Adhesion of small spheres

ISSN： 1478-6443

Source： Philosophical Magazine, Vol.89, Iss.11, 2009-04, pp. : 945-965

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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, z0, and introduced a parameter μ≡(RΔγ²/E²z0³)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.