Mechanics of nanoparticle adhesion â A continuum approach

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32J. TomasFigure 8. Particle contact forces for titania powder (median particle diameter d 50,3 = 610 nm, specificsurface area A S,m = 12 m 2 /g, surface moisture X W = 0.4%, temperature = 20°C) according to thelinear model Eq. (72), non-linear model Eq. (79) for instantaneous consolidation t = 0 and the linearmodel for time consolidation t = 24 h (Eq. (73)). The powder surface moisture X W = 0.4% is accuratelyanalyzed with Karl–Fischer titration. This is equivalent to an idealized mono-molecular adsorptionlayer being in equilibrium with an ambient air temperature of 20°C and 50% humidity.ure of irreversible particle contact stiffness or softness. A shallow slope designatesa low adhesion level F H ≈ F H0 because of stiff particle contacts, but a largeslope means soft contacts, or consequently, a cohesive powder flow behavior [91,147, 149]. The contact flattening may be additionally dependent on time or displacementrate (Section 2.1.3). Thus, the contact reacts softer and, consequently,the adhesion level is higher than before. This new adhesion force slope κ vis ismodified by the viscoplastic contact repulsion coefficient κ p,t , which includes certainviscoplastic microflow at the contact (Table 2 and Fig. 8),κκ + κF = ⋅ F + ⋅FAt ,p p,tHtot ,κ −κ −κ H0κ −κ −κNAt , p pt , At , p pt ,( 1 κ )= + ⋅ F + κ ⋅Fvis H0 vis N(73)with the so-called total viscoplastic contact consolidation coefficient κ vis that includesthe elastic-plastic κ p and the viscoplastic contributions κ p,t of contact flattening,
Mechanics of nanoparticle adhesion — A continuum approach 33κvisκp+ κp,t=κ −κ −κA,t p p,t(74)F=0 r 1,2Eqs. (72) and (73) consider also the flattening response of soft particle contactsat normal force F N = 0 caused by the adhesion force κ⋅F H0 (Krupp [56]) andκ vis ⋅F H0 . Hence, the adhesion force F H0 represents the sphere–sphere contact withoutany contact deformation at minimum particle–surface separation a F=0 . Thisinitial adhesion force F H0 may also include a characteristic nanometer-sizedheight or radius of a rigid spherical asperity a < h > a F=0 ), the contribution of the plate,second term in Eq. (76), can be neglected and the adhesion force may be describedas the sphere-sphere contact [98].Rabinovich and co-workers [113–115] have used the root mean square (RMS)roughness from AFM measurements and the average peak-to-peak distance betweenthese asperities λ r to calculate the interaction between a smooth sphere anda surface with nanoscale roughness profile (index Ra):
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Mechanics of nanoparticle adhesion â A continuum approach