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2 APM Proceedings

MICROMECHANICS OF FINE PARTICLEADHESION - CONTACT MODELS AND

ENERGY ABSORPTION

Juergen [email protected]

Abstract

The mechanical product behaviour of dry, ultrafine cohesive powders(d < 10µm) is characterized by insufficient flowability and large compress-ibility. These powders show a wide variety of typical flow problems thatcause insufficient apparatus and system reliability of processing plants. Con-sequently, a comparatively large energy input is necessary to promote thenon-rapid frictional shear flow in powder handling practice. Thus, it is veryessential to understand the fundamentals of particle adhesion with respect toproduct quality assessment and process performance in particle technology.

Comprehensive models are shown that describe the elastic-plastic force-displacement and frictional moment-angle behaviour of adhesive contacts ofisotropic smooth spheres. Using the model stiff particles with soft contacts,a sphere-sphere interaction of van der Waals forces without any contact de-formation describes the stiff attractive term. The soft micro-contact responsegenerates a flattened contact, i.e. plate-plate interaction, and increasing ad-hesion. These increasing adhesion forces between particles directly dependon this frozen irreversible deformation. Thus, the adhesion force is foundto be load dependent. It essentially contributes to the tangential forces inan elastic-plastic frictional contact with partially sticking within the contactplane and microslip. The load dependent rolling resistance and torque ofmobilized frictional contact rotation (spin) are also shown.

Next, the consequences of unloading and reloading paths are discussedwith respect to energy absorption. The total energy absorption comprisescontributions by elastic-dissipative hysteresis due to microslip within the con-tact plane and by fully developed friction work when the friction limits ofdisplacements are exceeded. With increasing contact flattening by normalload, these friction limits, hysteresis and friction work increase. This knowl-edge is used to improve the apparatus design and the reliability of processsystems in particle technology.

MICROMECHANICS OF FINE PARTICLE ADHESION - CONTACT MODELSAND ENERGY ABSORPTION 3

1 Introdution

In particle processing and product handling of fine (d < 100µm), ultrafine (d <10µm) and nanosized particles (d < 0.1µm), the well-known flow problems of drycohesive powders in process apparatuses or storage and transportation containersinclude bridging, channelling, widely spread residence time distribution associatedwith time consolidation or caking effects, chemical conversions and deteriorationof bioparticles. Avalanching effects and oscillating mass flow rates in conveyorslead to feeding and dosing problems. Finally, insufficient apparatus and systemreliability of powder processing plants are also related to these adhesion problems.The challenge is to understand the fundamentals of particle adhesion with respect toproduct quality assessment, process performance and control in powder technology,i.e., in particle conversion, formulation and handling.

This micromechanical philosophy in powder mechanics has been steadily con-tinued by the author [1-11].The fundamentals of the macroscopic cohesive powderconsolidation and frictional flow have been related to the microscopic interactionsbetween the particles that are in contact [5-10]. In this context, particle adhesion iscaused by surface and field forces (van der Waals, electrostatic and magnetic forces),by material bridges between particle surfaces (liquid and solid bridges [1-3], floccu-lants) and by mechanical interlocking of extreme particle shapes. A comprehensiveliterature review may be read in previous papers [8, 11, 12].

It is worth to note here that van der Waals forces - the focus of this paper - actat the surfaces of dry ultrafine particles. They are dominant and approximately 104

- 106 times the gravitational force. The state of arts in constitutive modelling ofelastic, elastic-adhesion, elastic-dissipative, plastic-adhesion and plastic-dissipativecontact deformation response of a single isotropic contact of two smooth sphereswas discussed in previous papers [6, 8, 11, 12] and is not shown here.

The dominant microscopic effect of variable adhesion forces at particle contactshas been the physical basis of universal models for particle adhesion that includesthe elastic-plastic and viscoplastic particle contact behaviour with hysteresis andconsequently, energy dissipation. Various contact elastic-plastic deformation pathsfor non-rapid loading, unloading and detachment have been discussed. As a result,the varying adhesion forces between particles directly depend on this frozen irre-versible deformation. Thus, the adhesion force has been found to be load dependent[6-11].

