Mechanics of nanoparticle adhesion — A continuum approach

24J. TomasIntroducing the particle center approach of the two particles Eq. (6) in Eq. (55),a very useful linear force–displacement model approach is obtained again for κ A ≈constant:( )F + F = π ⋅r ⋅ p ⋅ κ −κ ⋅h(58)N H0 1,2 f A p KBut if one considers the contact area ratio of Eq. (63), a slightly nonlinear (progressivelyincreasing) curve is obtained. Using the elastic–plastic contact consolidationcoefficient κ due to definition (Eq. (71)) one can also write:π ⋅r⋅ p ⋅κF F h1+κ1,2 f AN+H0= ⋅KThe curve of this model is shown in Fig. 2 for titania powder which wasrecalculated from material data and shear test data [147, 149]. The slope of thisplastic curve is a measure of irreversible particle contact stiffness or softness, Eq.(34). Because of particle adhesion impact, the transition point for plastic yieldingY is shifted to the left compared with the rough calculation of the displacementlimit h K,f by Eq. (30).The previous contact model may be supplemented by viscoplastic stress-strainbehavior, i.e., strain-rate dependence on initial yield stress. For elastic–viscoplasticcontact, one obtains deformation with Eqs. (36) and (58) (κ A ≈ constant):F + F = π ⋅r ⋅η ⋅ κ −κ ⋅h(60)( )N H0 1,2 K A p,t K(59)A dimensionless viscoplastic contact repulsion coefficient κ p,t is introduced asthe ratio of the van der Waals attraction to viscoplastic repulsion effects which areadditionally acting in the contact after attaining the maximum pressure for yielding.κp,tp=η ⋅hKVdWThe consequences for the variation in adhesion force are discussed in Section2.3.3 [147].2.3.2. Unloading and reloading hysteresis and contact detachmentBetween the points U – E (see Fig. 3), the contact recovers elastically along anextended Hertzian parabolic curve, Eq. (7), down to the perfect plastic displacement,h K,E , obtained in combination with Eq. (58):K32= − ⋅K,E K,U K,f K,U(61)h h h h (62)