The JKR theory relates the interfacial-force-induced contact deformation to the thermodynamic work of adhesion between solids, and provides a theoretical... 

The Hertz theory of contact mechanics has been extended, as in the JKR theory, to describe the equilibrium contact of adhering elastic solids. The JKR formalism has been generalized and extended by Maugis and coworkers to describe certain dynamic elastic contacts. These theoretical developments in contact mechanics are reviewed and summarized in Section 3. Section 3.1 deals with the equilibrium theories of elastic contacts (e.g. Hertz theory, JKR theory, layered bodies, and so on), and the related developments. In Section 3.2, we review some of the work of Maugis and coworkers. 

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. 

The surfaces of all materials interact through van der Waals interactions and other interactions. These interfacial forces, which are attractive in most cases, result in the deformation of the solid bodies in contact. In practice, the radius of the contact zone is higher than the radius predicted by the Hertzian theory (Eq. 7). Johnson et al. [6] modified the Hertzian theory to account for the interfacial interactions, and developed a new theory of contact mechanics, widely known as the JKR theory. In the following section, we discuss the details of the JKR theory. The details of the derivation may be obtained elsewhere [6,20,21]. 

When the surfaces are in contact due to the action of the attractive interfacial forces, a finite tensile load is required to separate the bodies from adhesive contact. This tensile load is called the pull-off force (P ). According to the JKR theory, the pull-off force is related to the thermodynamic work of adhesion (W) and the radius of curvature (/ ). 

Fig. 4. Schematic of the JKR treatment of contact mechanics calculations. The point (, a, P) corresponds to the actual state under the action of interfacial forces and applied load P. P is the equivalent Hertzian load corresponding to contact radius a between the two surfaces, ( o, P) and (S], a, P ) are the Hertzian contact points. The net stored elastic energy and displacement S are calculated as the difference of steps 1 and 2.

To account for some of the shortcomings of the JKR theory, Derjaguin and coworkers [19] developed an alternative theory, known as the DMT theory. According to the DMT theory, the attractive force between the surfaces has a finite range and acts outside the contact zone, where the surface shape is assumed to be Hertzian and not deformed by the effect of the interfacial forces. The predictions of the DMT theory are significantly different compared to the JKR theory. 

There was some argument in the literature over the relative merits and demerits of the JKR and the DMT theories [23-26], but the controversy has now been satisfactorily resolved. A critical comparison of the JKR and DMT theories can be obtained from the literature [23-30]. According to Tabor [23], JKR theory is valid when the dimensionless parameter given by Eq. 25 exceeds a value of about five. 

It has been also shown that when a thin polymer film is directly coated onto a substrate with a low modulus ( < 10 MPa), if the contact radius to layer thickness ratio is large (afh> 20), the surface layer will make a negligible contribution to the stiffness of the system and the layered solid system acts as a homogeneous half-space of substrate material while the surface and interfacial properties are governed by those of the layer [32,33]. The extension of the JKR theory to such layered bodies has two important implications. Firstly, hard and opaque materials can be coated on soft and clear substrates which deform more readily by small surface forces. Secondly, viscoelastic materials can be coated on soft elastic substrates, thereby reducing their time-dependent effects. 

The JKR theory, much like the Hertz theory, assumes a parabolic approximation for the profile of sphere, which is valid for small ratios of contact radii to the sphere s radius. Maugis [34] has shown that for small particles on a soft substrate, this ratio could be so large that such parabolic approximation is no longer valid. Under such conditions, the use of exact expression for the sphere profile is necessary for the applicability of the JKR theory, which is expressed as... 

A study by Shull et al. [35]. on the adhesion of soft and relatively thin elastomeric lenses to glass substrates shows that the compliance (C = S/P) predicted by the JKR theory is larger than the actual value by a constant factor. 