To continue these new constitutive models, the consequences of elastic-dissipativeas well as elastic-plastic, frictional unloading and reloading paths of normal andtangential forces, rolling and torsional moments are discussed with respect to theenergy absorption [13]. This energy absorption comprises contributions by elastic-dissipative hysteresis due to microslip within the contact plane and by fully devel-oped friction work when the friction limits of displacements are exceeded duringcontact sliding, particle rolling or rotation. Therefore one may expect that with in-creasing contact flattening by normal load the friction limits, hysteresis and frictionwork increase.

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2 Constitutive models for elastic-plastic, dissipa-

tive behaviour and load dependent adhesion

This paper is intended to focus on the new model of isotropic, stiff, and linearelastic, spheres that are approaching to soft contacts by attractive adhesion forcesof smooth surfaces. The soft or compliant, elastic-plastic contact displacement asresponse is assumed to be small hK/d � 1 compared to the size (diameter) of thestiff particle. The particles may have the mechanical properties of limestone, whichis widely used in powder mechanics as standard material. Its material stiffness isso large that the volume deformation is negligible. During surface stressing thesestiff particles are not so much deformed that they undergo certain changes of theparticle shapes.

2.1 Particle Contact Constitutive Model for Normal Load-ing

A typical normal force-displacement diagram for elastic-plastic contact behaviour isdemonstrated in Fig. 1, calculated by the equations collected in Table 1. The zero-point of this diagram hK = 0 is equivalent to the characteristic adhesion separationof direct contact a0. This characteristic adhesion separation for the direct contactof spheres is of a molecular scale (atomic centre-to-centre distance) and amounts toabout a0 = 0.3−0.4nm. With respect to large specific surfaces of ultrafine particles,this separation a0 depends on the properties of liquid-equivalent packed adsorbedwater layers. Consequently, more or less mobile adsorption layers due to condensedhumidity of ambient air influence the particle contact behaviour.

After approaching FN ∝ a−2 (with a = a0 − hK) from an infinite distance -∞ to a minimum separation a = a0, Fig. 1a), the smooth sphere-sphere contactwithout any contact deformation is formed by the short-range attractive adhesionforce FN = −FH0 (the so-called jump in). But this direct contact is not in a state offorce equilibrium and, as the response, is elastically deformed with an approximatedcircular contact area, Fig. 1b). A local plate-plate-contact is formed from theprevious contact point. When this Hertz or DMT curve intersects the abscissa thetotal force equilibrium FN = 0 is obtained. With increasing external normal loadthis soft contact starts at a pressure pf with plastic yielding at the point Y. Thisyield point Y is located here below the abscissa, i.e., the contact force equilibriumFN = 0 includes a certain elastic or elastic-plastic deformation as response of effectiveadhesion force. The maximum pressure pf within the contact plane cannot beexceeded and results in a combined elastic-plastic yield limit of the flattened plate-plate contact with an annular elastic zone and a circular centre. A confined plasticfield with a micro-yield surface is formed inside of the contact circle.

The elastic-plastic yield limit for loading results in a linear function, line Y-Uin Fig. 1c). This elastic-plastic yield limit cannot be crossed. Between the elastic-plastic yield limit and the adhesion limit the elastic domain is located. Any loadFN yields an increasing displacement hK. But, if one would unload, beginning atarbitrary point U, the elastically deformed, annular contact zone recovers along aparabolic curve U-A. The reloading would run along equivalent curves from point


Figure 1: Calculated normal force - displacement diagram of contact flattening oflimestone particles, modelled as smooth mono-disperse spheres, median diameterd50 = 1.2µm. For convenience, pressure and compression are defined as positivebut tension and extension are negative.

A to point U forward to the displacement hK,U as well. If the adhesion limit atpoint A is reached then the contact plates detach with the increasing distance a =

a0 + hK,A − hK, Fig. 1d).It is worth to note here that the secant unload stiffness between hK,A and hK,U

(450−820N/m) increases with increasing contact flattening hK,U and with increasingload FN,U [12]. Thus, with increasing external load applied at any process, e.g. inpowder compaction, the particle contacts become stiffer and stiffer and approachthe solid behaviour of a compressed briquette or tablet.

