Mechanics of nanoparticle adhesion â A continuum approach

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36J. Tomas( )*W =−8⋅E r ⋅ h −h15diss 1, 2 K , U K,A5( ) ( )+ π ⋅r ⋅ p ⋅κ ⋅h −κ⋅ h −h ⋅ h −h12 , f A K, U p K, U K, A K, U KA ,(81)Additionally, the specific or mass-related energy absorption includes the averageparticle mass m = 4/3⋅πr ρ a characteristic contact number in the bulk3P 1,2 spowder (coordination number k ≈ π/ε [13]) and the dissipative work W m,diss =kW ⋅ / m:dissPW5/2* −KU KA= −Eh h, ,mdiss ,20⋅ε⋅ρ⋅rs 12 ,( , , )3⋅π⋅ p ⋅ h −h+ ⋅⋅ − ⋅ −32( )f K U K Aκ h κ h h2 A K, U p K, U K,A⋅r⋅ε⋅ρ12 , s(82)A specific energy absorption of 3 to 85 µJ/g was dissipated during a singleunloading–reloading cycle in the titania bulk powder with an average pressure ofonly σ M,st = 2 to 18 kPa (or major principal stress σ 1 = 4 to 33 kPa) [147, 149].3. PARTICLE IMPACT AND CONTACT DISPLACEMENT RESPONSEIn a shear zone, when two particles (particle 2 is assumed to be fixed) come intocontact and collide, the velocity of particle 1 is reduced gradually. Part of the initialkinetic energy is radiated into both particles as elastic waves. Now the contactforce reaches a maximum value (maximum de-acceleration) and the particle velocityis reduced to zero.hKW 2 E r h h 4 E r h (83)* 3 * 5el= ⋅1,2⋅Kd3 K= ⋅15 1,2⋅K03With the particle mass m = ρ ⋅ 4 ⋅π ⋅r, the correlation between particle velocityv 1 and center approach h K is1,2 s 3 1,2obtained: = E h⋅ *2 K13⋅π ⋅ρrs 12 ,v5/2(84)In the recovery stage the stored elastic energy is released and converted intokinetic energy and the particle moves with the rebound velocity v 1,R into the oppositedirection.
Mechanics of nanoparticle adhesion — A continuum approach 37The so-called impact number or coefficient of restitution e=Fˆˆ1,R/ F1indicatesthe impact force ratio of the contact decompression phase after impact and thecontact compression phase during impact, e = 0 for perfect plastic, 0 < e < 1 forelastic–plastic, e = 1 for perfect elastic behavior, see examples in Refs. [29, 126,130, 132]. Thus e 2 < 1 characterizes the energy dissipation (W diss is the inelastic2deformation work of particle contact, Ekin,1 = mP ⋅ v1 / 2 is the kinetic energy ofparticle 1 before impact):2 kin,1−disse = E W(85)Ekin,1In terms of a certain probability of particle adhesion inside of the contact zonea critical velocity (index H) as the stick/bounce criterion was derived by Thornton(index Th) [60] who used the JKR model: = ⋅ d ⋅ E 21.871⋅FH,JKR3⋅FH,JKR1,H,Th m2P*v1/3(86)For an impact velocity v 1 > v 1,H particle bounce occurs and the coefficient ofrestitution is obtained as [60]:2 21,Rv1,H12 2v1 v1ve = = −(87)Even if the impact velocity v 1 is 10-times higher than the critical sticking velocityv 1,H,Th the coefficient of restitution is 0.995 [60].But in terms of combined elastic–plastic deformation the kinetic energy ismainly dissipated. If one uses the center approach h K,f of Eq. (30) the critical impactvelocity v 1,f for incipient plastic yield (index f) is calculated from Eq. (84) as[131–133]:vπ⋅ p pf= ⋅ E 3⋅ρsf1,f *2(88)The critical velocity v 1,H to stick or to adhere the particles with a plastic contactdeformation was derived by Hiller (Index HL) [126]:2( 1−e ) 1/ 21H,sls1,H,HL= ⋅ ⋅2 2e d π ⋅aF= 0⋅ 6⋅ρs⋅pfvC(89)
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Mechanics of nanoparticle adhesion â A continuum approach