JKR theory

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. 

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 (/ ). 

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. 

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... 

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. 

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. 

SFA has been traditionally used to measure the forces between modified mica surfaces. Before the JKR theory was developed, Israelachvili and Tabor [57] measured the force versus distance (F vs. d) profile and pull-off force (Pf) between steric acid monolayers assembled on mica surfaces. The authors calculated the surface energy of these monolayers from the Hamaker constant determined from the F versus d data. In a later paper on the measurement of forces between surfaces immersed in a variety of electrolytic solutions, Israelachvili [93] reported that the interfacial energies in aqueous electrolytes varies over a wide range (0.01-10 mJ/m-). In this work Israelachvili found that the adhesion energies depended on pH, type of cation, and the crystallographic orientation of mica. 

As indicated, an implicit assumption of the JKR theory is that there are no interactions outside the contact radius. More specifically, the energy arguments used in the development of the JKR theory do not allow specific locations of the adhesion forces to be determined except that they must be associated with the contact line where the two surfaces of the particle and substrate become joined. Adhesion-induced stresses act at the surface and not a result of action-at-a-distance interatomic forces. This results in a stress singularity at the circumference of the contact radius [41]. The validity of this assumption was first questioned by Derjaguin et al. [42], who proposed an alternative model of adhesion (commonly referred to as the DMT theory ). Needless to say, the predictions of the JKR and DMT models are vastly different, as discussed by Tabor [41]. 

As can be seen, the DMT detachment force is greater than that predicted by the JKR theory. 

Upon comparison of Eqs. 29 and 36, it is readily apparent that both theories predict the same power law dependence of the contact radius on particle radius and elastic moduli. However, the actual value of the contact radius predicted by the JKR theory is that predicted by the DMT model. This implies that, for a given contact radius, the work of adhesion would have to be six times as great in the DMT theory than in the JKR model. It should be apparent that it is both necessary and important to establish which theory correctly describes a system. 

Lest one be lulled into a false sense that, assuming that the JKR theory properly describes particle adhesion within its regime, DeMejo et al. [56] also reported that, for soda-lime glass particles with radii less than about 5 p.m, the contact radius varied, not as the predicted but, rather, as Similar results were reported for other systems including polystyrene spheres on polyurethane [58], as shown in Fig. 2, and for glass particles having radii between about 1 and 100 p,m on a highly compliant, plasticized polyurethane substrate [59] as illustrated in Fig. 3. 

Alternatively, if detachment is associated with a brittle failure, then one must first determine if the fracture followed an elastic loading where an elastic model such as the JKR theory is appropriate or if it follows a plastic or elastic-plastic loading. In this latter case, the force needed to detach the particle from the substrate depends on the specific properties of the materials and the details of the deformations [63]. 

As previously discussed, the JKR theory predicts that the detachment force is independent of the Young s modulus. Yet despite that, when Gady et al. [117] measured the detachment force of polystyrene particles from two elastomeric substrates having Young s moduli of 3.8 and 320 MPa, respectively, they found that the detachment force from only the more compliant substrate agreed with the predicted value. The force needed to separate the particle from the more rigid substrate was about a factor of 20 lower. Estimates of the penetration depth revealed that the particles would penetrate into the more compliant substrate more deeply than the heights of the asperities. Thus, in that case, the spherical particle approximation would be reasonable. On the other hand, the penetration depth... 

The force p needed to compress a single asperity and the displacement 8 of its tip relative to the undeformed region of the substrate was calculated using JKR theory and determined to be related to the radius of curvature of the asperity and the contact radius a by... 

Treating an asperity as an independent particle, JKR theory states that the force Ps needed to effect detachment of a spherical asperity from a planar substrate is given by... 

Several models have been proposed to predict adhesion force—the maximum force required to pull off the surfaces. Among these, the JKR theory is one receiving the greatest attention [2], which says that for an elastic spherical body in contact with a semi-infinite plane, the adhesion force can be estimated by... 

In the limit of large, softer solids in vapor pressure closer to the value marking the onset of capillary condensation, the generalized Hertz and the original JKR theories are found to be qualitatively identical. However, the contact area for zero applied load will in general be different, since it is dependent upon the nature of the source of adhesion ... 

TABLE 1 Single Chemical Bond Forces (in pN) for Every Tip-Substrate Combination, Calculated on the Basis of the JKR Theory of Adhesion Mechanics ... 

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