2.2 Load Dependent Adhesion Force

The slopes of elastic-plastic yield and adhesion limits in Fig. 1 characterise thecontact softness or compliance. If one eliminates the centre approach hK of theloading and unloading functions, a linear function for the contact pull-off forceFN = −FH at the detachment point A is obtained [9, 10].

The dimensionless, so-called elastic-plastic contact consolidation coefficient κ de-termines the slope of adhesion force FH influenced by predominant plastic contactfailure. This displacement or flattening coefficient κ characterizes the irreversibleparticle contact stiffness or softness as well. A shallow slope implies low adhesionlevel FH ≈ FH0 because of stiff particle contacts, but a large slope means soft con-tacts, or in other words, cohesive powder flow behaviour in the macroscale [7]. The

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Figure 2: Tangential force - displacement diagram for limestone particles modelledas smooth mono-disperse spheres. Both the non-linear elastic curves by partiallysticking and microslip within the flattened contact plane [12] and the straight lines ofelastic secant stiffness are demonstrated. But the non-linear elastic hysteresis curvesare not drawn here. One may consider the strongly increasing level of friction limitsdue to the adhesion within the flattened contact plane.

total adhesion force FH consists of a stiff contribution FH0 and a soft, displacementinfluenced term κ · (FH0 + FN). Thus, Eq. (11) can be interpreted as a generallinear constitutive contact model, i.e. linear in forces, but non-linear in materialcharacteristics.

2.3 Elastic-Plastic, Frictional Tangential Force

The Coulomb friction limits of tangential forces are described by the coefficient ofinternal friction µi. These limits for contact sliding, Eq. (15) in Table 2, depend onthe elastic-plastic contact consolidation, i.e., elastic-plastic flattening by the normalforce FN and the variable (positive) adhesion force FH(FN) as well, Eq. (11).

The contact loses its elastic tangential stiffness at kT,H = 0 and completely mobi-lized contact sliding is obtained for the friction limit of displacement δ = δC,H underload dependent adhesion force FH(FN), Eq. (11), see grey points for yield in Fig. 2.Compared to the contact radius rK = 6− 14nm, the elastic range is very small andlimited by the tangential displacement δC,H < 0.06nm. At these Coulomb frictionlimits (grey points, index C) the elastic behaviour is transmitted into the frictionalbehaviour of contact sliding shown by constant tangential force FT,C,H, Eq. (15).The tangential force-displacement diagram Fig. 2 is calculated by the equations ofTable 2.


Figure 3: Rolling resistance force (at the centre of mass) - rolling angle diagramfor limestone particles modelled as smooth mono-disperse spheres. The non-linearelastic curves by partially sticking and microslip within the flattened contact planeand the straight lines of elastic secant stiffness are shown. The hysteresis by partiallysticking and microslip within the flattened contact plane is demonstrated here onlyfor the largest normal force. One may consider the strongly increasing level offriction limits due to the adhesion within the flattened contact plane.

2.4 Elastic-Plastic, Frictional Rolling Resistance of LoadDependent Adhesive Contact

The frictional rolling resistance force of smooth soft spheres acts at the centre ofmass and can be considered by a tilting moment relation of the force pair around thegrey pivot at the perimeter of contact circle. As the response of contact flatteninga lever arm of contact radius rK with respect to FN is generated that is equilibratedby rolling resistance FR acting perpendicular to direction of FN with the lever armr− hK/2, see in Fig. 3 the rectangular lines of the sketch right below:

The linear elastic range is very small restricted by the limit of rolling angleγC,H = 8 · 10−5 − 1.9 · 10−4. At these limits the elastic behaviour is transmittedinto the contact rolling shown by constant rolling resistance force FR,C,H 6= f(γ) thatdepends on the positive adhesion force FH(FN), see Table 3.

2.5 Elastic-Plastic, Frictional Spinning of Load DependentAdhesive Contact

The new constitutive torsional moment - rotation angle functions for elastic-plastic,frictional behaviour are shown in Fig. 4 and Table 4. The linear elastic range is verysmall and restricted by a constant friction limit of rotation angle φC,H = 0.0076.At this Coulomb friction limit φC,H the elastic behaviour is transmitted into thecontact spinning shown by constant torque that also depends on adhesion force.

