Mechanics of nanoparticle adhesion — A continuum approach

dentation t < / ( ⋅p)Mechanics of nanoparticle adhesion — A continuum approach 31η κ , e.g., t < 5 d for the titania used as a very cohesiveKfpowder (specific surface area A S,m = 12 m 2 /g, with a certain water adsorption capacity).All the material parameters are collected in Table 2.A viscoelastic relaxation in the particle contact may be added as a timedependentfunction of the average modulus of elasticity E * , Yang [61] and Krupp[49] (t relax is the characteristic relaxation time):1 = 1 + 1 − 1 ⋅exp−* * * *E E ( ) ( 0) ∞t →∞ E0t = E∞( tt )The slopes of the elastic–plastic, viscoelastic–viscoplastic yield and adhesionboundaries as well as the unloading and reloading curves, which include a certainrelaxation effect, are influenced by the increasing softness or compliance of thespherical particle contact with loading time (Fig. 6). This model system includesall the essential constitutive functions of the authors named before [41, 55, 57, 60,61]. A survey of the essential contact force-displacement models is given in Fig. 7and Table 1.Obviously, contact deformation and adhesion forces are stochastically distributedmaterial functions. Usually one may focus here only on the characteristic oraveraged values of these constitutive functions.2.3.4. Adhesion force modelStarting with all these force-displacement functions one turns to an adhesion andnormal force correlation to find out the physical basis of strength-stress relationsin continuum mechanics [13, 14, 122, 149]. Replacing the contact area in Eq.(38), the following force–force relation is directly obtained:relaxp F + FVdW H0 NFH = FH0 + pVdW⋅ AK = FH0+ ⋅p2 1A pAf pl VdW+ ⋅ −3 3 AKpfp(69)(70)Therefore, with a so-called elastic–plastic contact consolidation coefficient κ,κpκ =(71)κ −κa linear model for the adhesion force F H as function of normal force F N is obtained(Fig. 8):F κF κF F F( 1 κ )ApH= ⋅H0+ ⋅N= + ⋅H0+ κ ⋅κ NA−κp κA −κp(72)The dimensionless strain characteristic κ is given by the slope of adhesionforce F H which is influenced by predominant plastic contact failure. It is a meas-