Mechanics of nanoparticle adhesion â A continuum approach

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18J. TomasAdditionally, the rate-dependent, perfect viscoplastic deformation (at the pointof yielding) expressed by contact viscosity η K times indentation rate h Kis assumedto be equivalent to yield strength p f multiplied by indentation height incrementh Kp ⋅ h = η ⋅h (35)f K K Kand one obtains again a linear model regarding strain rate:F = η ⋅ A = π ⋅d ⋅η⋅h(36)Nvis , K K 12 , K KAn attractive viscous force is observed, e.g., for capillary numbersCa = η ⋅ h / σ > 1 when comparatively strong bonds of (low-viscous) liquidK K lgbridges are extended with negative velocity –h [71–73].KConsequently, the particle material parameters: contact micro-yield strength p fand viscosity η K are measures of irreversible particle contact stiffness or softness.Both plastic and viscous contact yield effects were intensified by mobile adsorptionlayers on the surfaces. The sum of deformation increments results in the energydissipation. For larger particle contact areas A K , the conventional linear elasticand constant plastic behavior is expected.Now, what are the consequences of small contact flattening with respect to avarying, i.e., load or pre-history-dependent adhesion?2.2. Particle contact consolidation by varying adhesion forceKrupp [49] and Sperling [48, 56] developed a model for the increase of adhesionforce F H (index H) of the contact. This considerable effect is called here as “consolidation”and is expressed as the sum of adhesion force F H0 according to Eq.(17) plus an attractive/repulsive force contribution due to irreversible plastic flatteningof the spheres (p f is the repulsive “microhardness” or micro-yield strengthof the softer contact material of the two particles, σ ss /a 0 is the attractive contactpressure, index ss represents solid–vacuum–solid interaction): 2⋅σFH = 4⋅π⋅r 12 ,⋅σss⋅ 1+ a ⋅ pss0 f(37)Dahneke [52] modified this adhesion model by the van der Waals force withoutany contact deformation F H0 plus an attractive van der Waals pressure (force perunit surface) p VdW contribution due to partially increasing flattening of the sphereswhich form a circular contact area A K (C H is the Hamaker constant based on interactingmolecule pair additivity [69, 75]):
Mechanics of nanoparticle adhesion — A continuum approach 19H 12 ,KH=H0+K⋅ VdW= ⋅ 12 +6 a a00F F A pC ⋅r 2⋅h ⋅ (38)The distance a 0 denotes a characteristic adhesion separation. If stiff molecularinteractions are provided (no compression of electron sheath), this separation a 0was assumed to be constant during contact loading. By addition the elastic repulsionof the solid material according to Hertz, Eq. (7), to this attraction force, Eq.(38), and by deriving the total force F tot with respect to h K , the maximum adhesionforce was obtained as absolute valueF2CH⋅r12 ,2⋅CH⋅r12,Hmax ,= ⋅ 1+2 26⋅ a * 70 27⋅E ⋅a0which occurs at the center approach of the spheres [52]:h2CH⋅r12,Kmax ,=* 2 69⋅E⋅a0, (39)(40)But as mentioned before, this increase of contact area with elastic deformationdoes not lead to a significant increase of attractive adhesion force. The reversibleelastic repulsion restitutes always the initial contact configuration. The practicalexperience with the mechanical behavior of fine powders shows that an increaseof adhesion force is influenced by an irreversible or “frozen” contact flatteningwhich depends on the external force F N [57].Generally, if this external compressive normal force F N is acting at a single softcontact of two isotropic, stiff, smooth, mono-disperse spheres the previous contactpoint is deformed to a contact area, Fig. 1a to Fig. 2c, and the adhesion force betweenthese two partners increases, see in Fig. 3 the so-called “adhesion boundary”for incipient contact detachment. During this surface stressing the rigid particleis not so much deformed that it undergoes a certain change of the particleshape. In contrast, soft particle matter such as biological cells or macromolecularorganic material do not behave so.For soft contacts Rumpf et al. [62] have developed a constitutive model approachto describe the linear increase of adhesion force F H , mainly for plastic contactdeformation:( ) pVdW pVdWFH = 1+ ⋅ FH0 + ⋅ FN = 1+ κp ⋅ FH0 + κp⋅F(41)N pf pfWith analogous prerequisites and derivation, Molerus [14] obtained an equivalentexpression:
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Mechanics of nanoparticle adhesion â A continuum approach