Mechanics of nanoparticle adhesion — A continuum approach

Mechanics of nanoparticle adhesion — A continuum approach 210.017 for lactose, κ p = 0.016 for CaCO 3 ) with centrifuge tests [92] as well as bySingh et al. [97] (κ p = 0.12 for poly(methylmethacrylate), κ p ≈ 0 for very hardsapphire, α-Al 2 O 3 ) with an Atomic Force Microscope (AFM). The two methodsare compared with rigid and rough glass spheres (d = 0.1–10 µm), without anycontact deformation, by Hoffmann et al. [98]. Additionally, using the isostatictensile strength σ 0 determined by powder shear tests [91, 122, 147, 149, 151], thisadhesion level is of the same order of magnitude as the average of centrifuge tests(see Spindler et al. [99]).The enhancement of adhesion force F H due to pre-consolidation was confirmedby Tabor [30], Maugis [85, 86] and Visser [110]. Also, Maugis and Pollock [58]found that separation was always brittle (index br) with a small initial slope ofpull-off force, dF N,Z,br /dF N (F N,Z,br ≈ – F H ), for a comparatively small surface energyσ ss of the rigid sphere–gold plate contact (index ss). In contrast, a pull-offforce F N,Z,br proportional to FNwas obtained from the JKR theory [58] for thefull plastic range of high loading and brittle separation of the contact (Table 1):FF* NN,Z,br=−σss⋅E⋅3π ⋅ pf(46)Additionally, a load-dependent adhesion force was also experimentally confirmedin wet environment of the particle contact by Butt and co-workers [78, 79]and Higashitani and co-workers [87] with AFM measurements.The dominant plastic contact deformation of surface asperities during thechemical–mechanical polishing process of silicon wafers was also recognized,e.g., by Rimai and Busnaina [111] and Ahmadi and Xia [141]. These particlesurfacecontacts and, consequently, asperity stressing by simultaneous normalpressure and shearing, contact deformation, microcrack initiation and propagation,and microfracture of brittle silicon asperity peaks affect directly the polishingperformance. Thus the Coulomb friction becomes dominant also in a wet environment.2.3. Variation in adhesion due to non-elastic contact consolidation2.3.1. Elastic–plastic force–displacement modelAll interparticle forces can be expressed in terms of a single potential functionFi =±∂Ui( hi)/∂hiand thus are superposed. This is valid only for a conservativesystem in which the work done by the force F i versus distance h i is not dissipatedas heat, but remains in the form of mechanical energy, simply in terms of irreversibledeformation, e.g., initiation of nanoscale distortions, dislocations or latticestacking faults. The overall potential function may be written as the sum ofthe potential energies of a single contact i and all particle pairs j. Minimizing thispotential function ∂U / ∂ h = 0 one obtains the potential-force balance.ij ijij