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Figure 4: Torsional moment - rotation angle diagram for limestone particles mod-elled as smooth mono-disperse spheres. Both the non-linear elastic curves by par-tially sticking and microslip within the flattened contact plane [12] and the straightlines of elastic secant stiffness are demonstrated. But the non-linear elastic hystere-sis curves are not drawn here. Again, one has to consider the strongly increasinglevel of friction limits due to the adhesion within the flattened contact plane.

2.6 Essential Constitutive Particle Parameters

From these elastic-plastic and frictional force - displacement laws, force - force equa-tions and moment - angle models one can conclude that six independent, physicalmaterial parameters are necessary to describe the micromechanics of particle adhe-sion. The following material data of ideally assumed, smooth mono-disperse particleswith the mechanical properties of limestone are used:

(1) Median particle diameter d50,3 = 1.2µm and solid density ρs = 2740kg/m3,(2) Hamaker constant CH,sls = 3.810−20J. The characteristic adhesion force of rigidsphere-sphere contact FH0 = −2.64nN was back calculated from powder shear tests.(3) The plastic micro-yield strength pf = 300N/mm2, equilibrium centre separationfor dipole interaction a0 = 0.336nm, elastic-plastic contact area coefficient κA =

5/6, plastic repulsion coefficient κp = 0.153, elastic-plastic contact consolidationcoefficient κ = 0.224, (4) modulus of elasticity E = 150kN/mm2, (5) Poisson ratioν = 0.28, shear modulus G∗ = 34kN/mm2 and (6) contact friction coefficientµi = 0.76.

These material data of ideally assumed, smooth mono-disperse particles with themechanical properties of limestone are used to calculate all the curves of Fig. 1 toFig. 7.


Figure 5: Force and moment - normal force diagram of limestone particles to com-pare the load dependent elastic-plastic adhesion limit FH as positive force and theload dependent elastic-plastic friction limits of tangential force FT,C,H, rolling mo-ment MR,C,H and torsional moment Mto,C,H. The microscopic normal load FN isequivalent to macroscopically moderate average pressures σM,st = 1 to 25kPa (ormajor principal stresses σ1 = 2 to 40kPa), frequently applied in powder handling.

3 Comparison of load dependent adhesion and

friction limits

It is worth to note here that the load dependent adhesion and the friction limitsdetermine the energy absorption within the contact plane. Thus, diagram Fig. 5 isshown to compare the sensitivity of load dependent adhesion FH(FN) on the frictionlimits of sliding, rolling and torsion of these four stressing modes.

The tangential force or maximum resistance FT,C,H(FN) to let the contact beginto slide is larger than to pull-off and separate the particles by maximum adhesionforce FH(FN). The microscopic rolling moment or maximum resistance MR,C,H(FN)

to let roll the contact is larger than to let rotate or twist the particles by maximumtorsional moment Mto,C,H(FN).

4 Engergy absorption of adhesive elastic-plastic

and frictional contact

All necessary force-displacement and moment-angle relations are derived in formof algebraic functions so that they can be analytically integrated to obtain themechanical work [13]. Next, the load dependent specific hysteresis and friction work

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Figure 6: Particle mass related hysteresis work - normal force diagram. The singlecontact behaviour is shown to compare the load dependent maximum work of elastichysteresis of normal deformation Wm,N,diss, tangential microslip Wm,T,max, rollingfriction Wm,R,max and torsional microslip Wm,to,max.

of the four stressing modes of one particle contact are compared and briefly discussedfor normal forces from 3to130nN.

4.1 Comparison of the specific hysteresis work of the fourstressing modes

First, the sensitivity of load dependent adhesion FH(FN) on the maximum specificwork of elastic-dissipative hysteresis (during unloading and reloading) of normalloading, sliding, rolling and spinning (torsion) is demonstrated for the four stressingmodes in Fig. 6. The specific energy input that is required to compensate the energyabsorption within one loop of the elastic hysteresis for unload/reload in the normaland tangential direction is nearly in the same order Wm,N,diss ≈ Wm,T,max. But incontrast to this, the maximum specific hysteresis work necessary for one loop torsionWm,to,max is much larger than for rolling Wm,R,max.