They argue that the Hertzian load (Ph) is not signifieantly affected by the finite size effeets, therefore the JKR expression relating the load to the contact radius and adhesion energy (Eq. 11) should still be valid. Using a combined analytical and computational approach, Hui et al. [36] found that a correction given by Shull et al. for the eompression of such thin lenses was accurate for moderately large eontact radius... 

The JKR theory is essentially an equilibrium balance of energy released due to interfacial bond formation and the stored elastic energy. For simple elastic solids the deformation as a function of load, according to the JKR theory is given by... 

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. 

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.

Fig. 8. (ii) Geometry and interferometry in the SFA. The distance between the surfaces is determined from the wavelengths of FECO. (a) The PECO fringes when the surfaces are in contact. The separation profile, D versus r, can be measured from the fringe profile, and compared to that predicted by the JKR theory of contact mechanics, (b) The FECO when the surfaces are separated. By measuring the wavelengths of the fringes when the surfaces are in contact and when they are separated, we can determine the distance between the two surfaces. 

Given the importance of surface and interfacial energies in determining the interfacial adhesion between materials, and the unreliability of the contact angle methods to predict the surface energetics of solids, it has become necessary to develop a new class of theoretical and experimental tools to measure the surface and interfacial energetics of solids. Thia new class of methods is based on the recent developments in the theories of contact mechanics, particularly the JKR theory. 

Johnson and coworkers [6], in their original paper on the JKR theory, reported the measurements of surface energies and interfacial adhesion of soft elastomeric materials. Israelachvili and coworkers [68,69], and Tirrell and coworkers [62, 63,70,88-90] used the SFA to measure the surface energies of self-assembled monolayers and polymer films, respectively. Chaudhury and coworkers [47-50] adapted the JKR technique to measure the surface energies and interfacial adhesion between self-assembled monolayers. More recently, Mangipudi and coworkers [55] modified the JKR technique to measure the surface energies of glassy polymers. All these measurements are reviewed in this section. 

JKR-type mea.surements on soft elastomers. In their original paper, Johnson and coworkers [6] reported on adhesion measurements between soft materials. 

Fig. 10. Normalized contact radius as a function of normalized load for gelatin spheres in contact with poly(methyl methacrylate). Shown here arc some of the early data obtained using the JKR method by Johnson and coworkers. (Reproduced with permission from ref. [6. )...

In an attempt to determine the applicability of JKR and DMT theories, Lee [91] measured the no-load contact radius of crosslinked silicone rubber spheres in contact with a glass slide as a function of their radii of curvature (R) and elastic moduli (K). In these experiments, Lee found that a thin layer of silicone gel transferred onto the glass slide. From a plot of versus R, using Eq. 13 of the JKR theory, Lee determined that the work of adhesion was about 70 7 mJ/m". a value in clo.se agreement with that determined by Johnson and coworkers 6 using Eqs. 11 and 16. 

JKR type mea.surement.s on monolayers depo.sited on. soft elastomers. The recent interest in the JKR experiments has been stimulated by the work of Chaudhury and coworkers [47-50J. In a 1991 paper, Chaudhury and White-sides [47] reported their extensive studies on the measurement of interfacial work of adhesion and surface energies of elastomeric solids. The motivation for this work was to study the physico-organic chemistry of solid surfaces and interfaces. 

In a separate study using the JKR technique, Chaudhury and Owen [48,49] attempted to understand the correlation between the contact adhesion hysteresis and the phase state of the monolayers films. In these studies, Chaudhury and Owen prepared self-assembled layers of hydrolyzed hexadecyltrichlorosilane (HTS) on oxidized PDMS surfaces at varying degrees of coverage by vapor phase adsorption. The phase state of the monolayers changes from crystalline (solidlike) to amoiphous (liquid-like) as the surface coverage (0s) decreases. It was found that contact adhesion hysteresis was the highest for the most closely packed... 

Fractional coverage (% Phase state y,v by JKR method (mJ/m-) Contact angle hysteresis (mJ/m ) ... 

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