Mechanics of nanoparticle adhesion â A continuum approach

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28J. Tomas[58], Sadd [55] and, especially, Thornton [53, 60]. A complete survey of loading,unloading, reloading, dissipation and detachment behaviors of titania is shown inFig. 5 as a combination of Fig. 1a to Fig. 4e. This approach may be expressedhere in terms of engineering mechanics of macroscopic continua [1, 2] as the history-dependentcontact behavior.2.3.3. Viscoplastic contact behavior and time dependent consolidationAn elastic-plastic contact may be additionally deformed during the indentationtime, e.g., by viscoplastic flow (Section 2.1.3). Thus, the adhesion force increaseswith interaction time [32, 49, 77, 128]. This time-dependent consolidation behavior(index t) of particle contacts in a powder bulk was previously described by aparallel series (summation) of adhesion forces, see Table 1, last line marked withTomas [122–125, 146–149]. This method refers more to incipient sintering orcontact fusion of a thermally-sensitive particle material [62] without interstitialadsorption layers. This micro-process is very temperature sensitive [122, 124,125, 146].Additionally, the increasing adhesion may be considered in terms of a sequenceof rheological models as the sum of resistances due to plastic and viscoplastic repulsionκ p + κ p,t , line 5 in Table 2. Hence the repulsion effect of “cold” viscousflow of comparatively strongly-bonded adsorption layers on the particle surface istaken into consideration. This rheological model is valid only for a short-term in-Table 2.Material parameters for characteristic adhesion force functions F H (F N ) in Fig. 8Constitutive model of contactdeformationRepulsion coefficientConstitutive models ofcombined contact deformationContact area ratioContact consolidationcoefficientIntersection with F N -axis(abscissa)Instantaneous contactconsolidationplasticκpCVdWHsls ,p= p =3f 6 ⋅ π ⋅ a F = 0 ⋅ p felastic–plastic2AplTime-dependent consolidationviscoplasticκpt ,pVdW= ⋅ tηKelastic–plastic and viscoplasticκA= + κ3At ,= +3 ⋅ ( Apl+ A3el )3 ⋅ ( Apl + Avis + Ael)κpκ =κ −κApF a h pNZ ,≈−π ⋅F=0 ⋅r⋅12 , f≠ f( CHsls, )κFvis2Aplκp+ κp,t=κ −κ −κNZtot , ,+ AAt , p pt ,( CHsls, )visπ ⋅a ⋅h ⋅ p≈−1+ p ⋅t / η≠ fF=0 r12, ffK
Mechanics of nanoparticle adhesion — A continuum approach 29Figure 6. Characteristic elastic–plastic, viscoelastic–viscoplastic particle contact deformations (titania,primary particles d = 20–300 nm, surface diameter d S = 200 nm, median particle diameterd 50,3 = 610 nm, specific surface area A S,m = 12 m 2 /g, solid density ρ s = 3870 kg/m 3 , surface moistureX W = 0.4%, temperature θ = 20°C, loading time t = 24 h). The material data, modulus of elasticityE = 50 kN/mm 2 , modulus of relaxation E ∞ = 25 kN/mm 2 , relaxation time t relax = 24 h, plastic microyieldstrength p f = 400 N/mm 2 , contact viscosity η K = 1.8·10 14 Pa·s, Poisson ratio ν = 0.28, Hamakerconstant C H,sls = 12.6·10 -20 J, equilibrium separation for dipole interaction a F=0 = 0.336 nm, contactarea ratio κ A = 5/6 are assumed as appropriate for the characteristic contact properties. The plasticrepulsion coefficient κ p = 0.44 and viscoplastic repulsion coefficient κ p,t = 0.09 are recalculatedfrom shear-test data in a powder continuum [147, 149].
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Mechanics of nanoparticle adhesion â A continuum approach