4.2 Comparison of the specific detachment and frictionwork of the four stressing modes

The influence of normal load and load dependent adhesion on the work of contactdetachment and friction work during sliding, rolling and spinning is shown for thefour stressing modes in Fig. 7.


Figure 7: Particle mass related friction work - normal force diagram. The singlecontact behaviour is shown to compare the load dependent maximum work of con-tact detachment in the normal direction Wm,N,A, tangential sliding Wm,T,C, rollingfriction Wm,R,C and spinning Wm,to,C [12, 13]

The smallest values of energy absorption are calculated for the direct contactapproach and separation in the normal direction, i.e. the specific detachment workWm,N,A = 2−50µJ/g. If one compares maximum (or selected) specific friction workof sliding WT,C, rolling WR,C, and torsion WT,C, only the factors are different, namely2·µi, 1 and 2/3·µi [12, 13]. These comparatively small differences in the mass relatedenergy absorption of different loading/unloading procedures are compared in Fig. 7with respect to tangential sliding, rolling and spinning (torsion).

4.3 Comparison of the local energy densities of surface ac-tivation

The logical consequence of the model stiff particles with soft contacts is that onlythe mass fraction of the deformed contact matter is considered to describe the localintensity of energy absorption during mechanical stressing of two particles. Forthe sake of simplicity, the averaged mass of the locally deformed contact zone isgeometrically approximated by the caps of spheres 1 and 2 with their heights ordisplacements hK,1 = hK,2 = hK,U/2 [13]. The same factors 2 · µi, 1 and 2/3 · µiare found as well when one compares the characteristic (maximum) local energydensities of surface activation by contact sliding, particle rolling and torsion, Eqs.(35), (36) and (37) in Table 5.

The enormous local energy densities of these various stressing modes are found tobe in the orders of magnitude of 0.02 to 3MJ/mol. These theoretically estimatedmolar energies are directly compared with the lattice enthalpies of various ionic

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crystals of 0.7 − 15MJ/mol. The values are also located within the range of theenergy accumulated in lattice dislocations of about 100−1000kJ/mol and are morethen the sublimation (evaporation) enthalpies of solids of about 200 − 500kJ/mol.It is worth to note here that the local activation energy of tribochemical reactionswithin the contact zone nearly amounts to the same orders of magnitude and variesfrom 62 to 744kJ/mol [15, 16].

This local energy consumption is meaningful for surface activation, particle con-version, product formulation, macroscopic non-rapid frictional shear flow of cohesiveultrafine powders and the agglomerate disintegration in powder processing and han-dling. Thus, the micromechanical approach is qualitatively and quantitatively con-sistent and conclusive concerning the constitutive force-displacement and moment-angle laws and the estimated maximum specific energy absorption.

5 Conclusions and outlook

By the model stiff particles with soft contacts and the contact force equilibrium,universal models have been graphically demonstrated that include the elastic-plasticparticle contact behaviour, load-unload hysteresis and a history dependent adhesionforce function. The microscopic load dependent adhesion effect on the friction limitsof single particle contacts macroscopically leads to the significant influence of pre-consolidation stress on the frictional flow of ultrafine cohesive powders that is wellknown in powder handling practice. Thus, the physical models are used for theadvanced data evaluation of various powder product properties concerning particlesize distribution (nanoparticles to granules), moisture content (dry, moist and wet)and material properties (minerals, chemicals, pigments, waste, plastics, food etc.).The contact models are very meaningful to describe the frictional shear flow ofcohesive and compressible ultrafine powders. For this purpose, all the normal andtangential forces as well as angular and trajectorial moments of particles have to bebalanced to simulate their dynamics with physically realistic material data by thediscrete element method (DEM), see Tykhoniuk et al. [14].

The consequences of elastic-dissipative as well as elastic-plastic, frictional unload-ing and reloading paths of normal and tangential forces and rolling and torsionalmoments for the load dependent energy absorption and the friction work have beendiscussed and compared. The different quantities between the averaged and localintensity of the energy absorption have been explained for single stressing events ofcontact compression, sliding, rolling and torsion. The averaged energy densities ofthese various stressing modes have been found to be in the orders of magnitude ofabout 0.4−6mJ/g. But the enormous local energy densities amount to 0.2−30kJ/g,or in molar units 0.02− 3MJ/mol.

