Adhesion of small spheres

Abstract

Bradley 1 calculated the adhesive force between rigid spheres to be 2πRΔγ, where Δγ is the surface energy of the spheres. Johnson et al. (JKR) [2] calculated the adhesive force between elastic spheres to be (3/2) πRΔγ and independent of the elastic modulus. Derjaguin et al. [3] published an alternative theory for elastic spheres (DMT theory), and concluded that Bradley's value for the pull-off force was the correct one. Tabor [4] explained the discrepancy in terms of the range of action of the surface forces, z ₀, and introduced a parameter μ≡(RΔγ²/E²z ₀³)^1/3, determining which result is applicable. Subsequently, detailed calculations by Derjaguin and his colleagues [5] and others, assuming a surface force law based on the Lennard-Jones 6–12 potential law, covered the full range of the Tabor parameter. Greenwood and Johnson [6] presented a map delineating the regions of applicability of the different theories. Yao et al. [7] repeated the numerical calculations but using an exact sphere shape instead of the usual paraboloidal approximation. They found that the pull-off force could be less than one-tenth of the JKR value, depending on the value of a ‘strength limit’ σ₀/E, and modified the Johnson and Greenwood map correspondingly. Yao et al.'s numerical calculations for contact between an exact sphere and an elastic half-space are repeated and their values confirmed: but it is shown that the drastic reductions found occur only for spheres that are smaller than atomic dimensions. The limitations imposed by large strain elasticity and by the ‘Derjaguin approximation’ are discussed.

Keywords:

• contact mechanics
• surface energy
• adhesion
• JKR theory

Acknowledgement

I am most grateful to Prof. M. Ciavarella of the Politechnico di Bari for useful discussions, and for bringing to my attention the papers by Maugis, and by Lin and Chen.

Notes

Notes

1. The use of the Derjaguin approximation necessarily implies that the tangential tractions are zero: see 'Extension of Bradley's theory’ below.

2. Bradley refers to them as “molecules”, Rayleigh 18 as “particles”. It is not clear which is the safer term. I would prefer “charges”, but this might imply that the law of force is the electrostatic inverse square law.

3. The first explicit statement I can find is Muller et al.5.

4. The author has belatedly discovered a paper by Wu 19 in which the present problem has been solved using the Argento surface tractions. The (minor) differences in results are being investigated.

5. An idea perhaps envisaged by Derjaguin 17: “das ist so lange berechtigt, wie die Durchmesser dieser Flächenelemente wesentlich grösser sind als die molekulare Wirkungsradius”. But at that time there was no question of elastic deformation.

6. For comparison, using 6-pt Gaussian, − 0.250045 using 8-pt Gaussian.

7. Johnson 21 also gives Steuermann's equations for the pressure, repeating the misprint in Steuermann's paper of an additional factor n.