Mechanics of nanoparticle adhesion â A continuum approach

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20J. TomaspF = F + ⋅ F = F + ⋅ F(42)VdWκH H0 p N H0 p NfThe adhesion force F H0 without additional consolidation (F N = 0) can be approachedas a single rigid sphere–sphere contact (Fig. 1a). But, if this particlecontact is soft enough the contact is flattened by an external normal force F N to aplate–plate contact (Fig. 2c). The coefficient κ p describes a dimensionless ratio ofattractive van der Waals pressure p VdW for a plate–plate model, Eq. (19), to repulsiveparticle micro-hardness p f which is temperature sensitive:p Cκ (43)VdWH,slsp= =p 3f 6 ⋅ π ⋅ aF= 0 ⋅ pfThis is referred to here as a plastic repulsion coefficient. The Hamaker constantC H,sls for solid–liquid–solid interaction (index sls) according to Lifshitz’ theory[70] is related to continuous media which depends on their permittivities (dielectricconstants) and refractive indices [75]. The characteristic adhesion separationfor a direct contact is of a molecular scale (atomic center-to-center distance) andcan be estimated for a molecular force equilibrium (a = a F=0 ) or interaction potentialminimum [75, 76, 91]. Its magnitude is about a F=0 ≈ 0.3–0.4 nm. This separationdepends mainly on the properties of liquid-equivalent packed adsorbed waterlayers. This particle contact behavior is influenced by mobile adsorption layersdue to molecular rearrangement. The minimum separation a F=0 is assumed to beconstant during loading and unloading for technologically relevant powder pressuresσ < 100 kPa (Fig. 2c).For a very hard contact this plastic repulsion coefficient is infinitely small, i.e.,κ p ≈ 0, and for a soft contact κ p → 1.If the contact circle radius r K is small compared to the particle diameter d, theelastic and plastic contact displacements can be combined and expressed with theannular elastic A el and circular plastic A pl contact area ratio [57]:pF F FVdWH=H0+ ⋅N2 A elpf⋅ 1+ ⋅ 3 Apl(44)For a perfect plastic contact displacement A el → 0 and one obtains again Eq.(42):F ≈ F + κ ⋅ F(45)H H0 p NThis linear enhancement of adhesion force F H with increasing preconsolidationforce F N , Eqs. (41), (42) and (45), was experimentally confirmedfor micrometer sized particles, e.g., by Schütz [94, 95] (κ p = 0.3 for limestone)and Newton [96] (κ p = 0.333 for poly(ethylene glycol), κ p = 0.076 for starch, κ p =
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
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Mechanics of nanoparticle adhesion â A continuum approach