In this context, the light emission due to mechano-luminescence effects by elastic,elastic-plastic contact deformations and, consequently lattice dislocations, crystalphase conversions and particle breakages can be explained qualitatively at inten-sive and multiple stressing of particles in grinding processes with large local energydensities, see Aman and Tomas [17].

This microscopic energy absorption macroscopically leads to the significant influ-ence of pre-consolidation stress on the non-rapid frictional flow of ultrafine cohesive


powders and to the remarkable macroscopic energy consumption that is well-knownin powder storage and handling practice. This is also meaningful for the agglomer-ate disintegration in powder processing and generates inelastic contact and particledeformations, surface defects, surface asperity abrasion, particle-wall abrasion andmicro-cracking up to particle breakage. Those agglomerate disintegration effects areundesired and cause product damage and quality reduction, see Antonyuk et al.[18].

The micromechanical interaction rules of particle adhesion are necessary to de-scribe the pre-consolidation dependent, mechanical flow and consolidation behaviourof ultrafine cohesive powders and to simulate the dynamics of packed particle bedsin powder processing, i.e. at agglomeration, disintegration, size reduction, powderstorage and flow. At present, with this advanced knowledge an improved apparatusdesign is accomplished for industrial partners in process industries.

References

[1] J. Tomas and H. Schubert, Modelling of the strength and flow behaviour of moistand soluble bulk materials, Aufbereitungs-Technik, Vol. 23, pp. 507-515, 1982.

[2] J. Tomas, Modelling the time consolidation processes of bulk materials - prob-lems and pre-liminary solutions, Intern. Symp. Reliable Flow of ParticulateSolids II, Oslo 1993, pp. 335-372.

[3] J. Tomas, Zum Verfestigungsprozess von Schuettguetern - Mikroprozesse,Kinetikmodelle und Anwendungen, Chemie-Ingenieur-Technik, Vol. 69, pp. 455-467, 1997.

[4] J. Tomas, Particle adhesion fundamentals and bulk powder consolidation, Intern.Symp. Reli-able Flow of Particulate Solids III, Porsgrunn 1999, pp. 641 - 656.

[5] J. Tomas, Particle adhesion fundamentals and bulk powder consolidation, pow-der handling and processing, Vol. 12, pp. 131-138, 2000.

[6] J. Tomas, Assessment of mechanical properties of cohesive particulate solids -part 1: particle contact constitutive model, Particulate Science and Technology,Vol. 19, pp. 95-110, 2001.

[7] J. Tomas, Assessment of mechanical properties of cohesive particulate solids -part 2: powder flow criteria, Particulate Science and Technology, Vol. 19, pp.111-129, 2001.

[8] J. Tomas, The mechanics of dry, cohesive powders, powder handling and pro-cessing, Vol. 15, pp. 296-314, 2003.

[9] J. Tomas, Product Design of Cohesive Powders - Mechanical properties, com-pression and flow behaviour. Chemical Engineering and Technology, Vol. 27, pp.605-618, 2004.

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[10] J. Tomas, Fundamentals of Cohesive Powder Consolidation and Flow, GranularMatter, Vol. 5, pp. 75-86, 2004.

[11] J. Tomas, Mechanics of Nanoparticle Adhesion - a Continuum Approach, in:K.L. Mittal, Par-ticles on Surfaces 8: Detection, Adhesion and Removal, VSPUtrecht, 2003, pp. 183-229.

[12] J. Tomas, Mechanics of Nanoparticle Adhesion - a Continuum Approach, in:K.L. Mittal, Par-ticles on Surfaces 8: Detection, Adhesion and Removal, VSPUtrecht, 2003, pp. 183-229.

[13] J. Tomas, Adhesion of ultrafine particles - Energy absorption at contact, Chem-ical Engineering Science, Vol. 62, pp. 5925 - 5939, 2007.

