Mechanics of nanoparticle adhesion — A continuum approach

Mechanics of nanoparticle adhesion — A continuum approach 39This can be rearranged if one uses the dimensionless plastic repulsion coefficientκ p according to Eq. (43) to obtain the following simple expression:v1,H,HL2( − ) 1/ 21 e a 6⋅p= ⋅ ⋅κ⋅2eF = 0 fdpρs(90)Unfortunately, Eq. (90) does not include the increase of “soft” contact flatteningresponse h K by increasing particle impact velocity v 1 . Now this dominant energyabsorption W diss during particle impact stressing, beginning at any unloadingpoint U, is considered approximately as a trapezium-shaped area between elastic–plastic yield boundary and adhesion boundary for the contact of particles 1 and 2in the force–displacement diagram of Fig. 9. With the contribution of the work ofadhesion W A to separate this contact from equilibrium separation a F=0 to infinity,the energy balance gives (A K is the contact area):m212 ,2 2( )hKU, hKU,⋅ v − v = F ( h ) d h + −F ( h ) dh1 1R , N K K N,Z K KhKf, hKf,+ A ⋅ −p ( a) daK∞aF=0VdW( )3⋅pv v h h h a2 2 f2 21−1,R= ⋅ κA ⋅2K,U−K,f+ κp ⋅K,U⋅= 04⋅Fρs⋅r1,2(91)(92)The difference in characteristic impact velocities results directly in a center approach,h K,U , expressed by the unloading point U. The response of this contactdisplacement h K,U is a consolidation force, F N,U . Additionally, a certain preconsolidationlevel, F N,U , in a shear zone may affect the sticking/bounce probability.If the rebound velocity v 1,R = 0 the two particles will adhere. Consequently,the critical sticking velocity v 1,H is obtained without any additional losses, e.g.,due to elastic wave propagation:2 2( )3⋅pv = ⋅κ⋅ h − h + κ ⋅h ⋅af1,H 2 A K,U K,f p K,U 04 F =⋅ρs⋅r1,2(93)For example using data from Refs. [147, 149], this critical sticking velocity liesbetween 0.2 and 1 m/s for titania, curve in the sticking velocity–displacementdiagram in Fig. 9, which is equivalent to an average pressure level σ M,st = 2 to 18kPa (or major principal stress σ 1 = 4 to 33 kPa) [151]. These calculation results ofparticle adhesion are in agreement with the practical experiences in powder handlingand transportation, e.g., undesired powder blocking at conveyer transferchutes. In terms of powder flow the behavior after multiple stressing of soft de-