Mechanics of nanoparticle adhesion — A continuum approach

12J. TomasThus, with ψ = 1 the ratio of the initial tangential stiffnessdFT*kT= = 4⋅G ⋅ rdδK(15)to the initial normal stiffness according to Eq. (12) is:kkTN( −ν)2⋅1=2 −νHence this ratio ranges from unity, for ν = 0, to 2/3, for ν = 0.5 [63], which isdifferent from the common linear elastic behavior of a cylindrical rod.2.1.2. Elastic displacement of an adhesion contactThe adhesion in the normal loaded contact of spheres with elastic displacementwill be additionally shown. For fine and stiff particles, the Derjaguin, Muller andToporov (DMT) model [43, 65, 66] predicts that half of the interaction forceF H,DMT /2 occurs outside in the annular area which is located at the perimeterclosed by the contact, Eq. (17). This is in contrast to the Johnson, Kendall andRoberts (JKR) model [67], which assumes that all the interactions occur withinthe contact radius of the particles. The median adhesion force F H,DMT (indexH,DMT) of a direct spherical contact can be expressed in terms of the work ofadhesion W A , conventional surface energy γ A or surface tension σ sls asWA = 2⋅ γA= 2⋅ σsls. The index sls means particle surface-adsorption layers (withliquid equivalent mechanical behavior) – particle surface interaction. If only molecularinteractions with separations near the contact contribute to the adhesionforce then the so-called Derjaguin approximation [43] is validFHDMT , sls 12 ,(16)= 4⋅π ⋅σ⋅ r , (17)which corresponds to Bradley’s formula [44]. This surface tension σ sls equals halfthe energy needed to separate two flat surfaces from an equilibrium contact distancea F=0 to infinity [75]:σ∞1CHsls,sls=− ⋅ pVdW ( a) da=2 2a24⋅π⋅aF= 0 (18)F = 0The adhesion force per unit planar surface area or attractive pressure p VdWwhich is used here to describe the van der Waals interactions at contact is equivalentto a theoretical bond strength and can simply calculated as [75] (e.g., p VdW≈ 3–600 MPa):pC4⋅σHsls ,slsVdW= =36⋅π ⋅aaF = 0 F = 0(19)