[14] R. Tykhoniuk, J. Tomas, S. Luding, M. Kappl, L. Heim and H.-J. Butt, Ul-trafine cohesive powders: from interparticle contacts to continuum behaviour.Chemical Engineering Science, Vol. 62, pp. 2843-2864, 2007.

[15] G. Heinicke, Tribochemistry, Akademie-Verlag, Berlin 1984.

[16] K. Tkacova, H. Heegn and N. Stevulova, Energy transfer and conversion duringcomminution and mechanical activation, Intern. J. of Mineral Processing, Vol.40, pp. 17-31, 1993.

[17] S. Aman and J. Tomas, Mechanoluminescence of quartz particles during grind-ing in stirred media mills, Powder Technology, Vol. 146, pp. 147-153, 2004.

[18] S. Antonyuk, J. Tomas, S. Heinrich and L. Mrl, Breakage behaviour of sphericalgranulates by compression. Chemical Engineering Science, Vol. 60, pp. 4031-4044, 2005.

Juergen Tomas, The Otto-von-Guericke-University, Mechnical Process Engineering,Universitaetsplatz 2, D - 39106 Magdeburg, Germany


Microprocess Model Equation No.

Approach−∞ < hK ≤ 0

sphere-sphere- model

FN = −FH0 = − CH,slsr1,2

6(a0−hK)2 = −FH0a

20

(a0−hK)2 (1)

Elastic0 ≤ hK ≤ hH,f

Hertz andDMT

FN = 23E∗√r1,2h

3K − FH0 (2)

Yield limithK,f ≤ hK ≤hK,U

displacement FN = πr1,2pf(κA − κp)hK − FH0 (3)

contact arearatio

κA = 23

+ 13

Apl

Apl+Ael= 1− 1

3(hK,f

hK)

13 (4)

repulsion co-efficient

κp = pVdW

pf= CH,sls

6πa30pf

= 4σsls

a0pf(5)

Unload hK,A ≤hK ≤ hK,U

Elastic recov-ery, U-A

FN = 23E∗√r1,2(hK − hK,A)3 −

πr1,2κppfhK,A − FH0

(6)

contact de-tachment

hK,A,(1) = hK,U −

3

√hK,f,el−pl

(hK,U + κhK,A,(0)

)2 (7)

Reload hK,A ≤hK ≤K,U

elastic, non-linear A-U

FN = −23E∗√r1,2 (hK,U − hK)

3+

πpfr1,2 (κA − κp)hK,U − FH0

(8)

Adhesion limit Plate-platemodel

FN = −πr1,2pVdWhK − FH0 (9)

Detachment−∞ < hK ≤hK,A

Plate-platemodel

FN(hK) = −FH0a

20

(a0+hK,A−hK)2 −πr1,2pVdWhK,A

(a0+hK,A−hK)3a30

(10)

Resulting max-imum adesionforce

Load depen-dent adhesion

FH = FH0 + κ (FN + FH0) (11)

consolidationcoeff.

κ =κp

κA−κp=

pVdW/pf

2/3+1/3Apl/(Apl+Ael)−pVdW/pf

(12)

Table 1: Normal force - displacement laws of contact normal loading FN(hK) and vander Waals adhesion force. The essential symbols are explained in section 2.6 (median

particle radius of surface curvature r1,2 = d50

4and r1,2 =

(1r1

+ 1r2

)−1

, respectively ,

averaged stiffness E∗ = 2 ·[

(1−ν21)

E1+

(1−ν22)

E2

]−1) [6, 12].

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Mircoprocess Model Equation No.

contact stiffness load depen-dent

kT,H = dFT

dδ= 4G∗rK(1− δ

δC,H)

12 (13)

Initial stiffness Load depen-dent

kT,H0 = dFT

dδ

∣∣δ=0

= 4G∗rK =

4G∗√r1,2hK(14)

Friction limitskT,H = 0

Force FT,C,H = µi(1+ κ)(FH0 + FN) (15)

Displacement δC,H = 3µi

8G∗

√πpfκA (1+ κ) (FN + FH0) (16)

Loading 0 ≤ δ ≤δC,H

Force - dis-placement

FT = FT,C,H

[1− (1− δ

δC,H)

32

](17)

Unloading−δC,H ≤ δ ≤ δC,H

Frictionalbehaviour

FT = FT,U − 2FT,C,H

[1− (1− δU−δ

2δC,H)

32

](18)

Reloading−δC,H ≤ δ ≤ δC,H

Frictionalbehaviour

FT = −FT,re+2FT,C,H

[1− (1− δ+δre

2δC,H)

32

](19)

Table 2: Tangential force - displacement laws FT (δ) with load dependent adhesion

[12] (averaged shear modulus G∗ = 2[2−ν1

G1+ 2−ν2

G2

]−1, shear modulus Gi = Ei

2(1+νi)

i =1,2).

Microprocess Model Equation No.Contact stiffness load depen-

dentkR,H =16G

π(4−3ν)(1+κ)(FH0+FN)

κApf

(1− γ

γC,H

)2 (20)

Initial stiffness Load depen-dent

kR,H,0 = dFR

dγ

∣∣∣γ=0

=

16Gπ(4−3ν)

(1+κ)(FH0+FN)κApf

(21)

Friction limits kR,H = 0 Rollingresistance

FR,C,H = µR (1+ κ) (FH0 + FN) =√(1+κ)3(FH0+FN)3

πr21,2pfκA (22)

Rolling an-gle

γC,H =3(4−3ν)16Gr1,2

√πκApf (1+ κ) (FN + FH0)

(23)

Loading 0 ≤ γ ≤ γC,H Force - dis-placement

FR = FR,C,H

[1−

(1− γ

γC,H

)3](24)

Unloading −γC,H ≤γ ≤ γC,H

frictional be-haviour

FR = FR,U −

2FR,C.H

[1−

(1− γU−γ

2γC,H

)3] (25)

Reloading −γC,H ≤γ ≤ γC,H


FR = −FR,re +

2FR,C,H

[1−

(1− γre+γ

2γC,H

)3] (26)

Table 3:

Rolling resistance force

- rolling angle laws FR(γ) with load dependent adhesion [12]


Micro-process

Model Equation No.

contactstiffness

load dependent kto,H = dMto

dφ=

8Gr3K3

[2(1− Mto

Mto,C,H

)−1/2

− 1

]−1

(27)

Initial stiff-ness

Load dependent kto,H0 = dMto

dφ

∣∣∣Mto=0

= 2(1 −

υ)r1,2 (1+ κ) (FH0 + FN)

(28)

FrictionlimitskTo,H = 0

Torsional moment Mto,C,H = 2µi

3

√(1+κ)3(FN+FH0)3

πκApf(29)

Rotation angle φC,H = 3πµiκApf

4G(30)

Loading0 ≤ φ ≤φC,H

Moment-angle Mto =

4Mto,C,H

[√3φφC,H

+ 1− 3φ4φC,H

− 1] (31)

Unloading−φC,H ≤φ ≤ φγC,H


Mto = Mto,U −

8Mto,C,H

[√3(φU−φ)2φC,H

+ 1− 3(φU−φ)8φC,H

− 1] (32)

Reloading−φC,H ≤φ ≤ φγC,H


Mto = −Mto,re +

8Mto,C,H

[√3(φre+φ)2φC,H

+ 1− 3(φre+φ)8φC,H

− 1] (33)

Table 4: Torsional momentrotation angle lawsMto(φ) with load dependent adhesion[12].

Microprocess Work No. WM = M̃Wm

in kJ/mol

Compression,detachment

Wm,N,act = 2pf

ρs

[κA + κp

a0

hK,U+ 2FH0a0

πr1,2pfh2K,U

](34) 20− 40

Sliding Wm,T,C,act = 8µi · κApf

ρs

√r1,2

hK,U(35) 1000− 3000

Rolling Wm,R,C,act = 4κApf

ρs

√r1,2

hK,U(36) 660− 2000

Spinning Wm,T,C,act = 8µiκA

3pf

ρs

√r1,2

hK,U(37) 330− 1000

Table 5: Activation energy of locally deformed contact zone (M̃ = 100g/mol molec-ular mass of CaCO3).

2 apm proceedings¤ge/tomas_106-p-1142.pdfwater layers. consequently, more or less mobile adsorption...

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