Mechanics of nanoparticle adhesion — A continuum approach

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: